DAMAGE BOOK TEXT PROBLEM WITHIN THE BOOK ONLY CO >; 00 166473 J^ 5 ;o ARITHMETIC i JR THE USE OF SCHOOLS AND COLLEGES BY JADAV CHANDRA CHAKRAVARTI, M. A., LATE PROFESSOR OF MATHEMATICS, MAHOMEDAN ANGLOORIENTXU COLLEGE, ALIGARH T. J op- Sixty-sixth Edition S AN Y AL & Co,, 25, ROY BAGAN STREET SOLE AGENTS P. C. CHAKRAVARTI & BROTHERS, 74, Btchu Chatter jee Street* 1920 All Bights Reserved PUBLISHED BY BEJOY KUMAR MAITRA OF Messrs. SANYAL & Co. AND PRINTED BY HARX CHURN RAKSHIT AT THE BHARAT MIHIR 25, ROY BAGAN STREET, CALCUTTA. PREFACE. THIS work has been written with the view of providing 'book for class use in our Schools and Colleges, which shall sun the capacities of the young beginner and at the same time meet the requirements of the advanced student. So far as has been possible within the necessary limits of the book, I have carefully avoided laying down arbitrary rules and have endeavoured to establish the leading proposi- tions of the science of Arithmetic by a process of simple reasoning, being fully convinced that a mere ) ^Al facility in manipulating figures, sufficient the* e for the calculations necessary in every-day lift conducive to a healthy development of the K i c " . I have accordingly explained the processes of At K means of specimen examples fully wor> 1 rut, and ii3 ry division of the subject I have V^.^i *n simple prh. ;>les and have tried to proceed by ^at^.d natural steps to those of a more complex natu;j. Compound quantities have been assigned a somewhat earlier place than is usually given them ; in ether parts of the subject however there is but little departure from the com* mon order. Decimals have been treated as a natural ex- tension of the common system of notation ; but the principles of vulgar fractions have been made use of here and there for purposes of explanation. The method adopted for the addi- tion and subtraction of Recurring Decimals requires no con- version and re-conversion to vulgar fractions* A little more ( * ) space than usual has been devoted to the subject to Pro* blems, and I venture to hope that I have been able to make* ft simpler and more attractive by means of careful arrange- ment and classification. Although I have adopted the Uni- tary Method (a method so simple in its application and so suitable for young learners) in the section on Problems, 1 have not abandoned the Rule of Three as some writers have done, because I do not consider it to be a misleading process, if properly understood. The sections on Stocks and other branches of Commercial Arithmetic I have tried to make in some degree complete. And I may add that although the book contains nothing that might strictly be called original, yet it will be found to differ in many ways from any existing text-boo u ^n the subject. contains a large number of examples for exer- cise been worked out several times from the prir ,, jt it would be presumptuous to hope that t ^ A escaped notice. I shall be grateful to Teachers at> ^tudents for any correction that they may send me. I have to thank some friends for valuable criticism and advice and also for correcting .. d revising many of the proof- sheets. I have the pleasure of expressing my thanks also to some of the students of the M. A.-O. College, Aligarh, foi great assistance in verifying the answers to many of the examples. ALIGARH, N. W, P., . J. C. C. January -, 1890. PREFACE TO THE SECOND EDITION. This edition has been carefully revised and the few errora that crept into the former edition have been corrected. I have inserted some explanatory matter and a few new examples in certain parts of the book. The book has been further en- larged by the insertion of the Punjab and Allahabad Univer- sity Entrance Examination Papers. A few examples have been slightly altered for securing neat answers. These addi- tions and alterations will, however, be no hindrance to both the editions being used together in the same class. ALIGARH, December \ 1890, } J. C C. PREFACE TO THE SIXTH EDITION, In this edition the work has again been carefully revis/ and many important additions and a few slight alterat have been made. The following Exercises have been\ creased : 76, 106, 107, 115, 116, 117, 119, 132, 140. A Exercise ( i?4a ) has been inserted ; this relate* twenty-seven Sections of the boofc, and may b=> as those Sections have been read, S p entirely re-written and considerably matter has been subjoined at t^ form of an Appendix. These ' increase the usefulness o f worthy of the approba'* ALIGARH, ) August, i8or ( 4 ) PREFACE TO THE SIXTIETH EDITION. In the last edition some notes and examples were inserted in the Section on Approximation and also a few examples in the Section on the Metric System. SIRAJGANJ, 1 J C. C. 1917. / PREFACE TO THE SIXTY-SIXTH EDITION. In this edition the section on Approximation has been enlarged and improved, and the section on Metric System has been revised and corrected on the basis of the latest determination of the length of the metre in inches. SIRAJGANJ, \ November, 1920. J J * ' CONTENTS. Sect. I. INTRODUCTION ,.,...* /fl. THE METHOD OF REPRESENTING NUMBERS BY FIGURES * III. ADDITION 9 IV. SUBTRACTION . . ... 14 V. MULTIPLICATION *9 DIVISION 26 PROPOSITIONS IN THE FUNDAMENTAL OPERATIONS 30 MISCELLANEOUS EXAMPLES (Simple Rules) . . 37 MEASURES OF MONEY AND REDUCTION . . 40 COMPOUND ADDITION .~~~^ 45 COMPOUND SUBTRACTION \ 47 COMPOUND MULTIPLICATION COMPOUND DIVISION 48 49 MEASURES OF WEIGHT 53 XIV. MEASURES OF LENGTH . , . . . 5 8 XV. MEASURES OF AREA 60 XVI. MEASURES OF SOLIDITY AND CAPACITY . . 64 XVII. MEASURES OF TIME, ANGLES, NUMBER, AND APO- THECARIES' WEIGHT 65 MISCELLANEOUS EXAMPLES (Compound Rules) . -$8 XVIII. BARTER, GAIN AND LOSS, FTr .- , , ...(? FACTORS AND PRIME KTTmpyps HIGHEST COMMON FACTOR .. - LOWEST COMMON MULTIPLE r . :- FRACTIONS ,- ~m_ , , - . - 'MISCELLANEOUS EXAMPLES (Fractions) . XXIII. COMPLEX FRACTIONS . XXIV. FRACTIONAL MEASURES - _,. MISCELLANEOUS EXAMPLES (Fractional Measures) XXV. DECIMALS -- n XXVI. RECURRIKO DECIMALS MEASURES MISCELLANEOUS EXAMPLES (Decimals) PRACTICE ROOT 137 W 1< CUBE ROOT 161 MEASUREMENT OF AREA 164 MEASUREMENT OF SOLIDITY .... 172 DUODECIMALS 176 PROBLEMS AND THE UNITARY METHOD . 178 BANKRUPTCIES, RATING, TAXING, ETC. . . 189 PROBLEMS RELATING TO WORK DONE IN A CERTAIN TIME - , , . Sect. XXVII. XXVIII. XXIX. XXX. XXXI. XXXII. XXXIII. XXXIV. XXXV. PROBLEMS RELATING TO CLOCKS PROBLEMS CONCERNING TIME AND DISTANCED RACES AND GAMES OF SKILL . . . .203 , CHAIN RULE 205 'XXXVI. .COMPLEX PROBLEMS 207 XXXVI I? RATIO AND PROPORTION . . _ , ^-(21 2 XXXVIII> RULE OF THREE .217 XXXIX* DOUBLE RULE OF THREE 220 MISCELLANEOUS EXAMPLES (On sect, i-xxxix.) . 222 XL. DIVISION INTO PROPORTIONAL PARTS . . 2JI XLIy FELLOWSHIP OR PAPTOT.PQTTTP^ , r , (Zjf XLII. ALLIGATION 2^7 XLIII. AVERAGE VAT.TTTT. l - .,.. .1 - . . W9 XLIV:, PERCENTAGE ^ >. , , - ,... r-^40 XLV. COMMISSION, BROKERAGE, PREMIUM XLVIJ PROFIT AND LOSS ^_^ ^- . r- XLVII* 1 DIMPLE INTEREST^. ^ .- 1 , ^ XLVIII> COMPOUND INTEREST ^ ^ , ^ XLIA PRESENT WORTH AND DISCOUNTNL^. L. EQUATION OF PAYMENTS ..... 267 LI. STOCKS 268 ( Hi ) Sect. LI I. EXCHANGE , , 276 LIII. METRIC ^SYSTEM AND DECIMAL COINAGE. , . 278 LIV. INVOICES ^ND ACCOUNTS 289 LV. PROBLEMS IN HIGHER ARITHMETIC 290 EXAMPLES FOR EXERCISE (First Series) , . . 298 EXAMPLES FOR EXERCISE (Second Series) . . , 305 PROBLEMS 333 UNIVERSITY EXAMINATION PAPERS .... 355 ANSWERS TO EXAMPLES 454 APPENDIX I 535 APPENDIX II 535 NOTE. The easier part of the Section on Problems may be taken at a much earlier stage than is indicated by its position in the book ; and Examples 36 an$ 37 may be omitted on the first reading. [ I ] TABLES OF MEASURES. for further information turn to the pages referred tc\ English Money Table. [Page 40.] 4 Farthings (?.) make I Penny (id.). 12 Pence ... i Shilling (is. or i/-). 20 Shillings ... i Pound or Sovereign (i) 2 Shillings*" i Florin. 5 Shillings *i Crown. 21 Shillings i Guinea. 27 Shillings =i Moidore. Indian Money Table. [Page 41.] 3 Pies (f.) make i Pice. 4 Pice or 12 Pies ... i Anna (la.). 16 Annas ... i Rupee (Ri). 15 Rupees ... i Pound or Sovereign (i). English Jewellers 1 or Troy Weight. [Page 53.] (Chiefly used for weighing gold, silver an)* jewels.) 24 Grains (gr.) make I Pennyweight (i dwt). 20 Pennyweights ... i Ounce (i oz.). 12 Ounces ... i Pound (i lb.). So that a Pound Troy 5760 Grains. English Standard or Avoirdupois Weight. [Page 54.] 16 Drams (dr.) make I Ounce (i 6z.). 1 6 Ounces ... I Pound (i lb.). 28 Pounds ... I Quarter (i (jr.). 4 Quarters .. I Hundredweight (i cwt.). 20 Hundredweights ,... i Ton (i ton). A stone (st.) 14 lb. A Pound Avoir; 7000 Grains Troy. Indian Bazar Weight. [Page 55.] 4 Sikis make I Tola. 5 Sikis make z Kancha (Powa-chatak), 4 Kanchas or 5 Tolas... I Chatak (i ch.). 16 Chataks ... i ; Seer. 40 Seers ... x, Maund (i md.). 4 Chataks i Powa. 4 Powas i Seer. 5 Seers - i Punshury. 8 Funshuries I Maund, Madras Local Weight. [Pag* $6.] 3 Tolas make i Pollum. 8 Poilums i Seer. 5 Seers or 40 Poilums... i Viss. 8 Viss i Maund. 20 Maunds i Candy or Barum. A Madras maund - 25 Ib. Avoir. Bombay Local Weight. \Pag4 57.] 4 Dhans 8 Raktikas 4 Mashas 72 Tanks 40 Seers 20 Maunds make Raktika. Masha. Tank. Seer. Maund. Candy. A Bombay maund - 28 Ib. Avoir. English Linear Measure. [Page 58.] ,12 Inches (in.) JFeet yards 40 Poles or 220 yards 8 Furlongs or 1760 yards 3 Miles i Pole - 9 Inches 2 Spans or 1 8 Inches /. 2 Cubits 6 Feet - 4 Poles or 22 Yards joo Links make i Foot (i ft.). ... i Yard (i yd.). ... i Pole, Rod or Perch (i po.) ... I Furlong (i fur.). I Mile (i mi.). i League (i lea.). 5 yd. I ft. 6 in. I Span. i Cubit (HatK). i yard. I Fathom. i Chainl x Chain/ Used in land surveying. The following Table is used by tailors : Inches 4 Nails 4 Quarters 5 Quarters Nail (GtrraX). Quarter (Span). Yard. 11 [ 3 1 English Square Measure. [Page 61.] 144 Square Inches (sq. in.) make I Square Foot (f sq. ft.)* Square Feet ... i Square Yard (i sq. yd.). 3oJ Square Yards ... I Square Pole, Rod or Perch. 40 Square Poles ... i Rood (i ro.). [(i sq. po.). or 4 8 4 4 o STSrf. } - 'Acre(rac.). 640 Acres ... i Square Mile (i sq. mi.). A square chain 22x22 sq. yards or 484 sq. yards. .". 10 sq. chains i acre. i sq. pole -30 sq. yd. 2 ft. 36 in. *** For Indian LAND MEASURES see pages 63 and 64. Measures of Solidity (English) [Page 64.] 4728 Cubic Inches make I Cubic Foot (i cu. ft.). 27 Cubic Feet ... I Cubic Yard (i cu. yd.). Measures of Capacity. (English) [Page 64.] 4 Gills make Pint (i pt.). 2 Pints Quart (i qt.). 4 Quarts Gallon (i gall.). 2 Gallons Peck (i pk.). \ 4 Pecks Bushel (i bus.). 8 Bushels Quarter (i qr.). 5 Quarters ... Load (i Id.). 2 Loads Last (i last). Also 2 quarts pottle (i pot.). 2 bushels * strike (i str.). 4 bushels coomb ( i coomb). } A Barrel contains 36 gallons. For dry goods only Note. A gallon of distilled water weighs exactly 10 Ib. Avoir. A pint of water weighs a pound and a quarter. [A gallon contains 277*274 cubic inches]. A cubic foot of water weighs about looo oz. Avoir. Measures of Time. (English} [Page 65.) 60 Seconds (sec.) make Minute (i min.). 60 Minutes Hour (i hr.). 24 Hours Day (i da.). 7 Days Week (i wk.). 365 Days Year (i yr.). 366 Days Leap-year, loo Years Century. [ 4 3 Measures of Angles. [Page 66.] 60 Seconds (60") make I Minute (i') 60 Minutes ... I Degree (i*). 90 Degrees ... I Right Angle (l rt. gle.). Measures of Number. [Page 67.] 12 Units make 12 Dozen 12 Gross 20 Units Also 24 Sheets of paper 2o Quires 10 Reams Dozen. Gross. Great Gross. Score (Kurri). Quire. Ream. Bale. Apothecaries' Weight. [Page 67.] (i) Measures of Weight. Druggists use \btgrain to weigh small quantities and \h& found and ounce Avoir, to weigh large quantities. Some physicians in prescribing use the following table : 20 Grains make i Scruple (i scr.). 3 Scruples ... I Drachm (i dr.). 8 Drachms ... I Ounce Troy. (ii) Measures of Capacity. 60 Minims (m.) or drops make I Fluid drachm (fl. dr.). 8 Fluid drachms ... I Fluid ounce (fl. oz.). 20 Fluid ounces ... I Pint (O.). 8 Pints ... i Gallon (C.). Note. Since a pint of water weighs a pound and a quarter; the weight of a fluid ounce of distilled water is an ounce avoir. ARITHMETIC. I. INTRODUCTION. 1. A quantity is anything which may be regarded as being made up of parts like the whole. [Ham&lin Smith. Thus, a sum of money, the length of a rod, the weight of a sack of rice, a number of men, are quantities. 2. A quantity is called a unit quantity [or simply a unit} when it is used for the purpose of comparing the magnitudes of other quantities of the same kind. [J. B. Lock. Thus, a rupee is used as the unit of money when we speak of a certain sum as three rupees. A boy is the unit when we speak of a certain class in a school as containing fifteen boys. 3. That which indicates the magnitude of a quantity relatively to its unit is called a number. Thus, the number three indicates the relative magnitude of the quantity three rupees as compared with its unit a rupee. 4. The Measure or numerical value of a quantity is the number which expresses how many times the unit is contained in the quantity. Thus, if we use a yard as the unit of length, and speak of a certain length as five yards, the number five is the measure or numerical value of that length. Note. The numerical value of a quantity indicates its relative magnitude. The absolute magnitude of a quantity is indicated by its numerical value and unit together. 5. A number is called an abstract number, when it is not attached lo any particular unit ; as, four, five, seven. 6. A number is called a concrete number, when it is attached to some particular unit ; as, four horses, five men, seven yards. 7. Arithmetic is a part of the Science which teaches the use of numbers. II. THE METHOD OF REPRESENTING NUMBERS BY FIGURES. 8. In Arithmetic we f epresent all numbers by means of the ten symbols or figures i, 2, 3, 4, 5, 6, 7, 8, 9, o, called digits. The first nine of these figures are called the significant digits ; the last is called zero, cipher or nought * C. A. I 2 ARITHMETIC 9. Numbers from one to nine are represented by the nine significant digits taken in order. Thus one two three four five six seven eight nine 1234567 89 10. All higher nuntbers are represented by two or more of the figures, the following convention being adopted : It is agreed that in a line of figures, the figure in the first place towards the right shall have its simple value? and shall represent so many units ; the figure in the second place from the right shall have ten times its simple value, and shall represent so many tensGJ units, or tens ; the figure in the third place shall have ten times the value it should have in the second place or one hundred times its simple value, and shall represent so many tens of tens, or hundreds, of units, or hundreds ; thus 435 shall express one hundred times four units, together with ten times three units and also five units more ; or in other words, it shall express four hundreds, three tens and five units : and so on, the value of a figure increasing tenfold at each step of removal towards the left. 11. The following table, called the Numeration Table, gives the respective names of places of figures representing a number. billions, ns. CA 1 1J S-! S'S . |'!jA (A *0 S IA CA g 1L i|:l .2 vs B g-S .2 O rj rS rt* 'rS O ** >n ' - Iff fi Hundreds of Tens of thou: Thousands oi Hundreds of Tens of billio Billions. Hundreds of Tens of thous Thousands ol Hundreds of Tens of millu Millions. Hundreds of Tens of thous Thousands. Hundreds. Tens. Units. 9 3 7 6 5 4 3 2 i 987 6 5 4 32 I *The value of a figure which it hA when it stands by itself it called its simple or intrinsic value. The value of a figure which it has in consequence of its position in a line of figures is called its local or -accidental value* NUMERATION 3 The periods which follow those in the above table are trillionsj quadrillions, quintillions, sextillions, septillions, octillions, etc. 12. The symbol o has no value in itself and represents no number. In a line of figures, o in the first place (towards the right) indicates the absence of units ; in the second place, absence of tens ; in the third place, absence of hundreds ; and so on. Thus 30 represents three tens and no units \ 400 represents four hundreds, no tens, also no units ; 309 represents three hundreds, no tens } and nine units, 13. It appears then, that numbers from one to nine are re- presented by one figure ; numbers from ten to ninety-nine are represented by two figures ; numbers from one hundred to nim hundred and ninety-nine are represented by three figures ; numbers from one thousand to nine thousand^ nine hundred and ninety-nint are represented by four figures ; and so on. 14. The method above explained of representing numbers by means of ten figures and their combinations was invented by the Hindus. But Europeans call it the Arabic Notation because it was introduced into Europe by the Arabs who had learnt it from the Hindus. NUMERATION. 15. Numeration is the art of reading a number expressed in figures. Art. 9 enables the learner to read the numbers expressed by one figure ; and the following table will enable him to read the numbers expressed by two figures. 10 ten 23 twenty-three 36 thirty-six 1 1 eleven 24 twenty-four 37 thirty-seven 12 twelve 25 twenty-five 38 thirty-eight 13 thirteen 26 twenty-six 39 thirty-nine 14 fourteen 27 twenty-seven 40 forty 15 fifteen 28 twenty-eight 41 forty-one 1 6 sixteen 29 twenty-nine 42 forty-two 17 seventeen 30 thirty 43 forty-three 1 8 eighteen 31 thirty-one 44 forty-four 19 nineteen 32 thirty-two 45 forty-five 20 twenty 33 thirty-three 46 forty-six 21 twenty-one 34 thirty-four 47 forty-seven 32 twenty-two 35 thirty-five 48 forty-eight 4 ARITHMETIC 49 forty-nine 66 sixty-six 83 eighty-three 50 fifty 67 sixty-seven 84 eighty-four 51 fifty-one 68 sixty-eight 85 eighty-five 52 fifty-two 69 sixty-nine 86 eighty-six 53 fifty-three 70 seventy 87 eighty-seven 54 fifty-four 71 seventy-one 88 eighty-eight 55 fifty-five 72 seventy-two 89 eighty-nine 56 fifty-six 73 seventy-three 90 ninety 57 fifty-seven 74 seventy-four 91 ninety-one 58 fifty-eight 75 seventy-five 92 ninety-two 59 fifty-nine 76 seventy-six 93 ninety-three 60 sixty 77 seventy-seven 94 ninety-four 61 sixty-one 78 seventy-eight 95 ninety-five 62 sixty-two 79 seventy-nine 96 ninety-six 63 sixty-three 80 eighty 97 ninety-seven 64 sixty-four 81 eighty-one 98 ninety-eight 65 sixty-five 82 eighty-two 99 ninety-nine 16. When a number is expressed by three figures, the third figure from the right is read as so many hundred) the two remain- ing figures being read together as in the above table. Thus the number expressed by 100 is read one hundred ; the number expressed by 340 is read three hundred and forty ; the number expressed by 452 is read/<?r hundred and fifty-two ; the number expressed by 607 is read six hundred and seven. H. If a number is expressed by more than three figuresi divide the line of figures by commas into periods of three figures each, commencing from the right ; and read the first period (towards the right) as in Art. 16, read the second period as so many thousand) the third period as million) the fourth as thousand) the fifth as billion) the sixth as thousand) and so on. The periods must be read off from left to right in order. Thus 2,435 j s read 'two thousandfanx hundred and thirty-five 1 ; 23,204 is read 'twenty-three thousand) two hundred and four' ; 234,021 is read 'two hundred and thirty-four thousand 'and twenty-one' ; 324,103,200 is read 'three hundred and twenty-four million^ one hundred and three thousand) two hundred 1 ; 56)204,340,432,004 is read 'thirty-six billion) two hundred and four thousand) three hundred and forty million^ four hundred and thirty-two thousand and four*. i,boo represents a thousand ; 1,000,000 represents a million ; 1 ,000,000,000,000 represents a billion. NOTATION 5 EXAMPLES. 1. * To be done first orally -, then in writing. Express each of the following numbers in words : 1. 10 ; 16 ; 48 ; 99 J 76 ; 43 ; 5 J 3i 5 62. 2. loo ; III ; 902 ; 620 ; 300 ; 103 ; 234 ; 130. 3. 9216 ; 5409 ; 5004 ; ion ; 1210 ; 9000 ; 9999. 4. 12345 J 20103 ; 40040 ; 50001 ; 90600 ; 89346. 5. 500000 ; 708900 ; 102030 ; 309809 ; 379586. 6. 7234651 ; 7090709 ; 9000000 ; 7800040; 3567891. 7. 32567892 ; 34083092 ; 90009000 ; 55500055. 8. 789345621 ; 390085000 ; 222000000. 9. 7009056700 ; 3259287891 ; 8070088200. 10. 32500094001 ; 308506008230 ; 1357986428123. 11. What is the local value of each of the significant digits in the numbers, 72, 359, 4203, 70809, 1300450789 and 3079004078023 ? 12. What does each of the zeros in the numbers 20103, 307005060 and 300508230509 indicate ? 18. Express in words the least number of five figures and tbe greatest number of four figures. NOTATION. 18. Notation is the art of representing by figures a number expressed in words. The method is as follows : % Begin at the left hand, and put down the required figures in the places necessary to express the number, according to the Numera- tion Table ; and fill up the vacant places, if any, with ciphers. Thus, to represent by figures the number, five million^ twenty eight thousand^ three hundred and four > we put down 5 in the place of millions or in the seventh place from the right, 2 in the place of tens of thousands or in the fifth place, 8 in the place of thousands or in the fourth place, 3 in the place of hundreds or in the third place, and 4 in the place of units or in the first place ; and then we fill up the sixth and second places with ciphers ; and the number expressed in figures is 5028304. EXAMPLES. 8. State in figures : 1. Thirteen ; seventeen ; nineteen ; twelve ; eleven. 6 ARITHMETIC 2. Twenty-three ; thirty-four ; forty ; twenty-seven. 8. Seventy-seven ; ninety ; eighty-four ; sixty-three, 4. Three hundred and forty-two ; four hundred and eighty-six 5. five hundred and four ; nine hundred. 6. Two hundred and three ; four hundred and thirty ; five hundred and fifty-five ; four hundred. 6. Eight hundred and ninety-two ; seven hundred and four ; six hundred and forty ; five hundred and twelve. 7. Seven thousand, eight hundred and thirty-five ; nine thou- sand and twenty-eight ; six thousand and nine ; four thousand ; six thousand and eighty-five. 8. Five thousand, nine hundred and ninety-two ; eight thou- sand and seventy-four ; two thousand and three ; four thousand and forty ; three thousand, four hundred and three. 9. Twelve hundred ; eighty thousand and eight ; eighteen thousand, four hundred and fifty-four ; thirty-six thousand and twelve ; ninety thousand, 10. Twenty thousand and seventy ; thirty thousand and eight ; fifty-four thousand, four hundred ; sixteen thousand and four. 11. Four hundred and five thousand ; eight hundred thousand and forty ; seven hundred and two thousand and seventy-four. 12. Three million, nine hundred and four ; nine million, four hundred ; fifteen million and fifty ; one hundred and eight million, three thousand and four ; four million and five thousand. 13. Five thousand million, seven hundred thousand and twenty- eight ; three hundred and fifteen thousand seven hundred and sixty-four million, nine thousand and three. 14. Three billion and fifty ; four hundred and five billion, tea million, twenty thousand and seven ; one billion, one million, one thousand ; six billion and six. 15. Five hundred and twelve billion, two hundred and fifty-five thousand seven hundred and sixty-two million, seven hundred and thirteen thousand, four hundred and seventy-three. 16. Twelve billion and twelve ; seven hundred billion, seven hundred thousand and seven hundred ; three billion, three million, three thousand, three hundred and three. 17. Seven thousand three hundred and five billion, five hundred and two million, six thousand and twenty-four ; forty-seven billion, forty-seven million, forty-seven thousand and forty-seven. 18. State in figures the least number of seven figures and the greatest number of five figures. THE INDIAN METHOD OF NUMERATION 7 10. One boy wrote 70007007 and another wrote 777 when told to write 'seven thousand, seven hundred and seven* in figures ; what mistakes did they commit ? THE INDIAN METHOD OF NUMERATION. 19. The following is'the Indian Numeration Table in common use -S b*; . *! d i< . w H cj C3 n3 w XJ u * Q "d HH 198, 76, 54, 321 The above number is read thus : One hundred and ninety-eight crores, seventy-six lacs, fifty-four thousand, three hundred and twenty-one. Note. The Hindu names of places of figures are as follow : akcty dasha, shata t sahasra> oyut> laksha (lac), niyut, coti (crore), arbud,padma, kharba>nikharba, mohqpadma^sanku.jaladhi^antya^ madhya, parardhya* EXAMPLES. 3. Express in words according to the Indian Numeration : 1- 345543 ; 3020050 ; 7990570 ; 7050304. 2. 12345678 ; 305750080 ; 45000000. 3. 230078001 ; 7080904080 ; 3794857612. 4. 8274057009 ; 3500001230 ; 3103705040. 6. 1234567890 ; 6000789000 ; 5010702009. Express in figures : 6. One lac, fourteen thousand ; seventy-eight lacs ; fifteen lacs, four thousand and thirty ; seven lacs and seven. 7. One crore, five hundred ; twenty-eight crores, three lacs and four ; twenty crores ; one crore, one lac, one thousand and one. 8. Three hundred crores, five lacs, four thousand ; one hup- dred nd one crore, one lac, one hundred and one. 8 ARITHMETIC 9. Three hundred and twenty-eight crores, seventeen lacs, forty-five thousand, seven hundred and fifteen. 10. Seven hundred and five crores, seventeen lacs, twenty- four thousand, seven hundred and thirty-eight. 11. How many thousands are in a lac ? How many lacs in a million ? How many millions in a crore ? 12. Read according to the Indian numeration the number one hundred and three million, twenty-eight thousand, four hun- dred and one. 13. Read according to the English numeration the number one hundred and three crores, seven lacs, seven hundred and four. THE ROMAN SYSTEM OF NOTATION. 20. In this system the symbols chiefly employe djare I, V, X, L, C, D and M which represent i, 5, 10, 50, 100, 500 "and 1000 res- pectively. Again a bar placed over a letter increases its value a, thousand-fold ; thus X represents 10,000. The following table will explain the method of representing any number by means of the above symbols. ~ CD 400 D 500 DC 600 DCC 700 DCCC 800 CM 900 M fooo MCD 1400 MCM 1900 I I XI II XXX 30 II 2 XII 12 XL 40 III IV 3 4 XIII XIV 13 14 L LX V 5 XV 15 LXX 70 VI 6 XVI 16 LXXX 80 VII 7 XVII 17 xc 90 VIII 8 XVIII 18 C 100 IX 9 XIX 19 CC 200 X 10 XX 20 ccc 300 MDCCCLXXXIX 1889 DLXDCCXLII MM 2000 560742 Express in Arabic notation : 1. VI. 2. IX. 3. XLIX. 6. LXXV. 6. CCLXIV. 7. DCIX. 8. MCMXC.10. LXX. 4. XC1X, 8. DCLXIV. 11, MMDCCLXIV. Express in Roman notation : 12. 44. 13. 66. 14. 79. 16. 149. 17. 43 6 - 18- 990- 16. 83. 20. 5670. fcl. 3M9. 22. 45973. 23. loooooo. ADDITION III. ADDITION. 1. Addition is the method of finding a single number which is equal to two or more given numbers taken together. The given numbers are called summands, and the single number obtained by adding them is called their sum or amount. . The sign 4- signifies thart the two numbers between which it is placed are to be added. Thus, 7-4-2 signifies that 2 is to be added to 7. The sign + is called the plus sign, and 7 + 2 is read "seven plus two". The sign stands for the words "is equal to" or "equals." Thus, 2 + 3=5 states that the sum of 2 and 3 is equal to 5. The sign = is called the sign of equality, and 2 + 3=5 is reac * " two plus three is equal to five" or "two plus three equals five." 23. The numbers one, two, three, four, five, etc.* being taken in order, if we add the number one to any one of them, we get the number next following : thus 1+1=2 ; 2+i3 ; 3 + l* a 4; and so*on. We obtain the sum of 5 and 3 thus : 5+3=5+2+1 -5+ 1 + 1 + 1 =6+1+1 = 8. Results thus obtained are registered in the following table, called the Addition Table, which the learner should commit to memory. I and 2 and 3 and 4 and 5 and 6 and 7 and 8 and 9 and I are 2 I are 3 i are 4! i are 5 i are 6 i are 7 i are 8 i are 9 i are 10 2 ... 3 2 ... 4 2 ... 5J2 ... 6 2 ... 7 2 ... 8 2 ... 9 2 ...10 2 ...II 3 - 4 3 - 5 3 ... 6| 3 ... 7 3 ... 8 3 -. 9 3 ...10 3 ...ii 3 ...12 4 *.. 5 4 ... 6 4 ... 714 - 8 4 .- 9 4 ...10 4 ...ii 4 ...I2J4 ...13 5 ...6 6 ... 7 6 ::: \ 5 -. 8 5 ... 9 6 ... 9l6 ...10 5 ...10 6 ...ii 5 ...ii 6 ...12 5 ...12 6 ...13 5 .-13 6 ...14 5 -14 6 ...ic 7 ... 8 7 - 9 7 ...ioJ7 ...ii 7 ...12 7 ...13 7 ...147 -IS 7 ...16 8 ... 9 8 ...10 8 ...ii|8 ...12 8 ...13 8 ...14 8 ...15 8 ...16 8 ...17 9 ...10 9 ...ii 9 ...12 9 ...13 9 ...14 9 ...IS 9 ...16 9 ...17 9 ...18 Example. Add 7 + 8 + 9 + 8. Process: 7 + 8i5 ; i5 + 924 ; 24 + 8 = 32. Arts. Note. As facility in mental addition is the basis of all accurate facility in the subsequent processes of Arithmetic, the pupil should have a sufficient number of exercises in mental addition before he proceeds further. The use of fingers should be strictly prohibited. 10 ARITHMETIC EXERCISES IK MENTAL ADDITION. Af. B. The following exercises are not considered sufficient ; they are i ntended only to show the nature of the questions that might be asked. 1. What is the sum of (a) 2 and 9 ; 3 and 4 ; 8 and 7 ; 7 and 5 ; 9 and 9 ; 9 and 7 ; 3 and 7 ; 8 and 5 ; 9 and 6 ; 6 and 8 ; 8 and 9 ; 7 and 3 ? () 10 and 7 ; 20 and 8 ; 30 and 6 ; 50 and 9 ; 70 and 5 ? (c) ii and 6 ; 12 and 7 ; 26 and 4 ; 36 and 3 ; 72 and 7 ? *(*0 15 and 7 ; 16 and 8 ; 22 and 9 ; 37 and 6 ; 85 and 9 ; 43 and 8 ; 49 and 9 ; 28 and 7 ; 68 and 7 ; 98 and 7 ; 99 and 9? 2. Add (a) 5 to 7, to 17, to 27, to 37, etc. (b) 7 to 9, to 19, to 29, to 39, etc. (<;) 8 to 8, to 1 8, to 28, to 38, etc. 8. (a) How much do i and 2 make ? 3 and 2 ? 5 and 2 ? etc. (6) How much do 2 and 3 make ? 5 and 3 ? 8 and 3 ? etc. (<r) How much do 3 and 5 make ? 8 and 5 ? 13 and 5 ? etc* N. B. When the pupil has acquired a little facility the above question- may, with advantage, be put in the following form : 4. Count by increments of 6 starting at 4. Answer. 4, 10, 16, 22, 28, 34, etc. 6. I have 10 marbles in one hand and 7 in the other ; how- many marbles have I in all ? 6. Twelve articles make a dozen ; how many in two dozen ? 7. Ram had 19 marbles and he has won 8 ; how many marbles- has he now ? * 8. I have purchased a table for 16 rupees and a chair for 7 rupees ; how many rupees have I spent in all ? 9. If mangoes are selling at the rate of 13 for the rupee, how many shall you get for two rupees ? 10. John bought 25 mangoes and 9 oranges ; how many fruits tf id he buy in all ? *The following process in mental addition may be recommended fop beginners : =20 + 2 = 22. But the process should be abandoned as soon as facility in addition has- been acquired* ADDITION U 11. You are 13 years old ; your brother is 7 years older than you ; what is the age of your brother ? 12. If I give you 20 rupees I shall have 1 5 rupees left in my purse ; how many rupees have I ? 13. A boy has lost 8 marbles and has 27 left ; how many had he at first ? 14. You have 23 marbles in your pocket ; I give you 9 ; how many have you now in all ? 15. A man bought 35 maunds of rice on a certain day, and 9 maunds on the next day ; how many maunds did he buy in all ? 16. A man's age is 47 years ; how old will he be 7 years hence ? 17. If you buy 56 mangoes and your brother 8 more than you* how many does your brother buy ? 18. What is the number from which if I take 1 5 there will remain 60 ? 19. A man bought a table for 75 rupees and gained 5 rupees by selling it ; for how many rupees did he sell it ? 20. A man gave 19 rupees to his wife, 7 rupees to his son and 4 rupees to his daughter ; how many rupees did he give away in all ? 21. What is the united length of five roads which are i, 2, 3, 4 and 5 miles long respectively ? 22. I bought a book for 6 annas and a bottle of ink for 4 annas more than the book ; how much did I spend in all ? 23. A man sold 9 oranges to A 9 to B 7 more than to A : how many did he sell in all ? 24. Ram bought 2 mangoes at 4 annas each and 8 oranges at one anna each ; how much did he pay to the fruit-seller ? 26. From a rope are cut off first 27 yards, then 8 yards, and there are 7 yards left ; what was the length of the rope ? 24. In the case of large numbers the process of M addition is as follows : Example. Add together 378, 409 and 56. We write down the numbers, one under another, thus 378 409 _ 843 placing units under units, tens under tens, hundreds under hun- dreds, and so on and then draw a line under the lowest line of 12 ARITHMETIC figures. Under this line we place the sum which is found in the following way : We first add the units, thus (8+9 + 6) units = 23 units =2 tens + 3 units ; we place the 3 under the column of units and carry on the 2 tens for adding to the column of tens. Next we add the tensi thus (2+7+0+5) tens =14 tens =i hundred +4 tens ; we place the 4 under the column of tens and carry on the i hundred for adding to the column of hundreds. We then add the hundreds, thus (1 + 3 + 4) hundreds = 8 hundreds ; and we place the 8 under the column of hundreds. " Mental Process : 8, 17, 23; carry 2, 9, 14 ; carry I, 4> 8. EXAMPLES. 5. N. B. Sums should be dictated and the pupils required to read out the answers in words. The same sum may be given several times by altering the order of the summands. Add together I. 3 2. 6 3. 8 4. 7 5. 8 59759 98988 47 799 6. 56 7. 73 8. 40 9. 90 10. 79 42 26 37 j|o 84 II. 375 12. 879 13. 79 14. 986 16. 984 208 82 40 742 76 740 190 673 999 9,40 16. 7643 17. 429 18. 3098 19. 4807 248 7 207 309 5004 84 40 4 1134 9476 329 500 20. 28 21. 58073 22. 839 23. 38756 4007 9705 2058 50952 350 368 476 78095 9 78000 8205 34560 302 29 4746o ' 32308 24. 89763 26. 38760 26. 467895 27, 79 25964 5^07 58009 3025 73896 304 5555 329 58926 19 795073 876502 32157 7 567982 39879 98756 374 368000 300 ADDITION 13 28. 9038 29. 7 ' 30. 3578924 31. 9357350 30054 7000007 5893679 2984721 5028 34003 8279563 8305902 76 404040 9528789 7650729 9 36ooo 3474923 8472038 938050 38 8 i?34^? $679824 Find the sum of 32^ 804*^056, 48, 397834 and 909. 33. 73568, 9340, 8654, 76, 703 and 98. 34. 74,^*79048, 309, 8000386, 43 and 3002. 35. 300, 785, 897634, 12345, 207 and 20708. Find the value of 36. 432398 + 7867 + 83989 + 7030. 37. 70 4- 8200 4- 7396 4- 5678920 + 97 + 2. 38. 3 + 309 + 29 + 307895 + 32534-500. k j 39. 87 + 9800000 + 80234+10201+34567 + 9. 10. 3456 + 456 + 56+6 + 76000 + 984530789. 41. Add together the following numbers : seventy-nine ; three thousand, four hundred and fifty ; sixty-six thousand, six hundred and ninety-four ; four thousand and four ; eighty, 42. Find the total of six hundred and ninety-two ; four lacs, forty-five thousand and seven ; ninety-eight lacs, seven hundred ; forty-five ; seven. 43. Find the amount of seven hundred and forty-six million, seventy-four thousand, nine hundred and sixty-two ; eighty-six thousand, five hundred and four ; twelve million, seven thousand and three ; ninety-one ; seven million and seven. 44. How much are nineteen + seven lacs, seven thousand and seven + three hundred and four crores, seventy-four lacs and twenty-nine + eight crores, eight lacs, eight thousand and eight + seven thousand, seven hundred and forty-two + six + three lacs, four hundred and seven ? 45. Find the amount of 76, 378046, 30567, 8, 9345, 300009, 3708, 309* 37805892, 28, 7923000 and 342. 46. What is the number from which if 3457 be taken 479 is left ? 47. A man was born in 1856 ; in what year will he be 34 years of age ? 48. January has 31 days, February 28, March 31, April 30, May 31, June 30, July 31, August 31, September 30, October 31, November 30 and December 31 ; how many days are there in the whole vear ? 14 ARITHMETIC 49. State how many boys are in a school in which there are 125 in the first class, 87 in the second, 96 in the third, 107 in the fourth, 70 in the fifth and 256 in the other classes. 50. A garden contains 327 mango trees, 704 cocoanut trees, 456 date trees, 528 orange trees and only 25 tamarind trees : how many trees are there in all ? 51. A certain town contains 87,903 Hindus, 48,093 Maho- medans, 723 Europeans, 1,309 Eurasians and 159 other races: what is the total population of the town ? 52. A gentleman bought three pieces of land in a town for 9,700 rupees ; he built a house on one piece at a cost of 7,825 rupees, another on the second piece at a cost of 21,750 rupees, and a third on the remaining piece at a cost of 2,729 rupees : what sum did he spend in all ? 53. We imported 53,89,082 maunds of salt in January 1885 ; 7,09,280 maunds in February and 10,94,803 maunds in March : what was the entire weight imported in the first 3 months of 1885 ? 54. I bought four baskets of mangoes ; the first contained 246 mangoes ; the second 319 ; the third 19 more than the second ; and the fourth as many as the first and second together : how many mangoes did I buy ? 55. What is the number from which if I first take 70835 and then 85679, there will remain 7040 ? IV. SUBTRACTION. 25. Subtraction is the method of finding the number which Is left when the smaller of two given numbers is taken from the greater. The greater of the two given numbers is called the minuend) the less is called the subtrahend, and the number found by sub- traction is called the remainder or difference. The sign - , placed between two numbers, signifies that the second number is to be subtracted from the first. Thus 7-4 signifies that 4 is to be subtracted from 7. The sign - is called the minus sign, and 7-4 is read "seven minus four." 6. It follows from the definition of subtraction that it is the process of finding the number which must be codded to a given number to make a larger given number. Hence subtraction is sometimes called complementary addition. We are able to subtract a small number from another) from the known results of the Addition Table. Example. 7-43$ because 4+3-7. SUBTRACTION 15 EXERCISES IN MENTAL SUBTRACTION, 1. Take 3 from 8 ; 4 from 9 ; 5 from 7 ; 6 from 9 ; 5 from 8. 2. What is the difference between 10 and 6 ; 12 and 8 ; 16 and 9 ; 13 and 7 ; n and 6 ; 16 and 8 ; 18 and 9 ; 15 and 7 ; 17 and 8 ? 3. How many does 7 leave from 28 ; 5 from 27 ; 6 from 56 ; 7 from 99 ; 3 from 57 ; 8 from 88 ; 6 from 49 ; 4 from 26 ? 4. Subtract 9 from 22 ; 8 from 35 ; 7 from 42 ; 6 from 51 ; 5 from 60 ; 4 from 73 ; 8 from 86 ; 9 from 92 ; 5 from 81. 5. (a) What remains when we take 6 from 30, 6 from 24, 6 from 1 8, 6 from 12, 6 from 6 ? (b) What remains when we take 7 from 100, 7 from 93, 7 from 86, etc. ? (c) Count by decrements of 6 commencing at 100. Ans. ico, 94, 88, etc. 6. Take 7 from the sum of 5 and 6 ; 9 from the sum of 6 and 8 ; 6 from the sum of 5 and 4 ; 8 from the sum of 6 and 7. 7. A boy who had 1 5 marbles has lost 8 : how many has he left ? 8. I have 17 rupees in my purse ; if I give you 9 rupees, how many rupees shall I have left ? 9. Your brother's age is 14 years ; you are 5 years younger than he : how old are you ? 10. In a class there are 19 boys on the roll ; on a certain day 6 boys were absent : how many were present ? 11. A man had 16 rupees ; he gave 7 rupees to his wife and the rest to his son : how much did the son get ? 12. A man bought a table for 19 rupees and sold it for 2.5 rupees : how much did he gain ? 13. There are 37 mangoes on a tree ; if 8 be plucked, how many will be left ? 14. Ram has 48 marbles ; if Gopal had 9 more than what he now has, he would have as many as Ram : how many has Gopal ? 15. I have 16 marbles ; John has 28 ; how many more should I get to have as many as John ? 87. In the case of large numbers the process of subtraction is as follows : Example i. Subtract 34 from 86. We place the smaller number under the greater) as in 86 Addition. We now take 4 units from 6 units, and set down 34 the result, which is 2 units, under the column of units ; 52 16 ARITHMETIC next, we take 3 tens from 8 tens, and set down the result, 5 tens,, under the column of tens. Thus the remainder obtained is 52. Example 2. Subtract 368 from 952. Here, proceeding as in the previous example, we meet 95 2 with the difficulty of taking a greater digit from a less, and 3^8 to get over this difficulty we avail ourselves of the following 584 principle, usually termed borrowing : The minuend and subtrahend may be increased by the same number 'without altering their difference ; and we reason thus : We cannot take 8 units from 2 units ; we therefore add 10 units to the 2 units, making 12 units, and we take 8 units from the 12 units, and set down the result, 4 units, under the column of units. Having increased the upper number by 10 units, we add, by way of compensation, i ten to the lower number, changing 6 tens into 7 tens. We have now to take 7 tens from 5 tens, and as we cannot do so, we add 10 tens to the 5 tens, making 1 5 tens, and we take 7 tens from the 1 5 tens, and set down the result, 8 tens, under the column of tens. Having increased the upper number by 10 tens, we add, by way of compensation, i hundred to the lower number, changing 3 hundreds into 4 hundreds. We now take 4 hundreds from 9 hundreds, and set down the result, 5 hundreds, under the column of hundreds. Note. Instead of the above process it will be practically con- venient to determine how much must be added to the subtrahend to make up the minuend. Example. Subtract 576 from 829. We are to find the number which being added to 576 makes up 829. We place the smaller number under the greater, as in Addition. We now see that 6 units + 3 units =9 units; we therefore set down the 3 under the column of units : next, 7 tens + 5 829 tens 12 tens ; we set down the 6 under the column of tens, $76 and carry i hundred : then, (14-5) hundreds -4- 2 hundreds = 8 253 hundreds ; we set down the 2 under the column of hundreds. Mental Process : 6 and 3 are 9 ; 7 and 5 are 12 ; carry i, 6 and 2 are 8. EXAMPLES. 6. Perform the following subtractions : L 78 2. 95 3. 356 4. 789 5. 7825 25 43 13i 246 3504 SUBTRACTION I? 6. 11. 16. 21. 64 7. 97 8. 39 4 795 12. 480 13. 606 390 5380 17. 54090 18. 739 7073 86 , Z* 977 Z2? 84321 53789 708093 20503 9. 94 1' 93 85 6p 14. 843 15. 904 384 58? 19. 85858 20. 54321 58585 12345 20004 22. 789356 23. 17325 99999 26. 82439-76893. 28. 790256-82789. 3O. i oooooo 999999. 82. 780004-389210. 24. 805400 25. 7000203 70053 500956 27. 93406-7990. 29. 80000-76438. 81. 777770-88889. 33. 100956-39897. 34. What number must be added to each of the following numbers to make the sum equal to a million ? 19, 305,9475, 99446$ and 435oo. 35. What number must be taken from 93867 to leave 903 ? 36. By how much does a lac exceed twenty-nine? 37. By how much is a crore greater than one thousand and one ? 38. By how much is seventy-nine less than ten thousand ? 39. The Duke of Wellington was born in 1769 and died in 1852 ; how old was he at his death ? 40. Sir Isaac Newton died in 1727 aged 85 years : when was he born ? 41. Mount Everest is 29,100 feet high ; Kinchinjunga is 28,177 feet high : by how many feet is the former higher than the latter ? 42. If the receipts of a railway company are 3,98,450 rupees and the expenses 2,80,769 rupees, what are the profits ? 43. A merchant bought goods for 3,000 rupees and sold them tor 3,325 rupees : how much did he gain ? 44. If I had 540 rupees more than I have, I should be able to clear a debt of 10,000 rupees : how much have I ? 45. The sum of two numbers is 93875, and the greater number is 77359 : what is the smaller number ? 46. The smaller of two numbers is 3799, and their sum is 780900 : what is the greater number ? 47. What number must be subtracted from 7389 that the remainder may be 999 ? C. A. 2 1 8 ARITHMETIC 48. Find the difference between the sum and difference of a million and a thousand. 40. A has 39)876 rupees ; ^has 3,758 rupees less than A ; and C has 876 rupees less than B j how much has C ? 50. A boy when told to write 'three thousand, four hundred and five' in figures wrote 30004005 ; how much more did he write ? 61. A boy wrote 500403 when he was told to write 'fifty lacs, four thousand and three' in figures ; how much less did he write ? 8. The number to which the sign + is prefixed is called a positive number ; and the number to which the sign is prefixed is called a negative number. If no sign is prefixed to a number it is to be considered as positive. Numbers connected by the sign + or are called terms. The most convenient method of finding the value of an expres- sion (in which several numbers are connected by the sign + or - ) is to find the sums of the positive and negative numbers separately and then to take their difference. Example. Find the value of 273-369+821-403. Now, 273 + 821 = 1094 ; and 369 + 403^772 ; .'. the result required 1094 - 772 322. EXAMPLES. 7. Find the value of each of the following expressions : L 973-724 + 209. 2. 78965-B795-7386. 8. 8703-7955 + 3002-1030. 4. 1600-924-300-88. * 6. 94567 + 3285-77777-304-4-64. 6. To 753-98 + 7 I first add 329, and then take the difference of 720 and 699 from the sum ; what is the result ? 7. By how much is the difference of 7203 and 4980 less than their sum ? 8. By how much does the sum of 7985 - 899 and 7003 exceed their difference ? 9. The greater of two numbers is 94047, and their difference is 909+350 ; what is the other ? 10. What number must be added to 329+408-540 that the sum may be one lac ? MULTIPLICATION V. MULTIPLICATION. 9. Multiplication is a short method of finding the sum of a certain number of repetitions of a given number. The number to be repeated is said to be multiplied^ the number which indicates how often it is to be repeated. Thus? when 4 is multiplied by 3, the result is 4 + 44-4 or 12. The number which is multiplied is called the multiplicand the number by which it is multiplied is called the multiplier and the resulting number is called the product. The sign of multiplication is x . Thus 7x4 signifies that 7 is to be multiplied by 4, and is read "seven into four" or "four times seven." Sometimes a dot ( . ) is used instead of x . 3O. The multiplier and the multiplicand may be interchanged ithout altering the value of the product. Thus 3x4 = 4x3 ; for, 4 = 34-3-1-3 + 3 = 12, and 4x3=4-4-4 + 4= 12. The multiplier and multiplicand are called factors of the product. 81. The following Multiplication Tables must be committed to memory by the pupil. First Table. wi i 2 3 4 5 6 7 8 9 10 Once i 2 3 4 5 6 7 8 9 10 Twice 2 4 6 8 10 12 H 16 18 20 Thrice 3 6 9 12 15 18 21 24 27 30 4 times 4 8 12 16 20 24 28 32 36 40 5 times 5 10 15 20 25 30 35 40 "48" 45 50 6 times 6 12 18 24 30 36 42 54 60 7 times 7 14 21 28 35 42 , 49 Jl 64 63 70 8 times 8 16 24 32 40 48 56 72 80' 9 times 9 18 27 36 45 54 63 72 Si 90 10 times 10 20 30 40 50 60 70 80 90 ICO ARITHMETIC Second Table. I 2 3 4 5 6 7 . 8 9 10 II timea II 22 33 44 55 66 77 88 99 no 12 times 12 24 36 48 60 72 84 96 108 120 13 times 13 26 39 52 65 78 91 104 7 130 14 times 14 28 42 56 70 84 98 112 126 140 15 times 15 30 45 60 75 90 105 120 135 ISO 1 6 times 16 32 48 64 80 96 >II2 128 144 160 17 times 17 34 Si 68 8S 102 119 136 153 170 18 times 18 36 54 72 90 108 126 144 162 1 80 19 times 19 38 57 76 95 114 133 152 171 190 20 times 20 40 60 80 100 120 I 4 160 I 80 200 Third Table. ii 12 13 14 15 16 17 18 19 20 II times 121 132 143 154 165 176 187 198 209 220 12 times 144 156 168 180 192 204 216 228 240 13 times 169 182 195 208 221 234 247 260 14 times 196 210 224 2 3 8 252 266 280 15 times 225 240 255 270 285 300 1 6 times 256 272 288 34 320 17 times 289 306 323 340 1 8 times 324 342 360 19 times 361 380 20 times 400 MULTIPLICATION 21 EXERCISES ON THE MULTIPLICATION TABLE. (Oral.) 1. How much is 7 times 6 ? 8 times 9 ? 12 times 12 ? etc. 2. Multiply 12 by 8 ; 9 by 7 ; 16 by 9 ; etc. 3. What is the product of 9 and 9 ? of 16 and 6 ? etc. 4. What is the sum of 6 repeated 9 times ? 15 repeated 8 times ? etc. 5.' What number is as great as 10 times n ? 7 times 9 ? etc. 6. If 9 boys have 6 marbles each, how many have they all together ? 7. How many rupees are there in 12 boxes, each containing 1 1 rupees ? 8. Sixteen annas make a rupee ; how many annas are there In 5 rupees ? 9. Fifteen boys sit on each form in a school, and there arc fifteen forms ; how many boys are there ? 10. The multiplicand is n and the multiplier is 13 ; what is the product ? 11. The factors of a product are 9 and 19 ; what is the product ? 12. When mangoes are 20 for a rupee, how many can you buy for 5 rupees ? 13. There are 7 days in a week ; how many days are there in 8 weeks ? 14. In a house of 4 stories there are 1 5 rooms on each story ; how many rooms are there in the house ? 16. If a cow be worth 1 5 rupees, how much will you have to pay for 9 cows ? 16. On a page of a book there are 17 lines, and each line con- tains 19 letters ; how many letters are there in the page ? 17. By how much is 7 times 1 1 less than 90 ? 18. By how much is 3 times 16 greater than 35 ? 19. What number exceeds 9 times 9 by 19 ? 20. How many legs ha e 7 horses and 3 cows got altogether ? 3. We now proceed Jo show how large numbers are multiplied. Example. Multiply 2095 by 3. We arrange the numbers thus : 2095 3 6285 product. *2 ARITHMETIC The product is found in the following way : 3 times 5 units is i5 units ; we set down 5 in the place of units, and carry on I for adding to tens : next, 3 times 9 tens is 27 tens } and adding I carried, the result is 28 tens ; we set down 8 in the place of tens, and carry on 2 for adding to hundreds : next, 3 times o is o, * and adding 2 carried, the result is^ 2 hundreds ; we set down 2 in the place of hundreds : then, 3 times 2 thousands is thousands ; and we set down 6 in the place of thousands. Thus the product is 6285. Mental Process : 3 times 5, i5 ; carry I, 3 times 9, 28 ; carry 2, 2 ; 3 times 2, 6. JV. B. The student will see that the above short process is substantially the same as the following extended process of addition. 2095 2095 2095 6285 EXAMPLES. 8. Multiply 1. 23 by 2. 2. 32 by 3. 3. 21 by 4. * 39 by 5. 6. 47 by 6. 6. 58 by 9. 7. 98 by 8. 8. 76 by g. 9, 85 by 9. 10. 329 by 3. 11. 405 by 7. 12. 879 by 9. 13. 3245 by 6. 14. 7089 by 5. 15. 9206 by 8. 16. 78956 by 4. 17. 89035 by 7. 18. 85503 by 9, 19. 34079 by 2, 3, 4, 5, 6, 7, 8, 9. 20. Find the value of 725 + 7254-725 + 725 + 725. 33. If we write a cipher to the right of a number its value is increased tenfold : hence, when we multiply a number by 10, the product is obtained by annexing o to the number. Thus 23 x 10 230. Similarly, when we multiply a number by 100, iooo,..,the product is obtained by annexing oo, ooo,... to the number. Also, if we have to multiply a number by 30, we may first multiply it by 3, and then annex o to the result ; the final result will be the product required. So also, if we have to multiply by 300, we may first multiply by 3 and then annex oo to the result. o ; for o+o+oo. MULTIPLICATION 23 Example. Multiply 329 by 600. Process : 329 600 197400 Ans. EXAMPLES. 9. Find the product of !- 359 by 30. 2. 7035 by 40. 3. 3905 by 50. 4. 703 by 600. 6. 39 by 900. 6. 8229 by 700. 7. 3005 by 8000. 8. 9004 by 9000. 9. 30503 by 6000. 10. 7295 by 90, 800, 7000, 60000, 500000. 34. It is clear from the definition of multiplication that, if we have to multiply a number by 5, we may multiply it separately by 2 and 3, and then add the two results ; the final result will be the product required : if we have to multiply a number by 23 we may multiply it separately by 3 and 20, and then add the two results. Example I. Multiply 728 by 329. (A) 728 (B) 728 329 _3?9 6 5 52= product by 9. 6552 14560-= 20. 1456 218400= 300. 2184 2395 12 = product by 329. 239512 Here, to obtain the product of 728 by 329, we multiply 728 by 9, 20 and 300 separately, and add the three results. The partial products are found by the methods explained in the two preceding articles. In practice we do not annex the zeros in multiplying by 20 and 300 (because they have no effect in the addition which we perform afterwards) and our work stands as at (B). OBSERVE that the multiplier must be placed under the multi- plicand as in Addition ; also that, in all cases, the first figure on the right of each partial product must be placed in the same vertical column with the figure by which the product is obtained. Note 1. We may multiply by the figures of the multiplier ID any order we like, bearing in mind the foregoing observation. (I) 728 (2) 728 329 329 1456 by 2. 2184 by 3. 2184 by 3. 1456 by 2. 6 552 by 9. 6552 by fr 239512 239512 ARITHMETIC Note 2. When the multiplier or multiplicand or both end with ciphers, it is convenient first to omit them in working and then to annex as many ciphers to the product as have been omitted. Example 2. Multiply 37008 by 4203 ; 4309 by 12300 ; 290 by 243 ; and 40300 by 4370. (0 37008 4203 111024 74016 148032 155544624 (2) 4309 12300 1292^" 86l8 4309 53000700 (3) 290 2 43_ 87 116 58 70470 (4) 40300 __ 2821 1209 1612 176111000 EXAMPLES. 10. Perform the following multiplications : 904 x 98. 8762 x 904. 8463x340. 90407 x 6050. 820078x90072. 17. 19. 21. 23. 25. 1. 375X54. 2. 4. 4972 x 345. 5. 7. 708 x 708. 8. 10. 89025x8007. 11. 13. 863400 x 70600. 14. 16. 8573056 x 900082. 18. 9876507x39421. 20. 8976543x978653, 22. 307650x90060. 24. 830038 x 700208. 26. 35756x6570002. 3. 740 x 69. 6. 8072 x 972. 9. 8239x5009.- 12. 123456x70809. 15. 480390x8907. 7^90250x3009000. 3700x809025000. 370304x6070370. 784692x80075. 3257650x3257650. 209030 x 400800600. 27. Obtain the following products by using one line of multiplica- tion only ; 28. 4329x11. 29. 3809x12. . 30. 7204x13. 31. 7082x14. 32. 4890x15. 33. 8789x16. 34. 13570x17. 35. 28070x18. 36. 4356x19. 87. There are 192 pies in a rupee ; how many pies are there in 3705 rupees ? 38. A book contains 579 pages, and each page contains 3749 letters ; how many letters are there in the whole book ? 39. If the price of one cottah of land in Calcutta be 975 rupees, what is the price of 325 cottahs ? 40. If 29390 persons cross the Hughly Bridge daily, how many cross in a year of 365 days ? MULTIPLICATION 25 41. What is the weight of 739 bags of rice, each weighing 28 mauads ? 42. How many rupees must be paid for 6 elephants at 3479 rupees each, and 16 horses at 765 rupees each ? 43. A cistern has a leak by which 78 tolas of water come out per hour ; if the full cistern is emptied in 48 hours, how many tolas of water does the cistern hold ? 35. Example. Find the continued product 28 of 28, 8 and 3. _J8 We multiply 28 by 8, and the product by 3, the 224 final result being 672. _3 672 Ans> EXAMPLES. 11. Find the following continued products : I. 27x8x2. 2. 703x85x79. 3. 8050x70x30. 4. 59x85x76x5. 6. 3205x9x8x5. 6. 99x88x77x66, 7. How much is twice nine times seventy-three ? 8. A day contains 24 hours, an hour contains 60 minutes, and a minute contains 60 seconds ; how many seconds are there in a day ? 9. 5 tolas make a chatak ; 16 chataks make a seer ; 40 seers make a maund ; how many tolas are there in a maund ? 10. A book contains 339 pages, each page contains 27 lines, and each line contains 45 letters ; how many letters are there in the whole book ? II. How many mangoes are there on a tree which has 29 branches, each branch containing 325 mangoes ? 12. In a railway train there are 46 carriages ; each carriage has 6 compartments ; and each compartment contains 8 persons : how many persons are there in the train ? 36. The second^ third) fourth^... power of a number is the product of two, three^four^.. factors each equal to that number. Thus the second power of 2 = 2x2=4; the third Rr ^ r of 2 2x2x2 = 8. The second power of a number is r^ed its square, the third power its cube. The number itseu is often called \tsjirst power. The symbol 4* is used to express 4 x 4 ; also, 4* is used to express 4x4x4; and so on. The small figures 2, 3, are called indices or exponents of the powers. The process of finding any power of a number is called involution. 26 ARITHMETIC EXAMPLES. I Find the square of 1- I* 2, 3, 4, 5ii9> 20. 2. 24. 4. 68. 5. 100. 7. 248. 8. 729. Find the cube of 10. i, 2, 3, 4,.. .19, 20. 11. 93. 13. 879. 14. 555- 16. Find the value of 25* -1 h 40* - 1 :2 8 -M 3. 50. 6. 112. 9. 874. 12. loo. 15. 309. VI. DIVISION. 37. Division is the operation by which we find how ofteuv one given number, called the Divisor, must be subtracted from another given number, called the Dividend, so that the Be- mainder, if any, may be less than the first given number. The number of times the subtraction is performed is called the Quotient. It will be found that 7 units can be subtracted from 30 units, 4 times, and that then 2 units out of 30 remain over. Hence, when 30 is divided by 7, 30 is the dividend^ 7 is the divisor, the quotient is 4 and the remainder is 2. The sign of division is +. Thus 304-7 signifies that 30 is to be divided by 7, and is read "30 divided by 7" or simply "30 by 7". The symbol ty is also used to denote the same operation of division. 38. It follows from the definition of division that Divisor x Quotient + Remainder Dividend. When there is no remainder the division is said to be exact. In this case "division may be explained as the inverse of multiplica- tion, the quotient being the number whose product by the divisor is the dividend. 39. By division we break up a number (dividend) into equal parts : if the divisor represents the magnitude of a part, the quotient gives the number of the parts ; if the divisor represents the number of the 'parts, the quotient gives the magnitude of one of the parts, Example I. 30 oranges are divided among boys so that each boy gets 7 oranges ; how many boys get a share ? (Ans. 4 boyst 2 oranges remainder.) DIVISION 27 Example 2. 30 oranges are divided equally among 7 boys ; how many does each boy get ? (Ans. 4 oranges each, 2 oranges remainder.) N. B. The teacher should explain how in both of these cases the result may be obtained by repeated subtractions. 4O. The division of numbers not greater than 400 by numbers not greater than 20 is effected by means of the Multiplication Table. Example. Divide 59 by 7. Here, we have to find how often 7 may be subtracted from 59* or in other words, how many times 7 is contained in 59. We may find the quotient and the remaindel' by successive subtractions of 7 from 59. But we are saved the trouble of repeated subtractions by using a known result of the Multiplication Table. Thus, since 8 times 7 is 56, 59-1-7 gives 8 as quotient and 3 as remainder. EXERCISES IN MENTAL DIVISION. 1. How many times is 5 contained in 20 ? 8 in 72 ? 9 in 54 ? 14 in 14 ? 16 in 128 ? etc. 2. How many times can you subtract 7 from 56 ? 6 from 48 ? 9 from 8 1 ? 18 from 306 ? etc. 3. Divide 84 into 7 equal parts ; 104 into* 13 equal parts ; etc. 4. What is the fourth part of 36 ? sixth part of 54 ? twelfth part of 108 ? etc. 5. In 54 how many times 4, and how many over ? how many times 5, and how many over ? etc. 6. What is the remainder when 7 is subtracted as often as possible from 64 ? 6 from 42 ? 8 from 84 ? etc. 7. Find the quotient and remainder when 43 is diyided by 6 ; 70 by 8 ; 85 by 9 ; 190 by 16 ; etc. 8. How many times does the fourth part of 72 contain 3 ? fifth part of 70 contain 7 ? etc. 9. 135 mangoes were divided equally among 1 5 boys; how many did each get ? T 10. 54 oranges are distributed equally among the children of a family, and each one gets 9 ; how many children are there in the family ? 11. There are 16 annas in a rupee ; how many rupees are there in 144 annas ? 12. I bought a dozen chairs for 72 rupees ; what is the price of a stogie chair ? *3 ARITHMETIC 13. How many yards of cloth at 12 annas each can I buy for annas ? 14. How many dogs have 80 legs ? 41. When the dividend and divisor are any numbers, the process of division is as follows : Example. Divide, 88909 by 24. The form of the operation is 24 ) 88909 ( 3704 Quotient. 72 169 ,168 109 J* 3 Remainder. The explanation is this : We first take 8, and we find that 24 is not contained in it : we therefore take 88 and find how often 24 is contained in 88, and as it is contained three times, we set down 3 as the first figure in the quotient ; then multiply 24 by 3 and subtract the result 72 from 88 : to the remainder 16 we bring down the next figure in the dividend ; then, as 24 is contained in 169 seven times, we set down 7 as the second figure in the quotient ; then multiply 24 by* 7 and subtract the result 168 from 169 i to the remainder I we bring down the next figure in the dividend ; then, as 24 is not contained in 10 Ve set down as the third figure in the quotient and bring down 9, the next figure in the dividend ; then, as 24 is contained in 109 four times, we set down 4 as the fourth figure in the quotient ; then multiply 24 by 4 and subtract the result 96 from 109. We thus obtain 3704 as quotient and 13 as remainder. N. B. The student will see that in the above process what we really do is this : from 2 4 ) 88909 ( 3000 the dividend we first subtract 3000 times 24, I^opo next from the remainder we subtract 700 16909 ( 700 times 24, and then from the second remainder 16800 we subtract 4 times 24; we therefore altogether jf^ ( 4 subtract (3000 + 700+4) or 3704 times 24 from ^5 88909. The form of this extended operation is ~ ^ ~ - ^ shown at the side. Remr ' J 3 3704 Qt. EXAMPLES. 13. Divide L 376 by 2. 2. 9234 by 2. 3. 7085 by 2. 4. 7000 by 3. 6. 8025 by 3. 6. 90126 by 3. DIVISION 20. 7. 82045 by 4. 8. 10. 12345 by 5. 11. 13. 90403 by 6. 14. 16. 3789 by 7. 17. 19- 38474 by 8. 20, 22. 72124 by 9. 23. 25. 38972 by 10. 26, 28. 77777 by II. 29. 31. 38956 by 26. 32. 34. i oooo by 59. 35. 37. 35896 by 88. 38. 40. 97856 by 141. 41. 43. 89089 by 5 5 5. 44. 46. 398406 by 879. 47. 49. 809345 by 3456. 50. 62. 2080400 by 5456. 54. 47946387 by 7207. 56. 123456789 by 98765. 58, 1080924890 by 72034. 60. 38407890901 by 90735. 62. 297506823 by 708076. 64. 7801849202713 by 926. 45678 by 4- 77777 by 5. 78934 by 6. 32480 by 7. 16042 by 8. 78000 by 9. 32000 by io 57084 by 19. 96100 by 48. 10020 by 74. 28923 by 329. 26 534 by 584; 30321 by 681. 999999 by 8888. 7766334 by 7634 9997770 by 3906. 987654321 by 8642. 187654321 by 12345. 1200730092 by 897324. 208900563000 by 870056. 567892314670 by 8976867. 32813 by 4. 100200 by 5. 87345 by 6. 45986 by 7. 34509 by 8. 90001 by 9. 24560 by 10. 39042 by 16. 72043 by 37- 707070 by 62. 47500 by 91. 13013 by 269. 36780 by 628. 700000 by 991. 3270457 by 1002. 51. . 53. 55. 57. 59. 61. 9. 12. 15. 18. 21. 24. 27. 30. 33. 36. 39. 42. 45. 48. 63. 65. 9876 540456789 by 99. 357435 J on e of them s> 66. The product of two numbers is 705 ; what is the other ? 67. How many men will receive 113 rupees each out of rupees ? 68. How often must 817 be taken to make up 431376 ? 69. What number multiplied by 493 will produce 6409 ? 70. I subtract 3405 from 780953, then subtract 3405 from remainder^ and so on : how often can I do this ? 71. The quotient is 307, the divisor 98 and the remainder what is the dividend ? 72. The population of a certain town is 345330, and of 45 dies annually ; how many die in a year ? 73. A gentleman's yearly income is 19500 rupees ; how much must he spend per week so that he may neither save nor borrow ? (There are 52 weeks in a 'year.) 4068 the 29 one out 30 ARITHMETIC 74. A ship sails 125 miles a day ; how long will it take to sail a distance of 32000 miles ? 75. 2750 bottles are to be packed in boxes, each holding 125 bottles : how many boxes will be required ? SH&RT DIVISION. 4/5. The process of division may be greatly shortened' when the divisor does not exceed 20. Example. Divide 8259 by 6. 6 ) 8259 Quot. 1376, rem. 3. We draw a line under the dividend, and under this we set down the successive figures of the quotient, the multiplication, subtrac- tion, etc. being performed mentally. * EXAMPLES. 14. Divide, employing Short Division, i. 4. 7. 10. 13. 16. 19. -6, ...19, 20 separately. 20. Work examples i to 30 of Examples 13 by Short Division. VII. PROPOSITIONS IN THE FUNDAMENTAL OPERATIONS. ^ 43. To find the sum of any number of the natural numbers beginning with i. Rule. Multiply the last number by the next higher number } and divide the result by 2. Example i. Add together 1+2 + 3+4+. ..+ 15. Here the last number is 1 5, and the next higher number is 16 ; their product is 240 : therefore the sum required 240 -r-2> 120. Example 2. Add together 21 + 22+23+, ..4-35. Here, add together the numbers from i to 35, and also the num- bers from i to 20 ; and subtract the latter sum from the fofrmer. 34561 by 2. 12792 by 5. 34567 by 8. 580046 by ii. 450782 by 14. 3890457 by 17. Each of 3456789, 2. 78930 by 3. 3. 5. 23057 by 6. 6. 8. 19870 by 9. 9. 11. 807040 by 12. 12. 14. 743080 by 1 5. 16. 17. 8207305 by 18. 18. 80704030 and 987654321 80358 by 4. 98400 by 7. 34567 by 10. 135689 by 13 935862 by 16 12345678 by by 2, 3, 4, 19. 5 PROPOSITIONS IN THE FUNDAMENTAL OPERATIONS 31 44. Given the sum and difference of two numbers, to find the numbers. 1 Rule. To get the greater number^ add the sum and difference, and divide the result by 2. To get the smaller number^ subtract the difference from the sum, and divide the result by 2. Example I. The sum of two numbers is 40 and their difference Is 16 ; what is the greater number ? Process : 40 + i6 56 ; 56+2 = 28 Ans. Example 2. The sum of two numbers is 59 and their difference -is II ; what is the smaller number ? Process : 59-11=48 ; 48-7- 2=- 24 Ans. [EXAMPLES. \*.\ Find the value of "^ %, 1. i+2 + 3-f.. . + 20. '. v 'V 2. I+2 + 3 + ... + 30. 3. 1+2 + 3 + .. . + 45. ^' r 4. I+2 + 3 + ... + 75. 6. 1+2 + 3 + .. . + ioo. ^'^ 6. 7 + 8+9 + . .. + 50. 7. 40 + 41 + 42 + . .. + 90. 8. 100 + 101 + 102 + ..4+ 200. 9. The sum of two numbers is 376, and their difference is 114 ; what is the greater number ? 10. Find the greater of two numbers, of which the sum is 89251 and the difference is 385. 11. The sum of two numbers is 83957, and their difference is 74821 ; what is the smaller number ? 12. Find the smaller of two numbers, of which the sum is 79358 and the difference is 3456. 13. The sum of two numbers is 8527 and their difference is 729 ; find the numbers. 14. Find the two numbers, of which the sum is 10000 and the difference is 888. 45. Multiplication by factors. Example i. Multiply 329 by 35. Here 35 7 x 5. Process : 329 11515 32 ARITHMETIC Example 2. Multiply 1725 by 217, and by 72!) making in each case only two partial multiplications. (i) 1725 (2) X 725 217 721 12075 12075 36225 -36225 374325 Ans. 1243725 Ans. Here, we multiply by 7> and by 21 ; but the product by 21 is obtained by multiplying the first product by 3. 46. Abbreviated methods of Multiplication. (a) To multiply a number by 5, annex o to the number, and divide the result by 2. Thus, 172 x 5 = I72o^-286o. Exantyle. Multiply 172 by 15. 2 ) 1720 product by 10. ... .. (i) 860 - product by 5 (2) Adding (i) and (2), 2580 product by 15. (b) To multiply a number by 25, annex oo to the number, and divide the result by 4. Thus, 38 x 2 5 - 3800 4- 4 = 9 50. Example i. Multiply 38 by 35. 4)3oo 950 product by 25 (i) 380 product by 10 (2) Adding (i) and (2), 1330 product by 35. Example 2. Multiply 38 by 75. 4)3oo = product by ico (i) 950 product by 25 (2) Subtracting (2) from (i), 2850 product by 75. (c) To multiply a number by 125, annex coo to the number >, and divide the result by 8. Thus, 89 x 12 5 =890004- 8 11125. (4) To multiply a number by 9, 99, 999, 9999, ..., annex a* many o's as there are 9's in the multiplier, and from the result sub- tract the number itself. Thus, 345 x 99 =* 34500-345 =-341 5 5. (e) To multiply by a number which differs but little from io> loo, looo, loooo, ..., we employ a method similar to the above. Exampl*. Multiply 345 by 998. 345x1000 -345000 345x2 690 By subtraction, 344310 PROPOSITIONS IN THE FUNDAMENTAL OPERATIONS 33 47. Abbreviated method of squaring a given number. If the given number contains two figures : To and from the given number add and subtract the unit figure ; multiply the two results together, and to the product add the square of the unit figure. If the given number contains three (or more) figures, take from the end two (or more) figures instead of the unit figure. Example I. Find the square of 47. 47 + 7=54 J 47-7=40 ; ^ 54x40=21^0; 7*=49; 47 2 = 2160 + 49 =2209. Example 2. Find the square of 346. 346 + 46 = 392 ; 346-46 = 300; 392 x 300 = 1 17600 5 346 2 = 1 1 7600 + 46 2 . Now, 46+6 = 52 ; 46-6=40 ; 52x40=2080; 6 2 =36 ; Hence 346 2 97 1 1 EXAMPLES. 16. j ' \& Multiply, using factors not greater than 20, 1. 728 by 24. 2. 8025 by 42. 3. 9345 by 72. 4. 921 by 144. 6. 872 by 280. 6. 742 by 128. Obtain the following products by two lines of multiplication only. 7. 7925x328. 8. 825x729. 9. 3842x321. 1O. 392x366. 11. 526x848. 12. 734x4812. 13. Obtain the product of 23 56 by 125255 by three lines of multiplication. 14. Multiply 8273 by 147497 making only three partial multi- plications. Obtain the following products by the method of Art. 46. 16. 19. 23. 27. 80. 83. I 86. 40. 725x5. 16. 729x25. 20. 207x125. 24. 421x998. 739x50. 709x75- gfed, by the met 35- * 325. 329x5. 17. 812x5. 92x25-. 21. 98x125. 112x99. 26. 282x999. 28. 4268x980. 31. 371 xi 5. 34. 304x15* od of Art. 47, the square of 37. 55- 38. 86. 465. 42. 779. 18. 22. 26. 20. 32. 36. 84x25. 125x125. 204x9999. 827x9997. 892x35. 789x75. 39. 97. 43. 896. C, A. 3 34 ARITHMETIC 48. Division by factors. Example I. Divide 1 5792 by 48. Here 48 8 x 6. Process : 8 ) i 5792 6 ) 974 329 quotient. Example 2. Divide 934 by 24. (A) . 4 ) 934 - ' 6) 233.. .2 3?-5 The quotient is 38. The remainder =5 groups of 4 units 4- 2 units =20 + 2 = 22. (B) 4)934 3) 233... 2 2) T7...2 . qt. 38...I Remainder = 2 + (2x4) 4- (1x4x3) =22. In all cases, The true remainder** ist R + (2nd Rx ist dvr.) + (3rd R x ist dvr. x 2nd dvr.) + etc. 40. Abbreviated methods of Division. (1) To divide a number by 10, 100, 1000, , cut off one> two, three, , figures from the right of the number ; the figures cut off will give the Remainder and the remaining figures the Quotient. Thus, when we divide 53274 by 100, the quotient is 532, and the remainder is 74. (2) To divide by any number ending with ciphers, cut off the ciphers from the divisor and as many figures from the right of the dividend ; then divide the remaining figures of the dividend by the remaining figures of the divisor, and to the remainder annex fhe figures cut off from the dividend to get the total remainder. Thus f ' if we have to divide 3754 by 700, we divide 37 by 7, which gives 5 as quotient and 2 as remainder ; the total remainder is 254. (3) To divide a number by 5, 15, 35 or 45, multiply the number by 2 and divide the result by 10, 30, 70 or 90 (by the above method) ; divide the remainder by 2 to get the true remainder. Thus to divide 78 by 5, we multiply 78 by 2, getting 156 as the result ; this divided by 10 gives 15 as quotient and 6 as remain- der ; the true remainder is 6 -~ 2 or 3. Hence 78 divided by 5 gives 1 5 as quotient and 3 as remainder. (4) To divide a number by 25 or 75, multiply the number by 4 and divide the result by loo or 300 ; divide the remainder by 4 to get the true remainder. (5) To divide a number by 125, multiply the number by 8 and divide the result by 1000 ; divide the remainder by 8 to get the true remainder. PROPOSITIONS IN THE FUNDAMENTAL OPERATIONS ^EXAMPLES. 15LJ In the following examples employ Short Division. 1. 9364-24. 2. 736^32. 4. 2856-42. 5. 33I2-M44' 7. 3892072. 8. 234564-63. 10. 820344-121. 11. 7045684-240. 13. I234564-7 3 . 14. 9876544-480. Divide by the method of Art. 49 : 16. 3894 -Mo. 17. 34564-100. 19.' 82746 -Moo. 20. 893464-1000. 22. 38924-30. 23. 78924-50. 25. 73568-7-1900. 26. 7368944-16000. 28. 354693-1-2900. 29. 76892464-790. 31. 3784-5. 32. 46894-5. 34. 7845-^25. 35. 827694-25. 37. 837644-125. 38. 1378914-125. 40. 374 -MS- 41. 789^35- 43. 12344-75- 44. 13944-65. 3, 1890-4$. 6. 82744-25. 9. 748294-99. 12. 824506 4- 8*. 15. 8888884-5*. 18. 893454-1000. 21. 1234564-10000. 24. 984674-800. 27. 98765434-12600. 30. 92345874-3400. 33. 12764-5. 36. 1378924-25. 39. 3792-M25. 42. 9214-45- 45. 9246*85. 50. The process of multiplication and subtraction may be combined in a question like the following : Example. Subtract 7 times 347 from 3283. Mental process : 7 times 7 is 49 ; 49 and 4 are 53 ; carry 5 and 7 times 4 are 33 ; 33 andf 5 are 38 ; carry 3 and 7 times 3 are 24 ; 24 and 8 are 32. 3283 347 7' 854 Note. The above method may be very advantageously era. .ployed in the process of division. Example. Divide 8422 by 34. Here, by the method of the above example) we multiply 34 by 2, subtract the product from 84 and set down only the remainder 16 ; and so on. 34 ) 8422 (247 162 EXAMPLES. Subtract L 329 x 8 from 4827, 3. 3798 x 6 from 894670, 18. 2. 732x9 from 82170. 4. 9378x7 from 369812. g6 ARITHMETIC 5. 7384 xn from 100000. 6. 369 x 12 from 89468. Add 7. 389*41039. 8. 894x910786. 9. 7345 x 12 to 3940. 10. 39874 to 329 x 16. In the following examples use the method of Art. 50. 11. 37934-76. 12. 388754-329. 13. 824564-729. 14. 760820-7-378. 15. 3456789-^3246. 16. 345789-^3982> CASTING OUT THE NINES. 51. The following method called "casting out the nines" (s frequently employed in testing the correctness of the result of multiplication. Divide the sum of the digits in the multiplicand by 9 and set down the remainder ; do the same thing with the multiplier ; mul- tiply the two remainders together^ divide the result by 9, and set down the remainder j then if the multiplication has been performed correctly, the last remainder will be the same as the remainder obtained by dividing the sum of the digits in the product by 9. Example. 1 86 x 47 * 8742. The sum of digits in 1 86 = 15; 15-7-9 gives rem. 6 ; the sum of digits in 47 11; 11-7-9 gives rem. 2 ; 6x2 = 12; 12 4-9 gives rem. 3. Sum of digits in 8742 2i ; 214-9 gives rem, 3. N. B. This test will fail if such a mistake has been committed, as does not affect the 'sum of the digits of the product, or, increases or .decreases it by 9 or a multiple of 9. EXAMPLES. 19. Multiply, and test the result of multiplying L 37 56 by 738- 2. 8943 by 826. 3. 3789 by 989. 4, 30804 by 3080. 5. 78093 by 8034, 6. 73980 by 3001. 7. 39400 by 3900. 8. 80307 5 by 390. 0. 823794 by 8234. 51s. In a chain of operations of addition and subtraction the order of the operations is from left to right. Thus 8-54-4-2 means that 5 is to be subtracted from 8, then 4 is to be added to the result, and then 2 is to be subtracted from the last result But we shall get the same result if we subtract the sum of the jiegative terms from the sum of the positive terms ; and this method is often more convenient. MISCELLANEOUS EXAMPLES 37 In a chain of operations of multiplication and division the order of the operations is from left to right. Thus 24 x 4-7-2 means that 24 is to be multiplied by 4, and then the result is to be divided by 2; 244x2 means that 24 is to be divided by 4, and then the result is to be multiplied by 2 ; and 244-7-2 means that 24 is to- be divided by 4, and then the result is to be divided by 2. When an expression contains all (or some of) the signs + x, + , the mMj&iotiw*M&<L>^^ addition and subtraction. Thus, in 7-6 2+5x3, 6 must be divided by 2 before subtraction, and 5 must be multiplied by before addition. Example i. 8+2x6+2+ 3 = 4x6-7-24-3 = 244-2-7-3 = 12 + 3 Example?. 7 + 2x6-5-4-124-6 = 7 + 12-7-4- 2 =7+3-2 = 10-2 o o. EXAMPLES. 19a. Find the value of each of the following expressions : *1. 6x7+3. 2. 16-8x3. 3. 204-5-1-2. 4. 104-5x34-2. 6. 6x54-3x2. 6. 8x6-7-4-7-3, 7. 7x3+5x2. 8. 16-7-2-3x2. 9. 8+2-6 + 3. 10. 6x5-8+4. 11. 9 + 6+2-8. 12. 9-6+2 + 8, 13. 12+4+3 + 7-2x4. 14. 7x6-3x4-4x5. 15. 7x8x9-12x3-18. 16. 18+2-6+3+14+2. 17, io 2 -7X3+6 2 +3 2 . 18. 828 + 18-100+ 5* +23 19. 639+9x3-720+8+15-53x2 + 22+2x9. 20. 204x3+4+630+7x2+3-4x4x9+2-47x3. MISCELLANEOUS EXAMPLES. 20. 1. What number must be added to 3452 to make 6000 ? 2. What number must be taken from 3021 to leave 999 ? 3. The sum of two numbers is 8920) and the smaller number is 309 ; what is the greater number ? 4. The difference between two numbers is 379, and the greater number is 1000 ; what is the smaller ? 38 ARITHMETIC 6. The difference between two numbers is 79, and the smalle? number is 709 ; what is the greater number ? 6. What is the difference between the least number of five figures and the greatest number of three figures ? 7. The dividend is 3792, the quotient 12 and the remainder o ; find the divisor. 8. What number multiplied by 304 will produce 3344 ? 9. The divisor is 321, the "quotient II and the remainder 260 ; find the dividend. 10. What is the divisor when the dividend is 345, the ramain- der 5 and the quotient 20 ? Hr Find the sum of all the numbers of 3 digits, which you can form with the figures 3, o, 4. 12. Find the difference between the greatest and the least numbers of 4 digits, that you can form with the figures 3, 2, 7, 8. 13. There are two numbers, of which the product is 7243491, and the greater number is 34007 ; find the difference between the two numbers. 14. Find the sum of the products, two and two, of 369, 217 and 648. 15. How many times can 23 be subtracted from 920550, and what will be the final remainder ? 10. The product of two numbers is 173432, and half of one of them is 163 j what is the other ? 17. The product of two numbers is 123904, and double of one of them is 1408 ; what is the other ? 18. How many times in succession must 201 be added to 3166 to make the final sum 10000 ? 19. How much must be added to the product of 75 and 83 to give the product of 75 and 85 ? How much must be subtracted ta give the product of 74 and 83 ? 20. How often does the sum of 3692 and 2769 contain their difference ? 21. What number multiplied by 37 will give the same product as 185 multiplied by 309 ? 22. In a division sum the divisor is 5 times and the quotient is 6 times the remainder which is 73 ; what is the dividend ? 23. If, in dividing a number by 105, the operation be per- formed by short division by employing 1 factors 3, 5, 7 in succession, and the several remainders be 2, 4, 5, what is the complete remainder ? MISCELLANEOUS EXAMPLES 39 24. If when a number is divided continuously by 7, 8 and 9 the remainders are 5j 3 and 6 respectively, what would be the remain- der if the same number were divided by the continued product of 7, 8 and 9? 25. The quotient is 702, the remainder is 24, and the divisor 7 more than the sum of both ; what is the dividend ? 26. The sum of two numbers is 205, and one of them exceeds the other by 7 ; what are the numbers ? 27. Your age is 12 years ; your brother's age is 19 years ; what will be your brother's age when you are 16 years old'? 28. Find the sum of three numbers, the first of which is made up of 3908 and 78904, the second of which exceeds the first by 17401 and the third exceeds the difference of the other two by 7809. 29. There are two numbers ; the less is 94567, and the other exceeds it by 327 ; what is their sum ? 30. I have 3290 rupees in cash, 75000 rupees in Government promissory notes ; I owe 3525 rupees to A and 25 rupees less to B: how much am I worth ? * 3L The sum of two numbers is 729, the less is 57 ; what is then diffeience ? t>2. What number must be subtracted from the product of 329 and 412 to make it equal to their sum ? 33. A man sold 260 rhangoes at 2 pice each, and 50 oranges at the rate of two for a pice ; how many pice did he get in all ? 34. Obtain the product of 3749 by 216636 by three lines of multiplication. 35. Multiply 7384 by 42428 in three lines. 36. If I had 300 rupees more, I could have paid a debt of 750 rupees and have 25 rupees over ; how much have I ? 37. In a game of cricket A, B and C together score 134 runs ; B and C together score 76 runs ; and A and C together score loo runs ; find the number of runs scored by each. 38. A and B together have 79 rupees, C has 49 rupees less than what A and B together have, and B has 9 rupees more than C ; find what each has. 39. I bought a dog for 25 rupees, a cat for 15 rupees less, and a horse for 30 rupees more than twice the price of the cat and dog ; how much did I spend in all ? 40. A man, after selling oranges to three purchasers, found that he had a rupee worth left ; if he had sold 5 more oranges to each purchaser he would have only 3 left : at what rate per rupee did he ell the oranges ? 40 ARITHMETIC 41. A cistern has two pipes attached to it ; by one of the pipes 24 seers of water enter into the cistern per minute, and by the other 14 seers go out in the same time : how much water will there be in the cistern if both the pipes are left open for 6 minutes ? Also find how much the cistern holds if the empty cistern be filled in 10 minutes when both the pipes are open. 42. A gentleman's monthly income amounts to 250 rupees? and his monthly expenses amount to 175 rupees ; how much will he be able to save at the end of 2 years ? A year = 12 months.] 43. A man's age is 59 years ; his brother is 7 years older than he, and his sister 12 years younger than his brother : what was the man's age when his sister was born ? 44. A man was 30 years old when his eldest son was born ; how old will his son be when he is 40 years old, and what will be the man's age when the son is 40 years old ? 45. Find a number such that if it be added 12 times to 60 the sum will be 780. 46. The distance from Calcutta to Goalundo is 152 miles ; a train starts from Calcutta at 7 A.M., and runs towards Goalundo at the rate of 19 miles an hour : at what o'clock will it arrve there ? ^47. Take any number, subtract from it the sum of its dibits ; tfie result will be divisible by 9 without remainder. 48. If any number and the sum of its digits be each divi Jed by 9, the remainders will be equal. 49. Take any number, multiply it by 2, add 16 to the producti divide the sum by 2, subtract the original number from the quo- tient ; the remainder will be 8. 50. The product of any three consecutive numbers is divisible by 6 without remainder. VIII. MEASURES OF MONEY AND REDUCTION. 5. In practice it is found convenient to use large units for measuring large quantities, and small units for measuring small quantities. Thus, we say that the price of a table is 20 rupees f the price of a book is 10 annas ; the price of a toy is 3 pice. A list of the relative magnitudes of the various units used for the measurement of quantities of the same kind is called a Table. 53. English Money Table. 4 Farthings (g.) make I Penny (id.). 12 Pence ... i Shilling (is. or i/-). 20 Shillings ... i Pound or Sovereign (i). 2 Shillings I Florin. 5 Shillings *i Crown. 21 Shillings i Guinea. 27 Shillings i Moidore. MEASURES OF MONEY 41 Note, i, 2 and 3 farthings are usually indicated by J< }^. and f */. respectively. The following coins are now in circulation in England : Copper coins : a farthing, a half-penny, a penny. Silver coins : a threepenny piece, a fourpenny piece (or groaf}) a sixpence (or tester}^ a shilling, a florin, a half-crown, a crown. Gold coins : a ha'lf-sovereign, a sovereign. The following gold coins are now obsolete but were in circula- tion at various periods in England : a noble (6s. 8</.), an angel <ioj.), a half-guinea (los. 6dl), a mark or merk (13^. 4^.), a guinea (21*.), a carolus (235.), a jacobus (25$.), a moidore (27*.). The standard of gold coin in England is 22 parts of pure gold and 2 parts of copper, melted together. Each of these 24 parts is called a carat. Pure gold is said to be 24 carats fine and standard gold 22 carats fine. From a pound Troy of standard gold there arc coined 46 J$ sovereigns, or 46. 14*. 6d. The standard of silver coin is 37 parts of pure silver and 3 parts of copper. From a pound Troy of standard silver there are coined 66 shillings. In copper coin- age 24 pennies are coined from one pound Avoirdupois of copper. Gold coinage is the standard in England. Silver coinage is not a legal tender for more than 40 j., nor is copper coinage for more than lid. 54. Indian Money Table. 3 Pies (p.} make i Pice. 4 Pice or 12 Pies ... i Anna (10.). 1 6 Annas .. i Rupee (Bi). 15 Rupees ... i Pound or Sovereign A gold Mohur is a gold coin whose weight is equal to that of a rupee. Its value in silver money is variable. In paying Doctors 1 fees a gold mohur means Ri6, and in paying Lawyers' fees, 15 Sicca Rupees = 16 Current Rupees. 100 Raes of Bombay A quarter-rupee. i oo Cents of Ceylon i Rupee. A Pagoda of Madras 83. 8a. Copper coins : a pie, a half-pice, a pice, a double-pice. Nickel coins : an anna piece, a two-anna piece. Silver coins : a two-anna piece, a four-anna piece or quarter* rupee, an eight-anna piece or half-rupee) a rupee. Gold coins : a gold mohur, a sovereign, a half-sovereign. The standard of gold or silver coin in India is 1 1 parts of pure gold or silver and I part of alloy. The weight of a rupee or of a gold tnohur*i8o grains Troy. A double-pice weighs 200 grains Troy. 42 ARITHMETIC Gold coinage (except the Sovereign) is not a legal tender in India ; the rupee and half-rupee are legal tender for any amount, other silver coins and the nickel and copper coins being a legal tender for the fractions of a rupee only. The British Sovereign ( = Ri5) is now in circulation in India, IJ. I2#. ; id.^ia. ; i?. = i pice ; Ri = i.y. ^d. In keeping accounts in Bengali the following Table is in com* mon use : 4 Cowries make 5 Gandas 4 Buris or 20 Gandas ... 4 Pans 4 Chouks Ganda. Buri, Paisa or Pice. Pan or Anna. Chouk or Quarter-rupee. Kahan or Rupee. One Cowry =3 Krantis=4 Kags=5 Tals = 7 Dwips9 Dantis -27 Jabs - 80 Tils. The following Table gives the sub-divisions of a pice, as used in Behar, United Provinces and the Punjab : 2 Addhis = I Damri ; 2 Damris = I Chidam ; 2 Chidams = I Adhela ; 2 Adhelas i Paisa or Pice. REDUCTION. 55. A quantity expressed by means of a single unit is called a simple quantity. A quantity expressed by means of more than one unit is called a compound quantity. Thus, B; is a simple quantity ; 3. 4<z. 3^. is a compound quantity. Reduction is the process by which we express (i) a simple or a compound quantity in terms of a lower unit, or (2) a simple quantity in terms of higher units. 50. I. DESCENDING REDUCTION. Example I. Reduce 834. 70. 6p. to pies. Since Ri=*i6fl., &34=(34x 16)0. = 544*1. R34- 7- sa 544^ + 7. Again, since ia.-i2/5., 55ia.(55ix R34. 7 a. 6>. - 66 1 2p. + 6p. 66 1 8/. Ans. In practice the operations of multiplication and addition are- combined, and the process stands thus : R. a. p. 34 7 . 6 55i* 12 Ans. REDUCTION 43 Example 2. Reduce 3. 75. 4J0 7 . to farthings. Process : . s. d. 3 7 . 4* 20 I* 4_ 3234?. Ans. EXAMPLES. 21. Reduce to annas : L 39. 2. 8104. 3. 87208. 4. 83698. 6. 87. 90. 8. 823. 40. 7. 837. 120. 8. 851. 140. Reduce to pies : 9. 8309. 10. 8740. 11. 83402. 12. 8201.90. 13. 8112.100. 14. 8704.130. 15. 827. 00. $p. 10. 839. 120. 0^. 17. 867. 150. up, Reduce (i) to pice and (ii) to pies : 18. 83. oa. 2 pice. 19, 87. 130. i pice. 20. 89. 140, 3 pice, Reduce 21. 83705 to half-rupees. 22. 8408 to quarter-rupees. 23. 878. 140. to two-anna pieces. 24. 83. 20, to double-pice, 26. 830. 70. to half-pice. 26. 87. 80. 6>. to pice. Reduce to shillings : 27. /720. 28. ^240. 29. 709, 30. ^305. 31. 20. $j. 32. 26. i2s. 33. 30. 17*. 34. 35. igs* Reduce to pence : 35. ^35- 36. 670. 37. ^7020. 38. ^45-iu- 39. ^50.135. 40. ,76. 15*. 41. 3. 125. 6d. 42. g. os. lod. 43. 7. i6s. nrf. Reduce to farthings : 44. 1000. 45. ^305. 175. 46. 7. 12s. 90*. 47. 3- 7*. 3fc* 48. 7. os. 9$^. 49. 2. i6s. of* 7 . Reduce (i) to crowns, (ii) to sixpences and (iii) to fourpences : 60. 9.$s* 61. 10. los. 52. 44 ARITHMETIC Reduce 63. 2. js. 6d. to half-crowns. 54. 3. 3*. <)d. to threepencei. 65. 300 half-crowns to farthings. 56. 56 guineas to half-pence. 67. If the price of an orange be one pice, how many can you buy for Ri. 90. ? 68. A debt of 2. 75. j^d. is to be paid in farthings ; how many will be required ? 69. How many one-anna books can be bought with 7. 13^. ? 60. For how many children can a treat be provided with ft 1 3. 124. at 40. a head ? 61. I gave away i. 135. to a number of beggars, giving a penny to each ; how many beggars were there ? 57. II. ASCENDING REDUCTION. Example i. Reduce 1995 pies to R. a. p. Process : 12 ) I995A . rem. . rem. Rio. Answer. Rio. 60. 3^. Example 2. Reduce 1 5723 farthings to . s. d. Process : 4 ) J_5_723?- 12) 393q<& + 3?. rem. 20 ) 3271. +6rf. rem. ;i6 +7.?. rem. Answer. 16. 7*. Reduce to R. a. /*. i. joooo pies. 4. 3948 pies. 7. 30303 pies. 10. looopice. 13. 7082 half-pice. Reduce to . s. d. 16. 376 pence. 19. icoo farthings. 22. 8040 farthings. EXAMPLES. 22. 2. 30793 P^s. 6. 7823 pies. 8. 47474 pies. 11. 3785 pice. 14. 8936 half-pice. 3. 6. i mi pies. 9. loooi pies. 12, 3082 pice. 16. 3840 double-pice. 17. 702$ pence. 18. 20. 10008 farthings. 2L 23. 7929 far things. 24. %Q2o pence. 3333 farthings. COMPOUND ADDITION 4S 25. 379 half-pence. 26. 3940 threepences. 27. 27 guineas. 28. 390 half-crowns. 29. 396 sixpences. 30. 30 moidore* 31* I paid one pice to each of 960 beggars ; how many rupees did I spend ? 32. How much money will be required to buy 300 quarter-anna* post cards ? 33. If you buy 720 oranges at one farthing each, how many shillings shall you have to pay to the fruit-seller ? IX. COMPOUND ADDITION. 58. The following example will illustrate the method of add* ing together compound quantities. Example. Add together ,3. 7*. 4^., 8. 2s. 7j^., 9. and 2. i2s. 8</. We first add the farthings, and we find that there are 7 farthings ; and this being equivalent to id. + ^q.) we place } under the column of farthings and carry id. Next we add the pence, and we find that there are (with id. carried) 29 pence, and this being equivalent to 2J.-f5*/., 24 . 2 . 5 j Ans t we place 5 under the column of pence and carry 2j. And so on. S. d. 3 7 4: 8 . 9 19 . 9; 2 . 12 . 8; EXAMPLES. 23. a. pice. 1. 3 , 2 7 3 9 . 2 6 3 a. A 6. 9 9 10 4 7 . o 13 . II R. a. p. 9. 9 12 . 3 15 7 - i 9 . 2 10 . 2 3 S . 7 . o a. pice. 2. 8 3 12 . i 14 . 2 10 3 a. A 6. 12 . 10 7 7 II . II 14 . 8 R. 10. 12 7 20 31 J2 a. pice. a. pice. 3. 12 . 3 4. 13 . 2 7 i 10 . 3 13 2 9 15 3 8 . I a. 7. 7 12 14 13 6 7 10 4 11. a. A 8. 8 3 9 . ii 15 7 12 9 R. a. t- 22 . 12 33 13 14 14 o 3 9 2 n . 7 II 4 6 12. ARITHMETIC 18. 21. R. a. pice, R. a. A R. a, A 13 7 3 13. 8 7 9 14. 100 . 13 4 107 . 13 2 ii . II . ii 29 - 7 8 39 12 . I 309 . 14 8 7 12 3 7 . O . 3 39 . o . 10 309 O II 19 14 o 604 . 8 . 4 76. 7 9 12 . 8 . i 89 13 4 770 . 7 7 317 - 9 2 824 7 2 86 . 9 10 R. a. R. a. A R. a. 8 . 8 . 8 16. 349 . 15 . 4 17, 896 . 9 8 17 . 4 7 1207 13 8 64. ii . 2 309 - 12 . ii 740 9 . 6 42 . 9 II 1234 13 10 39 . 4 9 4276 . 13 4 239 8 . 9 123 . 12 . ii 7624 . 3 7 26 . 4 3 8 7 10 72 . 8 . 3 7 3 6 1286 13 7 726 . 12 . 10 29 14 5 836 9 2 3725 . 7 8 100 . 7 - 8 63 . 10 . 8 346 . 10 . 5 R. a. A R. a. R. a. 76 . 9 7 19- 374 , 12 3 20. 3846 . 9 ii 1249 . 12 . 3 483 13 7 8463 . ii 9 3400 . IS - 8 7682 . 14 6 768 . 10 2 343 . 9 300 . is 4 968 . 13 6 82 . 8 . o 82 . ii 10 39 - 4 7 7 9 4 4 . 10 8 46. 6 743 ii . 10 92 . o 9 7 9 9 376 . 13 ii 7 . 4 5 8 . 12 3 8824 . 6 . 5 89 . 7' 8 12 . 14 4 7286 . 5 4 345 . 9 2 10 . 8 8 510 . 10 . 9876 3 6 346 3 7 36 . 7 2 4242 . 8 ii 789 2 6 9 - 9 9 123 . 6 3 1234 I 4 982 . 2 . I 99 5 . 9 5678 . 7 2 S. d. s. d. s. A 7 12 3 22. 39 18 . [0 23. loo . IS 9 19 19 7 76 2 . 9 376 . 3 3 100 . 13 9 300 17 - .489- 14 7 76. 7 8 49 16 . 8 39- 4 6 304 . 8 2 4 3. 6 4* 9 8 -. COMPOUND SUBTRACTION 47 24. 392 1396 300 39 4 7892 s. 8 9 7 13 19 12 10 d. 3l & 2 4i . s. < 25. 9 12 oi 72 4 8| 3^4 17 7t 4782 6 2 400 19 3i 92 13 4f 4 6 6| 26. s. d. 346 19 3 46 12 A. 39 13 6 4 8 7 9 12 0; 13 14 4' 5 12 o -27. S. S. 3 4 5- 28. 300 i O1 i 29. 432 9 9 13 527 12 14 19 13 10; 7 3' 29 3i 4 5 7 13 2 2i 5: 73 t 820 ' 70 12 . ri 13 . of 14. <}$ 5 fj 8 5 15 7: 8 IS 2 8 9 61 6 19 95 r 9 16. 3j 5 12 o; \ 81 12 ii 12 17 4 300 15 ii k 390 II i 329 18. 7i X. COMPOUND SUBTRACTION. 59. The method of subtracting one compound quantity from another is as follows : Example. Subtract R7. ga. 6p. from Ri2. 30. qp. Here we have to find the quantity which being added to R7. ga. 6p. makes up R. a. p. Ri2. $a. gp. We see that bp. + Sp.=gp. ; 12 . 3 . 9 we therefore put down 3 under the column 7 . 9 . 6 of pies. Next, ga. + 10a.=i9^.Ri + 3. ; R4 . 10 we put down 10 under the column of annas and carry* R i for adding to the rupees of the subtrahend ; Ri (ftiraV?</; + R7 + R4=Ri2 ; and we place 4 under the column <of rupees. EXAMPLES. 24. Subtract 1. R7. ga. 2 pice from Ri3. I2a. 3 2. R28. I2. 3/5/ from RSO. 9^. 2 5. R3. 9^r. i /^/^ from 13. 4#. 4. 6. R39- 134. gp. from ^79. I2a. bp. 6. RS. 7. ^>. from Ri3. 7. ^8. Ri4. 141. 3^. from RiS. I2a. 9. R9. 7^. 6^. from Ri3. 30, 3^. Ri3. 120. 7^. from R29, 69. 15^2. 2^, from R8a 8 a. 48 ARITHMETIC 10. R9I. I2tf. li^. from R 150. oa. jp. 11. R726. 15*. 5^. from Riooo. 130. 4 12. Rio9. 100. 3^. from RIIO. 00. 9^. 13. 7. 17* 9^- from 17. ,13. 13^. 8f</. from ^15. 17^. oj</. 18. 37. is. 6J< from ^49. os. 3d. 19. ,96. 4J. io|^. from ^104. oj. o^^/. 20. ,102. 19^. iij^. from 105. js. of//. 21. 67. iu. 4J^ from 98. 6s. 2%J. 22. 98. i8j. 4K from 908. sj. 2j^. 23. 275. 15*. 5W. from 743. os. $ld. 24. 492. i8.f. 8|^f. from 611. 175. 2\d. XI. COMPOUND MULTIPLICATION. 60. Compound Multiplication is a short method of finding the sum of a certain number of repetitions of a given compound quantity. The process is as follows : Example. Multiply R$. I2a. 4^. by 7, and by 35. 7 times 4^. = 28/5. 20. -f 4^. ; set down ^- * & 4 and carry 2. 7 times 120. = 84^., which 5 . 12 . 4 with 20. 0m>*/=86a. = R5-f-6tf. ; set _ 7 down 6 and carry 5. 7 times Rs = R35 ; 840 .6.4 Ans. this with RS carried gives 840 ; and setting down this, the required product is R4O, 60. 4^. Note. To multiply by 35 we multiply first by 7 and the pro duct by 5. EXAMPLES. 25. Multiply 1. R3. 8a. ^pice by 3, 5, 7. 2. R9. 12*. 6/. by 5, 7, 9. 3. R39- I4* A by 11, 13, 16. 4. ^29. 185. gd. by 3, 7, 9. 5- ;37- I5J- 4l^.by6, 8, 13. 6. 40. 75. id^d. by 5, 9, 12. [In the following examples use the method of multiplication by factors.] 7. Ra. 4*. ipice by 21, 32, 25. 8. RSQ, 12*. $*. by 56, 99, 100, COMPOUND DIVISION 49 9. 48. 130. 8/. by 125, 121, 144. 10. 34. 16^. $\d. by 81, 64, 800. 11. ^48. 13.?. old. by 99, 72, 420. Find the value of 12. 9 things at 30. 4^. each. 13. 56 things at 82. 40. each. 14. 8 1 things at 2s. 6d. each. 15. loo things at 7*. 6$< each. 16. 1000 yards of broadcloth at 85. 7 a. 6f. per yard. 17. 700 copies of a book at 7*. *i\d* each. 18. 3000 maunds of wheat at R3. 50. 6p. per maund. 61. When the multiplier is a large number and cannot be split ap into factors, the following method should be used. Example. Process : Multiply 812. t- by 473- 8. a. p. 12 . 8 . 7 10 125 . 5 10 10 1253 . 10 4 4 Multiplying 3rd line by 7, Multiplying ist line by 3; Adding last 3 results, 5014 , 9 . 4 product by 400. 877 8 . 10 70. 37 9 9 3- 85929 . ii . ii product by 47 3, EXAMPLES . Multiply 1. 83. 4*. 2 pice by 23, 37. 83. 13*. 6/. by 421, 704. 4. 7s. 6d. by 511, 112. 6. us. old. by 753; 829. v . A gentleman spends 87. 8 he spend in a year of 365 days ? 3. 6. 7. 9. 2. 87. 12*. o/. by 37, 47. 4. 82. I2a. $p. by 2175, 3070* 6- 3- 9-f- &d. by 3684, 1237. 8. 7. os. i%d. by mi, 1231. , <)p. every day ; how much does 1O. Find the cost of 503 maunds of rice at 83. ga. $p. per maund. XII. COMPOUND DIVISION. 69. The process of dividing a compound quantity by an abstract number, that is, of dividing it into a given number of equal parts, is as follows : C. A. 4 50 ARITHMETIC Example I. Divide 8138. 30. 8138-7-29 gives R4 as quo- tient and 622 as remainder ; this remainder, together with 3 5 5#. -7- 29 gives 120. as quo- tient and ja. as remainder ; this remainder, together with #--87 A: 87/.-T-29 gives 3A as tient and no remainder. quo- Hence the quotient is &4. . by 29. R. a. A 29 ) 138 . 3 . 3( 116 16 29 ) 355 ( 29 ~ 12 29 ) 87 ( 3^. 87 The quotient is R4 120, EXAMPLES. Divide 1. R72. 30. 3 pice by 23. 2. 8286. I la. i^/Vtf by 59. 3. 8455. 14^, 7A by 61. 4. 8850. 14^. 4^. by 79. 6. 81025. 6a. 8/>. by 80. 6. 8583. 6a. 6/. by 98, 7. 84981. loa. 3;*. by 325. 8. 85049. I2a. $p. by 499, ,97- 9^- i<* by 29. 10. ^29. 6^. id. by 52. 11. ,1279. 13*. 8|df. by 23. 12. ^4476. 75. /< by 83. 13. ;94- 17-r- *H by 279. 14. ,860. o^. 7j./. by 365* In the 10 following examples use the method of Short Division* 15. 813. 150. 8/.-^2. 16. 8225. 13*7. 8/. 'T- 4- 17. 8728. 140. #.-5- 5. IS. 81007. IQO. 2p.-, r7- 19. 8329. 110. 4^.~8. 20. 81243. 80.^9. 21. 29. 7*. 6i^.~-3. 22. ^333. 19^. 3^.-7-< ). 23. 378. l6j. iot/.-7-8. 24. ,3781. oj. 9|^.-5- 9- Employ the method of Division by Factors in the 6 following examples. 25. 827. 1 00. 24. 26. 8160. 00. 3^. -7-49. 27. 8323.20.^.^56. 28. 8683.2^.6^-54.' 29. 3522. is. 7^.-f28. 30. ^543,115.^-42. 31. The price of 140 quires of paper is 832. 13*. ; 6nd the price of one quire. COMPOUND DIVISION 51 32. If 55 copies of a book are sold for 34. 6*., what is the price of a single copy ? 33. If the cost of 2880 articles be 480, what is the cost ol one article ? 34. A man's wages for 30 days are ,5. 5^. ; what does he earn per day ? Note. When the divisor is 10, 100, 1000, ......... , the following method should be used. Example 2. Divide 81345. 130. \p* by ico. The division in each step is R. a. p. effected by cutting off the two 100 ) 13,45 .13.4 (813. 70. 4^. Ans. figures from the right ; the 16 figures cut off give the re- a% 7 33 mainder and the remaining S I2 figures give the quotient. (See Art. 49, CO.] EXAMPLES. 8. Divide 1, 8i35- 120. 6/5. by 10. 2. 8376. 20. &$. by 10. 3. 8279. 110. by 100. 4. 81245. 130. 4p. by 100. 6. 84067. iia . &$. by loo. 6. 86100. 80. 4^ by ioa 7. 8203. 20. by looo. 8. 82135. 60. 8/^. by 1000. 9. ^438. 6s. 8, d. by 10. 10. ^227. i6s. 80*. by JO, 11. ^511. 2s. lid. by loo. 12. ^3007. 5^. 10^. by 1000. Example 3, Divide 897. 20. <#. into 31 equal parts. 8. 0. A 3i ) 97 2 9 ( = B 3 . 93 4 16 31 ) 66 (20. 62 4 12 31 ) 57 (IA 31 26 Here we have a remainder (26^.) after division, and we observe that if the quotient, 83. 20, i/,, be multiplied by the divisor the 5 ARITHMETIC product will be less than the dividend by 26p. ; again, if 3. 2a. ip. be multiplied by the divisor the product will be greater than the dividend by (31 26)^. or $p. The last therefore is nearest to- the correct result. Hence to the nearest pie the resuk is RULE. When there is a remainder after division, the quotient or the quotient increased by ip. is the result correct to the nearest pie, according as the divisor is greater or less than twice the number of pies in the remainder. If the divisor is equal to twice that num- ber, both the results are equally correct. EXAMPLES. 29. Find, to the nearest pie, the result of dividing 1. R35. la. 8p. by 7. 2. R49- 120. #. by io< 3. R67. 130. up. by 41. 4. R327. 80. 6p. by 100* 6. R427. loa. jp. by 56. 6. R394- iia. 2p. by 100. 7. R727. 150. iqp. by 67. 8. 6,923. 140. by ioa Find, to the nearest farthing, the result of dividing 9. 27. 17* 9^- bv 5- 10 - 4 2 - l8 * 3i^- by 10. 11. 333. iqs. $d. by 29. 12. ^498. 1 $s. o\d. by locx 18. 557- *6* u\d. by 210. 14. 876. 12*. by 300, Divide 16. R49I2. 8a. $p. by 24. 10. R789S. 40. ty. by 55. 17. R47&92 by 731. 18. R98765. 9^. ip. by looo^ 19. 7829 by 539. 20. 85632. los. lod. by 670, 03. To divide a compound quantity by another of the same kind, that is, to find how many times the latter is contained in the former, we proceed^as in the following example : Example. How many times is Ri. 2a. 3p. contained in* R26. 30. 9^. ? We reduce the compound quantities to the same expressed lowest denomination, and then proceed as in Simple Division. Ri. 20. 3p.2i()p. ; R26. 3^. ^.-5037^. Now 5037*219=23. Ri. 2a. $p. is contained in R26. 3^. gp.) 23 times. Note. The method of Art. ^62 is called partition and the above method is called quotition. MEASURES OF WEIGHT 53 EXAMPLES. 30. How many times is 1. 15. 7 a. 3/. contained in Ri39. la. #. ? &. R20. 120. 6/5. ... ?.. ... R$i i. na. 6/5, ? 3.. 853. 100. gp Ri288. 2a. ? 4 - 30- 7* 3i<* 637. 135. \\d. ? 6. ^17. 12s. 4J</. ^986. I4J. 2d. ? Find the quotient and remainder in the division of 6. R2ii. 15^. xo/5. by R7. *ja. 7/5. 7. R376. Sa. fp. by Ri7. I2a. 3/5. 8. R304. 15^. 9^. by R7. Sa. gp. 9. 784. 17 j. ii^/. by ^23. igs. z\d. 10. ^976 by ^9- 9* 9K 11. Divide R994. 130. 3^. into equal parts, each of which is qual to Ri7. fa. $p. 12. Divide ^286. 3^. 2( into parts, each equal to ^i. us. l%d. 13. How many maunds of flour, at R4. 8. 3^. per maundj can be bought for 81354. na. ? 14. How many rupees of is. 4^. each are equivalent to 235. ioj. 9^/.? 15. A servant whose pay for a day is 20. 6/5., is fined 9/5. if he comes in late, and at the end of 20 days he receives R2. I2a. gf. ; how often was he late ? 16. Multiply Ri8957. 130. by R 189. ga. #. ; and divide the same sum by the same sum. Shew that one of these operations is absurd and impossible, and perform the other. XIII. MEASURES OF WEIGHT. 64. English Jewellers' or Troy Weight. (Chiefly used for weighing gold) silver and jewels.) 24 Grains ($r.) make I Pennyweight (i dwt.). 20 Pennyweights ... i Ounce (i oz.). 12 Ounces ... i Pound (i lb.). So that a Pound Troy 5760 Grains. Diamonds ^and other precious stones are weighed by carats^ acfil5Sfaf Weighing about $\ grains. 54 ARITHMETIC EXAMPLES. 31. Reduce to grains : 1. 207 Ib. 2. 29 Ib. 8 oz. 3. 3 Ib. 9 oz. 13 dwt. 15 gr. 4. 28 Ib. 7 oz. 1 5 dwt. 6. 5 5 Ib. 6 oz. 9 dwt. 6. 7 Ib. 3 oz. 4 dwt. 9 gr Reduce to Ib., etc. : 7. 7845 gr. 8. 8923 gr. 9. 57892 gr. 10. looooo gr, Addition. oz. dwt. gr. oz. dwt. gr. Ib. oz. dwt. gr. 11. 3 . 17 . 23 12. n . 13 . 21 13. 3 . 10 . 7 . 9 9. 12. 7 9 . 2 . 19 4-3-9.3 7 7 IS 8 . 17 . 13 7 . 7 8 . 12 ^ji_JL:_ 2 , Ai_Ii-^-J* 8 . 9-3-13 14. Subtract 3 oz. 16 dwt. 14 gr. from 6 oz. 13 dwt, 12 gr. 16. Subtract 7 Ib. 9 oz. 8 dwt. 20 gr. from 10 Ib. 4 oz. 3 dwt. 4 gr, 16. Multiply 3 oz. 5 dwt. 16 gr. by 5, 32, 427. 17. Divide I5lb. II oz. 13 dwt. 8^r. by 23, and by 9 oz. II dwt. 1 6 gr. 18. Find the weight of 24 gold necklaces each weighing 2 oz, 7 dwt. 12 gr. 19. If 64 gold rings of equal weight are made of I Ib. of gold* find the weight of each. 20. How many gold rings, each weighing 7 dwt. 12 gr., can be made out of i Ib. o oz. 15 dwt. of gold ? 65. English Standard or Avoirdupois Weight. 16 Drams (dr.) make Ounce (i oz.). 16 Ounces ... Pound (i Ib.). 2.8 Pounds ... Quarter (i (jr.). 4 Quarters ... Hundredweight (i cwt.). 20 Hundredweights ... Ton (i ton). A stone (st.) 14 Ib. A Pound Avoir. 7000 Grains Troy. A stone of butcher's meat * 8 Ib. A cental of corn 100 Ib. A sack of coals 2 cwt. A peck of flour 14 Ib. A barrel of flour 196 Ib. A sack of flour28o Ib. A barrel of gunpowder 100 Ib. A pack of wool 240 ttx A pocket of hops 168 Ib. A quartern loaf * 4 Ib. EXAMPLES. 33. Reduce to drams : L 7 tons 13 cwt 2. 2 tons 2 cwt. 2 qr. MEASURES OF WEIGHT 55 8. 3 tons 9 cwt. 3 qr. 21 Ib. 9 oz. 4. 9 tons 7 cwt. 6. 2 tons 3 cwt. i qr. 6. 2 cwt. 3 qr. 20 Ib. II oz. 12 dr. Reduce to tons> etc. : 7. 999999 dr. 8. 123456 dr. 9. 90000 gr. 10. I billion gr. Addition. Ib. oz. dr. i 11. Ib. oz. dr. qr. Ib. oz. tons cwt. qr. Ib. 7 7 10 12, 13 . 21 . 3 13. i . 16 3 19 9 9 7 7 8 . 7 2 . 8 3 12 . IS . 6 8 . 19 . 8 12 . o . 25 3 . 12 . 12 9 2 . 2 2 4 . i . 7 4 . 4 3 21 . 3 4 4 7 . 2 . 9 14. Subtract 7 Ib. 8 oz. 9 dr. from 10 Ib. 12 oz. 15 dr. 15. Subtract 2 tons 13 cwt. 3 qr, 12 Ib. from 9 tons 2 cwt. l qr. 2 Ib. ' 16. Multiply 7 cwt. 3 qr. 12 Ib. 9 oz. 2 dr. by 7, 88, 329. 17. Divide 2 tons 10 cwt. 2 qr. 8 Ib. i oz. by 29, and by II Ib. 5 oz. 4 dr. 18. Find the weight of 625 iron balls, each weighing 7 Ib. 8 oz. 19. The total weight of 56 bales of cotton is 7 tons I cwt. ; what is the weight of each bale ? 20. How many pick-axes, each weighing 4 Ib. 6 oz., can be made from i ton 10 cwt. of iron ? 21. Which is heavier, a pound of gold or a pound of feathers ? 22. How many pounds Troy are equal to 144 pounds Avoir. ? 66. Indian Bazar Weight. 4 Sikis make 5 Sikis 4 Kanchas or 5 Tolas 16 Chataks 40 Seers Tola. Kancha (Powa-chatak^ Chatak (i ch.). Seer. Maund (i md.). 4 Chataks i Powa. 4 Powas i Seer. 5 Seers i Punshury. 8 Punshuries i Maund. The weight of a rupee is called a tola. The standard sect -80 tolas. A tola 180 grains Troy. One maund of Bazar weight * 100 Ib. Troy*- 82$ Ib. Avoir. 35 Seer 3 72 Ib. Avoir, i Ib. Avoir. + the weight of a double-pice (200 gr.)=half a seer. 3 Factory maunds **2 cwt. 49 Bazar maunds* at 36 cwt.* 8 * 11 ^ Factory _mAimdg^i-catt i md. 14 seers 7i ch. ARITHMETIC EXAMPLES. 33. Reduce (i) to kanchas, (ii) to tolas : 1. 3 md. 7 seers 3 ch. 3. i md. 34 seers 15 6. 35 seers 3 powas. Reduce to md., etc. 7. 4664 kanchas. 9. 39855 tolas. md. seers ch. 11, 3- 8 3 8 . 12 2 . 29 9 - 36 7 7 12. 7 IS 3 i 2. 2 md. 20 seers 12 ch :h. 4. 2 md . 16 seers 2 pofras. 6. 2 md. 6 punshuries. 8. 3333 kanchas. 10. 100000 tolas. Addition. md seers ch. md seers ch. kanchas 13 22 . 7 13. 3 8 . 7 . i y 36 13 37 12 . 8 . 2 12 21 . 8 8 29 . 9 I 4 32 9 29 36 . 13 . 3 2 20 . 2 2 4 10 . 2 14. Subtract 3 md. 29 seers 7 ch. from 8 md. 17 seers 4 ch. 15. Subtract 2 md. 37 seers 12 ch. 2 kanchas from 10 md. 29 seers 7 ch. 16. Multiply 5 seers 10 ch. 3 kanchas by 9, 42, 2153. 17. Divide 71 md. n seers 9 ch. by 73, and by 2 md. 34 seers ich. 18. Find the weight of 273 bags of rice, each bag weighing 2 md. 7 seers 3 ch. 19. If 44 bottles of equal size hold I md. 5 seers 6 ch. of ink, how much does one bottle hold ? 20. 657 md. of flour are to be packed into bags holding I md, i seer i ch. each ; how many bags will be required ? 21. How many grains of gold are there in a plate weighing I seer 5 ch. ? 22. If 3 chataks of gold be made into 36 equal rings, how many grains will each ring weigh ? 6T. 3 Tolas 8 Pollums 5 Seers or 40 Pollums SViss 20 Maunds Madras Local Weight. make Pollum. Seer. Viss. ' Maund. Candy or Barum. A Madras maund25 Ib. Avoir. MEASURES OF WEIGHT 57 EXAMPLES. 34. Reduce to tolas : I. 6 pollums 2 tolas. 2. 2 md. 3 viss. 3. 3 md. 7 viss 4 seers. 4. 7 md. 3 seers. 6. 2 candies 7 md. 6. 3 candies 1 5 md. 5 viss. Reduce to candies, etc. : 7. 4281 seers. 8. 5182 pollums. 9. 70000 tolas. 1O. 925761013$. Addition. seers poll, tolas md. viss seers can. md. viss poll. II. 3 . 7 . 2 12. 7 - 5 3 13. 7 *5 5 9 1. 6. i 8.3.2 16.7.12 4.5.2 9.6,4 21, 9. 2. 23 2.0.1 2.7.1 56 . 3 . o . 36 14. Subtract 3 md. 3 viss 3 seers 3 poll, from 7 md. 7 viss 1 seers I poll. 15. Subtract 28 can. 17 md. 6 viss 3 seers 2 poll, from 40 can. f2 md. 18. Multiply 3 md. 2 viss 3 seers 2 poll, by 7, 72, 231. 17. Divide 36 can. 17 md. 4 viss by 59, and by 18 md. 3 viss 2 seers 4 poll. 18. Find the weight of 128 bags of rice each weighing 2 md, 3 viss 23 poll. 19. If 320 horses eat 18 can. 9 md. of corn in a certain timc how much does one horse eat ? 20. 9 candies of rice were distributed among a number of beggarsi each of whom received i viss 2 seers 4 poll. ; how many beggars were there ? 21. How many grains are there in a seer ? 68. Bombay Local Weight. 4 Dhans m 8 Raktikas 4 Mashas 72 Tanks 40 Seers 20 Maunds ake 1 Raktika. i Masha. i Tank, i Seer. I Maund. i Candy. A Bombay maund 28 Ib. Avoir. ARITHMETIC 2 md. 2 seers 7 tanks. 3 md. 16 seers 36 tanks EXAMPLES. 35. Reduce to dhans : L 10 candies. 2. 2 md. 7 seers. 3. 4. 3 can. 3 md. 6. 3 seers 30 tanks. 6. Reduce to candies, etc. : 7, 60000 tanks. 8. 78625 tanks. 9. 7000000 dhans. 10, In one billion dhans, how many candies, md., etc.'? Addition. md. seers tanks 17 IS 57 7 . 29 . 61 15 o-33 5 - 31 - 4 seers tanks mashas 11. 37 15 i 12. 29 . 17 . 2 7 . 26 . o 9 35 - 3 can. md. seers tanks 13. i . 9 . 29 . 25 2 . 12 . 37 . 35 3 - 13 2I 56 4 7 - 5 - 6 4 14. Subtract 3 md. 7 seers 13 tanks from 3 can. 7 md. 15. Subtract I can. 13 md. 29 seers 69 tanks from 9 can. 2 md. 16. Multiply 3 md. 15 seers 25 tanks by 5, 36, 231. 17. Divide 7 can. I md. 12 seers 56 tanks by 37, and by 14 seers 9 tanks 2 mashas. 18. Find the weight of 312 bags of rice each weighing I md. 7 seers 15 tanks. 19. If 144 bullocks eat 7 can. 7 md. 26 seers of hay in a certain time, how much does one bullock eat ? 20. 17 candies of rice are distributed among a number of beg- gars, giving to each 2 seers 9 tanks ; how many beggars get a share ?* XIV. MEASURES OF LENGTH. 69. English Linear Measure. 12 Inches (in.) make i Foot (i ft.). 3 Feet ... I 5$ Yards ... i 40 Poles or 220 yards ... I 8 Furlongs or 1760 yards ... I 3 Miles ... i i Pole 9 Inches t Spans or 1 8 Inches /. 2 Cubits 6 Feet 4 Poles or 22 Yards loo Links Yard (i yd.). Pole, Rod or Perch (i po.X Furlong (i fur.). Mile (i mi.). League (i lea.). 5 yd. i ft. 6 in. Span. Cubit (Hath\ Yard. Fathom. Chafo} Used in bad-surveying. MEASURES OF LENGTH 59 The following Table is used by tailors : 2j Inches I Nail (Girrah). 4 Nails I Quarter (Span). 4 Quarters I Yard. 5 Quarters = i Eli. The following are also used sometimes : 72 points 12 lines 3 barley-corns 3 inches 4 inches 5 feet 1 20 fathoms 6080 feet 60 knots inch, inch, inch. palm. hand (used in measuring horses), pace. cable's length. knot or geographical mile. degree of latitude. N. B. In measuring land in Bengal, 4 cubits = I cottah ; 20 cottah* = 1 bigha. 70. In reducing poles to yards, we multiply the number of poles by 11, and divide the product by 2. In the converse operation, we multiply the number of yards by 2, and divide the product by II. Example i. Reduce 2 mi. 2 fur. 9 po. 3 yd. I ft. to inches. mi. fur. po. yd. ft. Process : 2.2.9.3.1 _8 "18 fur. 40 729 po. II 2)8019 half-ytf 4009 yd. + i ft. 6 in. rem. [V a half-yd. - 1 ft. 6 InJ 3 yd. I ft. added. 4012 yd. 2 ft. 6 in. 3 12038 ft. 12 144462 in. Ans. Note. In reducing miles or furlongs to yards, it is convenient to reduce them at once to yards, unless we are prevented by the form of the question, as in the above example. Half-yards may be reduced directly to inches by multiplying the number of half- yards by 1 8 (V a half-yard i8 inches.) <O ARITHMETIC Example 2. Reduce 201381 inches to miles. Process : 12 ) 201381 in. 3 ) 16781 ft. -f 9 in. 5593 yd. *2 ft. 2; ii )lil86 half-yd. 4 ) IQl6 PO. + 10 half-yd. 8 ) 25 fur. + i6po. 3 mi. +i fur. /. 201381 in. = 3 mi. I fur. 16 po. 10 half-yd. 2 ft. 9 in, *3 mi. i fur. 16 pa 5 yd. 2 ft. 9 in. 3 mi. I fur. 17 po. I ft. 3 in. [Y 5 yd. I ft. 6 in I po.) If in a result the yd., ft. and inches exceed 5 yd. I ft. 6 in., we roust substitute I po. for this. EXAMPLES. 36. Reduce to inches : 1. 125 yd. 2. 5 fur. 3. 3 mi. 4, 2 Lea, 6. 2 mi. 7 fur. 2 po. 6. 3 mi. 2 fur. 20 po. 7. 3 lea. 5 fur. 1 1 po. 8. 3 po. 4 yd. 2 ft. 9. 5 po. 3 yd. I ft. 10. 7 po. 2 yd. 9 in. 11. 2 mi. 7 fur. 13 po. 4 yd. 12. 2 lea. 6 fur. 20 po. 3 yd. I ft. 6 in. Reduce to miles, furlongs, poles, etc. : 13. 1 56 yd. 14. 202yd. 15. 107 yd. 16. 196yd. 17. 1234 in. 18. 5890 ft. 19. 73212 in. 20. 80021 in. 21. icoo in. 22. 10000 ft. 23. 234567 in. 24. 987654 in. Reduce 25. 7 fathoms to inches. 26. 3 cubits I span to inches. 27. 3 yd. I cubit to inches. 28. 5 ells to nails. 29. 2 ells I qr. to nails. 30. 1000 nails to ells. 81. How many links are there in a mile ? XV. MEASURES OF AREA. 71. A square inch, is a square whose side is an inch in length. MEASURES OF AREA 6* English Square Measure. 144 Square Inches (sq. in.) make I Square Foot (l sq. ft.). 9 Square Feet ... I Square Yard (i sq. yd.). 30 \ Square Yards ... I Square Pole, Rod or Perch,. 40 Square Poles ... I Rood (i ro.). [(i sq. po.). 640 Acres ... I Square Mile (i sq. mi.), A square chain 22 x 22 sq. yards or 484 sq. yards. 10 sq. chains I acre. I sq. pole 30 sq. yd. 2 ft. 36 in. 7/5. In reducing sq. poles to sq. yards, we multiply the number of sq. poles by 121, and divide the product by 4. In the converse operation, we multiply the number of sq. yards by 4, and divide the product by 121. Example I. Reduce 2 ac. I ro. 13 sq. po. 12 sq. yd. 7 sq. ft. ta sq, inches. ac. ro. po. yd. ft. Process : 2 . i . 13 . 12 . 7 4 9 ro. 4? 373 s q* P 4103 11 4 ) 45133 quarter sq. yd. 36 sq. in.)> 11283 sq. yd. + 2 sq. ft. 36 sq. in.[V a qr. sq, yd. = 2 sq. ft* 12 sq. yd. 7 sq. ft. added 11295 sq. yd. 9 sq. ft. 36 sq. in 9 * 101664 sq. ft. 12 1219968 14639652 sq. in. Ans. [The learner should note that, I qr. sq yd. 2 sq. ft 36 sq. in, ; tqr. sq. yd.4 sq. ft. 72 jq, in. ; and 3 qr. sq. yd. 6 sq, fu too sq. m. j. ^2 ARITHMETIC Note. In reducing acres or roods to sq. yards, Jt is convenient to reduce them at once to sq. yards, unless we are prevented by the form of the question. Quarter sq. yards may be reduced directly to sq. inches by multiplying the number of quarter sq. yards by 18 x 18 (V a quarter sq. yd.=a sq. cubit = 18 x ib sq. in.). Example 2. Reduce 8753067 sq. inches to acres. fi2 } 8?t; Process ; 144 \ ( ~~ I2 i f 27 sq. in. 9 )6o785...2' 6753 sq. yd. 4- 8 sq. ft. 4 sq. yd. ")24J5...7| 29qr.sq.yd. 4 ) 223...2J 4 ) 5 ro. +23 sq. po. I ac. + 1 ro. /, The result I ac. I ro. 23 po. 29 qr. yd. 8 ft. 27 in. = i ac. I ro. 23 po. 7 yd. I qr. yd. 8 ft. 27 in. = i ac. i ro. 23 po. 7 yd. 10 ft. 63 in. I ac. i ro. 23 po. 8 yd. I ft. 63 in. If in a result the sq. yd., ft. and inches exceed 30 sq. yd. 2 ft, 36 in. we must substitute i sq. po. for this. EXAMPLES. 37. Reduce to sq. inches : 1. 23 sq. yd. 2. 3 roods. 3. 120 ac. 4. 2 sq. miles. 5. 7 ac. 2 ro. 8 po. 6. 12 ac. 3 ro. 20 po. 7. i ac. i ro. I po, 8. 3 sq. po. 7 yd. 7 ft. 9. 5 sq. po. 3 yd. 2 ft. 10. 7 sq. po. 20 yd. 36 in. 11. 2 ac. 3 ro. 7 po. 17 yd, 12. 3 ac. 2 ro. 17 po. 9 yd. 2 ft. 72 in. Reduce to acres, ^>ods, sq. poles, etc. : 13. 365 sq. yd. 14. 740 sq. yd. 15. 971 sq. yd. 18. loco sq. yd. 17. 7824 sq. yd. 18. 37821 sq. yd. 19. 93456 sq.ft. 20. 87893 sq.ft.' 21. 7234 sq. in. 22. 78934 sq. in. 23. 987650 sq. in. 24. 9876543 sq, in, Reduce 25. 7 sq. chains to sq. inches. 26, One million sq. links to sq. yards. MEASURES OF AREA 63 73. Land Measure of Bengal. I Square Cubit makes I Ganda (i ga.). 20 Gandas make I Chatak (i ch.). 16 Chataks ... I Cpttah (i cot.). 20 Cottahs ... i Bigha (i bi.). i bigha 1600 sq. yards. 121 bighas 40 acres. 1936 bighas = i sq. mile. iV bighas = 3 bi. 8 ch. EXAMPLES. 38. Reduce to gandas : I. 3 bi. 12 cot. 12 ch. 2. 12 cot. 9 ch. 5 ga. 3. 6 bi. II cot. ii ch. 4. 19 bi. 7 cot. 8 ch. 6. 19 cot. 15 ch. 19 ga. 6. 15 bi. 15 cot. 15 ch. Reduce to bighas, etc. : 7, 431 ch. 8. 728 ga. 0. 7892 ga. 10, loooo ga* 74. Land Measure of the United Provinces. 20 Kachwansi make I Biswansi. 20 Biswansi .-, I Biswa. 20 Biswas ... I Bigha. i guj ilahi = 33 inches. 60 guj ilahi 5 5 yards. I bigha ==(60x60) sq. guj ilahi =(5 5 x 55) sq. yards, = 3025 sq. yards. * 74a. Land Measure of the Punjab. 9 Square Karam or 9 Sarsai make i Maria. 20 Marias ... i Kanal. 4 Kanals ... I Bigha. 2 Bighas ... i Ghuma. i karam = 3 cubits. I bigha i 620 sq, yards. 75. Land Measure of Madras. 144 Square Inches make I Square foot. 2400 Square feet ... i Ground or M anal, 24 Grounds I Cawny. 484 Cawnies ... i Square Mile, 121 cawnies 160 acres. 6 4 76. ARITHMETIC Land Measure of Bombay. 39i Square Cubits make 20 Kathis 20 Pands 6 Bighas 20 Rukehs Kathi. Pand. Bigha. Rukeh Chahur. XVI. MEASURES OF SOLIDITY AND CAPACITY. 77. A cube is a solid figure contained by six equal squares- A cubic inch is a cube whose edge is an inch in length. 1. a, Measures of Solidity. 1728 Cubic Inches make 27 Cubic Feet A ton of shipping (English.) I Cubic Foot (i cu. ft.). i Cubic Yard (i cu. yd.). 42 cubic feet. EXAMPLES. 39. Reduce 3, 7, 12, i6> 2O> 39 cu. yd. to cu. in. Reduce 123456, 987654 cu. in. to cu. yd. Measures of Capacity. (English.) 4 Gills make I Pint (i pt.). 2 Pints Quart (i qt.). 4 Quarts Gallon (i gall.). 2 Gallons Peck (i pk.). i 4 Pecks Bushel (i bus.). 8 Bushels Quarter (i qr.). 5- Quarters 2 Loads Load (i Id.). Last (i last). For dry goods only. 2 quarts pottle (i pot.). 2 bushels strike (i str.). 4 bushels coomb (i coomb). Also A Barrel contains 36 gallons. A half barrel (18 gallons) is called a kilderkin^ and a' quarter barrel (9 gallons) a firkin. A hogshead (ti\\&.) of ale contains i \ barrels or 54 gallons; a butt of ale 3 barrels and a pipe 6 barrels. The terms hogshead, butt &&&pipe are also used in measuring they are different for different kinds of wine. MEASURES OF TIME, ANGLES AND NUMBER 6$ Note. A gallon of distilled water weighs exactly 10 Ib. Avoir, A Pint of water weighs a pound and a quarter. [A gallon contains 277*274 cubic inches.] A cubic foot of water weighs about 1000 oz. Avoir. EXAMPLES. 40. Reduce to gills : L 12 gall. 2 qt. I pt. 2. 2 barrels 16 gall. 3. I barrel n gall 4. 6 bus. 2 pk. I gall. 6. 4 qr. 4 bus. 2 pk. 6. I Id. 3 qr. 7 bus, 7. 7 lasts I Id. 3 qr. 8. 2 lasts 4 qr. 5 bus. 9. 20 lasts I Id. 4 qr. Reduce to barrels, gallons, etc. : 1O. looo gills. 11. 2073 gills. 12. 3400 gills. 13. 7225 gills. Reduce to lasts, loads, quarters, etc. : 14. 3000 gills. 15. 1 500 gills. 16. 25000 gills. 17. 98765 gills. 18. What is the weight of 2 gall. 2 qt. of water ? 19. Give in pounds Avoir, the weight of 2 cu. yd. 2 cu. ft. of water. 20. How many pottles are there in a coomb ? How many in a strike ? MEASURES OF TIME, ANGLES, NUMBER, AND APOTHECARIES' WEIGHT. 79. Measures of Time. (English.) 60 Seconds (sec.) make i Minute (i min.). 60 Minutes i Hour (i hr.). 24 Hours 7 Days 365 Days 366 Days 100 Years Day (i da.). WeekU wk.). Year (l yr\ Leap-year. Century. Note 1. Each day is considered to commence at midnight Note 2. In rough calculations a month is taken to consist of 30 days. But the 12 months, called Calendar Monthsi into which the year is divided, are of variable length. February has 28 days (and in Leap-year 29), Thirty days have September, April, June and November. The other months have 31 days each. CN A. $ 66 ARITHMETIC Note. 3. If the number of a particular year is divisible by 4 it is a Leap-year ; but centuries not divisible by 400 are not Leap-years. Thus 1888, 1732, 1600 are Leap-years ; 1887, 17391 1800 are common years. [The solar year consists of 365*242218 mean solar days (or 365 da 5 hr. 48 min. 48 sec. very nearly) or nearly 365^ days ; hence to make the civil year correspond with the solar, we take 3 consecutive years of 365 days and a fourth, called leap-year , of 366 days, those being leap-years of which the numbers are divisible by 4. But in this way we insert loo days in 400 years, which is too much, for '242218 x 400 is 96*8872 or 97 days nearly ; to make the necessary correction centuries not divisible by 400 are taken as common years.] Note 4. The year contains 52 weeks and i day (V 52x7 + 1 365)) but in calculating the income of men paid by the week, it is customary to consider the year to consist of 52 weeks. EXAMPLES. 41. Reduce to seconds : L 7 hr. 12 min. 3 sec. 2. 7 da. 9 hr. 10 min. 3. 2 wk. 3 da. 12 hr. Reduce to weeks, days, hours, etc. : 4. 5000 sec. 6. 98765 sec. 6. One lac sec. 7. One million sec. Find the number of days (including one only of the two days named) from 8. 3rd Jan. to 7th April 1887. 9. 2oth Jan. to 2oth May 1888. 10. May loth '87 to Jan. 9th '88. 11. Aug. ist '80 to March ist '8z, 12. 2ist Feb. to 7th Dec. 1700. 13. 3oth Dec. '83 to 3oth March '86. 14. The ist January 1880 was on Monday ; what day of the tfeek was June 2oth of the same year ? 15. The 9th of December 1845 was on Sunday ; what day of the week was ist January 1847 ? 80. Measures of Angles. 60 Seconds (60") make I Minute (i') 60 Minutes ... I Degree (i e ). 90 Degrees ... i Right Angle (i rt. gle,), EXAMPLES. 42. Reduce to seconds : 1. 7*. 17'. 27". 2. 240. 25'. 35". 3. 4 rt gie. Reduce to right angles, degrees, etc. ; 4. 4000", 6. 37956"- & 7000'. 7. 8256'. 8, 987654", APOTHECARIES 1 WEIGHT 81. Measures of Number. 12 Units make 12 Dozen 12 Gross 20 Units Also 24 Sheets of paper 20 Quires 10 Reams Dozen. Gross. Great Gtoss. Score (Kurrf Quire. Ream. Bale. EXAMPLES. 4S. 1. In 50 reams of paper, how many sheets ? 2. How many reams, quires, etc. are there in fifty thousand sheets of paper ? 3. How many scores are there in 5 great gro^b ? 82. Apothecaries' Weight. (i) Measures of Weight. Druggists use the grain to weigh small quantities and the pound and ounce Avoir, to weigh large quantities. Some physi- cians in prescribing use the following table : 20 Grains make i Scruple (i srr.). 3 Scruples ... I Drachm ^i dr.;. 8 Drachms ... I Ounce Troy. (ii) Measures of Capacity. -60 Minims (m.) or drops make I Fluid Drachm (fl. dr.), 8 Fluid drachms ... I Fluid ounce (fl. oz.). 20 Fluid ounces ... i Pint (O.). 8 Pints ... i Gallon (C.). A teaspoonful i Fluid drachm. A dessertspoonful == 2^ Fluid d vtclims. A tablespoonful 4 Fluid drachms. Note. Since a pint of water weighs a pound and a quarter, the weight of a fluid ounce of distilled water is an ounce Avoir. EXAMPLES. 44. Reduce 1. 2 oz. 2 dr. 2 scr. to grains. 2. 3 oz. 3 dr. 12 gr. to grains. 3, 2 O. 12 fl. oz. to minims. 4. 2 C. 4 O. to minims, 6, 7 C. 7 O. 15 fl. oz. 5 fl. dr. 9 m. to mm<ms. 6g ARITHMETIC MISCELLANEOUS EXAMPLES. 45. L A girl can paper 2 pins in a second ; how many pins caa she paper in a working day of 8 hours 30 minutes ? 2. Find the price of 3 md. 7 seers of milk at 2. 6/J. per seer. 3. Find the value of 12 Ib. 7 oz. of gold at 3. 15*. \\d. per oz. 4. A train travels 19 mi. 7 fur. 30 po. per hour ; how far will It travel in 24 hours ? 5. A fruit-seller sold 210 oranges at i pice each, 76 apples at I anna each, and 55 mangoes at la. 6^. each ; how much did he realise from the sale ? 6. How many cwt. of coal will supply 64 fires for 3 weeks, each fire consuming I cwt. 2 qr. i Ib. per day ? 7. If the cost of 9 md. be 8480, what is the cost of a chatak ? 8. If the cost of a tdn be .203, what is the cost of a pound ? 9. How many shot, each weighing 2 oz. 3 dr., will make up a heap weighing one ton ? 10. How many parcels, each weighing I md. 10 seers, can be made up of goods weighing 132 md., and what weight will remain over ? 11. How many jars, each containing 2 gall. 3 qt. i pt. 3 gills, can be filled out of a cask containing 285 gallons ? 12. How many pieces of rope, each 2 ft. 9 in. long, can be cut off a length of 1760 yards, and what length will remain over ? 13. A train travels 45 miles in 2 hours ; how many yards does it travel in a second ? 14. A man gave R7. ga. 6/. to each of 24 men, and then had R6. 7 a. <#*. left ; how much had he at first ? 16. A has R3. 70. 9/. more than B, B has R2. 8a. 3^. less than d and C has Rl2 ; how much has A ? 16. If a man's net annual income be Rl78$6. 40., how much may he spend per day and per week to the nearest pie, so as not to run into debt ? [Reckon 52 weeks and 365 days to the year.] 17. If the daily income of a man be R3. 40. o/., how much, can he spend per day that he may save R239. %a. 6/. in a year ? 18. If a man spends R$. 3#. 3^. daily, how much will he be able to save out of an annual income of R24oo ? 19. How much to the nearest farthing can a person spend daily if he wishes to save ^300 out of an annual income of ^700 ? 20. A gentleman's gross annual income is R3ooo ; he pays- 872. 30. annually in taxes : what must his daily expenses be that he may save RioSo in a year ? MISCELLANEOUS EXAMPLES 6$ 21. A man spends 7. 80. 9^. daily, and saves Riooo a year ; what is his annual income ? 22. A clerk received ^114 . 7 . 6 as pay in 1888 ; how much is that per day ? 23. A man was born on the loth of January 1832 ; what was bis age on the I7th of April 1888 ? 24. I distribute R3oo among boys, giving a rupee, a half-rupee) a quarter-rupee and a, two-anna piece to each ; how many boys get a share ? 25. Sound travels 1125 ft. in a second ; if a gun is fired at a distance of 1875 yards, what time must elapse between the seeing of the flash and the hearing of the report ? 26. How many steps does a soldier whose stride is 2 ft. 8 in. take in walking 2 miles ? 27. If a soldier takes 3240 strides in walking I mile 1030 yards, what is the length of his stride ? 28. The circumference of a bicycle wheel is 12 ft. 7 in. ; how many complete revolutions does it make in going 10 miles ? 29. A certain sum of money was divided into 18 equal parts j each part was &4. 8a. $fi. and there \\ as R2. 70. 6p. over : find the sum. 30. A man earned R3$. 9<z. 6/. in January and R49. 80. 9^. in February ; he spent R26. 30. 3^. each month : how much did he save in the two months ? 31. A man earns i. js. 6d. per week, and pays js. 6d. every fourth week to his club ; what is his net income in a year of 52 weeks ? * 32. How many complete yards are there in the united length of 7 benches, each 7 ft. 7 in. long ? 33. A man spends in 4 months as much as he earns in 3 .months ; what does he save out of an annual income of R ( 275o. 8a. ? 34. A and B together have 56. 12$. 6</. ; A has $. 171. gd. more than B ; how much has A ? 35. The earnings of a man and his 2 sons amount to ,600 a year, and 'their expenses to 400 ; if the balance be divided equally how much will each receive ? 36. How many quart bottles can be filled out of a cask con- taining 2 cwt. I qr. 8 Ib. of water ? 37. The 1st of January 1881 was on Monday ; find the num- ber of Mondays in that year. 38. A vessel which holds to gallons weighs when empty go Ib. ; what is the weight of the vessel when full of water ? 70 ARITHMETIC 89. Your father was 25 yr. 7 mo. 10 da. old when you were born ; your sister was born when your father was 21 yr. 9 mo. 8 da, old : how old is your sister now if your age is 12 yr. 6 mo. ? 40. Four dollars, 3 half-guineas, 5 half-crowns and 6 florins amount to ^3. 12s. 8dl ; what is the value of a dollar ? 41. Two pieces of cloth of equal length cost ^3. o. 9 and 2. 5. o respectively ; the price of the first was 3^. tfad* per yd. : what was the price of the second per yard ? 42. A merchant bought 350 Ib. Avoir, of lead, and sold it by Troy weight ; how many pounds Avoir, did he gain ? 43. A shop-keeper's weight was deficient 3 tolas to a seer ; what quantity would he defraud his customers in selling 8 maunds ? 44. Fifty bags of rice are bought for R8oo. i2a. 6p. at 83. 3*. 3^. per maund ; what is the weight of each bag ? 45. Light travels 186500 miles in a second ; what time does it take in travelling from the sun to the earth, a distance of 92877000 miles ? 46. The small wheel of a tricycle makes 330 revolutions more than the large wheel in passing over a mile ; if the circumference of the large wheel be 8 ft., what is the circumference of the other ? 47. A weekly newspaper was numbered 4 on the 7th January 1885 ; when was it numbered 40 ? 48. A daily newspaper which is published on week days only was numbered 20 on Monday the I3th January 1884 ; on what date was it numbered 120 ? 49. A person travelled 120 miles by railway at 15 miles an hour, 120 miles by road at 8 miles an hour, and 60 miles by bullock cart at 2 miles an hour ; how long did he take ? 60. Supposing that the distance of the sun from the earth is 91776000 miles, and that light travels from the former to the latter in 7 min. 58 sec., find the velocity of light per second. 61. The value of a mark being iy. 4</., and that of a dollar 4$. 2*/.,how many half-crowns are there in 9 marks + 12 dollars ? 62. A person laid out .43. gs. $d. in spirits at 5*. fa. a gallon, some of which leaked out in the carriage ; he sold the remainder for ^54, at the rate of js. 6d. a gallon : how many gallons leaked out? 63. A wheel makes 600 revolutions in passing over I mile 40 yards ; what is its circumference ? 64. Divide R6$. 100. equally among 8 men, 12 women and 30 children ; supposing the children to have received their shares. BARTER, GAIN AND LOSS, ETC. 7* and the men to have given up their shares to the women, how much would each woman receive ? 65. How many times will a church-clock, which chimes the quarters, strike and chime in February 1900 ? 56. How many times does the 29th day of the month occut in 400 consecutive years ? 57. The circumferences of the large and small wheels of a tri- cycle are 13 ft. 9 in. and 3 ft. 4 in. respectively ; how many more turns will the latter have made than the former when the tricycle has gone a distance of 1 5 miles ? 68. If for every Ri rent paid to his landlord a man pays in addition i anna for gas, how much will he have left out of an annual income of R3ooo if he lives in a house whose monthly rent is R2O ? 59. After measuring 40 yards of a rope it was discovered that the measuring yard was an inch too long ; what was the true length measured ? 60. One man is 30 years 17 weeks and 5 days old, and another Is 26 years 9 weeks and 3 days old ; a third is just as much younger than the first as he is older than the second : what is the age of the third ? XVIII. BARTER, GAIN AND LOSS, ETC. 83. Barter. Example. How many seers of sugar at 40. bp. a seer must a grocer give in exchange for 9 Ib. of tea at Ri. 20. a lb.? Cost of 9 lb. of tea = Ri. 2#. X9 Rio. za. The number of seers of sugar rqd. = Rio. 2a. + 4a. 6^. = 36. EXAMPLES. 46. 1. How many pounds of tea at Ri. 4^. a pound must be given In exchange for 40 yards of silk at R2. ioa. a yard ? 2. How many dollars of $s. id. each can be obtained for 100 rupees of is. lod. each ? 3. If 48 yards of ribbon be given in exchange for 2 maunds of brown-sugar at 3 annas a seer, what is the price of the ribbon per yard ? 4 A man exchanges 45 sheep at 2 . 5 . 9 each and 37 pigs at 3 .13-6 each for 13 oxen at ^17 .6.6 each, the difference being paid or received in money ; how much does he pay or receive ? 5. Seven pounds of tea at Ri . 3 . 6 a pound and 13 pounds of coffee are given in exchange for 15 maunds of wheat at Ri. 13 . 3 a maund ; find the price of a pound of coffee. 72 ARITHMETIC 84. Gain and I*OBB. Example. If 2 5 yards of cloth are bought at 7-r. 6d. a yard and sold at &s. gd. a yard) how much is gained ? Profit on each yard8j. gd.~7s. 6d.**is. $& /. Total profit** u. 3^x25^1. us. $d. EXAMPLES. 4T. 1. A man gives 15 maunds of rice worth R3. Sa. a maund, and receives in exchange 22 maunds of flour worth R2. Sa. a maund ; does he gain or lose, and by how much ? 2. A man buys 150 yards of cloth at Ri. la. $p. per yard, and sells at Ri. 3#. 6p. per yard ; what does he gain altogether ? 3. A grocer bought a chest of tea containing 320 Ib. for 405, and sold it at Ri. 50. gp. per Ib. ; what did he gain ? 4. Twenty-nine sheep are bought atlRs. 8. each ; 15 of them are sold at R6. <\a. each, and the rest at R$. 40. each ; find the gain, 6. A grocer buys 1 5 maunds of sugar at 40. 6/. a seer, and sells at Ri3. 40. 6p. a maund ; what is his gain ? 6. Out of 2 md. 1 5 seers of milk, boughtifor R6. 90. gp., 7 seers are lost by leakage ; what is gained by selling the remainder at la. (>p. a seer ? 7. A cwt. of sugar is bought for Ri4. ga. 6/., and is sold for Ri6. 5*. 6p. ; what is the gain per Ib. ? 8. A grocer bought I cwt I qr. of sugar for i. i$s., and gained lu. 8*/. by retailing it ; at what rate per Ib. was it sold ? 9. A merchant bought 40 gallons of wine, and lost ^5 by selling it for ^37 ; at what rate per gallon did he buy it ? 10. A dealer bought wheat at 38^. gd. perjqr. ; he subsequently sold it at 2. os. $d. per qr., and made* a profit of ^i. i6s. altogether ; how many quarters did he buy and sell ? 11. ^ A man buys 45 yards of silk at 6$. 6d. per yard, 1 5 yards of which being damaged, he sells at 5^. per yard ; at what price must he sell the rest so as to gain i . 12 . 6 altogether ? 12. A grocer buys 200 Ib. of tea at Ri. 20. per Ib., and sells one-half of it at Ri. 30. per Ib. ; at what rate must he sell the remainder so as to gain R2$ on the whole ? 13. If 7$. bd. be lost by selling an article for ^3, what would have been gained or lost by selling it for 4 ? 14. I sold some goods weighing 13 cwt. 2 qr. 9 Ib., for " ib. ~~ ----- 72 . 17 . 7$> gaining thereby ^d. per ib. How much should I have gained per Ib. if I had sold them at ^5 . 12 . o per cwt. ? BARTER, GAIN AND LOSS, ETC. 73 15. A tradesman buys a piece of cloth 50 yards in length for 1^40. 100. ; at what price per yard must he sell it (i) that he may gain 5#. per yard, (ii) that he may gain 818. 120. on the whole ? 85. Mixtures. Example i. If 3 maunds of rice at 82. 80, per maund be mixed with 5 maunds at 83. 2<z. per maundi find the price of the mixture. | Cost of 3 md. at 82. 8a. R2. 80. x 3=87. 8a. ; cost of 5 md. at 83. 20. 83. 2a. x 5 = 815. loo, ; .*. cost of 8 md. of mixture 87. 8a. + 815. ioa. 823. 2a. .". Cost of i md. of mixture = 823. 2a.-r8. 82. 140. 3^. .*. Price required 82. 14^. j$. per maund. Example 2. How much water must be added to 12 gallons of beer at IO.T. a gallon, to reduce the price to 8j. a gallon ? The price of the whole mixture at 8j. a gallon must be equal to the price of 12 gallons of beer at IQJ. a gallon. Therefore, if we divide the price of 12 gallons of beer at IO.T. a gallon by 8j. we shall get the number of gallons in the mixture. Price of 12 gallons of beer = ioj. x 12 =* 120^. ; .*. number of gallons in the mixture i2OJ.-r8j.l5 ; .*. number of gallons of water added =I5-I23. EXAMPLES. 48. 1. A mixture is made of 7 seers of sugar at 40. 60. per seeri 2 seers at 40. per seer and 3 seers at 30. 6/>. ; find the value per seer of the mixture. 2. A man bought 3 qr. of wheat at 30^. per qr. and 9 qr. at 26s. per qr. ; he mixed them, and sold the mixture at 3^. 7\d. per bushel ; how much did he gain ? 3. To 20 seers of milk, bought at la. qp. a seer, 5 seers of water are added, and the mixture is sold at 20. per seer ; how much is gained ? 4. A merchant buys 1 5 md. of sugar at 89. 80. per md., 18 md, at 89. 4a. per md. and 10 md. at 89 per md., and pays 84. 20. for carriage ; he mixes them : at what price per md. must he sell the mixed sugar so as not to lose by the sale ? 6. 10 Ib. of coffee are mixed with 2 Ib. of chicory ; if the mixture be worth is. lid. per Ib., and the chicory 3</. per Ib., what is the value per Ib. of the pure coffee ? 6. A grocer mixes 36 Ib. of tea at 2*. 4^. per Ib. with 48 Ib, 74 ARITHMETIC at is. io^d. per Ib. ; at what price per Ib. must he sell the mixture so as to gain 13^. 6d. on his outlay ? 7. A woman buys 8 dozen eggs at 2^. per dozen, and 12 dozen more at i^d. per dozen ; at what price per dozen must she sell the whole so as to gain ^d. per dozen ? 8. How much water must be mixed with 36 seers of milk at ifl. 9^. per seer, so as to reduce the price to la. 6p. per seer ? 9. How many pounds of tea-dust (worth nothing) must a grocer mix with 20 Ib. of tea at 2s. 6tt. per Ib., to enable him to sell the mixture at 2s. per Ib. and gain at the same time 8j. on the transaction ? 10. A grocer buys 30 Ib. of tea at 25. a pound and 50 Ib. of tea at 2s. 8d. a pound, and having mixed them sells 40 Ib. of the mixture at is. \d. ; at what price per Ib. must he sell the remainder that he may neither gain nor lose ? 86. Division of Money. Example i. Divide Ri3. 9*. among A> B and C so that A may have I2a. 3^. more than Z?, and B Ri. 2a. Q/. more than C. B is to have Ri. za. typ. more than C, and A is to have I2a, $p, + Ri. 20. 9^. more than C ; if we take away these sums to be subsequently given to B and A respectively, the remaining portions of their share will be each equal to the share of C. R. a. p. R. a. p. 1.2.9 13 . 9 . o 12 31 I 2 . 9/ ._._ 3 ) lo ._7 . 3 _ R3 . J, . g 3.7. 9 = ^s share ; .". 4 . 10 . 6 = j5's share ; and 5.6. 9 = yfs share. EXAMPLES. 49. 1. Divide R39. jot. <$. between A and #, so that A may get &? 4<* 3^ more than B* 2. Divide ^28. 7^. 6d. between A and B, so that A may receive 3. 14^ 3^- less than B. 8. Divide R357. I4a. 6/. among 15 men, giving Rn. 140. 9/fc more to each of two of them than to each of the others. 4. Divide R679 among 27 men and 5 women, so that a man may get R6 less than a woman. 6. Divide R39. 4#. 6p. among A^ B and C 9 so that A ma? receive R3 more than /?, and B R4 more than C. BARTER, GAIN AND LOSS, ETC. 75 6. Divide R32Q. 70. o/. among A, B and C, so that A may get R7 more than /?, and B R2 less than C. "7' ^95 ioj. is divided among 8 men, 7 women and 6 boys, so that each man receives los. more than each womA, and each woman lew. more than each boy ; find how much the men receive. Example 2. Divide R$9. 6a. among 3 men, 5 women and 6 boys, so that each man may receive three times as much, and eack woman twice as much, as a boy. R. a. 3 men = 9 boys (5 ) 59 . 6 5 women = 10 ... 1 5 ) II . 14 6 boys = 6 ... 2 . 6 = each boy's share ; 25 .*. 4 . 12= ... woman's... and 7 . 2 ... man's ,~ EXAMPLES. 50. 1. Divide Ri5. 90. 6/>. between a boy and a girl, so that the boy may receive twice as much as the girl. 2. Divide RSI. 30. between A, B and C in such a manner that ^4's share may be 3 times, and tfs twice, Cs. 3. Divide Rioo among 3 men, 5 women and 10 boys, so that each man may receive 4 times as much as a boy, and each woman twice as much as a boy. 4. Divide ^ii . 15 . 4$ among A, B and C, so that A may receive twice as much as B, and B twice as much as C. 6. Divide ^10 .7.6 among 3 persons, so that one may re* ceive twice as much as each of the others. 6. Divide R39. 7 a. gp. between A and B> so that A may re- ceive Ri. 140. $p. more than twice the amount to be received by B. Example 3. Divide R28 into an equal number of rupees, half- rupees and quarter-rupees. A rupee -Ha half-rupee + a qr. -rupee = Ri+ 80. -t-4#.Ri. 120. .*. The number of each kind of coin=R28-rRi. 120. i6. EXAMPLES. 51. 1. Divide R22. 80. into an equal number of rupees, half-rupees, quarter-rupees and two-anna pieces. 2. Divide ^17 into an equal number of sovereigns, half- sovereigns, half-crowns, shillings and sixpences. 76 ARITHMETIC 3. A box contains an equal number of crowns, shillings and pennies ; the total amount in the box is $. 13^. : find the number of each. 4. Rioo^s divided among an equal number of men, women and boys ; each man receives 82. 80., each woman R2 and each boy Ri. i2a. : find the number of men, women or boys. 6. A bag contains a certain number of rupees, twice as many half-rupees, and 4 times as many quarter-rupees ; the whole sum amounts to 33 : find the number of eacr* 6. Among how many children may R6o be divided so that each child may receive a rupee, an eight-anna piece, a four-anna piece and a two-anna piece ? 87. Example. A and B together have Ri3. &*., B and C together have KS. 8#., A and C together have RII. 80. ; how much has .4 ? 813. 80. 4- RII. 8<z.= twice A's money + #'s money -f <7s money; but R8. Sa. = j&'s money -f^s money. /. (Ri3. 8#. + Rn. 80.-R8. 80.)orRi 6. 8#.= twicer's money; .*. A's money = Ri6. 8a.~-2 = R8, 4#. Or thus*: (Ri3. 8a. + R8. 8<z. + Rii. 80.) or R33. 8*.=twice A's money + twice B*s money -f twice Cs money ; * (&33 8a.-r2) or Ri6. i2a.=*A's money + ,#'s money -fCs money; t>ut R8. 8^.-^s money + Cs money ; .*. A's money =Ri6. I2^.~R8. 8^. = fi8. 4^. EXAMPLES. 52. 1. A and B together have R6. oa. 3^., B and C together have &4, 15^. 9^., A and C together have RS. 15^. ; how much has A ? 2. A and B together have R24. 10., B and C together have 19. 15*., A and C together have R23. 120. ; find how much B has. 3. A horse and a cow are together worth RIOI, a cow and a sheep are together worth RSI, a, horse and a sheep are together worth R8i ; find the price of a horse, of a cow and of a sheep. 4. A mark and a gulden are together worth 2^. i \\d^ a gulden and a rouble are together worth 5^. i J*/., a rouble and a mark are together worth 45. i%d. ; find the value of a mark| of a gulden and of a rouble. B. A man and a woman together have RSO. 70. 6/5., the woman and a boy together have R2o. 8<z., the man and the boy together have R25. 90. 6^. ; find how much the man, the woman and the tx>y together have. FACTORS AND PRIME NUMBERS XIX. FACTORS AND PRIME NUMBERS. 88. If one number divides another exactly ', the first is said to be a factor (or sub-multiple) of the second, and the second is said to be a multiple of the first. Thus 5 is a factor of 1 5, and 15 is a multiple of 5. In speaking of the factors of a number we exclude the number one or unity^ which may be said to be a factor of any number. \N. B. In the present section the word divisible is used in the sense- of exactly divisible."} 89. An even number is a number divisible by 2. An Odd number is a number not divisible by 2. 90. Criteria of Divisibility : A number is divisible by 2 when its last figure is o, or an even digit ; as 310, 54 : 4 when its last two figures represent a number divisible by 4 ; as 300, 320, 324 : >8 when its last three figures represent a number divisible by 8 ; as 2000, 3400, 3240, 3816 : 5 when its last figure is o or 5 ; as 370, 345 : 10 when its last figure is o : 3 when the sum of its digits is divisible by 3 ; as 126, 402 : 9 when the sum of its digits is divisible by 9 ; as 477, 801 : 11 when the difference between the sum of its digits in the odd places and the sum of its digits in the even places is either o, or divisible by n ; as 34672, 582934. To determine whether a number is divisible by 7, n, or 13 we have the following rule : Divide the figures of the number into groups containing three each, as far as possible, counting from right to left. Add the alternate groups, and subtract the smaller sum from the greater ; then if the remainder is o or is divisible by 7, n, or 13, the number itself is also divisible by 7, or by II, or by 13. Thus 98126 is divisible by 7, but not by n or by 13: for 126-98*28 which is divisible by 7, but not by n or by 13. 91. If a number is divisible separately by two numbers which- have no common factor, it is also divisible by their product. If a number is divisible by 3 (or 9), any other number ex- pressed by the same digits is also divisible bv ^ for o). ARITHMETIC If each of two numbers is divisible by a third number, their sum (and difference) is also divisible by the third. If a number is divisible by another, any multiple of the first is also divisible by the second. If each of two numbers is divisible by a third number, then the sum (and difference) of any multiple of the first and any multiple of the second is also divisible by the third number. , J \ EXAMPLES. _JK3L \ Determine whether the following numbers are divisible by 2, 3, 4> 5)8,9, 10 or 11 : 1. 138. 2. 945. 3. 684. 4. 420. 5. 8844, 6. 7942. 7. 1230. 8. 1772. 9. 2311. 10. 3475. 11, 8976. 12. 7128. 13. 12345. 14- 9^765. 15. 35600. 16. 23000. 17. 709281. 18. 777777. 19. 989898. 20. 1234567890. Determine whether the following numbers are divisible by 7, 11 or 13: 21. 99120. 22. 89133. 23. 67119. 24. 555555- 35. 433378. 26. 4123210. 27. 5573454S- 28. 123789666. Determine whether the following numbers are divisible by 6, 12 or 30 : 29. 372. 30. 948. 31. 7740. 32. 3725. 33. What is the least number which being added to 2311 will make the sum divisible (i) by 3, (ii) by 4 ? 34. What is the least number which being subtracted from 70031 will make the remainder divisible (i)by 5,(ii)by 8, (lii)by 9 ? 35. What number is the same multiple of n as 3705 is of 15 ? 9. A prime number or a prime is a number which is not divisible by any number (except itself and unity). I, 2, 3, 5, 7, ii| 13, etc. are jfrr/wommbers. A compositeTnumber is a number which has factors each greater than I. 4, 6, 8, 9, 10, 12, etc. are composite numbers. 93, To ascertain what numbers are primes. (i) To find the prime numbers in a series of numbers, 1, 2, 3,..., cancel every second number after 2, every third number after 3i every fifth number after 5, and so on ; the remaining numbers will FACTORS AND PRIME NUMBERS be primes. [In finding the primes in any series of numbers, we need not divide by any prime number whose square is greater than the largest number in the series.] (ii) To determine whether a given number is a prime, divide the number successively by the primes 2, 3, 5, 7, u, etc. ; if there is a remainder in each case the given number is a prime. [It is not necesspry to try a divisor whose square is greater than the given number.] Note. From Art. 90 it will appear that the units' figure ol every prime number (except 2 and 5) must be i, 3, 7 or 9. Hence any given number (not being 2 or 5) need only be examined when its units' figure is i, 3, 7 or 9 ; and in such a case we need not try the divisors 2 and 5. 93a. The following is a list of PRIME NUMBERS between i and 1009. I 59 139 233 337 439 557 653 769 883 2 61 149 239 347 443 563 659 773 887 3 67 151 241 349 449 569 66 1 787 907 5 7i 157 251 353 457 57L 673 797 911 7 73 163 257 359 461 577 677 809 919 ii 79 167 26 3 367 463 587 683 811 929 13 83 173 269 373 467 593 691 821 937 *7 89 179 271 379 479 599 701 823 941 19 97' 181 277 383 487 60 1 709 827 947 23 101 191 28l 389 491 607 719 829 953 29 103 193 28 3 397 499 613 727 839 967 31 107 197 293 401 503 617 733 853 971 37 109 199 307 409 509 619 739 8 .57 977 4i H3 211 3H 419 521 631 743 859 983 43 127 22 3 313 421 523 641 75i 863 991 47 131 22 7 317 43i 541 643 757 877 997 53 137 229 331 433 547 647 761 881 1009 94. Every composite number can be resolved into factors which are all primes. Note. A number has only one set of prime factors. Example. Find the prime factors of 4452. We divide the number successively (and in each 2)4452 case as often as possible) by those of the primes 2, 3, 2)2226 5> 7i u J 3> > that c <* n fa used as divisors^ until we come to a quotient which is a prime number. Thus 4452 2 x 2 x 3x 7 x 53, ~~*53 80 ARITHMETIC EXAMPLES. 54. Find the prime factors of L 8. 2. 12. 3. 1 8. 4. 24. 5. 27 6, 32. 7. 48. 8. 50. 9. 63. 10. 64. 11. 80. 12. 88. 13. 99., 14. 100. 15. 108. 16. 176. 17. 117. 18. 288. 19. 495. 2O. 625. 21. 999. 22. 1050. 23. 1296. 24. 1760. 25. 2000. 26. 3650. 27. 5760. 28. 2457. 29. 13824. 30. 200100. Determine which of the following numbers are primes, an<S find the prime factors of those which are composite : 31. 29. 32. 61. 33. 81. 34. 79. 35. 97. 36. 107. . 37. 113. 38. 207. 39. 227. 40. 349. 41. 3751- 42. 507, 43. 4573- 44. 619. 45. 713* 48, 997. 47. 6539. 48. 1793- 49. 509. 5O. 1363. Find the number of primes between 61. i and 30. 52. 10 and 50. 53. 20 and 70. 54. By what prime numbers may 37 be divided, so that the remainder may be 2 ? 65. By what prime numbers may 109 be divided, so that the remainder may be 4 ? 66. By what numbers may 29 be divided, so that the remain* der may be 5 ? XX. HIGHEST COMMON FACTOR. 95. A common factor of two or more numbers is a number which divides each of them exactly. Thus, each of the numbers 2, 3 and 6, is a common factor of 12 and 1 8. The Highest Common Factor (H. C. !F.) of two or more numbers is the highest number which divides each of them exactly. Thus, 6 is the H. C. F. of 12 and 18. Note. Two numbers are said to be prime to each other wheir they have no common factor. M B. The term measure is often used as synonymous with/ac/of, and Greatest Common Measure instead of fcfrto* common factor. HIGHEST COMMON FACTOR 8 1 9& The H. C. F. of two or more numbers is (he product of all their common prime -factors. Example i. Find the H. C. F. of 18 and 30. i82X3*X3 ; 30-2x3x5. The factors common to the two numbers are 2 and 3 ; hence the H. C. F. required 2X 3 6. Note. In finding the H. C. F. it is not necessary to find the prime factors of all the numbers. It is sufficient to find the prime factors of one of the numbers, and to form the product of those that divide each of the remaining numbers exactly. Example 2. Find the H. C. F. of 84, 140 and 168. Now, 84 =-2x2x3x7 ; and we find that each of the remaining numbers is divisible by 2x2x7, but not by 3 ; therefore the H. C. F. required -2x2x7 = 28. /EXAMPLES. 55. Find, by the method of factors, the H. C. F. of 1. 9 and 24. 2. 20 and 48. 3. 35 and 80. 4. 126 and 144. 5. 90 and 325. 6. 252 and 348. 7. 150 and 375. 8. 256 and 788. 9. 480 and 792. 1O. I5>35 120 - H- 16,24,140. 12. 90, 125, 342. 13. 224, 336, 728. 14. 625, 750, 1225. 15. 868, 3164, 4228. 97. The following rule gives the most convenient method of finding the H. C. F. of two numbers : Divide the greater number by the less, the divisor by the re- mainder, then the second divisor by the second remainder, and so on, until there is no remainder ; the last divisor is the H. C. F. required. Example I. Find the H. C. F. of 384 and 1296. Process : 384 ) 1296 ( 3 1152 144 ) 3&1 (2 288 96 ) 144 ( I 96 f 48 ) 96 ( a 96 .'. The H. G. F. required is 48. C. A. 6 02 ARITHMETIC Note. When the H. C. F. of three or more numbers is requir- ed, we first find the H. C. F. of any two of them and then find the H. C. F. of this result and another number, and so on, through all the given numbers ; the last result is the H. C. F. required. Example 2. Find the greatest number that will divide 50 and 60 leavin'g the remainders 8 and 4 respectively. 50-842 ; 60- 4 =-56. .'. The number required the H. C. F. of 42 and 56 14. EXAMPLES. Find the H. C. F. of 50. 1. 48 and 144. 2. 76 and 238. 3. 92 and 772. 4. 252,348. 6. 493)899- 6. 620, 2108. 7. 2121,1313. 8. 429)715- 9. 377) 1131. 10. 1379) 2401. 11. 266, 2793. 12. 3775, 10000. 13. 6023,15466. 14. 5865,69180. 15. 4081, 5141. 16. 3556, 3444. 17. 5187, 5850. 18. 6441, 10283. 19. 13667, 14186. 20. 43365, 44688. 21. 11050, 35581. 22. 12321, 54345. 23. 6327, 23997. 24. 13202, 146083. 25. 5325, 8307. 26. 9945, 50609. 27. 4155,24720. 28. 109056, 179712. 29. 218707, 826769. 30. 123456, 987654. Are the following prime to each other ? 31. 403 and 527. 32. 3370, 2703. 33. 387, 9234. 34. 1726, 1623. 35. 3890, 8275. 36. 3486,9448. 37, 211,2701. 38. 5789,7337. 39. 9367, 14501. Find the G. C. M. of 40. 703037 and 5134083. 41. 271469,30599. 42, 805, 1311, 1978. 43. 204, 1190, 1445. 44. 1617, 123, 789. 45. 1300,725,870. 48. 723)807,735. 47. 504, 2394, 2835. 48. 1190,1445,2006. 49. 13338) 14136, 15903. 50. 314) 570, 618, 720, 51. 602, 7394, 876, 92458. 52 What is the largest sum of money which is contained in E6. 40. and 7. 80. exactly ? 53. What is the largest sum of money which will divide j. TS. bd. and 13. 17*. 9</. exactly ? 64. Find the greatest number that will divide 728 and 900, leaving remainders 8 and 4 respectively, LOWEST COMMON MULTIPLE 83 56. Find the greatest number that will divide 261, 933 and 1381, leaving the remainder 5 in each case. 56. Is there any number that will divide 620 and 730, leaving the remainders 3 and 7 respectively ? 57. Two vats contain respectively 540 and 720 gallons ; find the vessel of greatest capacity that will empty off both vats. 58. Two masses of gold weighing 4427 and 7219 tolas respec- tively are each to be made into coins of the same size ; what is the weight of the largest possible coin ? 59. A labourer was engaged for a certain number of days for R2. 8<z., but being absent on some of those days he was paid only Ri. I2a. ; prove that his daily wages could not be more than 4 annas. 60. A woman bought a certain number of eggs for 150. 6#,> and sold some of them without profit for 50. 6p. ; shew that she had still left at least 20 eggs. XXI. LOWEST COMMON MULTIPLE. 08. A common multiple of two or more numbers is a number which is exactly divisible by each of them. The Lowest Common Multiple (L. C. M.) of two or more numbers is the lowest number which is exactly divisible by e'ach of them. Thus, each of the numbers 12, 24 and 36, is a common multiple of 3, 4 and 6 ; but 12 is their lowest common multiple. 00. The product of two numbers is equal to the product of their /f. C. F. and L. C. M. Thus, 2 is the H. C. F. and 12 is the L. C. M. of 4 and 6 ; and 4x6=2 x 12. Hence we have the following rule for finding the L. C. M, of two numbers : Divide one of the numbers by the //. C. F. and multiply the quotient thus obtained by the other. Example. Find the L. C. M. of 38 and 57. The H. C. F. of 38 and 57 = 19 ; 38~i92. .'. The L. C. M. required = 2 x 57 114. Note. When the L. C. M. of three or more numbers is required, we find the L. C. M. of any two of the numbers, and then find the L. C. M. of this result and a third number, and so on ; the last result being the L. C. M. required. 84 ARITHMETIC EXAMPLES. 57. Find the L. C. M. of L 12 and 32. 2. 76 and 98. 3. 81 and 99. 4. 320, 704, 6. 117, 192. 6. 1224, 1696. 7. 224, 336. 8. ^54, 806. 9. 957, 1001. 1O. 845, 899. It 779,1197. 12. 1287,6281. 13. 76,96,106. 14. 629, 851, 253. 15. 265, 385, 495. 16. 300, 906, 708. 17. Resolve 210 and 385 into their prime factors, and hence obtain their L. C. M. 18. Find the L. C. M. of 44, 54 and 72 by resolving them into their prime factors. 19. Find the L. C. M. of R3. ga. 4/*, and 7. loa. 3^. 20. The H. C. F. and L. C. M. of two numbers are 16 and 192 respectively ; one of the numbers is 48 : find the other. 21. The H. C. F. and L. C. M. of two numbers are 10 and 30030 respectively ; one of the numbers is 770 : what is the other ? 10O. The following rule gives the most convenient method of finding the L. C. M. of several small numbers : Place the numbers side by side in a line ; divide by any one of the prime numbers 2, 3, 5, 7, n, which will divide any two at least of the given numbers exactly ; set down the quotients thus obtained and the undivided numbers side by side ; and proceed in. this way until you get a line of numbers which are prime to one another. The continued product of all the divisors and the num- bers in the last line will be the L. C. M. required. Example I. Find the L. C. M. of 12, 18, 20 and 105. Process : 2 ) 12, 18,20, 105 2 ) 6, 9, 10, 105 3 ) 3 9, 5> IQ5 5 ) i, 3, 5> 35 i,3> i> 7 A L. C. M. 2x2x3x5x3x7 1260. Note. Work may be shortened by rejecting, at any stage, from the line^ any one of the numbers, which is a factor of any other number in the same line. Thus, if it is required to find the L. C. M. of 6, 12, 15, 30 and 4o> it will be sufficient to find the L. C. M. of 12, 30 andjo. LOWEST COMMON MULTIPLE 8$ Example 2. Find the least number which when divided by 12, 1 6 and 18, will leave in each case a remainder 5. The L. C. M. of 12, 16 and i8i44. .*. The number required 144 + 5 149. EXAMPLES. 58. Find the L. C. M. of 1. 6, 8, 16. 2. 12, 16, 24. 3. 5, 18, 16, 9. 4. 9, 4, 18, 6. 5. 12, 15, 18, 24, 56. 6. 15, 16, 20, 28, 42. 7. 22, 17, 33, 25, 85. 8. 8, 9, 12, 18, 30. 9. 6, 15, 27, 35, 45. 10. 28, 36, 54, 72, 90- 11. 24, 10, 32, 45, 25. 12. 9, 18, 24, 72, 144- 13. 51, 187, 153, 165. 14. 33, 55, 60, 80, 90. 15. 22, 88, 132, 198. 16. 17, 51, 119, 210. 17. 50* 338, 675) 702, 975- I 8 - 2 4, 35> 5 2 > 60, 91* 108. 19. 315,156,126,108,91. 2O. 27,87,203,261,189. 2L 126,145,87,210,585. 22. 2,3,4,5)6,7,8,9>io. 23. ,-2, 4, 6, 8, 10, 12, 14, 16. 24. 15, 16, 18, 20, 24, 25, 27, 30, 25. 24, 35, 52, 60, 91, 108, 126, 156, 315. 26. Find thfe least number which when divided by 12, 18 and 30, gives the same remainder 9 in each case. 27. Find the least number which when divided by 128 and 96 will leave in each case the same remainder 5. 28. Find the least number which being increased by 3, will be exactly divisible by 24, 36 and 48. 29. Find the smallest number of sq. inches which contains an exact number of sq. feet or of sq. cubits. 30. What is the smallest sum of money that can be paid ID pounds, or in guineas, or in moidores ? 31. Five bells toll at intervals of 3, 5, 7, 8 and 10 seconds respectively, beginning together ; after what interval of time will they again toll together ? 32. Three men journey 10, 15 and 18 miles a day respectively j find the least distance which would occupy each of them a com* plete number of days. 33. Two round pillars are 14 yd. I ft. 9 in. and 18 yd. 2 ft 3 in. respectively in circumference ; find the shortest rope that can be wrapped round each an exact number of times. 86 ARITHMETIC 34. A heap of shot when made up into groups of 28, 32 and 42| leaves always a remainder 5 ; find the least number of shot such heap can contain. 36. Find the least number which is divisible by all the nunV bers from i to 20 inclusive. 36. The circumferences of the wheels of a carriage are 6 ft. 3 in. and 9 ft. ; what is the least distance in which both the wheels will make an exact number of revolutions ? XXII. FRACTIONS. 101. When a quantity is composed solely of entire units, its measure is called a whole number or an integer. [In sections II XXI the word number has been used in the sense of a whole numberl\ When a quantity is composed of one or more equal parts of the anit its measure is called a fractional number or a fraction. Example. Two-thirds is a fraction ; for two-thirds of the unit Indicates a quantity which is composed of two equal parts, three of which make up the unit. 102. The number of equal parts, into which the unit is divided, Is called the denominator of the fraction ; and the number of such parts taken to make up the quantity is called the numer- ator of the fraction. The numerator and denominator are called terms of the fraction. A fraction is represented by writing the numerator above the denominator, with a horizontal line between them. Thus, $ represents the fraction of which the numerator is 4, and the denominator is 7. Such symbols are called Fraction-symbols or Fractions. Note 1. The symbol \ is read one-half \ \ is read one-third \ | is read two-thirds ; J is read one-fourth ; is read three-fourths ; and so on. A fraction expressed in the above notation is called a Vulgar Fraction. Example. *| of a yard* indicates a quantity which is composed of two equal parts, three of which make up one yard ; that is, '} of a yard' 2 feet. Note 2, We should get the same result whether we divide a yard (or any other unit) into 3 equal parts and take 2 such parts, or divide 2 yards (or twice that other unit) into 3 equal parts and FRACTIONS 87 take one of these parts. A fraction may thus be considered to ex- press the quotient of the numerator by the denominator. Hence \ is often read '2 divided by 3.' EXAMPLES. 59. Write down the value of 1. i of Ri. 2. i* 3. </. 4, 7. 10. 13. Jb of a md. A f a ft- A f u. t^j mile. 5. 8. 11. 14. AofRi. A f an anna. AofRx. seer. e. 9. 12. 15. JJ of a yd, A ton. tV s q- ft* 16, 19. TTJ CWt. A of 3 ft. 3 m. 17. 2O. f of 15*. ft of 7 J*/. 18. 21. f of Rx. 5*. A f i hr. 5^1 1O3. If the numerator and denominator of a fraction are each multiplied by the same number, the value of the fraction is not altered. For, consider the fractions $ and || : the first indicates that the unit is divided into 3 equal parts and 2 of these parts are taken ; the second indicates that the unit is divided into 36 equal parts and 14 of these parts are taken. Now, a part in the former case is obviously equal to 12 parts in the latter case : consequently 2 parts (taken) in the former case 24 parts (taken) in the latter case. Illustration : % of a yard 2 ft. ; and of a yard =24 in.=*2 ft, Corollary. If the numerator and denominator of a fraction are each divided by the same number, the value of the fraction is not altered. 104. A whole number may be expressed as a fraction with any given denominator. Thus for example, 3 } \ \ J = et c. 105. A given fraction can be transformed into another fraction of which the denominator is any multiple of the given denominator. Example. Transform f into a fraction with the denominator 12. 12 3x4 ; hence f 3$$=*iV Ans * EXAMPLES. 6O. L Express each of the whole numbers 2, 5, 7, 10 as a fraction with denominator 9. 2. Change xi to fractions having 2, 9, n, 25 and 35 for their denominators. 88 ARITHMETIC 3. Express 21, 76 and 159 as fractions with denominators 5) 9 and 75 respectively. 4. Express f and J each as a fraction with denominators 12, 1 8, 96 and 600. 6. Find fractions equal to J, f , , ft, J&, having 90 for their denominator. 6. Transform ,%, jf and $ into equivalent fractions whose denominators shall be n, 5 and 10 respectively. 7. Express f f , |g, J and JJ> each as a fraction with the denominator 6. IO6. A fraction is said to be in its lowest terms when its numerator and denominator have no common factor. Example i. Reduce ff g to its lowest terms. We divide the numerator and denominator by their H. C. F, which is 210. Thus iig-H8$H8-l ** Note. In reducing a fraction to its lowest terms, it is con- venient first to remove any factors common to both numerator and denominator, that can be found by inspection or by the application of the tests of divisibility (Art. 90). Example 2. Reduce Jf to its lowest terms. Process: f- Ans * * 14 & 14 Here, first 78 and 84 are divided by 2, giving quotients 39 and 42 ; next 39 and 42 are divided by 3, giving quotients 13 and 14 which are prime to each other ; hence the answer is }J . Example 3. Reduce by cancelling to their lowest terms : 74 2 It should be borne in mind that when a factor is conceited^ it is replaced by I and not by a FRACTIONS 89 EXAMPLES. 61. Reduce to lowest terms : 1. f. 2. &. 3- i- * * M- 6. i?. 7. fj. 8. g. 9. Jf. 10. , 1L fj. 12. *& 13. }J- I*- ff. 15. fg. 16. H. 17. ff. 18- B 5 J- I - AV 20. ft. EXAMPLES. 61a. Reduce to their lowest terms : 1. }f. 2. Jf- 3. *J. 4. *$. 5. 6. AV 7. HI 8. T %. 9. Jtl. 10. 11- iWu- 12. H. 13. T W*. 14. -iWi- 15. 10. IMf 17. H*5- is- i*S$- i - J8?*- 2 . 21. iffj. 22. fH|. 23. fi|$. 24. }f?f. 25. 26. Slff- 27. Sltf. 28. fj&f. 29. IJ^Jf. 30. "31. Htf 32. $$&. 33. tffU- 34. ftHfff- 86. EXAMPLES. 61b. Reduce by cancelling to their simplest forms : m. 2. mi 3. 5. iftHH. e. 8. *W^/. e. 10. Wffifr 11- AWa^- 12. 107. A mixerf number is composed of a whole number and a fraction, as 3|. This stands for 3+f, and is read VAr two-fifths.' A mixed number can be expressed as a fraction. Example, Express 4$ as a fraction. For, 12 thirds of the unit and 2 thirds of the unit make (12 + 3] or 14 thirds of the unit. Hence the rule : Multiply the whole number by the denomi- nator of the fractional part ; add the result to the numerator of that part for the new numerator, and retain the same denominator, 90 ARITHMETIC EXAMPLES. 62. Express the following mixed numbers as fractions : 1. 3*. 2. 7 |. 3. 9 &. 4. 8&. 6. 5*. 3. 7iin. 7. I2 7 8 a . 8. 2o/ . 9, 39^, 10. 9i&. 11. 29*S . 12. 76W- 13. 25}*. 14. in*??. 15. 99JJ. 17. 8AA. 18. 22&\. 19. 4 o*&T. 20. 4** 1O8. A proper fraction is one, of which the numerator is less than the denominator, as $. An improper fraction is one, of which the numerator is equal to or greater than the denominator, as , |. An improper fraction is either equal to an integer or a mixed number. tlxample* Reduce V an <* V to whole or mixed numbers. Hence the rule : Divide the numerator by the denominator ; the quotient will be the integral part of the mixed number ; the remainder will be the numerator, and the denominator of the gtven fraction the denominator, of the fractional part. (i) 7)21 Hence ^=" 3, rem. o. (ii) 6)29 Hence ^~ 4, rem. 1O9. The reciprocal of a fraction is a fraction formed b interchanging its terms ; thus the reciprocal of f is f , of 4 (or $) is EXAMPLES. Express as whole or mixed numbers : L J. 2. J. 3. #. 4. V- 6- e, ^. 7. V- 8 - ?i- 9 - *8- 10. If. 11. !J. 12. ft. 13. . 14. If. 16. ff 16. W- 17. !8i- 18. W- 19- W 20. H|. Express the reciprocals of the following fractions as whole or mixed numbers : 2L ,#&. 22. ,},. 23. fa 24. //. 25. ,|8 . 26. fg?. 27. ^Wr. 28. T ^. 29. #>. 80. FRACTIONS 9* 11O. Two or more given fractions may be reduced to equiva- lent fractions having the lowest common denominator. Example. Reduce J, & and A to equivalent fractions bavin? the lowest common denominator. The denominators are 9, 12 and 10 ; their L. C. M. is 180. 180-5- 9-20, /. t 1804-12-15, :. A Hence $, A and A- A% iVo and iVb respectively ; and these latter have the lowest common denominator. EXAMPLES. 64. Reduce to equivalent fractions having the least common denominator : 1. iandf. 2. A and A- 3. A and ^. 4. & , J. 5. ,f,i 6. ,?,J. 7. |,J,A- 8. Ai & 4- e. AAiii. 10. A.&.iJ* IL .* 12. AiAA- 13. 3i.4i,6t. 14. 2,i,J. 15- 3) 5) A- le - *i 3i> 2 > i- 17. 3, J, 4, J. is. i,J,i,i,J. 19, A,*iAAii. 20. i,|,f5,iV 21. J8, ,,,} 22. 2,2j, T v^,H. 23. A. At Ai Ai iJif 24. 2, 3*1 7i, Ait- ' 25. *,*'***. 26. 37i,2i,*i|. 27. x, *, 1, i, A> 2A. 111. Of two fractions having a common denominator the greater is that which has the greater numerator. Thus, of the fractions A an ^ A> tne fomier i s obviously greater, Of two fractions having a common numerator the greater is that which has the less denominator. Thus, of the fractions f and f the former is greater. Note. In comparing values of fractions, they must be reduced to equivalent fractions having the L. C. D. or L. C. EXAMPLES, o Whicb is greater, * -------------- L | orf? 2. A or A? 3. A or 4. tforig? 6. {orf? 6. f? or ARITHMETIC Find the greatest and the least of the following fractions : Arrange in order of magnitude : 13. i, f , A- 14. <& if, &. 15- |, J, 1}. 1 A 18 8 47 T"7 21 28 91 1ft 2.T 118 1415 " TT^J J3> S"* *f 55) aCf) So* J.O. 6"T| 3X1 > XlFff* ADDITION AND SUBTRACTION OF FRACTIONS. 11$. Addition. The sum of fractions having a common denominator is a fraction whose numerator is the sum of the numerators, and whose denominator is the common denominator, of the original fractions (see Art. 107). When fractions to be added have different denominators, they must be reduced to equivalent fractions having the L. C. D. Example I. Add together f and |. Process : |-f f +$ I f tAs =V sss2 t dns. Example 2. Add together J, and $. The L. C. M. of 2, 6 and 9 18. a&^iA.*!! Ans. Note. The sum should always be expressed in its lowest terms ; and if an improper fraction, should be reduced to a mixed number. EXAMPLES. 60. Add together 1. i>if 2. ?, ?,f 3, i, $, f. 4. /T> & A e i T u J8> A- e * 10. \,\. 11. !,|. 12. Simplify 19. J + & + A- 20. HHi 21.' 22. 1+J+A+A- 23. } + T T 5+/ 5 + A 24. f +$ + &+ 26. A + fHt+i. 26. Jj+H+i+A' 27. B Jii + ISJ+Wf- 29. ^^f|j4-H- 30. ADDITION AND SUBTRACTION OF FRACTIONS 93 113. In adding mixed numbers it is convenient to proceed as In the following example : Example. Add together 24, 3J and 7|. Process : 24+3^+7$ ~i Ans. N. B. It is also convenient t6 reduce improper fractions to mixed numbers. EXAMPLES, 67. Add 1. 34+4J. 2. 7i+6f- 3. 5& + 7i- ' 4. 5. Si+Si+^iV 6. 7} + 8 i+i4iV. 7. 8. 3i + 9$ + ii- 9 - 11. 3H4A+6A+iiV 13. 3A + ^ a +-W 14 - W+W+4. 15. 2j+3 + i4+- le - ]7 -l^^i 4- J-P Q^ + -IftA^ 18 19. io + 3* + W + Jt. 20. B. a. P' / J. d. yd. ft. in. 21.' 7 . 9 . 2-i 22. i 9 2 8 23. 7 . i . 3f 5 10 . 7J4 2 . o . 5 2 . 2 . zl 13 - 14 3 . 7 M 3 . . 7l' 2 . 7 'o}| i . o . 3s 2 . I . sA Ib. oz. dr. oz. dwt. gr- hr. min. sec. 24. I . 7 . 74 25. 3 . 10 26. 3 . 20 . Q| 2 . 3$ y 7 . o 8 7 . 22 . i9s 3 4 . > COCO M * ;4 ' 8 2 3 7 .0* . 4 4 5 . 7 34 29& 34A 114. Subtraction. TJje method of subtraction of fractions is similar to that of addition. Example i. Subtract f from f. Process : f-f-JjA-f. ^fj. Example 2. Subtract f from . The L. C. M. of 8 and 6-24- 94 ARITHMETIC EXAMPLES. 68. Perform the following subtractions : 1. fg-JJ. 2. V-^- 3. -J. 4. J-J. 6. I-A- 6. A-&- 7. A-A- 8 - i 8 a-T&. a. -ft. 10. A-A- 11- *&-& 12. 18J-M1- 13. |-|. 14. 7i-2*. 15. lA-iJ. 16- t-tt- 17. 4-xJ. 18. 2f~2j. 19. 7fi-7A- 20. JI-A. 21. I- ^5. 22. i-A- 23. i -A- 24. i-Jg. 115. The following examples are important. Example I. Subtract 3? from 7$ . Process : 7fi-3$*7i0-3la s =7 Example 2. Subtract 2| from 4}. Process : 4l 2| 4/r-2jJ3}| Example 3. Subtract ^ from 7. Process: 7- A a = s 6 + i~ 1 ^=6 + / 2 6 1 T 5 . Example 4. Subtract 3i from 9. Process : Q-Si-^-i-S + i-i-S + i^Si- Ans * EXAMPLES. 09. Perform the following subtractions : 1. **- Si- , 2. 9j-7i- 3. 3i-. 4. si-i. 6. I2f-7i- i7A-iH- 7. 8H-2& 8 - xoJf- 9. 5 J-2i 10. 71-3*. 11. 8^~7A. 12. 23^- 13. 5-2- 14 - 12*1-312. 16. X 34i-24f 16. SPA 17. 39i-288?.18. 9A-2H. 19. 7 J-|. 20. ic^- 21. 3-t 22. 7~|. 23. 9 -ft 24. xo- 25. I2-3|. 26. I7-4A- . i8-4A- 28. 2O- Simplify 29. 2|+3t-4i. 30. 7I+9A-IOJ. 31. 3H4J-H- 32. !7i-3i-7*. 33. 9A-8i+3t. 34. i2^-7i- 2 J. 85. 8-aJ+7*-3iV 36. 7-3H- MULTIPLICATION AND DlVlSlON OP FRACTIONS 95 37. 39. 41. 42. 43. 44. 45. 46. H-7H9-2*. Subtract &2. 1305. Subtract 87. loa. Subtract R2. 13*. Subtract ^3. 17*. Subtract ^4. 7J. 38. 7-8+8-f 40. 3 J+4l-5l from Ri3. 9**. dp. from Rio. 70. 3^. from R7. 2a. $p. from 14. 7J- from ,10. os. >. Subtract 4. js. -$%d. from ;io. os. 2*fad. \. Subtract 7 yd. 2 ft. 9J in. from 14 yd. o ft. 3^ in. MULTIPLICATION AND DIVISION OF FRACTIONS. 116. To multiply a fraction by a whole number, we. multiply the numerator by that number, leaving the denominator unchanged. Thus Example i. Example 2. 23fx 5-23X 5 + f x 5- Example 3. Multiply Jft by 57. Since Jft-x-^B, i%x 57^57- A T o s - Example 4. Multiply Since 99i%= s 100 L 5. 9. 13. 17. 21. 25. 29. 33. 36. Multiply A by 10. A by 21. EXAMPLES. 70. 2. 6. 10. 14. 3lby4. 18. 2|{byi2. 22. 3tJ by 54. 26. Jtfo by 29. 30. 99tf& by 9. 99i by 32. 18 by 1 5. 7. & by 36. 11. & by 144. 15. 6| by 7. 19. SA by 12. 23. 4llby249. 27. T 9 o 8 aby 3 9. 34. 9Aby39- 37. 9& T ffby2i. 11 by 1 1. 15 by 30- If by 51. II by 570. 7? by 9. 295 by II. 18? by 303. II by 70, 8A by 12. 35. 38. 4. 8. 12. 16. 20. 24. 32. 999i%fc by 23," 3I9||S by 20. ARITHMETIC 39. 41. 43. R7- 3*. 3iA by 7- 4s. o&d. by n. 40. gs. i if yd. by 9. 42. R8. 3*. 4fe>- by 6. 44. 3. os. 7&& by 12. 117. To divide a fraction by a whole number, we multiply the denominator by the whole number, leaving the numerator unchanged. Thus f -5-5 #$* ft : for, a part of the unit in ft is one-fifth of a part in $, and since the same number of parts is taken in both cases, ft is one-fifth of f . Example I. 72~^o ^p""io aa '5rx\ij ass ]y><5' sas 'j. Example 2. Divide 3759$ by 5. Process : 5 ) 3759$ 751, 4frem. Note. In the division of integers by integers, the complete quotients can always be obtained by the aid of fractions. Thus for example, Divide 1. 5. 9. 13. ft by 12. W8 by 5. H b y *35 2. 6. 10. . 14. I* by 28. 7. Air by 42. 11. fj by 160. 15. I by 7. 4 i | by 22. 8. HJby88. 12. ffi by 95. 16. iVi by 54- 17. 7i by 4. 18. 3l by 9- 19. 3| by 85. 20. 4|by ii. 21. x6ibyis . 22. 4} by 57. 23. 3i by 21. 24. 2$ by 40. 25. 2i3i by 5 . 26. 734 by 6. 27. 7i3f by 4. 28. 100} J by 1 5. 29. 33348 by 21. 30, 356^ by 33- 31. 999ft by 16. 32. 729lf by 19- 33. 324$ by 15. 34. 39i by 24. 35. RlO. 12CI. 2\&. by 8. 36. R22. 13*. 3i# by 9. 87. 20. 7s. 6f< by n. 38. ^99- 19* ' Ufa t. by 13, Obtain the complete quotient in the division of 89. 720 by 9. 40. 1 346 by 7. 41. 1000 by 23. 42. 1234 by n 43. 29. 7*. by B7. 30. 44. 82. 140. 6p. by I*. #. 46. 728. in. by 3. 7*. 46. ^100. 7*. 6$<L by 135. 8rf. 47. iUo. 80. j#. by 8. 48. 13.12*. 6 by 11. MULTIPLICATION AND DIVISION OF FRACTIONS 97 49. 8420. 7 a. <#. by 13. 5O. Rioo. 3*. n^. by 16. 51. 17. ijs. 7d. by 5. 62. 59. 19*. ud. by 15. 118. The definition of multiplication which we have given in Art. 29 implies that the multiplier is a whole number, and it is not applicable when the multiplier is a fraction. We therefore give below the general definition of multiplication. Def. To multiply one given number by another is to perform upon the number multiplied that operation which is performed upon unity to obtain the multiplier. Since I is repeated three times to obtain the number 3, to multiply a number by 3 is to repeat that number three times. Again, since I is divided into 3 equal parts and two of these parts are taken to obtain the number f , to multiply a number by is to divide that number into three equal parts and take two of these parts ; that is; to multiply a number by $ we have to divide the number by 3 and multiply the result by 2. Example. Multiply f by $. Since ^ 7 =- g | T ; and Tf| T X2=|? ; /. *xH = &- Ans. Hence the rule : To multiply one fraction by another, multiply the numerators for the numerator of the product, and multiply the denominators for its denominator. [N. B. This rule holds good for the continued product of three or more fractions.] Note. Hence it is clear that J x #=$ x f . 119. A compound fraction is a fraction of a fraction ; as I off The compound fraction, of $, means that we are to divide | (regarded as a whole) into 3 equal parts and take 2 of these parts, Hence \ of is equivalent to x , *>., to x f. Example. Simplify 3} of 9f . 3} of 9l-tfx9l-xy M B. Before effecting the multiplication, common factors should be removed from the numerator and denominator. EXAMPLES. 78. Multiply L fbyj. 2. ibyi 3. f by |. C. A. 7 98 ARITHMETIC 4. ifbyV. 5. ffbytt. 6. Jf by |g. 7. Hbyff 8. |f by ft. 9. 5* by &. 10. 3fby. 11. by2|. 12. If by 3*. 13. 4Aby7i 14. 7fbysJ. 15. af by i &. 16. 4*by3&- 17 - 2|by3f 18. 3i T i by 2$. 19. 5&bysJ. 20 - 3fby4|- 21. 2$ by 4*. Simplify 22. 34 of 2?. 23. iof44of3j. 24. 24 of 3 J of 4 25. lofijxyf 26. 4ix}of4 5 V 27. i|x2JX3{. 28. | of 2^x3! of 9. 29. 3Sof2|x4X7*. 30. JofAofff- 31. 3^x 32. 4ix2jxi*of2k 33. Jof 34. jxJx^x 5 4 T xii 35. J of ^ of i of I off. 36, 2jof3ixiJof2 1 %xiJ. 37. $ of 9x74x4! of J of |. 10, Example. Reduce 29 poles to inches. Process : 29 po. _5* 145 -29x5. JL4i ===2 9^ 2 ) ^o 29x4. 1594 yd. _ 3 12 5742 in. EXAMPLES. 73. Reduce to inches : 1. 7 po. 2. 13 po. 3. 29 po. 4. 39 po. 6. 49 po. 6. 4 fur. 39 po. 5 yd. 7. 10 mi. 5 fur. o po. 3 yd. Reduce to sq. inches : 8. 7 sq. po. 9. 13 sq. po. 10. 29 sq. po. 11. 39 sq. po, 12. 49 sq. po. 13. 9 ac. 2 ro. 7 po. 14. I sq. mi. 3 ac. 10 po. 1/81. Division by a fraction is the inverse of multiplication. To divide by | is to find that number which being multiplied by \ gives as the product. But x| being multiplied by f gives f as the product (V |xf=i) ; therefore f -rf = xf ; and hence H. C. F. AND L. C. M. OF FRACTIONS 99 we have the rule : Multiply the dividend by the reciprocal of the divisor. Example I. 8^3? Example 2. If f of a number is 4, what is the number ? Here the product of the number (required) by f is 4 ; .*. the number required = 4 ~-ff x j s * 8 ^ is=a 6|. EXAMPLES. * Divide 1. fbyf. 2. fbyf. 3. A by . 4. {$ by 5. 3iby2i. 6. 7iby T V 7. ^ by i^ T . 8. Jf by *?. !. 9. n**byi 10. i6fbyi2i. 11. ?| by A. 12. n^by 12] 13. I2f by I J. 14. 13^ by 2^. 15. 10$ by }?. 16. 9 by J !o* 17. I 4 ?bysf. 18. iiiby7i 19. 10 by 7i 20. 76 by 28*. 21. f of4iby7iof 3 f. 22. 3 Jx6| by ijx 14. 23. 4i-f7i 1 nby4|-2j. 24. 3|of3Hy7-3i 25. g- of a number is 14 ; what is the number ? 26. 3f of a number is 2| ; what is the number ? 27. Find the number, f of which is of J. 28. 3^ of 4^ of a number is 7 ; find the number. 29. of | of a number is 3^ of 10 ; what is the number ? 30. Which is greater, the quotient of 3^ by 6J or the continued product of f j | and f ? ^. C. ft AND L. C. M. OF FRACTIONS. The definitions which we have given of the H. C. F. and L. C. M. of two or more whole numbers will also be applicable when the given numbers are fractions, provided that we understand by exact division, that the complete quotients must be integers, Rule. To find the H. C. F. or the L. C. M. of fractions, reduce them to their least common denominator ; then find the H. C. F. or the L. C. M. of the new numerators, and write it over the common denominator. Example I. Find the H. C. F. and L. C. M. of f , z\ an The given fractions are equivalent to J, f $, Jj ; the H. C. F. of 12, 40 and 15-1, and their L. C. M.-i2o \ .". the H. C. F. required^ ; and the L. C. M. required %i=^ 7 i. 100 ARITHMETIC The following rules will be found practically more convenient. (i) The H. C. F. of two or more fractions in their lowest term* is a fraction whose numerator is the H. C. F. of their numeratorsi and whose denominator is the L. C. M. of their denominators. (ii) The L. C. M. of two or more fractions in their lowest terms is a fraction whpse numerator is the L. C. M. of their numerators, and whose denominator is the H. C. F. of their denominators. Example 2. Find the H. C. F. and L. C. M. of &, 2 and $. The given fractions when reduced to their lowest terms are equal to J, f and . (i) H. C. F. of numerators =* I , and L. C . M . of denominators 36 .". the H. C. F. required 3 V (ii) L. C. M. of numerators 8, and H. C. F. of denominators = I; .". the L. C. M. required = f = 8. EXAMPLES. 75. Find the H. C. F. and L. C. M. of 1. \ and J. 2, ^ and j. 3. $ 8 T and Jf . 7. U>|, 3i 8 - MJ, if. 9. 2i,3i,4j. 10. 3, , 10$. 11. W> If, 4. 12. if, 2{ft, 5J. 13. What is the greatest length which is contained a whole number of times exactly in both 7^ feet and 4j feet ? 14. Find the least number which, when divided by each of the fractions $, & and f 5 gives a whole number as quotient in each case. 15. Four bells commence tolling together ; they toll at inter- vals of i, i, i$ and i j seconds respectively ; after what interval will they toll together again ? MISCELLANEOUS EXAMPLES. 16. 1. What number must be added to 3$ of | that the sum may 2. What must we take from 3$ to leave 2j ? 3. From what must 4$ be taken to leave f of j ? . 4. What number multiplied by f + gives the product f -| ? 6. By what do we divide $ f if the quotient is 8 ? 6. How many times does $+ contain J- J ? MISCELLANEOUS EXAMPLES IOI 7. What number do we divide by 7$, if the quotient is 2} ? 8. If the divisor be f , and quotient f of the divisor, what te the dividend ? 9. Find the price of 217 Ib. at t&d. per Ib. 10. Find the cost of 325 maunds at R2. qa. 4^. per maund. 11. Find the weight of 125 boxes, each 7f Ib. 12. R;2o is iff of what amount ? 13. Find the sum of money, f of which is ^30. 14. Which is the greatest, 4$- 3$, 4i* 3i> 4i~ 3i or 4$+3J ? 15. What number is that from which if you subtract i- J, and to the remainder add of , the sum will be + ? 16. Find the least fraction which being added to $ shall make the result an integer. 17. A gives B J of his money ; B gives C J of what he re- ceives ; and C gives D \ of what he receives ; what fraction of A*s money does D receive ? 18. If I lose | of my money, what fraction of it have I left ? [The fraction I - $.] 18a. f of a post are imbedded in mud, & are in the water, and ti ft. are above the surface ; what is the length of the post ? [t + Ta=T 7 & J * -T 7 s~i^ -' ft of the post6 ft. ; and /. the length of the post = 6 ft. 4-&6 xA ft.2o ft.) 19. A book contains 25 pages, and a boy has read 15 of them ; what fraction of the whole has he yet to read ? 20. A sum of money is divided among three persons, A> B and C. A receives f of it, and B receives \ . How much does C get ? 21. A man owns ^ of an estate, and sells \ of his share ; what fraction of the estate does he still own ? 22. A merchant owned || of a ship, and sold ^ of his share j what part of the whole ship had he left ? 23. If I give away & of my money, and then f of what re- mains, how much of the whole is left ? 24. One-fifth of an estate is left to the eldest son, J to the second, and f of the remainder to the third ; how much was over ? 25. At his first game a person loses i of his money, at the second $ of the remainder, at the third of the rest ; what fraction of his original money has he4eft ? 26. When i\ of $ of a loaf of bread has been eaten, how much of the loaf will be left? 10* ARITHMETIC 27. After paying f of a bill) 24 is stiJJ due ; what was the amount of the bill ? 28. A person expends of his income in board and lodging, in clothing and ^ in charity, and saves ,318. What is his income 29. A boy after giving away J of his pocket-money to one companion, and of the remainder to another, has 2s. left. How much had he at first ? * 30. A man travelled ^ of his journey by coach, ^ by rail, and walked the remaining 9 miles ; how far did he go ? 31. One-tenth of a rod is coloured red, one-twentieth orange, one-thirtieth yellow, one-fortieth green, one-fiftieth blue, one-sixtieth Indigo, and the remainder which is 302 inches long, violet. Find the length of the rod. 32. Of a certain dynasty of the kings were of the same name, J of another, J of another, T \ of a fourth, and there were 5 besides. How many kings were there of each name ? 33. How many whole cakes would be wanted for 100 children If each has a third of a cake ? 34. By what number should f J^ be multiplied so as to pro- duce the least possible integer ? 35. simplify 5-r. 4 tons 15 cwt. 36. How often may $ be subtracted from 7, so as to leave a remainder not less than 3 ? 37. From a rope 20 ft. long, as many pieces as possible are cut off, each 2f ft. long ; what fraction of the latter length will be left ? 38. A cistern has two pipes attached to it, one to supply and one to draw off. The first can supply j of a gallon, and the second can draw off $ of a gallon, per minute. If both the pipes are opened when the cistern contains 81 gallons, how soon will the cistern be empty ? 39. The double and fourth part of a number, added together, give 7$ as the result ; what is the number ? 40. Find the number, of which the eighth part exceeds the tenth part by 7}. 41. What are the nearest integers to 12} and I7f ? Give reasons for your answer. 42. A number of mangoes is to be divided amongst 3 persons so that one may get A of it, another A> and the third the remainder ; what must the number at least be that this may be done without cutting any of the mangoes ? COMPLEX FRACTIONS 103 COMPLEX FRACTIONS. 123. A simple fraction is one, in which the numerator and denominator are both whole numbers ; as $, f . A complex fraction is one, in which the numerator or deno- minator or both are not whole numbers ; as * 2. 3| f+ij 5' 2^ 4f 3i of 2f ' Note. ^| is read <$ divided by 4?', or <3 by 4?. 4f Complex fractions can always be simplified as in the following examples : M B. The work within the brackets may be omitted in practice. Note. There is another method of simplifying complex frac- tions, which is explained by the following example. Example 5. Simplify We multiply the terms of the fraction by 12, the L. C, M, of the denominators 2, 3, 4 and 6. Thus the fraction -i*^-! 4 . 9+10 19 < EXAMPLES. Tjr. Simplify L 7' 2 ' 8? 8 * If 4 * I' 6 ' 4i' e^7^n.8?^ 9 iii 3rJ ' Si ' ' I3A' ' 2 4/i' ' j " " 3--i f 1L JL,. 12 . 2iA. 13 . fa* 14. ^Zt. io 4 ARITHMETIC ... 2f.f-f. ,0. J-jf-fJ. .L 22. . 23. 24. H*; 1 I_ l 236 2l" t "3l' t Example. Simplify the continued fraction, i 7 Process: 3 + ^ -3+ hr"""3+ ^ 7 V 7 2 - ;H 4+ 6i| 4+ r 3 ^ 204 ^204" EXAMPLES. 78. 2 Simplify " ^ 2. 2-- 3 + 3-1 6. 3+ e. 7+ 4+; i -5- 3 ~4+i 7. 10. -, 8. 9. 6-- 2+- 3 + 4+i 11. 6+^ i 2 + -. 12. 4 2 + 2-f- I- 3+1 2 + COMPLEX FRACTIONS IO$ !<>. The following examples of simplification are important. Examples HHHix x=J = 2j. Example*. \ 4-$x}=x xi-ft. Example 3. $x-^f = f xx=$. Example 4. 2 x i^-J x f -^ f-f- f = f x J x x f x f x etc. In the above examples the operations of division are converted into those of multiplication by inverting the fractions which are preceded by the sign of division ; since division by a fraction is equivalent to multiplication by its reciprocal. Note. In simplifying an expression, a compound fraction must be treated as a single number. The difference in meaning between -5-J of J and i-r-Jx should be noticed. but EXAMPLES. 79. Simplify 1. I* 54-5-23. 2. iJ-nJ-Mi. 3 s . 4. 2f-Hxi T * T . 5. 2jxf-rlf. 6. 7. iHi*X2j--2|. 8. JxRJx*-H-i. 0. 10. J-f-JxHKixi 11. 3i-^2iof6J. 12. 13. 2^^34x 4 J. 14. 2ixj-^3iofiJ. 16. 10. 2iof|~3lxiJ. 17. 18. 2of |4-3iofi. 19. 20. 2ixj-r3jxiJ. 21. 22. i-r-2jof 3irxiJ. 23. ii-r2jx 3J of 24. ii x 2j x 3K i J of 2j o . Convention of Signs : ^When an expression contains all (or some of) the signs +, -, x , and -;-, the multiplication and division are to be worked before the addition and subtraction. Example, f + 2 x HJ-i = | + f xixf-i= + i-i= 4 -i-3j. EXAMPLES. 80. Simplify L I*of 3 i-Aof3f. 2 - 2 |xt-f7ix^. 3. |^if-^ 3l \. i-i. 6, 2f+i| of ^-ij. 106 ARITHMETIC 9. 2*of3*-iHf off. 10. 3fof4f4.5f-.af 11. |of4t+I-s-A-f. 12. 18. f+f of KJof A- 14 16. | fi!-*ofi-K5. 16. 17. if of3j4-Aof3|of 3H4i off- 18. 4* + 5f *8-2o*x3* of A* of a .F BRACKETS. When an expression is enclosed in a bracket ( ), { } or [ 1 or placed under a vinculum , the whole expression Is affected by the sign that precedes or follows the bracket or vinculum. Thus, 2-K3+4) means that 2 is to be divided by the sum of 3 and 4. (2 + 3) x 4 means that the sum of 2 and 3 is to be multiplied by 4. i3-(3-f*5) means that the sum of 3 and 5 is to be subtracted from 13. 7 ~(34- 4 -2) means that the difference between 4 and 2 is to be added to 3, and the result to be subtracted from 7. Hence to simplify an expression like the above, we are to perform the operations indicated inside the brackets before per- forming operations indicated outside the brackets. Note. In a product the sign of multiplication is often omitted when one or more of the factors are enclosed in brackets. Thus, 3(5-4) means 3* (5 -4) 5 (3 + 2)(4 - 2) means (3 + 2) x (4 - 2 )- A bracket may be removed if it is preceded by the sign + : thus 8-4-(7-54-2)8+7- 5-4-2. A bracket preceded by the sign may also be removed if the sign of every term within the bracket is changed, namely + to and - to + : thus 8-(7- Example. Simplify 7~[f +{2$-(ifc-J)}l The expression or(ii) 7- etc THE USE OF BRACKETS IOf EXAMPLES. 81. Simplify 1. 3-(i+i|). 2. 4-(3i-i). 3. (3-iA)of3f 4. (3 - i A) x 3f - 1 A- 6. 3 - i A<3* - i-A). & (3 - i AX3* - 7. (3+iA)+3f-iA- 8 - 9. (3 + iA)-(3f-iA). 10. U. 6 + jii+(f-J)J. 12. 13. 6-{iJ-(|-J)K 14. 15. i7*-<8Ka*-i). 18- 17- 9i-[7i+{4-(5-2)H 18. 19. 3^[2 + 3^-<4+5^(2j-i)}]. 20. 8L 5j-[ 2 H{i-i(J-Pj)}]. 22. 189a. Example. Simplify ?-t". 4 n The expression . I 16 _Z+5?-39 4 16 16 284-SQ-39 " 16 48 "i8 3. Ans. EXAMPLES, Simplify 1 3i-4<rfi-* 7\ (3i-2i)of(if-' ' ^ 108 ARITHMETIC 5, JZ ^"^S 7 + ~ 6. <(A+i)x(3-K(i++ -=- off of 9. 20). 10. 6 + 6 17 2L + t+J THE USE OF BRACKETS 23. 5 24. 8-8x 2*- Li. 25. I2i2L : 6-1 2 + 37. 28. 2 + 29. 30. 4-r 4-f of 12 32. 83, 35. 3 L_x2 + L f tt+A '" 24- I - A-TT 2 X 1 + 110 ARITHMETIC XXIV. FRACTIONAL MEASURES. 130. Example I. Find the value of j of Ry. 8<z. 3). To multiply the compound quantity by j, we divide it by 4 and multiply the quotient by 3, thus : R. a. p. 4)7 . 8 . 3 14 . of _ _ _____ __ R5 . 10 . 2^ Ans. N. B. If we have to multiply by 5f , we multiply first by f (as in the above example) and then under the result set down the product by 5, and add the two results. If we have to multiply by 6f, &*., by V we divide by 4, and multiply the quotient by 27, using factors. Note 1. To divide a compound quantity by fi we divide It by 3 and multiply the quotient by 4. Example 2. Find the value of \\ of i J of Ri. \\ of ij of Ri-f off of RiRf =-A R. a. p. 3)5 o o Ri . 10 . 8 Example 3. Find the value of ^ of 17. 75. 6d.+$ A fi7.7.6-^~ 7 6 x 5 = ^i.8. xiixs- 7 /. the value required io . II . A second form of operation is as follows : A of *I7. 7. 6+ of ^ -^i .8 .nix 5 + =^7 -4 -9i +3.6,8 =;io . iu. $\d. Ans. Note 2. When we have to multiply or divide a compound quantity by a fraction, the terms of which are large numbers} it is generally better to adapt the following method, FRACTIONAL MEASURES III Example 4. Find the value of \\\ of Rio. 20. 6/. Process : f Jj of Rio . 2 . 6 = f \\ of 1950^. = iio2 T 7 x ^.=9ia. lo^^.^Rs. 1 1 a. io^j^. An** EXAMPLES. 83. Find the value of 1. f of RS. 70. 6p. 2. J of R2. 3. | of R3. 2a. 4. $ of Rig. 30. 6p. 5. |ofR3. 40. 6. f of 120. 7. vi of ^92. 19*. lid. 8. | of .70. 45. 9. 10. sfof RI2.90. 8A 11. RH&ii 12 - 13. 4 of ,2. iu. 7jY/. 14. 4i\ of ^9. 15. ^ 16. Ri3. 120. 9/.x3$. 17. Ri3. 130. 6p. x ] 18. i. 75. 6d. x |J. 19. .10. i 20. R25. 120. 9A-r 7 \V "21. ^100. 22. 3| of I cwt. i qr 4 I Ib. 23. 2$ of 128 yd. 2 ft. 7 in. 24. T 6 5 of i hr. i min. i sec. 25. ^ of 3 bus. 2 pk. I gall. 26. 3^ of 3J of Ri2. ga. $p. 27. J of $ of i J of R7. 30. 28. 2j of 6| of R7. ga. 3/*. + 7i of Ri. 3^. 4^. 29. j of 4-J of 2. 12s. 6d. - 7^ of /i. 6^. 6^/. 30. ^7jJ + f of i5J. + 7J.-f;| + 4? of 31. Ri3j- 3j of 7a. - R2. 40. ~H 32. jft of R2. ga. + i J I of R 7 . 8. + $& of R9. - 33. j of | of ;i+f off of 2s. 6^/. + | of : 34. f of I of Ri 4- 1 of f of 3*. #. -f- f of 35. J} of *i +| of 2 guineas- f of 3*. 9^. + } of : 36. | of a guinea 4- 1 of a crown -J of 3^. 6d. 37. f of R7. 8. 6^. - 1 of 7^1. *jp. 4- -r of ^~ 38. -^ r of R8. 9. + 3 -^ of ^ of R9. o*. 39. (3t-5-3i) of 3- 9*. of* +{ of 2 7 *-~| of 5*. 40. Arrange f of R;, {$ of R6.no. and R in order of magnitude. 4L f of 1} of a sum of money is 7. js. yd. ; find the sum. 112 ARITHMETIC 42. What is the sum, of which is R3- 9- 3A ? 43. From f of a certain sum of money when | of RS. 70. is subtracted the remainder is Ri. la. if. ; find the sum. 44. Find the value of of of R 5 . V 'A O . I 45. Simplify -- of t + i s \ of - ofi5f. + 131. To express one quantity as \^ fraction of another. Example I. Express 13**. 4^. as the fraction of Ri. Notei. R 7 . Example 2. Express R2. la. io/. as the fraction of R3. 2a. Example 3. Express S of R 2 - 3^. as the fraction of | of R8. _, f . |ofR2. 3<2. 4x35 2x35x4 280 The fract.cn - -i-A _. . Note 2. The above questions may be put in any of the following forms : (1) Express R2 as the fraction of R5. (2) Reduce R2 to the fraction of R5. (3) What part is R2 of R$ ? (4) What fraction is R2 of R$ ? (5) How many times is RS contained in R2 ? (6) What is the measure of R2 when the unit is R$ ? (7) Express R2 in terms of R$ as unit. Reduce \ of R$ + i of R2. 30. to the fraction of The 15^. 191 ^ax Sox 4 + 3x3^ x 3 = _95j ______ j 191X12 191X12 12* FRACTIONAL MEASURES S A V 113 r c { , 1. Express 83. 4^. as the fraction of Ri. 2. Express ga. gp. as the fraction of la. 3. Express R$. 50. as the fraction of its highest denomination, 4. Express 7J-. 6d. as the fraction of its highest denomination. 5. Express 7. los. 6d. in pounds. 6. Express js. 4^0*. in shillings. 7. Reduce R/. $a. 4$. to the fraction of Ri. 8. Reduce 3. 6s. 8d. to the fraction of ^i. 9. Reduce Sa. qp. to the fraction of RS. loa. Sp. 10. Reduce I2s. *>\d. to the fraction of i. 3*. $d. 11. What part is RQ. 3. 4A of' 1 Rio. 6a. 4p. ? 12. What part is 27 Ib. 12 oz. 15 dr. of 3 cwt. 3 qr. 21 ib. ? 13. What part of I md. 38 seers is 7 seers 5 ch. ? 14. What part of 6 mi. is 2 mi. 441 yd. I ft. ? 15. What fraction is 12s. io\d. of ^10 ? 16. What fraction is 5 gall. 2 qt. i pt. of 10 gall. 2 qt. I pt. ? 17. What fraction of a guinea is 75. 6 fa?. ? 18. What fraction of a ton is 12 Ib. 12 oz. ? 19. How many times is R7. &z. $\p. contained in R6. Sa. ? 20. How many times is 3 da. 7 hr. 8 min. contained in 8 da. ' hr. 3 min. ? 21. What fraction is i$s. ioj|^. of 2. 95. 7^. ? 22. What fraction is 5$ guineas of 10% ? 23. What fraction of 2j yd. is 2\ ft. ? 24. How many times does 8 Ib. 10 oz. 19 dwt. 9 gr, contain Ib. Troy ? 26. Explbss R20. 7. o^. as the fraction of 70. 9^. 26. Express ^20. 75. 9^. as the fraction of 7*. 9</. 27. Express | of R2. 7 a. %p. as the fraction of R7. 28. Express if of R8 as the fraction of Rio. io<*. ic^. 29. Express J of ^3. 6s. id. as the fraction of 9. ?s. 6d. 30. Reduce of is. i\d. to the fraction of a crown. 31. Reduce $% of 8^. qd. to the fraction of 3. 32. Reduce & of R7. ga. to the fraction of R9. 7 a. Sp. 83. Express ? of R2. 30. as the fraction of i of RS. C A. 8 114 ARITHMETIC 34. Express 3$ of Ri. ga. as the fraction of Jf of R;. 80. 36, Reduce f of i of is. id. to the fraction of \ of a guinea, 36. Reduce f of of Rio. 100. ic^, to the fraction of i\ of RS. 37. What part of \ of 3 md. 19 seers 8 ch. is 18 seers 7 ch. ? 38. What part of of 7 cwt. 7 Ib. is of a stone ? 39. What fraction of 2^ of } of 2 tons is f of 3 cwt. 2 Ib. ? 40. What fraction of a furlong is f of 7^ of i6 yards ? 4L How many times is f of 7 Ib. 7 oz. 7 dr. contained in of a quarter ? 42. What fraction of f x of a foot is a pole ? 43. What fraction is f of a gallon of of a pint ? 44. Express f of i hr. 1 5 min. as the fraction of i day. 46. Express 5 fathoms as the fraction of $ of 3^ of a pole, 46. What fraction of 7 > of 30. 13*. 2J</. is (8}-3f ) of 5. gs. ni<*? 47. Express R7|~i of R6 as the fraction of Rio. ga. 48. Reduce iVr-~ iW to the fraction of I2J. io</. 49. Reduce R7*~ f of R7 to the fraction of RS. 50. Express } of ^i -f of 2U. as the fraction of ioj. 6^ BL Express f of 12*. 6</. -4-} of i6j. 6</. as the fraction of ^i,| 62. Express ^ of ji. loj. + S of 5^. 4^-8^ of -~ of 55. 3J<* as the fraction 0*2^. i^. 63. What fraction off of 275. is 3 | of {f of ^i~? of 51.} ? \MISCELLANEOUS EXAMPLES. 85. 1. Express" the difference between the greatestand least of the fractions, ^ v & and | as the fraction of the other. 2. A clerk commenced work at a salary of RSO a month, which was each month increased by \ of that of the preceding month ; what was his third month's salary ? 3. A gives away f of R$o. He gives J of this to JB, f of it to C, and the remainder to D. How much does each get ? 4. A sum of money is divided among 3 men. If the first has J of it, the second ^, and the third the remainder which is /2, 7* 44<, what is the entire sum divided ? MISCELLANEOUS EXAMPLES 115 5. A has 14. 7 a. 4$^., and has 3f times as much as B ; what has? 6. A person owes a guinea to each of 3 creditors ; to one he pays \ of his debt, to another f and to the third | ; what sum will he be still owing altogether ? 7. After taking out of a purse f of its contents, $ of the re- mainder were found to be 13^. 5-Jk/. ; what sum did the purse contain at first ? 8. A post is divided into 3 parts : the first part is J of the whole length, the second % of the first, and the third is 3 ft. 6 in. : find the length of the post. 9. Five brothers join in paying a sum of money ; the eldest pays J of it, and the others pay the remainder in equal shares* and thereby each of them pays 6,20. fa. 7$p. less than the eldest brother ; what is the sum of money ? 10. Find the sum of money that shall be the same part of 3, los. that 2 Ib. 3 oz. avoir, is of 3 Ib. 2 oz. 11. What is the sum of money which is the same fraction of R2. la. that 7 yd. I ft. is of II yd, ? 12. What fraction of Ri. 130. ;/. must be added to ^| of s (l + f) of la - 4#* to make the sum equal to Si ? 13. If the American dollar be equal to f^ what fraction is f of a dollar of f of a guinea ? 14. Reduce the difference between i Ib. avoir, and I Ib. troy to the fraction of f of I Ib. avoir. 15. Reduce the sum of f of i, f of is. and | of id. to the fraction of j of a guinea. 16. A cask contains 35 gall. 2 qt. i pt. of wine ;what part of itlmust be taken out to fill 5 quart bottles ? 17. Find the greatest sum of money which is contained in each of of R3. 50. 4^., | of R;. 9^. fy. and f of 8a. 9^. a whole number of times. 18. Find the least sum of money that contains each off of fii. 3^ 3A> I of 8.2. 80. and f of 87. ga. 6fi. an integral number of times. 19. A sum of money increased by its fifth part amounts to 3, 1 50. ; what is the sum ? 20. What part of 5 units is | of a unit ? 21. Standard silver is coined at the rate of R2. 6a. io$p. per ounce ; find the least integral number of ounces that can be coined into an exact number of rupees. 116 ARITHMETIC 22. Find the least integral number of pounds avoir, that con- tains an exact number of ounces avoir, and of ounces troy. 23. From a rope 30 ft. long, as many pieces as possible arc cut off, each 3$ ft. long ; what fraction of the whole will be left ? XXV. DECIMALS. I3. In the ordinary system of notation the value of a figure decreases ten-fold at each step of removal from left to right ; thus, if a certain figure represents hundreds, the next figure to the right will represent tens ; and the next units. If by a natural extension of this system of notation we place figures to the right of the units' figure, the figure immediately to the right of it will represent tenths, the next figure will represent hundredths, the next thou- sandths, and so on. Thus 13 The number indicated is "twenty-one and two tenths, three hurtdredtns, four thousandths, five ten-thousandths." But in such a system of notation it is necessary to indicate clearly the position of the units' figure ; and it has been agreed that the figure to whose right a point ( ' ), called the decimal point, is placed shall be the units' figure ; and to distinguish this point from tlie one used as the sign of multiplication, it is placed towards the top of the figure. Thus 74*256 represents 74 units, 2 tenths, 5 hundredths, and 6 thousandths ; and is read "seventy-four decimal, two ,five, six." 74*056 represents 74 units, no tenths, 5 hundredths, and 6 thousandths ; and is read "seventy-four decimal, zero, five, six." 0*205 or "205 represents no units, 2 tenths, no hundredths, and $ thousandths ; and is read "decimal, two, zero, five." 133. A number expressed in the above notation is called a decimal or a decimal fraction. The part to the left of the point i s called the integral part, and the part to the right is* called the decimal part of the given number. DECIMALS 1 17 Note. Such numbers are called decimal fractions because each figure to the right of the decimal point represents a fraction which has some power of 10 as its denominator : thus 2*34 = 2 + ^+ ^$5. 134. The value of a decimal is not altered by annexing ciphers to the right of the last figure j thus, 2*3 5 = 2*3 50=- 2*3500 ; for, these ciphers do not alter the position of any of th$ othei figures relatively to the decimal point. Note. An integer may be expressed as a decimal by writing ciphers in the decimal part ; thus I2 = i2'oo. But the value of the decimal part of a number decreases ten- fold, a hundred-fold, , as we place one, two, , zeros imme- diately to the right of the decimal point. Thus *i is one-tenth ; *oi is one-hundredth ; ooi is one-thousandth ; and so on. 135. It will be observed that a decimal is multiplied by 10, loo, loop, , by removing the decimal point I, 2, 3, ....... places to the right ; and conversely, a decimal is divided by 10, 100, IOOC| , by removing the point i, 2, 3, , places to the left, Thus 20*31 = 2*031 x 10 203'I-~IO. EXAMPLES. 80. Express as decimals : 1. Three tenths. 2. Two and one hundredth. 3. Seven hundredths. 4. One tenth and four thousandths. 4>. Eight ten-thousandths. 6. Nine millionths. 7. Twelve and four hundredths and six hundred-thousandths, 8. One hundredth and three thousandths and five millionths. 9. One ten-thousandth and one hundred-millionth. 10. One hundred and five tenths and two thousandths. Multiply and divide each of the following numbers by 10, and by looo ; 11. 7. 12. 29. 13. '2. 14. '02. 15. 3*4. 16. 7*03- 17. 1*003. 18. '007, 19. 39*2. 20. 23-45. 21. 3000. 22, 123-2, 23. Write down the number which is ten-thousand times 'ooooi. 118 ARITHMETIC 24. Write down the number which is a millionth part of loooo, 25. How many tenths of an inch are there in 3*5, 7*05 and 4 inches respectively ? 26. How many tens-of-inches are there in 2*5, '6 and 3 inches respectively ? * 130. To convert a decimal into the equivalent vulgar fraction. Example. Express 71 and 2*017 as vulgar fractions. By the preceding Art., we have, (i) 71=71^-100==^; (ii) or,: Hence the rule : Write down the given number suppressing the decimal point for the numerator, and for the denominator write I followed by as many zeros as there are figures in the decimal part. 137. To convert a vulgar fraction having some power of 10 as Its denominator, into tl*e equivalent decimal. Example. Express j$, ^j% and ^J$o as decimals. (i) ft =12 -5- 10 =I'2. (ii) ^j=I24-IOO = *I2. (i) liJfo" 124-1000= -012. Hence the rule : Take the numerator and in it place the decimal point after as many figures (counting from the right) as there are zeros in the denominator. If the number of figures in the numerator be less than the number of zeros in the denominator prefix in the numerator the requisite number of zeros. EXAMPLES. 8T. Express as vulgar fractions in their lowest terms : 1. -4. 2. '83. 3. -04. 4. 1-5. 6. -074. 6. '0125. 7. '0025. 8. -075. 9. 2*88. 10. 725. 11. 4*00256. 12. 7*225. 13. '625. 14. '0625. 15. I'll. 16. '0006875. 17, 81-0005. 18. 6-4375. 19. 5*0096875. 20. 70*00004. Express as mixed numbers with the fractional parts in their lowest terms : 2L 2'5. 22. 7-25. 23. 8*125. 24, 175, DECIMALS 119 25. 2-025. 26. 3*05. 27. 9*0125. 28. 6-0075. 29. 3*0005. 30. 7*0675. 31. 12-225. 32. in. 83. 2-0001. 34. 1-2221875. 35. 1*0007225. 36. 12*08956640625, Express the following vulgar fractions as decimals : 37. /. 38. flfe. 39. %0-. 40. fjg. 4L T1 fc a . 42. T 5g^. 43. 1 ftfo 44. 45. ftftl 46. TT ^ ns . 47. *mP- 48. 49. 125 ten-thousandths. 50. 790 millionths. 188. The operations of addition, subtraction, multiplication and division of decimals are performed exactly in the same way as In the case of whole numbers. Hence it is an advantage to use decimals in preference to vulgar fractions . 139. Addition of Decimals. Example. Add together 72-305, 7*06 and 7896. We set down the decimals one under another, point under point ; thus 72-305 7-06 __ 80-1546 Ans. We then add as in the case of whole numbers, taking care to place the decimal point in the sum under the column of points. EXAMPLES. 88. Add together L 3-12, 12-023, '32, 47. 2. *oi, 30, 7*469. 3. 39'7 *o8, 3, 1-3022. 4. 1*3, '025, 79, '005. 5. 1*23, 2*345, 6*7891, -ooooi. 6. '04, '004, -93, '026. 7. 4*07* '089, 27012, 3*1398. 8. '0009, 900, 9-909. 9. 3'3j 10*70902, '004, -4, '12. 10. 7, '892, -oi, '098, 11. 700 + 32-7269 + '00903 + 3-4 + 263*86407. 12. *i + '00095 + 84*0563 + 7-3+325-65432. 13. 6*3 + 617*241 + '0078 + 37-045 + 8-6943 + -oi. 14. '74259+346-274 + 300+ lo-ooooi + '207. 16. '0705 + 705 + 7*05 + 20*00007 + *oi + '00043. 16. 840*004 + 7-2007 + R-oooo8 + ^300*03. 120 ARITHMETIC IV. ;7'542i2 + ;39'47 + ^-07078 + ^700. 18. 30 min. + *oo45 min. + 77o89 min. + 37685 min. 19. 329 ft. + *oi ft. + 3*1 ft. + '057 ft. + "308 ft. 20. 2*2 in. + 30*03 in. + '369 in. + 7072 in. + 8'ooo8 in. 140. Subtraction of Decimals. Example. Subtract 3*587 from 16*29. We arrange the numbers as in the case of addition ; thus 16*29 -3'.587 12*703 Ans. We then subtract as in the case of whole numbers, supposing a tero (or more where necessary) annexed to the right of the minuend, and taking care to place the decimal point in the re- mainder under the column of points. EXAMPLES. 89. Subtract 1. 37'039 from 44*123. 2. 7*0389 from 9*01. 3. "00078 from 1*1. 4. 100*389 from 300*09234. 6. 37*35 from 100. 6. 102 from 306*103. 7. '000725 from 'ooi. 8. '0001234 from '012. 9. '12345 from 7*6789123. 10. 3*1705 from 345*9875. 11. 7*325 from 8*025. 12. '9375 from 3*0005. 13. Rl'9999 from 8,9. 14. ^32*00051 from '33, Find the value of 15. 3789 + 7*002- '0079 + *i - 1 "oooo i. 18. 700 - '007 - 7078 - 3*1234 5 + '00025. 17. 100 - '0072 - 3*9345 - 12 - *i. 18. 2000 - ('079 + 3*67002 - 3*0012). 19. 1*345 - "072 - (3*123 - 30*321) + loo. 20. Is 3*1415926535 more accurately represented by 3*14159 or by 3*1416? 21. Is 2*718281828 more accurately represented by \27i82 or by 2*7 1 83? \ 141. Multiplication of Decimals. If we take any two decimals, convert them into vulgar fractions and multiply these latter together, we find that the numerator of DECIMALS the product is the product of the two given decimals with their decimal points suppressed, and that the denominator is i followed by as many ciphers as there are decimal places in the two given numbers ; and if now the product be reduced back to the equiva- lent decimal, it will contain as many decimal places as there are ciphers in the denominator. Hence we have the following rule for the Multiplication of Decimals : Multiply the given numbers as if they were integers, and mark off in the product a number of decimal places equal to the sum of the numbers of decimal places in the two factors. If the number of figures in the product be less than the number of decimal places in the two factors, prefix the requisite number of ciphers. Example. Multiply 13*325 by 3*2 and '00046 by 36. (i) I3-325 (ii) '00046 26650 t "276 I-R 42-6400 = = 42*64 ATIS* '01656 Ans, EXAMPLES. 9O. Multiply 1. 32*4 by 2*3. 2. 7-24 by 5. 3. 67*23 by '002. 4. 30*03 by 200. 5. 032 by -032. 6. 045 by '0072. 7. 800*008 by '035. 8. 34-12345 by 72. 9. 0202 by 2020. 10. 4030-4 by -0075- 11. 4*379 by -37. 12. 00125 by '25. 13. 10-607 by 402000. 14. 000625 by ^2800. 15. 725 by *ooo8. 16. 6400 by '00125. 17. 5*12 by 42*25. 18. 46*025 by I2'8, 19. 0064 by 'oi 2 5, 20. 00846 by '005. 21. 0078 53 by '00476 ^2. 56-875 by '0144. 23. 015625 by "0064. 24. '0204 by 40*2. 25. 700 by "005. 26. 79*235 by 39*02. 27. 40*25 by 30*04. 28. 12*8 by '0075. 29. 1*12005 by '12005. 30. 9*006 by 5*40005. 31. 2*5x2*5x2*5. 32. 25 x "25 x '25. 33. 05 x *o8 x *02. 34. 1*2 X 15 X '12. 35. ii x rix'ii. 36. 20X*2X*25. 37. 0005 x "005 x '05. 38. 7 x 7 x '07 x 7000. 39. *3 x '03 x '003 x 30. 40. 200 QX'C >055X2*5. Find the value of 41, (6*25) a -05) 8 . 42. (74 S-'c >o7)x-o35. 43. 7*6 - 3*7 x '009. 44. ('05 ;) 2 +4 5x20. 45. 7'5*75-75x*o75 +(7*| 5) 8 ~(7'5-"75)x < 07 5. 132 ARITHMETIC 14& Division of Decimals. I. When the Divisor is an Integer: Example I. Divide 808*9 by 25. Process : 2 5)808 '9(32 '3 56 Ans. Z5 58 "89 140 "Tso 150 Here we divide as in the case of whole numbers, taking ^care to place the decimal point in the quotient as soon as the division of the integral part is finished. If there is a remainder (as in the above case) after division, we affix a zero to the remainder, and divide. We treat all successive remainders in the same manner, and continue the division until the required number of decimal places in the quotient is obtained, or until there is no remainder. Note. The method of short division may be employed with advantage when the divisor does not exceed 20, or when the divisor can be expressed as the product of factors each less than 20, Example 2. Obtain the quotient to five places of decimals in the division of "025 by 7. Process : 7 ) '025 00357... Ans. II. When the Divisor is a decimal : Remove the decimal point in both the Divisor and Dividend as many places to the right as will make the divisor a whole number ; and then divide as in the preceding case. Note. Observe that removing the decimal point in the divisor and dividend an equal number of places to the right is equivalent to multiplying the divisor and dividend by the same number ; and that if the divisor and dividend be both multiplied by the same number the quotient is not altered. DECIMALS 123 Example 3. Divide 12*96 by ic*8. Here we divide 129*6 by 108 : 108 ) 129*6 ( 1*2 Ans. i*L 2~l6 216 Example 4. Divide 34*6 by *o8. Here we divide 3460* by 8 : 8 ) 3460' 432*5 Ans. 143. A vulgar fraction may be expressed as a decimal by dividing the numerator by the denominator. Example. Express as a decimal. Process : 8)5* '625 Ans, Note. The following results are useful : i-'S; J-'25;l='75;i='i2S. EXAMPLES. 91. Divide 1. 29*21 by 23. 2. 34*3 by 25. 3. 129-6 by 108. 4. "03096 by 72. 6. 4577 by 230. 6. '06227 by 1300, 7. '04009 by 1 520. 8. 3708 by 360. 9. '00281 by 1405. 10. 8357 by 488. 11. -001007 by 47500. 12. 431*376 by 8170 Divide, rinding the quotient as far as the fifth decimal place : 13. 42*5 by 23. 14. '0269 by 281. 16. 197 by 79. 16. '041326 by 101. 17. "0079 by 372. 18. 312 by 84. 19. 356*5 by 273. 2O. 6*5 by 342. 21. '0042 by 121. Find the quotient, by Short Division, to not more than 6 places of decimals, in the division of 22. 4*125 by 2. 23. 3*73 by 8. 24. '034 by 7. 25. 21*24 by 90. 20. 134 by II. 27. 36*7 by 16. 28. '04321 by^8o. 29. 8*567 by 13. 30. '01 by 6. Divide 3L "3125 by 'oj. 32. 8*454 by '024. 33. '5568 by 2*32. 124 ARITHMETIC 34. 6'33 by "0025. 35. 17*28 by '0144. 36. 4 by '00625. 37. '00281 by 1*405. 38. 177089 by 4735* 39. '00005 by '0000025. 40. 816 by "0004. 41. 84-375 by -00375. 42. 2874-465 by '0495. 43. '830676 by -000231. 44. 33*363 by '00275. 46. 7 by '0004. 46. '0007 by '0005. 47. 5*625 by '0000075. 48. '0603738028 by '0476. Find the quotient to five places of decimals : 49. 3*46 1 4- '027. 50. -3125-^*06. 61. '2--- -006. 52. -0007 5 3 -r '009. 63. 'ooooo r -~- '000043 1. 54. '5 76*91342. 66. 40oo-~*oooi2i, 56. '666666 -f "008. 67. '007-7- '00073. 58 - 4*006 54 -r 3 2 9'26 5. Employ Short Division in finding the quotient to not more than 6 places of decimals : 69. 28-r*o8. 60. 3*7 6 -f- -005. 61. '0076 -f- '003. 62. *oioi-f-*ooi6. 63. '000012 -7- '13. 64. 229^*007. 66. 39*4 -~ '007. 66. 4*767^*004. 87. 13*7 54- '012. 68. -02-M'i. 69. -03-7-1*4. 70. 3*4 ~ '009. Simplify -0075x2-1 ri8 x 3*o_4 0175 ' * -152 2-95' Convert into decimals : 74. J. 76. J. 76. f. 77. i 78. 79. Xl V 80. 3** 8L 9ft. 82. 3*- 88. Express as decimals as far as the fifth decimal place : 64. J. 85. J. 86. ?. 87. ^. 88. 89. if 90. 7A- 91 - 8 *V 92 10 i8- 9 3. Arrange in order of magnitude, by reducing to decimals as far as the fourth decimal place : 94. 1,1,1. 95. A, T ViV M. ftttift 97. A,A,. 98. *,,!*. 99. |,f,f Reduce to decimals : 100. ^ of "027. 10L '025 of 4$. 102. ^of|x8'36, 103. i of ^-'05 of 2^. RECURRING DECIMALS I2 144. H. C. F. and L. C. M. of Decimals. To find the H. C. F. or the L. C. M. of Decimals, affix ciphers (where necessary) so that all the given numbers may have the same number of decimal places ; then find the H. C. F. or the L. C. M, of them as if they were integers, and mark of! in the result as many decimal places as there are in each of the numbers, Example. Find the H. C. F. and L. C. M. of 3, 1*2 and '06. The given numbers are equivalent to 3*00, 1*20 and '06. The H. C. F. of 300, 120 and 6 = 6 ; their L. C. M.6oo. /. The H. C. F. required = '06 ; and the L. C. M. required = 6*00 =6. EXAMPLES. 93. Find the H. C. F. and L. C. M. of 1. 375 7*25. 2. 72-12, -03. , 3. -02, -4, -008. 4. 1*2, '24, 6. 5. r6, '04, '005. 6. 2-4, '36, 7*2. 7. *oS, -002, 'oooi. 8. 3*9, 6*6, 8*22. 9. *6, '09, i'S. 10. *i8, 2*4,60. 11. 20, 2*8, '25. 12. 1*5, '25, '075. XXVI. RECURRING DECIMALS. 145. In the process of reduction of vulgar fractions to decimals?, It will be found, in some cases, that the division does not terminate ; so that the quotient can be continued without limit. Example. Reduce \\ to a decimal. 55 )19; 3454545- 146. We can tell beforehand whether, in any particular case f the division will terminate or not. Let the fraction be in its lowest terms ; then if the prime-- factors of the denominator are each of them either 2 or 5, the- division will terminate ; and not otherwise. Thus (i) Afews) will produce a terminating decimal. (ii) /^(sxSys) will produce a non-torminating decimal. 126 ARITHMETIC EXAMPLES. 93. State, in each case, whether the equivalent decimal is terminat- ing or non-terminating : 1. J. 2. }. 3. I 4. ?}. 5. 5f. 6. 2*8. 7. !%. 8. M. 9. to- 10. f 1L 3*1- 12. A. 13. 7i- 14. ft. 15. ntf. 16. Write down those numbers between I and 20, which being denominators of fractions in their lowest terms, will produce non- terminating decimals. 147. In non-terminating decimals, certain digits must recur over and over again. Consider the fraction J. In the process of division the only remainders possible are i, 2, 3, 4, 5 ; consequently, after five steps at most, we must come to a remainder which has occurred before, and therefore from that point we must have a recurrence of the remainders, and therefore of the digits in the quotient, Example I . \ - '6666666 . . . Example 2. \\= -3454545... Note. It may be noticed here that division by 3 or 9 gives a period (See Art. 148) of one digit ; division by II, a period of two digits ; division by 7 or 13, a period of six digits. 148. Decimals in which certain digits recur are called recur- ring decimals. Note. A recurring decimal is also called a periodic, re- peating or circulating decimal. The whole body of digits which recur is called the period. Thus, in '6666... the period is 6 ; in '3454545... the period is 45. 149. In writing a recurring decimal we usually stop at the end of the first period and place dots over its first and last digits. Thus '666666 is written ' ; 373737 "37 '> '3454545 '345 ; 34576576 -34S7& A pure recurring decimal is one in which the Jjeriod com- mences immediately after the decimal point ; as, '& '37. A mixed recurring decimal is one in which one or more figures precede the period ; as, '345, * RECURRING DECIMALS 127 Note. It may be noticed that decimals equivalent to fractions with denominator 7 are all pure recurring decimals, all of which contain the same digits 142857. If these digits be ar- ranged in a circle, as in the annexed diagram, we may obtain the decimals equivalent respectively to ^, $, $, $, %, , by beginning in turn with I, 2, 4, 5, 7, 8, and reading off the remaining digits /n order in the direction of the arrow-heads. Thus ^"-'142857 ; *=*2857i4 ; $'=428571 ; and so on. EXAMPLES. 94. Express each of the following as a recurring decimal : 1. *. 2. f 3. f 4. I 5. If. 6. IS. 7. A. 8. I T { V 9. &. 10. sA. 1L W. 12. A- I 3 - WA^ 14 A 16- W- 16. %<*. 17. 5? 18. ioA- 10- 7A- 20. 9 A- 21. Wf. 22. W. 23. 4H 24. ?4JS. 25. 5 iJ. 26. 2-1-3. 27. 464-7. 28. 394-22. 29. 8463. 30. 44-^9. 3L *. 32. A. 33. fa. 34. ff ^. 35. 30. 8A- 37. 3iVV 38. ^ 39. $ 40. 41. i *2 4 'I I. 42. I4io'oi. 43. '34 "1 3. 44. . 07 45. 3 -. 46. 2+ 3 . 47. 7 + ^. 48. i + , *oon 1*1 2*3 "07 49. 3+4-- 60. 4 i. 61. 2. 62. '^. 13 "007 4i Si 150. In a given recurring decimal, the period may be sup- posed to begin at any point after the first repeating figure. Thus '3272727... - '327 "3272 = '32727 -etc. Again, the number of figures in the period of a recurring decimal may be doubled^ trel>ted,...vrithout altering the value of the decimal. Thus '3i7- '32727 --3272727 = etc. 151. Recurring decimals are said to be similar when they have the same number of non-recurring figures, and also the same 128 ARITHMETIC number of recurring figures. Thus '3 and *<5 are similar recurring decimals ; '327 and 2*4^ are similar. 152. Two or more given recurring decimals can always be made similar. Take the recurring decimals 2*3, '24$ and '25768. Now the highest mimber of non-recurring decimal places in any of these numbers is 2 ; and the numbers of figures in the periods respectively are i, 2, 3, the L. C. M. of which is 6. Therefore the given recurring decimals may be made similar by extending each of them to eight places of decimals, the first two places being non- recurring and the last six places being recurring. Thus 2-3 2-33333333 ; 24^ =* -24545454 ; 25768 = '25768768. EXAMPLES. 95. In each of the following recurring decimals begin the period at the fourth decimal place : L '2345. 2. '347(5. 3. '67. 4. '2345. 5. '00123. 6. -12345. ^ 7. -1234. 8. '123456. 9. Extend '34, '34 and '2^78 so that they may have the same number of figures in the period. 1O. Extend '162, '1234 and '3765 so that they may have the same number of recurring figures. Make the following sets of recurring decimals similar : 11- *2$, 7. 12. '345, 7<5, 72. 13. '307, 76*. 14. '07^, 7, '006123. 16. -233, -1234, -023. 16. -3, 7(5, 7236. 17. 7, '124, '24723. 18. 3-4, -2^8, -123. 19. 3*402, 7823, -31. 20. -423, -72, '1203. 153. To express a recurring decimal as a vulgar fraction Example i. -5 '55555... Now f 10 times - = 5*5555... and $ = -5555... Subtracting, 9 times '$ = 5 ; .'. -5-1. RECURRING DECIMALS I2Q Example 2. "2345 '23454545 Now, 10000 times 2345 and 100 times '2345= 23*4545. . Subtracting, 9900 times 2345 = 2345 -23 ; Example 3. 3*62 3*622222. . . New, 100 times 3*62 = 362*2222... and 10 times 3*62= 36*2221... Subtracting, 90 times 3*62 = 362-36 ; 154. Hence we deduce the following rule for reducing a recurring decimal to a vulgar fraction : For the numerator take the integral number formed by all the figures up to the end of the first period, subtracting the integral number formed by the figures (if any) that precede the first period ; for the denominator take the number formed by as many nines as there are figures in the period f followed by as many ciphers as there are figures between the decimal point and the first period. Example I. Find the vulgar fraction equivalent to *3. Process : *3 = $ = J. Ans. Example 2. Reduce "45 to a vulgar fraction. Process: '45 = Ji fir i = sJ. Ans. Example 3. Express *O476* as a vulgar fraction. Process : ^i-Wa^-^&^AYk- Ans - Example 4. Express '00271 as a vulgar fraction. Process: '00271=^1^5. Ans. Example 5. Express 2*37 as an improper fraction. Process : 2 *37 = *Ho aa =W=- 1 A 7 -. Ans. Example 6. Express 2*37 as a mixed number." Process : 2*372 + *37 = 2 + % = 2 + H = 2jJ. Ans. Note. It follows from the rule that *9=*J=*i ; similarly 'O9'i and '009= *oi ; and therefore 2*9=3, 2 '39 2*4, 2*3459 p-2'346 ; etc. Also '99=1, '999= *> "299 = '3 ; etc. Therefore when the recurring part contains the figure 9 only> the recurring part should be omitted and the preceding figure Increased by unity. C. A- 9 130 ARITHMETIC EXAMPLES. 90. vulgar fractions in their lowest terms : ~ '-" 3. -142857. 4. 7. -37^. 8. 12. 16. Express as L <?. 2. 6. -27. 6. -272. 9. '00785. 10. -00823. 13. 3-613. 14. 3*432. 17. '5925. 18 - '5- 21. -00123. 22. '01136. 26. -oo(575. 26. -024. 29. -00625. 30. -loooi. Reduce to improper fractions in their lowest terms : 33. 3'6. 34. 7'i8. 35. 1-34. 36. 37. 1-072. 38. 3-036. 39. 10-27$. 40. 41. 7-1236. 42. 7*63i. 43. 20-45906, 44. 46. 10-0227. 46. 13-94230769. 47. 11-001206. 48. 11. -001064. 15. 7-028. 19. 2*619047. 20. 23. -00729. 24. 27. -6378- 28. 31. 3-0607. 32. 769230. '032. -oi. 31*007. 10-2567, '38148. '2273. -02177. 276. 4-0686. loo'ooloi 49. Prove that 60. Prove that - 5 8* 61. Prove that 62. Prove that II i 2 3 4 5 6' 1 -076923 -153846 -230769 '307692 13" i 2 3 4 * ioi [202 -303 -404 -5pJ i " 2 "* 3 "" 4 ""5 " Express as non-recurring decimals : 63. "09. 64. -3679. 55. 1*69. 66. 'oocg. 57. -299. 58. 3-99. 69. 3-999. 60. 9-999. 155. Addition and Subtraction of Beourring Dooi- male, Rule for Addition : Make the decimals similar : add in the ttsual way and increase the last figure in the result by the figure (if any) carried from the first (to the left) column of the period ; then the sum will be a recurring decimal similar to the summands. Subtraction is effected in exactly the same way, the only differ- ence being that the last figure in the result in this case is dimi- nished (and not increased) by the figure carried RECURRING DECIMALS 131 Example i. Add together 2*37$, 'Siyj and 4-31. Process: 2-37$ -=2-37 75757 8173- '81 73173* 4'3* -4*31 7-50 307438 i 7-50 307489 Ans. Example 2. Add together 7*634 and '852. Process: 7-634 =7'63 44 85* -J85_ij 8-48 69 ^fw Example 3. Add together '768, '07 and 1*03. Process ; 763 76 8 07 = '07 7 roj -1*033 1*87 8 I i "87 9 i -88 An*. Example 4. Subtract '78372 from 4 '071. Process: 4*071 S = S 4*O7 171717 78372= 78 372372 3-28 "799345 i 3-28 799344 An*. Example 5. Subtract '86*2 from 6745. Process: 6745=6'; .,5 862 - -86 26 5'88 29 Am. EXAMPLES. OT. Perform the operations indicated below : i. 376 + 'oi. 2. 789 +'003. 8. ro4 + 2-o3+8'oi7. 4. 6. 3'45 + '^-H7i2. 6. 7. 2-82 + '034 + *ooii 8. 8-31 + -^+ -002. 9. lo'of + 'oco^ + 'i 10. I3 ARITHMETIC 11. '007 + *oi + '0123. 12. r 123 + 3'7<? + -4576, 13. 1*30103 4-9'y + 8-0934. 14. '003 + '063 + -603. 16. i'3 + -oi3 + '1234 + 97. 16. -oo4 + -37 + -234+rK 17. 7'3i2347o' + 1-6876525. 18. 74 + 3-001 +2-1234. 19. 72 + 3-0123 + -001234. 20. 1-34563 + 2*6543. 2L 3*I347 + 7'032 + '07 + 1*345 + '0079- 22. 1*370" + '23702 + -oooi + *(5+ -37. 23. 4'c>345 + 7'234 + 8i + '04567 + -Q3 + -12. 24. 376- "0072. 25. 4-1302 -1-052. 26. '4325 -'03764. 27. 2 -76 -'321. 28. 3*46 -'07234. 29. 3*4768 -roo4. 80. 7 -'23476. 31. -9- -oo89. 32. 9*468-3*123. 33. 2-4679 -'0034 5. 34. i--io2-'4<5. 35. 3*8972 - -0034. 86. -7284-*oi23. 37. 3*76-'i2345. 38. '1234 5 -'00037. 89. 789*073$ -1 8*00032 56. 4O. 30 -'37698034. 150. Multiplication and Division of Recurring Decimals. Rule. Reduce the decimals to vulgar fractions ; find the product or quotient as a vulgar fraction and reduce it back to the equi- valent decimal. But in the case of division, if the dividend and divisor are both recurring decimals, it will be generally convenient to make them similar before reducing to vulgar fractions. Example i. Multiply "69 by 7*3. Example 2. Divide U by 75. **75-* + /ifc-|x*-f-'8. Am. Example 3. Divide -732 by '027. EXAMPLES. 98. Find the value of 1. 'o3X*o6. 2. 4-8x*24. 3. '27x4-90. 4, 'fix 1-3. 5. 2*4 x -04. 0. 7-<?x6'7. 7. '3 + ^. 8. -34^-0032. 9. 8-024--OOJ4. 10. -34 56--- -2270". 11. 3'92-M'403. 12. -i 42857 -f--i, 13. -oSi^-346. 4. -0234^-28. 15. -3123 4- '604$. RECURRING DECIMALS 133 . Complex Fractions involving Decimals. Example. Simplify ^4+^1. '5x 'i -08 5 x-i -oft \*Yft t t i i t9 ..5 + 4 = 9. Ans. EXAMPLES. 00. Simplify, giving each answer in decimals, -0075 + 2' i 2 4'? 5 55?64 3 '003 x '05 "0175 " " '00032 " ' '0022 6-27 x 0-5 . ($ of T\y) x (7 $ of 2 1 j) (4 of |) x 8' 3 6- ofg) + i-4 5 4'2-3'i4 of 1*3 0*4, 6 1*83 + 2*04 1 + *5-~3& i'3 + 2'ioi '37 ofS'Ji" " 1*0025 + ' 0625-1^ ' *I2 Of ('0104- '002)+ '36 X '002 I2X 7 I2 8 . **%.*:**.+*****&. n6 '12$ 1-5 3-42 X . S4 . ioc/54 11 'ix'ix'i + 'oix'u* *_ui ,o J: * V *t \f 't _i_ /"*> \/ <-k' \X /^/" * * _- _ *2 X '2 X *2 -J- *02 X '02 X 'O2* " "OOOO35 ' 2*3 X . 2 i J 1* !75-'ii6 of ~ * . , . 1A 3^ lfi '07692^ 999 'ooi 13 J.^B 15 X*/. j ; ^S -5 - A ?S ~ - j 134 ARITHMETIC XXVII. DECIMAL MEASURES. 158. Example i. Reduce Rs'4 to pies. Process : R3'4 16 12 rty. Ans. Example 2. Find the value of 4*135 of 1. Process : 4*135 \ The 4 is not reduced to shillings. 20 j. 2700 y The 2j. is not reduced to pence. 12 i* 8=4 of ^1- Example 3. How many rupees, annas and pies are there IB 522 of R 5 ? Process : '522 ___ 5 16 . 976 12 /. '522 of RS - R2. 9. 9*12;*. Example 4. Find the value of '2^ of ^9. 7*. Process : ^9- 7*. 6</.225o</. '25 2250 125 .". '25 of ^9. 7J. 6//.^2. 61. Example 5. Find the value of '2J of Rio. 5*. Process : "23 ofRia 5a.-^j of Rio. 50. -etc, DECIMAL MEASURES 135 EXAMPLES. 100. Reduce L 147-15 to pies. 2. "0234375 of Ri to pies. 8 ,'134375 to pence. 4. '00375 of i to farthings. 6. '03125 of RS to pies. 6. '045 of 7 to farthings. 7. 8-23 to pies. 8. -07 of 5 to pence. 9. '895 cwt. to ounces. 10. 3*985 poles to inches. Express as compound quantities : 11. R7'325. 12. 3-35, 13. R2-02. 14. 2-575 of 150. 15. 3-45 of i6s. 16. -06 of Ri3*5. 17. 372 5 of 89-2. 18. -032 of 1 2 yd. 19. '234 ton. Find the value of 20. -625 of Ri. 40. 4^. 21. 725 of R9. 6a. 22. R9. 2a. x 1-35. 23. -6 of R7. 90. lop. 24. 3-9 of Rn. 90. 25. '079 of R35*5 26. '256 of 3. 4^. 9^. 27. -1875 of 9^.4^. 28. '0625 of 3*65. 29. R3. 3<*. %p. x 785. 30, 6 x 78125. 31. $s. 6\d. x -45. 32. 3 md. 7 seers 9 ch. x 3*24. 33. 2 tons 3 cwt. 2 qr, 8 Ib. x "65. 84. 3 po. 2 yd. i$ in. x 725. 35. I da. 3 hr. 3 min. 7 sec. x '$25. 86. 3'4ofR2. 40. 37. 'o3 of 3*. 6\d. 38. R7. ga.+'o6. 89. R3. 40. <#J.-7-*422. 4O. 7. &s. 2</.-J-'o44. 41. 11-1375 of R6. 80. --56 of R7. 8a. 42. '83 of R2. 80. + -6 of R4. I la. + 2-05 of R5. 43. -37 5 of R9 + '83 of 100. - & of 6/. 44. 'oi^ofR26o. 20. 6/.-K 351 ofRi3. 1 4 a. + 1-06033 of R7. 140. #, 46. '03125 of R2 + - 729 of R3A + ' 729 of R3|. 46. '6 3437 5 + '02 5 of 25j. + '325 of 30*. 47. 871875 of 8^. + 1*146875 of 6s. 80*. -'0625 of I guinea* 48. 6-83 of 3-8677083 H-5'8 of 2-4114583-4*375 of i'3- Arrange in order of magnitude : 49. ) of R3. 90., '025 of Rioo. ioa., '32 of RS. 80. 6O. "0034 of 1, -256 of is., 3^ of id. 51. What is the sum, 75 of which is R3. 90. ip. ? 52. J of 72 of a sum of money is 3*. (>d. ; what is '03 of the sum ? . Simplify " 62S f 136 ARITHMETIC 64. Simplify -426 of ?'| of -,- 3 .- of'* 47 * 4 '* of i. ijs. 6* oo 735 Hi 66. Multiply '892 of Ri6. 50. 4^. by 4*678. 66. Find the value of '857U2 of 2-0625 tons + '571428 of 3'375 cwt. + 714285 of 1*25 qr. + '2857i4 of 10*5 Ib. 67. Find the value of '69 of 1*5 md. + 'if of 2*25 7*75 md. + *4) of 7 md. 58. Find the greatest sum of money which is contained ID each of '25 of 55. 6d. and '05 of i a whole number of times. 150. The following examples illustrate the converse operation s Example I. Reduce 1000 pies to rupees. 5-2o83. Ans. J - 12x16 24 Example 2. Reduce i. 35. bd, to the decimal of i. .". the decimal 1*175. I Example 3. Express *3 of Ri. 30. 6^. as the decimal of 40. EXAMPLES. 101. Reduce L 3333 pies to rupees. 2. 8446^. to pounds. 3. 10000 Ib. to tons. 4. 90000 in. to miles. 6. 66666 sec. to days. 6. 39 guineas to pounds. Express each of the following as a decimal of its highest denomination : 7. la. gp. 8. R3. loa. &. 9. R5. 50. #. 10. 8j. (>d. 11. i. 3*. 8</. 12. ^7. 6^. 4^ 13. I md. 15 seers. 14. 3 cwt. 3$ qr. 15. 5 po. 4 yd. 16. 7 da. 5} hr. 17. i ac. 20 yd. 3 ft. 18. 7* . 2' . 20*. In the following examples, reduce the first of the two given quantities to the decimal of the second. 19, R3. 40. <#>. ; RS. 20, 7. los. MISCELLANEOUS EXAMPLES 137 21. ga. A$. ; I la. 3^. 22. R;. ga. lop. ; Ri2. 4*. # 23. 7J. 6</. ; 15*. 7^/. 24. .3. ioj. 9^. ; ^6. 2 25. f of/i.8j.6rf;ji. 26. J of 3. 9*. 4A ; &3- 27. '375 of Rio. loa. 10^. ; R3. 130. 3^. 28. 9*:. 8/. ; -38 of R3. 4*. 29, '35 of 7. 3*. &d. ; *o o 30. "003 of i. ; '7 of 9 j. 4^. 31. '25 of 30. 4/>. ; *o6 of R3. 32. 2| of 2. 6s. 5|rtf. ; ^18. 17*. iof^. 33. Express f of I2J. 6^. + '625 of 7J. 6^.- i 5o of i6j. 6d. as the decimal of 1. 34. Reduce of R-o5 + T 7 5 of40. + f of Ri to the decimal of RJJ. 36. Express '428571 of ^1*05 4- *38 of r$s. as the decimal of 36. Reduce '246 of 9^. 3</, + ' 259 of ^i. S^. + 'oi of ,3. 7S. bd. to the decimal of '03 of ^90. 37. Reduce '062435 of ^ioo-f7'4375 of ioj. 4- 1*356 of 7s. 6< + 2*784 of 2%d. to the decimal of ^29. los. l\d. 38. What decimal of R3. ga. must be added to '07 5 of 5a. 6/>. to make the sum equal to I anna ? 30. What decimal of ^6. los. must be taken from } of 9 that the remainder may be ^6. los. ? 40. Express ,874. 135. 4^. x 3*75 as the decimal of ,10000. MISCELLANEOUS EXAMPLES. . 10& 1. Give the local value of each of the significant digits In 02073. 2. Express the difference between 2*76 and 2*76, (i) by a cir- culating decimal, and (ii) by a vulgar fraction. 3. Express J(3i+2j~4) as a decimal, and '6 + ^of '025 + 3*06 as a vulgar fraction. 4. Reduce fc of 2*35 -r 1000 to a decimal. 6. Find the least number which must be subtracted from the sum of 2*36 and 3*002 that the remainder may be an integer. 6. Find the price of 321 yards of cloth at 1 1*25 annas per yard, 7. Find the total weight of 324 bags, each 13*75 lb. 8. By what decimal do we divide 3$ , if the quotient is 7*5 ? 9. R72o is *o8 of what amount ? 10. If the divisor be 2*36 and the quotient '125 of the divisor, what must the dividend be ? 138 ARITHMETIC 11. Divide 64*09 by 49*3, and arrange the divisor, dividend and quotient in order of magnitude. 12. If the diameter of a pice be 1*025 inches, how many must be placed in contact along a straight line to extend from Calcutta to Hughly, a distance of 24*6 miles ? 13. How often will a wheel, 2*75 yards in circumference, turn (ii a distance of 12*5 miles ? 14. A vessel holds 3*256 gallons ; how many times can it be filled from a cask of 96 gallons ? Will there be any remainder ? 16. How many times can you subtract 3*01 from 65*23, and what is the remainder ? 10, Express as a decimal the continued product of |, -~~ - 17. Express 21-43 crowns + 18*52 shillings in pence. 18. Subtract 4*42 cwt. from 7*28 tons. 19. Express 275 oz. + *o75 cwt m pounds. 20. Find the rent of 32*25 acres at ^1*025 per acre. 21. If the product of '064 and a certain number be divided by '00008, the quotient is 3404 ; find the number. 22. A book containing 219 leaves is 1*34 inches thick ; allow- ing "06 of an inch for the cover, find to 5 decimal places the thick- ness of the paper. 23. A roller 4*03 ft. in circumference makes 34*04 revolutions In passing from one end of a lawn to another : what is the length of the lawn ? 24. From a rod 2 yards long, portions each '063 of an yich in length are cut off ; how many such portions can be cut off, and what will be the length of the remaining piece ? 26. Find a decimal which shall differ from } by less than' 1D J 00 . 26. Multiply 9*036 by itself in two lines. 27. Multiply 37*056 by 12*10411 in three lines. 28. Find the least number of articles, costing &2 '37 5 each, that can be purchased for an integral number of rupees. 20. Find the smallest number of articles, costing 2. 6s. 2*37 d. each, that you can buy for an exact number of pounds. 30. A did "025 of a piece of work, and B '825 ; how much was left to be done ? 31. A boy, after giving away '8 of his pocket-money to ,one companion, and *o6 of the remainder to another, has 70. lop. left ; how much had he at first ? APPROXIMATION I3 32. A man received '3$ of '03 of a property, and sold '3 of his own share for R35o ; what would be the value of the whole property at the same rate ? 33. A gallon contains 277*274 cubic inches ; how many cubic yards are there in 200 bushels ? 34. A cubic foot of water weighs 62*35 Ib. avoir. ; what would be the error in calculating the weight of 30 cubic feet on the approximate supposition that a cubic foot of water weighs loco oz. ? 35. A is 75 times as old as B> and C 7$ times as old as B ;. A is 15 years old : how old is C ? 36. Four bells toll at intervals of i'3, 1*4, 1*5 and r6 seconds, beginning together ; after what interval will they toll together again ? 37. Find the largest sum of money which is contained in 37 5 and ^2*125 a whole number of times. 38. Divide RSO into two parts such that one part may be '6 of the other. 39. Divide ^52 between A, B^ C in such a manner that B may receive '3 of A, and C '3 of B. 40. Exp,,,of* + as a fr.c.io. XXVIII. APPROXIMATION. 16O. It is often inconvenient, and not always possible, to find an exact decimal equivalent to a proposed number. In such cases we may proceed to a few places of decimals and indicate by dots (...) that the work has not terminated. Thus \\ '95652... If however we wish to approximate to the result by terminating our work at any specified place, we should increase the last digit retained by I if the first digit rejected be 5 or greater than 5. Thus Jf =*'957 correct to three places of decimals or to the nearest thousandth ; also || '956 5 to four places. Note 1. It will be easily seen that the difference of '957 and '95652... is less than the difference of "95652... and '956 ; hence '957 represents '95652... more accurately than '956. It may be noticed that the approximate result is less than the actual result when the first figure rejected is less than 5, but greater when not less. 140 ARITHMETIC Note 2. Suppose that '36 is given as correct to two places of decimals. This has been obtained from the true value of the decimal by the addition or subtraction of a decimal which may be as great as '005 but not greater. Therefore the error in taking 36 for the decimal lies between + '005 and -'005, that is, the error is not greater than +'005 and not less than -'005. The actual error may be anything between +'005 and -'005. Hence the Limits of Error of a decimal correct to two places are "005. Similarly, the Limits of Error of a decimal correct to three places are '0005. And so on. Note 3. Sometimes approximate results are expressed true to a certain number of Significant Figures. Thus, 346271 correct to 5 significant figures = 346270, and correct to 4 significant figures = 3463:0 ; 7*9284 correct to 4 significant figures = 7*928, correct to 3 significant figures ==7*93, correct to 2 significant figures 7*9, and correct to I significant figure = 8 ; 4*00923 correct to 4 significant figures = 4*009, correct to 3 significant figures=4*oi, and correct to 2 significant figures = 4*0 or 4 ; "005293 correct to 3 significant figures = '00529, correct to 2 significant figures = '0053$ and correct to i significant figure = '005. CONTRACTED ADDITION AND SUBTRACTION. 161. The method of finding approximate results, />., results correct to a certain number of places of decimals, in addition and subtraction is illustrated by the following examples. Example I. Find the sum of "2367, "3178 and 1*62 correct to four places of decimals. We write down each decimal '2367 676 to 7 places, and obtain correctly '3178 178 5 decimal places in the sum ; the i '62 required result is then obtained ~2'I745J8...2'I746. Ans. by rejecting the fifth place. Example 2. Find the difference between '632 i and 'oo correct to five places of decimals. Process : '632 13)2 13 0088* 888 62324 3...=a'62324. Ans. Example 3. Find the sum of 72*65, 87968 and 4*02 true to significant figures. Process : 72*6565 87968 968 4*02 656 Ans. APPROXIMATION 14* Example 4. Find the value of i + + + ....., cbrrect to 3 places of decimals. irooo 000 I I 2 '500 000 I _ * 5 _ T AA AAA 1.2.3 3 *= lOO uUU I 166666 "041 666 1.2.3.4 4 I 041666 . _. ._.. == '008 333 I 008333 6 '001 388 1.2.3.4.5.6 I 001388 *ooo 198 1.2.3.4.5.6.7 I 000198 8 = *OOO|O24 1.2.3.4.5.6.7.8 I 000024 = *OOolOO2 1.2.3.4.5.6.7.8.9 9 and .'. the expression =171812... 1 7 1 8, to 3 places. Here we stop at ;^rT7V7~.;~Q~^ as in the decimals equivalent 1.2.3.4.5.0.7.0.9 to the succeeding fractions) the first six figures will be zeroes. EXAMPLES. 103. 1. Find the quotient of 40 divided by 19 correct to four places of decimals. 2. Obtain the decimal equivalent to tV correct to five places of decimals. *3. Find the value of '0312 + '023 i + *97o" correct to fourjplaces of decimals. 4. Find the sum of 72, 3*0123 and '001234 correct to three places of decimals. 6. Find the difference between '432 and '03764 correct to four places of decimals. 2 ARITHMETIC Find the value, correct to 2 places of decimals, of 6. i + A + T*B+T!toi+- 7. 8. Find the value, correct to 3 places of decimals, of 10- i+\+-?+*i+... 11. i+ 7 -+ 7 I a + 7 ^+- Find the value, correct to 5 places of decimals, of 12, ^s + ^sJ' + Cas)^... 13. i + -l +i j ;5 + L- i ii i * , [First express as decimals 2 , -4, tt , ...... , then divide the results respectively by i, 2, 3,..., and add.] ie I I I I II I I , I I , Find the value, correct to 3 places of decimals, of 15a " '-^-To^- 15b - - + -- 16. Express each of the following correct to four significant figures : (i) 378361; (ii) 735932; riii) -520681; (iv) 7*38512; (v) 2*00972; (vi) 2-00023; (vii) '034071 ; (viii) '0090628. 17. Express 3456792 correct to the nearest hundred^ and 80057123 correct to the nearest thousand. 18. Find the approximate value of 3*9281 (i) correct to the nearest unit) (2) correct to the nearest tenth, (3) correct to the nearest hundredth. 19. F ind a decimal that is within 'ooi of $. 20. Find a decimal that is within nyo\nJB of f f|. Note. The following algebraical method may be used with great advantage when each term of the series is the product of the preceding term by a constant proper fraction, positive or negative. APPROXIMATION I Example i. Find the value, correct to 4 decimal places, of i+ s| + + aV- Let S denote the sum of the series. Then S=,i + J-+ * +-*+... ; 25 25* 25 s multiplying both sides by J- 6 (the constant multiplier), we have js 5 " 2~ 5 + 27> + 2?'~ i ~- Hence, by subtraction, S- ^S i, or }*5-i ; Example 2 Find the value, correct to 3 decimal places, of I-i+i-i-K.; Let 5 denote the sum of the series. Then S-1-JfJ-J + ...i multiplying both sides by 4 (the constant multiplier), we have -K- - Hence, by subtraction, or EXAMPLES. 103 1;. Find the value, correct to 5 places of decimals, of i. 1+;+^+... 2. i +5 ^ + i+ i + . M 3< ^S^o"'"^**" 4 * I '20" I- 20""20' 1 ''" CONTRACTED MULTIPLICATION. I6. The following rule will shorten the process of multipli- cation when the product is required only to a certain number of decimal places. To multiply two decimals together, retaining say* 5 decimal places ; "Reverse the multiplier, strike out the decimal polrits^hd 144 ARITHMETIC place the multiplier under the multiplicand, so th at what was its- units' figure shall fall under the 5th decimal place of the multiplicand, placing ciphers, if necessary, so that every place of the multiplier shall have a figure above it. Proceed to multiply as usual, begin- ning each figure of the multiplier with the on** which is in the place to its right in the multiplicand : do not set down from this product but carry its nearest ten* to the next, and proceed. Place the first figures of all the lines under one another ; add as usual ; and mark off 5 places from the right for decimals." [De Morgan.] Example i. Multiply 7-2078 by 2*3072 to 5 places ; '00705328 by 12*30523 to 6 places ; and 29*82 by '00727 to 4 places of decimals. (i) 720780 (ii) 705328 (iii) 29820 27032 3250321 72700 1441560 "70533 2087 216234 14106 60 5045 2116 20 _J44 35 ^F67 V6 '62983 _ __. I - 086791 Note. The last figure in the product thus obtained may not be always correct, and to ensure its accuracy we must carry tb<* process one place farther than is required to be retained. Similar remark also applies to the method of d. vision explained in the next Article. Example 2. Multiply 'J4 by 4*57 retaining 5 decimal placeb, 4*03721 by '01207 correct to 5 decimal places, and 4086 by 2057 correct to the nearest thousand. (i) 3434343 (ii) 403721 (iii) 40860 1757574 7Q2I _75<>2 1373737 40372 81720 240404 8074 2043 I7I72 __ 282 f __28o 2 44 '04873 Ans. 8405^ thousand, 2 or 8405000. Ans. 1-63391$ Ans. * That is, carfjt I JMhe product is a number from to 14 : carry a if it is Jron^ 15^0 24 f catty 3 if ,it is from 25 to 34 ; tc. ; if tie product i* 4 oPfcss, we ignore it; APPROXIMATION 145 CONTRACTED DIVISION. 162 a. The following rule will shorten the process in division when the quotient is required to be correct only to a certain number of decimal places. Make the divisor a whole number ; and determine by inspection (or by taking one step in the ordinary way), how many figures there will be in the integral part of the quotient. In the divisor retain (from the left) as many figures as there are to be in the whole quotient integral part as well as decimal ; and strike off the rest. Proceed one step with this new divisor, but to the product of its first figure by the quotient-figure, carry the nearest ten from the preceding figure. Instead of bringing down a figure to the remain- der, strike off another figure from the divisor, and proceed as be- fore, until no figure is left in the divisor. If the number of figures in the divisor be less than the number of quotient-figures to be obtained, proceed in the ordinary way until the number of quotient-figures still to be obtained is one less than the number of figures in the divisor. As soon as this happens, instead of bringing down a figure to the remainder, strike off a figure from the end of the divisor, and then proceed as in the preceding case. When, by inspection, it ia found that there is no integral part of the quotient and there are ciphers just alter the decimal point in tbe quotient, subtract the number of ciphers from the total number or decimal places required, and take the remainder as the number of decimal places required in the quotient. Then proceed: as above. Example i. Divide 29-431542 by 3*25348 to 3 decimal places f and 6;3'i489 b Y '4*432 to 2 places. 0) 32,5,34$) 2943i54'2 (9*046 29281 150 130 20 (") 4 JA3.2) 67314890' ( 162470 41432 258828 2 48592 10236 ' 8286 1950 1657 293 290 3 C. A. 10 146 ARITHMETIC Example 2. Divide 400654 by 3292*65 to 5 places of decimals. 32^00)400-654(121 Ans. ~-ooi2i. 329 71 Here, of the 5 decimal figures the 66 first 2 being ciphers, we obtain the 5 remaining 3 figures by the contracted i method. Note. Perfect accuracy cannot always be expected in the results obtained by the contracted methods and they may sometimes differ slightly from those obtained by the ordinary long methods. EXAMPLES. Multiply L 21-1324 by -345721 103a. to 3 decimal places. 2. '32504 by 13-0254 to 3 3. '453 by -01694 to 4 4* 37576843 by 3'i4i 59 to 4 5. 71*032751 by 2-6719238 to 5 8. 65-00763 by -9876 to 5 7. '03281674 by 234781 to 6 * 8. '0008127 by 483*2716 to 6 8a. 4 562 by '07408 to 5 8b. 6-2438 by 38306 to 5 9. 4-683 by 14*293 to 3 10. r8i357 by -0785 to 6 lOa. -01385 by 6137 retaining 4 ... ... ... lOb. "346875 by -119808 retaining 4 IOC. 32-34 by -32056 correct to 3 10d. -342 by 3-253 correct to 3 100. '00926347 by 280-435 correct to 4 ... lOf. 421*619 by -547 correct to the nearest integer. lOg, 70870096 by 404 correct to the nearest million. Divide 1L 76*2307 by 47'i2345 to 3 decimal places, 12. 3*37o6 by 97846 to 3 ;. JL3. 32*791 by 26*67 to 3 APPROXIMATION 7 14. 378*325 by 30732 to 3 15. 367802 by 312*32 to 4 16. 728-389 by 376 to 4 17. 3892762 by 7*343 to 5 18. 2378934 by -00289 to 5 19. 13-2346891 by -01234031 to 6 20. 132*405678 by -000122134 to 7 20a. '5 by 7691342 to 4 ... 20b. '0003738028 by '0475 to 5 21. 3725 by 13-234 to 3 22. 1*82357 by -0785. to 6 23. -32165 by "35216 retaining 4 24. 1*59587 by 4 3062 correct to 3 162b, When an approximate decimal is multiplied by a number less than unity or divided by a number greater than unity, the error is obviously diminished in the result. This principle is made use of in the following examples. Example i. Find the continued (product of 127053, "003725 and 4*532 correct to 3 places of decimals. Take for the multiplicand 127053 which contains the largest number of significant figures. In the second multiplier, 4'532, shift the decimal point one place to the left so that the first significant figure may stand in the first decimal place and the multiplier becomes less than unity ; and make the compensating change in the multiplicand by shifting the decimal point one place to the rihght. Thus we have to find the product, 127 053 x -003725 x -4532. 127-053 5273 3812 889 25 6 "473 correct to 3 places. 2314 1892 237 _ _ 214 correct to 3 places. Am. 148 ARITHMETIC Example 2. Find the value of 0-34567 x '073456 0-67345 correct to 4 places of decimals. [C. U, 1918.] Shift the decimal point in the denominator and also in the numerator one place to the right so that the denominator may have one figure in the integral part and thus become greater than unity. Now we have to find the value of 0-34567 x 7 '34 56-1-6 7345. 34507 65437 241969 10370 1382 173 20 2*5391 correct to 4 places. 6 7 j 4^5 ) 2539 ro ( "37702 or '3770 correct to 4 places. Ans.. 202035 71875 47J4 2 4733 47H 19 12 6 EXAMPLES. 103b. Find the value, correct to 3 places of decimals, of 1. '023045X2X>3X i'32. 2. *i 5304 x 1025 x T2o6. 32-302x^354 4 -12345 x -51234 36-403 * ' '45123 '348662 * '285oix'6o8i75* [Hint : Change -348662 land '60817 5 to 3-48662 and 6*08175 respectively. Divide 3-48662 by '28501 correct^to 3 decimal places and the result by 6*08175 correctjto 3 decimal places ] Q '12345 * -23451 x -34512' SIMPLE PRACTICE \M9 XXIX. PRACTICE. \ 163. An aliquot part of a quantity is a quantity which can be expressed as a fraction of that quantity, having unity for its numerator. Thus 40., being J of Ri, is an aliquot part of Ri ; 2s. 6</., which is i of *> is an aliquot part of i. 164. Simple Practice is a convenient method of finding, by means of aliquot parts, the cost of a simple quantity, when the cost is given of the unit-quantity, in terms of which the simple quantity is expressed. Example. Find the value of 32 cwt. of wheat at 83. 8a. per cwt. Compound Practice is a convenient method of finding, by means of aliquot parts, the cost of a compound quantity, when the cost is given of one of the units, in terms of which the compound quantity is expressed. Example. Find the value of 7 cwt. 3 qr. of wheat at 83. 8a. per cwt. SIMPLE PRACTICE. 165. The following examples will explain the method of Simple Practice. Examples Find the price of 23 md. of rice at 83. 130. o/. per md. g a. A 23 . o . o = price at Ri per md. 3 8o.=^ ofRi. 69 II . . 8 o= price at R3 per md. 4a.=fi of 8*. 5 . 12 o=a 4** ,i Ia. = ; of 40. i . 7 0=* > io. ,, 6/.= f Of Itf. II 6=s 6^. #.= r of 6/. 5 9=3 >f ,, i t ,j R88 . 12 . 3price at R3. 13^. 9/. per md. Note 1. Since &3 . 13 . 9 is the difference between 4 and 20. 3^., a shorter method would be to find the price at 20. #. per md. and subtract, it from the price at R4 per md. 150 ARITHMETIC Thus 2*.-* Of Rl. . -4 of 2a. R. a. A 23 . o . o 4 92 . . 3 3 9 188 . 12 - 3 s R. a. A 23 . o . o 2 . 14 . o 5 . 9 price at R4 per md. 2*. 3^. 3 - price at R3. 13*. 9^. per mdc 9 price at 2<j. 3^. per md. Example 2. Find the cost of 9 articles at ^10. I2J. 6d. each, . s. d. 9.0. ocost at i each. I OS. 2S. ofi. > of los. 10 90 . o . oncost at .10 each. 4 . 10 . o= loj. 18 . o=* 4-6=,, 2S. . 12 . 6 cost at ;io. i2s. 6d. each. Note 2. Shorter thus : ioj.=* $ of 1 ; 2s. 6^.= J of los. Example 3. Find the value of I3^cwt. at R7. 100. 3^. per cwt 2a. 8a, ^ of Ri. } of 8a J of 2a. S r thus : * of Ri. | of 2 a. R. 13 . "i . o value at Ri per cwt. _7 o = value at R7 per cwt. o= 2a. ,) 94 6 1 . 8 . . 12 . . II . 3 I03 R. 13*5 7 . 2 . 4^ value at R7. loa. R. '1484375 16 3^. per cwt 94'5 675 1-6875 2109375 a. 2-37 5000 12 A 4*500 Rioyi484375 , SIMPLE PRACTICE 151 Example 4. Find the value of 42} things at i6j. z\d\ each. I or. $s. is. id. of ioj. of 5J. of is. oi2d. J. . 42 . 13 . 4 -value at i each. 21 . 6 8 value at los. each. 10 . 13 4 = 55. 2 . 2 8 - is. 7 il 2d. i 9*= 4^ io| J^. 34 12 = value at i6s. 7\d. each* EXAMPLES. 104. Find, by Practice, the cost 1. 400 at 83. 40. each. 3. 789 at la. 5. 439 at #. 7. 874 at 6a. 9. 939 at &2. iia. 11. 475 at 130. 6>. 13. 500 at 7. 3^. 15. 700 at loa. 4^. 17. 321 at 82. 50. 3^. 19. 366 at 7. na. gp. 21. 839 at ES. 130. 4^. 23. 454 at 15. ja. 10^. 25. 900 at 842. loa. ?&. 27. 768 at Ri9. ga. ^pice. 29. 8760 at &2i. 14^. i pice. 31. 555 atR89. 3^. 5^. 33. 8001 at R8o. %a. 8^. 35. 346^ at R8. loa. %p. 37. 703! at R29. 13^. 4^. 39. 821 j at &4i. ?a. $%p. 41. 600/3- at Ri2. I2a. ip. 43. 39-5 at RJ. 130. 4^. 45. 101*375 at Rio. ga. 6p. of the following articles : 2. 375 at 2. 5J. each. 4. 728 at 3</. 6. 399 at 4. 4*. 8. 723 at 15*. 10. 275 at 4d. 12. 342 at 2s. 6d. 14. 942 at 7J. 3</. 16, 374 at si</. 18. 230 at ^7. ioj. 6^. 2O. 767 at 10. 8s. 8^. 22. 339 at 145. io\d. 24. 900 at ^50. us. 26. 5013 at ^55. igs. 28. 1010 at ;n. iij. 30. 4596 at I2s. o|*/. 32. 3111 at 12. I2s. 34. 10000 at 7. 17 s. 36. 27! at ^8. i6s. 38. 301 J at 2. i$s. fid. 40. 442! at 76. 2s. A&d. 42. 249^ at ^20. 2s. 44. 847 5 at 2. iss.g 46. 10*875 at 2. 17 s. 152 ARITHMETIC o seers! 1 of I md. 2i'seers| J of 10 seers COMPOUND PRACTICE. 166. The method of Compound Practice is illustrated by the following Examples. Example i. Find the price of 15 md. 12$ seers at R2. 50. 3^. per md. R. a. p. 2.5.3 price of i md. 3 6 . 15 . 9 5 34 . 14 . 9 price of 15 md. 9 3l - i seers. 2 __3i| 2$ seers. 35 . 10 . 4jprice of 15 md. 12$ seers. Example 2. Find the cost of 2 tons 3 cwt. 3 qr. 5 lb. at 1 7 j. per cwt. /. J. d. 3 cwt. 43 cwt. 2qr. i qr. 99 4lb. ii I lb. [5 lb. 694 . 2 . io$f = costof2tons 3cwt. 3qr. Example 3. Find the value of 25 sacks of flour, each weighing 3 md. 10 seers, at 5. 80. per maund. R. a. p. 10 seers J of i md. 2 tons ?qr. i qr. 4 lb. i lb. 3 cwt. 43 cwt. i r Of I CWt. j r of 2 qr. i of I qr. r of4lb. 15 17 . 10 . COS 10 . o 4 634 47 68 1 7 3 . o . II . II . 18 19 ii 2 . o cos . vl )) . o * . 6 - . 3-n ' %- 5 . 8 . o . o . o value i value of 99 of I md. 3 rnd. 10 seers, i sack. 16 i . 8 . 6 17 . 14 . o 5 89 . 6 . o 5 446 , 14 . o value of 25 sacks* COMPOUND PRACTICE 153 EXAMPLES. 1O5. Find, by Practice, the value of 1. 7 md. 1 5 seers at 83. 7 a. %p. per md. 2. 9 md. 17$ seers at 84. loa. %p. per md. 3. 27 cwt. 2 qr. 7 Ib. at ^3. *js. 6J. per cwt. 4. ix tons 14 cwt. at .5. 17^. bd. per ton. 6. 17 tons 15 cwt. 2 qr. 21 Ib. at ^3. 15*. qd. per cwt. 8. 6 tons 3 cwt. 2 qr. 24 Ib. at 17^. jd. per cwt. 7. 2 tons 13 cwt. 3 qr. 7 Ib. at i. is. %d. per cwt. 8. 3 md. 27 seers 8 ch. at Rio. $a. 8/. per md. 9. 7 md. 1 8 seers 9 ch. at 813. 7 a. $p. per md. 10. 8 md. 3 seers 12 ch. at 3#. 4^. per seer. 11. i md. 17 seers 10 ch. at 7 a. 6p. per seer. 12. 4 cwt. 3 qr. 14 Ib. al 1. 13*. 4^1 per ton. 13. 7 cwt. 2 qr. 21 Ib. at 6 per ton. 14. 3 tons 17 cwt. 3 qr. 13 Ib. 12 oz. at i. iSs. gd. per cwt, 15. 3 md. 37 seers 12 ch. at js. 6d. per seer. * 16. 2 tons 7 cwt. I qr. 13 Ib. 14 oz. at 89. na. per qr. 17. 7 sacks of flour, each 3 md. 1 5 seers, at 87. loa. per md. 18. 24 bales of cotton, each 5 cwt. 2 qr., at i6s. *j\d. per cwt. 19. 35 chests of tea, each i md. 17 seers 9 ch., at R8o. 120, per md. 20. 321 boxes of coffee, each i cwt. 2 qr. 21 Ib., at 7. 18*. per cwt. 21. Find the total produce of a field of 3 ac. 3 ro. 25 po. at 3 qr. 6 bus. 2 pk. per acre. 22. Find the produce of 2 ac. 2 ro. 88 sq. yd. at 7 cwt. 3 qr, 14 Ib. per acre. 23. Find the price of 29 yd. 2 ft. 9 in. of silk at 75. ioj//. per yd, 24. Find the weight of 231 bales of cloth, each weighing 2 cwt. 2 qr/*l4 Ib. 25. Find the weight of 329 boxes, each weighing 7 md. 27 seers. 26. Find the tax on ^329. 15^. at is. *]\d. in the . 27. Find the tax on 83090. 8a. at la. \\p. in the 8. 28. Find the cost of 5 qr. 3 bus. 2 pk. of oats at 2. 14*. 46 er qr. 154 ARITHMETIC 29. Find the price of 12 gall. 3 qt. i$ pt. of milk at R3. 8a, per gallon. 30. Find the value of 225 cwt. at 21. $s. yd. per ton. 31. Find the value of 257 things, 10 of which cost R3. ga. 4p. 32. Find, to the nearest pie, the rent of 275*365 bighas at R$. fa. gp. per bigha. 33. Find the value of i ton 1 1 cwt. i qr. 1 1 Ib. at ^6*285 per ton. 34. Find the dividend on R5I46. I2a. at 140. 6/. in the R. 35. If a man's debts amount to 3792 5. 140., and he can pay only 30. 4^. for each rupee, how much do his creditors get ? XXX. SQUARE ROOT. I6T. A number is called the square root of its square. Thus 2 is the square root of 4 ; 3 is the square root of 9. The square root of a number is indicated by the symbol *J placed before it. Thus ^4 indicates the square root 0/4, that is, 2. 108. A number whose square root can be expressed exactly either by a whole number or by a fraction is called a perfect square. Note. It may be noticed that, no number, integral or decimal,. which ends with 2, or 3, or 7, or 8, is a perfect square. 169. When the square root of a whole number which is a perfect square does not exceed 20, we obtain it from the multipli- cation table. Thus from the table we know that the square root of 81 is 9 ; of 169 is 13. We have, however, a rule by which we can find the square root of any number consisting of more than two figures. 1TO. We observe that the square root of 100 is 10, of 10,000 is 100, of 1,000,000 is 1,000 ; and so on. Hence it follows that the square roots of numbers less than 100 consist of only one figure in their integral parts ; of numbers between 100 and 10,000, of >two figures in their integral parts ; of numbers between 10,000 and 1,000,000, of three figures in their integral parts; and so on. If then a point be placed over every second figure in any number be- ginning with the unite the number of points will be the same as the number of figures in the integral part of the square root. Thus the square root of 3136 consists of two figures in its integral part ; the square root of 15625; consists of three figures in its integral part. SQUARE ROOT 155 171. Now suppose we have to extract the square root of 3136. We first divide the number into periods of 3*3 ( 56 two figures each, by placing dots over every ?J _ second figure beginning with the units.* 106 ) 636 636 We then find the greatest number (5) whose square is contained in the first period ; this is the first figure of the root ; then subtract its square (25) from the first period and to the remainder (6) bring down the second period, thus getting 636 for the new dividend. Next, we divide this number omitting the last figure, by twice the part of the root already found (i.e., we divide 63 by 10), and annex the quotient (6) to the root and also to the trial divisor (10) ; then multiply the divisor as it now stands (106) by the figure of the root last found. Now, subtracting this product from 636, we have no remainder ; and we conclude that 56 is the square root of 3136. If there be more periods to be brought down, I 5^ 2 5 ( I 2 5 the above operation must be repeated, as in the \ ____ _ annexed example. 22 ) 56 44 245 ) 1225 1225 Here, after two figures in the root have been obtained, the re- mainder is 12 ; to this we bring down the third period, thus getting 1225 as the last dividend. We divide this number, last figure omitted, by twice the part of the root already found (*>., we divide 122 by 24), getting 5 as the quotient. We then annex 5 to the root and also to the trial divisor 24 : etc. In obtaining the second figure of the root by division we sometimes get a quotient which is too large. In such a case we find the root-figure by trial, as in the two following examples. (i) 225(15 Here, dividing 12 by 2, the quotient is 6. i Taking 6 as the required figure we find that 2 r j ^2! the product (26x6) is greater than 125. We I2 c then ta ke 5 which is found to be the required root-figure. (ii) 36i (19 i_ Here, division gives 13 which is obviously 29 ) 261 inadmissible. By trial we find 9 to be the 2^ x required root-figure. * N. B. Each period consists of the figure over which a dot is placed and the figure to its left. Here the first period is 31 and second 36. The fint period may consist of only one figure. v "* I 5 6 ARITHMETIC 173. When the trial divisor is greater than the number to be divided by it (or when the quotient is I but found too large) we set down o in the root, annex o to the divisor, bring down the next period, and proceed in the usual way. The two following examples are given for illustration. (i) 41209 ( 203 (ii) 4 403 ) 1209 1209 4461604 ( 2098 4 _____ 409 ) 4016 3681^ 4188)33504 33504 174. In the process of extracting the square root, a remainder is often left, which is greater than the divisor. In the following example the second remainder 35 is greater than the divisor 29. 39<5oi ( 199 i 29 ) 296 261 389 3501 EXAMPLES. 100. Find the square root of 1. 5. 9. 13. 17. 21. 25. 28. 441. 1024. 27225. 119025. 4937284. 82264900. 3226694416. 360117609604. 2. 6. 10. 14. 18. 22. 576. 6561. 54756. 193600. 2819041. 62504836. 3. 7. 11. 15. 19. 23. 26. 29. 6407522209. 295066240000. 729. 5625. 49284. 646416. IOO2OOI. 97535376. 24. 27. 30. 4. 8. 12. 16. 20. 96i. 92i6. 18225. 717409- 1522756. 21224449. 236144689. 15241578750190521. 31. A certain number of men spent 81681, each spending as many rupees as there were men ; how many men were there ? 32. A certain number of persons agree to subscribe as many pies each as there are subscribers ; the whole subscription being 33- $<* 4A How many subscribers were there ? 33. A gardener plants an orchard with 5776 trees and arranges theili so that the number of rows of the trees equals the number of trees in each row. How many rows were there ? SQUARE ROOT 157 34. A general having 11025 men under him, arranges them into a solid square. Find the number of men in the front, t 35. A general wishing to arrange his men, who were 63510 in number, into a solid square, found that there were 6 men over. How many men were there in the front ? 36. Find the least integer which must be subtracted, from 4230 in order to become a perfect square. 175. When a number, which is a perfect square, can be easily separated into prime factors, its square root may be found by inspection. Thus ^8100= ^2* x 5 2 x 3 2 x 3 2 =2 x 5 x 3 x 3 = 90. Example. What is the smallest whole number by which 1260 must be multiplied in order to become a perfect square ? Since I26o2 2 x 3 s x 5x7, .'. the number required** 5x7-35, EXAMPLES. 1O7. Find, by factors, the square root of 1. 900. 2. 1600. 3. 324. 4^ 576. 5. 1296. 6. 4096. 7. 1764. 8. 7056. 9. 11025. 10. 53361. 11. 99225. 12. 571536, 13. 27x12x14x56. 14. 182x77x66x39. 15. 609x290x165x154. 16. Find the smallest whole number by which 450 must be multiplied in order to become a perfect square. 17. Find the least number by which 2940 must be multiplied In order to become a perfect square. 18. Find the least number by which 968 must be divided in order to become a perfect square. 19. Find the least square number which is divisible by 10, by 1 6 and by 24. 20. What must be the least number of soldiers in a regiment, that will allow it to be drawn up 10, 15 or 25 deep, and also to be formed into a solid square ? 176. To find the square root of a Decimal Fraction. To find the square root of a decimal fraction we proceed as in the case of a whole number. In pointing) the first point mus v be 1 58 ARITHMETIC place4 pr supposed to be placed on the units' figure. In the root the decimal point must be placed immediately after the root-figures corresponding to the integral part of the number. We observe that if any decimal be squared there will be an even number of decimal places in the result. Consequently a deci- mal fraction (in its simplest form) to be a perfect square must have an even number of decimal places, and the number of decimal places in the root must be one-half of the number in the square. If the given decimal is not a perfect square (which is always the case when the decimal in its simplest form contains an odd number of decimal places) the square root will be a non-terminat- ing decimal ; and we can find the square root to any number of decimal places we like. In finding the square root of a decimal, the number of decimal places in it must be made even> by annexing ciphers, if necessary. Example I. Find the square roots of 11*9025 and '5625. 11-962*5 ( 3-45 Ans. '5625(75 Ans. _9 4? 64) 290 145)725 256 72j 685 ) 3425 3425 Example 2. Find the square root of '045 to three places of decimals. 045006 ( '212... Ans. Here, we are to have three 4 decimal places in the root ; 4I )~^ therefore in the given number, ^ we make the decimal places )~~obo "*' *44 56 Example 3. Find the square root of 3 to two places of decimals. $0606 ( 1*73... Ans. i 27 )200 SQUARE ROOT 1 59 EXAMPLES. 108. Find the square root of 1. 11-56. 2. 4*7089. 3. 39-0625. 4. 82-4464. 6. -0064. 6. -005329. 7. 1082-41. 8. 5774409. 9. '00053361. 10. '00002025. 11. 236*144689. 12. '804609. 13. '000003418801. 14. 1-002001. 15. 938703-06991561. Find to four places of decimals the square root of 16. 761-9. 17. 17- 18. 237-615. 19. 5. 20. 876-535. 21. 'I. 22. -5. 23. 23-1. 24. '9. 25. 20. 26. -016. 27. '00064. 28. 7. 29. 66. 30. 13. I7T. To find the square root of a Vulgar Fraction. The square root of a vulgar fraction is the square root of its numerator divided by the square root of its denominator. Example i. /''L-'^Lt V 25 ^25 5 Example 2. 6 Jio 3-1622... ~ If the denominator be not a perfect square \\ is advantageous to make it so by multiplication. tr *i 7 1 /ix6 tJ6 2*449... Example 4. - = = ~-' 4 8 - ~ 1 Example 5. Note. The square root of a fraction can also be found by reducing the fraction to a decimal and then extracting the square' root of the decimal. EXAMPLES. 109. Find the square root of I. i&. 2. 55I3A- 3. 3 2jJ. 4. 101,^. 6. J. 4$ 6. 2-7. 7. 28-4. 8. 3-36!. 9. 8-027. 10. '071. Find to 3 places of decimals the square root of II. I. 12, f. 13. |. 14. f. 16. &, 160 ARITHMETIC 16. -3. 17. '4i6\ 18. ~~ 3 . 19. . 20. &*. 5 2*5 012 21. Simplify \/(75i)x V(i'7)^ \/(2ff). 178. When more than half the number of figures of a square root has been obtained by the ordinary method, the remaining figures may be obtained by division only. Example I. To find the square root of 189475225. Here we find the first three 1894.7! 522 ( 137(65 Ans. figures in the ordinary way. i To find the remaining two 23 ) 89 figures by division, we take ^ Q twice the part of the root al- , ^ ready found, as the divisor ; ' ' we bring down one figure to 1869 the last remainder and divide ; 2 74 ) 1785 ( 65 then to the new remainder - 1 _44_ bring down the next figure 1412 and divide. The quotient thus 1370 obtained gives the two remain- "" '^ ing figures of the root. Note, Of course this process does not show whether the given number is a perfect square or not, but it is very useful in cases like the following. Example 2. Find the square root of 2 to seven places of decimals. Here we find 5 figures of 2* (1*41421135... Ans, the root by the ordinary method i__ and the remaining three by 24 ) lob division. 96 281 )~4oo 281 2824) 11900 11296 28282 ) 60400 56564^ 28284) 38360 ( 135 28284 100760 84852 159080 J4I42Q 17660 CUBE ROOT l6l EXAMPLES. 110. Find to 6 places of decimals the square root of L 5- 2. 17. 3. 761-9. 4. '0003841. 5. f. 6. 3. 7. -07. 8. -85. 9. 7619. 10. f. 11. 237-615. 12. 17. 13. f 14. 23-8369. 15. '000943. 16. 10. XXXI. CUBE ROOT. 179. A number is called the cube root of its cube. Thus 2 is the cube root of 8 ; 3 is the cube root of 27. The cube root of a number is indicated by the symbol V placed before it. Thus V& indicates the cube root 0/8, i.e.) 2. A number whose cube root can be expressed exactly either by a whole number or by a fraction is called a perfect cube. The cubes of i, 2, 3, 4, 5, 6, 7, 8, 9, are respectively i, 8, 27, 64, 125, 216, 343, 512, 729. [These results should be committed to memory.] ISO. The method of finding the cube root of a number is as follows : Example I. Find the cube root of 13824. 13824 (24 Ans. 8 2 2 X 300= 1200 2x30x4=* 240 16 1456 5824 5824 We divide the number into periods of 3 figures each, the number of dots indicating the number of figures in the cube root. We find that 2 is the highest number whose cube is less than the first period ; this then is the first figure of the root. We sub- tract the cube of 2 from the first period and to the remainder we bring down the second period. Next, we multiply the square of 2 (the first figure of the cube root) by 300 and set down the product 1200 ; this is the trial divisor. Dividing 5824 by this, the quotient is 4 5 this *s the second figure of the root. Now we multiply the fir$t figure of the cube root by 30 and this product by the second figure of the root, and set down C, A, II l62 ARITHMETIC the result under the trial divisor ; then set down under this the square of the second figure of the cube root. Adding these three we get 1456 as our divisor. We then multiply this by the second figure of the root and subtracting the product from 5824 we find that there is no remainder. Therefore we conclude that 24 is the cube root of 13824. If the cube root contains three or more figures the above process must be repeated. Example 2. Find the cube root of 33076161. Process : 33076161 ( 321 27 Arts. 3 x 30x2 32^x300 32 2700 1 80 4 6076 5768 2884 307200 960 I 308161 308161 308161 Note. Remarks of Arts. 172, 173 and 174 with regard to the process of extraction of the square root apply equally to the process of extraction of the cube root. EXAMPLES. 111. Find the cube root of 1. 1331. 2. 15625. 3. 46656. 4. 110592. 6. 117649. 6. 373248. 7. 2197. 8. 185193. 0. 704969. 10. 912673. 11. 15069223. 12. 105823817. 13. 843908625. 14. 873722816. 15. 219365327791. 16. 167284151. 17. 731189187729. 18. 10970645048. 19, 93162981941037* 20. 1371742108367626890260631. 181. A decimal fraction (in its simplest form) to be a perfect cube must have 3, 6, 9,... decimal places ; that is, the number of decimal places in it must be some multiple of 3. If the number of decimal places be not a multiple of 3, the cube root can be obtained to any number of decimal places we like. In extracting the cube root of a decimal, the number of decimal places must be \ade a multiple of 3 by annexing ciphers, if necessary. CUBE ROOT ' 163 The cube root of a vulgar fraction is the cube root of its numer- ator divided by the cube root of its denominator. EXAMPLES, lia. Find the cube root of 1. 17*576. 2. 132-651. 3. '493039- 4. 64481*201, 5. 18-609625, 6. -007645373. 7. '876467493. 8. -001030301. 9. j&V 1* izWffff- n. 49 A- 12- 755 8 $if- 13. -637. 14. 1587-962. 15. 3845^96. 1. 46AV 17. 2o|J. 18. 2-370. Find to three places of decimals the cube root of 19- 3'539- 20. 11. 21. 24. 22. 7-52. 23. '8. 24. -27. 25. -&. 26. |. 27. '0047. 28. 5$. 188. When at kast one more than ,&//" the number of figures In the cube root of a number has been found by the ordinary method, the remaining figures of the root may be found by division only. Note. In this case we take for the divisor 300 times the square of the part of the cube root already obtained and proceed exactly as in Art. 178. EXAMPLES. 113. Obtain to 6 places of decimals the cube root of 1. 3*539. 2. 24. 3. 7-52. 4. -002. 5. -003. 6. 1 8 T V 183. The fourth root of a number is found by taking the square root of the square root of the number. The sixth root of a number is found by taking the cube root of the square root of the number. The ninth root of a number is found by taking the cube root of the cube root of the number. EXAMPLES. 114. Find the fourth root of L 256. 2. 234256. 3. 1679616. 4, 1575-2961. I6 4 ARITHMETIC Find the sixth root of 5. 53I44I. '6. 308-915776, Find the ninth root of 8. 262144. 0. 1953125. 7. 24794911296, 10. 3000. XXXII. MEASUREMENT OF AREA. 184. In Arithmetic we consider the areas of rectangles only. Example. The floor, the ceiling and each wall of an ordinary room ; a sheet of paper ; each side of an ordinary box or brick ; all these are rectangular surfaces. The length and breadth of a rectangle are called its dimensions. 185. The unit Of area is a square whose side is the unit of length. Area or Surface is measured by the number of units of area which it contains ; just as a length is measured by the number of units of length which it contains. 180. To find the area of a rectangle. L& A BCD be a rectangle, of which the length AB is i yd. 2 ft., and the breadth A D is 3 ft. Then, if the unit of length be a foot, the measure of AB is 5 and of AD is 3. Divide AB and AD into 5 and 3 equal parts respectively, and through the points of division draw lines parallel to AD y AB respectively. Then the rectangle A BCD is divided into 5x3 equal squares, the side of each of which is a foot in length. Now, each of these squares is the unit of area ; therefore the measure of the area ABCD (which is the same as the number of these squares) is 5 x 3 or 1 5. .'. Area of ABCD**i$ sq. ft. And generally, is any rectangle, measure of area measure of length x measure of breadth ;, or, more briefly, area length x breadth. Whence, length area -5- breadth ; breadth area ~ length. MEASUREMENT OF AREA 165 Note. A square foot is a square whose side is a foot. Note the difference between "3 square feet" and "3 feet square." Three square feet denotes an area 3 times as large as a square foot ; three feet square denotes the area of a square whose side is 3 feet. Example I. Find the area of the floor of a room 10 ft. 6 in. long and 6 ft. 4 in. broad. Length of room *ioft. ; breadth =6J ft. ; /. area io x 6 \ sq. ft, = x sq.ft. *f * sq. ft. =66 sq. ft. 72 sq. in. Example 2. A rectangular court, 24 yards long and 16 yards broad} has within it a path of uniform breadth of 2 yards running round it ; find the area of the path. Area of court 24 x 16 sq. yd. 384sq. yd. The path takes off (2 + 2) yd. from the length and (2 + 2) yd. from the breadth ; /. length of inner court 20 yd., and breadth ... 12 yd. ; area 20 x 12 sq. yd. 240 sq. yd. .*. area of path -(384 -240) sq. yd. ~i44sq. yd. or thus : Length of the patk (24x2+ 12x2) yd. -72 yd. ; % the area of the path 72 x 2 sq. yd. -I44sq,yd. Example 3. Find the breadth of a courtyard 41 sq. ft. 80 sq. in. ta area, and 7 ft. 4 in. in length. 166 ARITHMETIC ?A) sq.ft. ~4i* sq. ft. ; length -7 J ft. /. breadth^ ft.-3Z4x 2 3 2 ft, 5 | ft. 5 ft. 8 in. Example 4. How many paving stones, each 2 ft. 8 in. long and 1 7 in. wide, will cover the courtyard in Ex. 3 ? Area of court * a *4i{ sq. ft. ; area of a stone ==2! x \\ sq. ft %* sq. ft. ; /. number of stones rqd. ^^r ^ x - n. Example 5. Find the cost of matting the room in Ex. i, at 3 annas per sq. ft. The cost may be found by Practice or by Compound Multi- plication. EXAMPLES. 115. Find the area of the rectangles having the following dimen- sions : L 15 ft. by 12 ft. 2. 20 ft. by 16 ft. 3. 13 ft. 6 in. by 8 ft. 8 in. 4. 9 ft. 10 in. by 6 ft. 7 in. 6. 10 ft. 7$ in. by 7 ft. 44 in. 6. 9 yd. 2 ft. by 7 yd. I ft. Find the breadth of a room whose 7. area =363 sq. ft, and length -33 ft. 8. area 6 sq. ft. 60 sq. in., and length =2 ft. 9 in. 9. area 5 ac. I ro. 36 po., and length = 267 yd. 2 ft. 10. area =94 sq. yd. 8 ft. 84 in., and length =32 yd. I ft. 8 In. 1L Find the area of a square field whose side is 32 ft. 8 in. 12. Find the area of a square room whose side is 3 yd. 2 ft. 3 in 13. How many paving stones, each \\ ft. by 9 in., would be required to pave a square courtyard whose side is 21 ft, ? 14. How many pieces of carpet, each 5 ft. long and 3 ft wide, will cover the floor of a room 20 ft. by 13 ft 6 in. ? 15. Find the cost of carpeting a room, 10 ft. 6 in. by 6 ft. 6 in,, at R2 per sq. ft MEASUREMENT OF AREA l6j 16. Find the cost of polishing a marble slab, 3 ft. 3 in. by 2 ft 6 in., at 2</. per sq. in. 17. A room, 20 ft. long, 16 ft. broad, has a stained border all round it 2 ft. wide ; what i s the area of the stained part ? 18. A rectangular piece of ground is 88 yards long and con- tains an acre ; it consists of a walk 6 ft. wide surrounding a grass* plot': find the area of the walk. 19. How many stone slabs, 3 ft. long, i ft. wide, are requisite for paving a path which encloses a rectangular garden half a mile long and quarter of a mile wide, the path being 6 ft. wide ? 20. A gravel path 5 ft. wide runs round a rectangular garden^ 100 yd. by 75 yd. ; find the cost of making it at 40. 6^. per sq. yd. 21. How many sq. yards of matting will be wanted to cover a room 31 ft. 6 in. by 22 ft. 6 in. ? What will be the cost at 4*/. per sq. yd. ? 22. If 1200 stones, each 2 feet square, will pave a court, find the area of the court. 23. The cost of varnishing the floor of a room, 24 ft. long, at 2s. 6d. per sq. yd., is $ ; find the breadth of the room. 24. A garden roller is 3 ft. 3 in. wide, and its circumference is 6 ft. 9 in. ; how many sq. ft. of ground does it pass over in one complete revolution ? 25. A sheet of paper is 20 in. long and 18 in. wide ; by how much must the width be narrowed to leave a surface of 2j sq. ft. ? 26. What length must be cut off a plank which is $J in. broad, that the area may be a sq. foot ? 27. A factory has 100 windows, 60 of which severally contain 8 panes, each 9 in. by 6 in. ; and the remainder severally contain 10 panes, each 2 ft. square ; find the cost of glazing the whole at 10 annas per sq. ft. 28. What must be the length of a piece of land, 1 5 yards wide* that can be exchanged for a piece of the same quality, measuring 20 yards each way ? 29. Find the area of the square which has the same perimeter as a rectangle whose length is 48 ft. and is 3 times its breadth. 30. How many flag-stones, each 576 ft. long and 4*15 ft. wide, are requisite for paving a cloister, which encloses a rectangular court, 4577 yd- long and 41*93 yd. wide, the cloister being 12*45 ft- w "k ? 31. A room measuring 42 ft. 6 in. by 22 ft. 9 in. inside, with walls 2 ft. 3 in. thick, is surrounded by a verandah 10 ft. 6 in. wide. Find the cost of paving this verandah with tiles measuring 4$ in, by 3 in., and costing 6 pie$ each. l6S ARITHMETIC 1ST. Example i. Find the length of the side of a square which contains 91 sq. ft. 121 sq. in. Area9i sq. ft. 121 sq. in. 13225 sq. in. ; /. length of side ^1322^ in. == 1 1 5 in. 9 ft. 7 in. Example 2. Find the diagonal of a rectangular field, 16 yd. long and 12 yd. wide. By Euclid I. 47, the diagonal ^16* + 12"* yd. *J 2 564- 144 yd. ^400 yd. = 20 yd. Example 3. The area of a room, which is twice as long as it Is broad is 26 sq. yd. 8 sq. ft. ; how long is it ? The room can be divided into two equal squares whose side is equal to the breadth of the room. Area of each square = 13 sq. yd. 4 sq. ft. I2I sq. ft. ; side of each square ^121 ft.ii ft. ; breadth of room = 1 1 ft * 3 yd. 2 ft. ; length of room = 7 yd. I ft. EXAMPLES. 116. 1. The area of a square field is 10 acres ; find the length of its side. 2. The area of a square room is 502 sq. ft. 73 sq. in. ; find the length of each side? 3. How many yards of fencing are required to enclose a square gardeft containing 4 ro. i po. 29 yd. f ft. ? 4. A rectangular field is 40 yards long and 30 yards broad ; find the distance from corner to corner. 5. What is the lenght of the diagonal of a square whose side is 4 yards ? 8. The area of a square is 900 sq. ft. ; what is the length of its diagonal ? 7. The area of the floor of a room is 162 sq. ft. ; its length is twice its breadth find its length. 8. Find the length of a rectangular field which is 3 times as long as it is broad and which contains 768 sq. yd. CARPETING AND PAPERING ifiQ 0. A room is half as long again as it is broad and its area Is -69*36 sq. yd. ; find its perimeter. 10. The sides of two squares contajn 77 yd. I ft. 9 in. and 7 yd. 2 ft. 4 in. respectively ; find the side of a square whose area is equal to the sum of the areas of the two squares. 188. Carpeting the floor and papering the walls o! -a room, Example i. Find the length of carpet i\ ft. wide, required for a room 28 ft. long, 20 ft. broad. The carpet which will cover the floor of a room has the same area as the floor. Area of floor 28 x 20 sq. ft. ; . e , 28 x 20 f 28 x 20 x 3 - . . length of carpet reqd.= -. ft. = ft. = 240 ft. = 80 yd. Example 2. Find the area of the four walls of a rectangular *room 20 ft. long, 15 ft. broad and 10 ft. high. The area of the four walls of a rectangular room is obtained by multiplying the circuit (*'.*., twice the sum of lerigth and breadth) of the room by the height of the room. The circuit (20 + 15) x 2 ft. =-70 ft. ; /. the area of walls 70 x 10 sq. ft. 700 sq. ft. To find the length of paper required to cover the walls, proceed as in the preceding example. Note. In estimating the length of paper required, deductions ^or doors, windows and fireplaces must be made. N. B. The cost of carpet or paper may be fotind by Practice or by Compound Multiplication. EXAMPLES. 117. Find the length of carpet required for rooms having the follow- Ing dimensions : L Room, 25 ft. long, 18 ft. broad ; carpet, 2 ft. 6 in. wide. 2. Room, 20 ft. long, 12 ft. 6 in. broad ; carpet, 27 in. wide. 3. Room, 30 j ft. long, 2oJ ft, broad ; carpet, 42 in. wide. Find the expense of carpeting a room, 4. 16 ft. by 10 ft., with carpet 3 ft. wide, at &2. 80. a yard I/O ARITHMETIC 6. 30 ft. 9 in. by 25 ft., with carpet 30 in. wide, at 4*. 6dft ft yard. Find the area of the walls of the following rectangular rooms : 6. Length 20 ft., breadth 16 ft, height 9 ft. 7. Length 15 ft. 6 in., breadth 12 ft., height 9 ft. 8. Length 21 ft. 7 in., breadth 16 ft. 5 in., height 3^ yd. Find the length of wall paper required for the following rooms : 9. 25 ft. long, 20 ft. wide, 12 ft. high ; paper 15 in. wide. 10. 14 ft. long, 10 ft. wide, 7 ft. high ; paper 14 in. wide. 11. 27 ft. long, 1 8 ft. wide, 10 ft. high ; with paper 16 in. wide, allowing for 2 doors each 7 ft. by 4 ft. 12. 28 ft. long, 20 ft. broad, 9j ft. high ; with paper 20 in. wide,, allowing for a door 6 ft. by 3^ ft. and a window 3 ft. by i\ ft. Find the expense of papering rooms whose dimensions are : 13. Length 21 ft., breadth 16 ft., height 10 ft. ; with paper 16 in. wide, at 40. a yard. 14. Length 50 ft, breadth 35 ft, height 15 ft. ; with paper 1 5 in. wide, at b<t. a yard. 15. Length 18 ft, breadth 16 ft., height 9 ft. ; with paper iq in. wide, at 9^. a yard, allowing for 3 doors each 6 ft. by 3$ ft, 1 windows 4 ft by 2j ft, and a fireplace 6 ft by 4 ft. 6 in. 16. How man) yards will remain out of 300 yards of matting 2 ft. 6 in. wide, after covering two floors, each 25 ft. 6 in. by 21 ft ? 17. A square room whose floor measures 56 sq. yd. 2 sq. ft. 36 sq. in., is loft. 4 in. high ; find the expense of whitewashing its ceiling and walls at 2p. per sq. yd. 18. The cost of covering the floor of a room, \2\ yd. by 8j yd., with carpet 2^ ft. wide, is ^30 . 14 . 7^ ; find the price of carpet per yard. 19. It costs 2. 5*. to paper a room 10 yd. long and 8 yd. wide, with paper i ft. wide, at 3^. per yard ; find the height of the room. 20. The cost of carpeting a room i6 ft long and I2j ft, broad) with carpet at 6s. per yard, is ^14. 17^. ; find the width of the carpet. 2L If a postage stamp be of an inch long and { of an inch broad, what will be the cost of covering the walls of a room which Is 15 ft long, 12 ft wide and 9ft high, with postage stamps, 6 pies each ? 22. What will be the cost of papering a room, 24 ft. long by to ft, broad and 8 ft. high, which has 2 doors each 7 ft. by 4 ft., LAND MEASUREMENT OF BENGAL 171 with paper 2 ft. wide, at 84 a piece ; the cost of putting it on being 4. per piece, and each piece being 4 yards long ? 23. The matting of a room, 3 times as long as broad, at 4 annas per sq. ft. cost R75 ; and the painting of the walls at 2 annas per sq. yd. cost R6. 6a. 2$^. ; what is the height of the room ? 24. Find the expense of lining a cistern 10 ft. long, 8 ft. broad and 3 ft. deep, with lead at Rio per cwt., which weighs 5 Ib. per sq. ft. 25. Find the cost of papering a room, 18 ft. long, 12 ft. broad and 10 ft. high, with paper 32 in. wide, at 6 annas a yard, allowing for a door 7 ft. by 4 ft., 3 windows each 4 ft. by 3 ft. and a panel- ling 2 ft, high round the floor. 26. A box with a lid is to be made of plank, one inch thick ; the external dimensions are to be 18 in., 12 in , and 9 in. : how many sq. ft. of plank will be required ? 27. The length of a room is 32$ ft. The cost of papering the walls at Ri. 140. per sq. yd. is R3o8. 2a. ; and the cost of car- peting the floor at R2. 40. per sq. yd. is Ri5o. 5*1. Find the height and width of the room. 28. Find the cost of whitewashing the ceiling and the inner and outer sides of the walls of a room, 20 ft. long, 12 ft. wide and I J ft. high, at I pie per sq. ft. ; the wails being i\ ft. thick and 3 ft. higher at the outside. LAND MEASUREMENT OF BENGAL, 189. If we have to find the area of a rectangular piece of land say, 14 bi. 3 cot. by 9 bi. 2 cot., we might proceed thus : Area *= 1 4s 8 a x 9^1 bi. (superficial) 128 J$$ bi.i28 bi. 15 cot. 4 ch. 16 ga. But such examples are usually worked by the following rule : Bigha multiplied by bigha gives bigha. Bigha cottah cottah. Cottah cottah dhooL Twenty dhools make a cottah. The truth of the rule will appear from the following considera- tions : I bi. xi bi. I bi. (superficial) ; I bi. xi cot. == I x^j bi. s\j bi. i cot. (superficial) ; I cot. x I cot. "^f x Jb bi. 5\j cot. I dhool. 1/2 ARITHMETIC By this method the above example will be worked thus : We multiply all the bi. cot. terms of the 1st line 14 3 (beginning with the 9 2 lowest) by all the terms 127 .7 (i4 bi. 3 cot.) x 9 bi. t>f the 2nd line (begin- _JL_?_'_6*-(i4bi. 3 cot.)x2 cot. ing with the highest). 128 .15. 6^(14 bi. 3 cot.)x(9 bi. 2 cot.) .", Area 128 bi. 15 cot. 6 dhools = 128 bi. I5$y cot. I28 bi. 15 cot. 4 ch. 16 ga. EXAMPLES. US. Find the area of the following rectangular fields : 1. 4 bi. by 3 bi. 2. 10 bi. 10 cot. by 5 bl. 3. 12 bi. 15 cot. by 8 bi. 10 cot. 4. 14 bi. 8 cot. by 14 bi. 8 cot. 6. 24 bi. 8 cot. by 14 bi. 13 cot. 6. 57 bi. 5 cot. by 42 bi. 8 cot. 7. 99 bi. 19 cot. by 49 bi. 19 cot. 8. 115 bi. 14 cot. by 105 bi. 7 cot. 9. 8 bi. by 3^ bi. 10. lof bi. by 15 cot. 11. 252 cubits by 164 cubits. 12. 408 cubits by 308 cubits. XXXIII. MEASUREMENT OF SOLIDITY. 190. In Arithmetic we consider the volumes of rectangular 0lids only. Example. A rectangular box, a brick, are rectangular solids. The length, breadth and thickness (or height or depth) of ft rectangular solid are called its dimensions. 191. The unit of volume is a cube each of venose edges Is the unit of length. Volume or cubic content is measured by the number of units of volume which it contains. 19$. To find the volume of a rectangular wlid or rectangular favallelopipid. MEASUREMENT OF SOLIDITY 17$ Let the annexed figure represent a rect- angular parallelepiped, of which the length AB is 4 ft., breadth BC is 3 ft. and thickness AD is 2 ft. Divide AB BC, AD respectively into 4, 3 and 2 equal parts, and through the points of division draw planes parallel to the sides. Then the solid X / B will be divided into a number of equal blocks, each of which is a cubic foot ; and since there are two layers, in each of which there are 4x3 blocks, we see that there are 4x3x2 blocks altogether} and the solid therefore contains 4x3x2 cubic feet. .*, The volume of the solid 4 x 3x2 cu. ft. And generally, in any rectangular solid. The measure of volume measure of length x measure of breadth x measure of thickness. Or, more briefly, Volume length x breadth x thickness. Whence, thickness volume -r (length x breadth) : etc. Example i. Find the cubic content of a rectangu" \\ block of marble whose dimensions are 3 ft. 2 in., 2 ft. 3 in. and I ft. 6 in, Volume 3jx2jx if cu. ft.ioiJ cu. ft. Example 2. How many bricks will be required to build a wall 20 ft. long, 10 ft. high and 2 ft. thick ; each brick with its share of the mortar being 6 in. long, 3 in. wide and 2 in. deep ? 20 x 10 x 2 -'9*oo. . , . . volume of the wall Number of br.cks- ^- Example 3. A rectangular cistern is 6 ft. long and 4 ft. broad $, what is the depth of water in it, when it contains 72 cubic feet or water ? _ , volume of water 72 f . ^ f . De P th areaofthebase6^ ft - =3 ft " Example 4. A box with a lid is to be made of half-an-incfe. plank ; its internal dimensions are to be 20 in., 15 in. and 9 In,. How many cu. in. of wood will be required ? J 74 ARITHMETIC The external dimensions of the box are 21 in., 16 in. and lo in. ; .'. its external volume 21 x i6x 10 cu. in.==336o cu. in. ; and Jts internal volume 20 x 15 X9 cu. in. = 2700 cu. in. .". Volume of wood required for the box =(3360 -2700) cu. in. =* 660 cu. in. We may obtain the area of the plank required by dividing the volume of the wood by the thickness of the plank. EXAMPLES. 119. Find the cubic contents of the rectangular solids having the following dimensions : 1. 10 ft., 8 ft., 5 ft. 2. 7j ft., Si ft., 4? ft- 3. 3 yd., 7 ft., 30 in. 4. 5 ft. 10 in., 3 ft., 6 in. 6. 7 yd. 2 ft. 9 in., 6 yd I ft. 3 in., 10 ft. 10 in. 6. Find the cubic content of a cube whose edge is 3$ ft. 7. How many pounds of water will fill a cistern 2 yd. long, 8 ft. broad and 9 in. deep, having given that a cu. ft. of water weighs 1000 oz. ? 8. How many bricks, each 9 in. by 6 in. by 4 in., are required for a wall 22 yd. long, 8 ft. high and 2 ft. 6 in. thick, leaving in it a doorway 6 ft. by 4 ft. 9. How many times can a bucket, holding 2 cu. ft. of water f be filled from a tank 30 ft. long, 25 ft. wide and 10 ft. deep ? 10. In what time will a cistern 16 ft. by 12 ft. by 10 ft., be filled by a pipe which discharges 40 cu. ft. of water per minute ? 11. How many sheets, each 4 ft. long, 2 ft. broad and J of an Inch thick, can be made from 4 cu. ft. of iron ? ( 12. Find the total weight of 27 sheets of copper, each 6ft. long, 4 ft. broad and J of an inch thick, a cubic foot of copper weighing 2 cwt. 13. How many times can a pint-bottle be filled from a cistern 138*637 in. by 70 in. by 10 in., having given that a gallon contains 277*274 cubic inches ? 14. A cu. inch of gold is hammered into a plate 6 in. square ; find the thickness of the plate as the decimal of an inch. 15. Water is flowing into a reservoir which is 5 ft. square ; how many cu. ft. of water will have flown in when the depth of water is 2\ ft. ? 18. A cistern, 12 ft. long and 8 ft. 6 in. broad, contains water ; how many ou. ft. of water must be drawn off to make the surface sink half an inch ? MEASUREMENT OF SOLIDITY 175 17. A room, 40 ft. io in. by 25 ft. \8 in., accommodates 100 persons ; what must be the height of the room if each person has 75if&cu.ft.ofair? 18. What length must be cut off a rectangular marble slab, i$ ft. broad and 8 in. thick, in order that it may contain 2 cu. ft. ? 19. Find the cost of digging a canal I mile long, 6 ft. wide and 5 ft. deep, at 4 annas per cu. yd. 20. A lake, whose area is 30 acres, is covered with ice 6 inches thick ; find the weight of the ice in tons, if a cubic foot of ice weigh 900 oz. avoir. 21. There are 1530 cu. ft. of air in a room 9 ft. high ; find the cost of carpeting it at Ri per sq. ft. 22. A square room, 10 ft. high, contains 4000 cu. ft. of air ; how many yards of paper, 2 ft. wide, will be required for covering its walls ? 23. A solid stack, 41 ft. 8 in. by 16 ft. 8 in. by 14 ft. 7 in., contains 125000 bricks, each 10 in. long and 3^ in. thick ; find the width of each brick. 24. A piece of ground is 100 yd. long and 75 yd. wide. To what uniform depth must it be excavated that the earth taken out may form an embankment of 25000 cubic yards, supposing the earth to be increased one-ninth in volume by removal ? 25. A box (with cover) is made of an-inch-and-a-half plank ; Its external dimensions are 4 ft., 3 ft. 6 in. and 2 ft. 3 in. : find the weight of the box, supposing a cu. ft. of the wood to weigh 36 Ib. 26. The roof of a verandah is supported by 16 teak beams, each 9 ft. long, 3 in. broad and 5 in. deep. If the weight of a cubic inch of teak is ^f of that of a cubic inch of water, and if a cubic foot of water weighs 1000 oz., find the weight in Ibs. of the timber in the verandah. 27. A crow wishing to quench its thirst came to a vessel which contained 28 cu. in. of water. The crow being unable to reach the water, picked up several small stones, each three quarters of a cubic inch in size, and let them drop into the vessel until the water came to the top of the vessel. If the size of the vessel was - such that it would exactly hold 73 cubic inches of water, find the number of stones dropped in by the crow. 28. The top of a tank is a rectangle whose sides are 15 ft. and 9 ft. ; it is of the same horizontal section throughout its depth. What must be its depth in order that it may contain 12960 gallons of water, one gallon cdtataining 277*274 cubic inches ? 29. A moat is to be dug all round a rectangular fort, 200 yd. long and 150 yd. broad ; it is to have vertical sides and to be 27 ft. 176 ARITHMETIC wide and 10 ft. deep throughout. Find the cost of digging it at 4 annas per cubic yard. 3O. A room, 21 ft. long by 13^ ft. wide, is surrounded by walls- l$ ft. thick and 14 ft. high. There are two doors each 4^ ft. by 6 ft, and one window 3 ft. by 4^ ft. Find (i) the cost of building the walls at the rate of 85. la. per cubic yard, and (ii) the number of bricks, each measuring 9 in. by 4 in. by 2^ in., required for the work. XXXIV. DUODECIMALS. 103. Duodecimals or Cross Multiplication is a method (similar to that of Art. 189) of finding areas and volumes, made use of by painters, bricklayers, etc., in measuring work. In duodecimals, the successive linear units are named and counted as follows : I foot 12 primes; I prime = 12 seconds ; i second = 12 thirds ; etc Note. A prime an inch. A second is often called a part. The successive superficial and solid units are named and counted' exactly in the same way as the linear units : Thus, I superficial foot 12 superficial primes; I supl. prime 12 supU seconds ; etc. I solid foot =12 solid primes ; I solid prime ~ 12 solid seconds ; etc* Primes, seconds, thirds, etc., are indicated by the accent (% ("), ('"), etc., respectively. The whole of the above statement may be briefly put thus : i linear foot \ i square foot f =i2' = i44"=*i728 / "20736 lir etc, i cubic foot J 194. We can easily convert quantities expressed in duodeci mals- to those expressed in feet and inches, and conversely; remembering that in linear measure the inch is the same as the prime, in square- measure, as the second) and in cubic measure, as the third. Example I. 2 ft, 3'. 4" 2 ft. 3'A=2 ft. 3$ in. Example 2. 3 sq. ft. 2'. 4". 3'"3 sq, ft. 28"^ =3 sq, ft 28J 1*. Example 3. 7 cu. ft. i'. 2". 5'". 6 iv =7 cu. ft. I73'Y* 7 en. ft. 1734 in - Conversely, Example 4. 4 yd. 3 ft. 2j in. -15 ft, 2'J- 15 ft. 2'. 4". DUODECIMALS 177 Example 5. 2 sq. ft. 19} in. 2 sq. ft. 19"! 2 sq. ft, i'. 7". 8'". Example 6. n cu. ft. loooj in.-u cu. ft. iooo"'J -ii cu. ft. 83". 4"'iii cu. ft. 6 r . II". 4'". 3*. EXAMPLES. Express in yards, feet and inches : 1. 12 ft. 7'. 5". 2. 20 ft. 8'. 3". 9"'. 3. 13 sq. ft. 9'. 3". 4. 22 sq. ft. 3'. 4". 8'". 6. 40 sq. ft. i'. o". 3'". 6. 2 sq. ft. 2'. 2". 2'". 2 lf . 7. 30 cu. ft. 3'. 4". 8. 74 cu. ft. 7'. 3". 4'". 9. 10 cu. ft. 2'. i". o'". 4*. 10. 3 cu. ft. 3'. 3". 3"'. 3**. 3*. Express in duodecimals : 11. 2 yd. 2 ft. 7 in. 12. II yd. i ft. 7$ in. 13. 8ft. i if in. 14. 10 ft. 9 Jin. 15. 6 sq. yd. 2 ft. 71 J in. 16. 7 sq. yd. 7 ft. 6o| in. 17. 2 cu. yd. 8 ft. 1 50$ in. 18 i cu. yd. I ft. 240$ in. 195. The following statements can be proved as in Art. 189* Feet into primes give (supl.) primes ; >? seconds seconds ; thirds thirds ; etc. Primes primes seconds ; seconds thirds ; etc. Seconds seconds fourths ; thirds fifths ; etc. Also (Supl.) feet into primes give (solid) primes ; ,t seconds seconds ; etc. primes primes seconds ; seconds thirds ; etc. Example *. Find the area of a rectangle 7 ft. 8 in. by 6 ft. 7 in. We multiply all the terms ft. ' f the multiplicand (commenc- 7 . 8 ing with the lowest) by all the 6 . 7 terms of the multiplier (com- 46 . o =(7 ft. 8') x 6 ft. mencing with the highest). 4.5. 8(7 ft - 8/ ) x ?' 50 . 5 . 8-(7ft.8')x(6ft7'X Area =50 sq. ft, 5'. 8" =50 sq. ft. 68" -50 sq. ft. 68 in. C. A. 12. I?8 ARITHMETIC Example 2. Find the capacity of a cubical vessel whose edge (s 2 ft 3 in. ft. ' * 3 (2ft 3')*2ft (2ft 3') x 3'. (2 ft 3') X (2 ft 3'). (5 sq. ft o' . 9")X2 ft (Ssq.fto' . 9") x (2 ft. 3'). 2 4~ . 3 6 6 . 9 5 2 . 3 9 10 I I . 3 6 2 . 3 II 4 - 8 . 3 675 Capac tyii cu. ft 4' - 8" . 3'"ii cu. ft. 675"'=:ii cu. ft EXAMPLES. 11. Find by Cross Multiplication the areas of the following rectangles : 2, 8 ft. 9 in. by 7 ft 8 in. 4. 16 ft. ii in. by 12 ft 10 in. 6, 1. 3 ft 4 in. by 2 ft. 3 in. 8. 12 ft 9 in. by to ft. 5 in. 6. 20 ft 7iin. by 15 ft. 4 in. 7. 13 ft. 8J in. by 7 ft i\ in. 9. 24 ft 65 in. by 9 ft. 3t in. 40 ft. 6 in. by 3 ft. 2j in. 8. 12 ft. 9! in. by 10 ft. 2| in. 10. 120 ft. 3j in. by 20 ft 5$ in, Find the volumes of the following rectangular solids : 11. 4 ft. 7 in. by 3 ft. 9 in. by 2 ft. 3 in. 12. 6 ft. 8 in. by 5 ft. 7 in. by 3 ft 5 in. 13. 10 ft. 8{ in. by 9 ft. 6 in. by 8 ft. 7 in. 14. 12 ft. 3$ in. by 7 ft. 4j in. by 5 ft. 2\ in, 15. 20 ft. 7$ in. by 15 ft 8f in. by 10 ft 2g in. N* B. For additional examples, see the two preceding sections. XXXV. PROBLEMS AND THE UNITARY MfeTHOD. 196. When the value, weight or length, etc., of any numbet of units is given, wfe can, by Compound Division, obtain the value f weight or length, etc., of one of the units. And when the value, weight or length, etc., of one unit is given, we can, by Compound Multiplication^ obtain the value, weight or length, etc,, of any number of units of the same kind. PROBLEMS AND THE|; UNITARY METHOD 179 I The solution by the application o f the two above principles is called the Unitary Method lor the Method of Reduction to the Unit The method will >e fully explained by the follow- ing examples. 197. Example i. If 9 articles cost 836, what is the cost of I article ? ' The cost of 9 articles -36, .* ............... i article R4. Ans. Example 2. If i Ib. of tea cosits 2J. 6^., what will 8 Ib. cost ? The cost of i Ibj. =*2s. 6d., ;i. Ans. ( EXAMPLES. 12. 1. If 7 articles cost R2. ioa. y what is the cost of i article ? 2 If 12 maunds of wheat cost 30, what will i maund cost ? 3. If 7$ yards of cloth cost RI. 14^., how much will I yd. cost ? 4. If the weight of 16 equal bags of rice be 40 maunds, what is the weight of I bag ? 6. If the length of a piece; o f cloth worth 18*. be 12 yards, what is the length of a piece of the same cloth worth i s. ? 6. If the rent of 13 acres of land is ,4. 17*., what is the rent of i acre ? , 7. If the income-tax on R2Co be RS . 3 , 4, what is the tax on Ri ? 8. If i chair costs &2. 12*., h| w much will 13 chairs cost ? 9. If i Ib. of sugar costs 7^., jwhat will 10 Ib. cost ? 10. If i bullock can plough $ bighas in a day, how many bighas can 11 bullocks plough in i a day ? 11. If a man walk 3J miles in I hour, how far does he walk In 9j hours ? ; 12. A servant's wages being 7.*. 6</. per week, how much ought he to receive for 7 weeks ? 13. If the railway fare for i mile is 2j^., what is the fare for 24 miles ? 14. If the carriage of i maund' for 150 miles cost &2, what will be the cost of the carriage of 10} \naunds for the same distance ? ISO ARITHMETIC Example 3. If 5 men can dc> a piece of work in 3 days, how long will it take i man to do it ? 5 men can do th & work in 3 days, /. i man (3x5) days, ^ *>., 15 days. Arts. p Example 4. If I man can do ^a piece of work in 21 days f in how many days can 3 men do it ? I man can do tr^e work in 21 days, .'. 3 men ,5 - days, 5 *.*., 7 days. Ans. f Note. In questions such as I the two above, it should be noticed that to an increase in the nur ^ber of workmen corresponds a diminution in the number of c) lays, and vice versa. 1. If 10 men can do a piece of work in 3 days, how long will it take one man to do it ? 2. If 12 men finish a piece o days could one man finish it ? 8. If 3 maunds of rice last 9 they last i person ? 4. If 7 cwt. can be carried ii be carried for the same sum ? 6. If 13 acres can be rented for how many months can i acre f work in 5 days, in how many persons 30 days, how long would DO miles for 3*., how far can i cwt. for 7 months for a certain be rented for the same sum ? 6. If i man can do a piece y>f work in 40^ days, how long will It take 9 men to do it ? ? 7. If 30 bushels feed 28 horses for a week, how many horses would they keep for 4 weeks ? " 8. If i man reap a field in i 8 days, how long will 4 men be doing it ? 0. A ship performs a voyagO in 55 days, sailing i knot an hour, how many days would she take to perform the same voyage sailing 5 knots an hour ? 10. If the carriages of 56 maipads for i mile cost a certain sum, how much will be carried 14 miUps for the same money ? 11. If 18 horses plough a fiel4^ n *5 days, how many horses will plough it in i day ? PROBLEMS AND THE UNITARY METHOD l8l 12. If 1 8 horses plough a field in 15 days, in how many days will i horse plough it ? 13. If i horse can be kept 8 days for R2. 80., for how many days can 4 horses be kept for the same sum ? 198. Each of the above questions requires either multiplica- tion or division for its solution. In the following questions the two processes are combined. Example i. If 3 yards of cloth cost 84. 8a., what will be the cost of 35 yards ? The cost of 3 yards = ^4. 8., .* i yard =R4- Sa.xfc, / 35 yards-R4. 8a.x4*, = R52. 80. Ans. In multiplying by 35 the method of multiplication by factors should be adopted. Example 2. How much must be paid for 17 maunds of sugar, when 8 maunds cost 74 ? The cost of 8 maunds ="^74, / I maund =74x4, ." 9 maunds = 74 x f , -83. 4*. ; ." 17 maunds Ri57. 40. (by addition). Here we avoid the multiplication by i*j which cannot befactonsed. Example 3. If 6 maunds of wheat cost 7. 8#., how much can be purchased for Ri2. 8a. ? 87. 8a. Ri2. 8a. I2oa. is the cost of 6 maunds, /. 4* 2 , /. 2000. 10 Ans. The artifice employed in this example should be carefully noted. We use here 400. as the unit common to 1200. and 2ooa. Example 4. If } of an estate be worth Rox>, what is the value of i of it? } of the estate is worth RQO, /. the estate is worth RQO x |, .*. } of the estate is worth RQO x J x J or R8o. Ans. 1 82 ARITHMETIC Example 5, Express i mile in metres, 32 metres being equal to 35 yards. 35 yards 32 metres, 5 yards ***ty metres, 1760 yards *=*&&$&& metres or 1609$ metres. EXAMPLES. 124. 1. If 30 bullocks cost B8io, what is the cost of 77 bullocks ? 2. If 5 cwt. cost R6. 4^., what is the cost of 16 cwt. ? 3. Find the value of 21 yd. of cloth when 44 yd. cost R33 4. If 7 pieces of cloth cost 350, what will 13 pieces cost ? 5. If 13 reams of paper cost 6. IO.T., what is the price of 21 reams ? 6. If 23 copies of a book cost 35. 150., how much will 31 copies cost ? 7. If the cost of 60 eggs be is. 3</., how many can be pur* chased for 55. ? 8. How many oranges can be bought for &2. 30. at the rate of 8a. 9^. a dozen ? 9. If 4 cwt. cost i. is. id.) what will 2 tons 8 cwt. cost ? 10. If 35 sheep produce 20 Ib. of wool, what would 63 sheep produce ? 11. If 42 men earn 83. 4. 6 for a day's work, what would na men earn ? 12. If the railway fare for 100 miles be &3. 8. 6, what is the fare for 275 miles ? 13. If 8 persons can be boarded for ^3, how many can be boarded for 7. los. ? 14. What is the value of 600 pins at the rate of 2d. per gross ? 16. If 7J Ib. cost 2J. 7^., what will ij cwt. cost ? 16. If | of a maund cost 83. I2a., find the cost of 3| seers. 17. If f of an estate be worth B27oo, what is the value of | of the estate ? 18. If A of a cargo be worth 357. 7-f.j what is the value of 5 of the cargo ? 19; The owner of '375 of a ship sold f of his share for R5040 ; find the value of '875 of the ship at the same rate. 20. A man lost | of his money, and then spent f of the remain- der ; after which he had 8120 left : how much did he lose ? PROBLEMS AND THE UNITARY METHOD 183 21. A gentleman possessing A of an estate sold ^ of ^ of his share for 241. 4/1. ; what would *2 of ^ of the estate sell for at the same rate ? 22. If a man walk 46 miles in 3 days, in how many days will he walk 115 miles ? 23. If the rent of 34 acres is 821. 40., what is the rent of 51 acres ? 24. A servant's wages being 10. 8$. per annum, how much ought she to receive for 7 weeks ? [i year= 52 weeks.] 25. A man's annual income is 84088 : what does he receive for 15 days ? [i year = 36 5 days.] 26. If 27 bus. 2^ pk. cost 10 . 7 . 2j, what is the cost of a bushel and a half ? 27. If 3 cwt. 3 qr. cost 6. 15^., what will be the cost of 2 cwt.? 28. A sack of potatoes weighs 89 seers ; if 6 such sacks cost R22. 40., what will be the ost of 22 seers ? 29. If 17 ac. 2 ro. 38 po. supply 3 horses, how many acres will supply 16 horses ? 30. If the carriage of 25 maunds for 500 miles cost RQ. 6a.j what weight can be carried the same distance for R8 ? 31. If a piece of land worth 8375 yield an income of R7. 8a. what should be the value of a piece of land which yields an income of Ri8. 120.? 32. If 3$ acres can be mown in 7 days, how long will it take to mow 9j acres ? 33. If 350 rupees weigh 9 lb., how many pounds will 625 rupees weigh ? 34. In a certain time the population of a town increased from 78960 to 82908 ; find by how many the population of another town of 92360 inhabitants would have increased at the same rate in the same time. 35. A man walks 4 miles in an hour ; how many yards does he walk in a minute ? 36. A railway train travels at the rate of 20 miles in i$ hours ; find the rate per minute. 37. An express train goes 10 times as fast as a man who walks 6 ft. in a second : how many miles per hour does it go ? 38. Express 7-fc miles in kilometres, 5 kilometres being equal to 5456 yards. 39. If 6\ grammes be equal to 105 grains, express a pound avoir, in grammes. 1 84 ARITHMETIC 40. Convert 3 . 7 . 6 to Indian money, given 8151. 41. Convert 7 tons to maunds, given 35 seers =-72 Ib. 42. Express 3^ dollars in Indian money, 9 dollars being equal to 20 rupees. 43. If 8 horses eat as much as 6 oxen, how many oxen will eat as much as 20 horses ? 44. If 4 men do as much work as 6 boys, how many men will do the work of 1 8 boys ? 45. If the price of 7 horses and 5 oxen is R52O, and that of an ox is R2o, find the price of a horse. 46. If the weight of 5 rupees and 3 pice is 1200 grains, and that of a rupee is 180 grains, find the weight of a pice. 47. If 8 horses and 20 sheep eat the grass of 7 acres in a certain time, how many acres will feed 10 horses and 24 sheep for the same time, supposing a horse to eat as much as 4 sheep ? 48. If 15 chairs and 2 tables cost 400, find the cost of 12 chairs and 3 tables, the cost of 10 cHairs being equal to that of 4 tables. 49. If the wages of 4 men be equal to those of 5 women, what will 8 women earn in a day, the daily earnings of 10 men being Bi. ga. ? 60. If a shop-keeper uses a weight of 15 oz. for I Ib., how much will a customer lose in buying 24 Ib. ? Example 6. If 35 men finish a piece of work in 8 days, how many men will finish it in 10 days ? In 8 days the work is done by 35 men, .*. ...10 ***- , or 28 men. Ans. Example 7. If the penny loaf weighs 12 oz. when wheat is 4 a quarter, what should it weigh when wheat is .4. i6s. a quarter ? 4 8oj. ; 4. i6s.g6s. When wheat is 8oj. a qr. the loaf weighs 12 oz., / 16*. (i2X 5) oz., ." 96* H** oz., or 10 oz. Ans. PROBLEMS AND THE UNITARY METHOD 185 Example 8. A garrison of 1200 men is provisioned for 60 days ; if after 15 days 300 men leave the garrison, how long will the remaining provisions last the men left ? The provisions left would last 1200 men 45 days, .". they would last 300 men (45 x 4) days, .". they would last 900 men *?* days, or 60 days. Ans, EXAMPLES. 125. 1. If 9 men can mow a field in 4 days, in how many days could 6 men mow the same field ? 2. If 12 horses can plough a field in 7 days, in how many days could 14 horses plough it ? 3. 'If 1 6 men finish a piece of work in 5 days, in how many days could lo men do it ? 4. If 25 men reap a field in 12 days, how many men could reap it in 20 days ? ^6. If 7 cwt. feed 1 5 horses for 8 days, how many horses would they feed 12 days ? 6. If 28 maunds can be carried 50 miles for a certain sum, what weight can be carried 125 miles for the same sum ? 7. If 16 bighas can be rented for 9 months for Bio, for how many months can 36 bighas be rented for the same sum ? 8. A man walks from Calcutta to Hugly in 6 hours, walking 4 miles an hour ; how long would he take if he rode at the rate of 9 miles an hour ? 9. If the twopenny loaf weighs 20 oz. when wheat is ,4. 16,1. a quarter, what should it weigh when wheat is ;8 a quarter ? 10. If the sixpenny loaf weighs 64 oz. when wheat is 6s. gd. a bushel, what is the price of wheat per bushel when the sixpenny loaf weighs 48 oz. ? 11. From a mass of silver I can make 64 plates weighing 3 oz. each, how many 4 oz. plates could I make from the same ? 12. A garrison of 1200 men has provisions for 75 days ; how long would they last if the garrison were reduced to 500 men ? 13. A fortress is provisioned for 4 weeks at the rate of 20 oz, day for each man : if only 12 oz, be served out daily for each man, how long can the place hold out ? 14. A garrison of loop men is provisioned for 70 days : if after 20 days the garrison is re-enforced by 200 men, how long will the remaining provisions last ? 15. If 7 men can mow a meadow in 7 days, working 10 hours 186 ARITHMETIC a day, how many additional hours a day must they work to do it in 5 days ? 10. If I borrow &3oo for 8 months, for how long should I lend B4OO in return ? 17. If it requires 27^ yd. of carpet 9 in. wide to cover a room, how many yards of carpet 7 in. wide will be necessary to cover the same room ? EXAMPLES. 126. 1. If 30 seers of corn feed 6 horses for 4 days, how rnany horses would they feed for 12 days ? 2. If 30 seers of corn feed 6 horses for 4 days, how many horses would 25 seers feed for the same time ? 3. If 30 seers of corn feed 6 horses for 4 days, for how many days would they feed 8 horses ? 4. If 30 seers of corn feed 6 horses for 4 days, for how many days would 52^ seers feed the same number of horses ? 6. If 30 seers of corn feed 6 horses for 4 days, how many seers will feed 10 horses for the same time ? 6. If 30 seers of corn feed 6 horses for 4 days, how many seers will feed the same number of horses for 9 days ? 7. If 20 men reap a field of 6 acres in 40 hours, in how many hours will 35 men reap the same field ? 8. If 20 men reap a field of 6 acres in 40 hours, how many men will reap the same field in 25 hours ? 9. If 20 men reap a field of 6 acres in 40 hours, how many acres will 35 men reap in the same time ? 10. If 20 men reap a field of 6 acres in 40 hours, how many men will reap i $ acres in the same time ? 11. If 20 men reap a field of 6 acres in 40 hours, how many acres will they reap in 55 hours ? 12. If 20 men reap a field of 6 acres in 40 hours, in how many hours will they reap a field of 8 acres ? 18. When rice is 3 per md., how many people can be fed for the same sum that would feed 90 people when rice is R2. 8a. per hid. ? 14. If I lb. of flour cost 9/. when wheat is 3 per md., what should be the price of a md. of wheat when I lb. of flour costs la. ? 15. How many yards of cloth worth 40. 6tf. per yard must be given in exchange for 30 yards at 3^. 6f. per yard ? PROBLEMS AND THE UNITARY METHOD 187 16. Find the length of a strip of land 20 yd. wide, that should be given in exchange for a piece measuring 40 yd. by 30 yd. 17. If 3 Ib. of tea cost as much as xo Ib. of sugar, how much tea should be given in exchange for 25 Ib. of sugar ? 18. A brewer receives 10 doz. of brandy in exchange for 4 barrels of ale worth 3. IQJ. a barrel ; what does the brandy cost him per bottle ? 19. A man contracts to perform a piece of work in 20 days and immediately employs upon it 16 men. At the end of 12 days the work is only half done ; what additional number of men must he employ to fulfil the contract ? 20. A merchant of Calcutta indented from London goods worth .640, and paid ;io for freight. If a rupee is equal to is. 9*/., for how many annas must he sell goods, for which he paid is. to the London manufacturer, in order to gain $o on the whole outlay ? 21. If a quantity of flour serve 36 men for 15 days at the rate of 12 oz. a day for each man, how many ounces a day will each man get, when the same quantity of flour serves 42 men for the same time ? 22. When grain is R2 per md. how many horses can be kept for the same sum that would keep 20 horses when grain is Ri. 8. per md. ? ^Example 9. If 10 men can do a piece of work in 12 days* working 7 hours a day, how many hours a day must 6 men work to do the same in 14 days ? 10 men can do the work in (12x7) hours, 2 ................................. (12X7X5) ........ , 6 ............. .". to complete the work in 14 days, they must work hours, or 10 hours a day. Example 10. If a number of men can dig a trench 210 yd. long, 3 wide and 2 deep, in. 5 days of II hours each, in h<m many days of 10 hours each, will they dig a trench 420 yd. long, 6 wide and 3 deep ? (210 x 3 x 2) cu. yd. is dug in 55 hours. I ....................... virf&TO hours, /. (420x6x3) ....................... JVAMM* hours, or 330 hours ; .'. the number of days required *& 33. l8S ARITHMETIC Example n. If 8 oxen or 6 horses eat the grass of a field in i o days, in how many days will 5 oxen and 4 horses eat it ? 8 oxen eat as much as 6 horses, .'. i ox eats | horses, .". 5 oxen eat ** horses, or *> horses ; *. 5 oxen and 4 horses eat as much as (^-4- 4) horses, or %* horses. Now, 6 horses eat the grass in lo days, i horse will eat 10x6 , /. V horses 1J^2U , or 7#f days. EXAMPLES. 1T. S 1. If 5 men can do a piece of work in 8 days, working 7 hours a day, how many men will do the same piece of work in 4f days, working 10 hours a day ? 2. If 9 men can do a piece of work in 7 days, working 10 hours a day, how many hours a day must 6 men work to do the same in 30 days ? 3. If 12 men can do a piece of work in 8 days of 7 hours each, in how many days of 6 hours each can 10 men do the same ? 4. If 20 masons build a wall, 50 ft. long, 2 ft. thick and 14 ft, 'high, in 12 days, in how many days will they build a wall, 55 it. long, 4 thick and 16 high ? 5. If 20 men dig a trench, loo yd. long, 5 wide and 3 deep, in 3 days, how many men will dig a trench 1 50 yd. long, 6 wide and 2 deep, in the same time ? 6. If 5 men reap a rectangular field, 200 ft. by 50 ft,, in 2 days of 10 hours each, in how many days of 8 h'ours each can they reap another, 300 ft. by 40 ft. ? 7. If 6 men or 8 boys can do a piece of work in 18 days, in how many days will 3 men and 5 boys do it ? 8. If 5 men, 7 women or 9 boys can dig a ditch in 15 days, in how many days can I man, I woman and I boy dig it ? 9. 4 men do as much work as 6 boys in the same time, and a piece of work in which 20 men and 15 boys are engaged take 25 days ; how many days would it take if 1 5 men and 20 boys were employed upon it ? 10. If 10 gas-burners, which are lighted 4 hours every evening for 1 5 days, consume a quantity of gas which costs &3, for how many days can 12 burners be lighted 5 hours every evening at the same cost ? BANKRUPTCIES, RATING, TAXING, ETC. l8$ 11. If a piece of matting, measu; ing 7 ft. 4 in. by 5 ft, cost R6. 140., what will be the cost of a piece of the same matting, measuring 10 ft. by 6 ft. 6 in. ? 12. If the cost of printing a book of 250 pages, with 21 lines on each page, and on an average 10 words in each line, be RI25* find the cost of printing a book of 300 pages, with 14 lines on each page and 8 words in each line. 13. If 8 men, working 7 hours a day, take 12 days to complete a piece of work, how long will 14 boys, working 6 hours a day* take to do the same work, the work of one man being equal to that of two boys in the same time ? 14. If the feeding of 8 horses and 20 sheep for a month cost Bioo, what will be the cost of feeding 6 horses and 50 sheep for a month, supposing that 2 horses eat as much as 15 sheep ? BANKRUPTCIES, RATING, TAXING^ ETC. 199. Example I. A bankrupt's debts are 7240, and his assets (i.e.) the value of his property) are 85430 ; how much can- he pay in the rupee ? In the place of 87240 he can pay 85430, .' fii I Rf ft8, or B|, or 12 annas ; he can pay 12 a. in the rupee. Example 2. A bankrupt's debt s amount to ^3720, and he pays j. in the pound ; what are his a; In the place of 1 : 3720 .. .". his assets are (3720x18 Example 3. A man pays an i; . in the rupee ; find his income. >sets 5 pays 18*., , (3720x18)*., * or 3348. icome-tax of 8125 at the rate of 8125=- 240000*. He pays $p. Hi, 2400 800; his inc oo. Example 4. After paying an a man has 780 left ; find his groi He has i$s. 6d. left out .* ........... is. .......... / ........... (780x20)*. .... ..... * /. his gross incc *9 ARITHMETIC Example 5. A man pa y s a n income-tax of 6^. in the rupee on 8 of his income ; how muc^ j n the rupee does he pay on his whole Income ? He pays 6/*. in the rupe e on 5 of his income, i.e., he pays Tff | Ia off of his income, or & of 'his income. But ^ of Ri=*4A ; .'. he pays 4^. in the rupee on h j s w hole income. Example 6. When inc- or e-tax is #. in the rupee a person has to pay Rao more than when the tax was 4^ in the rupee ; 'find his income. Difference of tax is \p. when the income is Ri, </. *(2oxi6x ^p R(2oxi6xi2), or R3840 ; .*. his inc EXAMP 1. Find the income-tax on f ',3600 at 5/. in the R. 2. How much will a poor-ra parish where the whole property ome is R384O. LES. 18. te of 2s. 6d. in the produce in a is rated at ^3768. $s. ? 3. Find the amount of road - ces s, at 6^. in the R, on a rental of 85500. 4. A bankrupt's debts are E much in the rupee can he pay ? 6. A bankrupts effects amoi are R36788 ; how much can he p 6. If a man has to pay of ,750, what is the rate of tax p 7. A bankrupt's debts are R rupee ; what are his assets ? ,7880, and his assets R4925 ; how .int to R6i3i .5.4? and his debts ay in tLe rupee ? 7 . 6 for income-tax on an income >er ? 3798, and he pays I2a. 6/. in the 8. A bankrupt's assets are i 2900, and he pays his creditors 14$. 6d. in the ; what do his de bts amount to ? 9. A man pays an inr --M x O f & 4O at the rate of 4^. in the rupee ; find his income r\ If I pay 16. income-tax, being at the rate of - the , what ? p P a yi r ix of 5/>. in the rupee a man has id 1 d. in the for income-tax has ss income ? \d. in the , and thereby lost > him ? PROBLEMS RELATING TO WORK 191 14. A man pays an income-tax of 4^. in the rupee on j of his income ; what rate per rupee does he pay on his whole income ? 16. A man pays an income-tax of 8^. in the rupee on f of his income ; what fraction of his whole income is paid as income-tax ? 16. When the income-tax is yd. in the pound a person has to pay 40 less than when the tax was t s, in the pound ; find his income. 17. When the income-tax is jd. in the pound a person has to P a Y 2$ more than when the tax was $d. in the pound ; find his income. PROBLEMS RELATING TO WORK DONE IN A CERTAIN TIME. 0O. Example I. A can do a piece of work in 7 days, and B can do it in 9 days ; how long will A and /?, working together, take to do the work ? A can do the work in 7 days, .*. A can do \ of it in I day ; B .......................... 9 ...... i /. B .......... * ................. ; .". A and B together can do (} + $) of it in one day, ." .................................... the whole in ?| days ; /. the lime required = ff days 3}$ days. Example 2. A and B together can perform a piece of work in 5 days, and A alone can do it in 8 days : what time will it take B to do it alone ? A and B can do the work in 5 days, .". they can do \ of it in I day ; A alone ........................... 8 ....... , .'. he ......... J ................. ; .". B alone can do (-J- J) of it in I day, > ...................... JB .................. > / ...................... the whole in ty days, or 13^ days. Ans. Example 3. A vessel can be filled by a pipe in 25 minutes, and it can be emptied by a % waste-pipe in 20 minutes ; if both the pipes be opened when the vessel is full, how soon will it be empty ? 1st pipe fills g of the vessel in i minute, 2nd pipe empties ^ .....* ........................... ; .*. when both pipes are open (A ~ A) f ^ e vessel is emptied in I minute, * lie ................. ..... ..... ........... ........ i .*. the whole will be emptied in 100 minutes. IQ2 ARITHMETIC Example 4. A and B can do a piece of work in 5 hours ; A and C in 4 hours ; B and C in 3j hours. In what time can A alone do it ? A and B can do $ in I hour ; A and C .......... J ............ ; .*. two men of A's strength, and B and C can do * + J in I hour ; but B and C can do f in I hour ; .*. two men of A's strength can do + J $ in I hour, or 1% in I hour ; .". A can do ^ in i hour ; /. A can do the whole in -^f- hours, or 12^$ hours. AHS. Example 5. A does of a piece of work in 20 days ; he then calls in B> and they finish the work in 3 days ; how long would B take to do the whole work by himself ? A does $ of the work in 20 days, .". A can do fa of the work in I day, .". A does fa of the work in 3 days, but A and B do \ of the work in 3 days, \ .'. B does (J A) of the work in 3 days, I *>> ......... A .............................. i .*. B can do A .................. * day, .*. B can do the whole work in - days, or 37 J days. ^4J. EXAMPLES. |9. V^ 1. ^4 can do a piece of work in 10 hours ; B can do it in & hours. In what time will they do it if they work together ? 2. If A does a piece of work in 4 days, which B can do in 5, and C can do in 6 ; in what time will they do it, all working together ? 3. A cistern can be filled by one pipe in 3^ hours, by a second in 3^ hours, and by a third in 5j hours ; in what time will it be filled by all the three in action together ? 4. A can reap a field in lo days ; B can reap it in 12 days ;. C can reap it in 15 days ; how long will it take them all together to reap it, and what part of the work will be done by each ? 5. A and B together can dig a trench in 4 days, and A alone can dig it in 6 days ; in how many days can B alone dig it ? 6. Two pipes, P and Q, together can fill a cistern in 20 minutes, and P alone in 30 minutes : how long would Q alone take ? PROBLEMS RELATING TO WORK 193 7. A vessel can be filled by one pipe in 8 minutes, by a second pipe in 10 minutes ; it can be emptied by a waste pipe in 12 minutes : in what time will the vessel be filled if all the three be opened at once ? 8. A vessel has 3 pipes connected with it, 2 to supply and i to draw off. The first alone can fill the vessel in 4^ hours, the second in 3 hours, and the third can empty it in i hours. If all the pipes be opened when the vessel is half-full, how soon will it be empty ? 9. A and B can do a piece of work in 6 days ; A and C in 5$ days ; B and C in 4 days. In what time could each do it ? 10. A and B can mow a field in 3^ days ; A and C in 4 days ; B and C in 5 days. In what time could they mow it, all working together ? 11. A does f of a piece of work in 9 days ; he tl^en calls in /?, and they finish the work in 6 days. How long would B take to do the whole work by himself ? 12. A does Y5 of a piece of work in I $ days ; he does the re- mainder with the assistance of B in 4 days. In what time could A and B together do it ? 13. A can do a piece of work in 16 days, B in 10 days ; A and B work at it together for 6 days, and then C finishes it in 3 days : in how many days could C have done it alone ? 14. A and B together can do a piece of work in 6 days, B alone could do it in 16 days. If B stops after 3 days, how long afterwards will A have finished the work ? 15. A and B can reap a field in 30 days, working together. After ii days, however, B is called off, and A finishes it by himself in 38 days more. In what time could each alone do the whole ? 16. A t B and C together can do a piece of work in 6 days, which B alone can do in 16 days, and B and C together can do in 10 days ; in how many days can A and B together do it ? 17. Five men can do a piece of work in 2 hours, which 7 women could do in 3 hours, or 9 children in 4 hours. How long would I man, I woman and I child together take to do the work *? 18. A can do a piece of work in 4 hours, B and C can do it in 3 hours, A and C can do it in 2 hours. How long would B alone take to do it ? 19. A and B together can do a piece of^work in 8 days ; B alone can do it in 12 days ; supposing B alone* works at it for 4 days, in how many more days could A alone finish it ? 20. Three taps, A, B and C, can fill a cistern in 10 min., 12 min. and 15 min. respectively. They are all turned on at once, C A. 13 194 ARITHMETIC but after i min. B and C are turned off. How many minutes longer will A take then to fill the cistern ? 21. Two pipes, A and B } can fill a cistern in 3 hours and 4 hours respectively ; a waste pipe C can empty it in 2. hours ; if these pipes be opened in order at 7, 8 and 9 o'clock, find when the cistern will be filled. 22* A piece of work was to be completed in 40 days ; a number of men employed upon it did only half the work in 24 days ; 16 more men were then set on, and the work was completed in the specified time : how many men were employed at first ? 23. A can do a certain work in the same time in which -ffand C together can do it. If A and B together could do it in 10 days, and C alone in 50 days, in what time could B alone do it ? 24. A and B can do a piece of work in 10 days, B and Cin 15 days, and A and C in 25 days ; they all work at it together for 4 days ; A then leaves, and B and C go on together for 5 days more, and then B leaves : in how many more days will C com- plete the work ? 25. A cistern can be filled by two pipes in 30 and 40 minutes respectively ; both the pipes were opened at once but after some time the first was shut up, and the cistern was filled in 10 minutes more. How long after the pipes had been opened was the first pipe shut up ? 26. A cistern has 3 pipes, A, B and C ; A and B can fill it in 2 and 3 hours respectively; C isi a waste pipe. If all the three pipes be opened at once ^ of the cistern will be filled up in 30 minutes. In what time can C empty the full cistern ? 27. Forty men can finish a piece of work in 40 days ; but if 5 men leave the work after every tenth day, in what time will the whole work be completed ? PROBLEMS RELATING TO CLOCKS. 01. Example I. Two clocks are at 12 noon ; one gains 40 seconds and the other loses jo seconds in 24 hours : after what interval will the one have gained 16 minutes on the other, and what time will each then show ? What will be the true time when the first clock indicates 3 P, M. on the following day ? (i) The one clock gains on the other (40 + 50) seconds in 24 hours ; *.*., it gains f min. in I day, 16 ......... l^fc days, or ^- days, or 10 days 16 hours (true time), PROBLEMS RELATING TO CLOCKS IQJ (ii) Now in 3f days the first clock gains ^x 40 sec. or 7 J min., and the second loses 4? x 5 sec - or 8f min. But the correct clock, at the end of the interval (/.*., 10 days 16 hours) will show 4 A. M. Therefore the first will show 4 h. 7^ min. A, M. ; and the second will show 3 h. 51$ min. A. M. (iii) From 12 noon to 3 P. M. on the following day there are 27 hours. 24 hr. 40 sec. of the first clock =i day of the correct clock f *'.*! W hr = i day , i hr -rffc da , /. 27 hr =%W'da Now Jfjtff' da. = i da. 2 hr. 59 2 5 1 Vi rnin. .*. When the first clock indicates 3 P. M. on the following day, the true time will be 2 h. S9^i^i rnm p - M - EXAMPLES. 130. 1. A watch which is 5 minutes too fast at 12 o'clock on Sunday gains 2 'mm. 1 5 sec. per day ; what time will it indicate at half past 2 P. M. on the following day Tuesday ? 2. A clock which is 10 minutes too fait at 9 A. M. on Monday loses 3 min. per day ; what time will it show at a quarter to 3 p. M, on the following Wednesday ? 3. One clock gains 2 minutes, and a second gains 3 minutes in 24 hours ; the first is put right at 12 o'clock on Tuesday, the second at 3 P. M. on the following Wednesday : when will they indicate the same time ? 4. Two clocks are exactly together at 8 A. M. on a certain day ; one loses 6 seconds and the other gains 10 seconds in 24 hours ; when will the one be half an hour before the other, and what time will each clock then show ? 6. A watch which shows correct time at noon on Tuesday gains 2$ min. a day : what is the correct time on the following Sunday when it is 9 A. M. by the watch ? 6. Two clocks strike 9 together on Monday morning ; on Tuesday morning one wants 10 minutes to n, when the other strikes 1 1. How much must the slower be put on, or the faster put back} that they may strike 9 together in the evening ? 196 ARITHMETIC 7. A clock which was 1*4 min. fast at a quarter to n P. M. on Dec. 2, was 8 min* slow at 9 A. M. on Dec. 7 ; when was it exactly right ? 8. A clock which was 1*2 min. fast at a quarter to 11 P. M. on Nov. 28, was exactly right at 11-30 P.M. the following day. How many minutes was it slow at a quarter to 2 P. M. on Dec. 7 ? '9. A clock which is 7^ min. fast on Tuesday at noon> is 4^ min. fast at midnight on the following Monday ; how much did it lose in a day ? 10. A watch which gains 7^ min. in a day is 12 minutes fast at midnight on Sunday. What will be the true time when the watch indicates 4-32 P. M. on Wednesday ? 11. Two clocks, of which one gains 3^ min. and the other loses i\ min. in 24 hours, were both within I min. of the true time, the former fast and the latter slow, at noon on Sunday last ; they now differ from one another by 1 5 min. : find the day of the week and the hour of the day. 12. A clock loses 2\ minutes a day ; how must the hands be placed at 9 A. M. so as to point to true time at noon ? 13. One clock gains 12^ minutes, and another gains 7^ minutes in 12 hours. They are set right at noon on Sunday. Determine the time indicated by each clock, when the one appears to have gained 2 if min. on the other. 14. A clock set accurately at i o'clock indicates 10 minutes to 6 at 6 o'clock : what is the true time when the clock indicates 6 o'clock ? 15. A watch is 73 seconds slow at noon on January 1st 1887 : how much must it gain daily that it may be 17^ seconds fast at noon on July 1st ? 16. A watch is set right at 10 p. M. on Sunday ; at 10 A. M. on Wednesday it is 5 minutes too fast ; what will be the true time when it is 2 P. M. by the watch on Friday ? 17. A watch which gains 5 minutes in 12 hours is put right on January 1st 1888 ; when will it again show correct time ? 18. A church-clock was 1 5 minutes too fast 10 days ago, and to-day at the same hour it is 15 minutes too slow : when did it show true time ? When will it again show true time ? 19. Two clocks, of which one gains and the other loses one minute in an hour, strike one o'clock together ; what will be the interval, measured by a correct clock, between their respective striking 2 ? PROBLEMS RELATING TO CLOCKS *97 Example 2. Find thd time between 4 and 5 o'clock when the hands of a clock are (i) together, (ii) at right angles, (iii) opposite to each other. Note. While the minute-hand passes over 60 minute-divisions the hour-hand passes over only 5. Therefore in 60 minutes the minute-hand gains 55 divisions on the hour-hand ; and therefore in 12 minutes the minute-hand gains 1 1 divisions on the hour-hand. At 4 o'clock the minute-hand is 20 divisions behind the other. (i) The two hands to be together between 4 and 5, the minute- hand has to gain 20 divisions on the hour-hand. The minute-hand gains n divisions in 12 minutes, ." ........................ . ......... i division in If ............ , .' .................................. 20 divisions in *%pf& ............ ; .". the time required is i- 2 ^^ min. or 2ij 9 T min. past 4, (ii) When the hands are at right angles there is a space of 15 minute-divisions between them. Between 4 and 5 this will happen twice ; first, when the minute-hand has gained 5 (*>., 20 15) divi- sions ; and secondly, when it has gained 35 (*.*., 204-15) divisions. The minute-hand gains 11 divisions in 12 minutes, .' .................................. i division in }? ........... , / .................................. 5 divisions in *f* ............ ; and ......... . ....................... 35 divisions in /. The two hands will be at right angles at *f {* min. or 5^ T min. past 4 ; and also at 1.2ft 3 -* min. or 38^ min. past 4. (iii) When the hands are opposite to each other, there is a space of 30 divisions between them. This will happen when the minute-hand has gained 50 (*.e., 20+30) divisions, The process will be similar to that in the preceding cases. The time is 54^ min. past 4. EXAMPLES. 131. At what time are the hands of a clock (i) coincident, (ii) at right angles, (iii) opposite each other, (iv) 12 divisions apart, (v) 22 divisions apart, between the hours of 1. 2 and 3 ? 2, 3 and 4 ? 3. 6 and 7 ? I9 ARITHMETIC 4. 12 and I ? 5. 7 and 8 ? 6. 10 and II ? 7. A watch is 10 minutes teo fast at noon ; it loses 2 min. in one hour : find the true time when its hands are at right angles between 2 and 3 o'clock. 8. A clock is 5 minutes too slow at I ; it gains I min. in an hour ; what is the true time when its hands are together for the fifth time after I o'clock ? 9. A clock is put right at 4 P.M. ; it gains i$ min. in an hour ; what is the true time when its hands are at right angles for the fourth time after 4 ? 10. A clock indicates correct time when its hands are together between 2 and 3 o'clock ; if it had been losing 2 min. every hour, what time did it indicate at 12 noon ? 11. A clock, in which the hour-hand has been displaced, shows the time to be 16 minutes past 3, and the two hands are together ; the time is between 3 and 4 o'clock. Find by how many minute- divisions the hand has been displaced. 12. If the hands of a clock come together every 63 minutes (true time), how much does the clock gain or lose in a day ? PROBLEMS CONCERNING TIME AND DISTANCE. Example i. A passenger train leaves Calcutta at 4 P.M. and travels at the rate of 20 miles an hour ; the mail train leaves Calcutta at 9 P.M. and travels, on a parallel line of rails, at the rate of 30 miles an hour : when and where will the second train overtake the first ? The first train has started 5 hours before the second ; and is therefore (20 X 5) or 100 miles away when the second train starts. Therefore the second train has to gain 100 miles on the first) at the rate of 10 (*>,, 3020) miles an hour. Second train gains 10 miles in I hour on the first, .* ............................ ico ........... 10 hours ............... ; .*. the time required is 10 hours after the second train starts ; and /. the second overtakes the first (30 x 10) or 300 miles from Calcutta. Example 2. A hare, pursued by a grayhound, is 30 yards before him at starting ; whilst the hare takes 4 leaps the dog takes 3 ; in one leap the hare goes i& yards, and the dog, 2$ yards ; how far will the hare have gone when she is caught by the hound ? PROBLEMS CONCERNING TIME AND DISTANCE 199 Whilst the hare runs (4x1$) yd., or 6 yd, the dog runs (3 x 2\) yd., or 7$ yd. Hence The dog gains li yd. whilst the hare runs 6 yd., .* 3. yd 12 yd., .* 30 yd 120 yd. ; .*. the required distance is 120 yd. Example 3. A starts from P to walk to Q, a distance of 5i| miles, at the rate of 3! miles an hour ; an hour later B starts from Q for P and walks at the rate of 4^ miles an hour : when and where will A meet B ? A has already gone 3! miles when B starts. Of the remaining 48 miles, A walks 3f and B walks 4^ in one hour ; that is, they together pass over (3f+4i) or 8 miles in one hour. Therefore 48 miles are passed over in ^ or 6 hours. Therefore A meets B in 6 hours after B started. And therefore they meet at a distance of 4jx6 or 25$ miles from Q. Example 4. Two trains, 77 yd. and 99 yd. long respectively! run at the rates of 25 and 20 miles an hour respectively on parallel rails in opposite directions : how long do they take to pass each other ? How long would they take to pass each other if they were running in the same direction ? How long would a person sitting in the first train take to pass the other ? (i) The two trains running in opposite directions will pass each other in the time in which (77+99) or 176 yards are passed over at the^rate of (25 + 20) or 45 miles an hour. Now, 45 miles are passed over in I hour, * i.e.) 45 x 1760 yd i. I hour, /. 176 yd aJa hour ; .". the time required =^3 hr., or 8 seconds. (ii) When the trains run in the same direction they pass each other in the time in which (77 + 99) or 176 yards are passed over at the rate of (25-20) or 5 miles an hour. The time required will be found to be 72 seconds. (iii) First, when the tr air/5 are running in opposite directions, a person sitting in the first train will pass the other in the time in which 99 yd. (i>., the length of the second train) are passed over at the rate of (25 + 20) or 45 miles an hour. The required time will be found to be 4$ seconds. Secondly f when the trains run in the same direction, 99 yd. are to be passed over at the rate ot (25 20) or 5 miles an hour. The required time is 40$ seconds. 200 ARITHMETIC Example 5. A man rows down a river 18 miles in 4 hours with the stream, and returns in 12 hours ; find the rate at which he rows, and the rate at which the stream flows. He rows 18 miles in 4 hours down the stream ; therefore he rows ^ or 4^ miles an hour down the stream. Again, he rows 18 miles in 12 hours up the stream ; therefore he rows ^f or i miles an hour up the stream. .*. 4$ miles an hour is the sum of the rate at which the man rows and the rate at which the stream flows ; and i miles an hour is their difference. Hence the rates are 3 miles and ij miles an hour respectively. Example 6. If a snail, on the average, creep 31 inches up a pole during 12 hours in the night, and slip down 16 inches during the 12 hours in the day, how many hours will he be in getting to the top of a pole 35 feet high ? Length of the pole=42o in. Now in 24 hours the snail creeps up (31 16) in. or 15 in. ; therefore in (24x26) hr. the snail creeps up (15x26) in. or 390 in.; therefore he has (420 390) in. or 30 in. more to get up. And he goes over 31 in. in 12 hr., and therefore over 30 in. in **$\^ hr. Therefore he reaches the top in (24x264-^^1^) hr., or in 635jf hours. [The number of days (26) has been so determined that (420 in. 15 in. x 26) may be equal to 31 in. or just less than 31 in.] EXAMPLES. 132. 1. One man takes 100 steps a minute, each 2 ft. long ; another walks 4 miles an hour ; if they start together, how soon will one of them be 38 yards ahead of the other ? 2. A person wishing to go from A to B walked for 4! hours at the rate of I mile in 2if min., he then rode for i6J hours three times as fast as he walked, and then had to travel by rail for 10 J hours three times as ( fast as he rode ; find the distance from A to B. 3. A train leaves Calcutta at 7-30 A. M. and' travels 25 miles an hour ; another train leaves Calcutta at noon and travels 49 miles an hour : when and.wher% will the second train overtake the first ? 4. A train going 30 miles an hour leaves Calcutta for A Hah a- had (600 miles) at 9 P. M.; another train going 40 miles an hour leaves Allahabad for Calcutta at the same time ; when and where will they piss each other ? 5* Two trains, each 88 yards long, are running in opposite directions on parallel rails, the first at 40 miles an hour, the PROBLEMS CONCERNING TIME AND DISTANCE 301 other at 35 miles an hour ; how long will they take to pass each other ? 6. In the above example, if the trains run in the same direc- tion, how long will a person sitting in the faster train take to pass the other ? 7. A man rows down a river 15 miles in 3 hours with the stream and returns in 7$ hours : find the rate at which he rows, and the rate at which the stream flows. 8. A ^man rows 12 miles in 5 hours against the stream, the rate of which is 4 miles an hour ; how long will he be rowing 15 miles with the stream ? 9. A policeman goes after a thief who has loo yards 1 start ; if the policeman run a mile in 6 minutes, and the thief a mile in 10 minutes, how far will the thief have gone before he is overtaken ? 10. A man starts at 7 A.M. and travels at the rate of 4| miles an hour ; at 8-15 A. M. a coach starts from the same place and follows the man, travelling at the rate of 6 miles an hour ; at what o'clock will the coach overtake the man ? 11. A starts from Allahabad to Cawnpore and walks at the rate of 5 miles an hour ; B starts from Cawnpore 3 hours later and walks towards Allahabad at the rate of 4$ miles an hour ; if they meet in u hours after B started, find the distance from Allahabad to Cawnpore. 12. A starts from Calcutta to Hugli (24 miles) at 6 A. M. walking 4 miles an hour ; B starts from Calcutta an hour later and reaches Hugli one hour before A ; where did they meet ? 13. A man walks to a town at the rate of 3^ miles an hour and rides back at the rate of 6 miles an hour ; how far has he walked, the whole time occupied having been 3 hours 10 minutes ? 14. A and B run a mile in opposite directions ; wljile A runs 6 yards B runs 5 ; B gets 9 seconds' start, during which time he runs 22^ yards ; find when he will pass A. 16. A train leaves Calcutta at 7 A. M. and reaches Burdwan at il A. M. ; another train leaves Burdwan at 8 A. M. and reaches Calcutta at 1030 A. M. : at what hour do they meet ? 16. A train starts from P for Q travelling 20 miles an hour ; i$ hours later another train starts from P and travelling at the rate of 30 miles an hour reaches Q 2^ hours before the first train : -find the distance from P to Q. 17. A horseman leaves Madras at 10 A.M. and in 5 hours over- takes a coach which left Madras at 9 A.M. If the coach had been ~2 miles farther on the road when the horseman started it would 202 ARITHMETIC have been overtaken in 7 hours, ^ind the rates of the horseman and the coach. 18. A and B start at the same time from Patna and Bankipora, and proceed towards each other at the rates of 3 and 4 miles per hour respectively. They meet when B has walked one mile farther than A. Find the distance between Patna and Bankipore, 18a. A, B and C start from the same place at intervals of an hour and walk at the rate of 3, 4 and 5 miles an hour respectively, A starts first, but when he is overtaken by B he returns towards the starting-place ; find the distance from the starting-place where he would meet C. 19. A man rides at the rate of I c miles an hour, but stops 5 minutes to change horses at the end of every 7th mile ; how long will he take to go a distance of 94 miles ? 20. A man rides at the rate of 10 miles an hour, but stops 10 minutes to change horses at the end of every I2th mile ; how long will he take to go a distance of 96 miles ? 21. If a gun fire 7 shots every 9 minutes, how many will it fire in an hour ? 22. A monkey, climbing up a greased pole, ascends 10 ft. and slips down 3 ft. in alternate minutes. If the pole is 63 ft. high, how long will it take him to reach the top ? 23. A vessel has 2 pipes attached to it, I to supply and I to draw off. The supply-pipe can fill the vessel in 40 minutes, and the waste-pipe can empty it in an hour. If the supply-pipe and waste-pipe are kept open in alternate minutes, in what time will the vessel be filled ? 24. A boy and a girl began to fill a cistern ; the boy brings a quart at the end of every 2 minutes and the girl brings a pint every 3 minutes. In what time will the cistern be filled, if it holds 4$ gallons ? v 803. Example. A^ B and C start from the same point and travel round an island 30 miles in circumference, /4 and B travelling in the same direction and C in the opposite direction. If A travels at the rate of 5, B at the rate of 7 and C at the rate of 8 miles an hour, in how many hours will they all come together again ? B gains 2 miles on A in I hour ; .". he gains 30 miles or com- plete circuit in -^ hr , that is, A and B are together at the end of every 15 hours. A and C together pass over 13 miles in i hr. ; .". they come together every \ \ hours. And therefore A, B and C will come together at the end of any number of hours which is a common multiple of 15 and f ; but the L. C. M. of 15 and ? is 30 : therefore A^B^C are first together at the end of 30 hours. RACES AND GAMES OF SKILL 2O3 EXAMPLES. 133. 1. A and B start together from the same point to walk round a circular course, 10 miles long ; A walks 4 miles and B 3 miles an hour. When will they next meet, (i) if they walk in the same direction, (ii) if they walk in opposite directions ? 2. A takes 3 hours and B takes 5 to walk round a park. If they start together, when will they next meet, supposing (i) that they walk in the same direction, (ii) that they walk in opposite directions ? 3. A, B) C start from the same point and travel in the same direction round an island 63 miles in circumference, A at the rate of 10, B at the rate of 12, and C at the rate of 16 miles a day ; in how many days will they come together again ? 4. A can go round an island in 1 5 days, B can go round it in 20 days and C in 25 days. If they start simultaneously from the same point, A and B travelling in one direction and C in the opposite direction, in how many days will they come together again ? In how many days will they come together again at the starting point ? 5. Three boys agree to start together from the same point and run round a circular park 6 miles in circumference ; they run at the rates of 3, 5 and 7 miles per hour respectively ; in how many hours will they come together again ? In what time will they come together again at the point from which they started ? RACES AND GAMES OF SKILL 04. Example i. A can beat B by 40 yards in a mile race \ B can beat C by 20 yards in a mile race ; if A and C run a mile} by how much will A win ? A can run 1760 yards while B runs 1720, .'. A if & B 40, /. A 1!AJ*U B I76o , [but B 1760 C 1740,] .'. A 1I4|AA C 1740, /. A 1760 C HAJ|a or I7oo& yards. /. A will win by (1760- 1700^) or 59& yards. Example 2. A can give B 20 yards and C 30 yards in a race of 200 yards ; how many yards can B give C in 300 yards ? 204 ARITHMETIC [Note. "A can give B 20 yards in 200 yards" means that in a race of 200 yards A can give B 20 yards' start. Consequently while A runs 200 yards B runs 180 yards.] While A runs 200 yards B runs 180, and A 200 C 170, ." B 180 C{ 170, / B 60 C A J ft , .* B 300 C H*ft or 283$ yards, .". B can give C (300-2834-) or i6J yards in 300. Example 3. In a game of skill A can give /?, and B can give C, 10 points out of a game of 50 ; how many should A give C ? [Note. "A can give B 10 points out of a game of 50" means that while A makes '50 points B can make (50-10) or 40 points.] C can make 40 points while B makes 50, . C 4 /? c .-. C 32 but A 50 /. C 32 .". A can give C (50 -32) or 18 B 40 B 40 A 50. points in 50. EXAMPLES. 134. 1. In a mile race A gives B 60 yards' start, and beats him by 28 yards. If A runs the mile in 5 minutes, how long will B take ? 2. In a mile race A can beat B by 40 yards, and B can beat Cby 40 yards ; how many yards' start can A give C that there may be a dead heat ? 3. A can give B 60 yards, and C 80 yards in a race of 500 yards ; by how much could B beat C in a mile race ? 4. A runs 15 yards while B runs 12 ; B runs 10 miles while C runs 12 : if C runs a mile in 10 minutes, what time will A take to do it ? 5. At a game of skill A can give B 15 points out of 50, and A can give C 10 points out of 40 : which is the better player, B or C, aod how many points can he give the other in 75 ? 6. A and B run a mile race ; A runs the whole course at the rate of 100 yards per minute ; B running at the rate of 80 yards per minute ipr 5 minutes, quickens his speed to 120 yards per minute : which wins ? by how much ? and by what time ? CHAIN RULE 20$ 7. In a game of billiards A can give B 10 points, and C 14 points in 50 ; how many can B give C so as to make an even match ? 8. A can give B 300 yards in I mile, and C can give B 700 yards in 2 miles ; if A and C run a mile, which will win and by how much ? 9. ^ can give B 100 yards' and 150 yards' start in a mile ; B can give C a start of 5 seconds in a mile : how long does each take to run half a mile ? 10. In a mile race A gives B 50 yards' start, and beats him by 38 yards ; B giving C 40 yards' start is beaten by 60 yards \ if A and C run over the same course, which will win and by how much ? 11. At a game of rackets A can pive B 8 points in 40, and B can give C 10 points in 50 ; how many points could A give C in 25? 12. A can give B 20 yards' and C 30 yards' start, while B can give C 2 seconds' start in a race of 250 yards ; how long does each take to run 100 yards ? 13. One boy runs 200 yards and another 180 yards in a minute. How many yards' start must the second have that they may run a dead heat in a mile race ? 14. In a game at fives A can gives B 3 points out of 1 5, and A can give C 7 points ; how many points can B give C so as to make an even match ? 15. A and B run a mile and A wins by half a minute. A and C run a mile and A beats C by 88 yards. B and C run and B wins by 20 seconds. In what time can each run a mile ? 16. A beats B by 20 yards, C beats D by 60 yards, and B beats D by 40 yards, in a mile race. If A and C run, which will- win and by how much ? CHAIN RULE. 205. Example I. If 8 rupees are worth 15 shillings, and 25 shillings are worth 6 dollars, how many dollars are equal to 45 rupees ? R8 -15*., .\ Ri V* 2 5 *. 6 dollars, .'. i^. 5 6 j dollars* 45 x -*- a 45 xj ff^ x A dollars, or 20 J dollars. *06 ARITHMETIC Example 2. If A in 3 days can do as much work as B in 4 days, and B in 5 days can do as much as C in 6 days, how long will A require to do a piece of work which C can do in 16 days ? What C can do in 6 da. B can do in 5 da., and B 4 .... A 3 * Jj *. I ... A .... 5 *. /. What C can do in 16 days -Scan do in i6x days, .' C A i6xfxdays or 10 days. EXAMPLES. 135. 1. If 25 rupees are worth 46 shillincs, 20 shillings are worth 25 francs, and 240 francs are worth 47 dollars, how many dollars are equivalent to 40 rupees ? 2. If R8i5J., 32o thalers, and 25 thalers93 francs, express a franc in Indian money. 8. If 72 carlini = 25 shillings, 4 shillings = 5 francs, and 8 scudi 45 francs, how many scudi are equal to 1296 carlini ? 4. If 5 chickens cost as much as 4 ducks, 6 ducks cost as much as 3 geese, and 7 geese cost as much as 5 turkeys, what is the price of a chicken when a turkey costs R8 ? 6. If 5 lb. of tea be worth 3 Ib. of coffee, 5 Ib. of coffee be worth 2 lb. of sugai , and 7 lb. of sugar be worth 30 lb. of rice, how many pounds of tea must be given in exchange for 20 lb. of rice ? 6. If 12 oxen eat as much as 29 sheep, 15 sheep eat as much as 25 hogs, 17 bogs eat as much as 3 camels, and 8 camels eat as much as 13 horses, how many horses will eat as much as 1632 oxen ? 7. If A can do as much work in 4 days as B can do in 5, and B can do a^ much in 6 days as C in 7 ; in what time will C do a piece of work which A can do in a week ? 8. If A en do as much work in i^ days as 2? can do in 2, and B can do as much in 2^ days as C in 3 ; in what time will A and B together do a piece of work which C can do in 10 days ? 9. While A does J of a piece of work B does J, and while B does \ C does J : in how many hours will C finish a piece of work which A finishes in 20 hours ? 10. If 3 ducks are worth 4 chickens, and 3 geese are worth 10 ducks, find the value of a goose, a pair of chickens being worth 40. 6/>, COMPLEX PROBLEMS 207 XXXVI. COMPLEX PROBLEMS. In the problems in the preceding section we have found the change in one quantity corresponding to the change in one other. In the following examples we shall have to find the change in one quantity corresponding to the changes in two others. Example i. If 15 horses can plough 12 acres in 10 days, in how many days can 9 horses plough 18 acres ? 15 horses can plough 12 acres in 10 days, i horse 12 acres in (xox 15) days, I horse I acre in 10 T x s 16 days, 9 horses I acre in *$ days, 9 horses 18 acres in ia ji ia days, or 25 days. Ans. Note. We might use 3 horses and 6 acres as common units with advantage. Thus : 15 horses can plough 12 acres in 10 days, 3 horses 12 acres in icx 5 days, 3 horses 6 acres in l^f* days, 9 horses 6 acres in -^ x x ^ d ys, " 9 horses 18 acres in 19 2 X X ^ X ~ days, or 25 days. Ans. Examples If 6 men earn Ri5 in 10 days, how much do 8 men earn in 7 days ? In 10 days 6 men earn Ri5, .'. In I day 6 men earn R{{ or Rf , /.In i day i man earns R^xs or RJ, /. In 7 days I man earns RJ, /. In 7 days 8 men earn R*f* or Ri4. Ans. Example 3. If 6 men can do a piece of work in 8 days, how many men can do a piece of work 4 times as great in \ of the time? The work can be done in 8 days by 6 men, / f 1 8 men, /. 4 times the work *... 72 men. Ans, Example 4. If the sixpenny loaf weigh 8 oz. when wheat is I5J. a bushel, what ought a bushel of wheat to be when the four- penny loaf weighs 12 oz. ? 208 ARITHMETIC Sixpenny loaf weighs 8 oz. when wheat is 15*. a bushel, .". penny loaf weighs 8 oz JJ * .*. penny loaf weighs i oz. 205 , /. fouirpenny loaf weighs i oz 80$ , .". fourpenny loaf weighs 12 oz 2$s. , or 6s. %d. a bushel. Example $. If 5 cannon, which fire 3 rounds in 5 minutes, kill 135 men in ij hours, how many cannon, which fire 5 rounds in 6 minutes, will kill 250 men in I hour ? In 54 rounds 135 men are killed by 5 cannon, .*. ... i round 135 5*54 j .". ... i round I man is -% 5 r > .'. ... 50 rounds i ifSifa i .*. ... 50 rounds 250 men are ^iliyflP 1 or 10 cannon. EXAMPLES. 136. 1. If 5 men earn 3 in 12 days, in how many days will 8 men- earn 4 ? 2. If 10 horses can plough 50 acres in 20 days, how many acres will 12 horses plough in 15 days ? 3. If 24 horses eat 9 bushels of corn in 21 days, for how many days will 33 bushels feed 7 horses ? 4. If 30 men can build a wall;2o ft. high in 15 days, how many men will it take to build one 25 ft. high in 7^ days ? 5. If 12 horses are fed for 17 days at a cost of Rno. 8#., how many horses can be fed for 27 days at a cost of Ri 17 ? 6. If 10 fires consume 75 maunds of coal in 14 days, in how many days will 18 fires consume 100 maunds ? 7. If the carriage of 10 md. 20 seers for 250 miles be 41. oa. 3^,, what should be paid for the carriage of 12 md. for 200 miles ? 8; If the wages of 13 men for 25 days amount to 20. 5*1., how many men must work for 16 days to receive B3o ? 9. What is a month's rent for n6| bighas of land, if &22. 8a. per annum be given for 9 bighas ? 10. If 14 person can|live on 81400 for 28 months, how long can 1 8 persons live on Ri35o ,? 11. If S pen dig a trench ;J yd. long in 21 days, how many men can dig a similar trench 20 yd. long in 35 days ? COMPLEX PROBLEMS 209 12. If 20 pumps can raise 1250 maunds of water in 5 hours, how many pumps can raise 750 maunds of water in 10 hours ? 13. If 20 men do a piece of work in 13 days, in what time can 15 men do another piece of work 2^ times as great ? 14. If 10 men do a piece of work in 8 days, how many men will do a piece of work, 4 times as great, in i of the time ? 16. If the fourpenny loaf weighs i o oz. when wheat is 50^. a quarter, what should a threepenny loaf weigh when wheat is 555. a quarter ? 16. If 3 Ib. loaf cost 8^. when corn is SQJ. per bushel, how much ought the 5 Ib. loaf to cost when corn is 36^. per bushel ? 17. If I get i Ib. weight of bread for 7^. when wheat is 15$. a bushel, what ought a bushel of wheat to be when I get 12 oz. of bread for $d. ? 18. If 14 men in 20 days of I2| hours each earn 6456. 4a., how many hours a day should 24 men work to earn 547. 8#, in 21 days, at the same rate ? 19. If 15 men can do a piece of work in 12 days of 6 hours each, how many men will it take to do 5 times the amount if they work 20 days of 10 hours each ? 20. Jf a man complete a journey of 1980 miles in 18 days, travelling 1 1 hours a day, in how many days would he travel 540 miles, going 6 hours a day at the same rate ? 21. When rice is R2. 8a. a maund, 10 men can be fed for 12 days at a certain cost ; how many men can be fed for 4 days at the same cost, when rice is &3 a maund ? 22. When flour is 84 a maund, 16 men can be fed for 5 days at a cost of R8 ; for how many days can 12 men be fed at a cost of Bio. 8<z., when flour is 3. 80. per maund ? 23. If 15 men can build a wall 270 ft. long, 5 high and 2 thick in 1 8 days, in how many days will 16 men build a wall 180 ft. long, 4 high and 3 thick ? 24. If 10 men working 6 hours a day dig a trench 105 ft. long, 4 wide and 2 deep in 6 days, how many hours a day must 264 men work in order to dig a trench 126 ft. long, 20 wide and n deep in 10 days ? 26. A garrison of 1200 men is provisioned for 50 days, allowing 10 oz. per man per day ; if it is reinforced by 300 men, to what must the daily allowance be reduced that the provisions may last the increased number of men 60 days ? 26. If the carriage of goods weighing 2 cwt. 3 qr. 6 Ib. for 300 miles cost 6. 10. 10, what will be the charge for carrying C. A. 14 210 ARITHMETIC 2 wagon-loads of the same, each weighing 14 cwt. o qr. 4 lb., 450 miles ? 27. If the gas for 6 burners, 6 hours every day, for 8 days cost 4. 8a., how many burners may be lighted 5 hours every evening for 10 days at the cost of R6. 40. ? 28. If 3 cannon, firing 4 rounds in 6 minutes, kill 250 men in half an hour, how many cannon, firing 3 rounds in 5 minutes, will kill 600 men in an hour ? 29. If 15 men can make an embankment, 966 yd. long, in 8 days, working io hours daily, how many men would be required to make an embankment, 575 yd. long, in 12 days, working 7i hours daily, 8 extra men being taken on during the last 2 days ? 30. If 50 men, working 8 hours a day, dig in 5 days, a trench of 275 cu. yd.; in how many days of 10 hours each could 40 men dig a trench of 330 cu, yd., when the hardness of the ground in the first case is twice that in the second, and 3 men of the former company can do the work of 4 men of the latter ? 31. If 6 men, working 8 hours a day, can mow 60 acres in 4 days ; in how many days will 4 men, two of whom work 10 hours and two 7 hours a day, mow 85 acres ? 32. If 6 men and 8 boys can reap a field of 1 5 acres in 4 days, how many acres will 7 men and 4 boys reap in 9 day^, two boys reaping as much as a man in the same time ? 33. If 4 horses eat as much as 18 sheep, and if 5 horses and 30 sheep can be kept for 15 days at a cost of RSI .3.6, at what cost can 7 horses and 1 5 sheep be kept for 20 days ? 34. The rent of a farm of 41$ acres for 39 months was R8o,. 60, ; what would be the area of another farm, the rent of which for 33 months was Rio3. 2a. t 4 acres of the latter being worth as much as 3 acres of the former ? 36. A vessel with a crew of 27 men, provisioned for 90 days at the rate of 22 oz a day per man, was, after vj days, forced by stress of weather to lie at anchor for a fortnight, at the end of which time 3 men died ; how must the provisions be apportioned that they may hold out the extra time ? 30. If 10 men or 1 6 boys, working 6 hours a day, can do a piece of work in 20 days, how many hours a day must 7 men and 8 boys work to do another piece of work 3 times ,as great in 15 days? 37. If 5 men, 8 women or 12 boys can do a piece of work in 16 days, working 7 hours a day, how many men, with the assistance of 4 women and 6 boys, will be able to do another piece of work 2j times as great in 35 days, ( working 5 hours a day ? COMPLEX PROBLEMS 211 07. The following problems are of a different class. Exantph i. The price of 5 horses and 6 oxen 'is R68o, that of 4 horses and 7 oxen is R6io ; find the price of an ox. The price of $ horses and 6 oxen=R68o, .*. 20 24 sa *R272O (0 Again 4 7 = R6io, / 20 35 RSOSO. ..(ii) /. The price of 1 1 1 oxen 3050 - 1*2720 [subtracting (i) from (ii] -R330 ; .". the price of I ox = R3O. Example 2. 3 men and 5 boys can do \ j of a piece of work in 3 days ; 4 men and 8 boys can do { of it in 2 days ; in what time can a boy do the whole work ? In 3 days 3 men and 5 boys can do JJ, /. ... I day 3 5 Jg, .'. ... i day 12 20 ||. ....... M (i) Again ... 2 days 4 8 , } j, .'. ... I day 4 8 ^, .*. ... I day 12 24 {. (ii) .". In i day 4 boys can do (|- J|) of the work, [subtracting (i) from (ii)] t\e.* 4 boys can do ^ of the work, / i boy can do ^j of the work, /. i boy can do the whole work in 30 days. EXAMPLES. 13*. / 1. If 9 horses and 7 cows cost 770, and 5 horses and 9 cows cost ft 5 30 ; find the price of a cow. 2. The price of 5 .maunds of flour and 6 maunds of rice is &39j and that of 7 maunds of flour and 4 maunds of rice is &37 ; find the price of one maund of flour and of one maund of rice. 3. If 10 rupees and II shillings weigh 2760 grains, and 8 rupees and 10 shillings weigh 2312^ grains, find the weight of a rupee and of a shilling. 4. If 7 sheep and 9 pigs cost 107, and 9 sheep and 7 pigs cost Rio i, how much will i sheep and i pig cost ? 5. The cost of 4 chairs and 5 tables is 8120, and that of 5 chairs and 4 tables Rio5 ; find the price of a chair and of a table. 212 ARITHMETIC 8. 2 men and 3 boys can dp f of a piece of work in 6 days ;, 3 men and 5 boys can do of it in 4 days. In what time can a boy do the whole work ? 7. 7 men and 8 boys can do a piece of work in 2 days ; 4 men and 12 boys can do f of the work in i day. In what time can a man do the work ? 8. 5 men and 6 boys can do f of a piece of work in 3 days ; 10 men and 18 boys can do the whole work in 2 days. In what time will a man and a boy be able to do double the work ? 9. If 6 men and 2 boys can reap 13 acres in 2 days, and 7 men and 5 beys can reap 33 acres in 4 days, how long will it take 2 men and 2 boys to reap 10 acres ? 10. If 2 boys and I man can do a piece of work in 4 hours, and 2 men and i boy can do the same in 3 hours, find in what times a man, a boy, and a man and a boy together, respectively, could do the same. 11. On a piece of work 4 men and 5 boys are employed, who do ^ of it in 6 days ; after this, i man and 2 boys more are put on, and more is done in 3 days ; how many more men must be put on to finish the work in one more day ? 12. A cistern containing 210 buckets may be filled by two pipes. When the first pipe has been open 4 and the second 5 hours, 90 buckefe of water were obtained. When the 1st was open 7 and the 2nd 3j hours, 126 buckets were obtained. In what time will the cistern be full, if both pipes work ? XXXVII. RATIO AND PROPORTION. * 5508. The ratio of one quantity to another of the same kind is that which expresses the relative greatness of the first quantity with respect to the second. Hence, the ratio of one quantity to another (of the same kind) is determined by \^\^ fraction whose numerator is the measure of the first quantity and whose denominator is the measure of the second quantity, both the quantities being expressed in terms of the same unit. Thus, the ratio of 3*. to $s. is determined by the fraction f ; of 2 yd. to 5 ft. by the fraction . The first of the two quantities forming a ratio is called the antecedent and the second is called the consequent of the ratio ; the two together are called the terms of the ratio. The ratio of 35. to 5* is written 3*. : 5*. Note. The inverse ratio of 35. to 55. is the ratio of 5^. to 35. RATIO AND PROPORTION 213 The value of a ratio does not depend -upon the nature of the quantities involved. Thus, the ratios, 2 yd. I 5 yd., 2s. I $.?., 2 Ib. I 5 lb., are all equal, each of these being determined by the fraction $. Hence, in investigating the properties of ratios, we usually consider the terms to be numbers, because numbers measure quantities of all kinds. 210. The value of a ratio is not altered by multiplying or dividing both its terms by the same number. Thus the ratios, 2 I 3, 4 I 6, 80 I 1 20, are all equal, 211. Ratios are compounded by taking the product of the antecedents for a new antecedent and the product of the conse- quents for a new consequent. Thus the ratio compounded of the ratios, 2 : 3 and 6:7 is 2x6:3x7 or 4 I 7. 1/5. Four quantities are said to be in proportion or pro- portionals when the ratio of the first to the second is equal to the ratio of the third to the fourth. Thus 3, 4, 9, 12 are in proportion : since the ratio of 3 to 4 is equal to the ratio of 9 to 12. N.B. When four quantities are in proportion, it is not necessary that all of them should be of the same kind ; it is only necessary that the first two should be of the same kind, as also the second two. The existence of proportion among the numbers is denoted thus : 3 I 49 : 12. which is read "3 to 4 equals 9 to 12" ; or thus : 3 I 4 \\ 9 I 12, which is read "3 is to 4 as 9 is to 12." Of triis proportion 3 and 12 are called the extremes, and 4 and 9, the means ; 12 is called a fourth, proportional to 3, 4 and 9. 13. When four quantities are in proportion so that first I second I : third \ fourth ; then also, second I first I : fourth .* third ; and fourth \ third II second I first. Also, if the quantities are all of the same kinds first '. third '. I second I fourth. 214 ARITHMETIC 214. When four numbers are in proportion, the product of the extremes is equal to the product of the means. For example, 3 \ 4=6 * 8, and we have 3x8=34x6* Hence also, an extreme product of the meansthe othet extreme ; and, a mean = product of the extremes * the other mean* 15. Three quantities of the same kind are said to be in continued proportion when the ratio of the first to the second is equal to the ratio of the second to the third. The second quantity is called a mean proportional between the first and third; and the third quantity is called a third proportional to the first and second. \ Thus, 2, 4 and 8 are in continued proportion ; for 2 I 4 = 4 I 8 ; 4 is a mean proportional between 2 and 8 ; and 8 is a third pro- portional to 2 and 4. It is obvious that the square of the mean proportional between two numbers is equal to their product. 16. Example I. Find a fourth proportional to 3, 9 and 4. 3 ! 94 I number required, .*. number required *&* 12. Example 2. Find the number which has the same ratio to 20 that 3 has to 5. 3 \ 5 number required I 20, .*. number required a *^ ft 12. Example 3. Find a mean proportional between 3 and 12. Square of the number required 3 x 12 36 ; .". the number required J 36 6. Example 4. A) B t C, D are quantities of the same kind j and the ratio of A to B is 3 I 4 , of B to C is 5 I 7> and of C to ZJ is 8 I 9. Find the ratio of A to D. A 3 B 5 , C 8 Now, j-J, ^and^--; . A B C $ $ % A 10 * K * x -4 x f " that is, A : D : : 10 : . RATIO AND PROPORTION 215 Note. We find the continued ratio of A, J3 9 C and A that is, we compare A^B^C and Z>, thus : A 7-1 9-1 B** 3 I 4> 1 We change the terms of the T A 2 I ratios in such a way that each 6 I "* I antecedent may be equal to l*"^?- TBJ the preceding consequent. B I C -30 : 40 : 56 : 63 ; which is read "A is to B is to C is to /? as 30 is to 40 is to 56 is to 63." And A) 9 C> D are said to be in proportion of or propor- tional to 30, 40, 56, 63. Example 5. A mixture (42 gallons) contains wine and water in the ratio of 5 to 2 ; find the quantities of wine and water in the mixture. If the mixture be divided into 7 (*>., 5+2) equal parts, 5 of the parts will be wine and 2 water. /. The quantity of wine = * 7 ^x 5 gallons =-30 gallons j and the quantity of water =*4x 2 gallons iz gallons. Example 6. A mixture (40 gallons) contains wine and water in the ratio of 3 to I ; how much water must be added to it that the ratio of wine to water may be 5 \ 2 ? We find, as in the preceding example, that the mixture contains 30 gall, wine and 10 gall, water. Now while the wine remains the same 30 gallons, the water is to be increased so that the ratio of wine to water may be 5 : 2 ; but 5 I 2=^30 I 12; /. (12-10) gall, or 2 gall, of water must be added. EXAMPLES. 138. Find the value of each of the following ratios in its siinplest form : 1. 15:21. 2. 839:^65. 3. $:$. lew, 4. 360 in. : 270 in. 6. 350 Ib. I 725 Ib. 6. 2. 5' I 3*. 7. 3i : 5f . 8. 2| I 4-3. 9. 3 yd. Z 7 ft. 6 in. Express in its simplest form the ratio compounded of the ratios, 10. 7:9 and 45 I 28. * 11. I I 2, 2 : 3 and 314.. 12. 2j : si and '3 : '25. 13. 4 : 7> 5 I 8 and 21 : 30. Compare the ratios, 14. 3:5 and 7 I 8. 15. 13 I 21 and 18 I 29. 16. 2 : 3, 3 I 4 and 4 I 5- '17. 3 I 7 5 I 9 and 7 : I* l6 ARITHMETIC Are the following in proportion ? 18. 6,11,18,33. 10. 5,7,20,27. 2O. R3, 82. 4*., 4, 3. Find a fourth proportional to 21. 7, 9 and 8. 22. 2$, 3 and 4^. 23. "2, -02 and '002. 24. 6-380, R570 and 12 Ib. 25. 4 yd., 2 yd. 2 ft and 2. 20. 12 acres, 27 ac. and 20 men. 27. 12 men, 9 men and 3. &8. 6 miles, 20 mi. and 9 hours. 29. 3 cwt., 84 Ib. and 1. 8j. Find a mean proportional between 30. 7 and 28. 31. 13 and 117. 32. 9464 and 5600. 33. / ff and 34. 2\ and s|. 35. '3 and '012. Find a third proportional to 36. 2j and 7^ 37. 7 and sf 38. R2 and Ri. 40. 39. Compare the rates of two trains, one of which runs 17 miles in 2 hours and the other 12^ miles in i\ hours. 40. A I B = $ : 4, B : C=| I ; find the ratio of A to C. 41. If A =! of'.#, and B=*2\ of C, find the ratio of A to C". 42. If, when A earns &4, B earns R$ ; and when B earns R6j C earns R7 ; and when C earns R8, /? earns R9 ; compare the earnings of A, B, C and D. 43. Two sums of money are proportional to 7 and 8 ; the first Is 2 ; what is the other ? 44. The weights of equal volumes of gold and water are as 37 is to 2. If a cu. ft. of water weigh 1000 oz., find the weight of a cu. ft. of gold. 45. The ratio of the circumference of a circle to its diameter is 22 I 7 ; find the circumference of a circle 10 ft. 6 in. in diameter. 46. One man adds 5 seers of water to 15 seers of milk, and another 3 seers of water to 12 seers of milk ; compare the amount pi milk in the two mixures. 47. While A makes a profit of 3, B makes 4 ; and while B makes a profit of ^5, C makes 6 ; if A makes a profit of 20* how much does C make in the same time ? 48. A mixture (50 gall.) contains wine and water in the ratio of 3 ! 2 ; find the quantities of wine and water in the mixture. 49. A mixture (30 gall.) contains wine and water in the ratio of 7 to 3 ; how much water must be added to it that the ratio of wine to water may be 3 I 7 ? 50. A greyhound pursues a hare and takes 4 leaps for every 5 leaps of the hare, but 3 leaps of the hound are equal to 4 of the hare ; compare the rates of hound and hare. RULE OF THREE 217 XXXVIII. RULE OF THREE. Problems which we have solved by the Unitary Method may also be solved by the method of finding a fourth proportional to three given quantities, Example i. Find the price of 12 maunds of sugar, when the price of 5 maunds is R6o. Here we observe that if the weight be increased 2, 3... times, the price will also be increased 2, 3. ..times ; therefore the ratio of the two weights is equal to the ratio of the two corresponding prices. Hence 5 md. '. 12 md. I I R6o I the answer ; /. the Example 2. If 12 men can do a piece of work in 5 days, in how many days will 15 men do it ? Here we observe that if the number of men be increased 2, 3... times, the number of days will be decreased 2, 3. ..times ; therefore the inverse ratio of the numbers of men is equal to the ratio of the corresponding numbers of days. Hence 15 men ; 12 men II 5 days I the answer ; .'. the answer *?$* days =*4*days. 18. The above method of solving a problem by finding a fourth proportional to three given quantities is commonly known by the name of Rule of Three. In the first problem we have an example of what is called the Rule of Three Direct, because there the direct ratio of the two weights is equal to the ratio of the corresponding prices. In the second problem we have an example of what is called the Rule of Three Inverse, because there* the Inverse ratio of the numbers of men is equal to the ratio of the corresponding numbers of days. 819. It is obvious that the second term in a proportion is greater or less than the first according as the fourth is greater or less than the third. Hence we may lay down the following genera) rule for arranging the terms in a Rule of Three question. Denote the answer by the letter x and place it for the 4th term ; and of the three given quantities place that which is of the same kind as the answer, for the 3rd term. Next from the nature of the question determine whether the answer will be greater or less than the third term, and place the greater or less of the two remaining quantities for the 2nd term according as the answer is greater or less than the 3rd term ; then place the remaining quantity for the first term. 2l3 ARITHMETIC Note. In working) the two first quantities in the proportion must be replaced by the numbers which measure them in terms of the same unit. Example i* If the third class railway fare for no miles is Ri. n. 6, what is the fare for 350 miles ? mi. mi. R. a. p. no 1 350 ii i . n . 6 1 *, f>.t ii : 35 II i . ii . 6 I .r ; Ri. ii .6x w _ R6o .2.6 R5 .7.6. Ans. Or thus : V Ri . n . 6 = 330^. ~R 5 .7.6. The latter method is the one more generally adopted. The learner should observe that the 3rd term being expressed in pies the answer obtain at the first instance is also in pies. Example 2. If a quantity of rice serve 100 men for 15 weeks, how many men will it serve 6 weeks ? weeks weeks men 6 I 15 II 100 I X) t.e. 9 2 I 5 1 1 100 I x ; men = 250 men. Ans. Example 3. A bankrupt's debts amount to ^1320, and his assets (*>., the value ofrhis property) are ^990, how much can he pay in the pound ? & & & 1320 I i II 990 I x, ;-i5*. An*. Example 4. A man, after paying an income-tax of 4^. in the rupee} has 4794 left ; what is his #rqss income ? p. p. a. 188 I 192 1 1 4794 I x, *>., 47 1 48 1 1 4794 :*; 6. Am. RULE OF THREE Example 5. If 8 oxen or 6 horses eat the grass of a field in 10 days, in how many days will 5 oxen and 4 horses eat it ? oxen oxen horses 8 : 5 :: 6 : *, .*. x *| 4 horses J. h orses. .", 5 oxen and 4 horses will eat as much as (* +4) or ty horses, horses horses days Now, * \ 6 \\ 10 I *, day s7| days. Example 6. ^4 can dp a piece of work in 7 days, and B can do it In 9 days ; how long will A and Z?, working together, take to do the work ? A can do | of the work and B can do J of the work in I day ; .'. A and B together can do (\ + %) or jf of the work in I day. work work day tt : i :: i : *, .'. *~?i days = 3j| days. Ans. Example 7. At what time between 2 and 3 o'clock are the hands of a clock at right angles to each other ? The minute-hand gains II divisions on the hour-hand in ll minutes ; and here it has to gain (10+15) or 2 5 divisions. div. div. min. ii : 25 :: 12 : x, .'. jrl^V* rnin.27 1 8 1 ' min. ; the two hands will be at right angles to each other at minutes past 2. Example 8. A can beat B by 40 yards in a mile race ; B can beat C by 20 yards in a mile race ; if A and C run a mile, by how much will A win ? While A runs 1760 yd., B runS 1720 ; and ....... B ...... 1760 yd., C ...... 1740. 1760 : 1720 :: 1740 : *, &A, 44 : 43 :: 1740 : *> .% While B runs 1720 yd., C runs 1700^ yd. ; but while B *uns 1720 yd., A runs 1760 yd. ; /, while A runs 1760 yd., C runs 1700^ yd. .% A will win by (i 760-1700^) or 59$ yd. " 320 ARITHMETIC Example 9. A starts from P to walk to g, a distance of 51 J miles, at the rate of 3| miles an hour ; an hour later B starts from Q for P and walks at the rate of 4j miles an hour : when and where will A meet B ? * A has already gone 3f miles when B starts. Of the remaining 48 miles, A walks 3} and B walks 4^ in one hour ; that is, they together pass over (3f + 4J) or 8 miles in one hour, miles miles hour 8 : 48 :: i : *, /. #=J^. hours =6 hours. .*. A meets B in 6 hours after B started. And therefore they tneet at a distance of 4^x6 or 25$ miles from Q. [For Examples for Exercise see Section xxxv,] XXXIX. DOUBLE RULE OF THREE. O. Complex problems which would require two or more applications of ttye Rule of Three are usually solved by a shorter method, commonly called the Double Rule of Three. The method will be best explained by means of examples. Example i. If 9 men can reap 6 acres in 10 days, how many men will reap 12 acres in 15 days ? 6 ! 12\ days 15 I ioj We denote the answer by x and place it for the 4th term, and place 9 men (which is of the same kind as the answer) for the 3rd term. We next take 6 acres and 12 acres (a pair of quantities of the same kind), and consider whether the answer will be greater or less than the 3rd term in the question "if 9 men can reap 6 acres, how many men will reap 12 acres, supposing the time to be the same in both cases ?" and we find that the answer will be greater; we therefore place 12 acres for the 2nd and 6 acres for the ist term. Then we take 10 days and 15 days (another -pair of quantities of the same kind), and consider whether the answer will be greater or less than the 3rd term in the question "if 9 men can reap in 10 days, how many men will reap in 1 5 days, supposing the number iof acres to be the same in both cases ?" and we find that the answer will be less ; we therefore place 10 days for the 2nd and 15 days for the ist term, under those already obtained. We now multiply the numbers in the ist term for the final ist term and the numbers in the 2nd term for the final 2nd term. Thus 6x15 i 12x10 :: 9 : #, Ans. acres 6 9 men DOUBLE RULE OF THREE 2|& Note. Each pair of quantities of the same kind should be replaced by their measures in terms of the same unit. Remark. Each additional pair of quantities of the same kind would be treated in a, like manner. Example 2. If 72 men can dig a trench, 324 yd. long, 12 yd. wide and 8 ft. deep, in 9 days of 12 hours each ; how many men can dig a trench, 1458 yd. long, 40 ft. wide and 3 yd. deep, in* 36 days of 9 hours each ? ft. long 324x3 ft. wide 12x3 ft. deep 8 days 36 hours 9 1458x3] 40 j 3 x 3 f : ; 72 men : . 9 I 12 Or better thus : cu. ft. (324X3)x(i2X3)x8 I (1458 x 3) x 40 x (3x3)"! hours 36 x 9 19x12 J . . 72 . #. Example 3. If 10 men can perform a piece of work in 24 days how many men will perform another piece of work 3 times as great in \ of the time ? work i ; j 24 . t Io days * .24 \ t } . 3x24x10 3x24x10x5 .. x** g^ - men = --- - - * men "150 men. An$< " a ~*t Example 4. If the sixpenny loaf weigh 8 oz. when wheat Is 15*. a bushel, what ought a bushel of wheat to be when the four- penny loaf weighs 12 o;z. ? pence 6 : 4 1 .. . ounces 12 : 8J " I5 *' ' ^ / *- 4 g|JJ*J.=^. = 6j. 8^. Ans. Example 5. If 5 cannon, which fire 3 rounds in 5 minutes,, kill 135 men in \\ hours, how many cannon, which fire 5 rounds in 6 minutes, will kill 250 men in i hour ? [The first 5 cannon, each firing 54 rounds, kill 135 men ; it is. required to find how many cannon, each firing 50 rounds* will kill 250 men.] rounds 50 I 54\ .. H men z 3S ! 250) - 5 cannon * * x 4 ttW8l 4 cannon xo cannon. A ns. 122 ARITHMETIC Examples in Double Rule of Three can be worked more conveniently in a little different manner. In this method the first work and second work are respectively taken for the third and fourth terms of the proportion, and the first cause and second cause respectively for the first and second terms ; Cor, the ratio of the two causes is equal to the ratio of the corresponding works. We shall apply the method to the first two of the foregoing examples, Example i. 9 men in 10 days will do the same amount of work as (9X 10) men will do in I day ; and x men in 15 days will do the same amount of work as (x x 1 5) men will do in i day. .*. QX 10 '. xx 15 I! 6 I 12, xx I5x6*9x lox 12, x ^^nr 1 * men =" * 2 men. A ns. Example 2. men 135 men. Ans* [For Examples for Exercise see Section xxxvi.] MISCELLANEOUS EXAMPLES. 139. L Find the least number which being added to 1409 will make the result divisible by 23. 2. A boy receiving 82. 40. a week has 8a. stopped every fourth week ; if there are 48 weeks in the school-year, how much does he get in 2 years ? 8. What are the prime factors in 4509oo45 and what is the smallest whole number by which it must be multiplied in order to become a perfect square ? 4. Find the least fraction which, being added to i+J-rJ -$xf ~J, shall make the result an integer. 6. Find, by Practice, the value of 37$ md. of sugar at 89. 13*. 6p. per md. 6. If 27 men can perform a piece of work in 15 days, how many men must be added to the number that the work may be finished in f of |he time ? 7. Find the greatest and least numbers of four digits exactly divisible by 34. 8. I distribute a sum of money among 32 men, giving 850. 70. 61* to the first) R$i. 70, 6^. to the next, &$2. 7 a. 6^. to the next, MISCELLANEOUS EXAMPLES 223 and so on, increasing the sum by Ri each time ; how much would each get if I divided the money equally ? 9,- Determine the least number, by which 378 must be multi- plied to produce a number exactly divisible by 336. 10. A screw advances "392 of an inch at each turn ; how many turns,, must be taken for it to advance 9*8 inches ? 11. Find, by Practice, the cost of 35 cwt. 2 qr. 7 Ib. at 7. iw. 4< per cwt. 12. If 12 iron bars, each 4 ft. long, 3 in. broad and 2 in. thick, weigh 576 Ib., how much will n weigh, each 6 ft. long, 4 in. broad and 3 in. thick ? 13. The population of a town is 5720, and there are 320 more men than women ; how many are there of each sex ? 14. A labourer, who works on week days only, earns 70. 9^. * a day ; supposing that the 1st of January 1885 was on a Sunday > find the amount of his earnings during the year. 15. Four bells ring at intervals of 3, 3^, 3^ and 3| seconds respectively, beginning together ; how often during 24 hours will the four bells ring together again ? 16. By what number must ^4- J- of J be multiplied in order to produce the least possible integer ? $ 17. A certain number of men subscribed ^63. os. 9*/., each subscribing as many pence as there were men ; how many men were there ? 18. If '428571 of a barrel of beer be worth 72 of 2. ios. t what is the value of "625 of the remainder ? 10. To the fourth part of a certain number I add 79, and obtain 100 as the sum ; what is the number ? 20. Divide EIOI. 150. 3^. among 20 men, giving to each of $ of them twice as much as to each of the others. 21. 720 gallons of cocoanut oil and 450 gallons of castor oil are to be put into an exact number of barrels, all of the same size, without mixing the two oils together ; find the least number of barrels required. 22. Express | of 7*. 6*/. + T25 of 5*--'54jof 9*. *d. as the decimal of 10. 23. The perimeter of a rectangle is 1 10 ft. ; the difference of two sides is 1 1 ft. : find its area as the decimal of an acre. 24. If a man can perform a journey of 170 miles in 4^ days of II hours each, in how many days of 8 j hours each, will he perform a journey of 470 miles ? 224 ARITHMETIC 25. To a certain number I add 3, and multiply the sum by 4, then divide the product by 5, and get 7 as quotient and i as remainder ; what is the number ? 26. A man bought 40 pieces of ribbon, all equally long, for 7* 80. at 2a. gp. a yard ; how many inches were there in each piece ? 27. What is the least debt in dollars (45. id. each) that can be paid in moidores ? 28. What is the capacity of a vessel, out of which, when it is half full, 4^ gallons being drawn, there remains J of the whole content ? 29. A square space, containing 113 sq. yd. 7 sq. ft, is to be lengthened by 3 ft. in one of its dimensions, and to be shortened by 3 ft. in the other ; what will then its area be ? 30. If a person walks 7 miles in 7,\ hours, how long will a second person take to walk 10 miles, supposing that the first walks 2j miles while the second walks 2j ? 31. Fourteen years ago a man was six times as old as his son whose present age is 20 years ; what is the present age of the father ? 32. A mtn buys 20 seers of milk at 30. 6p. per seer ; how much water must he add to it that he may gain Ri. 4<z. by selling the mixture at 30. per seer ? 33. I had coins of one kind weighing 2295 grains ; and of this I spent coins weighing 1035 grains ; show that a single coin cannot weigh more than 45 grains. 34. Two clocks begin to strike 12 together ; one strikes at an interval of 2'9io seconds, the other, of 2*083 sec^hds ; what decimal of a minute is there between their seventh strokes ? 36. Find the cost of painting the walls of a square room, 10 ft. high and 16 ft. long, with one door 8 ft, by 4 ft., and 2 windows^ each 5 ft. by 2, the amount saved by each window being fii. 140. What additional height would increase the cost by Ei2 ? 36. A merchant of Calcutta indented from London goods worth ^226, and paid ^34 for freight and packing. He sold half the goods at a gain of 2 annas per rupee ; at what gain per rupee must he sell the remainder that he may clear 500 on the whole outlay? [Rii*. 7i<l 37. Find the greatest fraction, the numerator of which is composed of 3, 5, i, o and the denominator of 3, 2, 8, o. 36, Two, persons buy 600 oranges each at 24 for a half-rupee ; MISCELLANEOUS EXAMPLES 325 one sells them at $a. 6f. a dozen, and the other at 80. $*. a score ; who gains more, and by how much ? 39. A number is exactly divisible by 7 and by 13, and it is known that the number is between 400 and 500; what is the number? 40. What fraction of J of a rupee is $ of 5 ; and what fraction of their sum is their difference ? 41. Find the length of the inner edge of a cubical cistern which will hold 256 Ib. of water, supposing that a cu. ft. of water weighs 1000 oz. 42. A person after paying an income-tax of i anna in the &, devotes <$ of the remainder of his income to purposes of charity, and finds that he has 5175 left ; what is his income ? 43. A person has a number of oranges to dispose of ; he sells half of what he has and one more to A, half of the remainder and one more to /?, half of the remainder and one more to C ; by which time he has disposed of all he had : how many had he at first ? 44. A certain number of men, twice as many women and three times as many children earned Ri6. za. in 3 days ; each man earned I2a., each woman 8<2. and each child 5*1. a day : how many women were there ? 45. Find the greatest weight that will measure (*>.) divide exactly) a Ib. avoir., and a Ib. troy. 46. If J| of a number exceed '83 of half the number by '2002, what must the number be ? 47. How many *bricks, 6 in. by 3 in. by 3 in., will be required for a wall, 16 ft. by 10 ft. by 2 ft., allowing j\ of the space for mortar ? 48. A creditor received on a debt of 3600 a dividend of 90. ic/fr. in the B ; and a further dividend of 6a. 8/>. upon the remainder. What did he receive altogether, and what fraction was it of the entire debt ? 49 A has 150, B has Ri2o ; if C has Bi6 mor* than what he has, then B and C together would have as much as A :ihow much has C ? 6p. Divide ^30. IQS. 8</. into two sums of money, one of which contains as many shillings as the other contains fourpences, 6L 378 oranges and 462 mangoes are to be distributed among boys so that each boy gets as many oranges and as many mangoes as any other boy ; find the largest possible number of boys, and the least possible number of fruits each boy may get. 52. What number is greater than its fifth part by $ ? C,A, 15 306 ARITHMETIC 63. Find how much card-board is required to make a cubical box and its cover f the edge of the box is 9 in., and the rim of the cover extends 3 in. deep down each side. 64. A work can be completed in 36 days by 30 men working 6 hoursfa day ; in what time would 18 men and 60 women, working 9 hoursia day, complete it, supposing that 3 men can do as much work as 15 women ? 65. A gentleman's monthly expenses are Ri 50 less than his income ; if his income be increased by Rioo a month and expenses decreased by 50, how much will he be able to save in a year ? 66, Three persons A t B, C start on a tour, each with 20 in his pocket, and agree to divide their expenses equally. When they return, A has ^3. us. gd., B has 2. 5^. and C has 17^. $d. What ought A and B to pay to C to settle their accounts ? 67.JJJT A man walks at the rate of 128 yards per minute ; find the least whole number of minutes he will take to walk over an exact number of miles. 68. Simplify (3-5- 2-3X3-5 + 2'3)-r 3-5 of 2-3x32-53. 69. The external dimensions of an open box are 5 ft., 4^ ft. and 3 ft. ; find the cost of painting the outside at 3 annas per sq. yd. What will be the cost of painting the inside at the same rate, if the box; is made of J-inch plank ? 60. Three men can do as much work as $ boys ; the wages of three boys are equal to those of two men. A work, on which 40 boys and 15 men are employed, takes 8 weeks and costs $$o ; how long would it take if 20 boys and 20 men were employed, and how much would it cost ? 61. What quantity of water must an inn-keeper add to a barrel of beer, which cost him $o, to reduce the price to i* S.T. a gallon ? 62. A certain number of men mow 4 acres in 3 hours, and a certain number of others mow 8 acres in 5 hours ; how long will they be mowing 1 1 acres, if they all work together ? 63. At 10 minutes to 2 in the afternoon a clock is 55 seconds slow, and at 6 in the evening it is 30 seconds slow : at what hour will it show true time ? 64. A train leaves Calcutta at 7 A. M. for Goalundo, 1 53 miles distant, and travels at the rate of 20 miles an hour ; another train leaves Goalundo for Calcutta at 1 1-30 A.M. and travels at the rate of 22 miles an hour ; when, and where, will the trains pass each other ? 66. A cistern, 6 ft. long, 5 ft. wide and 4 ft. deep, contains MISCELLANEOUS EXAMPLES 227 pulp for making paper. If of the volume of the pulp be lost in the process of drying, how many sheets of paper, 16 in. by 10 in.) will be obtained) if 400 sheets in thickness go to an inch ? 68. If 7 men and 5 boys can reap 168 acres in 18 days, how many days will 15 men and 5 boys take to ^reap 700 acres, one man being able to do three times as much work as a boy ? 67. Find the value of J of a guinea + 7 \ of Sj. 3</. + T fiF of 2. 1 5 s t ; and reduce the result to the fraction of a guinea and a half, 68. Two pipes, A and , fill a cistern in 25 and 30 minutes respectively. Both pipes being opened, find when the first must be closed that the cistern may be just filled in 15 minutes. 69. If ^ of a sheep be worth f of a rupee, and f of a sheep be worth ^ f of a cow, how much must be given for 106 cows ? 70. The cubic content of an open cistern, 6 ft. long and 4 ft broad, is 20 CIL ft. ; what will be the cost of lining the inside of it at is. per sq. ft. ? 71. Two persons walking at the rate of 3^ and 4 miles per hour respectively, set off from the same place in opposite directions to walk round a park, and meet in 20 minutes. Find the length of the path round the park. 72. If it takes 120 men to supply, in 5 days 1 work, a fortress with provisions for 5 months, when the garrison is 650 strong, how many will be required to supply it in 3 days for 4 months, after the garrison had been reduced by 130 men ? 73. A bag contains a certain number of shillings, twice as many sixpenny pieces and 3 times as many fourpenny pieces ; the whole sum amounts to 2 guineas : find the number of each. 74. A room, whose height is 9 ft., and length twice its breadth, takes 189 yards of paper, 2 ft. wide, for its four walls ; find its length. 75. A can do a piece of work in 20 days ; A and B together can do it in nj days. A works alone for 8 days, A and C together for 6 days, and B finishes it in 3 days. Find in what time B and C together could do it. 76. One clock gains 8 min., and another loses 4 min., in 24 hours. They are at right at noon on Sunday. Determine the time indicated by each clock when the one appears to have gained 12 minutes on the other. 77. The|whole time occupied by a train 1 10 yards long, travel- ling at the rate of 30 miles an hour, in crossing a bridge is 13 seconds ; find the length of the bridge. 228 ARITHMETIC 78. If a family of 9 persons spend 480 in 8 months, how much will serve a family (living upon the same scale) of 24 persons Cor 16 months ? 7. 6j. 8^. j-jof J-- jf 79. bimplify ^ ^ *Q_~fr of ( j_jj- 80. A room twice as long as it is broad is carpeted at 9^. a sq. yd., and the walls are painted at is. 6d. a sq. yd., the respective costs being ^44 2s* and ^8. %s. Find the dimensions of the room. 81. A cistern would be filled by a tap, A, in 3$ hours, or emp- tied by a tap,/?, in 3 hours. The cistern being half full, A is turned on at 8 o'clock, and B at 15 min. to 9 ; find when the cistern will again be half full. 82. If 2 guineas make 3 napoleons, and 15 rix-dollars make 4 napoleons, and 6 ducats make 7 rix-dollars, how many ducats are there in 490 ? 83. A person rows a distance of 3 miles down a stream in 40 minutes, but without the aid of the stream it would have taken him an hour ; what is the rate of the stieam per hour ? and how long would it take him to return against it ? 84. A boat propelled by 6 oars which take 25 strokes per minute travels at the rate of 7^ miles an hour ; find the rate of a boat propelled by 4 oars which take 32 strokes per minute ; the work done by each oar during one stroke in the latter case being a quarter as much again as in the former case. 85. A wagon, loaded with 1246 equal packages, weighs 26 tons 14 cwt. ; if the wagon itself weighs twice as much as the packages, find the weight of each package, 86. A did f of apiece of work in 6 hours, B did | of what remained in 2 hours and C finished it in half an hour. How long would they have been doing the whole if they had worked tegether ? 87. A clock loses 5 minutes a day. It shows correct time at noon on a Monday. After how many days will it again show correct time on a Monday ? 88. A privateer, running at the rate of 10 miles an hour, discovers a ship, 1 8 miles off, making way at the rate of 8 miles an hour ; how many miles can the ship run before she is overtaken ? 89. If the wages of 25 men amount to 766. loa. 8^. in 16 days, how many men must work 24 days to receive 103$, the daily wages of the latter being one-half those of the former ? MISCELLANEOUS EXAMPLES 23Q 9O 55 gallons of a mixture of wine and water contain 5 gallons more wine than water ; find the ratio of wine to water kn the mixture. tion of 4^ tons. 92. A can do half a piece of work in 3 hours, being twice as much as B can do ; A> B and C can together do the whole in 2\ hours ; in how many hours will C do a piece of work which 8 can do in 9 hours ? 93. How many seconds will a train, 184 feet in length, travel- ling at the rate of 21 miles an hour, take in passing another train, 223 feet long, proceeding in the same direction at the rate of 16 miles an hour ? 94. A can give B 20 yards' start in a mile race and can give C 40 yards' start ; how much start can B give C in a mile race ? '96. A ptete of work must be finished in 36 days, and 15 men are set to do it, working 9 hours a day ; but after 24 days it is found that only f of the work is done. If 3 additional men be then put on, how many hours a day will they all have to labour, in order to finish the work in time ? 96. Two equal wine glasses are filled with mixtures of wine and water in the ratios of 2 of wine to 3 of water and 3 of wine to 4 of water ; when the contents are mixed in a tumbler, find the strength of the mixture. 97. Divide &47 between A y B and C in such a manner that B may receive R2 more than 3 times, and C 83 more than 4 times the amount to be received by A. 98. At what times between 2 and 3 are the hands of a clock 5-J minute-divisions apart ? 99. Three boys agree to start together and run, until all come together again, round a circular court 15 yards in circumference. One runs at the rate of 6, the second, 7, and the third f 8 miles an hour. In how many seconds will the race end ? 100. In a game of skill A can give 2?, and B can give C, 10 points out of a game of 50 ; how many should A give C ? 101. If 7 cows and 20 sheep be worth 12, and 3 cows and 16 sheep be worth ,7, find the price of a cow and of a sheep. 102. Two equal wine glasses are respectively and J full of wine ; they are then filled up with water, and the contents mixed in a tumbler : find the ratio of wine to water in the tumbler. 230 ARITHMETIC 108. Express *<5 of 17. 8a. + *5 of i, 141. 6rf. as the fraction of 8170, a rupee being worth 2 shillings. 104. A can do a piece of work in 8 days, which B can destory in 3. A has worked 6 days, during the last 2 of which B has been destroying ; how many days must A now work alone in order to complete his task ? 105. A train 1 10 yd. long overtook a person walking along the line at the rate of 3 miles an hour, and passed him completely i 9 seconds ; afterwards it overtook another person and passed him in 9 seconds. At what rate was this second person walking ? 106. In a hundred yards 1 race A can give B four and C five yards' start ; if B were to race C, giving him i yard in a hundred, which would win ? 107. If 6 men and 2 boys can reap 1 3 acres in 2 days, and 7 men and 5 boys can reap 33 acres in 4 days, how long will it take 2 men and 2 boys to reap 10 acres ? 108. Gold and silver are mixed together in a mass of 30 oz,> so that for every 6 parts of gold there are 4 parts of silver. How much gold must be added to the mass, so that for every 5 parts of gold there may be 3 parts of silver ? 109. A publican bought 10 gallons of wine at /I .7.6 per gallon ; he mixed some water and filled quart bottles with it ; how much water must have been added, supposing that the cost price of the contents of each bottle was thereby reduced to 51. 8<sf. ? 110. If 12 oxen be worth 29 sheep, 15 sheep worth 25 hogs* 17 hogs worth 3 loads of wheat, and 8 loads of wheat worth 13 loads of barley ; how many loads of barley must be given for 340 oxen ? 111. A and B are two spouts attached to a cistern. A can fill it in jo min., and B can empty it in 15 min. If A and B be opened alternately for i minute each, in what time will the cistern be filled ? 112. A race course is one mile long ; A and B run a race and A wins by 80 yards ; A and C run over the same course and A wins by 20 seconds ; B and C run and B wins by 5 seconds. In what time can A run a mile ? 113. If I can walk a certain distance in 112 days when I rest 5 hours each day, how long will it take me to walk twice as far, if 1 walk twice as fast and rest twice as long each day ? 114. A cask contains 12 gallons of a mixture of wine and water in the ratio of 3 to i ; how much pf the mixture must be drawn off, and water substituted for the mixture in the cask to become half and half? DIVISION INTO PROPORTIONAL PARTS 23 1 115. A rectangular court is 50 yards long and 30 yards broad. It has paths joining the middle points of the opposite sides, o! 6 ft. in breath, and also has within it a path of the same breadth running all round it. The remainder is covered with grass. If the cost of the pavement be I s. %d. per sq. ft. and of the grass 3* P** sq. yd. i find the whole cost of laying out the court. 116. To complete a piece of work, A would take twice as long as B and C together, and B 3 times as long as A and C together ; A y Bj C" together can do it in 12 days. In what time could each do it by himself ? 117. A down-train usually travels at the rate of 30 miles an hour and meets an up-train 50 miles from the terminus. On one occasion, on account of an accident, it only went at the rate of 20 miles an hour and met the up-train 41 f miles from the lerminus. Find the speed of the up-train. 118. A can walk 5 miles an hour, arid the rates at which A and B walk are in the ratio of 7 to 6 ; how many seconds 1 start must A give B that he may just beat him in a 3-mile race ? 119. If 5 pumps, each having a length of stroke of 3 feet, working 1 5 hours a day for 5 days, empty the water out of a mine ; how many pumps with a length of stroke of 2j feet, working 10 hours a day for 12 days, will be required to empty the same mine ; the strokes of the former set of pumps being per- formed four times as fast as the other ? 120. If 7 horses and 12 cows cost as much as 10 horses and 6 cows compare the prices of a horse and a cow. XL. DIVISION INTO PROPORTIONAL PARTS. To divide a given quantity into proportional parts is to divide it into parts which shall be proportional to certain given numbers. Example i. Divide 873 among A, B, C, so that their shares may be in the proportion of 2, 3 and 4. If we divide 8873 into 9 (r *., 24-34-4) equal parts, then A will have 2, B -will have 3 and C will have 4 of these parts. Hence A's share -R 8 $ a x 2 8194. JB's share&apx3R29i. Cs share-Raj* X 4- R3S8. Example 2. Divide ^287 into parts proportional to i$, 2 and 3i. ii : 2 : si-f : * : tf - * : V : y-9 : 12 : 20. Now proceed as in the preceding example. 3* ARITHMETIC Example 3. A certain sum of money was divided between A, /?, C in the proportion of 5, 6 and 9 ; A received 4$ ; what was the sum divided ? Since 5 + 6 4- 9 20, if the whole sum were divided into 20 equal parts, A's share would contain 5 of these parts. Hence the value of one part */- ; .". the whole Example^. Divide ^50 among A, B, C so that B's share may be half as much again as A's, and Cs share f of A's and B's together. S's sharei$ of A's share ; .*. A*s share-H-5's sharew4's share + i^ of A's share (i -I- 1) of A's share*2| of A's share; Cs share $ of 2$ or A's share | of A*s share ; A's share : B's '. C's-i I i : $ ; etc. Example 5. Divide 52 into 3 parts snch that \ of the first part of the second part 5 times the third part. \ of the 2nd part a- of the 1st part, /, the 2nd part| of the 1st part. Again } 5 times the 3rd part*$ of the 1st part, the 3rd part**^ of the 1st part. .'. ist part I 2nd part I 3rd part. ist part : f of the ist part : t \ of the ist part =i : *: A; etc. Example 6. R82 is given to 5 men, 8 women and 10 boys, in such a way that a woman is to receive twice as much as a boy, and a man as much as a woman and a boy together ; what do the women receive ? 8 women receive as much^as 16 boys ; and 5 men receive as much as 5 women and 5 boys, or as 10 boys and 5 boys, or as 15 boys ; /. men's share I women's \ boys* 15 : 16 : 10 5 etc. Example 7. How many rupees, half-rupees and quartcr-rupees, of which the numbers are proportional to 3, 4 and 5, are together worth 850 ? DIVISION INTO PROPORTIONAL PAJLTS^ 233 9 \J - X/^y* Values of three groups of coins are / as 3 rupees ' 4 half- rupees I 5 quarter-rupees, or as 12 quarter-rupees I 8 quarter-rupees '. 5 quarter-rupees, or as 12 *. 8 ; 5 ; .". the amount in rupees " the amount in half-rupees = and the amount in qr.-rupees R$Jx 5**Rlo. Therefore there are 24 rupees, 32 half-rupees, and 40 qr.-rupees. Example 8. Divide ico between A, B, C, D, so that A's share I Z?'s2 I 3, 's I 's4 ; 5, and Cs I Z?s = 7 : 8. We find as in Ex. 4, Art. 216, that the shares of A, B y C, Z?are ^proportional to 56, 84, 105 and 120 ; etc. EXAMPLES. 14O. 1. Divide 815. loa. into parts proportional to i, 2, 3, 4. 2. Divide ;iS. 91. into parts proportional to 3, 2^, i, J. 3. Divide 26 tons in the proportion of 3*5, 2*25, 3^, 3}$. 4. Divide 532$ into parts which shall have the same ratio to one another as ^, , |, , J. 5. Divide ^4. 17^. 6J. into two parts one of which is f of the other. 6. A sum of money was divided into parts proportional to 3f , 4, 5*5 ; the smallest part was 830 ; what was the sum divided? 7. A sum of money was' divided between A, B, C, in proportion to their ages which were 10, 12, 13 years respectively ; A'a share was ^55 ; find the other shares. 8. Gunpowder is composed of saltpetre, sulphur and charcoal, in parts proportional 1075, 10 and 15; how many pounds oi charcoal are there in 6 cwt. of gunpowder ? 9. How much of the above gunpowder can be made with 25 Ib. of sulphur? 10. In a certain battle an army lost 4 men wounded and 2 killed out of every 25, and it mustered 38,000 men unhurt ; what was the number of men in the army at first ? 11.' Divide RQO between three persons, so that for every rupee given to the first, the second may get 12 annas and the third may get 8 annas. 12. Divide 8.36 between A, B and C, so that A gets } of 's share, and C gets f of A's share. > 234 ARITHMETIC 13. Divide 8360 among A, B, C, so that A may get 3 times as much as B, and B and C together $ as much as A. 14. Divide 32 between A, /?, C, so that ^4 may receive 3 times as much as /?, and C J of what ^4 and B together receive. 16. Divide 14 between A and /?, so that of *4's money may be equal to f of Z?'s. 16. Divide 30 into 3 parts such that of the first part| of the second of the third. 17. 821 is divided between A, B, C ; A's share is f of #'s ; it is also f of 2?s and Cs together ; find each one's share. 18. Divide i. 13*. 4\d. between A, B, C, D, so that A's share may be & of J^s, (?s share ^ of A's t and /?'s share the sum of A's and C's. 19. Divide ^3. 6s. between 5 men, 7 women and 10 boys, so that each woman may have $ of each man's share, and each boy f of each woman's share. 20. Rno is to be divided among 10 men, 16 women and 20 children ; if each man's share is to be equal to the shares of 2 women and the 16 women are to have twice as much as the 20 children, how much will each woman receive ? 21. A number of men, women and children are in the propor- tion of 3, 4, 5 ; divide ^3. $s. ^d among them, so that the shares of a man, a woman and a child may be proportional to 4, 3, i 22. Divide ^39 among A, B, C, so that A's share I Z?'s share 3 I 2i 2?a share I C's share 4 I 3. 23. A certain kind of brass is composed of copper, zinc, lead 1 and tin ; the ratio of copper to zinc is i I 2, of zinc to lead 3 I 5 and of lead to tin 7 I 8 ; find the quantity of zinc in i cwt. of the brass. 24. Four towns are to provide according to their population a contingent of 140 men. The populations of the towns are 1058, 1587, 2116 and 2645 respectively ; find the number of men to be provided by each town. 25* 700 coins consist of rupees, half-rupees and quarter-rupees ; the values of the rupees, the half-rupees and the quarter-rupees are as 2 1 3 I 5 ; find the number of the rupees. 26. How many rupees, eight-anna pieces and four-anna pieces, of which the numbers are proportional to 2$, 3 and 4, are together worth R8o ? 27. If 2 men do as much work as 5 women, and 6 women as- much as 10 children, divide a week's wages of 38 among 8 meo,. 9 women and 15 children. FELLOWSHIP OR P 28. The sum of three fractio- tlmes the second 1 8 times the 29. Divide 8142 among A A, B may get BS, and for eve* 30. Areas of circles ar' radii. Divide a circle concentric circles. 31. Iftheweif the ratio of n to per oz. avoir., find value to be that of 32. An estate 8 and 10. Find thi largest share woulc 7 33. A number in shares which are least be that this mi XLI. FE: 3. Suppose x trading, and that A \ RSOOO and C has R6c profit be divided ? It is obvious that tional to 3000, 5000 explained in the r The above : ship, the ca supposed to 8*4. ? that A ha' 5000 foi which tir Now, to be eqi for I mo 30000 cquivale profit mi which m Const cnt perio ITHMETIC **\l by the measure of the corres- mms of money must be express- ^ the several periods of time. *^t is called Compound ^v the several partners x eriods of time. ishes RSSO, B 500 i in 8320 profit ? ,tors, namely, 81200 a Ryoo ; what does th a joint capital of oo, jf the profit?, and B come is increased by capital to J Q . Find which A has 7$ annas ' partner, receives ^ rtion to the capital ; Capital of ,18000 ; ban C ; divide a for 5 months, \m o oxen for 5 oxen for at should e i6th of Capital of jcember. )0 and f nd a new e end of ALLIGATION 237 5 months E trebles his capital. The year's profits amount to Ri2oo ; how ought this to be divided ? 11. A and B start a business with capitals as 5 I 7. They withdraw respectively and j of their capitals at the end of 4 months. At the end of the year a profit of 226 is divided ; find A's share. 12. A and B entered into partnership with ^700 and ,600 respectively. After 3 months A withdrew \ of his stock but after 3 months more he put back f of what he had withdrawn. The profits at the end of the year are ^726 : how much of this should A receive ? 13. A and B start a business, A puts in double of what B puts. A withdraws of his stock at the end of 3 months but at the end of 7 months puts back J of what he has taken out, when- B takes out J of his stock. A receives 300 profits at the end of the year ; what does B receive ? 14. A and B hire a meadow for 6 months, A puts in 21 cows for 4 months ; how many can B put in for the remaining 2 months, if he pays ^ of what A pays ? XLII. ALLIGATION. 5. The following are examples of Alligation or the mixing of things of the same kind but of different qualities. Example i. How must a grocer mix teas at 2s. 6*/. a Ib. and 3J, 9< a Ib. so that the mixture may be worth 3$ a Ib. ? When the mixture is made and sold at $s. a Ib., each Ib. of the cheaper tea in it brings a gain of 6^., and each Ib. of the dearer tea brings a loss of yd. Therefore 9 Ib. of the cheaper tea brings a gain of 54^. and 6 Ib. of the dearer brings a loss of 54^. Hence,, in order that there may be neither any gain nor any loss, for every 9 Ib. of the cheaper tea we must take 6 Ib. of the dearer ; therefore the proportion is 9 parts to 6, that is, the tea* must be mixed in the inverse ratio of (he differences of the two prices and the mean price. Example 2. In what proportion should teas at 2s. 6</., 35., 45. 3</% and ^s. 9^., a Ib. be mixed to make a mixture worth 4*. a Ib. ? The first two prices are under, and the last two above, the mean price. We take equal quantities of the teas at the first two prices, and the mixture is worth 2s, <)d. a Ib. ; we also take equal quantities of the teas at the last two prices, and the mixture is worth 45. 6<# a Ib. Now we mix these two mixtures as in Ex. i, and we find 4 that these must be taken in the proportion of 6 to 15 or 2 to 5, Consequently the teas are mixed in the proportion of I, i, j. {. *38 ARITHMETIC Note. Instead of taking equal quantities we might take the teas in any proportion to make the first two mixtures ; and conse- quently an example of this kind (in which the number of ingredi- ents is more than two) may have an unlimited number of solutions. Example 3. In what ratio must a grocer mix sugar at 6a. per seer with sugar at 40. per seer so that by selling the mixture at 5#. 3^. per seer he may gam J of his outlay ? ij of the cost price of a seer of the mixture*- 50. 3^. ; .*. cost price of a seer of the mixture*- $a. 3^. 4- i40, 6fl. Now proceed- ing as in Ex. i, we find that sugar at 6a. per seer must be mixed with sugar at 4*. per seer in the ratio of (40. 6^.-4.) to (6a.-4a. 6/>.) i.e., of i to 3. EXAMPLES. 14fl. 1. How must sugar at 4a. per seer be mixed with sugar at 50. per seer to make a mixture worth 4*3. 3^. per seer ? 2. In what ratio must tea worth 2s. yd. per Ib. be mixed with tea worth 35. 8d, per Ib. to make a mixture worth 3$. per Ib. ? 3. Tea at 2s. 6d. per Ib. is mixed with tea at 45. 2d. per lb. f and the mixture is sold for 3$. 5^. a Ib. ; how were they mixed ? 4. In what ratio must a grocer mix coffee at $s. per Ib. with chicory at i<L so that by selling the mixture at is. per Ib. he may gain <r\ of his outlay ? 6. A grocer buys black tea at 2s. 6d. per Ib. and green tea at 31, 9*/. per Ib. ; how must he mix them so that by selling the mixture at jj. per Ib. he may gain of his outlay ? 6. In what proportion should water and wine at 12s. 6d. a gallon be mixed to reduce the price to 105. a gallon ? 7. Currants at id. per Ib. are mixed with currants at gd. per Ib. to make a mixture of 17 Ib. worth 7^. per Ib. ; how many pounds of each are taken ? 8. A person bought 60 md. of rice of two different sorts for 153. i2fl. 'I he better sort cost &3 per md. and the worse &2. 4#. per md. How many maunds were there of each sort ? 9. A liquid P is if times as heavy as water, and water is if times as heavy as another liquid Q ; how much of the liquid P must be added to 7 gallons of the liquid Q so that the mixture may weigh as much as an equal volume of water ? 10. A mass of gold and silver weighing 9 Ib. is worth ^3 18, 131. 6d. \ if the proportions of gold and silver in it were inter- changed, it would be worth ^129. los. 6d. ; supposing that the .price of geld is ^3. 171. io$d. per oz., find the proportion of gold and silver iu the mass, and the price of silver per or. AVERAGE VALUE 239 11. A merchant has wines worth 7^., gs., iis. and 155. a gallon respectively : how must he mix them to obtain a mixture worth los. a gallon, using equal parts of the first two kinds, and also equal parts of the last two kinds ? 12. In what proportion must a grocer mix teas at 2s. 6d., 3J. and 4*. 6d. per Ib. to make a mixture worth 4*. per lb,, using equal parts of the first two kinds ? 13. A man has whisky worth 22s. a gallon, and another lot worth iSs. a gallon ; equal quantities of these are mixed with water to obtain a mixture of 50 gallons worth i6j. a gallon ; find how much water the mixture contains. 14. A grocer buys teas at 2s. 6^., 3*. and 3*. gd. per lb. res- pectively : how must he mix them so as to obtain a mixture worth 3,?. 3< per lb., using the first two kinds in the proportion of 2 to 3 ? 16. A grocer wishes to mix teas at 2j., 35., 31. 6d. and 4*. per Ib. respectively ; how must he mix them (using the first two kinds in the proportion of 2 : 3, and the last two in the proportion of 3 I 4) so that by selling the mixture at 3^. 4^. per lb. ^ of the receipts may be clear profit ? XLIII. -AVERAGE VALUE. 6. The average or mean value of any number of quanti- ties of the same kind is their sum divided by the number of them. Example. Find the average age of four boys who are 10, u, 13 and 14 years old respectively. Average age-^ 10 *^- 1 -** 1 * years 12 years. EXAMPLES. 143. Find the average of the numbers, 1. 1,2,3,4,5. 2. 8,10,13,15,17,20. 3. 3i 7?, 81,9$, 10. 4. 1-3, 7-6, 8-9, 3-1, -8. 6. Find the average age of five boys who are 15, 13, 11, 9 and 8 years old respectively. 6. What was the averaee daily expenditure of a man in 1880, who spent 876 5 . 10 . 9 in the first half-year and R88i .5.3 in the last ? 7. The population of a town was 287 50 in 1870 and 30000 in 1880 ; find the average annual increase between the two dates. 8. Of 20 men 12 gain ,3. 7;. each and 8 men gain 2. 81. each ; what is the average gain per man ? 2^0 ARITHMETIC 9. Five men weighed respectively 8 st. 8 lb M 9 st. 4 lb., 10 st.,. 10 st. 10 lb. and II st. 6 lb.; what is the average weight per man ? 10. If 20 chairs are bought at &5 each, and 15 at 4. 8 a. each, and 15 more at &4 each, what is the average price of a chair ? 11. A train travels i mile in the first 10 min., i miles in the next 10 min., 2 miles in the next, i miles in the next, and r mile in the next : what is the average speed of the train per hour ? 12. The average weight of 6 men is 10 st. ; two of them weigh 9 st. 7 lb. each ; find the average weight of the others. 13. The average age of 8 men, 7 women and i boy is 45 years, that of the 8 men being 48 years and of the 7 women being 46 ; determine the age of the boy. 14. The average age of 5 children is 7 years, which is increas- ed by 6 years when the age of the father is included ; find the age of the father. 15. The average weight of 7 men is diminished by 3 lb. when one of them who weighs 10 stones is replaced by a fresh man ; find the weight of the new man. 16. The average age of a class of 20 boys is 12 years ; what will be the average age if 5 new boys receive admission in the class, whose average age is 7 years ? 17. If the chairs in Question 10, are sold so as to gain J of the cost price, what is the average selling price of a chair ? 18. The average price of a chair, a table and a cot is Ri9 ; the aveiage price of the table, the cot and a book-shelf is R22 ; if the price, of the book-shelf be Ri6, find the price of the chair. 19. The average temperature for Monday, Tuesday, Wednes- day and Thursday is 6o c '; the average for Tuesday, Wednesday, Thursday and Friday is 63* ; if the ratio of the temperatures for Monday and Friday be 21 I 25, find these tern pet dtures, XL1V. PERCENTAGE. 7. The term per centum or per cent, means for a hundred* Suppose that a trader who has a capital of ^4000 gains R2oo ; he gams RS for every hundred of his capital. This is expressed by saying that the trader's gain is 5 per cent. Note. The symbol % or the letters />. c* are used as an abbre- viation for the words per cent. Example i. What fraction of a number does 5 p. c. of it denpte ? 5 p. c. of a number * T $3 of the number**^ of the number* PERCENTAGE 341 Example 2. How much is 6J p. c. of R 320 ? The percentage- of &32O- of 310- 20. EXAMPLES. 144. What fractions are denoted by the following rates per cent. ? 1. I2|. 2. 33$. 3. J. 4. . 6. 125. Find the value of 6. 5 p. c. of 8700. 7. 7$ p. c. of 140. 8. j p. c. of 20. 9- 35% of 3480 men. 1O. J% of a sq. ft. 11. 8-5% of 50 cwt, 12. A man's income is 83000 a year ; if he spends 6J p. c. of it each month, how much does he save in a year ? 13. Five per cent, of the total population of a town are Englishmen ; the rest are Hindus ; if the population of the town be 37820, what is the number of Hindus ? 14. A man's income in 1871 was 500 ; in 1872 it was in- creased by 20 p. c. ; what was his income in 1872 ? 15. Find the difference between f of 870 and j p.j:. of 870. 16. A testator bequeathed by will f of his estat^ to his son, 60 p. c. of the remainder to his daughter, and the remainder to his widow : the son got ,75 more than the daughter. How much did the widow receive ? Example 3. What rate per cent, does the fraction denote ? r^_ f 3 3x100 agft 37^ The fract.cn, |.|__*-I- = i- ; .". rate per cent. 37$. Example 4. What per cent, of 840 is 83 ? /. rate per cent 7 j. EXAMPLES. 145. What rates per cent do the following fractions denote ? I. J> 2. J. 3. B V 4. f. 5. f 6. ,V 7. f. 8. . 9. }. 10. |g. What per cent of II. R26isRi3? 12. B4oisB8? 13. ^3 is 12*. ? C. A. 16 242 ARITHMETIC 14. '25 is } ? 16. } is 7 ? W- -6 is '3 ? 17. Of 3420 men in a town, 420 died ; what per cent, survived ? 18. Out of a debt of 8,2500, RIG.OO is paid ; what per cent, of the debt still remains unpaid ? 19. The number of boys in a school in January was 3 2 J ' February it increased to 360. Find the increase per cent. 20. A mass of gunpowder is made with 2 Ib. 5^ oz. of nitre, 5 oz. of sulphur and 7^ oz. of charcoal ; find the percentage composition of the powder. 21. Standard gold contains n parts pure gold out of 12 ; what per cent, is dross ? Example 5. Of what sum of money is R3O, 5 p. c. ? 5 p. c. of the sumR3o, or r8u of the sum R3o ; the sum - 30 x *$& =* R6oo. EXAMPLES. 146. Of what number is 1. 22, 10 p. c. ? 2. 57, 4} p. c, ? 3. 30, 120 p. c. ? 4. 81, i p. c. ? 6. 2$, 2 p. c, ? 8. 3, '27 p. c. ? 7. -A man spends 3250 a year, which is 66p. c. of his yearly income ; find his income. 8. A man spends 60 p. c. of his income and saves 82000 ; what is his income ? 9. The population of a town increased 7 p. c. from 1880 to 1883, and its population in the latter year was 13910 ; what was its population in 1880 ? 10. If a tax of 10 p. c on the income of a man yields 300, how much will an income-tax of 5 pies in the R produce ? MISCELLANEOUS EXAMPLES. 147. 1. The price of a bottle of red ink is 20 p. c, more than that of a bottle of black ink. If a bottle of red ink costs 12 annas, how much will a bottle of black ink cost ? 2. A trader in his first year gains 8 p. c. of his capital, but in the second year loses 10 p. c. of what he had at the end of the first year, and his capital is R224 less than at first ; find his original capital COMMISSION, BROKERAGE, PREMIUM . ?43 3. A trader's capital increased 10 p. c. every year ; at the end of 3 years it was 6050 ; what was his capital at first ? 4. In a mixed school 25 per cent, of the scholars are infants under 7, and the number of girls above 7 is f of the bojs above 7, and amounts to 36 ; find the number of children in the school. 5. A man spends 5 p. c. of his income in insuring life, and this part is exempted from income-tax ; his income-tax which is laid at 4 pies in the rupee, amounts to R3o. 50. ; find his gross income, 6. Three casks contain equal quantities of wine ; a mixture is formed by taking 25 p. c. of the first cask, 35 p. c. of the second and 45 p. c. of the third ; what per cent, of the whole quantity is taken ? 7. Two mixed schools have 90 and 120 children respectively ; in the first 60 p. c. and in the second 50 p. c. of the children are boys ; what per cent, of the children in the two schools are boys ? 8. In a town the numbers of male and female inhabitants are 3450 and 3020 respectively ; the decrease in the former is 10 p. c. while the increase in the latter is 5 p. c. Find the increase or decrease per cent, of the total population. 9. In a mixture of coffee and chicory the coffee is 40 per cent; to 500 Ib. of the mixture a quantity of chicory is added, and then the coffee is 36^ p. c. How many pounds of chicory are added ? 10. If A's income be 10 per cent, more than JB's, how much per cent, is Z?s income less than A's ? 11. A sells his goods 10 per cent, cheaper than Z?, and 10 per cent, dearer than C ; how much per cent, are Cs rates lower thank's? 12. The price of sugar being raised 10 p. c. by how much per cent, must a man reduce his consumption of that article so as not to increase his expenditure ? XLV, COMMISSION, BROKERAGE, PREMIUM. 228. Commission is the sum of money paid to an agent for Jauying^ojLseUing^ goods or property of any kind. It is usually a percentage upon the yalue_of goods ^bought or sold. The agent is sometimes called a broker, especially when he buys or sells Government Promissory Notes, Shares of Companies, etc., and the commission, brokerage. Premium is the sum of money paid to an Insurance Company which, in consideration thereof, undertakes to matte good a loss incurred through fire or shipwreck, or to pay a certain sum of money after a man's death to his relatives. The instrument containing the contract is called the Policy of Insurance ; and the stamp duty on the policy is called the Policy duty *44 ARITHMETIC Premium is usually a percentage upon the sum of money which the insurer or his relatives are to receive. Commission, Brokerage and Premium are therefore names given to ^percentage. in particular cases. Example i. An agent buys goods worth 8750, and receives a commission of 2$ per cent. ; how much does he get ? Commission^ of R75o& Ri8. I2a. Example 2. A cargo, valued at ^760, is to be insured at 5 p. c. premium ; what sum must be insured that, in case of loss, the value of cargo and the premium paid may be recovered ? If every /95 (;ioo-;5) be insured for .100, then in case of loss both the value of goods and premium paid will be recovered. Now since ,95 must be insured f6r ,100, :. 760 or ;8oo. Arts. EXAMPLES. 148. 1. A broker purchases goods worth 5000 ; what is his com- mission at 3^ per cent. ? 2. What is the cost of insuring cargo valued at 7000, the premium being 34 per cent. ? 3. A commission agent sells 720 bales of jute at R; per bale ; what commission does he receive at i per cent. ? 4. An agent buys a house for 6750, and receives commission at 3. I2a. per cent. ; what has his employer to pay altogether ? 5. A broker received J p. c. for buying Government Promis- sory Notes. His brokerage amounted to 3 5 ; what was the value of the Promissory Notes bought ? 6. A ship is insured for J of its value at if p. c., and the premium is 20 ; what is the ship worth? 7. The premium on a policy of insurance at 4 p. c. is E, 120 ;. find the amount of the policy. 8. How much must be paid to insure a cargo worth the premium being 251., policy duty is. 6d, and brokerage per 100 respectively ? 9. For what sum must a merchant insure a cargo worth 89760 at 2} p. c. so that in case of loss both the cargo and premium may be recovered ? PROFIT AND LOSS 24$ 10. Goods worth 7740 are insured at 3$ per cent., so that in case of loss both the value of goods and premium may be recovered ; find the amount of premium paid. 11. Cargo worth 5000 is to be insured, so that in case of loss its value and all the expenses connected with its insurance may be recovered. The premium is 2^ per cent, policy duty g per cent, and brokerage J per cent ; for what sum must the cargo be insured and what is the amount of the whole expense paid on insurance ? XLVI. PROFIT AND LOSS. . Under this head we estimate a profit or a loss, joqt afeso- y, but in relation to the^cost price, that is, as so much pej cent on the cost price. Example I. If chairs are bought at R$ each, and sold at RS. 90. each, what is the gain per cent.? The gain is 9. on 5 or 8o<*. ; and we have to find what per cent, of 800. is ga. Now, the fraction-i-- - _ 80 80x100 loo ioo .". the gain is nj per cent. Example 2. A horse is bought for R8o, and is sold at a profit of 25 p. c. , what does the profit amount to, and for how much is the horse sold ? Profit 2 5 p. c, of:R8o = T 3 5 tf of R8oR2o. /. The horse is sold for R8o + R2o, or Rioo. Example 3. Some goods are bought for 890 ; for how much must they be sold so as to gain 10 per cent. ? , The selling priceno p. c. of cost price I -}JgofR9o=R99. Example 4. By selling sugar at Ri2 per md. I gain 20 p. c. ; at what price per md, did I buy it ? 120 p. c. of the cost price selling price, or |Jg of the cost priceRl2 ; the cost price Ri2x}8JJ= Rio. Example 5. If 10 p. c. be lost by selling an article for R;2, for iiow much should it have been sold so as to gain $ per cent. ? 90 p. c. of the cost price R72, 15 .......... - .................. -&I2f 105 ........ ..... ....... . ........ -R84. Ans. 246 ARITHMETIC Example 6. By selling a house for ^69 there is a loss of 8 p. c.; what would be the loss or gain per cent, by selling it for ^69=^92 p. c. of the cost price, /. i -II ............................. , =104 .............................. .". There would be a gain of 4 per cent. EXAMPLES. 149. 1. I sell for &2o that for which I gave Ri6 ; what is my gain per cent. ? 2. At what rate, per cent, is the loss on selling for 11 . 9 . 8J what cost ^15.6.3? j 3. I sell 20 articles for the same money as I paid for 25 ; what do I gam per cent, on m,v outlay ? 4. If the selling price of f of a number of toys be equal to the cost price of the whole, find the profit per cent. V^ >itf 6. 70 gallons 01 wine are bought for ^50, ancrcf gallons are lost by leakage : the remainder is sold at is. io\d. a pint ; find the gain or loss per cent, on the outlay. 6. Certain articles are bought at 12. i$s. for 100, and are sold at 2\ guineas for a dozen ; find the gain or loss per cent. 7. A person by selling 48 yards of cjpth gained the cost of 16 yards ; find the gain per cent. ^Yi <<v % ^>!Qp 8. 320 maunds of rice were bought ai ft 5 per maund, and sold at a loss of 5 p. c. ; find the total loss and the selling price per seen 9. A merchant buys certain goods at 6 . 19 . 3 per cwt. and pays 15^. per ton for expenses ; at what price per Ib. must he sell them so as to gain 15 p c. on his total outlay ? 10. If oranges are bought at the rate of 15 for a rupee, how many must be sold for a rupee so as to gain 25 p, c. ? 11. The cost price of a book is js. 6d. ; if the expenses of sale be 5 p. c. upon this, and the profit 20 p. c., what would be the retail price ? 12. 24 gallons of ale are bought at 2s. a gallon and 30 gallons of porter at is. a gallon, and they are mixed together. If i| gallons of the mixture be lost by leakage, and 20 gallons sold at 2s. $d. a gallon, at what price per gallon must the remainder be sold to gain 20 p. c. on the whole outlay ? IS. A man having bought a quantity of tea for 875, sells i of it at a loss of 4 p. c. ; by what rate per cent, must he raise that PROFIT AND LOSS 347 selling price) in order that by selling the rest at the increased price he may gain 4 p. c". on his outlay ? 14. I bought note paper at the rate of 8 annas for 5 quires, and sold it so as to gain as much on the cost of 32 quires as 8 quires were sold for ; at what price did I sell the paper per quire ? 15. A horse is sold for 440, at a loss of 12 p. c. ; how much did it cost ? 16. A quantity of sugar is sold at 6a. gfl. per seer ; the gain is i2| p. c. and the total gain is Ri5. What is the quantity of sugar sold ? 17. If oranges are sold at the rate of n for the rupee, and the gain is 8J p. c., at what rate were they purchased ? 18. A bankrupt's stock was sold for 85205 at a loss of 17 per cent, on the cost price ; had the stock been sold in the ordinary course of trade it would have realized a profit of 20 per cent. How much was it sold under the trade price ? 19. A horse was sold for 8240 at a loss of 5^ p. c. ; for what should it have been sold to gain 26 p. c, ? 2O By selling tea at $s. per Ib. a grocer gains only 5 p. c. ; by how much must he raise the price so as to gain 15 p. c, ? 21. If by selling 7 mangoes for Ri . 2.4$ there be a profit of i6 per cent, at what price per dozen must they be sold to gain 20 per cent. ? 22. If a man lose 4 p. c. by selling oranges at the rate of 12 a rupee, how many a rupee must he sell them so as to gain 44 p. c.? 23. If by selling goods for 8141 there be a loss of 6 p. c., what will be the loss or gain per cent, by selling them for 8159 ? 24. Goods were sold for 837. 8a. with a gain of I2j p. c. ; what would have been gained or lost by selling them for 833. 80. ? 25. Tea which cost R6o per md. is retailed at 82. 8a. per seer, and there is a waste of 10 p.c. ; what is the rate of profit per cent. ? 26. Sulphuric acid worth 3</. per Ib. absorbs moisture and becomes 2$ p. c. heavier ; what is it then worth per Ib. ? 27. A merchant sells tea to a tradesman at a profit of 40 p. c.| but the latter becoming bankrupt pays only 12$. in the ; how much per cent, does the merchant gain or lose on his outlay ? 28. A tradesman's prices are 30 p. c. above the cost price ; if he allows his customers 10 p. c. on his bill, what profit does he make ? 29. How much per cent, must a tradesman add on to the cost price of his goods, that he may make 20 p. c profit after allowing his customers a reduction of 5 p. c. on his bill ? 30. The price of flour being raised 20 per cent, by how much 248 ARITHMETIC per cent, mast a man reduce his consumption of that article so as not to increase his expenditure ? 81. An article when sold at a gain of 5 p. c. yields Ri$ more than when sold at a loss of 5 p. c. ; what was its prime cost ? 32. A man sells an article at a loss of 10 p. c. ; if he had received R$ more, he would have gained 12^ p. c. What did the article cost him ? 33. A piece of cloth is sold for R4o. loa. at a profit of 30 p. c. If it had been sold at Ri. i2a. per yard, the profit would have been Ri2. Sa. ; how many yards are there in the piece ? 34. A man embarks his capital in three successive ventures. In the first he clears 80 p. c., and in each of the others he loses 15 p. c. ; what per cent, does he gain or lose on his original outlay ? 35. A boy buys a number of apples at 6 (or 40. and a third of the number at 4 for 2a. : at what rate must he sell them to gain 20 p. c. on his outlay ? Supposing his total profit to be &4, how many did he buy ? 36. How must a grocer mix teas at 3*. a Ib. and $s. Gd. a lb.. so that by selling the mixture at 3;. 8*/. a lb. he may gain 10 p. c. ? 37. I must sell my stock of sugar at 30. 6p. per lb. to gain 33i P er cent J by pixing it with an inferior sugar in the propor- tion of 4 to I I gain 33$ p. c, by selling at Ri. ga. 6/. for 7! lb. Find the cost of the inferior sugar per lb. 38. A grocer proposes to sell his tea at xo per cent, profit, but adulterates it by adding J of its weight of an inferior tea which cost him } of the price of the better ; what profit- per cent, does he make ? Also in what proportion must he mix the two kinds so as to gain 20 per cent.? 39. A merchant buys 1575 cubits of cloth. He sells of it at a gain of 6 p. c., i at a gain of 8 p. c., } at a gain of 12 p. c. and the rest at a loss of 3 p. c. If he had sold the whole at a gain of 5 p. c. he would have received Ri2o. iza. more than he did. What was the prime cost of a yard ? 40. How must wine at zos. a gallon and brandy at 45*. a gallon be mixed, so that by selling the mixture at 35$. a gallon there may be a gain of 1 5 p. c. on the price of the wine and 20 p. c. on the price of the brandy ? 41. A mixture ot two kinds of wine, at zos. and 25$, a gallon, is sold at a gain of 10 p. c. If the two kinds had been sold separately at a gam of 1 5 p. c. and 8 p. c. respectively, the total profit would have been the same. In what proportion were the two kinds of wine mixed together ? 42. A tradesman by means of false balance, defrauds to the extent of 10 p. c. in buying goods, and also defrauds in selling. What per cent, does he gain on his outlay by his dishonesty ? SIMPLE INTEREST 949 43, A man sells a house, at a loss, for 8400 ; had he sold it for &5oo his gain would have been f of his former loss ; find the cost price of the house. 44. A merchant has goods worth .300 ; he sells ore-third of them so as to lose 10 p. c. By how much per cent, should he raise that selling price in order to gain 10 p. c. on the whole ? XLVII. SIMPLE^NTREST. 30. Interest is money paid for the use of money lent. The money lent is called the Principal. The Amount is the sum of the principal and interest at the end of any time. The rate of interest is the money paid for the use of a certain sum for a certain time. Thus, if I borrow a sum of money on the condition that for the use of every rupee in the loan for a month I shall pay an interest of ^ anna, I am said to borrow at the rate of \ anna per rupee per month. Again, if I borrow on the condition that for the use of every Eioo in the loan for one year 1 shall pay an interest of &5, I am said to borrow at the rate of 5 per cent, per annum. Note. Per annum means^r a year. 231. When interest is calculated simply on the original prin- cipal it is called Simple Interest. Wote. 1. The term interest is generally used in the sense of simple interest. Example i. Find the simple interest on 24 for 5 months at \ anna per rupee per month. Interest on R I for i month M $a. / ................ R24 for i month / ................ R24 for 5 months Hence, to find the interest we multiply the principal by 5 and by ^f, that is, we multiply it by ^j. The work in practice should "stand thus : R. 24 _J 32) 120 (R3. 120. Ans. & 24 j6 384(12 3? I 4 64 150 ARITHMETIC EXAMPLES. 150. Find the simple interest on 1. Rs8 for 4 months at 6p. per rupee per month. 2. R76 for 9 months at 2 pice per rupee per month. 3. R240 for i year at 3^. per rupee per month. 4. R37S for 15 months at f anna per rupee per month. 6. R29 for 3 years 3 months at 2p. per rupee per month. 6. R720 for 1 8 months at qp. per rupee per month. Example 2. Find the simple interest on R728 for 5 years at 4 per cent, per annum. Interest on Rioo for I year =R4, ." Ri for i year / R728 for i year ] / R728 for 5 years = Ri45. ga. 7\p. Hence we deduce the following rule : Multiply the principal by the rate per cent, and by the number of years ) and divide the product by 100. The work should stand thus : R. We divide Ri456o by 100 by 728 cutting off the two figures on the . _4 right ; thus the quotient is Ri45 2912 and R6o is the remainder ; this 5^ remainder is equal to 9600. ; this 100 ) Ri45 60 divided by loo gives ga. as quo- S i6 tient and 6oa. as remainder ; a g ^ Q this remainder is equal to 720^. ; i-* this divided by 100 gives 7*2^. - as quotient. _ A 7v 2 o .'. Interest*' 10 *^ "- ** Note 2. The amount may be obtained by adding the interest to the principal. Thus the amount in the above example If the amount only is wanted we may also proceed thus Interest on Rioo for 5 years 4 p. c. =R2O. .'. The amount of Rioo in 5 8728 ............... -8873. SIMPLE INTEREST EXAMPLES. 151. is understood to be per annum unless 2. /soo for 4 yr. at 5 p. c. 4. ;i28 for 15 yr. at 3 p. c. 6. ;8oo for 3j yr. at 4 p. c. r. at 2|%. N. B. The rate per cent otherwise stated. Find the simple interest on 1. &2oo for 3 yr. at 4 p. c. 3. 8750 for 7 yr. at 6 p. c. 6. R450 for n yr. at 4^ p. c. Find the simple interest and the amount of 7. R49$. 4- for 2J yr. at 3%. 8. ,325. 5 ' for 4 9. 8225. Ha. 9A for 4 years at i per cent, per month. Find the amount only of 10. 8250 for 2 yr. at 7 p. c. 11. ^304 for 5 yr. at 4^ p. c. 12. 8335 f r 3^ years at j per cent, per month. 13. ^720. 8j. 6< for 2j years at 2$ per cent. JL4. ^329 9*. 4^. for 7^ years at 3j per cent. 15. ;22o for 7 months at 4} per cent. Note. 3. When the rate per cent, and the number of years (or either of them) are fractional numbers, it is convenient first to multiply these two, and then multiply the principal by the product. Example 3. Find the simple interest on 8345. loa. $p. for 2 years 6 months at sJ per cent. Now, 2 years 6 months =*2i years ; R. 345 - a. 10 A 3 5 1728 . 3 3 7 12097 . 6 9 3 8 ) 36292 . 4 3 R45.36 . 16 8 a. 5.84 12 4) IO u See Example 2. The interest -45. $a. -R45. $. 252 ARITHMETIC EXAMPLES. 15. N. JB. When the time is given in months and days, 12 months are reckoned to the year, and 30 days to the month. Find the simple interest on 1. R375 for 3^ years at 2 per cent. 2. ;*5o for 6| years at 3j per cent. & ;875 f r 3 years 4 months 15 days at sJ per cent. Find, to the nearest pie, the simple interest on 4, I&3C9. loa. 3A for 5 months 10 days at 4j per cent. 6. 3&2i. 150. 9^, for 2 years 9 months at 3^ per cent. 6. RIOI. 130. for i year 7 months 6 days at J per cent, per month. Note 4. When interest has to be calculated from one day of the year to another, it is customary to include one only of the days named. Example 4. Find the interest on ^320 from January 4th to May 3oth, at 3 per cent. Number of days 27 + 28 -I- 3 1 + 30 + 30 146 ; 146 days $5 of a yearf yr. ; and 3*! = $. 320 6 5 ZO _ s. 16 80 _ 12 ^960 .*. the interest ^3. i6j. Note 5. It should be noted that factors of 365 are 5 and 73. EXAMPLES. 153. jV. B. When the time is given in days or years and day$ t the yea* is taken to consist of 365 days. Find the simple interest on L ^400 from April 4th to June i6th at 3 p. c. 2. 750 from Feb. 23rd to Sep. 3oth at 4} p. c. SIMPLE INTEREST 2 93, 3. 321. 8a. from Dec. ioth., 1887^ to May 4th*, 1888, at 3? p.c, 4. ^847. 15$. from Jan. 1st to April ist at 2} p. c. 6. R349. 80. o/fr. from June ist to Oct. 4th at $J p. c. 6. 8309. 120. for i year 73 days at 2| p. c. >x 233. Inverse questions on Simple Interest. Example i. At />*/ */* per cent, will 425 amount to 476 in 3 years ? Interest on 8425 for 3yearsRsi, (i.e. &476-R425). / ................ Ri for 3 years / ................ Ri for i year ; ................ Rioo for i year R .'. the rate per cent. 4. EXAMPLES. 154. At what rate per cent, will 1. RSOO amount to R337. 80. in 5 years ? 2. R825 amount to RQQ5. 70, in 3 years ? 3. ,142. 10^. amount to ^163, 13*. u\d. in 4^ years ? 4. The interest on 22214. 4 a - amount to ^462. 120. gp. in> 7 months 10 days ? 6. A given sum of money double itself in 20 years ? 6. The interest on any sum of money be f ths of the amount in 20 years ? 7. The interest on 1368. i$s. become 14. 45. i\d. from July 5th to Nov. 2oth ? 8. At what rate per rupee per month will R25o amount to^ R3I2. 8a. in 8 months ? Example 2. In how many years will ^300 amount to 405 at 5 per cent. ? Interest on ^300 for i year M ; ft tjii 1 ;i5 ; and interest on ^300 for the required number of years ,405 ^300 The required number of years- ~~ 7 254 ARITHMETIC EXAMPLES. 155. In what time will 1. 8475 amount to 8532 at 4 p. c. ? 2. 8266 . 10 . 8 amount to 8293 . 5 . 4 at 3 p. c. ? 3. ^1451 .6.8 amount to ^1667 . 4 . 4f at 4j p. i, ? 4. In how many years and months will the interest on ^amount to 556 . 12 . Q| at 3j p. c. ? 6. In how many years, months and days will 8425 amount to 8474 . 3 . 8 at 5 p. c. ? 6. In how many days will the interest on 121 . 13. 4 amount to 2 . o . 5 at 6J p. c. ? 7. In how many years will a sum of money treble itself at 3|p c ^ 8. In what time will the interest on any sum of money at 6} p. c. be '1875 of the principal ? 9. In wh.it time will the interest on any sum of money at 5 p. c. J of the amount ? 10. On Feb. 1st. 1818, a person borrowed ^400 at 6} p. c., promising to return it as soon as the interest amounted to $ ; on what date did the loan expire ? 11. In how many months will 83200 amount to 84000 at .3 pies per rupee per month ? Example 3. What principal will amount to 81000 in 10 years at 2$ per cent.? Interest on 8100 for 10 years at 2^ p, c. =825 ; 8100 amounts to 8125 in 10 yr. at i\ p. c. Of the amount 8125 the principal 8ico, / ..................... Biooo .................. =8800. Ans. EXAMPLES. 156, What principal will amount to L. 8900 in 5 years at 4 per cent. ? 2. 84546 . 10 . 8 in 1} years at sj per cent. ? 3. ^190. 151. in 3 years at 4 per cent. ? * ^H53 9 4i in 3 years 7 months and 2$ per cent. ? 6. 8459 . 2 , 3 in 2 years 4 months and 12 days at 6} per cent. ? SIMPLE INTEREST 255 6. 8737. 8a. in ioo days at 3f per cent. ? 7. RSog at sf per cent, from April 2Oth to July 2nd ? 8. R255. 7a. 6p. in ij years at 3 pice per rupee per month ? What principal will produce 9. R37 v 8dr. 8/*. interest in 4 years 3 months at 3j per cent. ? 10. 23 . 7 . i interest in 15 years at 4$ per cent. ? 11. Find, to the nearest pie, the sum that must be invested at 3| per cent, for 13 years to amount to Riooo. 12. Find, to the nearest penny, the principal whose interest amounts to 100 in 2 years 5 moths and 10 days at 4 per cent. MISCELLANEOUS EXAMPLES. 157. 1. The interest on a sum of money at the end of 6 years is fths of the sum itself ; what rate per cent, was charged ? 2. A money-lender lent a sum of money for 3 years 7 months at i pice per rupee per month. At the end of the time he received Rioo3 . 14 . 6 ; what was the sum lent ? 3. A sum of monev increases by ^ of itself every year, and in 7 years it amounts to RQ02. 8#. ; find the sum. 4. ^275 increases by ^ of itself per year : how long will it take to amount to ^357. ios. ? 5. A sum of money amounts in 6 years at 5 per cent, simple -interest to R442 ; in how many years will it amount to RSIO ? 6. Rjoo is borrowed at the beginning of the year at a certain rate of interest and after 7 months R35o more is borrowed at half the previous rate. At the end of the year the interest on both loans is R34- 6/1. What is the rate of interest at *fc^b the first sum was borrowed ? ^""^ffc 7. What sum of money laid out at 3! per cen ^*^lBte|jB Rl interest a day ? '*^Si 8. The principal and interest for 5 years are toget^^9n&k and the interest 5* of the principal ; find the principal^^^^P rate per cent, per annum. l ' ;T^ 9. The principal and interest for a certain time at 3j per cent. are together ^450, and the interest is J of the principal" ; find the time. 10. What sum lent out at c per cent, will produce in 4j years the same amount of interest as RSOO, lent out at 6 per cent.| will produce in 4 years ? 11. If an investment of 75 becomes 78. 1 51. in 8 months, 356 ARITHMETIC what sum invested at the same rate of interest will become 17.7. 6d. in 10 months ? 12. A bequeaths to B a certain sum of money, which after paying a legacy duty of 10 per cent, yields an income of 810 when placed at interest of 3 per cent. Find the amount bequeathed, 13. A person who pays ^p. in the R. income-tax, finds that & fall of interest from 4 to 3| per cent, diminishes his net yearly income by 47. What is his capital ? 14. A sum of money doubles itself in 20 years ; in how many years would it treble itself ? XLVIII. COMPOUND INTEREST. 33. When interest^ as soon as it becomes due, is added to the principal, and interest charged upon the whole, it is called Compound Interest. Example. Find the compound interest on 8321. 8a. for 3 years- at 2j per cent, per annum. 8321. &*.R32i*5, and i\ p. c.2'5 p. c. R. 321-5 _ Division by 100 is 16075 effected by moving 6430 the decimal point 8-0375 int. for 1st year. two places to the 321*5 eft * 329*5375 =*amt. in i year. _ ?15 16476875 8*238437 5 int. for 2nd year. 329*5375 337 7 7 5937 5 amt. in 2 years. 16888796875 67555*8750 8-44439^437 5 int. for 3rd year. 3377759375 346-220335937 5 amt. in 3 years. 321*5 principal. 247203359375 = Total Interest which , 6-304^ COMPOUND INTEREST 257 Note 1. The compound interest might also be obtained by adding together the interest for the ist year, interest for the 2nd year and interest for the 3rd year. If the interest for the 2f years were required, it would be obtained by adding together the interest for the 1st year, interest for the 2nd year and f of the interest for the 3rd year. Note 2. If the interest is payable half-yearly ', the result may be obtained by finding the interest for double the number of years at half the rate per cent, EXAMPLES. 158. N. B. The interest is understood to be payable yearly unless other- wise stated. Find, to the nearest pie, the compound interest on 1. R4OO for 2 yr. at 5 p. c. 2. 6520 for 2 yr. at 4 p. c. 3. 8500 for T\ yr. at 3 p. c. 4. Riooo for 3 yr. at 4$ p. c. Find, to the nearest penny, the amount, at compound interest, of 5. ,650 in 3 yr. at 4 p. c. 6. ^320. 8*. in 2 yr. at 3$ p. c. 7. ;6oo in 2j yr. at 3 p. c. 8. .250 in 2f yr. at i J p. c. 9. Find the compound interest on 350 for i yr. at 4 p. c. per annum, the interest being payable half-yearly. 10. Find the compound interest on 200 for i\ yr. at 10 p. c. per annum, the interest being payable quarterly. 234. The following method of finding the Amount at com- pound interest is often useful. Example i. Find the amount, at compound interest* of $5000 in 3 years at 4 p. c. Amount of Rioo at the end of i yr.Rio4 J / ................ Ri ......................... -}*; / ................ any sum ........................ =&!$ * the sum. Also, amount of any sum at the end of 2 yr. Jg of the amount at the end of 1st yr. Similarly, amount in 3 years (}8$) 3 of that sum ; and so on. C A. 17 258 ARITHMETIC Hence } to find the amount of 85000 in 3 years, we have to multiply 85000 by (104)*, and divide the product by (ioo) g . Process : 8 5000 104 520000 104 208 52 54080000 104^ 21632 5408 85624*320000 =-amt. in 3 years, which = 85624. 50. i"44/>. Ans. Division by (ioo) 8 is effected by marking off 6 decimal places in the final product. Example 2. Find the amount of 8400 for 2 \ years at 6 per cent. compound interest. Amount = R 4 oo x jgg x )gg x 18S = etc. Example 3. What principal will amount to 551. 4a. in 2 years at 5 per cent, compound interest ? Principal x(} Principal = 8500. EXAMPLES. 159. Find, (by the method of Art 234) to the nearest pie, the amount, at compound interest, of 1. 81000 in 2 yr. at 5 p. c. 2. 300 in 3 yr. at 3 p. c. 3. R;oo in 2\ yr. at 4 p. c. 4. 750 in 3 yr. at 4$ p. c. 6. &2000 in 2j yr. at 4 p. c. 6. 84000 in 3 yr. at 3 p. c. 7. fti in ii yr. at 3i p. c. 8. Rio in 3j yr. at 3$ p. c. 9. R3000 in i J yr. at 6 p. c, per annum, interest being due half-yearly. 10. 8350 in if yr. at 4 p. c per annum, interest , being due quarterly. PRESENT WORTH AND DISCOUNT 2J9 What sum lent at compound interest will amount to 11. 100 in 2 yr. at 5 p. c. 12. 132* 6j. in 2 yr. at 5 p. c. ? 13 ^270. Ss. in 2 yr. at 4 p. c. ? 14, ^3413. 1 6^. in 2j yr. at 4 p. c. 15. ,1000 in 3j yr. at 6 p. c. ? 16. i in 3'J yr. at 8 p. c, ? MISCELLANEOUS EXAMPLES. 160. 1. Find the difference between the simple and compound interest on 8,500 for 3 years, at 4 p. c. 2. Prove that the amount at compound interest for 2 years at 2 per cent, is 1*0404 times the principal. * 3. Prove that the difference between the simple and compound interest for 3 years at 5 per cent, is "007625 times the principal. 4. The difference between the simple and compound interest on a certain sum of money for 2 years at 4 p. c. is Ri ; find the sum, 5. A person at the begiiining of each year lays aside Rioco, and employs the money at 5 p. c. compound interest ; how much will he be worth at the end of 3 years ? 6. The population of a town is 64000 and its annual increase is 10 per cent. ; what will be the number of its inhabitants at the end of 3 years ? 7. A merchant commenced with a certain capital, and gained annually at the rate of 30 per cent. At the end of 3 years he is worth R2I9/O. What was his original capital? 8. A money-lender borrows money at 4 per cent, per annum, and pays the interest at the end of the year ; he lends it at 6 per cent, per annum payable half-yearly, and receives the interest at (the end of the year ; by this means he gains Riq4. Sa. a year 5 how much money does he borrow ? XLIX. PRESENT WORTH AND DISCOUNT. 35. The Present Worth or Present Value of an amount due at the end of a given time is that sum which with its interest for the given time will be equal to the amount. Discount is the allowance made for the payment of a sum of money before it is due. From the definition of present worth, it follows that a debt which is due at some future period is equitably discharged by paying the present worth at once. Hence discount is equal to the interest on th* present worth. And Amount** Present Worthy Discount. 260 ARITHMETIC Example I. Find the present worth of R825, due 2\ year* hence, reckoning interest at 4 per cent. [N. B. This corresponds to Ex. 3, Art. 232.] Rioo amounts to Rno in 2$ years at 4 p. c, /. Present worth of Riio / ......................... Ri Ans. [ Discount R82 5 - R7 50 = R7 5. ] EXAMPLES. 161. Find the present worth of 1. R2O4, due 4 years hence, interest at 5 per cent. 2. 1518. I2a.j due in 4 years, at 5} per cent. 3. R3776. 40., due 18 months hence, at 4 per cent. 4. 1522, is. 6d., due 3 years hence, at 4$ per cent. 6. 1607. 185. 4^., due 4^ years hence, at 3 per cent. 6. ,1156. 2s. 8rf., due 3J years hence, at 4^ per cent. 7. Ri626, due 4 months 10 days hence, at 4$ per cent. 8. Ri83, due 25 days hence, at 4 per cent. 9. R2484?. 15^., due 3 years hence, at 7$ per cent, compound interest. 10. ,1050. I2.r. 6d?., due 2 years hence, at2j per cent, compound interest. Example 2. Find the discount on R5oo, due 4 years hence? interest being reckoned at 5 per cent. Interest on Rioo for 4 years at 5 p. c.~R2o. /. Discount on Ri2oR2o, = Rico. Ans. [Present worth = R6oo-RiooR5oo.] EXAMPLES. 162 Find the discount on 1. 355. 4*.> due 4 months hence, at 4j per cent, interest*. PRESENT WORTH AND DISCOUNT 26 1 H, ^2830. 3a. 4^., due 7 months hence, at 5 per cent. 3. R69OI. 140., due 9 months hence, at 3 per cent. 4, R298o. 6a. 8/>., due n months hence, at 4 per cent. 6. 370. 4* 8J^., due IS months hence, at 4$ per cent. 8* 2 7$* 6"j. &/. due ij years hence, at 4$ per cent. 7. ,241. I2J. 4<, due 146 days hence, at 4} per cent. 8. ;i2i. 15*., due 5 months hence, at 3$ per cent. 9. R52o8. I2a., due 3$ years hence, at 4^ per cent. 10. R25i6. 4., due 3 yr. 9 mo. 18 da. hence, at 6J per cent. 11. R6o77. 8<z. 6^., due 4 years hence, at 5 p. c. compound Interest. 12. ,413. 8.T. 9^., due 2 years hence, at 5 p. c. compound interest. 236. Inverse Questions. Example i. If the discount on R282. 8a. is 832. 8<s., reckoning interest at 4 per cent., when is the amount due ? \N. B. This corresponds to Ex. 2, Art.^ 232. ] Amount R282. 8a. ; discount =*R32. Sa. ; .". present worth /. Interest on R25o for the required number of years R32. 8. and interest on R25o for i year at 4 per cent.Rio ; .". the required number of years ^ 3i .*. The amount is due 3i years hence. EXAMPLES. 163. When is the sum due, if the 1. discount on RIOIO. ioa. at 5 per cent, interest is R9I. 2. discount on Ri5i8. 12^1. at 5} p. c. is R268. 120. ? $. discount on 520 . 17 . 6 at 4} p. c. is 70 . 17 6 ? 4. discount on ^5747 at 3$ p. c. is ^147 ? 6. present worth of R3S$o At 4 p. c. is 3500 ? 6. P. W. of Ri$94i. 6a. fy. at 3} p. c. is 13750 ? 7. P. W. of ^8776, 6*. io{^. at 2\ p. c. is ^8721. i6f. &/. 262 ARITHMETIC Example 2. If the discount on 8528. 120., due 3$ years hence, be 78. 12*., at what rate per cent, is the interest calculated ? [Jfc B. This corresponds to Ex. I, Art. 232.] AmountR528. 120. ; discount =B;8. I2a t ; /. present worth Interest on R4$o for 34 years = R78. 12 a. ; ^Q3 Ri for * 45 Ri for i ,, year = B Rate per cent, 5. EXAMPLES. 104. What is the rate of interest, if the 1. discount on R35o, due 2 years hence, is Rioo ? 2. discount on R748o, due 4 years hence, is R686 ? 3. discount',on^397 .*2 . 2|, due 4 years hence, is ^71 . 12 . 2\ r 4. discount on ^538 . 10 . 7^, due 2} years hence, is 37 17 3^ ? 5. present worth of Ri26o, due 4 years hence, is 81125 ? 6. P. W. of R2673. 20., due 3j years hence, is R2275 ? 7. P. W. of 2857. los.j due 12 J years hence, is ,2000 ? 37. Miscellaneous questions on F. W. and Discount. Example I. On what sum of money, due at the end of 2 years, does the discount, at 4 per cent., amount to R2O ? Here, interest on P. W. for 2 years R2o. Now, R8 is the interest for 2 years on Rioo, /. R20 ....................................... R250 ; /. the P. W.-R250 ; and .'. amountR27O. Am. Example 2. If the interest on RSOO at 5 per cent be equal to the discount on &57 5> when is the latter sum due ? PRESENT WORTH Alfp DISCOUNT $63 Here, R5ooP. W. of 8575 ; .'. R75interest on 8500. Now, the interest on R$oo f<> r the required number of year* R75, but the interest on 8500 for i year at 5 per cent. 825 ; /. the required number of years=|r~ 3. .". The sum is due 3 years hence. Example 3. The interest on a certain sum of money is and the discount on the same sum for the same time and at the same rate is 820 ; find the sum. Int. on the sum -Int. on P. W. + Int, on Disc. =Disc. on the sum 4- Int. on Disc. .". Int. on the sum -Disc, on the sum *= Int. on Disc. Hence R2 =Int. on 820, R22 ........... 220. Ans. Note. It should be carefully noted that the difference between the interest and discount on a sum of money for a certain time and at a certain rate is equal to the interest on that discount for that time and at that rate. EXAMPLES. 165 1. On what sum of money, due at the end of 16 months, docs the discount, at 4J per cent., amount to 6484. Sa. ? 2. If the discount on a certain sum of money, due 8 months hence, at i\ per cent., be R883. 10. 8, what is the sum ? 3. The discount on a certain sum of money, due at the end of 2j years, at 2| per cent., is ^32. los. : find the sum. 4. If the interest on R2275 at 3^ per cent, be equal to the discount on 2593. 8<z. for the same time and at the same rate, when is the latter sum due ? 5. If the interest on Soo at 3 per cent be equal to the discount on ^838, when is the latter sum due ? 6. If the interest on ^148 for 5 years is equal to the discount at the same rate on ^173. i8j., due 5 years hence, what is the rate of interest ? 7. The interest on a certain sum of money is Riao, and the discount on the same sum for the same time and at the same rate is Rioo ; find the sum. 8. The interest on a certain sum of money is 336, and the discount for the same time and at the same rate is 6300 ; find the sum. *$4 ARITHMETIC 9. The discount on a certain sum, due 2 years hence, is and the interest on the same sum for 2 years is 56. 40. : find the sum, and the rate per cent, per annum. 10. The interest on a certain sum, at 5 per cent., for a certain time is ^50, and the discount for the same time at the same rate Is 40 : find the sum, and the time. 11. If the difference between the interest and discount on a sum for 3 years at 3 per cent, be Ri, what is the sum ? 12. If the difference between the interest and discount on a certain sum of money for 9 months at 4 per cent, be 1 55,, find the sum. * 13. A offers for a house R8oo, and B offers 815 to be paid at the end of 4 months. Which is now the better offer, if the rate of interest is 5 per cent, per annum ? 14. A man buys 250 md. of sugar for R2$oo payable at the end of 6 months, and the same day sells them at Rio per md. ready money : what does he gain by the transaction, reckoning interest at 5 per cent, per annum ? 15. A tradesman marks his goods with two prices, one for ready money and the other for 6 months' credit : what ratio should the two prices bear to each other, allowing interest at 4 per cent. ? If the credit price of an article be RSO, what is the cash price ? 16. Five copies of a book can be bought for a certain sum payable at the end of a year and six copies of the same book can be bought for the same sum in ready money ; what is the rate of interest ? 17. The discount on R$5o for a certain time is RSO ; what is the discount on the same sum for twice that time ? 18. The interest on 720 for a certain time is 18 ; find the discount on the same sum for the same time. 19. If the discount on a sum of money, due 6 months hence, at 8 p. c. be 7. 10. nj ; find the P. W. of the sum. 20. A man bought an estate for 2000 and sold it immediate- ly for ^2287. los. payable at the end of 5 months. If the use of the money be reckoned at 4 per cent, per annum, what is now his gain per cent. ? 21. 259. 7-ff* is due 4 years hence and 173. 18;., 5 years hence : what sum at the present time is equivalent to both these sums, calculating interest at 3$ per cent. ? 22. What sum must be paid now in order that a person may receive R2ooo at the end of every year for the next 4 yearsj the rate of interest being 5 per cent. ? COMMERCIAL DISCOUNT . 26^ COMMERCIAL DISCOUNT. 38. A bill is a promise (in writing) to pay a certain sum of money at the end of a certain time. Example. Each of the following is a bill : a Bill of Ex- Change or Hundi (which is a document in which one person directs another to pay to him or to some other person, a sum of money at the end of a certain time) ; a Promissory Note (which is a document in which one person promises to pay another a sum of money at the end of a certain time). 39. When a banker or money-lender purchases a bill, that is, advances money at a certain rate per cent, on the security of a bill, instead of deducting discount he usually deducts interest for the time specified adding the 3 days of grace. The purchaser of a bill may sell it at any time before it is due. In this case also, the second purchaser deducts interest on the amount for the time the bill has still to run adding the three days of grace. Note 1. There is a custom, which has the force of law, by which a bill (if not payable on demand) always runs three day* (called the days of grace) beyond the time specified. Thus a bill drawn on the i$th January, at 3 months would be nominally due on the isth April, but actually due on the iSth. Moreover) calender months are always reckoned, so that a bill drawn on the 3ist January, at 3 months, would be nominally due on the 3Oth April and actually on the 3rd May. Note 2. In working an example the 3 days of grace should be added only when the information given in the question is suffi- cient to enable us to determine the exact number of days that must elapse before the bill falls due, and not otherwise. Example. A bill for ^505 drawn on the 7th March at 4 months Is discounted (i.e., sold) on the 28th April at 5 per cent. ; how much does the holder of the bill receive, interest being deducted ? The bill is nominally due on the yth but actually due on the loth July ; therefore the bill has still to run from 28th April to loth July, that is, for 73 days or $ of a year (including one only of the days named). Now, interest on 505 for } yr. at 5 p. c.^ 5 5 IO ^ 5sa 5 Ij /. The holder receives ^505-^5. ., *".*., 499. 191. Note 3. A banker in purchasing a bill obtains a small advan- tage by deducting interest instead of discount The mathematical discount is called True Discount 366 ARITHMETIC Banker's discount (/.*., interest) is called Commercial or Practical Discount. The bankers gain** 'the difference between the commercial and true discount. Note 4. In Arithmetic 'Discount 1 is always understood to mean true discount (and not commercial discount), Therefore in working examples true discount is always to be calculated unless commercial discount is expressly mentioned. 84O. A second kind of commercial discount (which has no reference to time) is the deduction which is made by a tradesman for immediate payment of his bill. Thus when a tradesman gives notice upon his bill that he will allow 10 per cent, discount for immediate payment, he deducts Rio for every Rioo in the amount of the bill. The calculation of this discount is therefore the same as of finding the simple interest on the amount of the bill for I year at 10 per cent. EXAMPLES. 166. 1. Find the difference between the commercial and true dis- count on a bill of R6oo2. 8#., due in 4 months, at 6J per cent. 2. A bill is drawn for ^250 on June I2th at 5 months, and is discounted on Sep. 3rd at 5 per cent. ; how much does the holder of the bill receive, banker's discount being allowed ? 3. Find the banker's discount on a bill of 730 drawn on July 3ist at 2 months and discounted on Sep. 3rd at 4 per cent. 4. What does a bill-discounttfr give as the present worth of a, bill for 91. 40. drawn on Sep. 4th at 5 months and discounted the same day at 6 per cent. ? 6. A bill of Ri82. 8#., nominally due on the I5th of May, is. discounted on the 23rd April of the same year at 3 per cent. ; what does the banker gain thereby ? 6. A bill is drawn for ,365 on March 31 st at 3 months and discounted on June I3th at 4 per cent. ; how much more was charged than the true discount ? 7. The difference between the commercial and true discount on a bill for 7$ months at 5 per cent, is &9 ; find the amount of the bill. 8. The amount of a tradesman's bill is R375 ; if he allows 10 per cent, discount, how much does he accept for immediate payment ? 9. A tradesman accepts 40 for immediate payment of a bill for ^50 ; what rate of discount does he allow ? EQUATION OF PAYMENTS 267 10. If the credit price of five copies of a book is equal to the cash price of six copies of the same book, what is the rate of discount ? [cf. Question 16, Ex. 165.] 11. A tradesman's prices are 25 p. c. above the cost price ; if he allows his customers a discount of 10 p. c. on his bill, what profit does he make ? 12. How much per cent, must a tradesman add on to the cost price of his goods, that he may make 20 per cent, profit after allowing his customers a discounLpf 10 p. c..on his bill ? L. EQUATION OF PAYMENTS. 841. When several sums are due from one person to another, payable at different times, we may be required to find the time at which they may all be paid together, so that neither the creditor nor the debtor may lose. The time so found is called the equated time of payment. We give below a rule for finding the equated time, which will be found sufficiently accurate for all practical purposes. Rule. Multiply each debt by the number of months [or days] after which it is due : then divide the sum of the products by the sum of the debts : the quotient will be the number of months [or days] in the equated time. Example. If 8400 be due from A to B at the end of 8 months f and R6oo at the end of 10 months, when may both sums be paid in a single payment ? Number of months in the equated time EXAMPLES. 107. 1. &2oo is due in 5 months and 400 in 8 months ; find the equated time of payment. 2. 450 is due 2 months hence, R4<x> is due 3 months hence and &25<> is due 4 months hence ; what is the equated time ? 3. Find the equated time of payment of ^600, one-half of which is due in 6 months, i in 9 months, and the rest in a year. 4. A owes B a debt payable in 4/3 months, but he pays J in 3 months, and i in 4 months : when ought the remainder to be paid ? 6. A owes B on the loth of April 900 due 40 days hence ; he pays 400 on the loth of May and 300 on the 2oth of the same month : on what date ought he to pay the rest ? 268 ARITHMETIC LI. STOCKS. * 24. Stock is the name given to the money borrowed by any Government to meet national expenses, or to the Capitals of Trading Companies. The money borrowed by a Government is called the National or Public Debt. The money lent to the Government is said to be in Government Securities or Government Promissory .Notes in India, and in the Funds in England. A part of the National Debt in England is called the Consolidated Annui- ties or Consols. When any Government raises capital by borrowing, it reserves to itself the option of paying off the principal at any future timci but promises to pay the interest at fixed periods. In India and England the interest is paid half-yearly. The capital of a Trading Company is divided into shares, generally of Rioo or 100 each ; those who join the company by buying one or more of these shares are called Shareholders. The shareholders are not required to pay the full price of their shares at once, but they have to pay it in instalments, as the business of the company progresses and Calls are made. The part of the capital of the company, which has thus been paid at any timci is called the Paid-up Capital. The profits of the company are divided periodically among the shareholders ; and the moneys thus received are called Dividends. When all the capital of a company has been subscribed and the company is in need of more capital, it is not usual to issue more shares like those issued at first. The company generally borrows money at a fixed rate of interest and agrees to pay the interest on this money before any dividend on the original shares is paid. Money so borrowed is called the Preference Stock of the company, the original capital being called the Ordinary Stock. The bonds which are given by Joint-Stock Companies, Munici- palities and similar other bodies for borrowed capital are called Debentures. 43. Stock is transferable by sale ; but its price varies from a variety of causes. When the market value of Rioo stock is Rioo cash, the stock is said to be at par ; when Rioo stock is sold for 98, it is said to be at a discount of 2 per cent, or, at 2 below par ; when it is sold for Rio2, it is said to be at a premium of 2 per cent, or, at 2 above par. Purchases and sales of stock are usually made through Brokers who generally charge J per cent, on the stock bought or sold. Thusi if the market value of Rioo stock is 897$, the purchaser has to pay R(97i+t) and the seller receives R(97i~i). STOCKS *6g Note. By "the 3 per cents." or "3 per cent, stock 11 is meant a stock, on Rioo (or 100) of which is paid a dividend of 83 (or 3) per annum. M B. Unless the brokerage is mentioned, it need not be taken into consideration in working examples in stocks. 44. Example i. What is the cost of Ri$oo stock in the- 4 per cents, at 97$, brokerage being J per cent. ? Cost of Rioo stock =R(975 + J)-R98, Ans. Example 2. How much stock at 97 i (brokerage included) caa be bought for R39O ? Amount of stock bought for R97i Rioo, ^R 100 * 390 ass ** Arts. W. B. It is obvious that we have nothing to do with the rate o interest in any of the two above examples. EXAMPLES. 168. 1. Find the cost of R2000 of 4 per cent, stock at 95. 2. Find the cost of ^250 in the 3 per cent, consols at 3 below par, brokerage being J p. c. 8. How much money can be obtained from the sale of R4$CQ stock in the Calcutta Municipal Debentures at Ri2 premium ? (Brokerage J p. c.) 4. Find the price of the 4 per cents, when R8oo stock can be purchased for R750. (B. J p. c.) 6. Find the price of the 4$ per cents, when Rl7oo is obtained from the sale of Ri6oo stock. (B. J p, c,) How much stock can be purchased by investing 6. Ri35Q in the 4 per cents, at Rio discount ? 7. Rso62. &a. in the 5 per cents, at I2| above par ? (B. J p. c.). 8. ^6909. iSj. in the consols at 92} ? (B. 2s. 6d. per cent) 0. A person lays out R$7 50 in the purchase of 4 per cent Govt Securities at 93! and afterwards sells at 95$ ; what profit does he e^ the :a ,usual brokerage being charged on each transaction ? 270 ARITHMETIC 10. A person buys ^1000 3 per cent, stock at 98$ , and sells out at 96$ ; how much does he lose by the transaction ? (B. J%). 11. A person bought Russian 5 per cent, stock at 72, and sold it when the price has risen to 75}, thereby clearing 65; 'how much money did he lay out ? 12. A person holds 4800 consols ; if he sells out at 87$ and Invests the proceeds in the i\ per cents, at 81, how much of the latter stock will he hold ? 13. A person invested ,5330 in the 3 per cents, at 91, and -when they had risen if per cent, he sold out and invested the money in the stock of the Dominion of Canada at 102 \ \ how much Canadian stock does he hold ? Example 3. What annual income will be derived from 83725 of 4i per cent, stock ? Income from Rioo stock * .............. 81 ....... N* B. This is merely a case of finding the interest, where the given fttock is the principal. Example 4. What annual income will be derived from 82042. >8a. invested in the 4 per cent. Govt. Securities at 102 (B. J%) ? Cost of Rioo stock .". Income on Rio2j money .' ................. 81 ......... ; ................ 82042$ : ......... =B*%V^=fi8o. Ans. Example 5. A person transfers 88000 stock from 4 per cent. <5ovt. Securities at 98$ to 6 per cent. Municipal Debentures at iSfA ; find the alteration in his income, the usual brokerage being charged on each transaction. Income from the 4 per cents, = R8ooox T J = R32o. Money obtained from the sale of 4 per cents. = RSooo x 9 ^. Income from Ri3i invested in 6 per cents.R6, ' ** ttl t*i*....*..M**(.. R 8000X98$ 131^x100 6o. .'. The alteration in income is 8360-8320, or 840 increase. STOCKS 271 Example 6. How much money must a person invest in the 4i per cent. Preference Stock of the O. E. Ry. Co. at 94$ (brokerage included) to obtain an annual income of R6oo ? Money to be invested for &4j income R94J, / .................................... R6oo ......... - 42 = R 12600. Atts. Example 7. Find the price of 4 per cent, stock when from the investment of 83900 a person obtains an annual income of Ri6o, brokerage being neglected. Cost of stock producing Ri6o income ==R 3900, : ................................... R4 .......... R97i. Arts. EXAMPLES. 169. 1. Find the half-yearly dividend on 3500 4 per cent, stock. 2. What annual income will be derived from ^37250 of 4$ per cent, stock, after paying an income-tax of 4^. in the R ? 3. What amount of 3J per cent, stock must be bought to produce a quarterly income of ^375 ? 4. What annual income will be derived from the investment of R59IO in the 4$ per cents, at 98$ ? (B. %.) 6. A person invests ,25935 in 3 per cent, stock at 90. If the first year's dividend be invested in the same stock at 91, and the dividend for the second year at 95, what will be his income for the third year ? 6. If I invest Ri642o in the E. I. Ry. Preference Stock which pays 5 per cent, and is at 102^, what will my clear income be, after paying an income-tax of 5^. in the R ? (B. % p. c.) 7. If I lay out R24oo in the 4^ per cents, at 96, and after receiving the half-year's dividend sell out when they have sunk to 94) how much do I gain ? 8. A person bought Bengal Bank shares at 113, and after receiving the half-year's dividend at the rate of 12 per cent, per annum sold out at H7ij and made a profit of Ri78. Sa. in all ; how many shares did he buy ? 9. If a person invest Ri88io in the 4 per cents, at 104$, at 272 ARITHMETIC what price must he sell out after receiving the half-year's dividend to make a profit of 450 ? 10. A person transfers 11000 from the 4 per cents* at 92 to the 5 per cents, at no ; find the alteration in his income. 11. How much stock can be purchased by the transfer of 4000 stock from the 3 per cents, at 90 to the 3$ per cents, at 96, and what change in annual income will be produced by the transfer ? 12. A person invested 85800 in the 5 per cent. Calcutta Municipal Debentures at par, and after receiving the half-yearly dividend he sells out at R2$ premium, and invests the entire proceeds in the 4 per cent. Government Securities at 95^ ; what change is made thereby in his income ? 18. A person laid out 814500 in the 3j per cents, at 72 J, and when they had fallen to 68 he sold out and invested the money in the 4 per cents, at 75$ ; find his gain or loss in income. 14. A person has an annual income of 8480 from stock in the 4 per cents. ; this stock he sells out at 95} and invests the money in a railway stock (paying 5 p. c.) at II9& ; find the alteration* in his income. (B. J p. c.) 16. How much money must a person invest in the 3 per cent, consols at 9i to obtain an annual income of 1000 ? (B. J p. c.) 16. How much must a person invest in the 4 per cents, at 93} in order to have a clear income of 8940 after paying an income-tax of 4^. in the R ? 17. How much 3 per cent, stock at par must a man sell in order to purchase enough 4 per cent, stock at II4*V to produce an income of 8252, a brokerage of p. c. being charged on each transaction ? 18. Find the price of the 4 per cents, when the investment of 83750 in them produces an income of 8160. 19. What is the price of the 4j per cents, when a man has an income of 8270 by investing 87800 in them ? (B. J p. c.) 20. A man invests ^1570 in the New 4 per cent. Egyptian Annuities, and has thereupon a clear annual income of 76, after paying an income-tax of is. in the ; find the price of the Annuities. (B. J p. c.) Example 8. What rate of interest is obtained on money invest- ed in the 4 per cents, at 795 ? (B. \ p. c.) Interest obtained on 880 money = 84, / 820 8i, / 8100 8s. /. Rate of interest obtained is 5 per cent. STOCKS 73 Example 9. At what price (including brokerage) would a person have to purchase the 4$ per cents, to get 5 per cent, for his money ? R5 interest on Rioo money, /. Ri .". the stock must be bought at 90. Example 10. What is the better stock to invest in, 4 per cents. at 95 or 4$ per cents, at 105 ? In the first case, interest on 95 money in the second case, ............... Rio5 It will be found that ^iif is greater than -fa \ and therefore the second is the better investment. Example i x. A person finds that if he invests his money in the 4 per cents, at 98 his income will be 42 less than if he invests it in the 5 per cents, at 112 ; find the sum to be invested, In the first case, income from Ri ^ in the second case, ................ Ri .*. difference of income from Ri^Rjfif Now, Riify 7 = difference of income from Ri, .'. Ri - .'. R42 - or Rio9;6. Ans. EXAMPLES. 170. What rate of interest is obtained by investing in the 1. 4 per cents, at 90 ? 2. 3 per cents, at 70 ? (B. J p. c.) 3. A person buys 800 3 per cent, consols at 85, and .500 more when they are at 97 ; how much per cent, will he get for his money after deducting an income-tax of jd. in the ? 4. What rate of interest do I get upon my moneyj if I buy Railway Shares of R75 each (which pay 4 per cent.) at 85 and pay an income-tax of 4^. in the R ? 6. At what price would a person have to purchase the 4 per cents, to get 5^ per cent, on his money ? 6. What is the price of stock, when the 4^ per cents, pay interest at the rate of 6 p. c. on the money invested ? (B. J p. c.). C. A. 18 274 ARITHMETIC #, When the 4 per cents, are at 88, what ought to be the price of the 4^ per cents, to give the same rate of interest ? 8. A man invested in the 4 per cents. ; if, after deducting an income-tax of 6fi. in the rupee, he obtained 4^ per cent, interest on the money invested, at what price did he buy ? 9. If Bank stock bought at 14 per cent, discount pay 6 per cent, on the investment, how much per cent, would it pay if it were bought at 28 per cent, premium ? 10. Which is the better investment, 4 per cents, at 82 or 5 per cents, at 102 ? 11. Which is the better stock to invest in, 3^ per cents, at 82$ or 4 per cents, at ioo I (B. p. c) 12. Find the difference per cent, in income between investing in the 4 per cents, at 88 and 4^ per cents, at 90. 13. A person finds that if he invests his money in the 4^ per cents, at 96 his income will be greater by Rio than if he invests it in the 4 per cents, at 88 ; find the money to be invested. 14. By investing a certain sum of money in the 3 per cents, at 75 a man gets $. 13. 4 less in income than he would get by investing the same sum in the *fo per cents, at 84 ; find the sum invested, MISCELLANEOUS EXAMPLES. 171. 1. A person invested money in the 4 per cents, when they were at 95, and some more when they were at 90 ; find the advantage per cent, of the second purchase over the first. 2. A person invests R 16600 in the 3 per cents, at 83, and when the funds have risen 7 per cent, he transfers } of his capital to railway stock at 67^ ; what dividend ought the latter to pay that he may thereby increase his income by RSO ? 3. Which is the better investment, 1256 in the 3^ per cents, at 87, or in the railway shares at ^89 per share, the dividends in the latter case being 3| per cent, on the sum invested ? 4. A person possesses 3200 3 per cents., which he sells at 99| ; he invests the proceeds in railway shares at ^56 a share, which shares pay 5 per cent, interest on 45, the amount paid on each share. By how much is his income altered by the transaction ? 6. A person has Rtooo stock in the 3 per cents, which he sells and re-invests in the 3$ per cents, at 8; an4 increases his income by RS ; find the price of the 3 per cents. 6. By selling ^1500 3 per cents, at 95 and re-investing it I increase my income by ^15 a year. If the dividend on the new shares is 8 per cent, what is the price of them ? 7. What sum must be invested in the 3 per cents* at 90 to amount in 23^ years at simple interest to 3210 cash ; the price of MISCELLANEOUS EXAMPLES 27$ the stock remaining unchanged ? How many years sooner would the amount be realized if the price of the stock rose to 96 ? 8. A gentleman in India has been receiving 12 per cent, on his capital ; he goes to England, invests it in the 3 per cents, at 94, and his income in England is .2400 a year ; what was his income in India ? Gi = Rio.) 9. How much 3 per cent, stock must be sold at 87^ to pay the present worth of Ri645 l \ a * due Io nionths hence, at 3} per cent.? 10. Municipal Debentures are at 119 when the Government Securities are at 93^, what should be their price when the Govern- ment Securities are at 71* ? 11. VVhat is the price of the 4 per cents, when ^ r of the sum invested is received as annual interest after deducting an income- tax of 4 pies in the rupee ? 12. A person invests 823800 partly in a 4^ per cent, stock at 97$ and partly in the 3 per cents, at par : if he holds twice as much 3 per cents as 4!, find the income that he obtains from the whole investment. 13. A man having money invested in the 3 per cents., from which he derives an income of ^864, sells out at 90, and invests in shares that pay 5 per cent, interest : if his income be now increased by ,336, at what price does he buy the shares ? 14. What sum must have I invested in the 3^ per cents, at 91 if, after investing ,4000 more in the 3 per cents, at 75, and paying an income-tax of yd. in the on my total gross receipts, I find my net income to be ^524. 5^. ? 15. A person who has a certain capital calculates that if he invest half his capital in the 3 per cents, at 90, and half in the 4 per cents, at par, his total income will be RIIOO ; what is his capital ? 16. A invests 3500 in buying equal amounts of 3 per cents, at 78^ and 6 per cents, at 109! . B invests the same sum, half in one stock and half in the other. Find (i) the difference in their incomes, (ii) the ratio of their rates of interest. 17. Four' per cents, are at 95, and 4^ per cents, are at 105. One person buys R2oo stock in eaqh, and another person invests R2oo in each : compare the rates of interest obtained by the two on their whole investments. 18. A shareholder receives one year a dividend of 10 per cent, on his stock and pays an income-tax of 4 pies in the rupee. The next year he receives a dividend of 12 per cent, and pays an income- tax of 5 pies in the rupee. If his income is 394 .5.4 more in the latter than in the former year, how much stock does he hold ? 19. 20 shares in a company are worth Ri6oo when the dividend is at the rate of 5 per cent. ; how many shares ought to be worth 960 when the dividend is at 6 per cent. ? 276 ARITHMETIC 20. A person invested 2800 in the purchase of 4 per cents, fct 90 and 44 per cents, at 95. If his total income is Bi 301 how much of each stock did he buy ? 21. A man invests ^1600 in the 4 per cent, stock at 80 and 7$ per cent, stock at 125 ; what sums must he invest in the respective stocks to make 5 j per cent, on his money ? 22. A person, by selling 4 per cents, at 87 and investing the proceeds in the 5 per cents, at 96, finds that his income is increased by Ri7 : how much 4 per cents, did he sell ? 23. 4 per cent, stock, bought at 95^, is held for 6 months at the end of which time the interest is paid ; it is then sold at the same price at which it was bought : find the rate per cent, per annum of interest obtained for the money used. (Usual brokerage). 24. A person invests 8255 in the 4 per cents, at 85, and sells part of his stock when they have risen 5 per cent, and the remain- der when they have fallen 8 per cent. ; he lost fin by the transac- tion : how much stock did he sell out at first ? 26. 5 per cent, stock is sold at 108, and with the proceeds 4 per cent, stock is bought at 91 J ; after a time 4 per cent, stock is sold at 95$ and the original stock purchased at 109, leaving a profit of Rlog on the transaction : find the amount of 5 per cents, sold. 26. If the 3 per cents, be at 95, and the Government offer to receive tenders for a loan of $,000,000, the lender to receive ; 5, 000,000 stock in the 3 per cents, together with a certain sum in the 3J per cents., what sum in the 3j per cents, ought the lender to accept ? 27. The present income of a railway company would justify a dividend of 6 per cent., if there were no preference share ; but as j5o,ooo of the stock consists of such shares which are guaranteed 7^ per cent, per annum, the ordinary shareholders get only 5 per cent. : find the amount of the ordinary stock of the company. 28. A person buys 6 per cent, bonds, the interest on which is payable yearly and which are to be paid off at par i year after the time of purchase ; if money be worth 5 per cent., what price should be given for the bonds ? LIL EXCHANGE. 245. Exchange means the giving or receiving a sum o! money of one country equal in value to a given sum of money of another country. The par of exchange between two countries denotes the intrinsic value of a coin of one country, as estimated in terms of a coin of the other country. EXCHANGE 27? The course of exchange is the actual or marketable value at any time of a coin of one country, as estimated in terms of a coin of the other country. Thus, the quantity of gold in the English Sovereign being 1*261 times the quantity of gold in the French Napoleon, at par of ex- change /i is equal to i'26i Napoleons ; but in the course of ex- change i may be equal in value to a little more or less than 1*261 Napoleons. Arbitration of exchange is the determination of the rate of exchange, called the arbitrated rate, between the first and last of a given number of places, when the rates of exchange between the first and second, the second and third, etc., of these places are known. 46. Money transactions between one country and another are usually carried on by means of Foreign Bills of Exchange or briefly Foreign Bills. The following is the usual mode of proceeding : Suppose 1 want to transmit 100 to a merchant in London. 1 go to a banker and buy a bill for the given amount, payable in London, at the current rate of exchange ; I then send the bill to the merchant in London, who presents it to the person on whom it is drawn and receives the amount. 47. The following table gives the principal foreign monetary systems. France *| Belgium V i franc ioo centimes ] Switzerland J Italy Spain . i lira . i peseta = 100 centesimi | ioo centimes i - * Greece . i drachme = loolepta I** 9 * a. Servia i dinar = TOO paras 1 Bulgaria i leva s ioo stotinkis j Roumania i ley ss ioo banis J Germany i mark ioo pfennige = n j Id. Austria i florin or gulden =1 ioo kreuzers **is. Hid. Turkey i Turkish pound a ioo piastres = i8j p. old. Holland I florin SB ioo cents = i s. Sd. Portugal I milreis =8 1000 reis 4^, 6d. Sweden j Norway > i crown ioo ore is. ol<* Denmark J Uiiited States i dollar ($) as ioo cents 4-f. 2d. Russia I rouble _ ioo kopecks Rl 12 ,3. China i taelio mace n ioo candareens &3< Japan ... i yen ioo sen -R2 .7-6, *78 ARITHMETIC Note. In the countries whose names have been printed in italics in the above table, as in India, the standard coins are silver \ in England the standard coin is gold ; hence the value of the Rupeei etc.> in English money varies with the amount of silver which can be bought for a gold sovereign. For some years past the value of silver as compared with gold has been steadily declining. A few years ago a Rupee was equal in value to about 2s.-, now it is equal to is. 4< Example I. Calculate the par of exchange between the sovereign and the rupee, supposing pure gold to be worth 15 times its weight of pure silver, having given that 46f$ sovereigns are coined from I Ib. troy of standard gold, ^ fine, and that a rupee weighs 1 80 grains of silver and is \% fine. _,, . . . 12x20x24 12x20x8x40 The sovereign weighs - - gu or . ----- gr. ; and therefore it contains ( 1 ^% x / x4 -^x^)gr. or Afti8**#*Ji gr . of pure gold. The rupee weighs 180 gr.; and therefore it contains (180 x }J) gr. or 165 gr. of pure silver, which is equivalent to *- gr. or n gr. of pure gold. Now the number of*rupees equivalent to a sovereign is the same as the number of times n gr. is contained in *&*&$M r Hence the sovereign = &fi Jfij|?T J1 rupees = 10*27. ..rupees. Example 2. Find the relation between the rupee and the shilling as determined from the intrinsic value of the two coins ; having given that a rupee weighs 180 grains, and is jj fine ; and that I Ib. troy of silver, fine, is coined into 66 shillings. We find, as in the preceding example, that the rupee contains 165 gr. of pure silver. The shilling contains (14*.|J*JA x J J) gr. or -*Vi* 1 r * f P ure silver .'. i rupee(i654-^ I x -f 1 ) shillings 2*043. ..shillings. Example 3. Exchange R$$o for English money at is. Sd. per rupee. ; Ri u. 8^., ^IJ. 8^x550 i6s. %d. Ans. Example 4. Determine the course of exchange between India and England) when Indian money is at a discount of 25 p. c., having given that at par I rupee 2 shillings. EXCHANGE 279 [Indian money being at a discount of 25 p. c. means that it Is worth 25 p. c. less English money than jt would be if it were at par.] At par Ri=2J., .'. at 25 p. c. disc. Ri=2j.- of 2s. -IJ. 6</. .". The course of exchange is is 6^. per Ri. Example^. If the rate of exchange between Calcutta and London is at is. 9^. per rupee, and that between London and Paris is at 25 francs per i, what is the arbitrated rate of exchange between Calcutta and Paris ? Ril,r. 9^. s =;BTj ! =B r <jX25 francs = 2 1 J V francs. (See Art. 205,) .". The required rate is 2^ francs per rupee. EXAMPLES, m. 1. Convert R.3782 to English money, the course of exchange being is. $\d. per R. 2. Exchange ^329. 7^. 6d. for Indian money at Rn. 40. per . 3. A spanish pistole is worth i$s. and an Austrian ducat gs. $d. ; how many ducats are equivalent to 226 pistoles ? 4. A French Napoleon or 2o-franc piece is worth ^79 ; find, to the nearest farthing, the value in English money of 123*21 francs. 5. A bill bought in Calcutta at is. 6d. a rupee, is sold in New York at 4s. $d. a dollar ; determine the course of exchange between New York and Calcutta. 6 If j3 2o thalers ; 25 thalers** 93 francs ; 27 francs 5 scudi ; and 62 scudii35 gulden ; how many gulden can I get in exchange for 11 ? 7. Find the arbitrated rate of exchange between Vienna and Calcutta in rupees for i florin, when the exchange between Calcutta and London is R3 for 5^., between London and Paris is 25 francs for i> between Paris and Berlin $ francs for 4 marks, and between Berlin and Vienna 2 marks for i florin. 8. If a thaler is equivalent to 40 kreuzers, to silber-groschen and half a gulden, and if 30 silber-groschen make a thaler and 60 kreuzers make a gulden, how many gulden are worth 8 thalers ? 9. If Ri in England exchanges for is. 5^., and if ^i in India exchanges for Ri3. $a. 6/., how much do you lose in RQ6o by the two exchanges ? 1O. , ,A person in Calcutta wishes to remit a debt of 240 dollars to New York . when the exchanges are i dollar** 2. 130.; Rl=M5. 6d. and 35.?. 6 dollars. Is it more 'advantageous for him to remit directly to New York or circuitously through London ? 280 ARITHMETIC 11. A merchant in London is indebted to one at St. Petersburg 15000 roubles : the exchange between St. Petersburg and London is 5orf. per rouble, between St. Petersburg and Amsterdam gid. Fl. per rouble, and between Amsterdam and London 36^. $d. Fl. per sterling. What difference will it make if the London merchant is drawn upon through Amsterdam or direct ? 12. If in London 1 get i for 25 francs 20 centimes, what shall I gain or lose per cent, by taking French money into Bavaria when the exchange is n gulden 40 kreuzers for i, and 8 gulden 20 kreuzers for a Napoleon? (i N apo. 20 fr. ; i fr.ioo cen- times ; i guld.6o kreuz.) 13. The Indian bazar matind is equal to 82? Ib. avoir., and the rupee is equal to 2s. If I md. of wheat cost &3, what will be the price in English money of I cwt. ? 14. Exchange 380 dollars for English money when it is at a discount of 5 per cent., given that at par i dollar =4$. id. 16. Exchange B66o for English money when it is at a premium of 10 per cent,, it being given that at par &i*is. io$d. 16. If India exchanges with England at a loss of 15 per cent, when the course of exchange is is. $d. per B, what is the par of exchange ? 17. A merchant in Calcutta wishes to remit to London 900, a rupee being equal to is. ; for what sum in English money must he draw his bill when bills on London are at a premium of 12 J per cent. ? 18. I pay RSIOOO to a bank for a bill of exchange payable in London. The rate of exchange is is. io$d. for the rupee, and the bank charges me 2 per cent, on the amount payable in England. How much will my agent in London receive ? 19. A person in London owes another in St. Petersburg 460 roubles, which must be remitted through Paris. He pays the requisite sum to his broker when the exchange between London and Paris is 23 francs for ^i, and between Paris and St. Petersburg 2 francs for one rouble. The remittance is delayed until the rates of exchange are 24 francs for i, and 3 francs for 2 roubles. What does the broker gain or lose by the transaction ? 20. The exchange of Calcutta on London at 3 months is is. 4}d. per R ; find the exchange at sight, reckoning 5 per cent, per annum. 21. Calculate the par of exchange between the gold mohur, weighing 1 80 grains, )J fine, and the U. S. eagle, weighing 258 grains, & fine. 22. Calculate the par of exchange between the Napoleon and the rupee, supposing pure gold to be worth 15 times its weight of pure silver ; being given that 16197$ grains of French standard THE METRIC SYSTEM 28l gold, A fine, is coined into 155 Napoleons, and that a rupee con- tains 1 80 grains of silver, JJ fine. 23. From 346$ grains of fine silver are coined 14 thalers ; find the value ot a thaler, when a pound troy of Indian standard silver, of which ii parts out of 12 are fine, is worth 32. 24. If I Ib. of English standard silver, of which 37 parts in 40 are pure silver be worth 62 *., find the value of a Hyderabad rupee which weighs 7 dwt. 17 gr., and has a fineness of 30 parts in 31. 25. The gold coinage of one nation contains I part of silver to ii parts of gold ; that of another nation, i part of silver to 23 parts of gold. It is found that 59 of the first weigh as much as 123 of the second. The intrinsic value of silver is one-sixteenth that of gold. Determine the par of exchange. LIII. METRIC SYSTEM AND DECIMAL COINAGE. 48. The Metric System of weights and measures, which originated in France, has been introduced to a greater or less extent into almost all the countries of Europe. It is also nearly always used in scientific treatises. The Tables of weights and measures in the metric system arc constructed upon one uniform principle, by attaching the following prefixes to each of the units. GREEK PREFIXES. LATIN PREFIXES. means loth part of. looth loooth Deea means 10 times. Deoi Heoto loo Centi Kilo loco Milli Myria 10000 In this system the fundamental unit of length is the Metre, whence the system is called the metric system. The metre is equal to 39*370. .. inches, and was originally taken to be the ten-millionth part of a quarter of the terrestrial meridian. An error has however been since found in the measurement of the terrestrial meridian, and the metre therefore is not exactly the length it was stated to be. The metre was computed to be 39"37079... inches. The latest determination makes it 39*370113... inches, but the last two figures are uncertain. i centimetre (cm.). I decimetre (dm.). 1 metre (m.). I decametre (Dm.). I hectometre (Hm.). I kilometre (Km.). I myriametre (Mm.). 10 millimetres (mm.) - 10 centimetres 10 decimetres * lo metres * 10 decametres 10 hectometres 10 kilometres 38a ARITHMETIC I metre =*39| inches nearly** about IT\ yards ; i cm.^f inches nearly; i dm =-4 inches nearly; I kilometre = about 5 furlongs. I inch 2* 5400... centimetres. Example. 23564 m. 7 dm. 9 cm. 8 mm. = 23564798 mm. = 2356479-8 cm.^235647'98 dm. =-2 3 5647 98 m. = 2356-4798 Dm, 235'64798 Hm. ==23*564798 Km.=2'356479S Mm.==2 Mm. 3 Km. 5 Hm. 6 Dm. 4798 m. The unit of area is the square metre. In measuring land the unit used is a square decametre, called an are, and the only multiple and submultiple used are \hihectare (=100 aresa square hectometre) and the centiare (^iJo of an are=-a square metre). TABLE. 100 sq. millimetres (mmq.) =* loo sq. centimetres loo sq. decimetres = loo sq, metres = 100 sq. decametres loo sq. hectometres = 100 sq. kilometres =* I centiare (ca.) sq. centimetre (cmq,). sq. decimetre (dmq.). sq. metre (mq.). sq. decametre (Dmq.). sq. hectometre (Hmq.). sq. kilometre (Kmq ). sq. myriametre (Mmq.). sq. metre. are (a.) [=i sq. decametre], hectare (ha.)[~i sq. Hm.] 100 centiares 100 ares I sq. metre 15 50*0 sq. inches; I sq. inch =6*4516 sq. centimetres* I are 1076*4 sq. feet nearly ; i hectare = 2^ acres roughly. Examples 2 Dmq. 64 mq. 9 dmq. 34 cmq. = 2640934 cmq. 26409*34 dmq. = 264*093 4 mq. 2*64O934 Dmq. = '026 409 34 Hmq. 0002640934 Kmq. Examples 73204 ca.=* 7 32*04 a. 7*3204 ha. = 7 ha. 32 a. 4ca. The unit of volume is the cubic metre. The multiples of the cubic metre are seldom used. In measuring wood the cubic metre is called a stere, and 10 steres make a decastere. TABLE. 1000 cu. millimetres * i cu. centimetre, looo cu. centimetres I cu. decimetre, looo cu. decimetres I cu. metre. I cu. metre = i stere ; 10 steres -i decastere. I cu. metre or stere 35*3 cu. feet (nearly), Example. 27*03567 cu- m."27o3$*67 cu, dm. * 2703 56 70 cu. cm* 27 cu. m. 35 cu. dm. 670 cu. cm. THE METRIC SYSTEM 283 The unit of capacity, both for liquids and dry goods, is the litre, and is equal to a cubic decimetre. TABLE. lo millilitres (ml.) I centilitre (cl.). lo centilitres = I decilitre (dl.). 10 decilitres I litre (lit.)- 10 litres = I decalitre (DL). 10 decalitres = I hectolitre (HI,). 10 hectolitres == i kilolitre (Kl.). Since i litre ~i cubic decimetre, 1000 litres =i kilolitre, and looo cubic decimetres = i cubic metre, .'. I kilolitre = i cubic metre. i litre = 6 1*024... cu. inches==i'759... pints = ij pints nearly; I kilolitre 35*317 cu. feet (nearly). Example. 3025*407 lit. = 3O254'o7 d 1. = 302 5407 cl. 302 5407 ml. 302*5407 Dl. = 3o*25407 Hl.3'025407 Kl. = 3 Kl. 2 Dl. 5 lit. 4 dl. 7 ml. The unit of weight is the gram (or gramme) which is the weight of a cubic centimetre of distilled water at its maximum density. TABLE. 10 milligrams (mg. i centigram (eg.). lo centigrams I decigram (dg.). 10 decigrams I gram (gr.). 10 grams = I decagram (Dg.). lo decagrams * I hectogram (Hg.). 10 hectograms i kilogram (Kg. or Kilo.). 10 kilograms I mynagram (Mg.). Since I litre =iooo cubic centimetres, and I kilogram =*iooo grams, .". the weight of a litre of water i kilogram. The weight of a kilolitre (i cubic metre) of water is 1000 kilograms and is called a tonneau de mer or mil Her. A guintal^ioo kilograms. I gram 1 5*432... grains or 15$ grains roughly; i ikilogram 2*2046... Ib. avoir. 2^ Ib. avoir, nearly. 4$a. The Metric units, with their relations with one another and their equivalents in British units, are collected together below for ready reference. I. Unit of Length is the Metre 3937oi 13.. .inches = 39ff inches nearly. II. Unit of Surface is the Are =i sq. decametre io76'4sq feet nearly 5*0 acre roughly. III. Unit of Volume is the Stere - 1 cu. metre = 35 '3 cu. feet nearly, 384 ARITHMETIC IV. Unit of Capacity \ ,both for liquids and dry goods, is the Litre i cu decimetre 6ro24...cu. inches **i 75Q. ..pints or if pints nearly *=> '2 gallons nearly. V. 'Unit of Weight is the Gram = the weight of a cu. centimetre of distilled water at 4C. = 15*4 32... grains = 15^ grain nearly = 0022... Ib. avoir. French Money. 10 centimes (c.) = I decime. io- decimes I franc (fr.). Accounts are kept in francs and centimes only ; thus "3278 francs" is read 32 francs 78 centimes, The Franc is a silver coin composed of 9 parts of silver and 1 part of copper, and weighs 5 grams. It is equal to 9|*f. nearly. The Napoleon is a gold coin 20 francs. THE PROPOSED DECIMAL COINAGE OF GREAT BRITAIN. 10 mils (m.)*i cent (c.) ; 10 cents I florin (f.) ; 10 florins -i. 49. The great advantage of a decimal system of weights and measures is, as we have seen, that a compound quantity can be reduced to a simple quantity, and vice versa^ without going through the processes of multiplication and division. Hence compound rules are replaced by the corresponding simple rules. Example i. Express 7 hectares 34 ares 6 centiares as a decimal of a sq. kilometre. 7 ba. 34 a. 6 ca. 73406 ca.- 73406 sq. metres -734*06 sq. decametres =7*3406 sq. hectometres 073406 sq. kilometres. Example 2. A wheel makes 1230 revolutions in passing over 2 kilometres 5 hectometres 9 metres 2 decimetres ; what is its circumference ? 2 Km. 5 Hm. 9 m. 2 dm. 2509*2 m. ; 2509-2^1230 2*04 ; /. the circumference reqd. 2*04 metres 2 metres 4 centimetres. Example 3. A cubic foot of alcohol weighs 94 Ib. ; find the weight of a litte in grams, supposing a litre to be equal to '035 cu. ft, and a gram 15*43 grains. Weight of a litre of alcohol '03 5x94 Ib. '035 x 94 x 7000 grains "035x94x7000 " 15-43 -1492*5,. .grams. THE METRIC SYSTEM 285 Example 4. Cloth is sold at 21 fr. 80 c. per metre ; what Is the corresponding price per yard in English money, if l be worth 25 fr. 25 c. ? t 1 metre39*37 inches.] i yard = 36 inches ^ metres ; . r j 36x2180 . f 36x2180 . . cost of I yard - ---- centimes*-/ --- J 39*37 * 39*37* 2525 - nearly. Example 5. Add together $. ;f. 2c. 3m., ,9. 2f. oc. 4m., and 3c. mils 3723 9204 73Q 13657 mils=;i3. 6f. $c. 7m. Ans. Example 6. Multiply 7f. 9c. 3m. by 32. mils 793 ""1586 25376 mils =^25* 3f. 7c. 6m. Ans. 350. We can easily decimalise a sum expressed in . s. d> and change decimal coinage into . s. d. Example i. Express 7. 15*. i\d. in decimal coinage. 4 ro 12 7'S 20 Example 2, Express 9. 3f. 9c. 8m. in . s. ^9*393 20 12 < 11*520 3f. 9C* 8iru 2S6 ARITHMETIC EXAMPLES. 172a. Reduce 1. 2305000 millimetres to kilometres. 2. 304007 centimetres to kilometres, etc. 3. 1203270 millimetres to decametres, etc. 4. 75 kilometres 7 decametres 3*05 metres to millimetres. 6. 3070-5086 decametres to kilometres, etc. 8. 23 sq. kilometres 8 sq. decametres 7 sq. metres to sq. metres. 7. 50 sq. kilometres 6 sq. hectometres 4 sq. metres to sq. decametres. 8. 40740 centiares to hectares, etc. 9. 8 hectares 7 ares to centiares. 10. 36*307 sq. hectometres to hectares, etc. 11. 3012035 cu. centimetres to cu. metres, etc. 12. 5 cu. metres 27 cu. decimetres 4 cu. centimetres to cu. millimetres. 13. 40700302 millilitres to kilolitres, etc. 14. 3040600 centigrams to myriagrams, etc. 15. 1375 centimes to francs, etc. 16. A man walks 7*92 kilometres in 2 hours ; how many metres does he walk in a second ? 17. The circumference of a bicycle wheel is 4 metres 8 centi- metres ; how many times will it revolve in going 16*83 kilometres ? 18. If 25 horses eat 676 kilo. 575 gr. of corn in 9 days, how long will 240 kilo. 560 gr. serve 16 horses ? 19. The weight of 226 equal parcels is I tonneau 921 kilo- grams ; find the weight of each. 20. If 27 decalitres 8 centilitres of wine cost 67 francs 52 cen- times, find the cost of 15 litres. 21. An estate containing 30 hectares 50 ares is divided into loco fields of equal area ; find the area of each. 22. How much wheat at 19 francs 55 centimes per hectolitre ought to be given in exchange for 312 hectolitres 80 litres of barley at I franc 25 centimes per decalitre ? 23. Express a yard in terms of the metrei supposing a metre to be equal to 39*37 inches. 24. Express a kilometre as a decimal of a mile, if a metre be J9'37 inches. THE METRIC SYSTEM 287 25. The standard height of the barometer is 760 mm. Find this height in inches, [i metre 39*3708 inches.] 26. Express a pound avoir, in grams, a gram being equal to I5'43 grams. 27. If a cubic iUch of air weigh '31 grains, what will be the weight in grams of a litre of air, having given that a cubic metre is equal to 35*3 cubic feet, and a gram I5'43 grams. 28 A gallon of watei weighs 10 Ib. ; find its volume in cubic centimetres, supposing kilogram to be equal to 2j Ib. 29. Mahogany is 55 Ib, to the cubic foot ; find the weight of a decastere <.f Mahogany in tonneaux ami k'lograms, supposing a cubic metre to be 35*3 cubic feet, and a kilogram 2^ Ib. 30. An inch is 2*54 centimetres, and a kilogram is 2*2 Ib. ; find the pressure of the atmosphere in grams per sq. centimetre, supposing it to be 15 Ib. avoir, to the square inch. 31. If a kilolitie be 220 gallons, find the value, in English money, of a pint of liquid which is worth 33 francs the decilitre, I2oo fiancs being equal to ^47. 32. A decimetre is equal to 3*937 inches, and a cubic inch of water weighs 252*45 grains. Express a kilogram in pounds avoir, correct to two decimal places. 33. A gallon contains 277*274 cubic inches, a cubic decimetre is 61 cub'C inches, and a kilogram is 2^ Ib. ; calculate the weight in pounds of a gallon of water. 34. The area of a room is 43*68 sq. metres, and its length is 832 centimetres ; find its breadth. 35. Find the length of a piece of ribbon which has a surface of 1575 sq. dm., and which is I'S cm. broad, 36. Find the value per metre of cloth, when a piece 37 m, 2 dm. 5 cm. long is worth 186 fr. 25 c. 37. Find the number of cubic centimetres (i) in a litre, (2) in a stere, 38. A plank is 3 m. long, 5 dm. broad and 2*5 cm. thick ; find the volume of the plank. 39. A room is 7*24 metres long, 4*21 metres broad, and contains 121921600 cubic centimetres of air ; find the height of the room. 40. Find the weight of a litre of water in grams. 41. Given that! Ib. 7000 grains, and I gram I 5 '4 grains, find the number of grams in an ounge, correct to 2 decimal places. 288 ARITHMETIC 42. Specific gravity of a substance being the ratio of the weight of any volume of the substance to the weight of the same volume of water, find the specific gravities of mercury and alcohol, having given that a decalitre of mercury weighs 136 kilograms and a centilitre of alcohol weighs 8 grams. 43. A plate of iron 55 cm. long and 43 cm. broad weighs 26901 grams. Find the thickness of the plate, if iron is 7*6 times as heavy as water. 44. If a gallon of water weighs 70,000 grains, and i kilogram *543 2 grains, how many times can a litre measure be filled from a 3-gallon cask and what decimal of a litre will be left in the cask ? 45. If the weight of a cubic foot of water 62^ lb., and I kilogram 2*2 lb., find the number of cubic feet in a cubic metre. 46. Reduce 5 m. 2 dm, 3 cm. to yards, feet and inches, taking I metre as equal to 39*37 inches. 47. Find the area of a field 145 metres long and 84 metres broad, and express it in ares. 48. A cistern 12 metres long and 7 metres wide holds 103656 kilograms of water. Find the depth of the water. 49. Given that a cubic foot of water weighs 1000 ounces, and an inch =2*54 centimetres, find the nearest whole number of grams in I lb. 60. Given that iron is 7*5 times as heavy as water, find the weight in kilograms of a sheet of iron 3*4 metres long, 2-5 metres broad and i centimetre thick. 51. A rectangular cistern 3*2 metres long and 2 % 6 metres broad has a capacity of 11040 litres. Find the depth of the cistern. 62. If i franc 9*4 annas, and I kilogram 1*07 seers, find in francs the price of a kilogram of an article which costs a lupee a seer. 63. A sq. yard '84 sq. metre, and ^125 francs. An estate measuring i&4& hectares is sold for five million francs. What is this in pounds per acre ? [i acre 4840 sq. yd.] 64. A room is 18 metres long and 9 metres broad. Find its area in square yards (correct to two places of decimals), taking a metre to be equal to 39*37 inches. 66. A room is 10 feet 6 inches long, and 5 feet 4 inches broad. Find its area in square metres, taking a metre as equal to 39*37 inches. 66. Given I metre 39*37 inches : (i) Express an inch in centimetres correct to two places of decimals INVOICES AND ACCOUNTS 289 (ii) Express a sq. metre in sq. inches correct to two places of decimals. (iii) Find the nearest whole number of cubic inches in a litre, (iv) Find the nearest whole number of litres in a cubic foot. 57. Given I inch = 2*5 centimetres approximately, the weight of a cubic foot of water ==62 -J Ih , and i Ib. 7000 grains, find the nearest whole number of grains in a gram. 58. Given i gram** 1 5 '43 grains approximately, and i tola i8o grains, express a seer of 80 tolas in grams, correct to two places of decimals. 69. Find the weight of a hectolitre of mercury which is 13*6 times as he ivy as water. 60. Given t grim = 15*432 grains, i Ib. -=7000 grains, and i seer = 14400 grains, shew that, approximately, 5 kilograms ==H Ib., and 14 Kilograms = i5 seers. 61. Given i metre = 39*37 inches, shew that 981 centimetres =32 feet nearly. 62. The driving wheel of a locomotive is 12*5 metres in cir- cumference, and it makes 2*5 revolutions in a second ; how long will it take to travel 100 miles, if i mile=r6 kilometres. 63. Find the value of 0-34 kik>L'r,mis+9*4 grams + 600 milligrams as the decimal of a pound ; given i gram = 15*432 grains, and i Ib. =37000 grains. 64. Find, to the nearest litre, the content of a tank 3*21 metres long, 2*15 metres broad, and 54 centimetres deep. LIV. INVOICES AND ACCOUNTS. ** 51. (i) Specimen of an Invoice. Calcutta, April 23, 1889. Charles Smith, Esq., Bought of William Moran & Co., 7, Bankshall Street. [| K. a. A 8 yd. of flannel at Ri. 4a. per yd. *! 10 o o 10 yd. of calico at 30. 6/. per yd. ... 1 2 3 o 2 pairs of gloves at Si. 90. a/, per pair I 3 3 6 R 15 1 6 1 6 C, A, 19 293 ARITHMETIC (i i ) Specimen of an Account. Calcutta, June 30, 1889. Charles Smith, Esq., To William Moran & Co., 7. Bankshall Street. 1889 R. a. A April 23) To goods, as per invoice 15 6 6 May 7, To ditto 3 7 3 I3> To ditto 9 o o June II, To ditto 7 6 R "28" _i_ 3 (iii) Specimen of a Detailed Account. Calcutta, June 30, 1889, Charles Smith, Esq., To William Moran & Co., 7, Bankshall Street. 1 889 R. a. A April 23, 8 yd. of flannel at Ri. 4a. per yd. ... 10 o if 10 yd. of calico at 30. 6p. per yd. 2 3 o 2 pairs of gloves at Ri. ga. gfi. per pair 3 3 6 May 7> 3 dozen stockings at R6 per doz. ... 18 o 13, 13 yd. of linen at 8#, 6/. per yd. 6 14 6 June 12, 20 yd. of carpet at R3. 8a. per yd. ... 70 o o 4 pairs of socks at Ri per pair 4 o R 114 T ^ ! - r Note. Invoices and Accounts are called Bills. Each sepa- rate entry in a bill is called an item. When an account is sent to a buyer it is said to be rendered. LV. PROBLEMS IN HIGHER ARITHMETIC. Example i. A person has a number of oranges to dispose of ; he sells half of what he has and 2 more to A, J of the remainder and 4 more to B } J of the remainder and 6 more to C ; by which time he has disposed of all he had. How many had he at first ? When he had given J of his oranges to C he had 6 left ; there- fore this is (i -|) or f of the number he had before C came, and therefore he had 6 x or 8 before C came ; therefore he had (8 +4) or 12 before he had given 4 oranges to B \ but this is the number PROBLEMS IN HIGHER ARITHMETIC 2QI he had left when he had given J of his oranges to B ; therefore this is (i J) or \ of the number he had before B came, and therefore he had 12 xf or 18 before B came ; therefore he had (18 + 2) or 20 before he had given 2 oranges to A ; but this is the number he had left when he had given \ of his oranges to A ; therefore he had 20 x 2 or 40 before A came : that is, he had 40 oranges at first. Example 2. The expenses of a family when rice is at 12 seers for a rupee are R8o a month ; when rice is at 15 seers for a rupee the expenses are 77 a month ; what will they be when rice is at 1 8 seers for a rupee ? The prices of a seer of rice in the three cases are R-jV, R^V and &iV respectively ; .'. the price of a seer is first reduced by ^(A""A) or *Vff anc * finally by RCiV-iV) or R^. Now, when the saving on a seer of rice is R^ the total saving is R(8o 77) or R3 ; .'. when the saving on a seer is R 3 V the total saving will be R|f or R$. /. The reqd. expenses = R( 80- 5)R7S. Or thus : When the saving on each seer of rice is R^ the total saving is &3 ; .'. the number of seers of rice required by the family per month -= R3 ~ R^ a = 1 80 ; and the price of 180 seers at 12 seers for a rupee is Ri5 ; .*. the other expenses of the family R(So- 1 5)=* 1*65. Again, the price of 180 seers at 18 seers for a rupee is Rio ; .'. the total expenses when rice is at 18 seers for a rupee will be R(65 + io) or R75, Example 3. A labourer was engaged for 36 days, on the agree- ment that for every day he worked he should have 40,, but that for every day he absented himself he would be fined 2a. He received R7. 8fl. at the end of the time ; how many days was he absent ? If he had worked all the 36 days he would have received Rg ; /. through absence he lost (R9-R7. 8a.) or Ri. Sat. But for each day of absence he actually loses (40. + 2a.) or 6a. ; .*. the number of days he was absent *=Ri. Sa. -r6a.*~4. Example 4. I have to be at a certain place in a certain time, and I find that if I walk at the rate of 4 miles per hour I shall be 5 minutes too late, and if at the rate of 5 miles per hour I shall be 10 minutes too soon ; what distance have 1 to go ? If I walk 4 miles an hour I require 15 minutes more time in going the distance than if I walk 5 miles an hour. And in walk- ing i mile I require 3 minutes more at the former rate than at the latter. Hence I have to go a distance of 5 (**., 15-7-3) miles, Example 5. I have a certain sum of money to be distributed among a certain number of boys, and I find that if I give S3 to each I shall spend R4 too little, but that if I give RS to each 1 shall spend R6 too much. How much have I to spend ? 392 ARITHMETIC If I give RS instead of R 3 to each I require R2 more per bead and (R44-R6) or Rio more on the whole ; .*. the number of boys= s Rio-7-R2=5 ; and /. I have to spend (Rsx 5 + ^4) or Rig. Example b. A Ib. of tea and 4 lb. of sugar cost 5c. ; but, if sugar were to rise 50 per cent, and tea 10 per ce \ , they w<,uld cost 6j 2d. ; find the cost of the tea and the sugar per lb If both tea and sugar were to rise 50 p. c., the co^t of i lb. of tea and 4 lb. .of sugar would be js. 6d. ; but tea rises only 10 p. c. ? /. 40 p. c. of the cost of a ib. of tea = 7J. 6//. ov. 2^. = \s. 4<i. : .*. the cost of a lb. of tea3^. ^d. .'. the cost of 4 lb of su^nr 5$. 3-f. 4d.~is. 8</. ; and .". i lb. of sugar costb 5^. Example 7. Three tramps meet together for a meal ; the first has 3 loaves, the second 2, and the third, who has his share of the bread, pays the other two %d. ; how ought they todiviHethe money? Each eats loaves ; .*. the first has given (3 J) loaves and the second (2 ) loaves to the third : .*. the 5^. given by the third ought to be divided in the ratio of (3 3) to (2- gj, *>., of 4 to I ; /. the first will take 4d. and the second id. Example 8. The sum of the ages of A and B is now 45 yeais, and their ages 5 years ago were as 3 is to 4 : find their present ages. 5 years ago the sum of the ages of A and B was 35 \ears ; if 35 years be divided in the ratio of 3 to 4, the parts are 15 years and 20 years. .'. The present age of A is (i 5 + 5/ or 20 y -ars, and that of B is (20+5) or 25 years. Example 9. A is twice as old as B, and 4 years older than C ; the sum of their ages is 71 years : find the age oi each. If C were as old as /*, the sum of the ages of A, H and C would be 75 years ; now, dividing 75 in the xatio of 2, i and 2, ^e find that the parts are 30, 15 and 30 ; .'. A's age is 30 years, ,#'s 15 years, and Cs (30-4) or 26 years. Example 10. A and B begin business with equal capitals. At the end of the year A has gamed l{6oo, and B ha^ lost ^ of his capital ; A has then twice as much as B* Find how much each had at first. (ft of B's capital) x 2 =* A's capital 4- R6oo, * (i 9 of^ >s capital)X2 , , .'. !or i|,of ^1's capital = />., A } $ capital 4- 1 of A : s capital =*^'s capital 4- R6oo, /. | of A's capital R6oo, A's PROBLEMS IN HIGHER ARITHMETIC 2Q3 Example, n. Divide 250 into two parts such that, 3 times the first part and 5 times the second part may be together equal to 950. 3 times the ist part + 5 times the 2nd part ==950 ; and the 1st part 4- the 2nd pait = 2$o, 3 times the ist part 4- 3 times the 2nd part*- 7 50 ; .*. 2 times the 2nd part = 200, [subtracting (ii) from (i)] .'. the 2nd part 100 ; and .'. the ist p c \rt = 2 50 100=150. Example 12. Mangoes are bought at Rio per 100 ; at what rate per TOO must they be sold that the gain on Rioo may be equal to the selling price of 2.50 mangoes ? Rioo is the cost price of TOCO mangoes ; .". (1000 250) or 750 mangoes must be sold for Rioo ; .*. the selling price of 100 mangoes Er ample 13. Two passengers going to the same place have 6 md. of luggage Between them, and are charged for excess of luggage Rrj. 8<*. and R3 respectively ; had the luegage all belonged to one person he would have been charged R8. 4^. for excess. How much is allowed free ? R4. 8'/. 4- R3 is the charge on 6 md. less twice tl e free allow- ance, and R8. 40. i c the charge on 6 md. less the free allowance ; /.the charge on fr^e allowance = RS. 4/2. (R_j.. S#. 4-R3) = i2^. /. (R8. 4". + i2#.) or R9 = charge on 6 md. ; .'. \ia. ^charge on $ md. Therefore ^ md. is allowed free. Exawple 14. Two guns are fired from the same p^ace after an Interval of 6 minutes, but a person approaching the place observes that 5 min. 51 sec. elapse between the reports ; what was his rate of progress, sound travelling 1125 ft. per second ? In 5 min. 51 sec. or 351 sec. the man travels a distance which sound will travel in (6 min.- 5 min. 51 sec.) or 9 sec. But is 9 sec. sound travels 1125 XQ ft. ; .'. in 351 sec. the man travels 1 125 x 9 ft.; .', in I hour the man travels *-\\\*\\ S?S8 ft miles or I9i|j miles. Example 15. R49 was divided amongst 150 children, each girl had 8a. and each boy 40. ; how many boys were there ? If 4. be given to each child, R37. 8#. will be spent, and the boys will have got their shares. The remaining sum, RII. Sa. must therefore be distributed amongst the girls only, giving 40. to each. Hence the number of girls is the same as the number of times 40. is contained in RII. Sa. ; therefore the number of girls is 46, and therefore the number of boys is 104. 294 ARITHMETIC This example may also be solved by the method of Art. 225., Thus : When #49 is divided amongst 150 children, each gets - 8 7 V a on the average. Hence the question may be put thus "Each boy is to have 40. and each girl 8.; in what ratio should they be mixed that each may have *W*<*. on the average?" Therefore by the method of Art. 225 we find that the ratio of the number of boys to the number of girls must be (S--% 2 -) I (W- 4) or 104 I 46, But 104 + 46=8150 ; .*. the number of boys 104, and the number of girls = 46. Example 16. A free-hold estate is bought at 20 years' pur- chase ; find the rate of interest obtained on the money invested. ["A free-hold estate is bought at 20 years' purchase" means that it is bought for 20 times the yeaily rent derived from the estate.] If the value of the estate is 20, the rent is Ri ; .". if the valua. of estate is Rioo, the rent is 5. Therefore the rate of interest obtained is 5 p. c. Example 17. If 36 oxen in four weeks eat up the grass on a field of 12 acres and what grows upon it during the time ; and 21 oxen eat up the same in 9 weeks ; how many oxen will it main- tain for 1 8 weeks, supposing the grass to grow uuiformly during the lime ? Origl. growth + 4 wk.'s growth maintains 36 ox for 4 wk. f .* ~.~ i ox for 144 wk. ; also, origl. growth + 9 wk's growth 21 ox, for 9 wk t| / i ox for 189 wk. Hence, subtracting 2nd line from the 4th, 5 wk.'s growth maintains I ox for 45 wk., I wk.'s growth ,.. I ox for 9 wk., 16 wk.'s growth i ox for 144 wk. ; but origl. growth + 4 wk.'s growth I ox for 144 wk. ; /. origl. growth =12 wk.'s. Now, I wk.'s growth maintains for 9 wk. I ox, I wk.'s growth for 1 8 wk. \ ox, (1 2 4- 1 8) or 30 wk.'s growth for 18 wk. 15 ox., *>,, origl. growth* 18 wk.'s growth for 18 wk. 15 ox. Answer. 15 oxen. EXAMPLES. 173. 1. A person has a number of oranges to dispose of ; he sells half of what he has and one more to A, half of the remainder and PROBLEMS IN HIGHER ARITHMETIC 2Q5 one more to Z?, half of the remainder and one more to C 9 and hall of the remainder and one more to D ; by which time he has dis- posed of all he had. How many had he at first ? 2. A thief having stolen some money from the palace of Siraj Uddowlah was caught on his way back by the head khoja who let him off on getting half the money and R2o more ; he was caught again by the sentry at the palace gate, who got a third of what he then possessed and Rio more ; lastly he was let off by the kotwalm his rounds on getting of what he still had and R6 more. The thief came home robbed of all he stole. How much did he steal ? 3. The expenses of a family, when rice is at 8 seers for a rupee, are 75 a month ; when rice is at 10 seers for a rupee, the expenses are &72 a month (other expenses remaining unaltered) : what will they be when rice is at 12 seers for a rupee ? 4. A labourer was engaged for 15 days, on the agreement that for every day he worked he should have 6#., but that for every day he absented himself he would be fined 2a. He received 4. 2a, at the end of the time ; how many days was he absent ? 6. I have to be at a certain place in a certain time, and I find that if 1 walk 3 miles an hour I shall be 10 min. too late, and if I walk 4 miles an hour I shall be 7^ min. too soon ; what distance have I to go ? 6. I have a certain sum of money to be distributed among a certain number of boys ; and I find that if 1 give R>2 to each I shall spend 84 too little, but if I give RS to each I shall spend R3 too much. How much have I to spend ? 7, I have a certain sum of money wherewith to buy a certain number of nuts, and I find that If I buy at the rate of 40 a penny I shall spend 5<y. too much, if 50 a penny, io*/. too little. How much have I to spend ? 8, A Ib. of tea and 3 Ib. of coffee cost 5*.; but, if .coffee were to rise 33^ p. c. and tea 50 p. c., they would cost 75. Find the cost of tea and coffee per Ib. 9. 3 Ib. of tea and 4 Ib. of sugar cost 8*. ; but, if sugar were to rise 25 p. c. and tea were to fall 25 p. c., they would cost ?s. Find the cost of tea and sugar per Ib. ( 1O. Three tramps meet together for a meal ; the first has 3 loaves, the second 4, and the third, who has his share of the bread, pays the other two 7 half-pence ; how ought they to divide the money ? 11. Two settlers in New Zealand own adjoining farms of 700 and 500 acres respectively. They unite their farms, taking at the same time a new partner who pays ^1200 on the understanding that ^ of the land will in future belong to each. How is the ^1200 to be divided between the original owners ? 296 ARITHMETIC 12. The sum of the ages of A, B and C is now 90 years, and their ages 10 years ago were as 3 '. 4 I 5 ; find their present ages. 13. A is twice as old as /?, and 5 years older than C ; the sum of their ages is 45 years ; find the age of each. 14. Divide R8o between A, B and C in such a manner that A may get 3 times as much as ./?, and B Rio more than C. 15. A and B begin business with equal capitals. At the end ^f the year A has gained ^130, and B has lost ^ of his capital ; A has then twice as much as B. Find how much each had at first. 16. A and B begin business with equal capitals. At the end of a certain time A has gamed J of his capital, and B has lost R2oo ; B has row of what A has. How much had each at first ? 17. Divide 155 into two parts such that, twice the first part and 3 times the second part may be together equal to 370. 18. Divide loointo two prms such that, \ of one part and \ of the other part may be together equal to 40. 19. Divide 350 into two parts such that, 3 times the first part and J of the second part may be together equal to 250. 20. Mangoes are bought at &5 per 100 ; at what rate per 100 must they be sold that the gain on tiioo may be equal to the selling price of 400 mangoes ? 21. Sugar is bought at 4<z. per seer ; at what rate per seer must it be sold that the gain on Rio may be equal to the selling price of 8 seers ? 22. Two passengers going to the same place had 8 md. of luggage between them, and were charged for excess of luggage R8 and 84 respectivelv ; had the luggage all belonged to one person he would have been charged Ri4 for excess. Find how much is allowed free, and how much luggage each had. 23. Two guns are fired from the same place after an interval of 10 minutes, but a person approaching the place observes that 9 min. 30 sec. elapse between the reports ; what was his rate of progress, sound travelling 1121 ft. per second ? 24. Two guns are fired from the same place at an interval of 15 minutes, but a person going away from the place hears the reports at an interval of 15 mm. 30 sec.; if sound travels 1125 ft. per second, find his rate of travelling per hour. 25. Two guns are fired from a place at an interval of 28 minutes, but a person approaching the place, at the rate of 13}? miles an hour, hears the reports at an interval of 27 min. 30 sec. Find the velocity of sound per second. 26. Cannons are fired at regular intervals in a town, and a person riding towards it at the rate of 9 miles an hour hears the PROBLEMS IN HIGHER ARITHMETIC 97 reports at intervals of 15 minutes ; at what intervals must the cannons have been fired, sound travelling 1120 ft. per second ? 27. Cannons are fired at intervals of 10 minutes in a town towards which a passsenger train is approaching at the rate of 30 miles an hour ; if sound travels 1136 ft. per ^econd, find at what intervals the reports will be heard by the passengeis. 28. R6o was distributed among 50 children, each girl had R2 and each boy Ri ; how many boys were there ? 29. 35 fruits, consisting of mangoes and oranges, were bought for R2. 8a. ;if the mangoes cost 2a. each and the oranges 6p. each, find the number of oranges bought. 30. A lump composed of gold and silver measures 6 cu. inches and weighs 100 oz. ; if a cu. inch of gold weighs 20 oz. and an equal bulk of silver 12 oz., find the weight of gold in the mixture. 31. 19 grain*? of gold or 12 grains of silver displace one grain of water. If a rinpr, composed of gold and silver, weighs 88 grains and displaces 5 grains of wa'er, how many grains of silver does it contain ? 32. A farmer has oxen worth 12. los. each, and sheep worth 2. $s. each ; the number of oxen and sheep being 35, and their value 191. los. Find the number he had of each. 33. if an income-tax of 7//. in the on all incomes below ;ioo a year, and of is. in the on all incomes above .100 a year realises .18750 on ^500000, how much is raised on incomes below 100 a year ? 34. How many years' purchase should be given for a free-hold estate so r", *o jet 5 per cent, for the money ? 35. An estate is bought at 25 years' purchase for R4o,ooo, one-fourth of the purchase-money remaining at mortgage at 6 per cent. The cost of collecting rents is Rioo per annum. What interest does the purchaser make on his investment ? 36. If r o o^p,n in 5 weeks eat up the grass on a field of 7 acres and what grows upon it during the time, and n oxen eat up the same in 4 weeks, how many weeks' growth is on the field ? 37. If 20 oxen in 4 weeks eat up the grass on a field of 4 acres and what grows upon it during the time ; and 17 oxen eat up the same in 10 weeks ; how many oxen will it maintain for 5 weeks, supposing the grass to grow uniformly during the time ? 38. In a certain meadow there is a crop of 525 stones of grass, which grows uniformly. If II oxen turned in would consume all the grass in 48 days, but 6 oxen would require 98 days, what weight of grass would each ox eat in a day ? 39. If 25 horses eat the grass of 35 acres of one field in n 2Q8 ARITHMETIC daysj in what time would 20 horses eat the grass of another field of 56 acres, where there is at first twice as much grass per acre as in the former field, the growth of the grass being neglected. What must be the ratio of the rates of the growth of the grass in that two fields so that your result may be accurately true r 40. A well is fed by a spring which flows continuously and uniformly into it. When there are 10,000 cu. ft. of water in the well, 7 men can empty it in 20 days ; and when there are 15,000 cu. ft, of water in the well, 5 men can empty it in 50 days. How many cu. ft. of water flow into the well in one day ? 41. A cistern has one supply-pipe (A) and 2 equal waste-pipes (/?, C) attached to it. A is opened, and when the cistern is partially filled B is also opened, and the cistern is emptied in 3 hours. Had C been opened along with B the cistern would have been emptied in i hour. How long after A was B opened ? 42. A cistern has two pipes attached to it, one to supply and one to draw off. If both the pipes are opened together, the cistern is filled in 9 hours ; but if the waste-pipe is opened one hour after the supply-pipe, the cistern is filled in 7 hours. In what time can the supply-pipe fill the empty cistern ? 43. A leaky cistern is filled in 5 hours with 30 pails of 3 gallons each, but in 3 hours with 20 pails of 4 gallons each, the pails being poured in at intervals. Find how much the cistern holds, and in what time the water would waste away. EXAMPLES FOR EXERCISE. W4a. ( First Series. ) 1. State in words 10030200720021. 2. Find the value of 66674 - 9^45 - 201 + 843 - 8761. 3. Reduce ^49. 6*. i\d. to farthings. 4. Find the prime factors of 51425. 6. Reduce Jjjdof to its lowest terms. 6. Find the sum and difference of 23*001 and "0414. 7. Find the value of f of ^7. 7 a. 7 p. 8. Write in words 3200103102 according to the Indian numeration. 9. The greatest prime number known is expressed by 1251*4- 2920* find the number. 10. What sum will remain when four bills, amounting to &5 . 7 . 6, &3 . 4 . 9, R2 . 15 . 3, and &io . 13 . 3 respectively, have been paid out of 25 ? EXAMPLES FOR EXERCISE 2Q9 11. Find the G. C. M. of 23791 and 8029. 12. Subtract I4 T V& from i6 3 \. 13. Multiply '038 by "0042, and divide '03217 by 6*25. 14. Find the value of '00625 of i. 16. Subtract one crore five lacs three thousand and twenty from twenty-nine million twelve thousand and four. 16. Multiply 765389 by 64164 in 3 lines 17. I go to town with $ is. 3<V. What have I left after buying a dozen chairs at i$s. j^d. each ? 18. Find the L. C. M. of 9669 and 16115. 19. Add together ]$> 3j, i^ and a T T . 20. Express as a decimal '0003 + 3Y/ir""' oo 849-t-$on 21. Reduce f of ^ of 195. 6d. to the fraction of of ^ of 2^. Express 944 in Roman notation, and CDXCIX in Arabic notation. 23. Multiply 387659 by 85672 in 3 lines. 24. How many cows at Rio. i^a. each can I buy with the proceeds of selling 87 horses at 8115. 2a. each ? 26. Simplify 26. Multiply '006134 by 80*032, and divide the result by '0032. 27. Reduce (8-riJ) of i/>. to the decimal of Ri. 4^. 28. If a rupee is worth 2s. o^/., and a dollar 4^. 4j*/., find the least number of rupees which makes an exact number of dollars. 29. What number multiplied by 76 will give the same product as 153 multiplied by 380 ? 80. Find the greatest number which will divide each of 3456, 16244 and 99225 without remainder. 81. Reduce 57 tons 9 cwt. i qr. 10 Ib. to drams. 82. Simplify fx|-r- 1 J of ij. 33. Find the least fraction which being added to J-J of $- j will make the sum an integer. 34. A did '0025 of a piece of work, and B 7855, How much was left undone ? 86. Find the cost of 3^125 yards at ^'375 a yard. 3CO ARITHMETIC 38. What number is the same multiple of 35 that 3456 is of 9 ? 37. If my income is R3$oo and I save ^507 a year, what is my average daily expenditure ? 38. Sin.pKfy W"^. ~!J () i ff~7 39. If the sum nf 2(f and 3^ be added to the product of 2$ and'J, by how much will the result differ from 28 ? 40. Reduc^ 3 2 W to a decimal. 41. Find the vulgar fraction equal to '27899. 42. Find ihe value of g of R3 . 7 . 6-K 375 of R6 . 8 . 6, 43. Find the least number wliicli being subtracted from 97856 will make the n suit divisible by 141. 44. Reduce 3 acres i lood 2 perches to square teet, 46. Arrange -f, 7, J in oider of magnitude. 46. Divide -!-J of 12 by g of f~i2. 47. Add 372 4- 'oo? -f '272?. 48. Reduce 'oj of Kj to the decimal of J of R 1*5. 49. Find the least number of weeks in which an exact number of half-guineas can be earned, the wag^s per week being 7*5 shillings. 50. What is the least number which bring added to 30321 will make the sum divisible by 63r ? 61. A bill of 6. is. lid. has to be paid by several persons ID equal shares ; if three of them together pay ^r. 135. 3<, how many are there to share the cost ? 52. Simplify 2^-riSi-* IS X2if. 53. Divide 352 95624 by "000504, 54. Express 1*4 1 '13 as a decimal. 55. Reduce '543 of 19^. 3^. to pence. 56. Find the greatest unit of time by means of which 2 hr. 3 min. and I hr. 4 min. 30 sec. can both be expressed as integers, 57. I multiply a number by 36 and divide the result by 12 and obtain 374 181 as quotient. What was the number ? 68. A and B together have 36. 130. 9/ M and A has 83. 30. 3/. more than B ; find how much B has. 69. Reduce T Vtfti 4 s to its lowest terms. 60. Express 3}J poles in poles, yards, etc. EXAMPLES FOR EXERCISE 301 61. What are the nearest integers to 8 T 9 , r and 7 T \ ? 62. Find the difference between the product and quotient of 5*312 by '0125. 63. Simplify (2-364- 1*697} + i'3 x (2 44-7' 5). 64. If in a division sum the divisor be 7 times and the quotient 5 times the remainder, what is the dividend when the remainder is 360- ? 65. Reduce 300,003,840 grains to pounds Troy, 66. Find thr cost of 1371*4 articles at R8. oa. *]\p. each. 67. Multiply 7+6 by 2^-jhf. 68. What fraction of a journey of 15 miles have I gone on reaching a place 6 miles distant ? 69. By what must 1550^3 be divided that the quotient may be 4595 ? 70. H a metre he 39*37 inches, how many metres make 3 miles ? 71. When 70^0400 i- divided by a certain number, the quotient is 381 and the remainder 1664 What is the number ? 72. Reduce 67501 inches to poles, etc. 73. If 2/5 tons cose Hg64. 5.;. 8A, what is the cost of i too ? 74. Simplify [ y 3- -ia 75. Divide equally amongst 5 boys f of 4. 2s. ijv/. 76. Divide 7029 by '0165. 77. What decimal of &3. 7*2. must b taken irom R4. 1501. to leave ii2'5 ? W, If when a number is divided c.int nuousiy by 5, 6 and 7> the remainders are 2, 3 and 4 iepecuvd>, what would be the remainder if the number were divded by 210 ? 79. If I md. cost &n. i^., find the cost of ^, of a md. 80. The ist of January 1893 was on a Sunday; on what day of the ueek will loth February fall m the year 1094 ? 81. Find the value of ^A^of^. ^8ff 7 | DJ2 82. If from a rope 7 ft. long as many pieces as possible are cut off, each i J ft. long, what traction of the whole will be left ? 302 ARITHMETIC 83. Reduce '142857 + ' 5714$ -'285714 to a vulgar fraction. 84. Simplify .-*- 85. Find a number such that if it be added 35 times to 25 the sum will be 25540. 86. If a person spends in 4 months as much as he earns in 3, how much can he* lay by annually, supposing that he earns ^250. loj. every 6 months ? 88. How many steps does a man whose length of pace is 32 Inches take in 4! miles ? 89. Divide 75445 by '00625. 90. How many inches are there in '1215625 of a mile ? 91. Subtract '432 of an acre from i\ roods, expressing the result in sq. yards and the decimal of a sq. yard. 92. A man buys 100 md. of rice ; he loses as much by selling 60 md. at &3 a md. as he }. T ains by selling the rest at &4. 4#. a md. Find the cost price of a md. 93. By what prime numbers may 109 be divided so that the remainder may be 4 ? 94. AddJttJ + JJtf + jfiJ. 96. How many times can '053 be subtracted from 14*578, and what will be the magnitude of the remainder ? 90. Express '236 of 40. 7/>. + '516 of ioa. as the decimal of Hi. 40. 87. Simplify ^^^1. r J '003 x '0005 98. Three bells toll at intervals of 1*2, 1*8 and 27 seconds respectively, beginning together ; how often will each toll before their tolling together again ? 99. The remainder after a division is 97, the quotient is 521, and the divisor is 9 more than the sum of both ; what is the dividend ? 100. Two pieces of cloth of the same length cost ^5. us. gd. and 7. 4^. respectively ; the price of the first was 3*. ijdf. per yard : what was the price of the second per yard ? 101. Divide } of of f of 42 by the sum of 2\ and 4$. EXAMPLES FOR EXERCISE 303 102. Simplify i[2-{ 2 -i(2 103. Reduce f to a decimal. 104. Multiply 28 8 by 25*3 and divide the product by 6*48. 105. The distance between two wickets was marked out for 22 yd., but the yard measure was ^ of an inch too short ; what was the actual distance ? 106. If a number of articles at 4. oa. 5^. each cost ^7059. 143. iij#., how many are there ? 107. Simplify of -~ of ~ * of 117. 108. Find the value of * X '**~ *' 74X -^ 4 of Ri. 4*. 426- -174 * 109. Subtract 5-142857 from 5*14^857. 110. Divide rco625 by I32'5 to five places of decimals. 111. Reduce 4 hr. 48 min. to the decimal of 6 hr. 112. A man owns T 8 ff of a house, and sells '1351 of his share ; what fraction oi the house does he still own ? 113. How many revolutions will be made by a wheel, which revolves at the rate of 243 revolutions in 3 min., while another wheel revolving 374 times in n min. makes 544 revolutions ? 114. Multiply 10 sq. yd. 4 ft. 76 in. by 132. 115. Reduce to its lowest terms 116. Find the least number which, when divided by each of vz> * 2 5> arj d "3 &i ves a whole number as quotient in each case. 117. Simplify ; * ~ri . 118. Find, to the nearest pie, the value of '1234 of Rl2*5. 110. A kilolitre contains 35-32 cubic feet, and a gallon contains 277*274 cubic inches ; find to the nearest integer the number of gallons in a kilolitre. 120. A farmer has 899 sheep and 493 lambs. How forms them into flocks, keeping sheep and lambs separate, and having the same number of animals in each flock. If these flocks are as large as possible, how many flocks will there be altogether ? 121. If 257 pounds of tea cost 34. 16*. 7i^-> fi ad the price of a pound to the nearest farthing. 304 ARITHMETIC 122. Simplify ^v ~ H~$ 123. How many whole cakes will be required for 50 children if each is to have 2% of i^ of 2f of ot {& ot la * a cake ? 124. Find the value of ^ -^ 5^A f ^ '37 5 + '04 125. Find the circulating decimal which will become 2 when multiplied by 2-J -7-4-5, 12 r< . A German mark is worth ^'04895 ; find to the nearest farthing the value of 3725*39 marks. 127. To a certain number I add 2, 1 multiply the sum by 4, I divide the product by 3, and 1 take 3 from the quotient ; the remainder is 17. What is the numbci ? 128. On what day ot the week will Feb. 10 fall in the year 1960 ? 129. Find the greatest prime number which used as divisor of 12260 v\ill ler.ve remainder 17. 130. Find the value of 2 ' 8 of ^ l -5 a '-f\ *2I V4. 2ft. 0/>. 131. What is the number whose halt exceeds its fifth part by 6? 132. Simplify '428571 x -49 x '20571 p8. 133. How many times does a ca'riaie wheel, whose circum- ference is 17*125 feet, turn round in a distance ot 12*45 miles ? 134. Determine t) e prime factors of 282660 and 40299, Hence deduce the G. C. M, and L. C. M. ot these numbers. 135. Find the least integer which, * hen divided by I 2 \ and 1 2, will ive a whole number as quotienc in each case. 136. Simplify f of ||- Jg of l| + {( 137. Reduce /y +v%v + v9v%vv to a decimal. 138. If a cu. yd. of clay make 460 bricks, each ioi cu. in., how much does clay contract in baking ? 139. Multiply 324-567 by I3'2i2 in 2 lines. 140. One pendulum oscillates 6 times in 3*2 seconds, and another pendulum 8 times in 3*6 seconds ; if started simultaneous!) r how often will they tick together in an hour ?> EXAMPLES FOR EXERCISE 30$ EXAMPLES FOR EXERCISE. l?4b. (Second Series.) 1. Write down the greatest and least numbers of four digits that you can form with the figures, 3, o, 2, i. 2. Simplify Jt3+i{3 + i(3+i|)}H*. 3. The telegraph posts on a railway line are 66 yards apart ; find the smallest number of miles that corresponds to an exact number of posts. 4. A bath is supplied with water from two pipes, one of which can fill it in 12$ min., the other in 15 min. ; there is also a dis- charging pipe which would empty it, when filled in lo mm. The first pipe is open alone for 4 min., and then the first and second open together for I min. ; if now the third pipe is opened as well, how long will it take to fill the bath ? 5. The wages of A and B together for 20 days amount to the same sum as the wages of A alone for 35 days. For how many days will this sum pay the wages of B alone ? 6. A cask contains 5 parts wine and 3 parts water ; how'much of the mixture must be drawn off and water substituted injorder that the resulting mixture may be half and half ? 7. A person borrows ^130 on the 5th of March, and pays back .133. i8j. on the loth October ; find the rate of interest charged. 8. The digits in the units' and lacs' places of a number are 3 and 8 respectively ; what will be the digits in the same places in the remainder when 99999 is subtracted from the number ? 9. A whole number diminished by of itself, when divided by 307 gives a quotient 12 and a remainder 96 ; what is the number ? 10. The length of a rectangular tennis-court is 5 yards longer than its breadth, and its perimeter is 130 yards ; find its area. 11. The train which leaves Calcutta at 4-30 P. M. arrives at Burdwan at 8 P. M. ; and the tiain which leaves Burdwan at 4-50 P. M. arrives in Calcutta at 8-30 p. M. : when do they pass each other ? 12. The rent of a farm consists of a fixed sum of money together with the value of a certain number of maunds of wheat ; when wheat is &2 a md. the rent is 840 ; when wheat is &2. 40. a md. the rent is &42. 8a. , What will be the rent when wheat is &2. i oa. a md. ? 13. Assuming that the circumference of a circle is to its dia- meter as 22 is to 7, and that the circumference of the earth is to its diameter as 160 metres to 167 feet, determine to 4 places of decimals the ratio of a metre to a foot. C. A. 20 306 ARITHMETIC 14. The interest on a given sum of money for one year is $. 8x. 4<, the compound interest for two years is 11. is. Find the rate per cent. 15. If when a number is divided continually by 5, 6 and 8 the remainders are 2, 3 and 4 respectively, what would be the remainder if the same number were divided by 240 ? 16. Divide 1255 by 1*004 and hence deduce the quotient of I2'55 by 1004 and '01255 by 1004000. 17. 1 bought a certain number of chairs for 45 ; also a certain number for &28. 2a. at the same rate ; find the greatest possible price of each chair. 18. A clock which gains 2j min. in a day, is 3 min. slow at noon on Sunday ; when will it show correct time, and what time will it indicate at 6 on Monday evening ? 19. A person bought 4 railway tickets to go 60 miles. Two were for the 1st class, one for the 2nd, and the fourth, a half first class ticket, for a child. The cost of a 2nd class ticket was $ of that of a first class, and the whole sum paid was i. n. 8. Find the price of each ticket, and the rate per mile for the first class. 20. There are two mixtures of wine and water, in the ratios of 3 I 2 and 4 I 5 respectively ; if one gallon of the first be mixed with 2 gallons of the second, what fraction of the resulting mixture will be wine ? 21. A book sent from England costs me (including is. 6d. postage) i6s. irf., my book-seller allowing me two pence in the shil- ling discount on the published price. What is the published price ? 22. What number is the same multiple of 7 that 3975 is of 15 ? 24. On laying down a bowling-green with sods 2 ft. by 9 in., it is found that it requires 120 sods to form one strip extending the whole length of the green f and that a man can lay down one strip and a half each day ; find the space laid down by 5 men in 2 days. 26. A can do a piece of work in 3 days, B can do 3 times as much in 8 days, and C $ times as much in 12 days. In what time will they do it together, supposing them to work at the rate of 9 hours a day ? 26. A farmer pays a corn-rent of 5 quarters of wheat and 3 quarters of barely, Winchester measure ; what is the money value of his rent, when wheat is at 60 r., and barley at 541. per quarter, Imperial measure ; 32 Imperial gallons being equal to 33 Winchester gallons ? EXAMPLES FOR EXERCISE JO7 27. Six coins of equal weight, made of gold and silver mixed, were melted together and re-cast, in one the gold and silver were in the ratio of 2 I 3 ; in two others, of 3 I 5 ; and in the rest, of 5 \ 4. In what ratio will the gold and silver be mixed in the new coins ? 28. A tradesman, selling goods for a certain price to be paid six months hence, offers to give one-tenth more of the same goods for the same price in ready money. What is the rate of discount ? 29. Find the greatest and least numbers of 6 digits which are -exactly divisible by 239. 30. There is a number, to which 3 is added and ^ of the result taken ; to this 5 is added and T \ of the result taken, giving i-J ; what is the number ? 31. Find all the numbers of 5 digits divisible by 9, which have unity for their first and last digits and 2 for their middle digit. State the principle upon which you proceed. 32. On a stream, B is intermediate to and equidistant from A and C ; a boat can go from A to B and back again in 5 hr. 15 min., and from A to C in 7 hr. How long would it take to go from C to A ? 33. If the price of bricks depends upon their magnitude, and if 100 bricks, of which the length, breadth and thickness are 16, 10 and 8 inches respectively, cost R,2. ga. t what will be the price of 921600 bricks which are one-fourth less in every dimension ? 34. There are two mixtures of wine and water, the quantities of wine in them being respectively '25 and 75 of the mixtures. If 2 gallons of the first be mixed with 3 gallons of the secondj what will be the ratio of wine to water in the compound ? 36. How much per cent, must be added to the cost price of goods that a profit of 20 per cent, may be made after throwing off a discount of 10 per cent, from the labelled price ? 36. Determine the least number, by which 616 must be multi- plied so as to produce a number exactly divisible by 770. 37. Multiply the sum of y\ and j; by rj, and add the result to the difference of 2*3^4 and 1*697. 38. The floor of a room is 50 ft. long and 40 ft. wide. Find the cost of supplying it with carpet, 2 ft. wide, at &3 per yard, and oil-cloth, 2 yards widei at Ri per yard ; the oil-cloth to be laid along the sides and ends a yard and a half wide, and the carpet to extend one foot over the oil-cloth everywhere. 30. On a certain evening half an hour after sunset a watch was set at 12 o'clock. The morning following it was 8 minutes 3 o8 ARITHMETIC past 4 by a common clock when it was 4 minutes past 8 by this- watch. Find the time of sunset the previous evening. 40. A has shares in an estate to the amount of ('i 5 -~ "36) of it. S has shares in the same estate to the amount of '472 of it. Find the difference in value between the properties of A and B, when oj<5 of the estate is worth 3 73*3- 41. Three equal glasses are filled with mixtures of spirit and water ; the proportion of spirit to water in each glasc is as follows i in the first glass as 2 *. 3, in the second glass as 3 I 4, and in the third as 4 I 5. The contents of the three glasses are emptied into a single vessel ; what is the proportion of spirit and water in it ? 42. If the true discount on a bill of 14641 be 4641 at 10 per cent, compound interest, how many years has the bill to run ? 43. Twenty-fifth part of a certain number is equal to the seventh part of 42 ; what is the number ? 44. Simplify AV4 of 6f + A)-5- 4 J of (6f + ). 45. A company of Sepoys proceed in 5 equal rows, and after sometime arrange themselves into 7 equal rows. Find the least number above looo, which the company may contain. 46. A is twice and B is just as good a workman as C. The three work together for two days, and then A works alone for half a day, and B for a day. How long would it have taken A and C to- gether to complete as much as the three will have thus performed ? 47. A steam-ship whose speed averages 14 miles an hour, reaches a certain port in 12 days ; how many days afterwards will a sailing vessel arrive, which started at the same time and sailed on an average 8 miles an hour ? 48* From a cask of wine J is drawn off and the cask is filled up with water ; J of the mixture is then drawn off and the cask is again filled up with water ; after this process has been repeated 4 times, what will be the ratio of wine to water in the resulting mixture ? 49. The sum of ^2100 is due in 4 years, but it is paid by instalments as follows : .275 at the end of 2 years, ^460 at the end of the 3rd year, ^500 at the end of the 4th year, and 6co at the end of the 5th year. What amount should be paid at the end of the 6th year, in order to clear off the balance, simple interest being reckoned at the rate of 5 per cent, per annum ? 60. Twenty times a certain number is equal to 7 times 40 ; what is the number ? 51. What is the less number of $hot, each i oz., that will weigh an integral number of pounds ? EXAMPLES FOR EXERCISE 309 52. A rod of brick work contains 306 cu. ft. ; find the cost of fouilding a brick wall, 68 yd. by 6 ft. by 2 ft. 2 in., at Bi8 per rod. 53. How long would a column of men, extending 3420 feet In length, take to march through a street, a mile long, at the rate of 58 paces in a minute, each pace being i\ feet? 64. 195 men are employed to work on a railway embankment, l\ miles long, which they are expected to finish in 4 weeks. But at the end of i week it is found that they have finished only 520 yards. How many more men must be engaged to finish it in the required time ? 56. A is a cask containing 125 gallons of wine ; B is another cask containing 175 gallons of water. 100 gallons are drawn from each, mixed together, and the casks are refilled with the mixture. This operation is once more repeated. Find the ratio of wine to water in each cask now. 56. A person who pays 5^. in the income-tax finds that a rise of interest from 6 to 6J per cent, increases his income by 23* los. What is his capital ? 67. From a certain number I take 320 ; to the remainder I add 24 ; 1 multiply the sum by 8, and find that the product is equal to the sum of 304 and 760 ; what is the number ? 58. What decimal of 2*25 units is "05 of a unit ? 59. A jar can be exactly filled by glasses holding 3 pints each it can be exactly emptied again by glasses holding 5 pints each given that the cap icity of the vessel is between II and 12 gallons find the exact capacity. 60. Two clocks are set right at noon on Monday. One loses and the other gains I min. a day. What time will be indicated by the latter, when the former points 10 h. 49iV m - p< M on the following Saturday ? * 61. Three gardeners working all day can plant a field in 10 days, but one of them having other employment can work only half time. How long will it take them to complete the work ? 62. One vessel contains 20 gallons of wine ; another contains 20 gallons of water. One gallon is taken from each, and poured into the other. This is done 3 times. Find the strength of the two mixtures. 63. A gentleman bequeaths 'his property to his children to be so divided that their shares shall be equal on their coming to age at 21, counting interest and discount at 5 per cent. He dies worth ^13240, leaving three children aged 23, 21 and 19 respectively. How much should each receive ? 3X0 ARITHMETIC 64. To a certain number I add 7> I multiply the sum by $* I divide the product by 9, and take 3 from the quotient ; the remainder is 12 : what is the number ? 65. Simplify C5 + 75)(2'5-^) 66. Find the weight in tons per sq. mile of a ram-fall of 7 inches, having given that a cu. ft. of water weighs 1000 oz. 67. A, B and C are employed on a piece of work. After 15 days A is discharged, \ of the work being done. B and C continue at the work, and alter 20 days more B is discharged f J- more of the work being done. C finishes the work in 30 days. In what time would the work have been done, if A and B had continued to work ? 68. If one man walks 165 miles in 6 days, how far will another man walk in 15 days, if the first man walks 3f miles in, the same time that the other man takes to .walk 4 miles ? 69. If 3 cubic inches of iron and 2 cubic inches of water weigh as much as 2 cubic inches of iron and 9 cubic inches of water ; find the ratio of the weight of a cubic inch of iron to that of a cubic inch of water, 70. I buy goods for B6oo, and sell them directly for R68o, giving three months' credit ; what is gained per cent, per annum ? 71. From the tenth part of a certain number I subtract 10, and find that the remainder is 10 ; what is the number ? 72. | of a number exceed the sum of its third and fourth parts by 26 ; what is the number ? 73. Two cog-wheels, having 75 and 130 teeth respectively, are working together ; after how many revolutions of the smaller wheel will the teeth which once touch, touch again ? 74. A train leaves P for g, at the same time that a train leaves Q for P ; the trains meet at the end of 6 hours, the train from P to Q having travelled 8 miles an hour more than the other. Find the rates of the trains, the distance from P to Q being 162 miles. 76. If 1000 rupees a month be equivalent to ^1112. los. a year, what is the value of a rupee in English money ? 76. Divide 20 among 2 men, 3 women and 4 children, so that each woman gets twice as much as a child, and each man as much as a woman and a child together. 77. If the interest of ^253. 2$. 6d. at 5 p. c. be equal to tho. discount on ^257. 6s. \o\d. for the same time and the same rate* when is the latter sum due ? EXAMPLES FOR EXERCISE 311 78. Find a number such that if it be subtracted 25 times from 7201 the remainder will be 951. 79. How many parcels of gold dust, each weighing 17*36 grains, can be made up out of i Ib, 2 oz. I dwt, 3 gr. ; and how much will remain over? 80. A room is 20 ft. lone* 15 ft. wide and 10 ft. high. There are in it 4 doors, each 7 ft. by 4. ft. ; the fireplace is 6 ft. wide and 4 ft. high ; a skirting 2 ft. deep runs round the walls. Find the expense of papering the room at 6 annas a sq. yd. 81. If the hands of a clock coincide every 65$ min, (true time), how much does the clock gain or lose in a day? 82. A can copy a certain manuscript in 17 hours by writing at the rate of 3 lines per minute ; B can copy the same in 24 hours. After 476 lines have been copied by A, in what time can B finish it ? 83. A town contains 12 Hindus to every 3 Mahomedans and to every 3 Christians ; if there are 4800 Hindus, find the number of Christians. 84. Two suras, each of ^138. 2J. 6d, being due, one at the present time and the other 12 months hence, how much ought to be paid 6 months hence to clear off both debts, interest being 4 p.c, per annum. 85. The difference between two numbers is 375, and one of them is 7809 ; what is the other ? 86. Simplify If 8* of {& of 3& + 6i of 3. o* 9*-4H of 3. 2s.}. 87. A fruit-seller has 1134 mangoes and 630 oranges. He forms them into heaps keeping the mangoes and oranges separate, and having the same number of fruits in each heap. If these heaps are as large as possible, how many fruits are there in each ? 88. A cistern, the cubic content of which is 360 cu. ft., has two pipes which can empty it in 3 and 4 hours respectively. It has also a third pipe with an orifice of I sq. ft.> through which water flows into the cistern at the rate of i yd. per minute. If all the three pipes be opened together when the cistern is full, in what time will it be emptied ? 89. If 4 men or 6 women can do a piece of work in 20 days, in what time will 3 men and 2 women do it ? On what supposition will the numerator of the fraction in your answer represent the number of hours they worked on the day to which the fraction refers ? 90. Divide 1140 among A, B, C, in such a way that A may get half as much again as B, and B half as much again as C 1 , 3^2 ARITHMETIC 91. A dealer buys 10 horses at 400 each, 8 horses at R$oo each and 4 horses at R6oo each. He keeps the horses for 6 months, during which time each costs 8.15 a month, and sells them clearing I2j p. c. on his original outlay after paying all his expenses. Find the average selling price of each horse. 92. A carriage and a horse are together worth Ri2oo ; if the carriage is worth R2oo more than the horse, how much is the horse worth ? 93. The population of a town is 60,000 ; if the births are I in 20, and the deaths I in 30 annually, what will the population become in one year? 94. A cistern, 9 ft. by 6 ft. by 5 ft., is emptied in 15 minutes by a pipe whose cross section is 36 sq. in. ; how fast does the water flow in the pipe ? 95. A race-course is 2j miles round. Four men start to walk round it. They walk at the rate of 3j, 3}, 4^ and 5 miles per hour. How long will it be before they all meet again at the starting point ? 96. 40 lb. troy of standard gold containing 1 1 parts in 12 of pure gold, is coined into 1869 sovereigns ; calculate in grains the weight of pure gold in a sovereign. 97. Divide &7. 50. into two parts, one of which is f of the other. 98. If mangoes be bought at the rate of 13 for a rupee s how must they be sold to gain 30 per cent. ? 99. A has 324 ; B has ^29 less than A ; and C* if he had 205 more than what he has, would have as much as the double of A and B together : how much has C ? 100. In how many years will the error amount to a day in considering the year to consist of 365^ days instead of 355*242218? 101. The circumferences of two wheels measure 168 and 401 inches respectively ; find the largest cogs which can be cut in each that they may work together. \ 102. The hands of a clock which \gains uniformly at the rate of 15" a day were set at sunset on the evening of the first of the month at 6 o'clock. The true time of sunrise on the 3rd was known to be a quarter to six, but the clock indicated a quarter past six. Find the error made in setting the clock on the ist. 103. A train travels 30 miles an hour when it does not stop, and 25 miles an hour including stoppages ; in what distance will the train lose one hour by stoppages ? 104. Divide 123 among A, B, C, so that as often as A gets R3 B shall get 2$, and as often as B gets 4 C shall get 3$. EXAMPLES FOR EXERCISE 313 105. A merchant buys 4000 maunds of ricef $ of which he sells at a gain of 5 p. c., J at a gain ofl 10 p. c., $ at a gain of 12 p. c., and the remainder at a gain of 16 p. c. If he had sold the whole at a gain of 1 1 p. c., he would have made 8.728 more. What was the cost of the rice per maund ? 106. A man sold 16 oranges to A, to B 4 more than to A, to C 5 less than to B \ had he sold 3 less to each he would have left only one-third of what he had ; find how many he had at first, 107. amplify F y -- 108. A room is 18 ft. long ; and the cost of carpeting it is 72. If the breadth of the room were 4 ft. less, the cost would be 854 } find the breadth of the room, 109. A can mow i\ acres of grass in 6 hours, and B 2j acres in 5^ hours ; in what time will they together mow a field of 10 acres, and how many acres will each mow ? 110. The cost of 12 md, of wheat and 10 md. of pram is Rjo when gram is at R2 per md. What is the price per md. of gram when 8 md. of rice and 6 md. of gram cost 34, the price of rice 'being J higher than that of wheat ? 111. Divide &2o. 40. among 5 persons so that the share of each (except the first) may be double of the shares of all who come before, 112. A merchant bought a so-gallon cask of wine for 741. Supposing it to have lost 4 gallons, at what price perdozen bottles (nine bottles holding a gallon) should he sell it in order to gain 15 p. c. upon the whole original cost ? 113. A man lost as much by selling 20 chests of tea at R62O per chest as he gained by selling 25 chests at 692 per chest ; what did each chest cost him ? 114. A man left his property to two sons and a daughter ; to the elder son he left i of his property, to the younger son , and to the daughter the rest, which was 4000 less than what the two sons together received : what was the entire property ? 115. Three lines of paling run side by side for a distance of 864 yards. The rails are respectively 4, 6 and 9 feet apart. How often will a person walking outside the palings, on looking across them, see three rails in a line ? 116. Three persons, A, B^ and C, who can walk respectively 2, 3, and 4 miles per hour, start from the same place P at intervals of an hour. A starts first, and as soon as B has caught him up, & returns to the station P ; find where he will meet C. 314 ARITHMETIC 117. A fraudulent tradesman uses a yard measure one inch too short ; what does he gain by his dishonesty in selling 20 yd, of cloth at Ri. 2a. per yard ? 118. A 9 By C had each a cup of tea, containing 4 oz , 5 oz. and 6 oz. respectively. They blended their teas and then refilled their cups from the mixture ; how much of the teas of A and B are contained in Cs cup ? 119. If by selling wine at R6 per gallon I lose 25 per cent., at what price must I sell it to gain 25 p. c. ? 120. A man, having lived at the rate of 300 a year for 6 years, finds himself in debt, and reduces his expenditure to ^250 a year ; he is out of debt in 4 years : what is his income ? 121. Express the sum of '571428 of a viss, | of * of -~~~ o! o 3f 3o4 a maund and i 8 8 i%k f a cwt - as a decimal of one ton. [One viss 3 lb. 2 oz. ; one maund 82f lb.] 122. A rectangular cisternj 12 ft. long, 10 ft. wide and 4 ft 3 in. deep, is filled with liquid which weighs 2040 lb. How much deep must another cistern be, which will hold 196 lb. of the same liquid, its length being 7 ft. and width 3 ft. 6 in. ? 123. .A can run 100 yd. in 12 sec.i and B in 13 sec. How much start in distance must A give B m order that they may run a dead heat ? 124. The Fort-Barracks are lighted with gas from 100 burners. Find the cost of lighting them per night of 10 hours, at the rate of BSJ for loco cu. ft. of gas, assuming that for the first 3 hours each burner consumes i cu. in. per second, and during the remain- der of the night the light is so reduced that the consumption of gas by each burner is only J of that quantity per second. 125 1 20 coins consist of crowns, half-crowns and florins ; the values of the crowns, half-crowns and florins are as 25 \ 10 '. 6 ; how many half-crowns are there ? 126. A merchant sells 60 md. of rice at a profit of 8 p. c. and' 94 md. at a profit of 10 p. c. ; if he had sold the whole at a profit of 9 p. c. he would have received 17 annas less than he actually; did : how much per md. did he pay for the rice ? 127. A man having a certain number of mangoes to dispose of, sells half of what he has and one more to A, half of the remain- der and one more to B, half of the remainder and one more to C* half of the remainder and one more to D ; by which time he has only one left ; find how many he had at first. EXAMPLES FOR EXERCISE 315 128. Simplify f +$-8 of 2lJ + -06474358$. 129. A dollar being worth 45. 2d. and a rouble 3*. ijrf., find the sum of money which can be paid by an exact number of either dollars or roubles, the number of roubles exceeding the number of dollars by 20. 130. A can do a piece of work in 15 days, B in 12 days and C in 10 days. All begin together ; A leaves after 3 days, and B leaves 2 days before the work is done. How long did the work last ? 131. A tank is 300 yd. long and 150 yd. broad ; with what velocity per second must water flow into it through an aperture 2 ft. broad and i J ft. deep, that the level may be raised I ft. in 9 hours ? 132. The height of the top of a flag-staff standing on a tower is no ft., and the height of the tower is 6 ft. more than 12 times the length of the flag-staff ; what is the length of the flag-staff? 133. A merchant buys some cloth at such a price that by selling it at &4. 6a. per yd. he will gain 5 p. c. on his outlay. What percentage will he gain or lose if the cloth be sold at&3. 140, per yd. ? 134. I wish to buy an equal number of 3 kinds of toys, worth respectively is. 9 is. 6d and is. bd. each ; how many can I get for 10? 135. In a book on Arithmetic an example was printed thus : " Add together ^, , 1, ^," the denominator of one fraction bein accidentally omitted. The answer given at the end of the book was jJJ ; required the miss- ing denominator. 186. Find the side of a square courtyard, the expense of paving which at 3*. 9^. per sq. yd. was ^42. 3$. gd. 137. A and B start at the same time from Calcutta to Hugli and from Hugli to Calcutta respectively, each walking at the rate of 4 miles an hour. After meeting /?, A increases his rate to 4^ miles an hour, and arrives at Hugli in ij hours from that time. After meeting A, B reduces his rate to 3j miles an hour. In what time will he reach Calcutta ? 138. If the rent of a farm of 24 acres be ^39, what will be the rent of another farm of 36 acres, 5 acres of the (ormer being worth 6 acres of the latter ? 139. A purse contain 8 . 7,. II, made up of pennies, shillings, half-crowns and crowns, the numbers of which are proportional- 3l6 ARITHMETIC to 7, 3> 2 and 5 respectively ; how many of each coin are there in the purse ? 140. Calculate the profit per cent made by a book-seller, assuming that he pays us. ^d. for a 16-shillmg book and receives 25 copies for 24. 141. A person mixes together 10 lb. of tea at Ri. 4#. a lb., 12 lb. atRi. 6a. y and 14 Ib. at Hi. 8a. He reserves 6 Ib. of the mixture for himself and sells the remainder at Ri. i$t. 4A a lb. How much does he gain in money ? 142. Multiply '047321 and 12172-8144, using only 3 lines of multiplication. 143. Three men, the length of whose strides are 2 ft. 6 in., 3 ft. and 3 ft. 6 in., walk a mile. How often do they step together ? 144. A and B start on a bicycle race. A has 10 minutes' start, during which he goes 2j miles ; B rides at the rate of 16 miles an hour. Which will win in a race of 40 miles ? 145. If 3 soldiers or 10 coolies can dig 150 cu. ft. of earth in 5 days, how many coolies must be employed to assist 7 soldiers in removing 580 cu. ft. of earth so as to get it done in 4 days ? 146. I2J. 3j</. is divided among men, women and children whose numbers are proportional to 3, 5 and 7 respectively ; if a man receives 5.}^., a woman $\d. and a child 2j*/,, find the number of men. 147. .An article was sold so as to gain 5 p. c. on its cost price. If it had been bought at 5 p. c. less, and sold for is. less, lo p. c, would have been gained. Find (he cost price. 148. A wine merchant bought 7 gallons of wine at 17*. a gallon and 5 gallons at 155. a gallon ; he mixed the whole and added some water. The whole mixture he put into quart bottles which cost him 8j. 6d. and sold each bottle at 44. and gained l. l?s. 6d. on the whole. How much water did he mix ? 149. Find the value of of i + J of ,140. los. 6//. + $of 213. 150. The weight of water contained in a rectangular cistern, 8 ft. long, 7 ft. wide, is 93^ cwt. Find the depth of water in the cistern, supposing a cu. ft. of water to weigh 1000 oz. 151. 25 men are employed to do a piece of work, who could finish it in 2qdays ; but the men drop off by 5 at the end of every 10 days : in what time will the work be finished ? 152. If 48 men, working 8 hours a day for one week, can dig a trench 235 ft. long, 40 ft. wide and 28 ft. deep ; in what time can EXAMPLES FOR EXERCISE 3*7 12 men, working 10 hours a day, form a railway cutting of 131,600 cu. yards ? [A week 6 working days.] 153. The sum of areas of two circles, of which the diameters are as 3 is to 4, is equal to the area of another circle 10 ft. in diameter ; find the diameters of the two circles, having given that areas of circles are to one another as the squares of their diameters. 164. A merchant sells sugar to a tradesman at a profit of 50 per cent. ; but the tradesman becoming bankrupt pays only 5 annas in the rupee. How much per cent, does the merchant gain or lose by the sale ? * 155. How many parcels of 6 Ib. and 8 Ib. each can a grocer make out of a hogshead of sugar, weighing 4 cwt. 3 qr. 14 Ib., so as to have the same number of parcels of each sort ? 156. A had los. in his purse, and B having paid A 2 x ~ of *\ , i. us. 6d. finds that he has remaining ^ of the sum which A now has ; what had B at first ? 157. A number is exactly divisible by n ; but when divided by 5, 6 or 8 leaves always the remainder i : find the least number which satisfies these conditions. 158. A boat's crew row over a course of 2j miles against a stream, which flows at the rate of 3 miles an hour, in 30 minutes, The usual rate of the stream \s one mile an hour. Find the time which the boat would take in the usual state of the river. 159. If the cost of n miles of iron rails be 55000 when iron is selling at 6.95 a ton, what will be the cost of 19 miles of the same rails when iron is selling at 105 a ton ? 160. A circular plate of gold, 10 in. in diameter and 2 in, thick, is melted and formed into two other circular plates, each I in. thick, whose diameters are as 3 to 4 ; find the diameters. 161. A man buys goods for 750, and sells \ of them at a lo^ of 4 p. c. ; by what increase per cent, must he raise that selling price in order that by selling the rest at the increased rate he may gain 4 p. c. on the whole transaction ? 162. A person gives 53 guineas for 184 gallons of wine ; how much water must he add to it, if he wishes to sell it at 5*?* 3^- & gallon and make a profit of 7 half-guineas ? 163. A vessel containing 21*84375 gallons of water is emptied by a pitcher which contains when full '078125 gallon. How many times can the pitcher be filled entirely, and what fraction of a pint will it contain when the last quantity of water is poured into it ? 3t8 ARITHMETIC 164. A room is 8 yd. long ; the cost of carpeting it is 894. 8*., and that of papering is R86. loa. If the breadth of the room were i yd. more and its height I ft. less, the cost of carpeting would be iliio. 4# while the cost of papering would remain the same. Find the breadth and height of the room. 165. A and B run a race ; A has a start of 40 yd., and sets off 5 min. before /?, at the rate of 10 miles an hour. How soon will B overtake him if his rate of running is 12 miles per hour ? 166. If the gas for 5 burners, lighted 5 hours every evening 'for 10 days, cost RS. 120., what will be the coat of 75 burners which are lighted 4 hours every evening for 15 days ? 167. Find the three highest integral numbers whose sum is under a thousand, so that the first may be $ of the second and second of the third. 168. A tradesman sells one kind of sugar at 30. per seer and loses 20 p. c., and another kind at $a. pel seer and gains 25 p. c. He mixes the two together in equal proportions and sells the mixture at 6a. per seer. What is now his gain per cent. ? 169. Two equal. sums are divided, the one among 36 men, and the other among a certain number of women ; each man received Ri. 4. and each woman 10 annas less ; how many women were there ? 171. Three equal circular wheels revolve round a common horizontal axis ; the first makes a revolution in 5^ minutes, the second in 2? minutes, and the third in Zr minutes. Three marks, one in each wheel, are in a horizontal line at a certain moment. What is the shortest interval after which they will be in a hori- zontal lin e again ? 172. A can do a piece of work in 6 hours, B in 8 hours and C in 10 hours ; how long will it take C to complete a piece of work, of which has been done by A working 7 hours and B working 8 hours ? 173. A walks i\ miles in 40 min., taking exactly a yard each step ; in what time will B walk 4 miles when his stride is 40 in, and he takes 21 steps while A takes 22 ? 174. Three persons, A, B, C, a^ree to pay their hotel bills in the proportion 4 .* 5 .' 6. A pays the first day's bill which amounts to i. 55. 5*/. ; B the second which amounts to i. i6s. id. ; and C the ihird which amounts to i. iSs. 6d. ; how must they settle their accounts ? 175. A person bought a French watch bearing a duty of EXAMPLES FOR EXERCISE 3IQ 25 per cent., and sold it at a loss of 5 per cent. ; had he sold it for ^3 more, he would have cleared r per cent, on his bargain. What had the French maker for the watch ? 178. An equal number of men, women and boys earn Ri6$ in 6 days. If a woman * ams I3<z. qp. a day, a man 8#. more, and a boy 8a. less, how many are there of each ? 177. What sum increased by J of $ of & of itself, amounts to .2463 ? 178. The length, width and depth of a cistern are 8 ft., 5 ft. 4 in. and 4ft. 6 in. respectively. How many gallons does it contain, having given that a cu. ft. of water weighs 1000 oz. and that a pint of water weighs a pound and a quarter ? 179. A and B are termini of a railway 144 miles long. A fast train starts from B at 9 A.M. ; another fast tram, travelling at the same rate, starts from A at 10 A.M. A slow train starts from B at 10-20 A.M ; the fast train from A meets the other fast train at 11-30 A.M. ; and the slow train at 12-32 P.M. Find the rates at which the trains travelled. 180. If Ri = u. iojY/., ;i=4'84 dollars, and I dollar 5'2 francs, find the value in francs of 10 lacs of rupees. 181. Three merchants, A, B, C, trading with a capital of 3850, find after a certain time that their respective shares are increased by ;66 .7.6, 59 .8.7 and 66 .13.11 ; how much did A sub- scribe to the original capital ? 182. A grocer buys 200 Ib. of tea, and sells 180 Ib. for the same amount that he gave for the whole. The rest he sells at a profit of 20 per cent, What is his gain per cent, on the whole outlay ? 183. The large wheel of an engine is 20 ft., and the small wheel 12 ft., in circumference. If the large wheel slips on an average 2 inches in every revolution, how many revolutions will the small wheel make more than the large one in going a distance of 12 mi. 1728 yd. ? 184. Calculate correctly to 7 places of decimals the value ol I+JL+JL + JLj. 9 3-9 8 5-9 5 7-9 T 185. The circumferences of the wheels of a carriage are 6& ft. and 8jk ft. ; what is the least distance in which both the wheels will simultaneously complete an integral number of revolutions ? How often will the lowest points of the two wheels at starting touch the ground together in 10 miles ? 186. In a 2oo-yd. race A beats B by 20 yd., and C by 40 yd. By how many yards can B beat C in a loo-yd, race ? 330 ARITHMETIC 187. On a piece of work 2 men and 5 boys are employed, who do $ of it in 6 days ; after this I man and I boy more are put on, and J more is done in 3 days ; how many more men roust now be put on if the work is to be completed in i day more ? 188. A, B) C invest capital to the amount of ^800, ^600 and ; A was to have f of the profits which amount to ,330 ; find C's share of the profits. 189. A tradesman defrauds his customers (i) by an adulteratior of the article to the extent of 7 per cent., (li) by using a balance which indicates i lb. when the amount in the other scale is really 15 oz. Which of the two practices is the more fraudulent, and to what extent is the customer cheated when he orders I lb. of the commodity ? 190. Find the distance between two towns when 8309. 50. 4^. is paid for the fare of 17 first class passengers at la. 8/fr. a mile, of 26 second class at la. 2p. a mile, and of 40 third class at S/>. a mile. 191. Find the. value of \ ^LIJ-S- *? "H 5 \ of ~^ of 1 2 S f 3i 3j f 77 J 4-*- 7 2 ft. 3 in. , . , , = of 24 weeks 4 days 19 hours. 192. How many poles of fencing are required to enclose a square park containing 27 ac. 12 po. i yd. ? 193. A, B) C can do a piece of work in .6, 8, and lo days respectively. They begin to work togethere ; A continues to work till it is finished, B leaving off 2 days, and C I day before the work is completed. In what time is the work finished ? 194. If the supply of a number of persons with bread at *]%d. the loaf for 31 days cost ^27. i8j., what will it cost to supply f of that number for 20 days at 6|</. the loaf ? 195. A, B) C purchase a farm for Rioooo, of which A pays R4COO ; they sell it so as to gain a certain sum, of which B takes &27S and C Bi7S ; find A's share of the profit, 196. One company guarantees to pay 5 per cent, on shares of looo rupees each ; another guarantees to pay 4! per cent, on shares f75 rupees each ; the price of the former is 1245 rupees and of the latter 85 rupees. Compare the rates of interest which the shares return to the purchasers. 197. If 5000 people took in hand to count a billion of sovereigns, and beginning their work at the commencement of the year 1852, could each count on the average 100 sovereigns in a minute (without intermission)) when would they finish their task ? EXAMPLES FOR EXERCISE 32 L 198. The total area of three estates is 1768 acres. If the areas of the two smaller estates be respectively three-fifths and two-thirds of that of the largest, find the acreage of each. 199. There are 3 pendulums, the first makes 35 beats in 36 seconds, the second 36 beats in 37 seconds, and the third 37 beats in 38 seconds. Supposing they commence together find how many times they will again beat coincidently in 24 hours. 200. Sound travels at the rate of 1142 ft. per second ; what is the distance of the thunder cloud, when the thunder succeeds the lightning at an interval of 9 seconds ? 201. If 4 men and 6 women can do a piece of work in 5 days, which 5 men and 10 children can do in 4 days, or 3 women and 4 children can do in 10 days ; find (i) how many men, (ii) how many women, (iii) how many children, could do the work in one day. 202. A and B enter into partnership ; A puts into the business RSOOO more than ./?, who, as acting partner, is to have a salary of 125 a month ; at the end of 2 years the gross profits computed at J of the capital per annum, are found to be 87000, from which 's salary is to be paid : find each one's share of the net profit. 203. The 3 per cents, are at 85 J ; what price should the 3^ per cents, bear, that an investment may be made with equal advantage in either stock ? And what interest would be derived by so investing 5ooo/. ? 204. Find the least sum of money that must be subtracted from ,660. 7-y, 4f. to make the remainder exactly divisible by 39. 205. What decimal must be added to 206. If gold can be beaten out so thin that one tola will form a leaf of 20 sq. yards, how many of these leaves will make up the thickness of a sheet of paper, the weight of a cu. inch of gold being 52^ tolas and 432 sheets of the paper in thickness going to an inch ? 207. A race-course is a mile long : A and B run a race and A wins by 10 yards ; C and D run over the same course and C wins by 30 yards ; B and D run over it and B wins by 20 yards \. if A and C run over it, which would win, and by how much ? 208. Four men are employed to reap a field and after work- ing 5 days they have cut 10 acres ; 2 more men are then put on, and the whole is finished in 3 more days. How many acres are there in the field ? 209. A) B and C are employed to do a piece of work for 8529 f\ A. 91 322 ARITHMETIC A and B together are supposed to do Jg of the work, and B and C together -jfa of the work : what should A be paid ? 210. If 16430 be invested in the Govt. 4j per cent, loan at 1061 what is the monthly income derived ? Supposing that the loan is paid off at par in 10 years, what would be the rate of simple interest on the sum invested ? 211. 120 tons of coal are purchased for ^87. 16. 9 ; find, to the nearest farthing, the price at which they must be retailed per ton so that no loss may be incurred ; and at that price what profit will accrue ? 212. Reduce to a decimal correct to 6 places : 1.3 3.3 s " 5-3' 7.3 r " 213. Find the greatest unit of time by means of which n hr. 31 min. 1 8 sec. and 23 hr. 4 min. 27 sec. can both be expressed as integers. 214. A man does f of a piece of work in 18 days, and then gets a boy to help him. They work together for 3 days, when the boy leaves, and the man finishes the work in 7j days more. How long would it take the boy to do the whole ? 215. If 10 horses and 98 sheep can be kept 9 days for 37 .17.6; what sum will keep 45 horses and 216 sheep for 40 days, supposing 5 horses to eat as much as 76 sheep ? 216. A starts businsss with 81200, and subsequently admits B who brings Hi 600. At the end of the year A receives \ of the profits ; when was B admitted ? 217. A man who has a certain capital calculates that if he invest it in 3j per cent, stock at 91, his income will be 2$ more than if he invest it in 3 percent, stock at 88. What is his capital? 218. A tradesman buys 200 Ib. of tea for i6 9 intending to gain one-fourth of his outlay by sale ; but two pounds' worth at this calculation being damaged, at what price shall he sell the remainder per Ib. to gain as much upon the whole outlay as he intended ? 219. Express (& + 2i)-(2f-ij)x {(5i x 7f)-ri6iV} in its simplest form. 220. The diagonal of a square court-yard is 100 ft. ; find the area, 221. Sound travels at the rate of 1140 feet a second. If a shot be fired from a ship moving at the rate of 10 miles an hour, how far will the ship have moved before the report is heard I4i miles off ? EXAMPLES FOR EXERCISE 32$ 222. The length of the minute-hand of a church clock is 5$ feet ; what distance will the end of it travel through in 35 daysj if 7 times the circumference of a circle be 22 times its diameter ? 223. Three men A, B, C, undertake to complete in 20 days a piece of work for 247. 8#. A furnishes 10 men for 8 days and 6 men for the remaining days ; B furnishes 7 men for 7 days and 12 men for 12 days ; C 1 furnishes 15 men who work on alternate days only until the work is completed. Find A's share of the sum. 224. A person having 8,500 in 4 percent. Govt. bonds sell? out when they are at 8J per cent, discount, and with the amount thus realised purchases 5 per cent, bonds which are at 6| per cent, premium : what does he gain or lose in annual income by the change ? 225. A contractor employs 100 men, 40 of whom work 10 hours on week days and only 5 honrs on Sunday ; the rest work 8 hours a day. If the wages of the former be 5^. per hour and of the latter 4^. per hour, what is the amount of wages paid in 4 weeks ? 226. Two chests of tea of the same size and quality are consigned to A, B^ C. A at first was to have $ of a chest, B f , and C the rest. But A, B purchase -ft, -ft of Cs share respectively. How much will each have ? 227. Find the side of the largest square tile, with which a court, 33 yd. i ft. 7 in. long and 20 yd, n in. broad, can be paved. 228. In a bicycle race of 2 miles over a circular course of I furlong, the winner in his last round overtook the second at a point in his 1 5th round. Their paces were as 159 to 149! At what distance was this point from the winning post ? 220. If 3 men can do as much as 7 boys in a day, how many days will it take 25 boys to finish a piece of work of which 12 men have done a quarter in 13 days ? 230. At B) C hold a pasture in common for which they pay Ri6 per month ; they put on it 70, 50 and 40 sheep respectively. A sells $ of his flock to B after 4 months, and after 3 months more C sells f of his to A. How much of the rent should each pay at the end of the year ? 231. A person bought 10 Bank of Madras shares at 1540 each and for 5 years got interest on his investment at the rate of 5^ per cent. He then sold his shares at a loss of 22^ per cent. How much did he make by the transaction, and what rate per cent, per annum had he for his money ? 232. A certain number of cows and twice as many sheep were bought for 94. 6a. ; the cows cost Rio. 3*2. bp. each and the sheep 4. So. 3A each : how many sheep were bought ? 324 ARITHMETIC 283. The master of a ship, worth ^5161. 3*. gd, is himself owner of f of of f of her. He sells her for of her value ; what is his own share ? 234. The height of a square room is one-half of its breadth, and the cubic content of the room is 108 cu. yd. ; find its dimensions. 235. Two pipes, A and 2?, would fill a cistern in 37 J min. and 45 min. respectively. Both pipes being opened, find when the second pipe must be turned off, that the cistern may be just filled In half an hour. 236. If 13 locomotive engines, each of 290 horse-power, woik- ing 1 1 hours a day for 7 days a week, can convey 73 1 5 tons of goods to a distance of 221 miles in a given period, how many hours' work a day for 6 days a week must be done by 7 locomotives of 319 horse-power each, in order to convey 4845 tons of similar goods to a distance of 1 54 miles in an equal period ? 237. How must teas at 2s. a Ib. and 2s. gd. a Ib. be mixed so that by selling the mixture at 2s. &J. a Ib. there may be a gain of id. per Ib. ? 238. If I sell 40 shares of R25o each in the Oriental Bank at 121 per cent, premium, how many shares of 8.1000 each in the Madras Bank at 72 per cent, premium can I buy, and how much will be left ? 230. Equal quantities of sugar, flour and rice were bought for &72O. ga ; the price of a md. of sugar is twice as much as that of a md. ot flour, and the price of a md. of flour is twice as much as that of a md. of rice : find the cost of the sugar. 240. Find the value of -^2*1 x 2i| of 1 2s. gld. 2-1742 278 241. A tea-merchant has a rectangular space for storing tea, It is I Si ft- long, io ft. broad and 9$ ft. high. He wishes to fill this space with packets of a cubical shape, all of the same size. What is the largest size of such cubical packets that can be made to fill it exactly, and what would be the number of such packets ? 242. A hare starts 40 yardsjbefore a greyhound and is not seen by him till she has been up 30 seconds. She runs at the rate of 12 and the hound at the rate of 1 15 miles an hour ; how long will the chase last, and what distance will the hound have run ? 243. If 3 men and 5 boys can reap 20 acres in 10 days, and if 5 men and 3 boys can reap 34 acres in 15 days, how many boys must assist 9 men, in order that they may reap 45 acres in 9 days ? 244. A grocer bought 60 Ib. of sugar of two different sorts for Ri6. 4a. The better sort cost 5^. per Ib., and the worss 4*, per Ib. Find how many pounds there were of each sort. EXAMPLES FOR EXERCISE 325 245. How much stock in the 3 per cents, must I sell to pay off a debt cf ^470, the price of the stock being 94^, and the commis- sion of J- on ;ioo of stock being also taken into consideration ? 246. How many four-anna pieces can be coined from 9 Ib. of -standard silver ? 247. Find, by Practice, the dividend on a debt of 347*> a * 13-f. l\d. in the . 248. The sides of a square are divided each into 8 equal parts, and lines are drawn through the points of division parallel to the sides. If the area of the square be 256 sq. ft, find the length of the side of each of the smaller squares, into which it is divided. 249. A and B run a mile race : at first A run 5 yards to &s 4, but after A has run half a mile he tires and runs 3 yards in She time in which he at first ran 5, B running at his original rate. Which wins, and by how much ? 250. If the carriage of 150 ft. of wood, that weighs 3 stones per ft. co^t R3o for 40 miles, how much will the carriage of 54 ft. of wood, that weighs 8 stones per ft., cost for 25 miles ? 251. A greengrocer sells potatoes at 2s. t 2s. 6d. and 3^. 6//. a bushel, selling equal quantities of the first two kinds ; what quan- tities of each kind does he sell, if the total quantity sold is 60 bushels, and if the average price obtained is 35. a bushel ? 262. A person invests 1250 gold mohurs in the Govt. five per cent, rupee stock at 105. The stock is converted subsequently to 4$ per cents, at 95. Find the difference in his income, each gold mohur being considered equivalent to 17. 263. If a person whose income is 8,1825 a year spend 844. io a week for the first 20 weeks, to what must he limit his daily expenditure for the rest of the year so as not to be in debt at the end of it ? 254. What number multiplied by itself will give 255. A cubical block of marble whose edge is 2 ft. is placed within a rectangular cistern 4 ft. long, 3 ft. wide and 2 ft. deep, which is then filled with water ; how many pounds of water must be taken out to reduce the surface 6 in. ? [A cu. ft. of water weighs Ib.] 256. A and B can do a piece of work in 2 days, but when B works half time the work is done in 4 days. Show that B is twice as good a workman as A. 257. If 2 men and 5 women can do a piece of work in 8 days x)f 9 hours each ; how long will it take 3 men and 6 women to do a piece of work twice as great, working 8 hours a day, the work of a man being double that of a woman ? 325 ARITHMETIC 268. Gold is 19 times as heavy as water, and copper 9 times, In what ratio should these metals be mixed that the mixture- may be 15 times as heavy as water ? 259. When the 3 per cents, were at 90 I found that by selling out and investing in the 4 per cents, at 95 I could improve my in* come by 8243, What was the amount of my stock in the 3 per cents. ? ^ 280. A person has in his drawer 1 5 piles of rupees, each con taining 20 ; his servant steals them and puts in their place 15 piles, each consisting of 19 double-pice with a rupee at the top. How much does the person lose ? 261. A person owes the sum of 831500, and 88500 ; and his property amounts to 814125 only. How much is he able to pay in the rupee ; and what is the loss upon the second debt ? 262. A rectangular piece of ground of 243 sq. yd. is one-third as broad as it is long ; what is the distance round it ? 263. A passenger train going 41 miles an hour, and 431 ft. long, overtakes a goods train on a parallel line of rails. The goods train is going 28 miles an hour, and is 713 ft. long. How long does the passenger train take in passing the other ? 264. The distance by rail from Turin to Venice is 420 kilo- metres, and the first-class fare is 56 lire ; find at the same rate in Indian money, the fare from Calcutta to Benares, a distance of 480 miles, reckoning 7 lire equal to 83 and 8 kilometres to 5 miles. 265. 40 lb. of coffee, at 2s. 6d. a lb., were mixed with a certain quantity of chicory at is. tyd. a lb., and the resulting mixture was worth 2s. a lb. How many pounds of chicory were there in the mixture ? 266. How much money must be invested in the 3 per cent, consols when they are at 92^, to produce the same income as- would be produced by 81520 invested in the 3$ per cents, at 95 ? 267. If 820 .7.6 be gained by selling an article for 879 . 10 . 9, how much would have been gained or lost by selling it for 859 .7.6? 268. Find by, Practice, to the nearest peniy, the rent of 3753675 acres at 2. igs. io\d. per acre. 269. Determine, by Duodecimals, the area of a rectangle whose adjacent sides are respectively 9 ft. 3J in. and 6. ft. 4^ in. 270. A can beat B by 5 yd. in a loo-yd. race, and B can beat C by i o yd. in a 2ooyd. race ; by how much can A beat C in a 40o-yd. race ? 271. If 210 coolies, in 7 days of 10 hours each, dig a channel, EXAMPLES FOR tXERCISE 327 I mile long, 6 feet broad and 2jfeet deep ; in how many days of 7 hours each should 35 coolies dig a channel, 660 feet long) 7$ feet broad and i\ feet deep ? And how many cubic feet does each cooly dig in an hour ? 272. The average of eleven results is ^o ; that of the first five is 25, and that of the last five is 28. Determine the sixth result. 273. What amount must be invested in the 4$ per cent, stock at io3|, in order to obtain, after deducting an income-tax of 3j per cent., a clear income of ^4000 a year ? 274. 4 thalers, 6 half-crowns and 8 florins amount to 2 ; what is the value of a thaler ? 275. A reduction in the income-tax diminishes a tax, which is Ri$ when the tax is 8 pies in the rupee, by 83 . 12 . o ; what is the diminished rate of the tax ? 276. The length of a room is twice its breadth and 4 times its height, and it contains 216 cu. yards of air ; find its length. 277. A can reap a field in 5 days, and B in 6 days, each working 1 1 hours a day ; in what time could they together reap it, working 10 hours a day ? 278. If 38 men working 6 hours a day can do a piece of work in 12 days, find in what time 57 men working 8 hours a day can do a piece of work twice as great, supposing 2 men of the first set to do as much work in I hour as 3 men of the second set can do in \\ hours. 279. The average weight of 5 men is 5 st. 7 Ib. ; the average weight is diminished by 7 Ib. when the weight of a boy is included : what is the weight of the boy ? 280. A share-holder in a commercial company receives one year a dividend of 5 per cent, on his shares. The next year he receives a dividend of 7^ per cent, and finds that he ia &4I2. 8a. richer. Find the amount of his shares. 281. To march at quick step is to take 108 paces of 2 ft. 8 in. per minute ; what rate ia this per hour ? 282. A society subscribed R2i. 50. 4^. to a charity, each member paying as many pies as there were members in the society ; find the number of members. 283. Find, by Duodecimals, the volume of a block of marble> 3 ft. 7 in. long, 2 ft. 3^ in. wide and i ft. 2j in. deep. 284. A train, 880 feet long, overtook a man walking along the line at the tate of 4 miles an hour, and passed him in 30 seconds ; the train reached the next station in 1 5 minutes after it had passed the man. In what time did the man reach the station ? 328 ARITHMETIC 285. If 40 men and 50 boys can do a piece of work in 6 days, working 6 hours a day, in how many days will 8 men and 20 boys do a piece of work half as large again, working 7 hours a day, assuming that a man does as much work in 3 hours as a boy in 5 hours ? 286. The average age of 8 men is increased by 2 years, when one of them, whose age is 24 years, is replaced by a fresh man ; what is the age of the new man ? 287. If the price of the 4 per cents, just before the payment ol a half-yearly dividend be 93, what ought to have been the price 3 months previously, supposing no change in the value of money to have taken place during that interval? 288. The weekly wages at a mill amount to 186. 4*. In the mill a certain number of women are employed at 2s. icW. a day, five times as many men at $s. bd. a day, and 6 times as many boys at 2j. 4*/. a day : how many men are employed ? 289. If the income-tax be 7*2?. in the in the first half of the year, and 3$</. in the second, what is the net income of a gentleman whose gross annual receipts are ,1542 . 10 . 6 ? 290. An open cistern, made of sheet iron a quarter of an inch thick, is internally 62^ in. long, 36 in. wide and 24 in. deep ; find the weight of the cistern when fill of water, if iron weighs 7 times as much as water and a cu. ft. of water weighs 1000 oz. 291. In a two-mile race A wins, B being 22 yd. behind, and C 106 yd. behind B. By how much would B beat C in a three mile race in which A does not run ? 292. If the wages of 1 8 coolies for a month amount to R8$ when rice is 24 seers per rupee, what ought the daily pay of a cooly be in proportion when the price of rice is 82. loa. 8^. per maund ? 293. A and B started on a race and ran a distance exactly together. Then B began to fail and gave up the race when he had run 56 yards farther, A having gone during the same time 320 yards. The average of the entire distances run by the two men was 1 1 88 yards. What distance had they run together ? 294. The ^23 shares of one company pay a dividend of 1 per share ; the 15 shares of another yield ^,7 2 5 per share. The market value of the former is ^24*92, of the latter 17. Compare the rates of interest returned to the purchasers. 295. A man bought 100 oranges at 2 a pice, and 100 more at 3 a pice, and mixed and sold the whole at 5 for 2 pice ; how much did he lose ? 296. Find, by Practice, the cost of fencing 3 mi. 3 fur. 1 80 yd* I ft 6 in. of road at ,479. i$s. per mile. EXAMPLES FOR EXERCISE 329 297. An open cistern, made of sheet iron $ inch thick, is -externally 10 in. long, 8 in. broad and 5^ in. deep ; find the price of the cistern at E>8 per cwt., if a cu. ft. of iron weighs 4$ cwt. 208. A does half as much work again as B in the same time, and B does one-third as much again as C ; working together they can do a certain work in c days ; but if after working 2 days A leaves off, how long will B and C take to finish it ? 299. When rice is 10 seers the rupee, 7 persons can bejed for 30 days at a certain cost. For how many days can 6 persons be fed at the same cost when rice ia 14 seers the rupee ? 300. If the daily wages of a labourer rise from 4/7. 9^, to 6a., what percentage of the increase in the price of food and other commodiiies will cause his position to be unaltered ? 301. A person buys 5 shares in a company, and sells three of them at a gain of 10 per cent, and the remaining two at a gam of r6f per cent. The gain on the latter sale is 2 . 19 . 7^ more than on the former. How much did he pay for each share ? 302. A man buys 25 seers of milk at la. 6fi. a seer, and sells it at la. $p. a seer, making a prout of 5 annas ; how many seers of water did he add to the milk ? 303. Now that the income-tax is 5 pies in the rupee, a person's net income is 6=374 per mensem ; what will it be when the income-tax is raised to 7 pies ? 3O4 Find, by Duodecimals, the area of a square whose side rs 12 ft. 8 in. 4 pt. 305. A train starts from A at 12 o'clock and runs towards C", Which is 100 miles distant, at the rate of 30 miles an hour ; at the same time the mail coach starts for C, from B, which is half way between A and C, and runs at 10 miles an hour ; at what distance from C will it be overtaken by the train ? 306. If 13 solid inches of copper balance 17 of iron, and 15 of iron balance 16 of tin. and 19 of tin balance 12 of zinc, how many solid inches of zinc balance 2470 solid inches of copper ? 307. If the income-tax be 6 pies in the rupee for the first half of the year and 3 per cent, in the second, what is the gross income of a gentleman whose net annual receipts amount to &1454. in. ? 308. What sum must a person invest in the 3 per cents, at 90, in order that by selling out 1000 stock when they have risen to 93|, and the remainder when they have fallen to 84^, and -investing the whole proceeds in the 4 per cents, at par he may increase his annual income by 9. 5^. ? 335 ARITHMETIC 809. Divide Ri 1 5. 20. among 20 boys and 25 girls, so that each boy may receive 12 annas more than each girl ; how much will each boy receive ? 310. Three-fifths of the square of a certain number is 126*15 what is the number ? 311. An open cistern whose capacity is 4320 gallons is exter- nally 14*1137 ft. lonpr, 10*25 ft. wide and 5*16 ft. deep ; the slides are i in. thick ; find the thickness of the bottom, having given that a gallon contains 277*274 cu. inches. 312. A and B walk a race of 10 miles ; A gives B 20 minutes 9 start ; A walks uniformly a mile in 17^ minutes and catches B at the 8th mile-stone : find by how much B lost in time and space. 313. If 17 men can build a wall loo yd. long, 12 ft. high and 2^ ft. thick, in 25 days, how many men will build a wall twice the size in half the time ? 314. In 1861 three towns had populations of 17650, 19600, 18760 respectively. In 1871 the population of the first had decreased 18 per cent., that of th^ second had increased 21 per cent,, while the population of the third had increased by 4690 ; find the change per cent, in the total population of the three towns. 315. A gentleman invests 8,5600 in the 5$ per cent. Govt. paper, and derives therefrom an annual income of 8275. At what premium was the 5^ per cent, paper at the time he invested ? 316. Find the circumference of the wheel of a locomotive, which makes 5 revolutions in a second, and which performs a journey of 30 miles in 44 minutes. 317. A man has an income of .200 a year ; an income-tax is established of yd. in the , while a duty of ikd. per Ib. is taken off sugar ; what must be his yearly consumption of sugar that he may just save his income-tax ? 318. A, 2?, C are three spouts attached to a cistern. A can fill it in 20 min., B in 30, and C can empty it in 40 mm. If A t B and C be opened successively for one minute each, in what time will the cistern be filled ? 319. A besieged garrison consists of 300 men, 120 women and 40 children, and has provisions enough for 200 men for 30 days. If a woman eats f as much, and a child as much, as a man, and if after 6 days 100 men with all the women and children escape, how long will the remaining provisions las the garrison ? 320. The price of rice being raised 5 o per cent., by how much per cent, must a house-holder reduce h consumption of that article so as not to increase his expenditure EXAMPLES FOR EXERCISE 33 1< 321. The owner of 4 per cent. Govt. paper, bringing in 8976 per annum, exchanges it for 5 per cent, paper. His annual interest is increased by &44- What is the increase or decrease of his, nominal capital ? 322. A bill on London for 175 drawn at 6 months after- sight, is purchased at Madras, the rate of exchange being is. od. the rupee. Four months before it becomes due, it is discounted in London at the rate of i\ per cent, (per annum) discount. What was paid for the bill in Madras, and what does it realise in London ? 323. A man laid out ^30. 155. in spirits which he bought at: I5J. a gallon ; he retailed them at 17$. 6d. a gallon, making a profit of 4. 55. : how many gallons must he have lost by leakage ? 324. Arrange >/2, */3 and | in order of magnitude. 325. Two trains, running at the rates of 25 and 20 miles an hour respectively on parallel rails in opposite directions, are observed to pass each other in 8 seconds, and when they are run- ning in the same direction at the same rates as before, a person , sitting in the faster train observes that he passes the other in 31^ seconds ; find the lengths of the trains. 328. If 6 dollars and 6 roubles are together worth i. i$s. Qd., and 4 dollars and 8 roubles are together worth i. us. S*/,, what is the value of 6 dollars and 8 roubles ? 327. In an examination A obtains 10 per cent, less than the minimum number of marks required for passing ; B obtains uj. per cent, less than A ; and C 41 1^ per cent, less than the number of marks obtained by A and B together. Does C pass or fail ? 328. I have 6500 to invest in public securities. Will it be most to my advantage to invest it in the 5 p. c. Govt. loan which is at xof per cent, discount, or to purchase at par Treasury Bills which bear an interest of 3 pies per cent, per diem ? Calculate the difference. 329. If the par of exchange be two English shillings for the Indian rupee, but if an Indian bill of exchange for 540. 120. be negotiated in London for ^51. IQJ., how much per cent, below par is the rate of exchange ? 330. On Monday January 3, 1888, a man commenced to sub- scribe for a daily pice paper (published on week days only) ; what had he spent by June I3th of the same year ? 33L A gentleman's income is diminished by ,150 ; but the income-tax being raised from 6d. to yd. in the 9 he pays the same amount of tax as before ; find his present income. 332, A and B start to run a race ; their speeds are as 17 to 18* 332 ARITHMETIC A runs 2j miles in 16 min. 41 sec. ; B finishes the course in 34 min. : determine the length of the course. 333. If c men and 8 boys reap 9 acres in 10 days, and 4 men and 4 boys reap 3 acres in 5 days, how many acres will 2 men and 3 boys reap in 7 days ? 334. To 432 gallons of a mixture of brandy and. rum, which contains 8| per cent, of brandy, some water is added, and the proportion of brandy in the mixture is thereby diminished to 7$ per cent. How much water is added ? 335. A person who has ^1900 Russian 4 per cent, stock sells out at 104 and devotes ,962. 13.9. 40?. to the purchase of 3 p. c. consols at 95, and lends the rest of the sum realised on mortgage. What interest must he ask for his money that his income may be the same as before ? 338. If the rate of interest for money be 3 per cent., what should be the rate of exchange for bills payable at sight in England when the rate for those payable 4 months after sight is 15. 8J</. per rupee ? 337. A merchant buys 60 yards of cloth ; he sells half of it at a gain of 3 annas per yard, and the remainder at a gain of 2 annas per yard, and realises 44. la. What was the cost price per yard ? 338. A man buys a number of mangoes for Rg, the price in 'pies of each mango being equal to the square root of the number purchased ; find the number purchased and the price of each. 339. A train which travels at the uniform rate of 30*8 ft. a second, leaves Madras at 7 A. M. ; at what distance from Madras will it meet a train which leaves Arconum for Madras at 7-20 A. M., and travels one-third faster than it does, the distance from Madras to Arconum being 42 miles ? 340. If 5 men, 2 women and 3 boys, or 6 men and 4 boys, can mow 3 acres in 5 days ; how many acres would 3 men, 2 women and one boy mow in n days, supposing a man to do as much work as 3 boys ? 341. A person loses in his first year 23 per cent, of hia capital, r but in the next year he gains 40 per cent, of what he had at the end of the first year, and his capital is now 720 more than it was at first ; find his original capital. 342. A person invested equal sums of money in the 3 per cents, at 97$, and in the 3$ per cents, at 102^ ; his resulting income was ^259. los. How much did he invest ? 343. A merchant in London receives two bills, drawn at 4 months after sight, each for 5000 ; one he discounts immediately! PROBLEMS 333 the rate of interest being 3 per cent, per annum ; the other he keeps till maturity ard then exchanges at the rate of is. <)d. per rupee, and finds that he was got as much as he did for the first bill. What was the rate of exchange when the first bill was discounted ? 344. A man, having bought 128 yards of cloth for R8o, sells one-fourth at a loss of 2 annas per yard ; by how much must he raise that selling price, in order that) by selling the rest at the increased rate, he may gain 2 annas per yard on the whole ? 345. Incomes below ^150 a year being subject to 5^. in the income-tax, and incomes above .150 to *jd. in the ; find what income above ^150 a man must have, that he may be just 7fdf. a year poorer than a man who has 149. los. a year. 346. A and B run a mile, and A wins by 160 yd. ; A and C run over the same course and A wins by 20 min. ; B and C run over it and B wins by 12 min. In what time can A run a mile ? 347. If 1 6 darics make 17 guineas, 19 guineas make 24 pis* toles, 31 pistoles make 38 sequins, then how many sequins are there in 1581 darics ? 348. What sum must be paid on the insurance of a cargo of the value of 33575. 4<?. so that in case of loss the cargo and all expenses of insurance may be recovered ? The premium is at the rate of 4725 per cent, policy duty 3$ annas per cent, and agent's commission J per cent. 349. A person has ^26041 of a 4 per cent, stock. He saves each year J of his income, which he invests at 4 per cent. What is his income in the 4th year ? 350. If gold be at a premium of 5 per cent, and a person buy< goods marked 300 rupees, and offer gold to the amount of 300 rupees, what change ought he to receive in notes, 5 per cent, being abated for ready payment ? PROBLEMS. 175. 1. By what number less than 1000 must 4389 be multiplied so that the last three figures (to the right) of the product may be 438? 2. If 5 cwt. 3 qr. 14 Ib. cost 6 per cwt., what will be the cost per pound when the cost of the whole has been reduced by 7. i6s. 8/ ? 8, On measuring a distance of 32 yards with a rod of a certain length it was found that the rod was contained 41 ti tries with half an inch over ; how many inches will there be over in measuring 44 yards with the same rod ? 334 ARITHMETIC 4. Find the least number above 1000, which when divided 'by 5 or by 6 or by 9, will leave the same remainder 3. 6. A bill of 100 was paid with guineas and half-crowns, and 48 more half-crowns than guineas were used ; find how many of each were paid. 6. A has twice as much money as B. They play together, and at the end of the first game B wins from A one-third of A's money ; what fraction of the sum which B now has must A win back in the second game that they may have exactly equal sums ? 7. What is the smallest whole number which is exactly divisible by i/g, 2 5 2 r and 3} ? 8. A pays 9 .3.4 more rates than B, their incomes being equal ; living in different towns they are rated at 2,5. and is. 4d. In the respectively ; what is their income ? 9. A pint of water weighs a pound and a quarter, and a cu. foot weighs 1000 oz. ; how many gallons are there in a cu. foot ? How many gallons will fill a cistern 5 ft. long, 2fr feet wide and 2 feet deep ? 10. A gallon contains 277*274 cu. in. ; a cu. ft. of water weighs 1000 oz. How many gallons weigh a ton ? and what is the weight of a pint ? 11. If 162 gallons fill a cistern 5$ ft. by 4i ft. by i ft., find the number of cu. inches in a pint. 12. If a cu. inch of water weighs 252*45 grains, which is the 'more accurate of the following rough statements : a cu. ft. of water weighs loco oz. a cu. yd. weighs f of a ton ? 13. If a decilitre be '052 gallon, find the value of a pint of liquid which is worth 2 francs the decilitre ; 1200 irancs being equal to ^49. 14. Three men are employed on a work, working respectively 8, 9, 10 hours per day, and receiving the same 'daily wages. After three days each works one hour a day more, and the work is finished in three days more. If the total sum paid for wages be 2 . 7 . 6J, how much of it should each receive ? 16. The sum of two numbers is 5760, and their difference is equal to one-third of the greater ; find the numbers. 10. Two casks contain equal quantiti es of beer ; from the first 34 quarts are drawn, and from the second 80 ; the quantity remain- ing in one cask is twice that in the other. How much did each cask originally contain ? 17. Shew that if the price in rupees of a cwt. of goods is dividid by 7, the result is the price in annas of a Ib. weight of the .goods, PROBLEMS 335 18. If 72 be divided among 5 men, 7 women and 13 boys so that 2 men receive as much as 5 boys, and 2 women as much as 3 boys, how much will each man, woman and boy receive ? 19. How many revolutions will be made by a wheel which revolves at the rate of 329 revolutions in 3 min. while another wheel revolving 431 times in 4 min. makes 2586 revolutions ? 20. If a train goes 22^ miles an hour, how many revolutions does the driving-wheel, 1 1 ft. in circumference, make in a second ? 21. A game licence costs 1 5^., and a cartridge 2//. A sports- man kills his bird once in 5 shots. If birds are worth zs. 6^. a brace, how many birds must be shot just to pay expenses ? 22. A vulgar fraction has for its numerator 1 57, and its nearest approximate value in thousandths is '370 ; what is the denominator ? 23. A man after a tour in England finds that he had spent every day half as many rupees as the total number of days he had been from home. His tour cost RiSoo, How many days did it occupy ? 24. A plate of metal is beaten to the thickness of J of an inch, and the weight of a circular medal cut from it, whose diameter is 1 1 inches, is ij oz. troy. If the same plate be beaten to the thickness of of an inch, what will be the weight of a medal cut out of it of the diameter of 1} inches (the areas of circles being proportional to the squares of their diameters) ? 25. It is said that 240,000 letters are posted in Berlin daily, 16*6 per cent, of which are town letters. This gives one letter for every 3 persons in Berlin ; what is its population ? 26. The French unit of linear measure in a metre equal to 39*371 English inches ; the square formed on aline of 10 metres (called an are) is the French unit of surface. Find the equivalents in English square measure, of a hectare (100 ares/. 27. A rectangular swimming bath is 60 ft. long and 40 ft. broad ; it can be filled by a supply-pipe in 5 days, and if 6,000 cubic feet of water be thrown in, the rest can be filled in 3 days 1 8 hours. Find the depth of the bath. 28. The debts of a bankrupt amount to 21345. 4. and his assets consist of property worth R9 167. loa. 8^. and an undiscount- ed bill of 5130 due 4 months hence, simple interest being reckon- ed at 4 p. c. per annum. How much in the rupee can he pay his creditors. 29. The diameter of the fore-wheel of a carriage is 1} ft. and that of the hind-wheel is 3 feet ; how far will the carriage have travelled when the fore-wheel has made 100 more revolutions than the hind-wheel ? (The circumference of a circle : diameter 1 1 3*1416 : i.) 336 ARITHMETIC 30. Tea at 4*. 3$</. per Ib. is mixed with tea at 3*. 7^. per Ib*. so that the mixture contains 72 per cent, of the former. Find the- weight of a chest of this mixture which is worth 6. i6s. lod. 31. A merchant buys China tea at 3?. 6d. per Ib. To improve the flavour he adds 2 oz. of Assam tea to every Ib. of China tea* and finds that the mixture costs him 4*. per Ib. How much per Ib, did he give for the Assam ? 32. Standard silver, of which in parts in 120 are pure silver* being worth RSI per Ib., find the value of a Sicca Rupee which weighs 7 dwt. 12 gr. and has a fineness of 979 parts in 1000. 33. A contract is to be finished in $ months and 17 days, and. 43 men are put on to work at once ; at the end of f of the time: it is found that only } of the work is done ; what extra number of hands will be required to complete the contract in the given time> the last employed men to work 12 hours a day, whilst the first 43 men work until the contract is completed only 10 hours a day ? 34. A man can do as much work in 4 hours as a woman in 6 hours, or as a boy in 9 hours ; how long will it take a boy to complete a piece of work, one-half of which has been done by a man working 10 hours and a woman working 16 hours ? 85. If a piece of cloth, 4 yd. long and 15 in. wide, cost R3. 2a. 9 how much should you give for another piece, 19 yd. long and 12 in. wide, every sq. in. of which is worth f of the value of a sq. ft. of the former ? 36. A person sets out to walk 26 miles ; for a quarter of the distance he goes at the rate of 5 miles an hour, for half the remain- ing distance at 4 miles an hour and 3 miles an hour for the other half. State the exact time occupied in the journey. 37. How often between 12 and i are the hands of a clock an integral number of minute-spaces apart ? 38. Two clocks begin striking the hour of noon together on a certain day, the interval between every two strokes being i" and 2" respectively. They gain i" and 2" respectively in every 24 hours* After what length of time will they end striking the hour of noon together ? 39. A and B start at the same time on a journey. A walks at the rate of 4 miles an hour, and B of 3 miles an hour. When A has gone half way, B gets a ride and goes at twice the rate of A, until he has ridden a distance equal to ^ of the whole journey be- yond the spot at which he passes A. B then walks the remainder of the journey, A having walked it all. Will A or B arrive first ? And what fraction of the whole journey will the other still have to travel ? 40. If 15 men can dig 600 cu, ft. of earth in 5 days, working PROBLEMS 337 S hours a day, how many men would be required to dig 1575 cu. ft. in 14 days, working 9 hours a day, supposing that a man who works 8 hours a day does in 25 hours the same amount of work that a man who works $ hours does in 26 ? 41. If 2i horses and 217 sheep can be kept 10 days for the same sum as it would cost to keep 9 horses and 60 sheep for 27 days, find how many sheep eat as much as 3 horses. 42. In running a four-mile race on a course half a mile round| A overlaps B at the middle of the 6th round. By what distance will A win ? 43. A and B start to run a race at 3 o'clock. The winner comes in at 6 j minutes past 3, beating the other by 40 yards. At 4 minutes past 3 the loser was 1140 yards from the winning-post. Fine! the length of the course, and the speed of the winner in miles per hour. 44. Five men do '6oo of a piece work in 2*12 hours, how long will 6 boys take to finish it, it being known that 3 men and 7 boys have done the whole of a similar piece of work in 3 hours ? 46. If 4 men earn as much in a day as 7 women, and one woman as much as 2 boys, and if 6 men, 10 women and 14 boys working together for 8 days earn 22, what will be the earnings of 8 men and 6 women working together for 10 days ? 46. The distance by Railway from Madras to Salem is 2o6| miles. A Passenger Train travelling 20 miles an hour leaves Madras at 7 A. M. ; and a Special Train at 10 A. M. the same day. At what rate must the latter travel, so as just to overtake the former at Jollarpett Junction (132 miles Irom Madras), and at what hour must a Goods Train leave Salem for Madras travelling IS miles an hour, so as to reach Jollarpett at the same time as the other Trains ? 47. Two trains measuring 330 ft. and 264 ft. respectively, run on parallel lines of rail. When travelling in opposite direc- tions they are observed to pass each other in 9 seconda t but when they are running in the same direction at the same rates as before the faster train passes the other in 27^ seconds. Find the speeds of the two trains in miles per hour. 48. A man near the sea-shore sees the flash of a gun fired from a vessel, steaming directly towards him, an4 hears the report in 1 5". He then walks towards the ship at the rate of 3 miles an hour, and sees a second flash 5 minutes after the first, and immediately stops ; the report follows in 10*5". P'ind the rate of the ship the velocity of sound being 1200 feet per second. 49. A soldier has 4 hours' leave of absence ; how far may he ride on a coach which travels 8 miles an hour, so as to return to the camp in time, walking at the rate of 4 miles an hour ? C, A, 22 338 ARITHMETIC 60. Two trains start at the same time, the one from Calcutta to Allahabad) the other from Allahabad to Calcutta. If they arrive at Allahabad and Calcutta respectively 5 hours and 20 hours after they passed of each other, show that one travels twice as fast as the other. 51. A cistern is provided with two pipes, A and B. A can fill it in 20 minutes, and B can empty it in 30 minutes. If A and B be kept open alternately if or one minute each, how soon will the cistern be filled ? 62. A t B) C are pipes attached to a cistern. A and B can fill the cistern in 20 and 30 minutes respectively, while C can empty it in 15 minutes. If A> B, C be kept open successively for one minute each, how soon will the cistern be filled ? 63. A train having to perform a journey of 150 miles, is obliged after 100 miles to reduce its speed by one-fifth. The result is that the train arrives at its destination half an hour behind time. What is its ordinary rate ? 54. A down Passenger Train, 176 yd. long, travelling at the rate of 20 miles an hour, meets at 7 A. M. an up Goods Train, 293 J yd. long, and passes it in 24 seconds. At 7-30 A. M. the down Passenger meets the up Mail, 88 yd. long, and passes it in 12 seconds. When will the Mail overtake the Goods ? 55. A and B start together from the same point on a walking match round a circular course. After half an hour A has walked 3 complete circuits, and B four and a half. Assuming that each walks with uniform speed, find when B next overtakes A, 66. A certain sum is to be divided among A, B and C. A i^ to have 30 less than the half, B is to have 10 less than the third part, and C is to have ,8 more than the fourth part. What does each get ? 67. 4212 is divided among A } B, C, so that A receives f as much as B and C together, and B $ of what A and C together recieve. Find how much each receives. 68. Two-thirds of a certain number of persons received i8df. each, and one-third received 2s. bd. each. The whole sum spent was 2. I5J. How many persons were there ? 59. A crew which can pull at the rate of 9 miles an hour, finds that it takes twice as long to come up a river as to go down ; at what number of miles an hour does the river flow ? 60. A t B t Care partners ; A whose money has been in the business for 4 months claims J of the profit ; 13 whose money has been in the business for 6 months claims j of the profits j C had ^1560 in the business for 8 months : how much money did A and B contribute to the business ? PROBLEMS 339 61. Two persons A and B rent a field. A puts on it 12 horses for 2\ months, 20 cows for 4 months and 50 sheep for 5 months ; B puts 1 8 horses for 3| months, 15 cows for 5 months and 40 sheep for 4^ months. If in one day 3 horses eat as much as 5 cows, and 6 cows as much as 10 sheep, what part of the rent should A pay ? 62. A can dig a trench in \ the time that B can ; B can dig it in | of the time that C can ; all together they can dig it in 6 days. Find the time it would take each of them alone. 63. For 5 guineas can be obtained either 12 Ib. of tea and J 5 Ib. of coffee, or 36 Ib. of tea and 9 Ib. of coffee ; find the price ot a pound of each, 64. Divide 48 into two parts such that if one part be multi- plied by 3 and the other by 5, the sum of the products shall be 180, 65. Divide 20 into two parts such that three times one part may be equal to twice the other part. 66. A decimetre is equal to 3*937 inches, and a cubic decimetre of water weighs i kilogram. If a cubic inch of water weighs 252*45 grains, express a kilogram in pounds avoir, correct to two decimal places. 67. Twenty gallons of liquid contain 60 per cent, of nitric acid and the rest water. How many gallons of water should be added to the mixture to lower the proportion of nitric acid to 40 per cent. ? 68. Divide Biooo among i man, 3 women and 36 children so that the man gets 4 times as much as each woman, and the women together get 12 times as much as each child. 69. Two men undertake to do a piece of work for ^40. One could do it alone in 5 days, the other in 8 days. With the help of a boy they finish it in 3 days. How should the money be divided ? 70. The sum of the ages of A and B is now 55 years, and their ages 10 years ago were as 4 is to 3 ; find the present ages, 71. A tradesman's prices are 20 p, c. above cost price ; what profit does he make, if he ahows his customers a discount of a penny in the shilling ? 72. Four apples are worth as much aa 5 plums, 3 pears as much as 7 apples, 8 apricots as much as 1 5 pear^, and 5 apples sell for 2d. I wish to buy an equal number of each of the four fruits, and to spend an exact number of pence : find the least sum I can spend. 73. The manufacturer of an article makes a profit of 20 per cent.| the whole-sale dealer, of 10 per cent., and the retail-dealer, of 5 per cent. What is the cost of the manufacture of an article which is retailed for &7. 80. 9A ? 340 ARITHMETIC 74, Two cogged wheels, of which one has 16 cogs and the Other 20, work in each other. If the latter turns 60 times in | of a minute, how often does the former turn in 16 seconds ? 76. The price of butter having risen 25 p. c., the daily allow- ance of each person in a family is reduced from i oz. to \ oz. If the monthly charge for butter is thenceforward 12^ , what was it before the changes were made ? 76. A bankrupt has book-debts equal in amount: to his liabi- lities, but on ^4000 of them he can recover only 1 55. in the ;, and the expenses of the bankruptcy are .200 ; if he pay 155-. i\tL in the ;, what is the amount of his liabilities ? 77. A ship 40 miles from the shore springs a leak which ad- mits 3| tons of water in 12 minutes. 60 tons would suffice to sink her, but the ship's pumps can throw out 12 tons of water in an hour. Find the average rate of sailing so that she may reach the shore just as she begins to sink. 78. Standard silver is formed by mixing II paits of fine silver with one of copper. How many rupees can be coined from I Ib. avoir, of fine silver, if i Ib. troy of standard silver is coined into 32 rupees ? 79. If 2j tolas of gold, 22 carats fine, be worth &49. 8*., of what fineness must gold be in order that \\ tolas of it may be worth 834- 8a. ? 80. A man having to walk 36 miles finds that in 3 hr. 20 mm. he has walked f of the remaining distance ; find his speed. 81. Supposing the alloy in a rupee to be ^ of the mass, and the coin to be worth 2 pice if it were all alloy, what would be its exact value if it were all pure silver ? 82. A mixture contains wine and water in the ratio of 3 \ 2 ; if it contains 3 gallons more wine than water, what is the quantity of wine in the mixture ? 83. 3 men and 6 boys can do 4 times as much work as a man and a boy can do, in the same time. Find the ratio of the works done by a man arid a boy in the same time. 84. A mixture is composed of 4 parts brandy and r part water ; one gallon of water is added, and the mixture contains 3 times as much brandy as water : find the quantity of brandy in the mixture. 85. A mixture contains wine and water in the ratio of 3 I 2, another contains wine and water in the ratio of 4 ; 5 ; how many gallons of the latter must be mixed with 3 gallons of the former that the resulting mixture may contain equal quantities of wine and water ? PROBLEMS 341 86. A, B and C are three vessels holding r, 2 and 4 gallons respectively. A is empty, B is full of water and C is full of wine. A is filled from B, B is replenished from (7, and then A is emptied into C. When this operation has been performed once more, what will be the ratio of the wine in B to the water in C ? 87. An alloy of silver is mixed with an alloy of gold in the ratio of 73 to '37 ; the quantitv of dross in the silver alloy is 12 parts in 100, and in the gold alloy 15 parts in 100 : compare the quantities of gold, silver and dross in the mixture. 88. A barters some sugar with B for flour which is worth 25. 3^. per stone, but uses a false stone weight of 13^ Ib. ; what value should B set upon his flour, that the exchange may be fair ? 89. If the work done by a man, a woman, and a child be in the ratio of 3, 2, i, and there be in a factory 24 men, 20 women and 16 children, whose weekly wages amount to R224, what will be the yearly wages of 27 men, 40 women and 15 children ? 90. A Ib. of tea and 3 Ib. of sugar cost R3, but if sugar rose 50 per cent, and tea 10 per cent., they would cost 83. Sa. ; find the prices per Ib. of tea and sugar. 01. A bankrupt has goods worth 9750 ; and had they rea- lised their full value, his creditors would have received 13 annas in the rupee ; but |ths were sold at 17*5 p. c , and the remainder at 2375 P- c., below this value. What sum did the goods fetch, and what dividend was paid ? 92. Gold is sold at the Mint at 3. ijs. gd. per oz., and is mixed with alloy, worth $s. 2d. per oz., m the ratio of n I i. If sovereigns be coined of this mixture, each weighing 5 dwt. 3*47 gr., what is the Mint profit per 100 sovereigns ? 93. A bag contains 160 coins consisting of half-crowns, shil- lings, sixpences and fourpences, and the values of the sums of money represented by each denomination of coin are the same ; how many of each are there ? 94. In sending 100 cheroots to England I paid freight f of their prime cost ; landing charges^ of their cost including freight ; and duty 7\ times their cost including freight and landing charges. Altogether the cheroots' duty paid, in London cost me ./, What did 1 give for them in Madras ? 95. A number of rupees is divided amongst four men. A receives of the whole, B of the remainder, C $ of what then remains, and the number of rupees given to D is the square root of the whole number to be divided. What sum does each receive ? 90. For | of the distance up a ghaut the rise is I foot in 24 (measured along the road) and for the remaining third the rise is I in 16. The top of the ghaut is 1,400 ft. above the bottom ; what is its length ? 342 ARITHMETIC 97. In a company of 100 people, of whom some are rich and some are poor, the rich subscribe and give la. yp. to each poor man ; this costs the rich men 7 a. ip. each : how many rich and how many poor men are there ? 98. Given that gold is worth 3. 17 s. iod. per oz., and silver 45. lod. per oz., and that the weights of equal volumes of gold and silver are as 19* ! ir ; find the volume of silver equal in value to a cubic inch of gold. 99. A tradesman bought a quantity of goods, and sold f of them at a profit of 10 p. c. ; the price rising, be got 1 1\ p. c. profit on the remainder, and on the whole gained 425 : what sum did he lay out ? 100. A publican buys two butts of wine, one for Ri2oo, and one for Rnoo ; he also buys a third and after mixing the three, retails the wine at &22. 8a. a dozen, making 12^ p. c. on his out- lay : supposing the number of dozens in a butt to be 52, find the price of the third butt. 101. A merchant sells 49 quarters of wheat at a profit of 7 p. c., and a certain number of quarters at a profit of II p. c. The cost price of a quarter of wheat beingf 3. 12s. 6d., he would have lost 2. los. gd. if he had sold the whole at a profit of 9 p. c. Find the total number of quarters of wheat sold by him. 102. The shares in a banking concern are Riooo each, 6426. I of a. are only paid up, and the shares are quoted in the market at 460. The dividend is ^j\ per share quarterly. A gentleman holds loo original shares. Find what interest he makes per cent. ; and how much per cent, would he make, if he sold out and invested in 4 per cent, Govt. stock at par ? 103. A person finds that if he invest a certain sum in railway shares paying 6 per share when the 100 share is at ^132, he will obtain 10. i6s. a year more for his money than if he invest in 3 per cent, consols at 93. What sum has he to invest ? 104. A person has ^24,180 to invest ; the 5^ per cent. Govt. loan being at 108 and the 6 per cent. Municipal loan of Ri,ooo being at 1020 ; find how he must divide his capital between the Govt. and Municipal loans, that he may obtain the same income from each. 105. A railway proprietor receives one year a dividend of 6 per cent, on his stock, and pays an income-tax of 4</. in the , The next year he receives a dividend of 6-J per cent, and pays an income-tax of 3*?. in the , and finds that his net income is ^249 more. How much railway stock does he hold ? 106. A man sold at 48 and 95 respectively $co ordinary stock in the A Railway paying a dividend at the rate of if and ;8oo PROBLEMS 343 preference stock in the B Railway paying a dividend of 4 per cent. He then invested J of the money in the Tramway Company where the ^24 share paying interest at 6 per cent, was at 6 premium ; i 50 in the C Railway w hich paid no interest ; and the remainder in Bank shares at par : what rate of interest must he receive from the Bank in order to increase his annual income by 12. $s. ? 107. There are two railway engines whose rates of motion may be represented by I and 75. Supposing the slower to have been 12 miles in advance of the faster train on the same line, how far would the faster train have to travel before it overtook the other ? 108. The value of I Ib. of gold is 20 times that of I Ib. of silver and the weights of equal volumes of gold and silver are as 19 I 10 ; find the value of a bar of silver equal in bulk to a bar of gold of value ^380. 109. A merchant owes a bill of 85,796, payable in 8 months and another of ^.7,822, payable in 12 months ; he takes up these two bills and gives in their place one for 13,716, payable in 12 months : what is the rate of interest per cent, per annum ? 110. A Calcutta merchant has to pay Rio, 5 12. 8a. to his agent in Bombay. What must he give for a bank draft to that amount, exchange being at looj ? 111. A man bequeaths his property amounting to 49,166 in such a way that i of his wife's share, of his eldest son's, f of his younger son's and i of his daughter's share are all equal Find the share of each. 112. A and B exchange goods ; A gives 13 cvvt. of hops, the retail price of which is 56^. per cwt. but in barter he rates them at $. B gives 10 barrles of beer, the retail price of which is is. a gallon, but the value of which he raises in proportion to the in- creased price of the hops. How much must B give in money ? 113. A person having to pay Rio, 572 two years hence, invests in the 4 per cent. Transfer loan to accumulate interest till the debt shall be paid, and also an equal sum the next year. Suppos- ing the investment to be made when paper is at 86J, and the price to remain the same ; what sum must be invested on each occasion that these may be just sufficient to pay the debt at the given time ? 114. A train has been travelling 20 miles an hour : the steam power is doubled, whilst from various causes the resistance of the train is increased by one-half. (The original steam power is three times the resistance). At what rate will the train now travel ? 116, A sailing vessel reaches Madras from Calcutta in 6 days,; a steamer whose speed is to that of the sailing vessel as 3 I 2 starts at the same time, but meets with detentions that average 6 hours daily. Which will reach Madras first ? And by how much ? 344 ARITHMETIC 116. A book containing between 900 and 1000 pages is divided into four parts, each part being divided into chapters. The whole number of pages in each of the four parts is the same. Each chapter in the first part contains 20 pages, each chapter in the second 40, each chapter in the third 60, and each chapter in the fourth 80. Find the whole number of chapters in the book. 117. A person buys a piece of land at ^25 an acre, and by selling it in allotments finds that the value is increased by one-half, so that, after reserving 20 acres for himself, he clears 200 on his purchase-money by the sale of the remainder. How many acres were there ? 118. Find how much rice a family requires monthly, when a reduction in the price from 7 to 10 measures for the rupee reduces the total monthly expenses from RSI^ to 30. 119. A barters sugar with B, for rice which is worth if annas a measure, but in weighing his sugar uses a false maund weight. B discovers this, and to make the exchange fair raises the price of his rice to 2\ annas a measure. Find the real weight of the false maund which A uses. 120. A person pays an income-tax of 4^. in the during the first half of the year and of 3d. in the during the second half, and finds that owing to an increase in his income he pays the same amount of tax for the second as for the first half of the year. If his gross income for the year is 700, find his net income. 121. The materials of an old building were sold for 1,500 upon condition that they should be removed within 30 days under a penalty of Bio per day for every day beyond 30 days. The pur- chaser employed 40 men at 3^ annas per day to do the work, and after selling the materials for ^2365, he cleared 8190 by his bar- gain. Find the number of days the men were at work. 122. A and B enter into partnership ; A supplies the whole of the capital, amounting to 45,000 upon condition that the profits are to be equally divided, and that B pays A interest on half the capital at 10 per cent, per annum but receives 6,120 per mensem for carrying on the concern. Find their total yearly pro- fits when -#'s share is equal to \ of A's share. 123. If the value of a rupee varies from i s. gd. to 1$. q\d. and of the franc from <)\d. to lod. ; find the maximum number of francs which it is always safe to give for 500. 124. If the volume of a spheres* Jx 3*1416 x the cube of the radius, find how many spherical balls each -J inch in diameter can be made out of a cubic inch of clay, and how much clay will remain over. 125. Paper-money is at a discount of 10 per cent. A man buys goods marked 27 (paper-money) and offers that sum in gold. How PROBLEMS 345 -much paper-money must he receive in change, 10 per cent, abate- ment being allowed for cash ? 126. A reservoir is to be emptied, the rate of discharge of the contents being diminished by 100 gallons every hour. The first half will be emptied in 3 hours, the second in 4 hours. How many gallons does the reservoir contain ? 127. What must be the least number of soldiers in a regiment to admit of its being drawn up 2, 3, 4, 6 or 8 deep, and also of its being formed into a solid square ? 128. A) B and C are partners. A receives of the profits, B and C dividing the remainder equally. A's income is increased by B>4oo when the rate of profit rises from 5 to 7 per cent. Find the capital of B. 129. How many years' purchase should be given for an estate so as to get 4 per cent, for the money ? 130. An agent has to receive a rent paid in corn from a tenant, and to deliver it to the landlord. At each payment he uses, so as to benefit himself, a false balance, such that 4 seers in one scale balance 5 seers in the other. Corn being worth R2. 8<2. a md., the value of his plunder is &4. What is the corn-rent ? 131. A zemindary is bought at 20 year's purchase for Rzyooo, one-third of the purchase-money remaining at mortgage at 9 per cent. The cost of collecting rents is 140 per annum. What interest does the purchaser make on his investment ? 132. A baker's outlay for flour is 70 per cent, of his gross receipts, and other trade expenses amount to \ of his receipts. The price of flour falls 50 per cent., and other trade expenses are theieby reduced 25 per cent. By how much should he now reduce the price of a ^d. loaf to make the same amount of profit ? 133. 1000 copies of a pice newspaper weigh \ of a maund, and when the paper duty was removed the profit on the receipts was increased 5 per cent. What was the duty per md. on paper ? 134. A horse was sold at a loss of 10 p. c.; if it were sold for &7o more there would have been a gain of 4 per cent. : for how much was the horse sold ? 135. A contractor sends in a tender of 7000 for a certain work ; a second sends in a tender of 6950, but stipulates to be paid &30oo at the end of a month ; find the difference between the tenders, supposing the work to be finished in 3 months, and money to be worth per cent, per month simple interest. 136. A labourer was engaged for 20 days, on the agreement that for every day he worked he should have 40., but that for every day he absented himself he would be fined la. He received fte. 134. at the end of the time : how many days was he absent ? 346 ARITHMETIC 137. A man was hired to do a certain amount of work, on the. condition that for every day he worked he should have 12**., but that for every day he absented himself he should lose 40, He worked 3 times as many days as he absented himself, and received on the whole Bio. How long was he doing the work ? 138. A grocer buys two maunds of sugar ; he sells one maund at a profit of 10 p. c., and the other which cost R2. 80. more, at a profit of 15 p. c. If the retail price per seer of the latter be !&<* more than that of the former, find the cost price of each maund. 139. A shop-keeper buys 2 md. of sugar, and i md. more of a superior kind, giving Ri. 8#. a md. more for the latter than the former. He retails it, when mixed, at 4 annas a seer, and makes a profit of 25 p. c. on his outlay. What did he give per md. for each kind of sugar ? 140. Two boys begin to count two equal piles of rupees. One counts 5 while the other counts 4. When the former has just finished the latter has 6 left. What is the number of rupees in each pile ? 141. The price of a yard of jean is of the price of 2\ yd. of longcloth ; and the weight of 5 yd. of jean is of the weight of 8 yd. of longcloth. If the price of 2 Ib. of jean be 3, what is the price of i^ Ib. of longcloth ? 142. Three tramps meet together for a meal : the first has 5 loaves, the second 3, and the third, who has his share of the bread, pa>s the other two 8 half-pence ; how ought they to divide- the money ? 143. A and B barter : A has 7 md. of flour worth 3. 8. a md. t but insists on having &3. 120. a md.: B has rice worth Ri. 5#. a measure, which he raises in price in proportion to A's demand. A receives 16 measures of rice ; what cash does he get besides ? 144. A and B barter : A has 200 Ib. of tea worth 2s. 6J. a Ib. but insists on 2j. gd. a Ib,: B has coffee worth is. gd. a Ib. : how much must he raise the price so that A gets $. 25. and 2 cwt. of coffee ? 145. A river 14 ft. deep, 182 yd. wide flows at the rate of 3 miles an hour (i) how many tons, (ii) how many gallons of water, pass a certain point per minute ? [A cu. ft. of water weighs 62$ Ib., a gallon contains 277^ cu. in.] 146. A four-wheeled carriage travels round on a circular railway. The circumferences of the two wheels of the carriage and of the two circles of rails are proportion al to 6, 7, 7000, 7014, Find the number of revolutions made by each of the four wheels in a complete circuit ? 147. Eleven boys fired 10 shots each at a target, and scored PROBLEMS 34T 286 ; 20 bull's-eyes were made and u misses ; how many centres and outers were there? (A bull's-eye scores 4, a centre 3, an outer 2). 148. The sum of i 77 is to be divided among 15 men, 20 women and 30 children, in such a manner that a man and a child may receive together as much as two women, and all the women may together receive 60 \ what will they each respectively receive ? 149. A owed B three-fourths of what B owed C ; to settle- matters, B gave &2 to A who then paid C ; what did B owe C ? 160. A man for 4 years spends RSOO a year more than his income. At the end of that time, he reduces his expenditure 3o per cent, and in 3 years pays off his debt and saves Riooo. What is his income ? 151. A tree grows 2 yards in its first year, and afterwards it grows each year i foot less than it did the previous year. The value of the tree at any time is equal to the number of rupees in- the square of the number of yards in its height ; find the value of the tree when it has done growing. 152. If standard gold, worth ^3. ijs. \^\d. per ounce be so- far alloyed as to be worth only ,3. i6j. i\d* per ounce, find the least integral number of sovereigns made of the alloyed gold> which shall be equal in value to an exact number made of the standard gold. 153. Find the least integral number of ounces of pure silver, worth 82. I4#. 6 1 6 ,^. per ounce, that, with the proper proportion* of alloy, can be coined into an exact number of rupees. 154. Mahogany is 50 Ib. to the cubic foot, water is 62^ lb., and iron is 7^ times as heavy as water ; what thickness of iron will weigh as much as a 6-inch plank of mahogany ? 155. A sum of 862 is to be divided among 10 men, 15 women, 8 boys and 12 girls. For every rupee that a man gets, a boy gets 6 annas, and for every half-rupee that a woman gets, a girl gets 2 annas. The whole money obtained by the boys is equal to that obtained by the girls. How much does each person get ? 156. A wooden closed box, made of ^-inch plank, is externally 15 in. long, 10 in. broad and 6 in. high. The box weighs 6 lb. when empty, and 80 lb. when filled with mercury. Compare the weights of equal bulks of the wood and mercury, 157. 8430 is divided among 45 persons consisting of men, women and children. The sums of the men's, women's and children's shares are as 12 1 15 I 16, but the individual shares of a. man, woman and children are as 61 5! 4. Find the number of men, women and children. 158* Bronze contains 91 per cent, of copper, 6 of zinc, and 3 348 ARITHMETIC of tin. A mass of bell-metal (consisting of copper and tin only) and bronze fused together is found to contain 88 per cent, of copper, 4*875 of zinc, and 7*125 of tin. Find the proportion of copper and tin in bell-metal. 159. An alloy contains 12 parts by weight of lead, 4 of anti- mony, and i of tin. How much of this alloy must be taken, and how much l a ad and tin added to it to make up 9 cwt. of type-metal consisting of 14 parts lead, 3 antimony and I tin ? 160. Three persons A, /?, C, finished a piece of work. A worked at it for 5 days, D for 7 days and C for 9 days. Their daily wages were as 4 '. 3 I 2, and the total earnings amounted to 7. 6a t What were the daily wages of each ? 181. Two passengers are charged for excess of luggage Hi. 80. and RS. 4<2. respectively. Had the luggage all belonged to one person he would have been charged 87. Sa. for excess. How much is allowed free, the charge for excess being iia. per md. ? 162. If the cost of making bread be one rupee per bushel of wheat, what is the price of wheat when the two-anna loaf is twice as large as it is when wheat is &5 a bushel ? 163. If the rate of wages vary as the price of rice, and if 57 men working for 35 days receive 405. $a. qp. when rice is sold at the rate of 136 measures for S39 ; find the price of rice per measure when 70 men working for 19 days receive &353. 4^. 6/>. 164. There is a leak in the bottom of a cistern. When the cistern was in thorough repair, it would be filled in i\ hours. It now takes half an hour longer, If the cistern is full, how long would it be in leaking itself empty ? 165. A can do f of a piece of work in j of the time in which B can do -f of it, and B can do J of it in of the time that it would take Ctodo another piece of work one-fourth as large again as the first. If C can finish the former piece of work in 10 hours, how long would it take A and B together to do it ? 166. A and B start on a journey at the same time. B travels at f of A's rate, and arrives 3 hr. 1 5 min. after him. In what time did each complete the whole journey ? 167. The expenses of a family when rice is at 20 seers for a ^rupee are 50 a month ; when rice is at 25 seers for a rupee the expenses are 848 a month ; what will they be when rice is at 30 seers for a rupee ? 168. A man who can walk down a ghaut at the rate of 4^ and up it at the rate of 3j miles an hour, descends and returns to his starting point after walking for 2 hours 4 minutes. How far did he walk? 169. An express train owing to a defect in the engine goes at PROBLEMS 349 | of its proper speed, and arrives at 6-49 P.M. instead of 5-55 P.M. ; at what hour did it start ? 170. A person going from Pondichery to Ootacamond travels 90 miles by steamer, 330 miles by rail and 30 miles by horse- transit. The journey occupies 30 hr. 50 min., and the rate of the train is 3 times that of the horse-transit and i times that of the steamer. Find the rate of the train. 171. A person walks from A to B at the rate of 3 miles an hour, and after transacting some business which occupies him an hour 3 returns to A by the tramway at the rate of 5 miles an hour. He then finds he has been absent 2 hours 20 minutes. Find the distance from A to B. 172. The expenses of a family, when rice is 12 seers for a rupee, are R$o a month ; when rice is 14 seers for a rupee, the expenses are 48 a month (other expenses remaining unaltered) ; what will they be when rice is at 16 seers per rupee ? 173. A bankrupt has book-debts equal in amount to his liabilities, but on R864O of such debts he can recover only 8& annas in the rupee, and on ^6300 Only 5$ annas in the rupee.~ After allowing 81054 . II . o for the expenses of bankruptcy, he finds that he can pay his creditors 12 annas in the rupee. Find the total amount of his debts. 174. A train starts with a certain number of passengers. At the first station it drops J of these and takes in 20 more. At the next it drops \ of the new total and takes 10 more. On reaching the third station there are 60 left. What number started ? 175. One pound troy of standard silver which contains 37 parts in 40 of fine silver is coined into 66 shillings. If the value of pure silver rises 10 per cent., what must be the reduction of pure silver in a shilling ? 178. A landlord has an estate worth R4oooo a year, but has to pay \ anna in the rupee on the gross income for taxes. He sells it at 20 years' purchase on the gross income, and invests the proceeds in the 4 per cents, at 95. What is the difference in his income ? 177. In firing at a mark A hits in 2 out of 4 shots, B in 3 put of 5, and C in 4 out of 7. The mark was hit 468 times. Supposing each to have fired the same number of shots, find how many hits each made and the total number of shots fired. 178. A shop-keeper buys sugar at Bi2. 80. a md. ; at what price must he sell it to gain 8 per cent,, and allow a purchaser 10 per cent* discount ? 179. In a manufactory loo coolies work for 4 days a week, but on the remaining 3 days some are absent ; the weekly wages of 350 ARITHMETIC the coolies are thus reduced in the ratio of 32 ' 35. Find the number of absentees. 180. The manager of a boarding house having already 50 boarders, finds that an addition of 10 increases the gross monthly expenditure by 20, but diminishes the average cost per head by Ri. What did the monthly expenses originally amount to ? 181. If 9 oz. of gold, 10 carats fine, and 5 oz., n carats fine, be mixed with 6 oz. of unknown finene ss, and the fineness of the resulting mixture be 12 carats, what was the unknown fineness ? 182. A tradesman's stock in trade is valued on January istj -1868, at .8,000, he has also .350 in cash and owes ^1,870 ; during the year his personal expenses, ^300, are paid out of the proceeds of his business, and on January ist, 1869, his stock is valued at ^7,950, he has 570 in cash and owes ,1,510. What is the whole profit on the year's transactions after deducting 5 per cent, interest on the capital with which he began the year ? 183. If 20 English navvies, each earning 3$. 6d. a day, can do the same piece of work in 15 days that it takes 28 foreign work- men, each earning 3 francs a day, to complete in 20 days ; taking the value of the franc at 100?., determine which class of workmen it is most profitable to employ. If a piece of work done by the navvies cost 3,000, what would be the cost of the same work done by foreign workmen ? 184. A merchant in New York wishes to remit to London 5110 dollars, a dollar being equal to 4*. 6d. English : for what sum in English money must he draw his bill when bills on London are at a premium of 9j per cent, ? 185. A person borrows ;ioo, and at the end of each year pays 25 to reduce the principal and to pay interest at 4 per cent, on the sum which has been standing against him through that year. How much will remain of the debt at the end of 3 years ? 186. If a metric system of area were adopted wherein i acre I rood 3 perches is represented by 5*12, express the unit of measurement in sq. yards and decimal parts of a sq. yd. 187. If gold weighs 19 times as much as water, and silver 12 times as much, find how many times heavier than water is a coin which contains 10 parts of gold and I of silver. 188. A certain reef of quartz when crushed yields *ooil per cent, of gold. If the working expenses amount to 62*5 per cent, of the gross receipts, and the net profit on each loo tons is 52. 105, ; find the number of grains in a sovereign. 189. A certain article of consumption is subject to a duty of 6s. per cwt. ; in consequence of a reduction in the duty the con- sumption increases one-half, but the revenue falls one-third. Find 'the duty per cwt. after the reduction. PROBLEMS 35Z 190. If the duty on a certain commodity were reduced 25 per cent,, by how much per cent, must the consumption be increased <hat the same revenue may be derived from it ? 191. If 2 cu. in. of gold together with 3 cu. in. of silver are equal in weight to 74 cu. in. of water, and the weights of equal volumes of gold and water be represented by the numbers 19 and I, what number represents the weight of an equal volume of silver ? 192. A farmer bought equal numbers of two kinds of sheep f one at 3 each, the other at 4 each. If he had expended his money equally in the two kinds he would have had 2 sheep more than he did ; find how many he bought. 193. A man travels 150 miles in 13 hours, partly by rail and partly bv steamer ; if he had gone all the way by rail, he would have ended his journey 8 hours sooner, and saved J of the time he was on steamer ; how far did he go by rail ? 194. In a distilling operation, during 3 hours the fluid contain- ed 70 per cent, of alcohol, during 2j hours 6p per cent., and during the remaining i hours 40 per cent. What is the average strength of the whole fluid distilled over, assuming that it came over at a uniform rate during the whole time ? 195. During a distillation the fluid that comes over in 3 conse- cutive hours contains 47, 35 and 20 per cent, of alcohol respectively. The rates at which it comes over during these 3 hours are in the ratios of 2, 3 and 4. What is the percentage of alcohol in the whole mixture ? 198. I bought a number of mangoes at 35 for R2. I divided the whole into two equal parts, one of which I sold at 17, and the other at 18 mangoes per Ri. I spent and received an integral number of rupees, but bought the least possible number of man- goes. How many did I buy ? 197. Find the cost in rupees of one mile of railway, which consists of two rails, each weighing 40 Ib. 'per yard, on wooden sleepers, weighing 70 Ib. each, placed 2 ft. 8 in apart. The rails cost in England 6 . 13 . o per ton and the sleepers 2^ 4^. each. The rate of freight is 1.5.0 per ton, and landing charges amount to R2. 8a. per ton. Rate of exchtnge is. 8</. per rupee. 198. The length of the E. B. Railway being 1 10 miles and the capital employed in its construction 1 5ooooo/,, what must be the gross annual traffic receipts per mile in order that a dividend of 5 per cent, may be paid to the share-holders after allowing 45 per cent, of the gross receipts for current expenditure ? 199. A person in India sells a bill on London for 358/1 pay- able at 3 months sight at the rate of n. iof</. per rupee. The purchaser requires payment on presentation ; what amount does ne receive after discount at 5 per cent, has been deducted ? 352 ARITHMETIC 200. A Guernsey pound contains 18 oz. avoir., and the Guernsey shilling contains 13 English pence. If a Guernsey pound of butter cost u. 6^., Guernsey money, what will be the price in English money of 2% lb. avoir. ? 201. A contractor employs a fixed number of men to com- plete a work. He may employ either of two kinds of workmen : the first at 26 s. 6d. per week each, the second at iSs. 6^. per week each : the work of the one of the former being to that of one of the latter as 5 to 4. If he finishes it as quickly as possible, he spends ^270 more than he would have done if he had finished it as cheaply as possible, but takes 4 weeks less time. What would it have cost if he had employed equal numbers of the two kinds of workmen ? 202. A manufactory turns out 50 tons of iron goods weekly, using up for that purpose 51 tons of iron at 6. i$s. per ton, 100 tons of coal at Us. 6d. per ton, and 45 worth of other materials ; rent, rates and taxes amount to ^219 annually ; wages and incr dental expenses to ^75 per week. At what price per cvvt. must the iron be sold in order that the works may gain 8 per cent, per annum on a capital of ,35000 ? [Reckon 52 weeks to the year.] 203. Two lumps, composed of gold, silver and copper, together weigh 10 oz. ; one lump contains gold 75 p. c. and silver 15 grains per oz,, the other contains gold 85 p. c. and silver 12 grains per oz. The total quantity of silver in the two lumps is 141 grains. If the two lumps are melted and formed into one, what per cent, of gold will it contain ? 204. The only three creditors of an insolvent whose assets amount to 100 and who can pay only $d. in the } agree among themselves to take dividends in the proportion of the number of ;. s. and d. respectively, contained in the amounts due to them. The dividends thus taken are in the proportion of 12 I 7 I 6, What are the amounts of their debts ? 206. At an examination \ of a class gains } of the maximum number of marks, ^ gains J, f gains , J gams j., and the rest J. The average number of marks gained by the whole class is 166 ; what is tbe maximum ? 206. A mass of gold and silver weighing 9 lb. is worth ,318. 13^. 6d.\ if the proportions of gold and silver in it were interchang- ed, it would he worth ^129. los. 6d. ; it is known that I oz. oLgold and 2 oz. of silver a're worth 4. 85. i\d. ; what is the price of gold and silver per oz, ? 207. A person shooting at a target, distant 550 yards, hears the bullet strike the target 4 seconds after he fires. A spectator, equally distant from the target and the shooter, hears the shot strike the target i\ seconds after he heard the report ; find the velocity of sound. PROBLEMS 353 208. A boatman rows 5 mi. with the tide in the time he would take to row 3 mi. against it ; but if the hourly velocity of the current were \ a mile, he would row twice as rapidly with the tide as against it. Find his power of rowing in still water, and the velocity of the current. 209. A messenger sets out at the rate of 30 miles a day, but falls off in his speed 4 miles daily. Four days afterwards another sets off from the same place on the same route, travelling 50 miles the first day but falling off like the first 4 miles daily. After what time will one overtake the other ? 210. Six months ago A invested 7620 in the 3 per cents, at 95 J, and six months hence he will receive ^4300 four per cents, at 127. What is the present value of his property ? 211. Two boats, A and B, row a race. A takes 4 strokes to B*s 5, but 6 of J5's are equal to 5 of A's. A starts in front of B at such a distance that B must take 10 strokes to row over it. How many strokes must B take before overtaking A ? 212. A, B and C run a mile race. A beats C by 76 Jf yards ; B beats Cby n seconds ; the pace of A is to that oi B as 45 I 44. In what time does each run the mile ? 213. Three boys begin to fill a cistern ; one brings a seer every minute, another 2 seers every 2 minutes, and the third 3 seers every 3 minutes. If the cistern holds 40 seers, in what time will it be filled ? 214. A sells his goods 10 per cent, cheaper than B, and 10 per cent, dearer than C ; how much would a customer of B save by taking Rioo worth of goods from C ? 215. Cannons are fired at intervals of 10 minutes in a town towards which a passenger train is approaching at the rate of 35 miles an hour; if sound travels 1142 feet per second, find at what intervals the reports will be heard by the passengers. 216. A man bought a horse and a carriage for &5oo, and sold the horse at a gain of 20 p. c. and the carriage at a loss of 10 p. c., thus gaining 2 p. c. on his whole outlay ; for how much was the horse bought ? 217. If 3 men and 5 women do a piece of work in 8 days, which 2 men and 6 children, or $ women and 3 children, can do in 12 days ; find the relative strength of men, women and children. 218. Three round balls revolve with equal velocities in three concentric circular grooves. They start from a position in which they are all in the same radius of the outermost circle. The innermost ball occupies 10 seconds in traversing its groove once. After what time will they all be again on a radius of the outermost circle, the tadii of the grooves being proportional to the numbers i, 3, 5 ? C. A. 23 354 ARITHMETIC 219. Two guns are fired at the same place after an interval of 21 minutes, but a person .approaching the place observes that 20 min. 14 sec. elapse between the reports ; what was his rate of progress, sound travelling 1125 feet per second ? 220. Ash saplings after 5 years' growth are worth is. 3^., and increase in value is. $d. each year afterwards. For their growth they require each twice as many square yards as the number of years they are intended to grow before cutting. A plantation is arranged so that each year the same number may b2 ready for cutting. Find the greatest annual income which can be obtained per acre, allowing 20 per cent, for expenses. EXAMINATION PAPERS. UNIVERSITY OF CALCUTTA. ENTRANCE PAPERS. 1858. 1. Multiply Ri8957. 130. by R568. n0. ; and divide the same sum by the same sum. Shew that one of these operations is absurd and im- possible and perform the other. 2. Find the value of the decimal '16854, and deduce the rule arith- metically or algebraically. 3. Extract the square roots of 3 and of '3 to 7 decimal places, and explain the rule that in integers the pointing off of the periods begins from the right hand, and in decimals from the left. 4. A plate of metal is beaten to the thickness of of an inch, and the weight of a circular medal cut from it, whose diameter is ij inches, is ij oz. Troy. If the same plate be beaten to the thickness of J of an inch, what will be the weight of a medal cut out of it of the diameter of if inches, (the a/eas of circles being proportional to the squares of their diameters) ? 1859, A. 1. What do you mean by a prime number, a factor^ a ratio ? Resolve 30 and 132 into their prime factors, and find their ratio in its simplest terms. 2. How much muslin at Rr. 5. 8/5. per yard is equal in value to 143 yards of cambric at R3. 130. 8/. per yard ? 3. Whether is the product of 2\ and 3^ or the product of 2\ and 3} the greater ? Extract the square root of the difference. 4. If a person get a bequest of f of an estate of 2,000 acres, and sell f of his share, how many acres does he retain ? Simplify the expression - 10+ - - T 2 + *V 5. Find, by Practice, the rent of 586 acres I rood 31 poles at 4. u. per acre. 6. A piece of land is 11*916 pole? broad, how long must it be to contain an acre ? Divide accurately 0x163 by 0*36. 7. How much must be paid for 1250 stock when it sells at 108 per cent. ? 1859, B. I. A man can count at the rate of loo a minute, how long will it take turn to count five hundred lakhs ? 56 ARITHMETIC 2. A shop-keeper purchased 250^ yards of cloth for Rgoo and paid expenses amounting to Rioj : what must he charge per yard in order to make a profit of 50 per cent. ? 3. Reduce '005 of a pound to the fraction of a penny, and extract the square root of '00006241. 4. Add together *|, -*, 9 and f of of ^. 5. State the rules for pointing in multiplication and division of decimals and multiply '256 by "0025 and divide '0036 by '4 and 4 by *ooooi. 1860. 1. If the price of bricks depends upon their magnitude and if 100 bricks, o/ which the length, breadth and thickness arc 16, 8 and 10 inches respectively, cost R2. 93., what will be the price of 921600 bricks which are one-fourth less in every dimension ? 2. Explain the method of pointing in extracting the square roots of whole numbers and decimals. Find the square root of 57,214,096 and also the square root of *5 to four places of decimals. 3. Simplify (i + f + J + Hl-Hf-f) and g 4. A tea-dealer buys a chest of tea containing 2 maunds and 16 seers, at R4 2a. per seer, and two chests more each containing 3 maunds and 24 seers, at R4. ioa. per seer : at what rate per seer must he sell the whole in order to gain 576 rupees ? 1861. _ , . . c .. 4f x 8A 6^ of 4| I. Express as a decimal fraction -~ - -* x -- J- 5 -. V - 2. Reduce 35. 6ef. to the decimal of 5, and -0234 to a vulgar fraction. 3. If an estate be worth ^2,374. i6s. per annum, and the land-tax be assessed at is. u?d. in the 9 what will be the net annual income ? 4. How much land may be rented for 1,716. los. 6</., if 3 acres are rented for 4. 13*. 4</. ? 5. Extract the square root of '00099856. 1862, 1. What is the difference between if - ^ and *o6 ? 5tr 3io 2. Reduce "14 of a pie to the fraction of a rupee, and find the value of "0875 of a pound sterling. 3. If the wages of 18 coolies for ajmonth amount to R85 when rice is 24 seers per rupee, what ought the daily pay of a coolie to be in proportion when the price of rice is R2, ioa. 8/. per maund ? CALCUTTA ENTRANCE PAPERS 357 4. A and B run a race. A has a start of 40 yards, and set off 5 minutes before B, at the rate of 10 miles an hour. How soon will B over- take him if his rate of running is 12 miles per hour ? 5. Extract the square root of i \* to 5 places of decimals. 1863. 1. Find the value in vulgar and decimal fractions of - . f, 7? x " 2. Find the fractional value of (2 '3797 9 4- 4 '22) ^(3*041 - "937). 3. The weight of five casks of coffee being 31 cwt. 3 qr. 13 lb., calculate the price at 90 shillings per cwt. 4. If a man can perform a journey of 170 miles in 4^ days of II hours each, in how many days of 8| hours will he perform a journey of 470 mUes ? 5. Extract the square root of 964 '226704. 6. What sum of money will produce ^"43 interest in 3i years, at 2} per cent, simple interest ? 1864. 1. How many paving stones, each measuring 14 in. by 12 in., are required to pave a verandah 70 ft. long and 9 ft. broad ? 2. Add together J, , t \, Jf , and 7 \. And simplify 3. Find the value of 17 cwt. 3 qr. 22 lb. at 4. 6s. 7dl per cwt. 4. Add together '0125 of a pound, '0625 of a shilling, and *5 of a penny ; and reduce us. *&d. to the decimal of a pound. 5. Extract the square root of '000196 and divide the result by 140. 6. A company guarantees to pay 5 per cent, on shares of Ri,ooo eacb ; another guarantees to pay 4! per cent, on shares of R75 each ; the price of the former is Ri,245 and of the latter R85. Compare the rates of interest which the shares return to purchasers. 1865. 1. Find the value of 1 1 f + 14! + 21 4 r + 32*| T , both by vulgar fractions and by decimals, showing that the two results coincide ; and reduce 25 Q - 36'. 45" to the decimal of 75*. 2. Find the product of the sum and difference of '0421 and '0029, and divide one-tenth of the square root of that product by ten times the continued product of '02, '03 and '07. 3. How many yards of matting 3-5 feet wide will cover the floor of a room 85 } ft. long, and 40-5 ft. broad; and how much will it cost at R2, loa 8/. per square yard ? 4. In the wages of 25 men amount to R766. loa. 8/, in 16 days, 353 ARITHMETIC how many men must work 24 days to receive Ri,O35, tQe daily wages of the latter being one-half those of the former ? 5. What principal in 9 years 73 days will amount to ftioo. 150.1 at 6* per cent, simple interest ? A bill for 5,035. 40. drawn on Sep- tember I2th at 5 months was discounted on January i6th at 4 per cent, fr what was the discount charged ? 1866, A. 1. Add together f v , -f yt f, -046875 and 1-23. Simplify ^75^ and 4^Si^o64 F y -0175 '00032 2. Find, by Practice, the value of I ton 5 cwt. 2 qr. 14 Ib. ?at 3. 15*. yd. per cwt. . 3. Find the square root of 1524*9025 and of 152*49025 to three places of decimals ; and the value of '6099 of i. $s. 3^. 4. Three gardeners working all day can plant a field in 10 days, but one of them having other employment can only work half time. How long will it take them to complete the work ? 5. Find the compound interest of ^"55 for one year, payable quarterly at 5 per cent, per annum. 1866, B. 1. Reduce 3. 45'. 36" -25 to the decimal of 36. Simplify (J + l ++-!)* oft of 2f. 2. Find the value of 6 cwt. 2 qr. 7 Ib. at ^3. 45. 6$d. per cwt. 3. Find the square of 0*0204 and the square root of 81757764 ; and divide one-tenth of the latter result by one hundred times the former. 4. Divide O'XOOI by 0*00039062 5,. and 10*01 by 390*625. 5. What is the expense of paving a rectangular verandah whose length is 42 feet and breadth 15 feet with Burdwan paving stones, 18 inches square, and which cost Ri5 per score ? 6. The 3 per cents, are at 854 ; what price should the 3^ per cents, bear, that an investment may be made with equal advantage in either stock ? And what interest would be derived by so investing 5ooo/. ? 1867. 1. The driving wheel of a locomotive is 226 inches in circumference, and makes 91 revolutions per minute ; at what rate per hour is the engine travelling ? 2. Divide the least common multiple of 156, 260, 720 and 429 by their greatest common measure, and find the square root of the quotient. 3. If a butcher buy 10 cwt. of beef at 44*. 4^. per cwt. and sell it at the rate of $\d. per Ib,, how much does he lose or gain ? CALCUTTA ENTRANCE PAPERS 319 4. Find the value of the following expressions : $l*Hx9*X3Jg d 1^x67 0*625 f i43* I2J. o^ +0*62$ of 71. l6s. od. I of 5175 5. Reduce i. $s. 6d. to the fraction of ^1,000, and JJ. l\d. to the fraction of ^isa IQJ., and express the results both as vulgar and decimal fractions. 6. If 450 amount to 523. los. in l year 8 months, calculate the rate per cent. 1868. i. Find the difference between 1*6 of 3-4 of jl'!25 and { of 3*6 of ,9*1125, and find the value of 6-27x0*5 ( of A) x (| of 2i j) (* of |) x 8-36^ ( of f )"+ 1 '4 " 2. Extract the square root of 153-140625, and of 3*3, each to three places of decimals. 3. If one man walks 165 miles in 6 days, how far will another man walk in 15 days if the first man walks 3| miles in the same time that the other man walks 4 miles ? 4. Three equal glasses are filled with a mixture of spirits and water ) the proportion of spiiits to water in each glass is as follows : in the first glass as 2:3, in the second as 3 : 4, and in the third as 4 : 5. The contents of the three glasses are poured into a single vessel ; what is the proportion of spirits to water in it ? 5. Find the interest on 350 from 3rd March to 28th December at 4} per cent, per annum. 6. How many yards of carpet 25 inches wide will be required foi a room 19 feet 7 inches long, and 1 8 feet 9 inches wide ? 1869. __, and reduce 4 hr. I min. 10} sec. to the decimal of a week. 2. Add together "062435 of loo/. +7 '4375 of los. + 1 '356 of 7*. 6<t. + 2*784 of 2^., and reduce the result to the fraction of 29/. los. l\d. 3. Divide '0007 by '035 and by 3500, and extract the square root of each quotient to four decimal places. 4. A room is 37 ft. 2 in. long, 25 ft 8 in. broad, and 22 ft. 6 in. high ; find the cost of covering its four walls with paper \\ yd. wide, at is. i%d. a yard., 5. In what time will 5637. 135. 4}< amount to 901. ijs. 4$< at 3! per cent. ? 360 ARITHMETIC 1870. 1. Find the cost of matting a room whose floor is 8 yards long by 7$ yards wide, with mats 2 ft. wide and 9^ ft. long, at the rate of 9 annas 2 pies per mat. If the same room be I$J ft. high, find how many cubic feet it will contain. 2. Distinguish between a vulgar fraction and a decimal fraction. Multiply 999JIU b y 999- State the rule for the multiplication of decimals, and apply it to point the produces in (i) i '23 x '0011 and (ii) 29000 x g oi. Divide # by -i 4 , and show that -^ = L 2 _3I?3. 41 4UHI 3. Find the square root of 197 4oJ and of 4?V the latter to four places of decimals. 4. Two gangs of six men and nine men are set to reap two fields of 35 and 45 acres respectively. The first gang complete their work in 12 days, in how many days will the second gang complete theirs ? 5. Find which is the better investment, 3^ per cent, stock at 98!, or 3} per cents, at 105. 6. Find how many rupees are equivalent to 200 /. at the rate of u* P er ru pee. 1871. 1. 6625 railway tickets were sold at a station, % ths of which were 9 annas each and the rest 5 annas each. What was the amount received for the tickets ? 2. Find the greatest and least of the fractions f , ,\, T \> iVir- Add together 2f of 2. 135. 6$d. and (3. 15*. 9fr*)-r6f and simplify ; 3. Divide '027 by I4'4 and 1208-04 by '017. Find the value of 11*1375 of R6. Sa. - "56 of 87. Sa. and reduce $a. 6/. to the decimal of R3 fa. 4. If the carriage of 9! md. for a distance of 80 miles be &3, how many miles should 130 md. be carried for R27. Sa. ? 5. What sum of money will amount to R376i. 140. in 3$ years at 4} per cent, per annum simple interest ? 1872. I. A merchant bought goods which cost him R9,8io. In the first day he sold to the amount of R992. Sa. 6p., in the second to that of Rl,992. Sa. 3/)., and in the next three days to an amount equal to twice the two former. Finding that he bad one-fourth of the goods left he calculated his profits in the five days. How much were they ? CALCUTTA ENTRANCE PAPERS '2. What fraction of Rio is R6. loa. Sp. ? Find the value off of R2. 8<z. + f of R4 lia. +2*05 of R5. Simplify 4 *of A- 3. Divide 27472 by '0544 ; find the value (correct to six placet of decimals) of (i) '^ a ' S - () 6*645 -5-3678 ; and extract the square troot of 951*1056. 4. Find, by Practice, the Cost of 15 md. 25 sr. zi ch. of oil at ioa, 3/J. per maund. 5. If the interest of Ri,ooo in 5 years be R25o, what will be the interest of R3,5oo for I year and 6 months ? 1873. z. Find the value of (i) JL a ilt x J+ (ii) 24f of Ri03. 70. 6p. If T \ of a maund is worth R45, what is the price of of a maund? 2. Reduce *J, to a decimal ; '019 to a vulgar fraction ; and 4'2-vi4 f i"3 of 4 . , - - --r- of - , 7. to its lowest terms. 1-3+2-102 '37 of 8-gi 3. What is the expense of matting a room 31 ft. 5 in. long by 20 ft. 4 in. wide, the mat costing 140. per 12 square hath (linear hath = i8 in.) ? 4. In what time will R8,5oo amount to Rz5,767. 80. at 4^ per cent. per annum ? 5. A person owes the sum of R3 1,500 and R8,5oo ; and his property only amounts to Ri4,i25. How much is he able to pay in the rupee ? and what is the loss upon the second debt ? 1874. z. Wn at fraction of f of a rupee is J of R5 ; and what proportion does their difference bear to their sum ? Divide 999*666 by '30036 ; and 2*3^428 by zo*2i42857. 2. When rice is zo seers the rupee, nine persons can be fed for 30 dayt tt a certain cost. For how many days can six persons be fed at the same cost when rice is 14 seers the rupee ? 3. A wooden box 3 ft. 8 in. long, 2 ft. 3 in. high, and 2 ft. 4 in. wide, is made of board one inch thick. Find the quantity of wood used ; and the cubical content of the box. 4. It is said that 240,000 letters are posted in Berlin daily, z6*6 pet 362 ARITHMETIC cent, of which are town letters. This gives one letter for every tares 'persons in Berlin ; what is its population ? 5. What sum will amount to a lakh of rupees in ten years at 5 pet cent, simple interest ? Find the discount on 81,308 due two years hence at 4^ per cent. per annum. 1875. ... 1 Slmphfy Find the value of A o f &I7- 6 * 4A + 3? of &- $ a ; and extract the square root of '049 to four places of dtcimals. 2. A person received on the death of his aunt Vv of ^ er property and spent '54 of it in paying off his debts ; what fraction of his aunt's property did he then possess ? 3. A room is 30 ft. long, 22 ft. wide, i8 ft. high, and has 5 doors ar>d 3 windows ; find the expense of colouring the walls at 30. per sq. yd., deducting 30 sq. ft. for each door and window. 4. Find the present worth of 19,021 due 4 years hence at 3f per Cent. 5. If ft 16,430 be invested in the Government 4^ per cent, loan at 106, what is the monthly income derived ? Supposing' that the loan is paid off at par in 10 years, what would be the rate of simple interest (per cent. per annum) on the sum invested ? 1876. ,. Simplify Find the value of & of Ri6. 140. -1*14 of R$. 00. $ of ftp. 60. 6> Reduce (i6'O5~6*2J)of a rupee to the decimal of R22. ia. 2. An equal number of men, women, and boys earned &39. 60. in 7 days. Each boy received 2a. a day, each woman 30. 6/., and each man 40. 6/. How many were there of each ? Find the square root of 531 '065 to five places of decimals. 3. How many yards of matting 2 ft. 4 in. wide will be required for a square room, whose side is 9 ft. 4 in. ? and what will be the price of it at 20. 3^. per yard ? Find the value of 33 cwt. 3 qr. 7 Ib. at 6. T$. &/. per cwt. 4. If 4,000 men have provisions for 190 days, and if after 30 days 800 men go away, find how long the remaining provisions will serve the onmber left. CALCUTTA ENTRANCE PAPERS 363 5. At what rate per cent., simple interest, will 1462. 8a. amount to 1*1725. 120. in 4 years ? 1877. I. Simplify - *1- f * -y-t^f - %, and find the value of ^ ' 4-f of 5* *iV- 1. is. 4 2. Find, by Practice, the value of 739! maunds of sugar at Ri23i. 40. per hundred maunds. 3. Find the discount on ^453. 1 5*. due 6 years hence at 3$ per cent. per annum. 4. A man sells 3 per cent, stock at 75, and invests the proceeds in 5 per cents. ; at what rate must he buy them in order that his income may be the same as belore ? 5. If 7 men and 5 boys can reap 168 acres in 1 8 days, how many days will 15 men and 5 boyb take to reap 700 acres, one man being able to do three times as much work as a boy ? 6. In a rectangular area, loo yards long and 50 yards broad, there are two paths crossing one another, each parallel to one side of the rectangle, and each 4 yards broad. Find the cost of paving the area with atone at 120. per square yard, and of covering the paths with gravel at 6a. per square yard. 1878. I. Calculate to three places of decimals the value of - * ^ . ' 2. Calculate to five places of decimals the square root of I -t-(*o67) 8 . 3. Reduce ft 48 3. 120. 6/. to the decimal of R 1,290. la. 4/. 4. Find the simple interest on R757. 4<. 3^. for 343 days at 3J per cent, per annum. J. Add together, T } , inVi, Tfiau T$I Express your answer as a decimal. 6. Find, by Practice, the value of 99 cwt. 3 qr. 27 Ib. at 5. 25. 607. per cwt. 1879, I. What is the local value of each of the figures composing the numHei 2. R49 was divided amongst 1 50 children, each girl had 80. and each boy 40. ; how many boys were there ? 3. Simplify : (a) 8- 364 ARITHMETIC (0 I56x-47i^27. (d) What decimal of 4. 3*. 4^. is t ^, of 5. 8s. 4^. ? 4. A tank 72 yards long, 50 yards broad, and II feet deep, is full of water ; how many times can each of 16 water-carts, length 5 ft., breadth 5 ft., and depth 27 inches, be filled from the tank before the water in it falls; 6 inches ? f 5. If 17 men can build a wall loo yards long, 12 ft. hi^h and 2$ ft. thick in 25 days, how many will build a wall twice the size in half the time ? 6. Find the change of income when a person transfers 2,616. 5*. from the 5 per cents, at 95^ to the 4 per cents, at 83, brokerage as usual. 7. In a game of skill A can give B, and B can give G, 10 points out of a game of 50 ; how many should A give C ? 1880. I. Express each of the figures composing the number 123*456 as a multiple or sub-multiple of 10. What fraction must be added to 2$ + yJli_ 2 j. o f ^ t ^ at t ^ e sum rr.ay be equal to 3 ? 2. (a) What fraction of | of 187. $a. is R28. 8a. ? (b) Of what sum of money will '325 be 13 ? (f) Extract the square root of 7 '0225. 3. Divide i 27. 8j. among 2 men, 3 women and 7 boys, giving each of the boys $ of what a woman receives and each of the men twice as much as a woman. 4. A leaky cistern is filled in 5 hours with 30 pails of 3 gallons each, but in 3 hours with 20 pails of 4 gallons each, the pails being poured in at intervals. Find how much the cistern holds, and in what time the water would waste away. 5. A race-course is a mile long ; A and B run a race and A wins by 10 yards ; C and D run over the same course and C wins by 30 yards ; B and D run over it and B wins by 20 yards ; if A and C run over it^which should win, and by how much ? 6. A tradesman puts two prices on his goods : one for ready money t the other for 6 month's credit, interest being calculated at 12^ per cent. per annum. If the credit price of an article be R26. 90., what is its cash price ? 1881. I. What do you mean by Multiplication ? Define quotient^ factor^ expression and dimension. CALCUTTA ENTRANCE PAPERS 365 2. Add together J, f , }, , $, and f ; and simplify 3. What decimal of R45 is 35. 2a. 6/. ? Find 'the value of -- Zl O f gi annas. 0015 * 4. Express 37*8463 as an improper vulgar fraction in its lowest terms j and find, correct to 4 places of decimals, the result of dividing the square root of this number by the square root of 1 1. 5. A man who has a certain capital calculates that if he invest |t in 3! per cent, stock at 91 his income will be ^25 more than if he invest it in 3 per cent, stock at 88. What is his capital ? 1882. 1. The quotient arising from the division of 6739546^ a certain number ib 1559 and the remainder is 3107 ; find the divisor. 2. Subtract f of $ of , 4 T of ^31. 3*. from f of T \ of $ of ,100/16*, 8*/., and express the remainder as the decimal of 10. 8j. 4<f. 3. Seven bells begin to strike simultaneously and strike at intervals of 2, 3, 5, 15, 21, 65, 77 seconds respectively. After what time will they again strike simultaneously, and how often will each have struck ? 4. (i) Simplify ~~-HA of of f|). (ii) Find the value of J to five places of decimals. 5 A besieged garrison consists of 300 men, 120 women, and 40 children, and has provisions enough for 200 men for 30 days. If a woman eats $ as much as a man, and a child as much, and if after 6 days 100 men with all the women and children escape, for how long will the remaining provisions last the garrison ? 6. A person begins to speculate with a certain sum of money ; in his first transaction he loses }th of this sum ; in his second he gains 10 pet cent, on his investment ; in his third he loses ^ths of the sum invested ; in his fourth he gains 66f per cent. If he then has ftio,ooo, with what sum did he start ? 1883. i. Divide 2 J + 8 X T - J of (7 J - 3i) by 1 1 + L_. 2. Divide the square root of 122*257249 by '36856 and multiply the quotient by the square root of '00062$. J66 ARITHMETIC 3. What decimal of a square yard is 9 square inches ? Add together 1*032 of RS., '64 of Rl'25 and *o8 of half a rupee. What is the value of ,10-5416? 4. Find, by Practice, the value of 6 tons 3 cwt. 21 Ib, 14 oz. at 3. IOJ. per ton. 5. If it costs R20O to build a wall 6 ft. high by I ft. 3 in. broad by 166 ft. 8 in. long, what will be the cost of building a wall 3! ft. by i| ft. by 115 ft. ? 6. When will the interest amount to the principal at 3^ per cent, per annum ? What will the interest on Ri5o at one anna per rupee per month amount to in 5 years, and how much is that rate per cent, per annum ? 1885. 1. Of what number is 2| the fth part ? By what fraction must ~- of + ~ --- f be divided in order to * t + i% 7z give a quotient = | ? _. ... *I2 of ('0104- 'oo2)-f*36x '002 2. Simplify -- ^T.7 2 -- ; and express your result as a fraction of *6. Reduce of i6j. 4}*?. to the decimal of l. gs. loJV. 3. What circulating decimal multiplied by ' will give 2 for a product? 4* If -428571 of a barrel of beer be worth '72 of 2. ios. 9 what is the value of '625 of the remainder ? 4. Find the price of 10 Ib. n oz. 16 dwt. 16 gr. of gold at 3. 175. per oz. Extract the square roots of 9$ and ~~ to 4 places. 5. If 27 men can perform a piece of work in 15 days, how many men must be added to the number that the work may be finished in three -fifths of the time ? I buy a horse for 40 and sell it for 45 at a credit of 8 months. What do I gain per cent., reckoning morey worth 6 per cent, per annum ? 6. Which is the better investment, bank stock paying 10 per cent, at 319 or 3 per cent, consols at 96 ? What will be the cost of 1,500 3 per cent, consols at 89!, brokerage being | per cent. ? What rate of interest will such investment obtain ? 1886. CALCUTTA ENTRANCE PAPERS #7 *.** SO. 3. Reduce i. us. io^/. to the fraction of j. l8j. 6%d. What fraction of 10 must be added to 16. los. $d. to make it 20 ? 4. What decimal of 9 mds, 20 seers is J of 7 mds. 5 seers ? Reduce 5J sq. yds. to the decimal of an acre. 5. Find the value, by Practice, of 2 tons 15 cwt. 35 Ib. at 13. 6s. &/ per ton. 6. What sum of money at 4 per cent, simple interest will secure the same income as R25475 at 4^ per cent. ? 7. If a rupee is equivalent to is. 6^/., what is the price of a sovereign in rupees ? If, after buying 250 sovereigns at this price, I sell them again when the rupee is equivalent to is. 6d., how much shall I gain or lose by ihe transaction ? 1887. I. Simplify : . (M i '83 +2 0416 + '3 -3* i -002 5 + -0625 -i T V * 2. Express f of 71. 6d. + 1*25 of 5*. - *545 of 95. 2d. as a decimal fraction of 10. 3. (a) Find, by Practice, the value of 5 tons 5 cwt. 2 qr. 17$ Ib. at $. 6s. &/. per ton. (d) Find the income on which the income-tax at 5^. per rupee Is 8:52. la. 4p. 4. If 50 men can do a piece of work in 12 days, working 8 hours a day, how many hours a day would 60 men have to work in order to do another piece of work twice as great in 16 days ? 5. If ft45o amount to 8:540 in 4 years at simple interest, what sum will amount to R637. 8a. in 5 years at the same rate ? 6. Extract the square root of 177 '1561, and of '2 to 3 decimal places. 1888. I Simplify I. bimpliiy 2. Divide 16*016 by '00143, and extract the square root of 1440*9616. 3. Add together 55*5002, 3'if, 4*506 and 75*271, and find the value of the following 7365 of ,3. 6s. & +'504 of 15. I2s. 6d. -f 2*102083 of^5 4, Find, by Practice, the value of 2 tons 7 cwt. 3 qr. n Ib. at 21* 12;. 6d. per cwt. 368 ARITHMETIC 5. A man can walk 600 miles in 35 days, resting 9 hours each day ; how long will he take to walk 375 miles if he rests 10 hours each day, and walks 1} times as fast as before ? 6. If the interest on money be one pie per rupee per month, what is the rate per cent, per annum ? A man holds 15^ shares of a bank, and receives 19. is. 3</. per quarts* . If the interest he receives be 5 per cent, per annum, find the value of a share. 1889. 1. Multiply '0069347 by 7439*6. 2. Divide 2100*006983 by 243*5846 correct to five places of decimals. 3. Find in any way the value of 1,347 cwt. 3 qr. and 21 Ib. at j3. 171. io\d. per cwt. 4. Extract the square root of I +(*o634) 8 to six places of decimals. 5. Find in English money the value of ft 100,000 at is. 4/1^. pe* rupee. 1890. I. Simplify 2? of -IJlr.?* - + 3 f +*& and find, by Practice, the I ~ II ~~ value of 3,049 articles at &7. 130. 7A each. 2. Divide 27*03 by '0037, and reduce *75-*ioi-*27 to a vulgaff fraction. 3. Find the cost of putting a fence round a square field whoie area is 13*225 acres at Ri. I2a. per yard. 4. A piece of work can be done in 72 days by 17 men wording together. If after 9 days of work those are joined by 4 others, in how many days will the work be finished ? 5. Find the price of 4$ per cent. Government Promissory Notes when an investment of 59,422. 8a. produces a monthly income ot 213. 122. 1891. I. Simplify the following expression : 4-- 2-- 2. Find the value of 2 '4607 x 'o6-3*75x '012 -f 2 '163 -Tr 1*03. 3. Find the value of 15 cwt. 3 qr. 9 Ib. at ft25 1 20. 7/. per cwt. 4. If a man walking at the rate of jj miles an hour, walks to a placer in 4 hours 30 minutes, how long will it take a man, walking at the rate of 3t miles an hour, to walk there and back ? CALCUTTA ENTRANCE PAPERS 369 5. A man invests a certain sum in 4} per cent. Government Paper mt 104. The price falling to 101, he sells out and loses R6oo by the transaction, exclusive of brokerage. Find the sum invested. 6. A gives B 10 yards' start and C 15 yard's start in a race of loo yards ; how much should B give C in 150 yards ? 1892. f Q imnlifv 3*- ' Simphfy 2. Find, to the nearest integer, the value of 39*37 x 760 x iv$96 i '293 x 12 3. Find the square roots of "097344, of '009604, and of '996004. 4. Find the interest on 10 lakhs of rupees for 10 days at 4} per cent. per annum. 5. 3,000, which I held in the Four per cents., was sold for me when they were at 82$ by a broker whose commission is | per cent. ; and the proceeds were re-invested by him in the Four and a half per cents, at 98}. What amount of the latter stock did he purchase ? 1883. I. Simplify : 2. Divide 1*84626 by 23 '4. Express '456 and '654 as vulgar fractions reduced to their lowest terms, and their sum as a circulating decimal. 3. Find the cost of 73 cwt 3 qr. 14 Ib. at 4. 13*. 6J. per cwt. 4. Distinguish between true discount and banker's discount. Find the former in the case of a bill for R3486. 6a. Sp. due 16 months hence, the rate of interest being 5} per cent, per annum. 5. A man invests ft 1 63000, part in Government 4 per cent, stock at 108, and the remainder in Municipal 5 per cent, debenture stock at 109$. Find how much he must invest in each in order that he may have an equal income from the two sources. 1894. 1. In a compound metal containing tin and copper only, the propor- tion of tin to copper is 7 7 5 1092-25. Find to the nearest, penny the value of 8 cwt. 3 qr. of it. Tin costs 140^ ; copper 8o/. per ton. 2. A rectangular court is 50 yards long and 30 yards broad. It has paths joining the middle points of the opposite sides of 6 feet in breadth mad also paths of the same breadth running all round it. The remainder is C* A. 24 570 ARITHMETIC covered with grass* If the cost of the pavement be if. &?. per square foot and the turf 3*. per square yard, find the cost of laying out the court. 3. Find the value of ^267187 j of 3 in shillings, pence, and decimal of a penny. 4. Find the square root of I -(0678) to four places of decimals. 5. At a cricket match, a contractor provided luncheon for 24, and fixed the price to gain 12$ per cent, on his outlay. Three persons were absent. The remaining 21 paid the fixed price, and the contractor lost 2 rupees. What was the charge ? 1895. 1. Find the square root of I -f}(*O345) s correctly to four places of decimals. 2. Find the sum of money which put out at simple interest at 2j per cent, per annum will in 134 days exactly produce 124. loa, i{\\lp. (A year contains 365 days.) 3. If one pound sterling be worth twenty -five francs and sixty cen- times ; and also worth six thalers and twenty stlber gcoschen ; how many francs and centimes is one thaler worth ? [Af. B. One thaler =30 silber groschen. One franc =IOO centimes.] 4. Simplify Hi 5. I invest Ri28o5 * n the four per cents, at 98$ , and when they have risen to 102 $ I sell out and invest in the 4^ per cents, at 105! ; what is the change in my income ? (Brokerage f per cent, on all transactions.) * Or convert HJf into a decimal fraction, pointing out accurately the recurring portion (if any). 1896. 1. What greatest number and what least number can be subtracted from 23759143 that the remainders may be divisible by 24, 35, 91, 130, and 150 ? 2. (!) Simplify (2) Divide '0023465 by -03125. 3. Extract the square root of 5f correct to 4 places of decimals. 4. Find the simple interest on 42 3 5. I2a. $$p. for 3 years and 7 months at 3} per cent, per annum. 5. If by selling a horse for R 1 100, I lose 18 per cent. ; how much per cent, should I have gained or lost, had it been sold for 1320 ? CALCUTTA ENTRANCE PAPERS 37! 6. A man invested the same sum in two different stocks, 3} per cent* Government Securities at 103$ and 4 per cent Municipal Debentures at 205 ; his income from one is R93 more than from the other ; what sum was invested in each stock ? 1897. I. Reduce of , cwt . 3 7 lb- 4, thc T* decimal of 2} tons. (a) Find the vulgar fraction equivalent to the recurring decimal 'ijj, without assuming any rule. 2. What do you understand by an aliquot part of a quantity ? Is an area equal to 15! sq. yd. an aliquot part of an acre ? Find, by Practice, the income-tax on Ri2$o. ioa. 8/. at the rate of 5 pies per R. 3. What is meant by the ratio of one quantity to another ? What is a 'proportion ? 320 people dine together 4 days a week, but on the remaining 3 dayt some are absent ; the consumption of food is thus reduced, for the whole week, in the ratio of 109 to 1 12. Find the number of absentees. 4. In what time will ft 3 5 46 amount to 8:7683 at 3} per cent, simple interest ? 5. A person has stock in the 34 per cent. Government Securities, which yields R2$56 a year. He sells out half of the stock at ioa|, and invests the proceeds in Howrah Mills shares at 153. What dividend ought the latter to pay that he may thereby increase his annual income by R330 ? 6. Extract the square root of 3*14159 to 4 decimal places. 1898. 1. What is that least number which being divided by 48, 64, 72, 80* 120 and 140 leaves the remainders 38, 54, 62, 70, no and 130 respectively? 2. (a) Simplify of | +*)* of H-. (b) What decimal of 2. 131. 4^. is '0625 of 2'6 of i. 6s. 8aT. ? 3. Extract the square root of 54756, also of (4*02)* to 4 places of decimals. 4. What sum will amount to R3OO in 3$ years at 6J per cent, pet annum simple interest. 5. A grocer buys 480 mds. of sugar for 6135 payable at the end of 3 months, and on the same day sells them at Ri2. na. per maund ready -(money. What per cent, does he gain or lose by the transaction, reckoning interest at 9 per cent per annum ? 373 ARITHMETIC 6. One-third of a certain capital is invested in the 3} per cent. Government Securities at 105, one-fourth in the 3 per cent. Government Securities at 97}, and the remainder in the 4} per cent. Calcutta Municipal Debentures at 112}. If the total annual income is 830, what is the capital ? 1809. I. Find the greatest number which will divide 1028, 1629 and 2130 leaving the remainders 3, 4 and 5 respectively. (3) Prove that '2J4=|f| without assuming the rule of converting a recurring decimal into a vulgar fraction. 3. Find, by Practice, or otherwise, the value of 7 tons 2 cwt. 2 qrs* at R3* 2a. per maund, assuming that I ton is equal to 27 J maunds. 4. Extract the square root of 51076 and of '051076. $. A grocer mixed 20 maunds of one kind of rice at R4 a maund with a certain quantity of a second kind of rice at 3. 8<z. a maund, and selling the mixed rice at ft$. 120. a maund, gained RIO.JJ Find how much rice he mixed, and the gain per cent, on his outlay. 6. Find the discount on Ri2i8 due six months hence at 3 per cent. per annum simple interest. 1900. I. What do you understand by the Greatest Common Measure and the Least Common Multiple of two or more whole numbers ? Nine belb begin to strike simultaneously and strike at intervals of I, 2, 3, 4, 5, 6, 7, 8, 9 seconds respectively. After what interval of time will they next strike limultaneously ? / \ Q5 M is<;, ' frfc-3* of 2* 2flrof4f + |ofnt 67 . . <)Sunphfy -x-- -- -.,* (3) Reduce '0416 to its equivalent vulgar fraction in its lowest terms, and explain the reason for the process you employ. 3. Find the value of (1-25)* +2-25 x(i-25) 9 +375 x(75) a + ('75)% without reducing the decimals to vulgar fractions. 4. The length, the breadth and the height of a room are 25 ft. 7 ins.* 90 ft. 5 ins. and 14 ft. respectively. Its walls are papered at 3*. 6J. a sq> yd. and its ceiling painted at is. id. a sq. ft^JFind the total cost 5. The subscriptions to a certain memorial fund amount to R$76. 90. and each person subscribed as manny annas as there were subscribers altogether. Find the number of subscribers. 6. Explain clearly what you mean by saying that the 3$ per cent. Government Securities are at 101. A person invests Ri9,7oo in the 3} per cent ^Government Securities at 98), and when they rise to ioij he sells out and invests the proceeds in the 4} per cent Calcutta Municipal Debentures at 114^. Find the hinge in his income. CALCUTTA ENTRANCE PAPERS 373 1901. 1. (a) Simplify ~^ + ~j of f~- '583 x '142857, expressing your answer as a decimal. () Reduce 3. 15*. 4< to the decimal of Rioo. [jfl^RlS-] 2. (a) What is meant by an aliquot part of a number ? Is 2} yds. an aliquot part of a mile ? (b) Find, by Practice, or otherwise, the value of 25 tons 15 cwt. 3 qrs. 17 Jib. at 2. 131. 4^. per ton. 3. If the fourpenny loaf weighs 3 Ib. 9 oz. when wheat is at 9* 4^ per bushel, what ought the sixpenny loaf to weigh when wheat is at iu. id. per bushel ? 4. (a) Define Interest. What do you understand by the expression Rate per tent, per annum ? (b) At what rate per cent, per annum simple interest will 20* amount to ^236. 135. 4^. in 4 years 7 months ? 5. Extract the square root of 7468 '4 164. 6. A man invests one-third of his capital in the 3} per cent. Govern- ment Securities at 96^, and the remaining two-thirds in the 4? per cent. Calcutta Municipal Debentures at 105$. If the difference of the two annual incomes be Ri997, find his capital. 1902, I. How can you ascertain whether a given vulgar fraction can be reduced to a terminating or a recurring decimal without actually convert- ing it into a decimal ? What kind of decimal will the fraction jHi* pioduce ? (b) Simplify . 2 cwt. 2 qrs. 21 Ib. 3 - 6 - 5 7+1 and reduce the result to the decimal of i'x. 2. The area of a rectangular field whose breadth is 500 yards is 100 acres. Find the cost of cultivating it at R3. 2a. 8/. per loo square yards and also the cost of fencing it round at R2. 8a. per yard. 3. If 12 men and 15 boys can do a piece of work in 30 days working 7} hours a day, how many boys must assist 21 men to do a piece of work twice as great in 25 days, working 9 hours a day ? (3 men = 5 boys.) 4. Extract the square roots of 5 T V and 76*195441* 5. (a) Define discount. (b) Find the discount on 700 due 3 years 4 months hence at 5 per cent, per annum simple interest. 374 ARITHMETIC 6. Which is the better investment, the 3J per cent. Government Securities at 95!, or the 4 per cent. Calcutta Municipal Debentures at ioi ? What will be the difference in the annual income by investing R22,I27 in each of them ? 1903. 1. (a) Simplify : 67 x '67 x '67 - 67 x '67 + '067 + -oi (3) What decimal of a mile is a yard ? 2. (a) What is meant by the aliquot part of a number ? Is an acre an aliquot part of a square mile ? (b) Find, by Practice, or otherwise, the price of 25 tons 12 cwt. 3 qrs. 17 J Ib. at 6/. 13*. $d. per ton. 3. Three taps A, B and C can fill a cistern in 5, 6 and 7$ minutes- respectively. They are all turned on at once ; but after one minute, A is turned off. How much longer will B and C take to fill the cistern ? 4. (a) Define the square root of a number. (b) Extract the square root of 10^1 and of 2? to four places of decimals. 5. A man buys wine at 5*. a gallon ; he mixes it with water, and by selling the mixture at 4;. a gallon gains 12\ per cent, on his outlay. How much water did each gallon of the mixture contain ? 6. (a) Define Present Worth. () A tradesman marks his goods with two prices, one for ready money and the other for 3 months' credit, allowing interest at 4} per cent. per annum. If the credit price be marked at 50. 90., what ought to be the cash price ? 1904. 1. Define the G. C. M. and the L. C. M. of two or more numbers. (a) Find the greatest number of six digits which is exactly divisible ty 27, 4S 72 and 96. 2. Write down the local value of each of the figures in the number 010203. 3. A can do a piece of work in 25 days, B in 20 days/ and C in 24 days. The three work together for 2 days, and then A and B leave ; but C continues, and after S| days is rejoined by A, who brings D along with him, and these three finish the remainder of the work la 3 days* In what time would D alone have done the whole work ? CALCUTTA ENTRANCE PAPERS 375 4. The area of a square cricket field is 9 ac. 3 ro. 8*16 po. ; find the length of a side. 5. Define Discount. (a) The difference between the interest and the discount on a certain sum for 3 years 4 months at 5 per cent, 'per annum is 16. l$s. $d. Find the sum. 6. A person invests a certain sum in the 3} per cent. Government Securities when they are at 97f : had he waited till they had fallen to 97^, he would have had Rs. 400 more of Government Securities. How much money did he invest, J per cent, being charged as brokerage in both cases ? 1905. 1. When is one number said to be a measure of another ? What is a Prime Number ? A man bought two heaps of mangoes, one for Rio. $a. and the other for Ri8. oo. 9/. If the price of each mango be the same, and not less than two and not more than three annas, find the total number of mangoes he bought. 2. (i) What is the meaning of J and of of } ? (2) Simplify : 3. Extract the square root of 19*951 and of | correct to three places of decimals. 4. Find the cost of paving a pathway 6 ft. wide, round and imme- diately outside a flower garden, 21 yds. long and 10 yds. broad, at $J pies per sq. yd. 5. Find the price of 35 mds. 13! srs. of rice at R3 2<x. per maund. If it is sold at the rate of R3. 3$a. per maund, what is the profit per cent. ? 6. I pay R45900 to a Bank for a Bill of Exchange payable in London. The rate of exchange is i j. 4aT. for the rupee and the Bank charges me 2 per cent, on the amount payable in England. How much will my agent in London receive ? 1906. 1. (l) When is one number said to be a multiple of another ? How can you ascertain by inspection whether a given number is a multiple of 3 ? (2) What is the greatest number consisting of five digits which can be added to 8321 so that the sum may be exactly divisible by 15, 20, 24, 27, 32 and 36 ? 2. (i) What is the meaning of of J ? Give an illustration. 376 ARITHMETIC (2) Simplify; (a) iS'9-4'1 3 The cost of matting a room 1 6 ft. broad and 12 ft. high at 30. per q. yd. is Ry. oa. 4^. What will be the cost of papering its walls at the lame rate, allowing for six doors, each 6 ft. by 3 ft. ? 4. Extract the square root of '027 and of f correct to four places of decimals. 5. A book sent from England costs me (including Ri. 2a. postage) Rl2. I a. But my bookseller allows me a discount of id. in the shilling on the published price. What is the published price in English money, the rate of exchange being I s. ^d. for the rupee ? 6. Define Present Worth. A man bought a horse for 30 guines and sold him immediately for 36. I*, payable at the end of 6 months. If interest be reckoned at 6 per cent, per annum, find his gain per cent, upon the transaction. 1907. 1. What do you understand by the G. C. M. and the L. C. M. of two or more integers ? What is a prime number ? Find the least number which is exactly divisible by 12, 34, 56 and 78. 2. Simplify : '2 X '2 X *2 + *O2 X -O2 X *O2 . 2 \ - I l6 * -6 x -6 x '6 + -06 x *o6 x -06 ~ 2-3+ \\ " Rs. 2. gg. 6p. l hr. 16 m. 4$ sec. * ^Rs. 3. I2<*. 2 hr. 7 m. 45 sec/ 3. Find the price of 8 mds. 16 srs. 2 chks. of rice at R5. ga. per maund. 4. How many paving stones, each of them I foot long and 9 in. wide* will be required for paving a street 30 ft. wide, surrounding the outside of a square grass plot, the area of the grass plot being 10 acres ? 5. If 8 men or 15 women can earn Ri2o in 30 days, how much can 21 men and 24 women earn in 45 days ? 6. The debts of a bankrupt amount to 2134. IQJ. 6d. and his assets consist of property worth gi6. $s. $d. and an undiscounted Bill of ^513 due 4 months hence, simple interest at 4 per cent. How much in the pound can he pay to his creditors ? 1008. I* (i) When can a vulgar fraction be converted into a terminating decimal ? What kind of decimal will the fraction \\\ produce ? (2) Simplify: CALCUTTA ENTRANCE PAPERS 377 iJ < A K^T- of 2 ' Iom ' 3 ^- of Rs. 2. 8a. * __ - 3 -6 +-15x4 -7-24 -2 -i 17*5625 of 2md. 2oJ si. 2. Find, by Practice, or otherwise, the value of 5 acres 3 roods 7 poles "Si sq. yds. of land at 161. 6s. Sd. per acre. 3. (a) The hands of a clock coincide after every 66 minutes of correct time. How much is the clock fast or slow in 24 hours ? {) A race course is 440 yds. long. A and B run a race and A wins by 5 yds. ; B and C run over the same course and B wins by 4 yds. ; C and D run over it and D wins by 16 yds. If A and D run over it which would win and by how much ? 4. (a) What number multiplied \xy itself will produce 4||J ? () Extract the square root of f- correct to four places of decimals* 5. A trader allows a discount of 5 per cent, to his customers. What price should he mark on an article the cost price of which is 712. 8a., so as to make a clear profit of 33$ p. c. on his outlay ? 6. A person invests rupees 44100 in the 3^ p. c. Government Securi- ties at 98 and when they rise to 98 he sells out and invests the proceeds in the 5 p. c. Calcutta Municipal Debentures at ilo}{. Find the altera- tion in his income. ALTERNATIVE QUESTIONS. 2. A reservoir is 25 ft. 5 in. long and 12 ft. 10 in. wide. How many gallons of water must be drawn off to make the surface sink I ft. ? (A cubic foot of water weighs 1000 'ounces and I gallon = 10 Ib. avoirdupois). 4. The discount on a certain sum due 2 years hence is R638. 8a. and the interest on the same sum for the same time is RyiS. $a. Find the sum and the rate per cent, per annum. 1909. 1. Multiply 62031 by 46189, and divide the product by 7429. 2. Simplify t (2) 2-142857! 4- '07692307 x 2-3. 3. Find, by Practice, the price of 28 bags of sugar, each weighing 3 cwt. 2 qrs. I Ib., at 9. 6a. &,\p. per cwt. Or, Extract the square root of 137769*395929. 4. The area of a square garden is 10 acres. On the inside of the garden and along four sides of it there is a gravel path 5 feet wide* Find the cost of constructing the path at I anna 6 pies per square foot 37*> ARITHMETIC Or, On what capital will the interest for 219 days at 4 per cent, per annum amount to 14. 2s. 6d. ? 5. Among a certain number of children 91509 mangoes and also 83721 oranges may be equally divided. How many are the children ? Give all possible answers. Or, What profit per cent, is made by selling an article at a certain price, if by selling at two-thirds of that price there would be a loss of 20 per cent. ? UNIVERSITY OF CALCUTTA. MATRICULATION PAPERS. 1910. COMPULSORY PAPER. 1. Multiply 407566 by 800209 ; and divide 507233438305 by 670549. Or, Find the G. C. M. of 253512 and 568512 ; and the L. C. M. of 432, 720, 1152. 2. Reduce to its simplest form : (0 - M ^p. Or, A contractor engaged to finish six miles of railway in 200 days, but after employing 140 men for 60 days he found that only one and a half miles were completed. How many additional men must be engaged that the work may be finished within the given time ? 3. (i) Find, by Practice or otherwise, the value of 458 things at Rs. 8. 5 as. 4 pies each. (2) In what time will a sum of money double itself at 6 per cent. simple interest per annum ? Or, The weight of a cubic inch of water is 253-17 grains and that of a~ cubic inch of air is -31 grains. Find to three places of decimals how many cubic inches of water weigh as much as one cubic foot of air. CALCUTTA MATRICULATION PAPERS ADDITIONAL PAPER. 1. Extract the square root of 6256586734489. Or, A cistern contains 243! cubic feet of water. Find the lenth of the side of a second cistern 4 ft. 4 in. deep, with a square base, which contains 4 times as much water as the first. 2. (i) Calculate, correct to three places of decimals, the value of i + i-f + - - + &c. to infinity. I'2 I'2'3 I'2'3'4 (2) A metre is defined to be the ten-millionth part of a quarter- of the circumference of the earth, and is equal to 39*37079 inches. Find the circumference of the earth in miles. 1911. COMPULSORY PAPER. 1. Multiply 87904563 by 7059089 ; and divide the product by 998875. Or, A square grass-plot whose side is 200 yards, is bordered on the out* side by a path 10 feet wide. Find the cost of gravelling the path at fts. 2. 8 as. per 100 square feet. 2. (i) Simplify : j4^ir ^4*857 f i^. (2) What decimal of a rupee is a pie ? Or, What decimal of an hour is a second ? 3. (i) Find the value of 5 mds. 25 seers 10 chts. of milk at Rs. 5, 10 as. 8 p. per maund. (2) What sum of money must be put out at 3} per cent, per- annum, simple interest, in order to amount to ^248. iSs. g<f. in 2\ years ? Or, A contractor undertakes to execute a certain work in a given time ; he employs 55 men, who work 9 hours daily ; when \ of the time has expired, he finds that only f of the work is done ; how many men musk he now employ II hours a day to fulfil his contract ? 38o ARITHMETIC ADDITIONAL PAPER. 1. Find the square root of 220191808516, Or, 26 1 9 '46 7 83041. Or, A general wishing to arrange his men, who were 335250 in number, into a solid square, found that there were 9 men over. How many men were there in the front ? 2. (I) Find a decimal that is within looooo 113 Or, Find the value (correct to five places of decimals) of (2) Assuming a metre to be 39! inches, find the nearest whole number of litres in one cubic foot. 1912. COMPULSORY PAPER. 1. Multiply 814703 by 703692 ; and divide 246741768 by 75318. Or, Reduce to its lowest terms 142593 514199" 2. (i) Reduce to the Simplest Vulgar Fraction a -46 -a -j6 4 j 'S + -12J 19* (2) Find the value of 3 cwt. 3 qrs. 14 lb. - Or 9 (1) At what rate per cent, simple interest will 440, 6*. Set. amount to 511. 17*' 9^' .in 5 years ? (2) Find the price of 12 maunds 8 seers 4 chhataks of Ghee at Rs. 36. 405. per maund. 3- If the wages of 45 women amount to ^207 in 48 days, how many men must work 16 days to receive 76. iy. 4**., the daily wages of a maa being double those of a woman ? CALCUTTA MATRICULATION PAPERS 381 Or, A rectangular Court-yard 100 feet long by 80 feet wide has within it a gravel path S feet wide running round it. Find the area of the path, and the cost of gravelling it at 50. 3^. per square yard. ADDITIONAL PAPER. 1. Find the square root of 137769-395929. A rectangular Court, three times as long as it is broad, is paved with 2028 stones, each i feet square. Find the length of the Court. 2. If a metre be 3 '2809 feet and the length of a line drawn on the earth from the North Pole to the Equator be 10,000,000 metres, find the circumference of the earth to the nearest mile. Or, Find, correct to five places of decimals, the value of '-+!. 1+I.L + I.J + I.L + .. 23 2 3 5 2 5 7 2* 9 2 1918. COMPULSORY PAPER. I. (i) Multiply 426042 by 90578. Or, Divide 5208465 by 754. (2) Find the G. C. M. of 253512 and 568512. Or, Find the L. C M. of 105, 135 and 210. 2. (i) Simplify J + f-f + 1-t-iV (2) Express in decimals the sum of Or, (1) Find what decimal of a maund is a chhatak. (2) Find the price of 432 pieces of cloth at Rs 5. Jas. 6p. each. 3. (i) If Rs. 750 amount to Rs. 873. I2<w. in 5 years and 6 month? t find the simple interestjper cent, per annum. (2) A can run 8 yards in the same time that B can run 9. They start together ; when B has ran 252 yards, how far behind is A ? 382 ARITHMETIC ADDITIONAL PAPER. . Find the square root of 29*192409. Or, Find the cost of fencing a square field of 10 acres at 6 as. 8 pies per yard. 2. A room is 20 metres in length and 10 metres in breadth. Find the number of square yards in the area of the floor, taking a metre as equal to 39*37 inches. Or, Define a prime number, and state all the prime numbers between 70 and 90. 1914. COMPULSORY PAPER. 1. Multiply 777 5 77 by 358, and express the result as a whole number and a proper fraction. 2. Find the G. C. M. of 7163 and 13091. Or, Find the L. C. M. of 48, 72, 80, 108, and 120. Or, Find the Price of 273 maunds, 33 seers, 7 chhataks of Ghee at v fts. 53. Sas. per maund. 4. Add together 0*022 of 1,0 '946 of a shilling, and 3*48 pence, and subtract the sum from 0*26 of a guinea. Express the answer in pence and the decimal of a penny. Or, Find what sum will amount to Rs. 723. oas. iQp. in 6 years and 3 months at 4j per cent, per annum, Simple interest. ADDITIONAL PAPER. 1. Extract the square root of 7 correct to 3 places of decimals. Or, Shew that 103 is a prime number. 2. Given one centimetre =0*3937 inches, find in square metres the m:ea of a floor whose length is 21 feet and breadth 10 feet 8 inches. CALCUTTA MATRICULATION PAPERS 383 1916. COMPULSORY PAPER. I. (I) Multiply 790463 by 95076. Or, Divide 277286112 by 35064, (2) Kind the G. C. M. of 253512 and 568512. Or, Find the L. C. M. of 125, 160, and 280. 5 3+-V 2. (i) Simplify 2- (2) Multiply 17*55 ty 4*004, and divide the product by 0*819. (The results are to be expressed in decimals. ) Or, ( I ) Express as a recurring decimal (2) Find the price of 729 slabs of marble at 7. na. $p. each. 3. (i) At what rate per cent, per annum (simple interest) will a sum of money double itself in 10 years ? (2) Find the cost of papering the walls of a room 12 ft. 6 in. long, 7 ft. 6 in. wide, and 12 ft. high, with half-anna postage stamps measuring \% inch by f inch. ADDITIONAL PAPER. 1. Find the square root of 170*485249. 2. The palace of the King of Babylon contained a thousand rect- angular court-yards, each 60 metres long and 54 metres broad. The court- yards were all paved with marble slabs, 18 inches long by 18 inches broad. Required the total number of slabs. (Metre = 39*37 inches. ) Or, Multiply 0*48785 by 0*85963 by a contracted method so as to obtain the result correct to five places of decimals. 1916. COMPULSORY PAPER. I. (i) Multiply 560789 by 987065. Or, Divide 823479885 by 9897. 384 ARITHMETIC (2) Find the G. C M. of 36176 and 85085. Or. Find the least whole number which is exactly divisible by i, 2, 3, 4>, 5, 6, 7, 8 and 9. a. (i) Simplify 3| a**7 .8$. 2i~*of* 204' (2) What decimal of a sovereign is a penny ? O, (1) Express 5 VsVr as a terminating decimal fraction. (2) Find the cost of 153 articles at i. 2s. Set. each. 3. (i) If I have to pay 2 pies as interest on one rupee for one month ; what is the rate per cent, per annum ? (2) If 24 men can dp a piece of work in 15 days, working 8 j hours a day, how many men will be required to do another piece of work twice as great in 17 days, working 6 hours a day ? ADDITIONAL PAPER. i. Find the square root of 0*0041409225. a. Express the value of 0*0437$ kilogram + 0*3775 gram + 0*72 milligram as the decimal of a pound Avoirdupois. [i gram =15*432 grains, and one pound Avoirdupois = 7000 grains.] Or 9 Divide 2*4494897 by 1*4142135 by a contracted method^ correct to three decimal places. 1917. COMPULSORY PAPER. i. (a) Multiply 783256 by 347816. Or 9 The quotient after division of a certain number by 372 is 273 and the re mainder is 237. Find the number. (b) Find the G. C M. of 31752 and 41580. Or, The circumferences of the fore-wheel and hind-wheel of a carriage are 9ft. II in. and 12 ft. 9 in. respectively. Find the least distance over which the carriage must travel in order that both the wheels may make a complete number of revolutions. a. (a) Simplify CALCUTTA MATRICULATION PAPERS 385 Or, Find the value of J of gs. io</. -} of 6s. gd. +$ of i. o*. 7*f. (Express the answer in shillings and pence.) n\ a- ir 'I7OI-fl6*2 (&) Simplify . * J -005 x -07 Or, Reduce < y to recurring decimals. 3. (a) Find the cost of 21 tons 5 cwt. 3 qrs. of coal at Rs. 5 pel ton. Or, Find the simple interest on Rs. 892 for 8 months at 6J per cent, pel annum. (b) By selling goods at Rs. 240 a merchant gains 25 per cent. How much would he gain per cent, by selling them at Rs. 216 ? Or, In an examination 5 2 P er cent, of the candidates fail in English and 42 per cent, fail ^ in Mathematics. If 17 per cent, fail both in English and Mathematics, find the percentage of those who pass in both subjects. 1917. ADDITIONAL PAPER. 1. Find the square root of 57592921. Or, Find to within one millimetre the length of the side of a square whose area is two square metres. 2. Calculate, to four places of decimals, the value of I+ I+ J_+_L_+ _.? + 2 2X4 2X4X6 2X4X6x8 Or, Divide '12345678 by "09876543, correct to four places of decimals. 1918 COMPULSORY PAPER. I. (I) EM*r, Multiply 390626 by 331779- Or, Find the G. C M. of 78657 and 90275. (2) A reservoir contains 218,703 gallons of water. How many cisterns, each holding 37 gallons, can be filled out of it, and how many gallons will be left in it when they are all full ? c. A. 25 386 ARITHMETIC (2) Simplify (I -4-0 362)* (0*31 +0*123 -0*0005). (Express the result in decimals.) 3. (i) Either, What will be the cost of paper 20 in. wide, at 3l< a yard, for the walls of a room 21 ft. long, 15 ft. wide, and 10 ft. high ? Or, Find the cost of 5 cwt. 2 qrs. 14 Ib. of butter at 2. 5*. 6cL per cwt. (2) Either, What sum of money will amount to Rs. 1,352. 40. in three years at 4$ per cent, simple interest ? Or, A garrison of 420 have food enough to last them 35 days. After 5 days they are reinforced by 210 men, bringing no food with them. How much longer will the food last ? 1918. ADDITIONAL PAPER. 1. (i) Find the square root of 1000014129. (2) Find the dimensions of a tank which is 2*56 metres deep and which holds 3,000 litres, the length of the tank being three times the width. 2. Either, Find the value of ~H .u-^-o. -- -f ..., correct to four places of decimals. 2 30 400 5000 r Or, Find the value of - * ' J-, correct to four places of 7345 decimals. 1919. COMPULSORY PAPER. I. Either, (i) Multiply 9080076 by 6700809. (2) Find the G. C. M. of 96577 and 448477. Or, (i) Divide 4599559845 by 90705. (2) Find the L. C. M. of 289, 323, and 361. 2. (i) Simplify - --f of R. i. loa. 8/. -0*125 of 0-16 of Rs. 23. * ir (2) What decimal of an hour is a second ? 3. (i) Find the price of 17 cwt. 3 qrs. 14 Ib. of sugar at 2. g per cwt, (2) In how many years will Rs. 5,000 amount to Rs. 6,xoOat cent, per annum simple interest ? CALCUTTA MATRICULATION PAPERS 387 ADDITIONAL PAPER. 1. Either, Multiply 5947*183 by 0*093187 by a contracted method GO as to retain four places of decimals only. Or, Find the value of the following series correct to three places of decimals : ill r i _ i r j. i . i I 1x5 1x5x9 1x5x9x13 1x5x9x13x17 '"""" 2. Either , Find the square root of o '08042896. Or, Find the cost of constructing a path 4 ft. wide round a rectangular courtyard 10 yds. long and 7 yds. broad, if each square foot costs 2a. 6p. 1920. COMPULSORY PAPER. 1. Either, (i) Multiply 80070430 by 34070080. (2) Find the G. C. M. of 47821 arid 68191. Or, (i) The dividend being 545322774 and the quotient 89706, find the divisor. (2) Find the least nnmber which is exactly divisible by vhe first nine integers. 2. (I) Simplify ^-3|-s. IO Hof*. (2) Express o'i6 of 2 cwt. 2 qrs. -f-o'i(5 of 2'6 cwt. as the fraction of one ton. Convert the fraction into a recurring decimal. 3. (i) Find the rent of 19 acres 3 roods 20 square poles of land at 4.' 55. per acre. (2) What sum will amount to Rs. 6375 in 5 years at 5 per cent, per annum simple interest ? ADDITIONAL PAPER. i. Either, Find by a contracted method the value of 0*53209853x0-43429448 correct to seven places of decimals. Or, Find correct to four places of decimals the value of V L " - . \/7 + */5 2- Either, A clock in the kitchen loses at the rate of 6*5 seconds an 'hour when the fire is alight, and gains at the rate of 3*9 seconds an hour when the fire is not burning ; but in the whole day it neither gains nor loses. How long in the twenty-four hours is the fire burning ? Or, 40 per cent, of the gross receipts of a tramway company is taken up in meeting the working expenses, 40 per cent, of the remainder goes to the $88 ARITHMETIC reserve fund, and the balance is paid away as dividends to shareholders at the rate of 3 1 ,- per cent, on their shares, the total value of which is Rs. 864000 ; find the amount of the gross receipts. II. UNIVERSITY OF MADRAS. ENTRANCE PAPERS. 1880. 1. The circumference of a circle being equal to 3! times its diameter,, find the diameter of an engine-wheel which makes three revolutions a. second when the engine is moving at 40 miles an hour ? 2. If 24 men build a wall 2^ miles long, 2 feet broad, and 6 feet high, in 146 days of 10 hours each, what length of wall 2} feet broad, and 5 wet high, will 15 men build in 365 days, working 8 hours a day ? 3. Express '345 of Ri6. oa. 8/. - '073 of R6. 40. op. as the decimal of R8. 9<** 3A 4. A person sold 86 measures of rice for Ri3- Ja. o/., thus gaining 25 per cent. ; and 154 measures at a profit of 10 per cent. Supposing he had sold the whole at a profit of 16 per cent., how much more would he have gained ? 5. The length of a room is 32^ feet. The cost of painting the walla at Rl. 140- per sq. yd. is R3o8. 2a. ; and the cost of carpeting the room at R2. 40. per sq. yd. is Ri5o. 50. Find the height and width of the room. 6. Extract the square root of 6095961. Also of '0062 to four places of decimals. 7. Five men start to walk round a race course, which is i| miles round. They walk at the fates of 3, 3!, 4 4? and $ miles per hour. How long will it be before they all meet again at the starting point ? 8. If R32,ooo, put out at compound interest, amount in 2 years to R34,279/J, what is the rate per cent. ? 9* A person leaves R6,78o to be divided among his 5 children and 4 brothers, so that after the legacy duty has been paid, each child's share shall be twice as great as each brother's share. The duty on a child's sharejis one per cent. , and on a brother's share 4 per cent. Find what amounts they respectively receive. 1881. 1. Find, by Practice, the cost of : 8 cwt. 3 qr. 12 Ib. at ft27. 40. 4^. per cwt. 7 mi. 5 fur. 165 yd. at R682. 70. 4^. per mile. 2, A loom measuring 42 feet 6 in. by 22 feet 9 in. inside, with walls 2 feet 3 in. thick, is surrounded by a verandah 10 feet 6 in. wide. Find the cost of paving this verandah with tiles measuring 4$ in. by 3 in., and costing R3 2a. per hundred. MADRAS ENTRANCE PAPERS 389 3. A bankrupt has book-debts equal in amount to his liabilities, but on ~R8,64O of such debts he can recover only 8fa. in the rupee, and on R6,3OO, only 5a. in the rupee. After allowing fti,o54. iia. for the expenses of bankruptcy, he finds he can pay his creditors I2a. in the rupee. Find the total amount of his debts. 4. Extract the square root of 2329^$. Also of T 8 to four places of decimals. 5. Reduce "036 ; '001875 ; "3909 ; and '925 to equivalent vulgar fractions in their lowest terms. 6. A sum of money in 10 years at 3 } per cent, simple interest amounts to 1*727. oa. 6p. In how many years would it amount to R84O. 2a. at 4 per cent. ? 7. Find the cost in rupees of one mile of railway, which consists of two rails each weighing 40 Ib. per yard on wooden sleepers weighing 70 Ib. each placed 2 ft. 8 in. apart. The rails cost in England 6. 135. per ton, and the sleepers 2s. ^\d. each. The rate of freight is i. 5*. per ton, and landing charges amount to ft2. 8a. per ton. Rate of exchange is. 8</. per rupee. 8. For what sum should a cargo worth R26,3I5 be insured at 7$ per cent., so that the owner may recover in case of loss the value both of cargo and the sum paid for insurance ? 9. Two trains measuring 330 feet and 264 feet respectively, run on parallel lines of rail. When travelling in opposite directions they are observed to pass each other in 9 seconds, but when they are running in the came direction at the same rates as before the faster train passes the other in 27^ seconds. Find the speeds of the two trains in miles per hour. 1882. I. What decimal fraction of a mile is 68l yd. o ft. 4 5 \ in. ? 3. The wheels of a cart are 13 ft. 6 in. in circumference. One breaks down, and is replaced by a new one, which is rather small. To test it, the owner makes a chalk mark on each wheel where it touches the ground, and tells his man to drive over a piece of level road, and to count the turns made by each wheel until the chalk marks next touch the ground at the came time. The man obeys ; but, when he returns to his master, can only recollect that one wheel made one more turn than the other. His master, however, measures the distance traversed by the cart, 360 yd., and thence Snds the circumference of the new wheel. What is it ? 4. Find the value of ~-_ / .~ r~ correct to three places of decimals. 5. (a) What is the smallest whole number which is divisible by 3$, 15, and 17 J without remainder ? () What is the greatest number which will divide 3051 and 2331, leaving remainders of 8 and 4 respectively ? 390 ARITHMETIC 6. The table below shows the marks gained at an examination in seven different subjects by a class of six boys A, B, C, D, , F. Complete the table so as to show, correct to one place of decimal : (a) What percentage of the total marks is gained by each boy ; (d) What percentage of the marks awardable in each subject is gained by the class ; (c) What percentage of the total marks is gained by the class. a ! . d 1 2 A ' OX) .9 I* ' ! 1 ! 1 fr rt n tuflg o M 1* >- . 3 i c .12 o d < W j W w o HI A ~ - . 33 27 12 95 79 I 63 3 B ~ - . 76 49 52 , 73 67 82 15 C ~- 48 69 43 61 58 85 21 /)_.... 53 41 27 1 91 61 47 23 71 62 39 ! 85 73 68 14 F _ ... _ 47 18 21 1 78 92 27 12 * 7. Divide 5 '89651 by 1375854, expressing the quotient as a decimal. 8. A Bank advances R 1,500 to a person on agreement that interest at the rate of 9 per cent, per annum shall be paid half-yearly for its use. The person fails to make any interest payment, and at the end of eighteen months, the Bank obtains judgment against him for the principal and compound interest at the rate and on the terms agreed to. Find to the nearest pie the amount he has to pay. 9. The roof of a verandah is supported by 16 teak beams, each 9 ft. long, 3 in. broad, and 5 in. deep. In the weight of a cubic inch of teak is $1 of that of a cubic inch of water, and if a cubic foot of water weighs 1,000 oz., find the weight in Ibs. of the timber in the verandah. 1883. 1. A cistern, whose capacity is 43,092 gallons, is to be filled with water by a pipe which conveys 23 gallons I qt. per minute. On account of a leakage the cistern is only just filled in 34 hours. What is the average amount of leakage per hour ? 2. I sold some goods, weighing 13 cwt. 2 qr. 9 Ib. for 72. 17 j. 7}^., gaming thereby 3 J</. per Ib. How much should I have gained per Ib. if I had sold them at $. 12s. per cwt. ? 3. If 40 men and 50 boys can do a piece of work in 6 days, working* 6 hours a day, in how many days will 8 men and 20 boys, do a piece, or work half as large again, working 7 hours a day, assuming that a mao does as much wcrk in 3 hours as a boy in 5 hours ? MADRAS ENTRANCE PAPERS 391 ^ 4. ^ Three equal circular wheels revolve round a common horizontal -axis with different velocities. The first makes a revolution in 5$ minutes, the second in 2f minutes, the third in 3^ minutes. Three marks, one in each wheel, are in a horizontal line at a certain moment. What is the shortest interval after which they will be in a horizontal line again ? 5. Find, by Practice, the cost of 475 tons of coal at 2. i6s. &/. per ton. If this is sold again for ^1,453. io*., what is the whole gain, and what the gain per cent. ? ^ 6. A and B start on a journey at the same time. B travels at yths of 4's rate, and arrives 3 hours 15 minutes after him. In what time did each complete the whole journey ? 7. If an investment of 75 becomes 78. 15*. in eight months, what sum, invested at the same rate of interest, will become 201. 175. bd. in ten months ? 8. Simplify the expression : ^ 3-39x3' 9. A and B started on a race and ran a certain distance exactly together. Then B began to fail and gave up the race when he had run 56 yards further, A having gone during the same time 320 yards. The average of the entire distances run by the two men was 1,1 88 yards. What distance bad they run together ? 1884. 2. Find, by Practice, the cost of 15 tons 1 1 cwt. 3 qr. 10 Ib. 8 or. at R93. 50. 4^. per ton. 3. Extract the square root of i*V to five places of decimals ; and divide 1-438 by -013, giving the result in decimals. 4- When the rupee is worth is. 7f</., what is the nearest sum ol Indian money equivalent to ^79. 35. *]%ct. ? 5. A tea-merchant has a rectangular space for storing tea. It is ic| ft. long, 10} ft. broad and 9} ft. high. He wishes to fill this space with packets of a cubical shape all of the same size. What is the largest size of such cubical packets that can be made to fill it exactly, and what would be the number of such packets ? 6. A starts in business at the beginning of the year with ft3,ooo. On March ist, he takes a partner B with R4,ooo. And on June 1st, he receives another partner C with 5,000. The profits at the end of the year amount to 8:1,480. What share of the profits should each partner receive ? And what is the fate per cent, per month of the profits on the capital invested ? 7. What sum of money must I invest at 4 per cent, compound interest so that I may gain &390. 3*. 2|A in 3 years ? 3Q2 ARITHMETIC 8. A tradesman has been accustomed to give his customers three months* credit, but wishes to introduce the ready money system into his business. For how much ready cash should he sell an article that he has hitherto sold for 8. 2s. 9 the rate of interest charged being 5 per cent, per annum ? 9. What rate per cent, will be received for money invested in 3} pel cent, stock at 84 ? 10. Find the cost of building the walls of a rectangular room, 20 ft. long, 16 ft. broad, and 10 ft. high, with a door 7 ft. by 4 ft., and a window 5 ft. by 3 ft., at 2\a. per cubic foot, the walls being 2 ft. thick. 1885. I. Explain how the value of a fraction is not altered when its numei ator and denominator are multiplied by the same number. 2. If the rupee is worth is. 6f</., express R6. 50. 4/. as a fraction of i ; and find the least number of rupees equal in value to an integral number of pounds. 3. State the rule for converting recurring decimals into vulgar frac- tions ; and find the value of 0*63 of 275 of 3. 2s. 6d. +0-285714' of 1-3 of 7> 5-f- 10^- - o*59 2 5 ^ 2. ids. 3</. 4. Find by any method the value of 5 cwt. 2 qr. 21 Ib. of goods at 3. 7*. 6d. per cwt. 5. The carriage of 1 7 cwt. for 52 miles on a certain railway is 8s. 4<f. ; find what will be the cost of carrying 4! cwt. for 300 miles on a rail- way on which the rate per mile is 9 per cent, lower. 6. A landlord pays I per cent, for collecting his rents and a tax of 7 pies in the rupee on what he receives after paying the collector. He has a clear rental of 8:1,831. Sa. Find his gross rental. 7. A grocer mixes four kinds of tea which cost him 5^., 41., 35., 2s. per Ib. respectively in the proportions of 2, 3, 4, 7 respectively. Find at what rate he must sell the mixture so as to gain 25 per cent, on the whole. 8. Define the terms interest, discount, and find in what time ^533. 6s. $d. will amount to 672 at 6} per cent, per annum simple interest. 9. A person invests 4800 in 4 per cent, stock at 96, and after a yeai sells out at 92 and invests the proceeds together with the interest for the year in stock at 967. How much stock does he then purchase ? 10. Find to four places of decimals the square root of Jf 5 anc * calcu- late the cost of surrounding with a fence a square field of 22} acres at 3<& per yard. XI. The population of a country increases at the rate of 7 per cent, every 10 years. What was the population 20 years ago of a country whose present population is 4,007,150 ? MADRAS ENTRANCE PAPERS 393 1886. (N.B. Answers in money must be stated in . s. d. or in R. a. p. -as the case may be, and not as fractions of i or of Ri.) Z. State and explain the rule for the multiplication of vulgar fractions. 2. Express 66. 14*. $\d. as the decimal of Ri,ooo, the rupee being worth is. 4%d. 3. Distinguish between pure and mixed recurring decimals. Find the value of 0-945 f ; 2 3 s - 6 *^- +0*37259 of i. Ss. i^d. 4. Find by any method the rent of 156 ac. 3 r. 24 p. n sq. yd. at a 4p* per acre. 5. A clock which gains 3 m. 56 s. in 24 hr. was set correctly at noon on the 1st of January 1884. Find to the nearest minute the next date at which it indicated correct time. 6. Twenty men are employed to make a tank 40 ft. long, 20 ft. broad, and 6 ft. deep. They work for 30 days and have just completed one-third of the work, when it is resolved to increase the length of the tank by 10 ft., the breadth by 4 ft. and the depth by 2 ft. How many additional men must be employed in order that the work may be completed in 30 days more ? 7. The difference between the simple and compound interest on a sum of money for 3 years at 5 per cent, is 7. 12s. 6d. Find the sum. 8. The capital of a certain railway is 1000000 in 20000 shares of 50 each, fully paid up. The gross annual receipts are 105000 of which 48 per cent, is absorbed in working expenses, 4600 goes to the reserve fund, and the remainder to pay dividend. Find what annual income a person will obtain from the investment of 4500 in the undertaking, the shares being at 62. los. g. Ice is manufactured for 6 pies a pound. Two-thirds of the quantity made is kept for sale at the factory and the remainder is sent to branch shops. If the average loss from melting of the former be 12 J per cent. and that of the latter be 25 per cent., find the gain on every ton made. IO. The average width and depth of a river at its mouth are 240 yd. -and 6 feet respectively, the average rate of flow is 3 miles per hour, and the amount of sediment per cubic foot of water discharged is ij cubic inches. Find the amount of sediment deposited annually ; and the depth of the deposit, supposing it spread uniformly (i.e., to the same depth throughout) over an area of 146 square miles. 1888. Simplify . P y 5*-4f 394 ARITHMETIC 3. Find the value of i^ of "01236 of 5. na. 8/. ; and taking the rupee as worth is. 4^., express the result as the decimal of one shilling. 4. Find by any method the value of 9 tons 17 cwt. 3 qr. 25 Ib. of coffee at 72. iSs. 4^. per ton. 5. When iron is at 3. 7J 6rf. a ton, the cost of laying a railway IO miles 2 fur. 20 po. in length with rails weighing 270 Ib. each is R67,5oo. Find the cost of laying a railway 25 miles 220 yd. long with rails of the same length weighing 500 Ib. each, when iron is at 3. 145. ^d. a ton. 6. Find the present value of ^482. 6s. io>\d. due 3 years hence at 5 per cent, compound interest. 7. When exchange is at the rate of is. e&d. per rupee, a person in Madras orders from a bookseller in England a parcel of books, the pub* lished price of which is $. The bookseller allows discount at the rate of 25 per cent, on the published price, but includes in his bill a charge of 13*. for packing, freight, &c. When the books arrive in India, a further sum or R2. 8a. has to be paid on account of landing charges and cost of delivery* If the books can be obtained from a bookseller in Madras at the rate of 9^ annas per shilling of the published piice, find how much the person loses by ordering from England. 8. A person holds forty 500 shares in a concern which pays dividend at the rate of 6 per cent, per annum. When the shares are at R675, be sells out and invests half the proceeds in 4 per cent stock at 90. With the other half he buys a house, for which he receives an annual rental cf Ri,440, subject to a deduction of 30. o,/. per rupee for repairs and taxes. Find the alteration in his annual income. 9. In a certain year a country produces 50,000,000 bushels of wheat* Of this quantity 40 per cent, is available for export at 3. 20. per bushel. In the following year the acreage under wheat has increased 20 per cent, but the yield per acre is only seven-eighths of what it was in the previous year, while the quantity required in the country has increased $ per cent. If at the same time the export price has fallen to &3 per bushel, find the increase in the value of the wheat availabe for export. 10. The population of a country is 33,264,000, and there are 99 malef> to 101 females. 2 out of every 1 1 boys and I out of every 33 girls of school-age are under instruction. If the boys of school-age form pne- seventh of the male population and the girls of school-age form one-seventh of the female population, find the total number of pupils under instruction. 1889. 2 simplify 2. Simplify 3. Multiply 41 '36514 by *ooi, expressing the result as a decimal; and find the value of '3472 of 1. 45. - <O38& of 2. 6s. $d. 4. Find by any method the cost of 79 ca. 17 m. 5 v. 25 pal. of salt R2I. ioa. 8/>. per candy. MADRAS ENTRANCE PAPERS 395 5. The cost of rice for a family of 2 adults and 3 children from January 1st, 1889, to December nth, 1889, both days inclusive, during which time rice was selling at 15*4 seers per rupee, was Ryo. ya. What will be the cost of rice for a family of 3 adults and 5 children from December igth, 1889, to May nth, 1890, both days inclusive, assuming that the price of rice will be 147 seers per rupee, and assuming also that the quantity required per day by each adult is the same in both cases, and that in both cases ths quantity required by a child is two-fifths of the quantity required by an adult ? 6. On what sum due I year 4 months hence does the true discount amount to ^100. i8j. 9^., simple interest being reckoned at 4! per cent, per annum ? 7. How much 3 per cent, stock must a person sell when the selling price is 91, in order that by investing the proceeds in the 4$ per cents, at 113! he may derive from the investment an annual income of RgSiy. 8a. ,. after paying income-tax at the rate of 5 pies per rupee ? 8. A and B can do a piece of work in 10 days, B and C in 15 days, and C and A in 20 days. They all work at it for 6 days ; then A leaves, and B and C go on together for 4 days more. If B then leaves, kow long will G take to complete the work ? 9. In a certain year the total amount received by a railway company for the carriage of passengers was R2y 51000. Of this sum 6 per cent, was contributed by first class passengers, 5 per cent, by second class, and the remainder by third class. The fares were 1 8, 6, and if pies per mile for first, second and third class passengers respectively. Assuming that the average distance travelled by each third class passenger was 36 miles, and the average distance travelled by each passenger of the other classes was 160 miles, find the total number of passengers carried during the year. 10. The length of a rectangular field is twice its breadth. If the rent of the field at 3. JS. 6d. an acre is ^151. 175. 6d , find the cost of surrounding it with a fence at ^\d. per yard. 11. Extract the cube root of 9 to five decimal places. 1890. I. Reduce 2149908480 sq. inches to acres, etc. If this is the area of a rectangle the length of which is 5 m. 7 fur. 5 p. I ft. 6 in., find its breadth. o> IT 5468 147 , c 6 '25 f Simplify ^ -f -^ -f- -?-' - 3 1- of of 7 JT ' . 2202 12303 441 JT 5'5 1-285714 3. Find the value of 237 candies 17 maunds 6 viss at R4ioo. la. \p. per candy. 4. 300 coolies aie set to build a tank-bund. In 14 weeks they have done vff of the work when rain stops the work for 4 weeks and washes away f of what they have done. At the end of that time the work is resumed with only 250 coolies. In what time from the commencement will the work be finished ? 3<# ARITHMETIC 5. Find the amount of 585937 5 for 3 years at 4^ per cent, per annum, reckoning compound interest. 6. Explain the difference between discount and interest. If the dis- count on 2830. 151. *j\d. be equal to the simple interest on 2784. TS. 6d. for the same time, find the time, the rate of interest being 5 per cent, pet annum. 7. A person invests 34539 in the 3 per cents, at 87. After receiving one year's dividend he sells out at 89. He then invests the whole in Railway stock, paying 5 per cent., at 115. What will the difference in his income be ? 8. A cistern 10 ft. 6 in. long by 7 ft. 6 in. wide by 3 ft. 4 in. high is lined inside with lead, 7 Ib. of which cover a square foot. Find the weight of the lead and its cost at 531. 4^. per cwt. 9. A cask contains 16 gallons of spirit. Two gallons are drawn off and the cask filled up with water. Two gallons are again drawn off and the cask filled up as before. This is done a third time. Compare the quantities of spirit and water remaining in the cask. 10. Find the square root of 379749833*583241. 1891. 2. Subtract 13 times Ri7. 6a. up. from 17 times Ri3- 6a. up. 3. &33O. 30. Tp. are to be divided among 193 persons, two of whom receive R2 each, and ten 3 each. The others receive equal shares. Find the value of each ihare. and simplify (without reduction to vulgar fractions if you can) 2-03 + 1 -345 +27 -34 + 16-2317. 5. How long will it take to walk round a square field 14 acres I rood 24 poles in extent at the rate of 3 miles an hour ? 6. Find the cost of white-washing a room 22 J ft. by 12 ft. and II ft, high, at one anna per square yard, making allowance for four windows each 4 ft. x 2.\ ft. and two doors each 8 ft. x 4 ft. Find also the cost of a carpet for the same room with 3 ft. border all round the carpet, costing &4 per square yard and the border R6 per square yard. 7. Find the compound interest on 3143. 6s. 8dT. for 3 years at 3 per cent, per annum. 8. A cistern can be filled by three pipes in 30, 40 and 60 minutes respectively, and emptied by an escape pipe in half an hour. The three taps are turned on at noon, but the escape pipe is at the same time accidentally left open and not closed for a quarter of an hour. At what time will the cistern be full ? 9. I purchase 16 Ib. of tea at is. Jet. per Ib., 14 at 2s. 2d. and 17 at is. %d. Seven pounds of the mixture becoming spoiled have to be sold at MADRAS ENTRANCE PAPERS 397* a low price, but by selling half the remainder at 2s. 4<f. per Ib. and the- other half at 2 j. 7 </., I eventually make a profit of 25 per cent, on the original outlay. At what price per pound was the spoiled tea sold ? 10. A person invests a sum of money in the 4 per cents, at 102. When they have risen to 104, he transfers R6ooo stock to another investment' paying 5 per cent, of which the shares are at 120. When the 4 per cents. fall to par, he transfers the reirainder to the 5 per cents, which are still at the same price and now finds his income R25 more per annum than it was at first. What was the sum originally invested ? 1892. 3. Find the value of '0416 of 33. 7s. 6d. - -0345 of 32. 135. i\d. and express R37I. 2a. 6p. as the decimal of a lakh of rupees. 4. Find by any method the cost of making a road 37 m, 6 f. 31 p. 3 yd. long at ft 1 785. 3. 4/. per mile. 5. Find the present value of 482. 6s. lojrf". due three years hence at' 5 per cent, per annum compound interest. 6. Extract the square root of 13*697142031225 to six places of decimals* 7. The annual rainfall of a district is 497 inches. Assuming that the fall is distributed uniformly over the district, and that a cubic foot of water weighs 62*5 Ib., find the weight in tons of the rain that falls through- out the year on a square mile. 8. When exchange is is. 2\d. per rupee, a Madras bookseller sends to a London publisher a bill for ^104 in payment of books ordered. Freight and landing charges amount to R37. Sa. The publisher allows the book- seller discount at the rate of 35 per cent, on the published price, and the latter sells the books at the rate of io annas per shilling of the published- price. Find how much he gains on the transaction. 9. In the year 1891, the cost of rice for a family of 2 adults and 4 children was R86. 70. 9/. In that year rice sold at 1 1 *2 seers per rupee, and each child received two-fifths of the amount given to an adult. Assuming that in 1893 the price of rice will be 13*5 seers per rupee, what will be the cost of rice for the same family from January 5 to August II, both days inclu- sive, if the allowance of each adult be increased by one-fourth and the allowance of each child be three-sevenths of that of an adult ? 10. The capital of a railway company amounts to Ri8,9o,oo,ooo of- which one-fourth is 5 per cent, preference stock and one-third 4$ per cent. preference stock. In a certain year the receipts are Rl, 81,50,000, and the working expenses amount to 55 per cent, of the receipts. Of the net receipts 1*5,40,000 are added to the reserve fund, and the remainder, after paying dividend on the preference stock, is divided among the ordinary- shareholders. What rate of interest will they receive ? 11. In the ten years from 1871 to 1881 the population of a country increased at the rate of 9*5 per cent., and in the ten years from 1881 to< 398 ARITHMETIC 1891 the rate of increase was io'5 per cent. If the population in 1891 was 31,023,759, find what it was in 1871. 1894. 2. Simplify V/ 3. Find the value of 2*04752 of 2. 2*. iff. - 1*734375 of 2. 6s. Sd. 4. Find by any method the value of 59 ca. 14 m. 7 v. 27 pal. of salt at R26. loa. 8/. per candy. 5. In a certain year the produce of a tea-estate was sold in London at an average rate of 9^/. per lb., and the amount realised was remitted at an average rate of exchange of is. 2\d. per rupee. In the following year the average price realised was only S^/. per lb., but the quantity sold exceeded by 12^ per cent, the quantity sold in the previous year and the average rate of exchange at which remittances were made fell to is. l$e?. If in this year the total amount realised from sales in London was R 1 05000, find how much was realised in the previous year. 6. A sum of money was invested for four years, interest payable an- nually. The rate of interest was 5 per cent, per annum for the first two years and 4 per cent, per annum for the last two ; and the amount at the end of four years was 1,164. IOJ 3l& What was the sum invested 1 7. Ice is manufactured for 2* pies per lb. and sold at 6 pies per lb. Of the total quantity made one-half is kept for sale at the factory, and the remainder sent to brarch shops. The loss from melting is 12\ percent, in the case of the former and 25 per cent, in the latter ; and the agents at the branch shops receive commission at the rate of 15 per cent, on the price of every pound sold by them. Find the profit on every ton of ice manufactured. 8. Two persons, A and B, set out together on a journey. They walked at the rate of 3 miles an hour ; and after they had proceeded for three quarters of a mile, B returned, walking at the same rate, to the place of starting. Here he was detained three quarters of an hour. Set-* ting out again he overtook A, who had been walking all the time, at the end of 2j hours from the second time of starting. At what rate did he walk? 9. A person sold 25 Bank of Madras shares and invested the proceeds in the Government 3^ per cents, when they were at 3^ premium. If his net annual income from the investment, after paying income-tax at the rate of 5^. in the rupee, be R876. ga. , find the price at which he sold each of his bank shares. 10. In the year 1891 the population of a country was 35640000 and there were 1025 females to every 1000 males. Of the total population 7*5 per cent, could read and write, but of the females only i per cent* could do so. Find what percentage of males could read and write. 11. Extract the square root of 81*13183159704101 to seven places of decimals. MADRAS ENTRANCE PAPERS 399 1895. jofi-ffof^-f >2 *-!of2| 2. 3. Find the value of '041962-7- "936, expressing the result as a decimal. 4. Find the value of 97 miles 5 furlongs 170 yards of wire at R34- <6a. o^. per mile. 5. When iron was at 2. 14*. 2d. per ton and the rupee at is. 7\d., fhe cost of laying a railway with iron rails weighing 50 Ibs. per yard was ^278,250. Find what will be the cost of relaying it with steel rails weighing 75 Ibs. per yard, when the price of steel is 3. 17$. 6</,, per ton and the rupee is at is. i\d. 6. The reservoir from which a certain city draws its water supply has a surface area of 2\ square miles. If the city has a population of 450,000, and if the average daily supply is at the rate of 20 gallons to each inhabitant, find what must be the average depth of the resevoir so that *when full it may contain a year's supply, (i gallon = 277 '274 cubic inches, and a year = 365 days.) 7. Find what sum will amount to 669. los. 3!^. in 2 years at 3| per cent, compound interest, payable annually. 8. The capital of a certain railway is 275 lakhs of rupees in shares of R5OO each, fully paid up. The gross receipts in a certain year amounted to 22 J lakhs, and the working expenses amounted to 48*4 per cent, of the gross receipts. Of the net receipts the sum of Ri 31, 400 was placed in the reserve fund and the remainder went to pay dividend. Find the amount of dividend received by a person holding 1500 shares, after deduct- ing income-tax at the rate of 5J>. per rupee. 9. Rupee silver is an alloy consisting of n parts of silver to I part of copper, and the weight of I rupee is 180 grains. If the price of silver is 2S. 6d. per oz. troy, the price of copper $\d. per Ib. troy, and if the rate of exchange is is. i\d. per rupee, find the total value in rupees of the silver and copper required for coining a lakh of rupees. 10. A merchant pays a lakh of rupees for a season's goods. He marks the goods 25 per cent, over prime cost, and from what he sells at this rate realises fti 12,500. At the end of the season he sells the remaining goods at reduced rates, one half at a reduction of 25 per cent, on the former prices, one quarter at a reduction of 50 per cent, and the remainder at a quarter of the former prices. If the expenses of the business amount to 12 per cent, of the sale receipts, what is his rate of profit on the transactions of the season ? 1 1. Find the dimensions of a cubical cistern having the same capacity s a tank 31 feet 6 inches long, 21 feet broad and I foot 9 inches deep. 1896. 3TV-2xj+2tof(2t-i) . Hof2A 400 ARITHMETIC 3. Prove that any number of pies can be expressed as thousandths of a rupee by multiplying the number by 5 and adding to the product ^ f itself. Apply this rule to express 6a. o./. as the decimal of a rupee, and verify the correctness of your result, obtaining it in another way. Find* the value of '0012 $76 of a lakh of rupees. 4. Find by any method the cost of constructing a railway 329 miles/ 5 fur. 176 yds. long at R77,386. 130. 4/. per mile. 5. A contractor undertook to finish a certain piece of work in 150 days. He employed 20 men, 30 women and 75 children j but at the end of 60 days, finding that only one-fourth of the work was done, he dismissed all the women and 50 of the children and employed more men. The work was then finished 5 days before the stipulated time. Assuming that 3 men could do as much as 5 women, and 2 women as much as 3 children, find how many additional men were employed. 6. Find the present value of Ri 1 1 5. 130. o/. due 2 years hence at 3j per cent, per annum compound interest. 7. How much 3} per cent, stock of 109} must a person sell in order that by investing the proceeds in 3 per cent, stock at 103^ he may derive from the investment an annual income of 6825. 8a., after paying income- tax at the rate of 5 pies in the rupee ? 8. A grocer imports sugar at 15.*. $d. percwt., the cost of which he lemits when exchange is at the rate of is. z\d. per rupee. Freight and landing charges amount to R38. 6a. per ton, and import duty at the rate of loa. 6p. per cwt. has also to be paid. At what rate per maund of 25 Ib. must the grocer sell the sugar so as to gain 12 per cent, on his total outlay ? 9. In a certain year the total value of the exports from the Presidency of Madras showed an increase of 12 '5 per cent, as compared with the total value of the exports for the previous year. Of the various items of export, coffee which in the first of these two years represented 13*59 per cent. o( the total value of the export, showed an increase of 7*5 per cent. What percentage did coffee represent of the total value of the exports in the second of ths two years ? 10. In a certain year the quantity of wheat raised in a country was 54,000,000 bushels. Of this one-third was available for export at ft2. 40. per bushel. In the following year the acreage under wheat showed an increase of 20 per cent., but the yield per acre was only three-fourths of what it was in the previous year, and of the total quantity of wheat raised only one-fourth was available for export. If the value of the wheat ex- ported in this year was ioi| lakhs of rupees less than in the previous year,, what was the export price of wheat per bushel ? 11. Extract the square root of 4985*067295890281 to 6 places of' decimals. 1897. - -* MADRAS ENTRANCE PAPERS 4^1 3. Find the value of -875 of JU* 5*- 4#-+ * 8 59375 fftl - 5* & - of 4. Find by any method the value of 21 tons. 17 cwt. 2 qr. 23 Ib. of coffee at 1547. 40. 8/. per ton. 5. When the price of grain is II measures per rupee, the cost of grain for 24 ponies for 31 days is Rl82. 2a. Assuming that 5 horses require as much as 8 ponies, find the cost of grain for 25 horses for 6 weeks when the price is io| measures per rupee. 6. The interest on a certain sum of money for 3 months at 5 per cent. per annum exceeds the true discount on the same sum due 3 months hence at the same rate of interest by iia. 3$. Find the sum. 7. A person holding sixty RSOO shares in a concern which* paid dividend at the rate of 5 per cent, per annum, sold out when the shares were at R625 and invested half the proceeds in 3 J per cent. Government pape* at 105. With the other half he bought a house, for which he received an annual rental of Ri92o, subject to a deduction of 40. 3^. per rupee for repairs and taxes. Find the alteration in his annual income. 8. 50 men, 100 women and 150 children, working for a certain time on a tank -bund earn together Rn8i. 40. If the wages of a man, a woman and a child be 40., 2a. 6/>. and la. 6p. per day respectively, find how- much is earned on the whole by each man, woman and child. 9. A merchant buys a quantity of tea at an average rate of I2a. 6p. per Ib. He assorts the tea into three kinds, which he sells at Ri. 2a., 143. and 9. per Ib. respectively. If in the process of assortment 2^ per cent. of the tea is lost, and if of what remains 36 per cent, is of the dearest kind and 24 per cent, of the cheapest, find the merchant's gain per cent, on the transaction. 10. A merchant in Madras owes 12,270 marks to a merchant in Hamburg. If exchange on Hamburg is at the rate of I -32 marks per rupee, while exchange on London is at the rate of is. tfd. per rupee, and the exchange between London and Hamburg is 20*45 marks per pound sterling, find, to the nearest pie, how much the merchant will gain by remitting, through London instead of direct. 11. The length of a field containing 21 ac. 3 ro. 25 sq. po. 3& sq. yds. is twice its breadth. Find the length of the field. 1898. -* 3. Find the value of 'iifti of R9- 130. 6/. and express the difference between if of R37. 130. 4/. and jf of R37. 2a. as the decimal of R$o. 4. Find the cost of paving a rectangular area 35 feet 9 inches lon,g and. 23 feet 6 inches broad aj the rate of R5. 130. 6>. per square yard. 5. Two watches, one of which gained at the rate of I minute 3^*0 seconds and the other lost at the rate of I minute 55*8 seconds daily, wers 402 , ARITHMETIC set correctly at noon on the 1st of January, 1896. When did the watches next indicate the same time and what time did etch indicate ? 6. Find what sum will amount to 7364. ioa. Q/. in two years at 3 per cent, per annum compound interest. 7. A person invested Ri6,5oo in 3^ per cent. Government paper at 96$, and an equal sum in bank shares of the nominal value of &5oo each. If, when the bank is paying dividend at the rate of &5p per share, his annual income from the bank shares exceeds his annual income from the Government paper by 87. 8a. , find what he paid for each of his bank shares. 8. A merchant at Madras imports 600 tons of English coal, the price of which at the pit mouth in 12s. 6d. per ton. The cost price of carriage and freight to Madras amounts to i6s. 6\d. per ton and the landing charges amount to 3. $a. 4/. per ton. If the merchant remits the price of the coal and the cost of carriage and freight to Madras when the ex- change is at the rate of if. 31$^. per rupee, at what price per maund of 8af Ib. avoirdupois must he sell the coal in Madras in order that he may gain ^3255 on the transaction ? 9. In 1896 the working expenses of a certain railway amounted to 50*8 per cent, of the gross earnings and the net earnings to 4*41 per cent. of the total capital expenditure. In the same year the working expenses of another railway amounted to $4 "9 per cent, of the gross earnings and the net earnings to 5*25 per cent, of the total capital expenditure. If the total capital expenditure on the former railway was 1222 $ lakhs of rupees, and if the gross earnings of the latter were four-fifths of the gross earnings of the former, what was the total capital expenditure on the latter ? 10. In a certain year 2*5 per cent, of the articles given out for delivery from post offices in the Presidency of Madras were returned undelivered. Next year there was an increase of 7*5 in the number of articles given out for delivery and an increase of 10*5 per cent, in the number of articles returned undelivered. If in this year the number of such articles was 1957176, find how many articles were given out for delivery in each year. 11. Extract the cube root of 1754*099916 to two places of decimals. 1899. 3. Divide '0003922130575 by '047729 ; and find the value of 2*02376 of 7. 140. # 4. Find by any method the value of 19 cds. 17 mds. 5 v. 25 pals, of sugar at 58. 50. 4/. per candy. 3. Find to the nearest pie the amount of 1750 for 3 years, reckon- ing compound interest at 3! per cent, per annum. 6. A person invested a sum of money in 3 per cent. Government paper at 91}, and when the price rose to 94} he sold out and invested the MADRAS ENTRANCE PAPERS 403 amount realised in the 4 per cents, at loij. If from this investment he derives an annual income of R2727. la. 4$. after paying income-tax ot the rate of 5 pies in the rupee, find what sum he invested in the 3 per cents. 7. A person A sets out from a place P to walk to a place Q. A quarter of an hour later a second person B sets out from P to walk to Q, but after walking half a mile returns to P, where he is detained 10 minutes. Again setting out from P he reaches Q 5 minutes after A. If A walks throughout at the rate of 3 miles an hour and B at the rate of 4 miles an hour, find the distance between P and Q. 8. In a certain year the total number of passengers carried on a railway was 12976200 and the receipts from the passenger traffic amount- ed to 45 lakhs of rupees. Of this sum 1*6 per cent, was contributed by first class passengers, 4 per cent, by second, and the remainder by third. If the fares for first, second and third class 'passengers were 12 pies, 4 pies and i$ pies per mile respectively, while the average distance travelled by each first class passenger was 60 miles, and the average distance travelled by each second class passenger was 40 miles, find what was the average distance travelled by each third class passenger. 9. A coffee merchant in India buys coffee for shipment to England, for which he pays on the average RII. 150. 3^. per maund of 25 Ib. The process of curing reduces the weight of the coffee 10 per cent, and for carriage, curing and freight the merchant has to pay at the rate of R6o per ton of cured coffee. If the coffee is sold in England at the rate of 925. 6d. per cwt. and if the amount realised from the sale of it is remitted when exchange is is. 4</. per rupee, find the merchant's gain per cent, on his outlay. 10. In a certain country the number of males who can read and write exceeds the number of females who can do so by 2459600. If the total female population is to the total male population as 1025 to 1000, and if 15 out of every 100 males and I out of every 100 females can read and write, find the total population of the country. 11. Find the edge of a cubical cistern having the same capacity as a rectangular cistern 14 feet 7 inches long, 12 feet 3 inches broad and 3 feet 9 inches deep. 1900. fraction in its lowest terms. 3. Find the value of -094921875 of Ri3. 50. 4/., and express ft2i. I34* 9A as the decimal of a lakh of rupees. 4. Find the cost of repairing a road 27 miles 6 furlongs 196 yards long at ft 1 786. 50. 8/. per mile. 5. When the price of grain was 10} measures per rupee the cost of grain for 6 horses and 12 ponies for 6 weeks was R282. loa. Assuming that the daily allowance for a pony was two-thirds of the daily allowance 404 ARITHMETIC for a horse, find what will be the cost of grain for 12 horses and 6 ponies for the months of April, May and June if the price during that period is 9 measures per rupee and the daily allowances for a horse and a pony are the same as before. 6. Find the present worth of 545. na. 8/. due 2 years hence at 4 per cent, per annum compound interest. 7. A sum of money is made up of rupees, half rupees, quarter rupees- and two-anna pieces. The number of rupees is ten times the number of half rupees, six times the number of quarter rupees and eight times the number of two-anna pieces, and the value of the rupees exceeds the value of all the other coins by R428. 8a. What is the sum ? 8. A merchant imports sugar at 17*. I id. per cwt M the cost of which be remits when exchange is at the rate of is. 4}^. per rupee. Freight and landing charges amount to R37. 80. per ton and import duty at the rate of 8a. 8/. per cwt. has also to be paid. If the merchant sells the sugar at the rate of 3. 140. 6p. per maund of 25 lb., find how much he gams per cent, on his total outlay. 9. The capital of a spinning company amounts to 15 lakhs of rupees in shares of 50 each, fully paid up. The gross value of the goods manufactured in a certain year was 31 '25 lakhs of rupees and the net profits amounted to 4 per cent, of this. If at the end of the year the sum of half a lakh of rupees was placed in the reserve fund and the remainder of the profits was distributed among the shareholders, find what rate of interest was received by a shareholder who at the beginning of the year bought shares at R62. 8a. each. 10. In two successive years the working expenses of a certain railway amounted to 48*25 per cent, and 47*1 per cent, of the gross receipts for these years respectively. If the net receipts for the second of the two years showed an increase of 3*5 per cent, on the net receipts for the first, tmd what was the rate of increase per cent, in the gross receipts. XI. A rectangular cistern has a capacity of one million gallons. The length of the cistern is twice its breadth and its depth is 7 feet 6 inches Taking a gallon to be 277*274 cubic inches,' find to the nearest inch the length of the cistern. 1901. 1. The gross earnings of a certain railway company iuring the first six months of the year 1899 were as follows : January, 61134267. 130. 5^. ; February, 1098763. ioa. lip. ; March, 1109835. 6a. gp. April, 1148239. 20. 6p. ; May, 1132470. 15*1. 8/. ; June, 1087493. i2a. 7^. Find (I) the gross earnings for these six months, (2) the average daily earnings. 2. Reduce to its simplest form the expression 3. Find the value of -018984375 of 66. ioa. 8/. and multiply -0465$ by '934, expieiiing the result as a decimal. MADRAS ENTRANCE PAPERS 4. Find by any method the value of 179 cds. 13 mds. 7 v. 33 pals, of salt at R23. $a. 4/. per candy. 5. During a certain period the yield of a tea estate containing 187 ac. I to. 28 po. was at the rate of 375 Ib. per acre per annum, and the average price realised for the tea produced was 7</. per Ib. During the same period the yield of another estate containing 257 ac. I ro. 660 sq. yds. was at the rate of 300 Ib. per acre per annum, and the average price realised was 8f </. per Ib. If the value of the tea produced on the former estate during the period in question was R2Q,452. 8a., and if the average rate of exchange was the same in both cases, what was the value of the tea produced on the latter estate ? 6. If the interest on ^1368. I2. for 45 days is R$. 140. 6>., what is the rate per cent, per annum (365 days) ? 7. A person owned house property yielding a rental of 1750 per mensem, of which he had to spend 37$- per cent, on repairs, management, taxes, etc. He sold the property, realising R365,75o which he invested in 3^ per cent. Government paper at 96^. If he has to pay income-tax at the rate of 5 pies in the rupee on the interest derived from the investment, find the alteration in his clear annual income. 8. A merchant wishing to clear out his old stock sold one lot of goods at a reduction of 12\ per cent., another at a reduction of 25 per cent, and a third at a reduction of 50 per cent, on the usual prices, He realised R682. 8tf., Rioi2. 8. and R45O for these lots respectively, and found that on the whole he had a loss of 2* per cent, on the price paid by him for the goods. What would he have lost or gained per cent, if he had sold all the goods at a reduction of 20 per cent, on the usual prices ? 9. In a certain year three-eighths of the quantity of wheat raised in a country was exported at R2. 6a. per bushel. In the following year the acreage under wheat showed an increase of 12$ per cent., but the yield pr acre was only five-sixths of what it was in the former year, and of the total amount of wheat raised only one-third was exported. If the value of this at R2. 8a. per bushel was 375 lakhs of rupees, find the value of the wheat exported in the first of the two years. 10. A merchant in Madras owing ^398. 5*. qd. to a merchant in London, finds that by remitting through Paris, instead of direct, he can save R;8. Sa. If exchange between Madras and Paris is at the rate of 171 francs per rupee and exchange between Paris and London at the rate of 25*2 francs per ^i, find the rate of exchange between London and Madras. 11. Extract the square root of 822655*9194245541 to five places of decimals. [The remainder must be written down.] III. UNIVERSITY OF BOMBAY. ENTRANCE PAPERS. 1865. i. Point and write in words, both according to the English and Indian numerations, the two numbers : 1234567654321. 5020040003060. 406 ARITHMETIC 2. Subtract R4$867. I2a. 6/. from R86325. 8a. 3/. How are the. numbers placed in subtraction ? 3. If a room is 28 feet long, 20 feet wide, 13 feet high, and thf* windows and doors take up half the walls, find the cost of papering at 120. a square yard. 4. How many square feet are there in 578 pieces of Grey Domestics 39 inches wide and 72 yards long ? and what is the price at R2O. 14^. per piece ? 5. Multiply J+i+J+i+J by i x ijx ij x ij x ij. 6. Reduce J 9t J T , J y to decimals. 7. If I sell 500, 4 per cents, at 93, and buy 5$ per cents, at 109* what is the change in my income ? 8. Divide a lakh of rupees between A, B and C, in the proportion f 2 > 3 4> and the, same amount between D, E and F in the proportion of *, land J. 9. If I sell 40 shares of R25O each in the Oriental Bank at 121 per cent, premium, how many shares of Ri,ooo each in the Madras Bank at 72 per cent, premium can I buy ? and how much will be left ? 10. A person travelled 120 miles by railway at 15 miles an hour, 120 by road at S miles an hour and 60 by bullock -cart at 2 miles an hour : how long did he take ? 11. Find the square root of I73388'96 and the cube root of 1860*867. 1866. 1. Represent in figures : Ninety-nine millions, ninety- nine thousand and ninety-nine. And by the old English method of numeration, eight billions, two hundred and seven thousand and five. Point and write in words 3x9680209078 and 20090060002, the first according to the Indian method and the second according to the English method of numeration. 2. Add together f of f and J+JJ -||, and explain why fractions must be reduced to a common denominator for the purpose of Addition and Subtraction. (a) What fraction must be divided by f to give a quotient JJ ? 3. A person who has of a mine sells f of his share for Ri,5oo ; what is the value of his share and of the whole mine ? 4. Explain why in reducing a fraction to a terminating decimal, the number of decimal places depends on the form of the denominator of the fraction and not on that of the numerator; 5. Reduce I cwt. 3 qr. 5 Ib. to the decimal of f of a ton. BOMBAY ENTRANCE PAPERS 407 6. Perform the operations indicated below : (i) 47-03-2-876843. (ii) 5776x2-003. (Hi) 62'5-M25'i25. (iv) 6 '25-7- '000125. (v) ^(2119-6816). 7. Define the terms : Stocks, shares, consols. State some of the circumstances which affect their value in the market. How much stock can be purchased by the transfer of Raoooooo from the 4 per cents, at 90, to the 5| per cents, at up ; and what change would be effected in the income derived from the two investments ? 8. Find, by Practice, the price of 549 yards at l8j. 9^/. a yd. 9. I bought cloth at 15^. a yard and lost 5 per cent, in selling ; what was it sold for ? 10. If a person owe Rioo payable in 2 months, and 750 payable in 7 months, what is the just time for the payment of the two debts ? 1867. 1. Give a demonstrative example, illustrative of the following axiom : - If the divisor be increased a certain number of times, the quotient U diminished in the same degree ; but if the divisor be diminished the quotient is increased. 2. Define prime and composite numbers. Resolve 54180 into prime factors. 3. Reduce -^~^~^\ to its simplest form. 4. Reduce \ to a circulating decimal ; and find the fraction equivalent to i'7oi6* 5. Find the product by contracted multiplication of 72 -49 and 10*87632 to three places of decimals. 6. If f of a maund of sugar cost Rio, what will | of a seer cost at the same rate ? Give answer in annas as well as in rupees. 7. Explain direct and inverse proportions. 8. 250 men are employed to work on a Railway embankment, a mile and a half long, which they are expected to finish in four weeks. But at the end of one week it is found that they have only finished 520 yards. How many more men must be engaged to finish it in the required time ? 9. What time must elapse between the time of placing 8:250 in the Government Savings' Bank and taking out the amount just as it goes over 300, supposing interest at 5 per cent, per annum, compound interest ? 10. In a school of 250 children, 44 per cent are learning Geography, 36 per cent, are learning Grammar, 12 per cent, cannot read, and 4 per cent have advanced as far as Algebra. What are the actual numbers of each ? 408 ARITHMETIC 11. Extract the square root of 6085, '00025 and ~. 5 2 '4 12. What is the cost of a marble slab, 6 ft. 3 in. long, 2 ft. 8 in. broad, and 4 in. thick at Ry. Sa. per cubic foot ? What is the weight of the slab, one cubic foot weighing 170 Ib. ? 1868. r. How many yards of matting 2 feet 3 inches wide will be required for a square room whose side is 18 feet 9 inches ? 2. What will be the cost of a Bill of Exchange on London foi 1364. 14^. 6d. at is. loJaT. per rupee ? v* x * 3. Reduce ?* x ($ x |) to its simplest form. J o 4. What is the difference between '67 and '07 ? 5. If an ounce of gold be worth 4 '18953, what is the value of '03753 Ib. ? 6. If A owns '24 of a ship, and B the rest, and the difference in the value of their shares is ^2876, what is the value of the whole ship ? 7. What sum must be invested in 5} per cent. Promissory Notes to produce a monthly income of R35O ? / 8. At what rate per cent, would R 17,200 amount to R 1 8,650 in 5 years ? 9. There are two schools, one containing 650 boys and the otbei 340 boys ; 5 per cent, of the former are generally absent and 7 *5 of the latter ; what is the average attendance in each ? 10 If 8 per cent, be gained by selling 218 yards of cloth for ^92. 131., at what price per yard must it be sold so as to gain 17 per cent. ? 11. If 400 men could do a piece of work in 3*4 days, how many men would do \ of the same work in 15 days ? 12. What is the value of a beam of timber whose length is 30 feet* breadth 3$ feet, and thickness 2\ feet, at 3*. 9^. per cubic foot ?, 13. Find the cube root of '4. 1 1869. 1. Find the G. C. M. of 2231 and 4656 ; and the L. C M. of 4, 9, 16, 28, 42. 2. Add together }, T V, Ai i? T t*. 3. Find the value of : BOMBAY ENTRANCE PAPERS 409 4. Convert into vulgar fractions the decimals '015625 and '01190476 and reduce the results to their lowest terms. 5. Reduce R6. 7fa. to the decimal of Rio. 6. Divide the sum of R328i. I2$a. among 4 persons in the proportion 0* 3> 5> 8, 9- 7. If 442 amount to 530. Ss. in 5 years, what is the rate per cent. of simple interest ? 8. Find the amount of 1,000 in 6 years, at 5 per cent, compound interest. 9. If 27 men take 15 days to mow 225 acres of grass, how long will 33 men take to mow 165 acres ? 10. A person has Rioo,ooo stock in Government 4 per cent. ; he sells out all his stock at 92^, he then re-invests the purchase money in Bank of Bombay Shares of 6500 each, at R625, which pay 6 per cent, per annum ; find the alteration in his income. 11. Find the square roots of 3129361 and 434 '027. 12. Show that the cube root of '637 is '3. 1870. 1. Write down in figures the following : Six hundred and fifty -four thousand three hundred and twenty -three billions, four thousand and twenty-one millions, fifty thousand three hundred and one. Express in words the number 1327875430029 according to the English and Hindu systems of numeration. 2. Find the value of 3&+4I+ iJt + 3AV. D0tn b V vulgar fractions and decimals, and show that the two results coincide. 3. Divide the difference of 7$ and 9| by their sum, multiply the quotient by & of 7}. 4. If an ounce of gold be worth 4 '602$ ; what is the value of a bat of gold weighing I '68J Ib. ? J. If a family of 9 persons spend R4,8oo in 8 months, how much will serve a family (living upon the same scale) of 24 persons for 16 months ? 6. Three equal glasses are filled up with mixtures of spirit and water ; ^he proportion of spirit to water in each glass is as follows : in the first glass as 2 : 3, in the second glass as 3 : 4, and in the third as 4 : 5. The contents of the three glasses are emptied into a single vessel ; what is the proportion of spirit and water in it ? 7. What are the weights of a sovereign and a shilling, the pound Troy of standard gold being coined into 46. 14*. 6</., and the pound of silver into 66 shillings ? 8. Find the interest on 215. I2J. for 3 years 8 months and 10 days at 4} per cent, per annum. 410 ARITHMETIC 9. A ship worth R9,ooo being entirely lost, of which one-fourth belonged to A> one-sixth to , and the remainder to C ; what loss will each sustain, supposing ^5,400 of the ship were insured ? 10. Extract the square roots to six places of decimals of '099 and of 3'3- 11. How much stock in the 3 per cents, must I sell to pay off a debt f ;55o, the price of the stock being 94 J, and commission of J on jioa of stock being also taken into consideration ? 1871. 1. The distance of the sun from the earth is ninety-one millions even hundred and seventy-six thousand miles, and light travels from the former to the latter in seven minutes and fifty-eight seconds ; find the- velocity of light per second. 2. Find the G. C M. of 441441 and 844272 and the L. C. M. of 7, ii, 21, 63, 91, 99, 117, 143- 3. Define a fraction, and prove that the value of a fraction is not altered if we multiply both its numerator and denominator by the same whole number. Bring JYff - * f ^ +-f } v 2igg x 3 ,Vk| cwt. to the fraction of lAt x 4t + Tff 4r/ J 4f tons. 4. State and prove the rules for reducing terminating and circulating decimals into their equivalent vulgar fractions. Ex. '03125 and 729. Find the value of '03125 of R2 + 72$ of R3xV -J- 729 of R4f . 5. If lo horses and 98 sheep can be kept 9 days for 37. 17*. 6d. ;. what sum will keep 45 horses and 216 sheep for 40 days supposing 5 horses to eat as much as 76 sheep ? 6. If the par of exchange be two English shillings for the Indian rupee, but if an Indian bill of exchange for R54O. 12 a. be negotiated in London for $i. IQS. ; how much per cent, below par is the rate of exchange ? 7. Distinguish between interest and discount. The interest on a certain sum of money^ for three years is 825, and the discount for the same time is 645, simple interest being reckoned in both cases. Find the rate per cent, per annum and the sum. 8. A person desires to paper his room with postage stamps ; the room is 14 feet 9 inches long, 9 feet 3 inches broad and 10 feet 6 inches high ; it contains two windows, each 5^ feet by 4 feet and 3 doors each 6 feet by 3 feet ; a postage stamp is ( inch long and } inch broad. Find the number of postage stamps required to cover the room. 9. A person invests 1,250 gold mohurs in the Government five pet cent rupee stock at 105. The stock is converted subsequently to 4} per cents, at 95. Find the difference in his income, each gold mohur being considered equivalent to Ri7. BOMBAY ENTRANCE PAPERS 41 i 10. A certain number of persons agree to subscribe as many pies eacfe as there are subscribers ; the whole subscription being R5,797- oa. ip, How many subscribers were there ? 1873. 1. Simplify:- * 2. Find the value of '375 of a guinea + '54 of 8j. 3^. + '027 of 2. 15;* and reduce the result to the fraction of a guinea and a half. 3. A man owns v of a ship and sells '3571428 of his share ; what fraction of the ship does he still own ? 4. If the income-tax be 6 pies in the rupee for the first half of ths year and 3 per cent, in the second, what is the gross income of a gentleman whose net annual receipts amount to 8:1,454. 10. ? 5. Five men do '6006 of a piece of work in 2*72 hours ; how long will 6 boys take to finish it, it being known that 3 men and 7 boys have done the whole piece of work in 3 hours ? 6. If the difference between the simple and compound interest of a. sum of money for 2 years at 5 per cent, be $. iSs. 9|^., find the sum. 7. When the three per cents, were at 90, I found that by selling out and investing in the 4 per cents, at 95 I could improve my income by 243.. What was the amount of my stock in the three per cent. ? 8. A gardener plants an orchard with 5776 trees and arranges them so that the number of rows of trees equals the number of trees in each row. How many rows were there ? 9. How many seconds will a train 184 feet in length, travelling at the rate of 21 miles an hour, take in passing another train 223 feet long proceeding in the same direction at the rate of 16 miles an hour ? 10. Find the cube root of 1879080904. 1874. I. Simplify the fraction : 2. Divide 8 '064 by { -846 +*f- of -2916}. 3. A man owns A of a house, and sells '1351 of his share ; what fraction of the house does he still own ? 4. In a subscription list one-half of the subscriptions are a guinea each, one-third a half-guinea each, and the 5 shilling subscriptions which- complete the list amount to 12 ; find the whole amount subscribed. 5. If the work done by a man, a woman, and a child be in the ratio of 3, 2, i, and there be in a factoiy 24 men, 20 women and 1 6 children, 412 ARITHMETIC whose weekly wages amount to R2O4 ; what will be the yearly wages of 27 men, 40 women and 15 children ? 6. The debts of a bankrupt amount to 2134. ips. 6ct. t and his assets consist of property worth ^"916. 15.?. 4^, and an undiscounted bill of ^513 due 4 months hence, simple interest being reckoned at 4 per cent. How much in the pound can he pay his creditors ? 7. A merchant buys 4,000 maunds of rice, one-fifth of which he sells at a gain of five per cent., one-fourth at a gain often per cent., one-half at a gain of twelve per cent., and the remainder at again of sixteen pet cent. If he had sold the whole at a gain of eleven per cent., he would have made R728 more. What was the cost of the rice per maund ? 8. The shares in a banking concern are Riooo each, 426. iofa. are only paid up, and the shares are quoted in the market at 460. The dividend is R7J per share quarterly. A gentleman holds 100 original shares. Find what interest he makes per cent. ; and what he would make and how much per cent., if he sold out and invested in 4 per cent. Govern- ment stock at par. 9. A and B are the termini of a Railway 144 miles long. A fast train starts from B at 9 h. o m. ; another fast train, travelling at the same rate, starts from A at 10 h. o m. A slow train starts from B at 10 h. 20 m. ; the fast train from A meets the other fast train at 1 1 h. 30 m., and the slow train at 12 h. 32 rru ; find the rates at which the trains travelled. 10. Arrange in order of magnitude : V(SO), V(344), V(*402). 1875. 1. Write out in words the following expressions : (a) 8271096. (3) 9032804. (<:) 319080259417. (d) 8004640. 2. What is the rule for the addition of concrete numbers ? Add to- gether 17 miles, 3 furlongs, 19 poles, 28 yards, 2 feet, 10 inches ; 4 miles, 3 furlongs, 8 poles, 7 yards, 2 feet and 9 inches. 3. Explain what is meant by the following words and give examples : Measure, Multiple^ Greatest Common Measure^ and Least Common Multiple. 4. How many acres are contained in three countries, of which the first comprises 723100 square miles, the second 12342, and the third 89704 square miles ? 5. Divide \ of f of \ of 42 by the sum of 2j and 4$. 6. What are continued fractions^ and when do you make use of them ? Find three fractions approximating to \\\. 7. Find the product of 17 '302 and '579 to three places of decimal*, by the rule of contracted Multiplication. 8. What sum will discharge a debt of R7,20O due a year and a half hence at 4 per cent, per annum ? BOMBAY ENTRANCE PAPERS 413 9. Find the square root of 745^29 and the cube root of 32768. 10. Divide a guinea between A, B, C 9 D, so that B's share is } more than A's, C*s \ more than 2Ps and /)'s J more than Cs. x I* How much stock can be purchased by the transfer of R2OOOO stock from the 3 per cents, at 90 to 3^ per cents, at 96 ; and what change will be effected in income by it ? 12. Required the number of square feet there are in a piece of slate 2\ feet \ in. in length, and ij feet $ in. in width. 1877. 1. Define the arithmetical terms: notation, numeration, unit, integer, fraction, abstract, concrete. Can you (i) multiply concrete numbers to- gether ? (2) divide a concrete number by a concrete number ? Give examples to illustrate the nature of such operations. 2. Two men A and B start together, and when A has gone a mile il * + i r-.V + * + * B has gone ,f of if of | of ?- -f of 71? of - *% a - of a mile : & *-* - i which is in advance of the other ? 3. Express the difference between '378 of 13*. lojaf. and '37$ of Get. as a fraction of 4. A Ib. of tea and 3 Ib. of sugar cost R3, but if sugar rose 50 per cent, and tea 10 per cent., they would cost ^3. 80. ; find the prices per Ib. of tea and sugar. 5. The circumferences of the wheels of a carriage are 6 T V feet and &j*5 feet ; what is the least distance in which both wheels will simultane- ously complete an integral number of revolutions ? How often will the lowest points of the two wheels at starting touch the ground together in to miles ? 6. A, B and C rent a field for R2,878. A puts in 12 horses for 5 months and 45 sheep for 3 months ; B puts in 15 oxen for 6 months and 54 sheep for two months ; C puts in 6 horses and 48 oxen for 3 months. Now, 4 horses and 3 sheep together eat as much as 5 oxen and I horse, and 2 oxen eat as much as 7 sheep ; how much of the rent should A, B, C, pay respectively ? 7. What sum of money will amount to 699/. 13;. 2*4^ in 2 years reckoning compound interest for the first year at 4 per cent, and for the second 3! per cent, per annum ? 8. A person finds that if he invest a certain sum in railway shares paying 6 per share when the 100 share is at ^132, he will obtain jxo. i6/. a year more for his money than if he invest in 3 per cent, consols at 93, What sum has he to invest ? 414 ARITHMETIC 9. Find the value of tj( '00139876) - *J( -000030664297). 10. A man near the sea-shore sees the flash of a gun fired from a vessel steaming directly towards him, and hears the report in 15". He then walks towards the ship at the rate of 3 miles an hour, and sees a second flash 5 minutes after the first, and immediately stops ; the report follows in io"*5. Find the rate of the ship, the velocity of the sound being 1,200 feet per second. 1878. 1. Seven men find a lump of gold weighing 13 Ib. 7 oz. Troy. What will be each man's share, supposing gold to be worth 3. 17*. per ounce ? 2. Simplify : 3. Find the value of : 387 of 8. 16*. 30?. +6J of Jf of 7*. %\d. +& of la. 4. What is the length of the edge of a cubical cistern which contains ^as much as a rectangular one whose edges are 154 ft. II in., 70 ft. 7 in. and 53 ft. I in. ? 5. In 1861 three towns had populations of 17650, 19600, 18760, respectively. In 1871 the population of the first had decreased 1 8 pet cent., that of the second had increased 21 per cent., while the population of the third had increased by 4690 ; find the change per cent, in the population of the third town. 6. A bankrupt has goods worth R975O ; and had they realised theii full value, his creditors would have received isa. in the rupee ; but $ths were sold at 17*5 per cent, and the remainder at 2375 per cent., below their value. What sum did the goods fetch, and what dividend was paid ? 7. What sum will amount to ^1,591. 135. 2*i6dT. in 3 years at com- pound interest ; the interest for the first, second and third years being 3, 1 and I per cent, respectively ? 8. Find the true discount on 2,750 due two years hence at 4j pel cent. 9. If 4 men earn as much in a day as 7 women, and one woman as much as 2 boys, and if 6 men, 10 women and 14 boys working togethei for 8 days earn 22, what will be the earnings of 8 men and 6 women working together for 10 days ? 10. A person having a certain sum of money to invest, finds that an investment in a railway stock bearing 5 per cent, interest at 117} will yield him 29 more annually than an investment in the 3 per cents* at 92}. How much money has he to invest ? 1879-80. I. Add the following numbers : Eighty-four thousand three-hundred end one ; nine hundred and thirty-three thousand ; forty-seven million* BOMBAY ENTRANCE PAPERS 415 tix thousand three hundred ; and subtract from the result two millions eighty-one thousand and eighty. 2. Explain the terms measure, common measure and greatest common -measure, and prove that every common measure of dividend and divisor Cs a measure of the remainder. Find the value of '45 of i. 3*. go*. + '257 of 11. 5*. 6a\ + '3125 3. of 5- 7 _ 9 1 4. Find the value of 1 4- 4 and also of J -f-| - T 5. If by selling wine at R6 per gallon I lose 25 per cent., at what .price must I sell it to gain 25 per cent. ? 6. A person borrows 130 on the 5th of March, and pays back 132. ZOs. 6d. on the i8th October ; find the rate of interest charged. 1880-81. 4. Simplify the following expressions : a+ - - ; ~r *;7 J and add together the results. -, J 534^3 *75 5 i 2. Three boys agree to start together and run, until all come to- gether again, round a circular court 15 yards in circumference. One runs at the rate of six, the second seven, and the third eight, miles an hour. In flow many seconds will the race end ? 3. If three soldiers or 10 coolies can dig 155 cubic feet of earth in X days, how many coolies must be employed to assist 7 soldiers in remov- ing 600 cubic feet of earth so as to get it done in 4 days ? 4. In what time will R2,25o amount to R2,$65 at 7 per cent, per annum ? 5. A merchant sells a lakh of rupees out of the four per cents, at 16 discount, and invests the proceeds while exchange is at 2s. id. in the three per cent, consols at 96. What income does he derive therefrom ? 1881-82. 1. If the income-tax be 7<f. in the pound in the first half of the year, *nd 3i< in the second, what is the net income of a gentleman whose gross annual receipts are , 1,542- ios. 6d. ? 2. A passenger train going 41 miles an hour, and 431 feet long, over- takes a goods train on a parallel line of rails. The goods train is going &8 miles an hour, and is 713 feet long. How long does the passenger train take in passing the other ? 3. Find the cost of painting the outside of a cubical box whose edge IB 3*5 feet, at 1*3 shillings per square yard. 4. A person invests 848,000 in the 4 per cents, at 80, and at the end "Of each year invests the dividend, which becomes due, in the same stock ; 416 ARITHMETIC supposing the funds to remain at 80 for 3 years, find bis dividend at the end of the third year. 5. Define discount. If the discount on R2,26i .5.4 due at the end of a year and a half be 128, what is the rate of interest ? 6. Find the square root of - ~ and the cubic root 05423564751, *Io 1882-83. 1. Find the value of ^596875, and reduce II poles 4 yards 4} inches- to the decimal of one mile. 2. A railway passenger counts the telegraph posts on the line as he passes them. If they are 58 yards apart and the train is going 48 miles per hour, how many will he pass per minute ? 3. Three men can do as much work as five boys ; the wages of three boys are equal to those of two men. A work on which 40 boys and 15 men are employed takes 8 weeks and costs ^350 ; how long would it take if 2O boys 20 men were employed, and how muoh would it cost ? 4. What sum will amount to 5431. 15*. nj< in 6 years at 4! per cent, simple interest ? 5. The sides of two squares contain 77 yards I foot 9 inches and 7 yards 2 feet 4 inches respectively ; find the side of a square whose area is equal to the sum of the areas of the two squares. 1883-84. I. (a) Express in figures : Sixteen billion, seventy-five million) forty thousand and two. (b) Simplify the expression (c) Find the value of 375 of 5*. 6d. +5-05 of $. u. 8^+5*07 of 7* 6V. +3-135 of 2 - ls - 3& 2. At the examination of a school T V of the children were presented in the 6th standard, i in the $th standard, J in the 4th, | in the 3rd, i in the and, and the remainder 107 in the 1st standard ; how many were presented altogether, and how many in each of the other standards ? 3. In a bicycle race of two miles over a circular course of^ I furlong, the winner in his last round overtook the second at a point in his fifteenth found. Their paces were as 159 to 149. At what distance was this point from the winning post ? 4. Find the expenses of an excursion, which includes 5782 miles of railway at $d. per mile, 517 miles of carriage at ioj</. per mile, 57 days of hotel keep at 14*. $<*. per day, allowing 5 guineas for extras. 5. Divide I *04 by '000078125 and prove your result by vulgar frac* tions. Find the square root of 8658-3025 and the cube root of 753 -571. BOMBAV ENTRANCE PAPERS 417 1884-86. 1. Reduce to a vulgar fraction '42857!. Divide 301 '6 by 416. Find the value of '475 of i + '42 of 2. 17 s. gd. 2. A merchant buys 1260 maunds of corn, one-fifth of which he sellj at a gain of 5 per cent., one-third at a gain of 8 per cent., and the remain- der at a gain of 12 per cent. If he had sold the whole at a gain of 10 per cent., he would have obtained ^22. 13*. more. What was the cost price per maund ? 3. A room, 10 ft. 6 in. high, 22 ft. long and 14 ft. broad, is painted up to one-third of the height and the remaining two-thirds papered. The painting is charged at 7$d. per square yard, the paper costs 5*. 2d. per square yard, and the work of papering is charged at 2d. per square yard. How much will the whole cost amount to ? 4. A person sells out ^3850 four per cent, stock at 104 and invests the proceeds in another stock at 143. If the dividend on this be 5J pel cent., what will be the change in his income ? 5. What must be the rate of interest in ordr that the discount OD 387. 75. 7i< payable at the end of 3 years may be 41. los. i\d. ? 1885-86. i. Reduce -I-? of 2 guineas + J y of ' of 4 crowns - ~- T -""-* of ji to the decimal of 5 half-guineas, and prove that - is greater than T \ and less than f . 2. A man contracts to perform a piece of work in 30 days and imme- diately employs 15 men on it ; at the end of 24 days the work is only half done. How many boys should be given to assist them that the contract may be fulfilled, each boy working two-fifths as much as each man ? 3. A person buys 80 tons of coal, and after selling them again at I/. &/. per sack finds that he has gained 4 ; had he sold them tor is. 40?. per sack he would have lost 6. Find the weight of each sack and the cost price per ton. 4. A field of 7 acres is sown with wheat, barley and maize, the areas of the crops being respectively as 2 : 3^ : 4y. If the values of an acre of each be also respectively in the same ratios, and an acre of wheat be worth 7, what is the worth of all the crops in the field ? 5. If the three pel cents, are at 92! and the four per cents, at 123!, in which should one invest ? And how much is one investing when the difference in income is a shilling ? 1886-87. I. Explain carefully the meaning of prim* number % facto* -, divisor^ measures, multiple. Resolve 5005 into its prime factors. Add together as decimals 8'J j8, 14*65651, '20568963. C. A, 27 41 5 ARITHMETIC 2. The circumference of the forewheel of a carriage is 6| feet and that of the hind wheel is I2f feet. How many feet must the carriage pass ovei before the wheels shall have made a complete number of revolutions ? 3. A vessel is filled with a liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup ? 4. (i) The surface of a cube is 308*16 square feet. Find the length of its edge. (ii) Extract the cube root of 45*698 to four places of decimals* 5. If the price of gold be 3. los. io|</. an ounce and a cubic inch of gold weigh 10 ounces, what is the price of the gold that would be required to gild a dome whose surface is 5000 square feet, the thickness of the gold gilding being '0002 of an inch ? 6. A person invests in 4 per cent. Government paper so as to receive 4 per cent, clear when the income-tax is 5 pies in the rupee. What per- centage will be received if the tax be increased to 7 pies in the rupee ? ^ 1887-88. 2. If 9 Ib. of rice cost as much as 4 Ib. of sugar, and 14 Ib. of sugai are worth as much as I* Ib. of tea, and 2 Ib. of tea are worth 5 Ib. of coffee, find the cost of 1 1 Ib. of coffee if z\ Ib. of rice cost b\d. 3. If Ri65. 140. lyVA be tne discount of a debt of 2820, simple interest being at the rate of 3| per cent., how many months before due was the debt paid ? 4 The price of gold is ^3. 17*. loj</. per oz. ; a composition of gold and silver weighing 18 Ib. is worth 637. 7.?., but if the proportions of gold and silver were interchanged, it would be worth only ^"259. u. Find the proportion of gold and silver in the composition, and the price of silver per oz. 5. By selling 4 dozen mangoes for 13 rupees, it was found that Ath of the outlay was gained^ ; what ought the retail price per mango to have been in order to have gained 60 per cent. ? 1889-90. ,; Simplify ' \ '5-H of ----- -25 - \ V -142857 of ifj 2. A rectangular cistern, whose length is equal to its breadth, Is ct feet deep and contains 5 tons of water. If a cubic foot of water weighs 1000 ounces, find the dimensions of the cistern. 3. A, B and Ccan walk at the rate of 3, 4, 5 miles an hour ; the* start from Poona at I, 2, 3 o'clock respectively ; when B catches A B lends him back with a message to C ; when will C get the message ? BOMBAY ENTRANCE PAPERS 419 4. If I borrow money at 3 per cent, per annum, interest payable f early, and lend it immediately at 5 per cent, per annum, interest payable faalf-yearly (receiving compound interest for the second half-year), and gain thereby at the end of the year R66o ; what was the sum of money which I borrowed ? 5. A person buys tea at 6 annas per seer and also some at 4 annas pei seer. In what proportions must he mix them so that by selling the mixture nt 5$ annas per seer he may gain 20 per cent, on each seer sold ? 1891-92. 1. Simplify : (i) 3*6428571 -( '009923 + '0102 - '000123). I4 J l"\ OOSO \/34*5744 - V9-663597 2. Two passengers have together 5 cwt. of luggage and are charged for the excess above the weights allowed 51. 2</. and 9$. io< respectively ; but if the luggage had all belonged to one of them he would have been charged 19*. 2d. How much luggage is each passenger allowed to cany free of charge, and how much luggage had each passenger ? 3. Two clocks A and /?, whose rates are uniform, at noon yesterday indicated 1 1 hrs. 55 min. A. M. and o hr. 2 m. p. M. respectively. A indi- cated the correct time at 9 P. M. yesterday and B at 6 A. M. this morning. When did A and B last agree and what time did they then indicate ? 4. A person borrows two equal sums of money at the same time at 5 per cent, and 3! per cent.^ simple interest respectively, and finds that if toe repays the former sum with interest on a certain date a year before the latter, he will have to pay in each case the same amount, viz., 736. Find the amounts borrowed. 1892-98. I. What decimal of a rupee is '964 pie ? Find the value of '97625 tupee. 2. How long will two examiners, working 8 hours a day, take to look over the answers to this paper, if four examiners, working 5 hours a day, can do it in 8 days ? 3. On a river, B is intermediate to^ and equidistant from A and C ; . boat can go from A to /?, and back, in 5 hours 15 minutes, and from A to C in 7 hours ; how long would it take to go from C to A ? 4. What income will a retired officer obtain in England, from one lakh of rupeei, Indian Government 4} per cent bonds, when for drawing and remitting it, his agents in India charge him 3 per cent., and exchange is at I*. 2}< for the rupee ? 4*0 ARITHMETIC 5. Three equal glasses are filled with a mixture of spirits and water, the proportion of spirits to water in each glass being as follows : In the first glass as 2 : 3, in the second 3 : 4, and m the third 4 : 5. The contents of the three glasses are poured into a single vessel ; what is the proportion of spirits to water in it ? 1893-94.. (Set in the Moffussil). 1. Divide each of the numbers 2,572,125 and 4,061,250 by 125 ; and express as a decimal the first quotient divided by the second. 2. Find, by Practice, the value of 5 yd. 22j in. at 2. is. 2cL a yard. 3. If the carriage of 2 cwt. I qr. and 1 8 Ib. of goods, for 56 miles,. be ji. I J., what weight can be carried at the same rate, 200 miles for 4. 31. 4<f. 4. A man invests .3,000 in the 5 P er cents. If after deducting an income-tax of &/. in the pound, the man's clear income is ; 174, what is the price of the 5 per cents. ? 5. A cistern is filled by two taps A and B in 4 hours and 6 hours respectively, and is emptied by a waste pipe C in 3 hours. When the cistern is half full, A and B are closed, and C is opened ; after one hour, B is turned on ; and after half an hour more, A is turned on. In what time after C is first opened, does the cistern become full ? 6. A person buys two kinds of tea, at 5*. a Ib. and 6s. a Ib. , res- pectively ; and after mixing them he sells the mixture at 6s. 6d a lb. 9 . thereby gaining 17 per cent. In what proportion does he mix them ? 1893-94. (Set in Bombay). I. Reduce to their simplest forms : 3 5- 2. Find, by Practice, the value of 9 cwt. 3 qr. 24 Ib. at 3. 5*. 8dL per cwt. 3. If 40 men, 60 women or 80 children can do a work in 6 months, io what time will io men, io women and io children do one-third of the work ? 4. A person invested 1,000 in the 3 per cents, at oof ; but the price rising to 91$, he sold out, and invested the proceeds in ths 3} per cents. at 97} ; find the increase in his income. 5. A cistern can be filled by two pipes, A and J8 9 in 12 minutes and 14 Minutes respectively, and can be emptied by a third, C 9 in 8 minutes. If all the taps be turned on at the same moment, what part of the cistern will remain unfilled at the end of 7 minutes ? BOMBAY ENTRANCE PAPERS 421 6. Two clocks point to 2 o'clock at the same instant on the afternoon of 2$th April ; one loses 7 seconds, and the other gains 8 seconds, in 24 hours ; when will one be half an hour before the other, and what time will each clock then shew ? 1894-95. 1. When the number representing the year is a multiple of four, it is a leap year, consisting of 366 days, except when this number is a multiple of 100, in which case it is an ordinary year, consisting of 365 days, but when the number is a multiple of 400, it is again a leap year ; on this supposition, calculate the number of days from the first January 1495 * 3 1st December 1894, both days inclusive. 2. A school of boys and girls consists of 453 children ; the numbef representing the boys is -52 of the number of the girls. How many boys were there ? 3. Two-thirds of a certain number of poor persons received is. 6d. each, and the rest 21. 6</. each ; the whole sum spent being 2. 155., how many poor persons were there ? 4. If 3 men and 5 women do a piece of work in 8 days, which 2 men and 7 children can do in 12 days, find how long 13 men, 14 children and 15 women will take to <io it. 5. A sells a house to B for R486o, thereby losing 19 per cent. ; B sells it to Cat a price which would have given A 17 per cent, profit. Find /?'s gain. 6. The compound interest on one rupee is one quarter of a rupee at the end of three years ; find the rate per cent, per annum, correct to two places of decimals ; and calculate exactly the compound interest at the end of 9 years. 1895-96. 1. When a fraction is reduced to its lowest terms, find the form of the denominator so that the fraction may be expressed as a non-recurring decimal. Reduce gy to decimals. 2. A field can be reaped by 10 women in 4 days, or by 6 boys in 10 days, or by 2 men in 12 days. One man, three boys and three women are employed. What is the total expense, if the wages of a man, a woman and a boy are 80., 50. and 30. respectively ? 3. The total fare for a journey of 504 miles, partly by main line and partly by branch line, was KlJ. iltf. 6/>. ; the rates per mile being 6/. on the main line and 8/. on the branch line. What distance was travelled on the branch line ? 4. A certain sum amounts to 1 86 rupees *9 annas and $?& pies in three years at compound interest, and the amount at the end of the third year is to that at the end of the fourth year as I : 1-142857. Find the original sum and the rate of interest* 4^2 ARITHMETIC 5. A person sells 1600 Russian stock at 75^, and invests the proceeds in railway stock at 120. The brokerage for selling Russian stocks is $ per cent, stock, and the expenses of buying railway stock are one per cent, on the actual value. What amount of stock did he buy ? 1896-97. X. When a vulgar fraction in its lowest terms is reduced to a decimal, whether recurring or non-recurring, prove that the number of decimal places in the period is never greatt r than the number representing the denominator diminished by one. Simplify i '6996x2 729 ; and prove that 9 19 29 39 9x19x29x39 2. Divide 12,540 among A* B and (7, so that A shall receive \ as much as B and C together, and B shall receive f $ of what A receives. 3. Two railway trains on adjacent parallel lines are running in opposite directions, one at the rate of 40 miles and the other of^ 30 miles an hour. Each has an engine and tender, and the first train has 12 carriages and the second 17. If the length of an engine and tender be 41 feet, the length of a carriage 32 feet, and the coupling spaces be each 5 feet, how much time will eUpse from the moment that the engines meet till the last carriage of the trains have passed each other ? 4. Distinguish clearly between true and. false discount. A banker's discount calculated for one year in 26 times his gain thereby. Find the rate per cent, of interest. 5. A person purchases ftio,ooo stock partly in the 4 per cents, at 108 and partly in the 3J per cents, at 104. He sells the former at 106 and the latter at io6J, and loses R35 by the transaction. How much took did he buy in the 4 per cents. ? 1897-98. 1. Define numerator and denominator of a fraction, and prove that by multiplying these by the same number, the value of the fraction is not altered. Simplify l-r[i-f i-r {i + i-r(i + i-f 2)}] ; and show that J. + _L_.L + _L X _L X _L 12 99 70 12 99 70 7 ,-S x + l. x i. + x i 17' 12 99 12 70 99 70 2. What is inverse proportion ? Give two illustrations of it. A contractor undertakes to dig a canal 12 miles long in 350 days, and employs 45 men ; he finds that in 200 days he has completed 4^ miles. How many additional men must he employ to get the undertaking finished* in time ? BOMBAY ENTRANCE PAPERS 423 3. Guns are fired at intervals of 10 seconds in a town towards which a passenger train is approaching at the rate of 30 miles per hour. If sound travels 1 144 feet per second, find at what intervals the reports will be heard by the passengers. 4. A sum of Ra85 put out to compound interest for 3 years produces R29. Ja. 4f \p. Find the rate per cent, of interest. 5. I invest a certain sum in the 3^ per cents, at 91, and 4000 sterling in the 3 per cents, at 75 ; after paying an income-tax of 7^. in the , my net income is ^524. 5*. What sum have I invested in the 3$ per cents. ? 1809-1900. 1. Explain the terms compositt number and common multiple. (a) Find the least number which must be added to seven thousand and one million nine hundred and seven thousand and sixty-one, in order that the sum may be a multiple of seven hundred and nine thousand four hundred and eighty. (3) Find the Least Common Multiple of 1160, 2948, 3886. 2. If in France the railway fare for a distance of 384 killometres^ is 25 -28 francs, how does this rate of charge compare with the English Parliamentary rate of id. per mile ? Given one metre = I yard 3^ inches, ,1=25 '2 francs. 3. A walks to a place at the rate of 4} miles per hour ; at 8 miles from his destination he meets B> and turns back with him (walking at /? J s rate) for a mile. If A is half an hour late at his destination, what is &'$ rate ? And at what rate should A have walked after parting with 2?, so as to arrive at the proper time ? 4. A trader's debts amount to 5 1 74. 15*. ; he has assets sufficient to pay his creditors 16*. 6d. in the pound. Some creditors, however, have the right to be paid in full, and in consequence the others receive only 15*. in the pound. Find how much is paid in full. 5. A man has an income of ^415 derived from capital invested in 4 per cent, stock ; he sells out his stock at 102, and re-invests the proceeds in 5 per cent, stock. What price must he pay for the latter, if his new income is ^425 ? 1900-1901. 1. (a) Prove that the Least Common Multiple of two numbers is equal to their product divided by their Greatest Common Measure. Find whether the rule is true for three numbers. (b) A company of soldiers is formed into 6 equal rows ; after a time it is re-arranged into 7 equal rows, and finally into 8 equal rows. Find the least number of soldiers above 900 which the company may contain. 2. Reduce the weight of S'dJxsMtt-rsHxrsM 4-i x -405 cubic feet of water to the decimal of a ton ; it being known that one cubia foot of water weighs 62*37 Ib. avoir. 4*4 ARITHMETIC 3. If 38 men working 6 hours a day do a piece of work in 12 days* find in what time 57 men working 8 hours a day can do a piece of work twice as great, supposing 2 men of the first set to do as much work in I hour as 3 men of the second set can do in I \ hours. 4. A person invested 8:15,147 in 4 per cent, stock, and R 12, 054 in 6 per cent, stock. When the stocks were at R86. la. and RiO2 respectively, what income did he derive from these investments ? He afterwards transferred at the above rates a certain sum of money from the 6 per cent, stock to the 4 per cent, stock and then found that the income from each stock was the same. How much stock had he finally in the 6 per cents. ? 5. (a) Find the square root of '0001083681. (b] A stone dropped down a shaft falls through a number of feet equal to 16*1 times the square of the number of seconds during which it is falling. Find to two places of decimals the number of seconds that the stone will take to reach the bottom of a mine 1104 yards deep. IV. THE PUNJAB UNIVERSITY. ENTRANCE PAPERS. 1875. 1. Write in figures one million, ten thousand and one. Subtract 397 from 1,163 and explain the process. 2. Shew that when any number is divided by nine the remaindei is the same as when the sum of the digits is divided by nine. 3. State the rules for the multiplication and division of vulgar fraction. What is a complex fraction ? and simplify (i) UV + fof7l>*}f,and(a) J-fyij. 4. What is the value of '3375 of an acre ? Reduce i. IOJ. 4</. to the decimal of two guineas. 5. Find the square root of 9,98,001 and that of 3*14159 to three place* of decimals. 6. If five pumps each having a length of stroke of 3 feet, working 15 hours a day for 5 days, empty the water out of a mine ; how many pumps with a length of stroke of 2\ feet, working 10 hours a day for 12 days, will be required to empty the same mine ; the strokes of the former pumps being performed four times as fast as those of the other ? 1876. 1. How many revolutions will a cart wheel of three feet six inches diameter make in going a distance of 6 miles, the ratio of the diameter of a circle to its circumference being given as I : 3*14159 ? 2. A piece of land measuring 48 ghumas 3 kanals and 17 mar la* of which 39 ghumas 4 kanals and 17 marlas are cultivated and the rest THE PUNJAB ENTRANCE PAPERS 425 uncultivated is sold at the rate of ft75/- a ghuma for cultivated and R35/- a ghuma for uncultivated land. What is the price of the whole ? 3. The revenue of a village containing 15,756 acres of cultivated land Is assessed at 13 annas an acre. What will the local rate of 6 per cent, an the land revenue payable by the village amount to ? 4. A bania purchases 1,526 maunds of grain at 36 seers for a rupee- He sells one half at 26 seers the rupee ; at what rate must he sell the remainder so as to clear 50 per cent, on the transaction ? 5. Find the interest on 24,485 rupees for I year and 131 days at 12 per cent, per annum. 6. A man hires a workman on this condition that for every day he worked he should get one rupee but that for every day he was absent he should be fined 12 annas. When 356 days were past the workman was to receive ftu8. How many days had he worked ? 1877. 1. If a pound of pure silver be worth 62 shillings, the shilling con- taining 222 parts of pure silver in 240, what will be the value in shillings of a rupee weighing 180 grains, the rupee containing 979 parts of pure ilver in 1,000 ? 2. (a) How much is '0125 of a day ? (If) Find the value of 3! + 4! + ill +3*W- Express the result both as vulgar and decimal fraction. 3. Divide '10724 by "003125 and extract the square root of the result to 3 places. 4. (a) What sum at simple interest will amount to R6,ooo in 6 years at 4 per cent, per annum ? (&) How much Government paper of the six per cent, can be bought for RSOO when the funds are at 94 and what dividend will be got an it yearly. 1878. 1. If 135 rupees 4 annas be divided equally amongst 24 persons what will each receive ? 2. Define a vulgar fraction. By how much does the difference of l^ and ^ fall short of their sum ? Express the defect as a decimal of 7. 3. (a) Subtract '03 from '6$ and divide the result by '102. () Shew that -%- = '14 159 nearly. 4. A room whose height is 1 1 feet and length twice its breadth takes 143 yards of paper 2 feet wide for its four walls ; how much carpet will it require ? 5. At what rate (simple interest) will 1,300 rupees amount to 1,381 rupees 4 annas in 15 months ? 4l6 ARITHMETIC 6. Find the square root of *l to 3 places of decimal. What number has *oi for its square root ? 1879. I. (a) Show by an example that if the numerator and denominator of a fraction be divided by the same number, the value of the fraction is not altered. (} Reduce to their lowest terms -ffff an( l iHJ anc * express their difference in decimal form. 3. One cubic inch of water weighs 253*17 grains while one cubic inch of air '31 grains ; find the number of cubic inches of water (to three places of decimals) that would be equivalent to one cubic foot of air. 4. (a) What portion of R34. Sa. is A of f of RSO - & of Rio J ? (6) Find (accurately to 4 places of decimals) the square root of "OOI. 5. A rectangular field measures 6 acres and 960 yards ; its length is 3 times its breadth ; find the distance between the diagonal angles. 1881. 1. Distinguish between a vulgar fraction and a decimal fraction and show how to reduce one to the other. 2. Divide the continued product of *O2 1, '0021 and 210 by that of 14 and *OO7 ; and extract the square root of 5'5 to four places ot decimals. 3. Express ; of a rupee to the decimal of a guinea ( = Rip})* 27 + 1 of '3 4. A person withdrew R$,ooo from a bank, which paid him interest at 5} per cent, and invested the money in the 6 per cent. Municipal Deben- ture at 103}* Find the change in his income. 1883. i. (a) Divide the difference of '4607 and '00809 by the difference of 6Uandi~~~ () Prove that ~ is greater than } and less than J. a. Divide J[3-f}{3+J(3-f-iJ)}]by -125. 3. (a) Shew that the value of a decimal is not altered by adding Iphers to the right hand side* (&} Find the value of 7'jJ x -36 - 2'34$ in vulgar fraction. 4. A railway train having travelled at | of its proper speed reaches its journey's end 2} hours behind time ; injvhat time should the journey ht? e been done ? TH3 PUNJAB ENTRANCE PAPERS 4^7 $. Five hundred boys are distributed in three houses ; the smallest house contains J? of the whole number and the largest contains Jf of thfr smallest ; what is the number in each ? 6. A person realises Ri 85500 by selling his 3J per cent, stock at 92}. He invests one-fifth of the realised money in the 4 per cents, at 96 and t he remainder in 3 per cents, at 90. What is the difference in his income by this transaction ? * ^ 1384. 1. Mu!uply and divide R625 by R25, if you think the operation* possible. Give your reasons. 2. State and explain the rules for multiplying and dividing one decimal number by another ; exemplify by multiplying '0256 by I '05 and '105 s successively, and dividing tne results by '00105. 3. Simplify 4. Extract the square root of H 1000 5. Find, by Practice, the value of 45 md. 22 sr. and 10 ch. of grain at Ri. 6a. per maund. 6. The assets of a bankrupt consist of R956o. 40., a bankshare o v> RI2OO quoted at IO7|, and an undiscounted bill of R3225, due 4 month* hence at 4 per cent, per annum simple interest ; his liabilities amount to 25014. Hov\ much in the rupee can he pay his creditors ? 7. Compare the ratios 'y and . 1885. 1. Simplify *i*t* + i!t*i of K-tti and find how many times '027 can be taken from 3*33. 2. Convert g into a decimal ; why is the result a terminating and not a recurring decimal ? Subtract "03 from '6 and divide the result by oo?. 3. Find, by Practice, the value of 12 maunds 8 seers 4 chataks of ghee at R;2. 8a. per maund. 4. A legacy of 1901. 5;. is to be distributed amongst a number of* persons, in such a way that each shall receive as many shillings as there a re persons ; what will be the portion of each ? 5. Find the Least Common Multiple of 35280 and 592704. What l> the smallest number of square yards which can be measured either by roods or square chains ? 428 ARITHMETIC 6. Four per cents, are offered at 98, five per cents, at Ri2o}{ 9 ^hich is the better investment ? How much is one investment when the difference of income is R$o ? 1880. I. Simplify >--?- s and extract the square root of the result 1-6 + 2*629 *to three places of decimals. to a decimal fraction correct to four places. Is there anything to suggest that the result will be terminating or recurring decimal ? 3. What fraction of 51,120. i$s. is 17 -975 of 71. 2s. 4. A clever housekeeper went out shopping and found that 2 cocoa- nuts were selling for the same price as 144 plums ; she bought half a dozen cocoanuts, exchanged one of them for 5 melons, and a couple of melons for 5 oranges ; she then gave 3 oranges for 42 limes, and finally secured a couple of plums for 5 limes. Has she gained or lost in buying the plums 7 5. Distinguish between Interest and Discount. Find the Interest and Discount of Ri,45o. Sa. for 3 years at 4} per cent, per annum, simple interest. 1887. 1. (a) Write in figures three billions, five millions, four hundred and nine thousand and sixty-two. () Write out measures of length and surface, both English and Indian. (c) ' Express an acre as the decimal of a bigka> a cubit being equi* valent to 18 inches. 2. Owing T * 7 of an estate I sold ^ T of f of my share for *ff- ; what is the value of - * f I f tne estate at the same rate ? 4f 3. A merchant having 100 maunds of grain sold 50 maunds at &9 per maund, and thereby gained 7| per cent. At what rate should he sell the remainder 50 that he may gain 10 per cent, on the whole ? 4. A merchant in trade successively admits three partners at the end of 3 months, 5 months, and 6 months respectively from the opening of the business. The capitals embarked by them were 400, 450, 480 and R49S respectively. After 6 months more, the profit was found to be Ri,ooo. Divide this rateably between the partners. 5. What sum of money invested in the 4 per cents, at par would rea- lise the same income as Rio,ooo invested in the 4! per cents, at 102 ? THE PUNJAB ENTRANCE PAPERS 6. Extract the square root of 0025 + 1-6 '4264-2 '62$ 1888. I. Simplify : _ __ _ I -A a-* 4-iJ 6J 2. Express the difference between '378 of 13*. io}< and '37$ of 161. &/. as a decimal of -426 x 3 7 ^ x ^ x ~^- of 1. 17*. 6^/ 3. Four men working together all day, can finish a piece of work ia II days, but one of them having other engagements can work only half time, and another only quarter time. How long will it take the men to> qomplete the work ? 4. A merchant sells his goods worth R5oo directly for R6oo giving three months' credit. Find his profit per cent., interest being calculated at 12 per Cent, per annum. 5. Find the value of - :_ correct to three places of decimals. I ~~ V'4 1889. 1. Express 80080080*0975 in words and give the local value of the digits. What decimal of R75 is R24. 2a. 6p. ? What is the least number which when divided by 22, by 88, by 132 and by 198 gives in each case remainder 7 ? 2. Why is the fraction | objectionable ? After walking 4j miles, a man has accomplished 2 i- ?i|J* + l * of J 1 1 of his journey ; how far has he still to walk ? 2j-i^) Of -- ~ t'7 *OII2 3. Add together ^ and -^ . Five bells which commence tolling! together, toll at intervals of l*2 9 i"5 !75 i '8, 2*1 seconds respectively ; after what interval will they again toll together ? 4. Define "present worth." A farmer buys 57 sheep for Ri2o, payable at the end of 12 months, and sells them directly at Ri. 120. ready money ; what does he lose by the transaction, supposing the interest of money to be 5 per cent. ? 5. Show which is greater J2 or ^3. Which is the better investment, 3 per cents, at 83} or 3} per central ) per cent* discount ? ARITHMETIC 1890. I. Simplify (a] (ft '47 -('5~ -0303) -0873 -(-0083 + -06)' 2. What part of f of 5 cwt. is i| of a ton ? Express '378 of i6s. 6d. as a decimal of '426 of i. i*js. 63. 3. A man bequeathed f v of his property to one son, 30 per cent, of the remainder to another, and the surplus to his widow. The difference of his son's legacies was ,784. How much did the widow receive ? 4. A ship with 1200 men on board had sufficient provisions to last 17 *weeks. The survivors of a wreck having been taken aboard, the provi- sions were consumed in 15 days. How many men were taken aboard 2 5. At what price must a person invest in the 4 per cent. Government ' Promissory Note, so that after paying income-tax at *he rate of 5 pies ID the rupee, he may receive 4^ per cent, on his investment ? 6. A and B travel together 120 miles by rail. A takes a return ticket for which he has to pay one fare and a halt Coming back they find that A has travelled cheaper than B by 40. zp. for every roo miles. Find the Care per mile. 1891. I. Simplify : (I) 2. Express 77 oz. + '075 cwt. as decimal of 2-25 of '27 of a ton. 3. A sum of money invested at 5 per cent per annum simple interest amounts in 6 years to 1*1326 ; in what time will it amount to 4. What is discount ? Distinguish between true and commercial discount. The interest on a certain sum at 5 per cent, per annum for a certain time is 50, and the discount at the same rate for the same time is 49. Find the sum and the time. 9 5. Nine gallons are drawn from a cask full of wine, it is then filled >mth water. Nine gallons of the mixture are drawn, and the cask is again filled with water. The quantity of wine, now left in the cask is to that of the water in it as 16 : 9. How much does the cask hold ? THE PUNJAB ENTRANCE PAPERS 431 1892. I. Find by how much the square root of 9+ differs from |{|. Which of these comes nearest to 2* Find the value of 3 A stream which flows at a uniform rate of I '109 miles an hour, is 26 7rds wide, the depth at a certain ferry being 6 feet ; how many gallons pass the ferry in a minute ? (Each gallon contains about 27 7 cubic inches). 4. A person invests 14970 in the purchase of 3 per cents, at QO and 3i per cents, at 97. His total income being 500, how much of each stock did he buy ? 5. A spirit merchant buys 80 gallons of whisky at 1 8*. per gallon, and 1 80 gallons more at 15;. per gallon, and mixes them. At what price must Cte sell the mixture to gain 8 per cent, upon his outlay ? 1893. V. Add; R. . As. P. 3436 12 2 5242 10 3 248 6 9 43i 13 5 5302 I* 4* 6789 8 i* 5001 15 6J 136854 7 2 298 9 4l 836993 I qf e. Multiply 319*9657 by '04286. 3. Find the value of r ~/\ correct to 5 places of decimals. tj\%~*~ V 2 ) 4. Calculate the income-tax on R666. IO annas 8 pies at 5 pies per rupee. 5, A local train which travels at the rate of twenty-four miles an hour* (eaves Lahore at twenty minutes past eight and reaches Amritsar at five minutes past ten the same morning. It stops at Mianmir for ten minutes find at each of three other stations for five minutes. Find the distance between Lahore and Amritsar. 1894. I. Convert f- and f into circulating decimals and point out the relation between the figures in th:ir periods* 43* ARITHMETIC 2. The sides of a rectangle are as 3 : 4 and the area is 1452 squaic feet. Find its length and breadth. 3. Exchange RyoSo for English money at is. 3$^. per rupee. 4. What is discount ? How is it commonly calculated ? If a sum of Rl,ooo becomes due three months hence, what is its present value as commonly calculated, and what as correctly calculated, interest being^ reckoned at 5 per cent. ? 5. Find the square root of lox correct to five places of decimals. 1895. I. Divide - \ r * by -,and reduce the quotient to a recurring decimal. 2. The Imperial gallon contains 277^27 cubic inches, and a cubic foot. of water at its maximum density weighs 62*42 Ib. ; find the weight of a pint of water correctly to two places of decimals. 3. The capital of a firm consists of ^713. 3*. ; ^964. 175. ; ^2391. 35.' ttbscribed by three partners ; divide 2231 among them in proportion ta their several capitals. 4. Find the square root of 5 correctly to seven places of decimals. $. The area of a rectangular field is f of an acre ; and its length i> twice its breadth ; determine the lengths of its sides approximately. 1896. I. Reduce to lowest terms 2. A cubic foot of copper weighs 560 Ib. It is rolled into a square - bar 40 feet long. An exact cube is cut from the bar. What is its weight* to four decimals of a pound ? 3. The area of a country is 32300000 acres. It consists of 3 kinds- of land the areas of which are in proportion to the numbers 2, 3 and . How many acres are there of each kind of land ? 4. If 3 per cent, stock is at 98^, how much money must be invested ' la the stock to yield an annual income of Ri2o ? 1897. I. The sum of 177 is to be divided among 15 men, 20 women and 30 children in such a manner that a man and a child may together receive as much as two women, and all the women may together receive ;6o. What will they respectively receive ? * Find the value of r~\ eott ^ to 7 P 1 *** 8 of decimals. - THE PUNJAB ENTRANCE PXPERS 433 3 A garrison of 700 men has provisions sufficient for 10 weeks. How long would they last if the garrison were reduced to 560 men ? 4. Find the least common multiple of 4}, $t 6| and 7f . 1898. ' 1. Find the cost of papering the walls of a room 22 feet long 18 feet wide and 20 feet high with rolls of paper 21 inches wide at R2. loa. pet roll of 12 linear yards. 2. Simplify : 3. A person holding ; 10,000 in the 3 per cents, sells out at 93! and invests the proceeds in 4 per cent, stock at loij. Find the change in hi- income, allowing per cent, commission in each transaction. 1899. i. The length of a hall is three times the breadth. The cost of white- washing the ceiling at 5V. per square yard is 4. 12s. T\d. and the cost of papering the walls at is. qd. per square yard is ^35. Find the height of the hall. 2. Show that the difference between the interest and the true discount on a given sum at a given rate for a given time is equal to the interest on the discount. 3. A man has $. I'js, consisting of sovereigns, half-crowns and shillings in the proportion of 2, 3, II. How many has he of each coin ? 4. Which is the better investment, the 3^ per cents, at 102 or the 3 per cents, at 97 ? 1900. 1. Find the square root of 400 1204 -090601. 2. Find the present worth of Rioooo due 8 years hence at 4$ per cent. 3. A rectangular courtyard, the sides of which are as 5 : n, costs Ri44. 60. for paving at ioa. 6p. per square yard. Find the length of its sides. 4. Show that compound interest reckoned quarterly at Ri. 30. 7^, per cent, is nearly equal to interest reckoned yearly at 5 per cent 1901. 1. Find the true discount on a bill for 721. 13*. &/. paid 73 days before due, the rate of interest being 3} per cent, per annum. 2. Divide each of the numbers 4061250 and 2572125 by 1251 and express the ratio of the quotients correctly to three places of decimals. C. A. 28 434 ARITHMETIC 3. A man buys eggs at if. 3< per dozen and sells them at us. & per hundred. Find his gain per cent. 4. There are four vessels of equal capacity : f of the first is filled with spirit, $ of the second, of the third, and of the last. The first is th^n filled with water and from this mixture the second is filled up, again from this second mixture the third is filled up and likewise the fourth from the third. What proportion of spirit to water is there in the fourth vessel ? 1902. 1. Define a prime number. Find the prime factors of 5S5t 5S5 2. A railway truck is 29 ft. 4 in. in length ; how many such truck* will be required to fill up the entire length of the line between Lahore and Amritsar, a distance of 32 miles ? 3. The difference between the simple and compound interest on a sum of money for 2 years at 5 per cent, per annum is fti2. Find the sum. 4. If 3 fowls and 4 pigeons cost &2. 30. 6/., and 5 fowls and a pigeons cost R2. I2a. t find what must be paid for 4 fowls and 3 pigeons. 5. A person sold 60 yards of cloth for &28* 20. gaining thereby the cost price of 9 yards. Find his gain per cent. 1903. 1. Show whether 983 is a prime number or not. The greatest common measure of two numbers is 373, and their least common multiple is 28721. Find the product of the two numbers. 2. A does \ of a piece of work in 3^ hours, B does \ of the remainder in I \ hours, and C finishes it in 5$ hours. How long would it have taken the three working together to do the work ? 3. Find the simple interest on 2,541. Sa. for 2 years 8 months at 70* per cent, per month. 4. Divide a sum of &345 I2fa. between A, J3 9 C, so that B may receive 25 per cent, more than A, and 20 per cent, more than C. 5. A bought 100 raaunds of wheat for 276. ga. , and sold it to B at a gain of 20 per cent. ; B sold it to C at a loss of 20 per cent. What price per maund did C pay for the wheat ? 1904. 1. Resolve 451584 into prime factors, and hence write down itt square root. Find the G. C. M. of the product of the first seven odd numbers and the product of the first eight even numbers. 2. Divide 3*14159 by 72, using factors, and finding the quotient correct to 3 decimal places* Find the product of 36*827 and 401*59 correct to a decimal places. THE PUNJAB ENTRANCE PAPERS 435 3. Find, by Practice, the price of 623 feet of piping at 5}. per foot. 4. A bought a bicycle for (275 and sold it to B at a gain of 2 annas in the rupee ; B sold it to C at a loss of 2| annas in the rupee. How much did C pay for it ? 5. Which of the fractions \\ and is nearer the exact value of Give reasons. , 1905. I. Find the value of: (i) (2 (ii) The square root of 8103060289. 2. The sum of 2,840. la. is to be divided between 7 men t II women, 5 boys, and 6 girls, so that for every R3. I2a. a man received A woman may get R2. 30., and for every R2. ioa. a woman received a boy may get Ri. 140., and a girl Ri. 2a. Find how much each person receives. 3. Find the difference between the interest and the oiscount on 8:25,078. 2d., the time being 21 months and the rate 4 per cent. 4. What will it cost to make a gravel walk 10 feet wide round the inside of the edge of a square field whose area is 10 acres, at 4^3. pet square yard ? 5. (i) The massacre at Cawnpore took place on the 2Sth June, 1857. What day of the week was it ? (ii) How many times in the course of the day do the bacds of a watch cross each other ? 1907. 1. Find the greatest number which will divide 16652, 10735 ac <* J 9$8, -and leave remainders 2, 5 and 7 respectively. 2. Find, by Practice, the value of 52 acres 3 roods 22 sq. poJes at ll$ . 12. 6 per acre. 3. What sum lent at compound interest will amount to R 16143-12-0 In 2\ years at 5 per cent, per annum ? 4. If 4 per cent, paper be at 1 10, what sum must I invest in order to iccure a monthly income of R374, after paying an income-tax of 5 pie in the rupee ? 5. Simplify the expressions : - 1908. I. The circumference of the iront wheel of a carriage is 6 J feet, *nd of the hind wheel 12$ feet. How many feet must the carriage pass over so that each wheel may make an exact number of complete n vo- lutions? 436 ARITHMETIC 2. Find the difference between 3 '141 59 and 3 + Also find the difference between theii squares. 3. A dealer bought a horse for no, and sold it the same day for 121. 15*., allowing the buyer 5 months' credit. Money being worth 3$ per cent, per annum, what was his gain per cent. ? 4. The total population of India is 294 millions, out of which 150 millions are males. Out of every i,oop males 98 can read and write, but only 5*3 per cent, of the total population can do so. Find the percent- age of the women of India who can read and write. 5. Prove that the L. C. M. of two given expressions may be found by, dividing their product by their H. C. F. 1909. 1. What part of Rl. 140. is T of - a * := ~ of A of Ri. Divide the difference between 5*5225 and the square of '075 by 126*1. 2. When 2\ tolas of gold can be purchased for ft 58. 6a. 6>. what should be paid for a tola of silver if its value is fixed in the ratio of I to 15$ to that of gold? 3. A 9 /?, and C could reap a field in 18 days ; /?, C, and D in 20 days 5. C, Dj and A in . 24 days ; and D, A, and B in 27 days. In what time would it be reaped by them all together ? 4. A bookseller began business on 1st January, 1908, with a capital of R8,ooo. On 1 5th September he was joined by a partner, who brought R 1 1, 500 to the business. At the end of December the profits were found to be Ri,654. Find, to the nearest anna, the share of each. 1910. I. What is the least number which when divided by 36, by 40, by 42,. gives in each case 5 as remainder ? 2. Express ^sV* as a decimal fraction. 3. Find, by Practice, the price of 37 cubic yards 3 cubic feet 280 cubic inches at R45. 8a. 6/. per cubic yard. 4. Explain what is meant by discount and present worth of a bin. Find the present worth and discount on a bill of 1,036. 45. due in 7J months, interest at 5i per cent. 5. A, B and C are partners in a business and their shares are in the proportion of } : \ : i. A withdraws half his capital at the end of 4. months, and after 8 months more a profit of R2,024 is divided. What U- <fg share ? THE PUNJAB ENTRANCE PAPERS 437 1911. 1. Define the following terms, and give examples to illustrate your definitions : Notation ; Numeration ; absolute and local values of digits. Write in words 2384751690. What is the local value of each of the significant digits in the following cumbers ? 92375, 247835. 2. Find the value of : - ^L_^? to 3 places of decimals. i- s/'4 Express -4^TV f ^33- l * s * 5l< as the fraction of 157. 17*. 8^. lOy-T T 3. What is meant by an aliquot part of a quantity ? Is $a. 4$. an -aliquot part of a Rupee ? Find, by Practice, the price of 256479 articles at 4. 12s. 6f d. per loo. 4. Define Present Worth and Discount. If the interest on RuSy. 80. at 3 per cent, is equal to the discount on 1193. 70. for the same time at the same rate, when is the latter sum due ? 5. A contractor undertook to build a house in 21 days and engaged 15 men to do the work. But after 10 days he found it necessary to engage 10 men more, and then he accomplished the work one day too soon. How many days behindhand would he have been if he had not engaged the 10 additional men ? 1912. I. Find the sum of 79368 added to itself 65937 times, and write the result in words. Find the number which will divide 5970 and 5260 and leave remainders -7 and 9 respectively. *. Shnplify W Which is the greater of 27*84 x '1481 and 3. State and illustrate the difference between direct proportion and inverse proportion. If 8 men and 12 boys can finish a piece of work in 12 days, in what time will 40 men and 45 boys finish another piece of work 3 times as great, supposing that 16 men can do as much work in 8 hours as 12 boys do in 24 hours f 4. A boy buys eggs at 9 for 4</. and sells them at zi for 5< What does he gain or lose per cent* ? 43$ ARITHMETIC The difference between the Simple Interest and the^Compound Interest on a certain sum of money for 2 years at 4 per cent." is R2o'. What is the sum ? 5. It is between 2 and 3 o'clock ; but a person looking at the clock, and mistaking the hour-hand for the minute-hand fancies that the time of the day is 57 minutes earlier than the reality. What is the true time ? 1913. 1. Explain what is meant by a prime number. Write down all the numbers between 108 and 120 which are prime. What is the least number which when divided by 12, 15, 20 or 54, leaves in each case a remainder of 4 ? 2. (a) Explain the meaning of | and T \, and show by a diagram that they are equal to one another. (6) Find the value of 3 '141 59 x '45078 correct to 4 places of decimal* (contracted method preferred). 3. Two men undertake to do a piece of work for Rs. 7. One can do it alone in 7 days, the other in 8 days. With the assistance of a boy they finish the work in 3 days. How should the money be divided ? 4. Exactly three years ago a man borrowed 3750 from a bank at 6 per cent, per annum. At the end of a year he paid the interest of that year and part of the loan, altogether ft 1200. Similarly he paid RSoo at the end of the second year. What sum must he now pay to clear oft the debt ? 5. The area of a square is 11370*32 square inches. Find the length fits diagonal. V. UNIVERSITY OF ALLAHABAD. ENTRANCE PAPERS. 1889. I. Define a fraction and shew that J=f . By how much does the difference of igV and A feU s ^ ort of tbeir sum ^ Express the defect as a decimal. * <> (&) Subtract '03 from X>3 and divide the result by '102. 3. Find the square root of *ooi to four places of decimals. What number has *i for its square root ? 4. What sum of money will amount to R 1,381 . 4 . o in 15 months at 5 per cent, per annum simple interest ? 5. How long will it take to walk along the four sides of a square field which contains 16 acres 401 square yards, at 3 miles an hour ? ALLAHABAD ENTRANCE PAPERS 439 6. A and B complete a piece of work in 8 days ; B and C do the same in 12 days ; and A % B and C finish it in 6 days. In how many days will A and C complete the work ? 7. A who travels 3$ miles an hour starts 2j hours before B who goes the same road at 4} miles an hour ; where will B overtake A ? 1890. 1. Multiply '347695 by 2*0026, and divide the product by "01905. 2. Simplify i| + 37-5i+2T-iiV. 3. Kind, by Practice or otherwise, the value of 2345 md. 27 seers 10 ch. of wheat at 3. 10. 8 per md. 4. Extract the square root of I - ('00135)* to 5 places of decimals. 5. The weight of a cu. in. of water is 253*17 grains, that of a cu. in. of air is '31 grains ; find to 3 places of decimals how many cu. in. of water are equal in weight to one cu. ft. of air. 6. On measuring a distance of 32 yd. with a rod of a certain length it \\as found that the rod was contained 41 times with $ an inch over. How many inches will there be over in measuring 44 yd. with the same rod ? 1891. i. Define "Notation", "Numeration" ; and prove that "three times four' "four times three". 2. Reduce to a single fraction : 4'ioo " of ' 7344 ' 3. The w me in a pipe when fall is worth ig. gs. gd. How much has leaked away if what is left is worth 9. l6s. T^fad* ? 4. In discounting a bill, what do you mean by "The Banker's profit" ? If the simple interest on ^923. iSs. 1 \d. amounts to 17. 9J. 3^. exactly in 138 days, what is the rate of interest per cent, per annum ? 5. Extract the square roots of 99,980,001 ; and of 6ojVv 1892. i. How is a fraction affected by adding the same number to the numerator and the denominator ? 1 -4- A Prove that ~ - is greater than } and less than J. 2. (a) Divide 3 + J{ 3 + i(3 + i|)H by -125. () Reduce }J and $ff f to their lowest terms and express their difference as a decimal. 3. Forty men finish a piece of work in 40 days, if 5 men leave the work after every tenth day, in what time will the whole work be completed ? 440 ARITHMETIC 4. Find the difference between the Simple Interest and Discount of ,330 in 4 years at 2j per cent, per annum. 5. Extract the square root of --- J ^ 1000 1893. 1. Two recurring decimals are added together ; prove that the numbei of digits in the period of the result cannot exceed the product of the numbers of the digits in the original periods. 2. Find the value of "54 of '30^2 of I mile 5 fur. 30 poles. 3. Multiply R2. anna I. by -. 4. Find, by Practice, the cost of 10 cwt. 3 qr. 23 Ib. 8 oz. at l* 5*. & per cwt. 5. A sum of money was divided amongst 5 people ; 4 of them received respectively '15, T \ '* I of tne whole, while the 5th received ; 105. 3*. 6d. What was the sum divided ? 6. An oz. of standard gold, one-twelfth of which is alloy, is worth 3. 171. io|< ; how many sovereigns would be coined from 36 Ib. 8 oz. of pure gold ? 7. Find the square root of 6246*057024 and of 7 ! 1894. i. (a) A multiplication sum having been worked is partially rubbed out ; the figures that remain are the entire multiplicand 999 and the last three digits 193 in the product. Restore the complete work. 2. (a) What decimal of Rioo must be added to f\&\ of 5. 10. 8. that the sum may be 10 annas ? (b) Extract the square root of 25-6. 3. Two trains start at the same time from Mirzapur and Delhi and proceed towards each other at the rates of 16 and 21 miles per howr respectively. When they meet it is found that one train has travelled 60 miles more than the other. Find the distance between the two stations. 4. Two years and six months ago, I borrowed a sum which with simple interest at 6 per cent, per annum now amounts to 8:638 .4.0. Find the sum. 1895. i. (a) Explain what is meant by the following terms : Prime factors ; common measure ; common multiple / lowest common multiple, ALLAHABAD ENTRANCE PAPERS 44! (b) A courtyard 452 feet long and 404 feet wide, is to be paved with square stones all of one size. What is the largest size which can be used ? 2. (a) Simplify 52? of i *f + *xj-}. (b) Find the square root of 3*1415926 to four places of decimals* 3. The difference between the Interest for 4 months, and the Discount, on a certain sum due in 4 months at 4 per cent. , is one rupee. What is the sum ? 4. A merchant sells silk of two qualities which cost him ft$. 50. 4/>. and R4. 40. 4^. per yard, respectively. The selling price of the latter is two-thirds that of the former, but the quantity sold is double and the merchant gains 25 per cent, on the whole. Calculate the selling price per yard of each. 5. A policeman goes after a thief who has loo yards' start ; if the policeman run a mile in six minutes, and the thief a mile in ten minutes ' how far will the thief have gone before he is overtaken ? 1896. I. Simplify : CM '*25X(*I75 Of -285714) 1 ' '0002$ 2. (a) Express of 75. & +1^25 of 55. - -545 of gs. 2d. as a decimal Traction of (b) Extract the square root of 40000*400001. 3. What is an aliquot part of a quantity ? Find, by Practice, the time of building a wall 27 yards long, I yard thick and 6 ft. high, of which one cubic yard is built in 3 hours 18 minutes and 45 seconds. 4. How far shall I ride with a friend who leaves Allahabad at 9 A* M. and will drive to Karchana which is 10 miles from Allahabad in one hour, that I may, by walking back at the rate of 4 miles an hour, reach home at 11-30 A. M. ? rA owes B fti435 due at the end of 4 months, 8:630 due at the end months, R86o due at the end of a year. B wants his money forth- with. What ought A to pay him reckoning interest at 7$ per cent. ? 1897. I. What is the largest number which divides both 2397 and 2491 without remainder ? What is the smallest number which is divisible by both of these numbers ? a* State and prove the rule for pointing in multiplication of decimals, why is tbt removal of the decimal point one place to the right equivalent 442 ARITHMETIC to multiplication by 10 ? Illustrate your answer by comparing the numbers 23 *oi 5 and 230*15. Find the square root of '08027. 3. A person lent another a sum of money for 72 days at 3 per cent, per annum. At the end of that time he received 293. I2s. o}</. What was the sum lent ? 4. The compound interest on a sum of money for 3 years at 5 per- cent, is ^331. os. 3< ; what is the simple interest ? 5. If a rupee is worth one shilling and three pence half-penny, and shilling is worth 1*25 francs, what is the value in francs of 1,365 rupees ? 1898. 1. Define measure of a number and find the G. C. M. of: (1) R2. 40. and 100. (2) fandf. Find the greatest number which will divide 13956 and 14565 and leave- a remainder 7 in each case. 2. Simplify : ia \ ('+)' <*) 3. Extract the square root of "-15 and calculate the difference between this square root and 3 + T VV2 to three places of decimals. 4. Find the cost in English money of travelling from Vienna to Trieste, a distance of 363 English miles, the average cost per German mile being 13 kreutxers. Given that I German mile =4 J English miles. 1=25 '5 francs. 375 francs = 105 kreutzers. 5. What is the present value of a legacy of 149. " 3<* due seveD fears hence at 2{ per cent, simple interest ? 1809. I. Simplify: % A of (i+5J)+f of A of (7-*f)-i and express { of fti. 50. as the decimal of Ri. 40. ALLAHABAD ENTRANCE PAPERS 443. 2. A number may be divided by 125 by multiplying it by 8, and then marking off the last three digits as decimals. Explain the reason for this | and divide 5335 by 125. 3. What is the meaning of an "aliquot part" ? Find, by Practice, the value of 24 tons 3 cwt. 2 qrs. 25 Ib. at 17. iif. 6tf. per ton. 4. A piece of work can be done in 72 days by 17 men working together. If after 9 days of work these are joined by 4 others, in how many days will the work be finished ? 5. Extract the square root of $ and.of '$ each to 4 places of decimals j and show that the square root of "4 is *6. 6. What is the difference between the interest on a bill of ; 138. 13*. 4. for 3 months at 4 per cent, per annum and the discount on the same- for a quarter of a year at the same rate ? 7. (a) A speculator sells at a profit of 50 per cent, but his purchaser fails and only pays 8a. in the rupee. How much per cent, does the speculator gain or lose by his venture ? (3) A person investing in the 4 per cents, receives 5 per cent, for his money. What is the price of stock ? 190O. 1. State the rules for multiplication and division of decimal fractions* Assuming that the surface of a sphere is 3*1416 times the square of ita diameter and that the earth is a sphere whose diameter is 8000 miles, find what fraction of the whole surface of the earth is the area of India which is 1350000 square miles. Express your result as a decimal fraction* 2. What are circulating decimals ? Distinguish between pure and> mixed circulating decimals. (a) Add together J, T j, T Vi /* anc * express the sum as a mixed- circulating decimal. (5) Reduce '0416 x 4 g 2 . * 7 of R8. 5*. to the fraction of I anna. 3. (a) Find, by Practice, the price of 100 bags of Rosa sugar, each weighing 4 seers 2 powas and 3 chataks, at 6a. o/. per seer. (3) Find the square root of JO'6i to three places of decimals. 4. What sum of money will amount to R3$28 in two years at 5 per- cent. compound interest, and what will it amount to in two more years ? 5. What monthly income will be derived from the investment of one* lakh of rupees in the 3} per cent Government of India paper at loo}{ ? 1901. I. (a) What is the greatest length which is contained a whole Dumber of times exactly in both 2$}{ feet and 21 A feet ? 444 ARITHMETIC (b) Find the value of ___ 2. (a) Express the difference between -9428571 and '57142 as vulgar fraction in its lowest terms. (6) Extract the square root of ^f ^-=- to five places of decimals. 0*03 3. In a two-mile race A wins, B being 22 yards behind and C 106 yards behind B. By how much would B beat C in a three-mile race. 4. What sum at compound interest will amount to 650 at the end of the first year and to R676 at the end of the second year ? 5. How much 3J per cent. Government Securities at 95i must be -sold in order to purchase enough 5 per cent. Calcutta Municipal Deben- tures at II9J to produce an annual income of 665, a brokerage of j pel cent, being charged on each transaction ? 1902. 1. Find the G. C. M. and also the L. C. M. of 49-383 and '142569. 2. Simplify J^x^ + ^xl^--.^ p y -075 i* 2-1 375 3. Find, by Practice, the value of 246! maunds of sugar at ft 1 3. 5a. 4/. per mauncl. 4. A and B have between them 132 horses ; '25 of X's= '142857 of /Ps. How many has each of them ? 5. Six men and five boys can do a piece of work in 7 days ; they rk at it till they have completed f of it ; then two of the men leave und two more boys come. How long will the work be in hand, if a boy does half as much work as a man ? 6. If I lend a friend 8:1,250 at 4 per cent, simple interest and tell him to keep it until principal and interest amount to R 1,666. ioa. 8/., 'how long will he have it ? 1003. 1. (a) How many lengths of 2} inches each can be cut from a rod 7} feet long, and what will be the length of the portion left ? (d) Reduce f of R4. la. 3^. to the fraction of { of R7. 140, S/, 2. (a) Divide '016085 by 3-125 ; and express I '4583 -M j as a decimal. (t>) Simplify : 5-5-081 4*9 --- ^\ - xS - ' ^3 4 '2 '33 3. A and B can do a piece of work in 12 days ; after working 2 days they are assisted by C 9 who works at the same rate as A, and the work 4s finished in 6} days more : in how many days would B alone do the work ? ALLAHABAD ENTRANCE PAPERS 449. 4. The 4 P. M. passenger train from Delhi to Tundla stops first at Ghaziabad, I2f miles distant, at 4-30 P. M. ; the whole journey is 127,); miles, and 20 per cent, of the time is expended in stoppages : at what time is the train due at Tundla ? 5. At what rate per cent, simple interest will 833. 50. 4^. amount to 952. io. 4/>. in 3 years and 2 months ? tt 1904. I. Simplify : -A -00281 x -0625 (6} 1-405 ' 2. (a) A bankrupt's liabilities are ^6,235. los. and he pays his creditors- 5*. 6d. in the pound. Find, by Practice, the amount of his assets* (b) Find the square root of 10*001 correct to four places of decimals, 3. If 3 p. c. more be gained by selling a horse for ^83. 5*. than by selling him for 81, what is the original price of the horse ? 4. What will Ri,ooo amount to in 3 years at 5 p. c. per annum compound interest ? 5. If the 3 per cent, consols are at 92$, what sum of money must be invested in this stock to get an annual income of ^630 brokerage being \ per cent. ? 1905. I. (a) Simplify : 4i (3) Find the value of -p correct to four places of decimals. 2. (a) Add together '175 of I ton, '83 of I cwti and '93 of I Ib. and reduce the sum to the decimal of 10 tons. (5) Find, by Practice, the rent of 3 acres I rood 27 poles of land at l l6s. 8^. per acre. 3. By selling a horse for 50 a man lost 4 per cent. ; find what would have been his gain or loss per cent, if it had been sold for R6o. 4. Find the discount on Ri,ooo due 3 months hence at 4 per cent* per annum. 5. A person transfers 1000 stock from the 4 per cents, at 90 to the 3 pet cents, at 72 : find the alteration in his income. 446 ARITHMETIC 1906. 1. A merchant has three kinds of wine : of the first kind 403 gallons, of the second 434 gallons, and of the third 465 gallons. What is the least number of full casks of equal size in which this can be stored without mixing ? 2. Find the sum of money that is the same fraction of 5 crowns that Rz. 8fl., is of ft2. 5*. 4^. H 3. A sum of money amounts in ip years at 4j per cent, simple interest to 2,972. 8a. In how many years will it amount to R4.356. 40. ? 4. Extract the square root of 15848361. 1907. 1. Is 823 a prime number ? Why is it unnecessary to try factors above 23 in answering the question ? 2. Show that to 3 figures v =% and that to 5 figures =1$* where v =3'i4i59265 3. Find the quotient of 68937825 by 72*6328 correct to four figures. 4. Find to 3 decimal places the square root of $. 1908. .1. Find the sum of the 21 odd numbers which follow 15432. 2. Reduce % + f + f + - V- - A to a fraction in its lowest terms. 3. Find all the prime numbers less than a hundred. 1909. z. A metre = 39 '3708 inches. Express '325 of a metre as a decimal of a yard (to six figures). 2. What will be the gain per cent, if mangoes bought at the rate ef six for 50. are sold at the rate of five for 6a. ? 3. In the first four months of 1906 the Indian Government sold BiJli .amounting to R97, 984, 3 1 1, obtaining 6,537,578 in exchange. Find the value of a rupee in English money to the nearest tenth of a penny. N. B. Use no more figures than are necessary to obtain a result to the degree of accuracy indicated. 4. A holder of ft55oo of 3$% Government paper sells at 9iJ and invests in 4% stock at zoz. If the brokerage is \ for the first and J foi the second, find the change in his income. ' 1910. z. Write in figures the number ninety-nine billion ninety-nine million ninety-nine thousand and ninety-nine. Zl^ + 4-*55x.o -0175 -00032 ALLAHABAD ENTRANCE PAPERS 447 a. Find the least integer exactly divisible by Si, 7j, and 9. Extract the square root of 76300225. 3. What sum put out at compound interest at 5 per cent, would amount in 3 years to 810. 6s. tyt. ? 1911. 1. Write in figures the number nine billion eighty-nine million nine thousand and ten. Simplify -442857! + -557i42 t 2285714 + 7714285 2. Define the terms yard and metre. If one inch is equal to 25 '4 millimetres, find the number of kilometres rn a mile. 3. Extract the square root of 1157428441. 1912. I. A hall is 10*01 metres high, 40 metres long, and 8*001 metres wide. Find the number of cubic millimetres it contains, and write your answet in words. ad (2) find 0-41375 of 2. los. 3. Find the price of 3 per cent, stock when an increase of income pi 5. 61. 3< is made by transferring to them a sum of 4, 37 5 3i per cent. stock at 95J. 1913. ^ I. Simplify : 2. Extract the eighth root of 214358881. 3. A man subscribes to a provident fund 4% of his income ; on the remainder he pays income-tax at 5 pies in rupee, and after this deduction he gives T ^ of the remainder in charity. Of the remainder he gives T V to bis mother, who thus receives ft 12 a month. Find the man's gross annual income. 4. An acre is 0*40467 hectare, and i is equal to 25*25 francs. An SRtate measuring 1927 hectares is sold for ten million one hundred thousand francs. What is this in pounds per acre ? 1914. I. (l) Simplify: .- 4*-3*' A 4 " 448 ARITHMETIC (2) Add together j of i 9 \ of u., and ? of irf., and express the sum as the decimal traction (correct to two places) of one guinea. 2* Find the square root of 25+^125 correct to three places of decimals. 3. A sells an article to B at a profit of 20 per cent. B sells it to C at a profit of 5 per cent. If C pays 70*., what did it cost A ? 4. I invest equal sums in a 4 per cent, stock and in a 3 per cent. stock and get 5 per cent, for my money ; the 4 per cents, are at 90 5 what is the price of the 3 per cents. ? 1915. 1. Find the square roots of (I) 3036-01, (2) ! to three places off decimals. 2. Prove that the product of any two numbers is equal to the producJ of their H. C. F. and their L. C. M. The L. C. M. of two numbers is 244188, and their H. C. F. is 84* If one of the numbers is 1428, find the other. 3. A rectangular lawn 51 ft. long is surrounded by a path which is 4 ft. 6 in. wide. If the path is 96 sq. yds. in area, find the breadth of the lawn. 4. If 766. 13*. 4^. is the discount on 4600 due in ^\ years, what la the rate per cent., at simple interest ? 1916. I. Simplify of 4 tons 7 cwt- 2. Carpet 2 ft. wide at 6s. get. per yd. for a room 25 ft. 4 in. wide costs ^30. Ss. , and paper I ft. Sin. wide at 4j*/. per yd. for its walls costs $. $s. (no allowance to be made for doors or windows). What is the height of the room ? 3. The manufacturer of an article makes a profit of 25 per cent* the wholesale dealer makes a profit of 20 per cent., and the retailer makes a profit of 28 per cent. What is the cost of production of an article retailed for 16 shillings ? 1917. I. (i) Simplify _ 2j 7 22i-it5* Tl i3 5 ALLAHABAD ENTRANCE PAPERS 449 (2) Find a fourth proportional to 1$, 0*09, ?V 5 anc * express the result as a decimal. 2. (i) Reduce 0-07 of i. $s. + 0*675 of 2. is. & +0-1875 of 8rf. to the decimal of lo t (2) Find the square root of 2 to four places of decimals. 3. A man buys milk at 2\d. per quart, dilutes it with water and sells the mixture at 3^. per quart. How much water is added to each quart ol milk if his profit is 60 per cent. ? 4. Find the present value of 845, due 2 years hence, compound interest being reckoned at 4 per cent, per annum. 1918. 1. (a) Find the number nearest to 100,000 that can be divided exactly by 2, 3, 4, 5, 6 and 7 respectively. (b) A man in India wishes to send to his son in England 300 a year in monthly instalments. How much will he have to pay monthly in rupees ; the value of i rupee in English money being o. is. 4-3?ci. ? 2. One revolution of the pedal crank drives a bicycle a distance equal to the circumference of a circle of 70 ins. diameter. How many revolu- tions does the crank make in travelling I mile ? If the wheels are 28 ins. in diameter how often do they revolve in the same distance? [IT ^-V 1 -] 3. One clock gains 25 sees, a day while another loses I minute a day. They are both set at the rigut time at 8 A. M. in August 15. On what day and at what time will they differ by I hour ? 1910. T. (a) Find all the prime numbers that divide both 1287 and 1144 without remainders. (l>) Simplify rn 3>+(4txtt). 12-32 - 7*56 lj 6J-(ijx)' m 20-35+3-45* 2. The inside measurements of a room are 42 ft. 6 ins. and 22 ft, 9 ins>. ; the walls are 2 ft. 3 ins. thick and there is a verandah all round to ft. 6 ins. wide. Find the cost of paving the verandah with tiles measuring 4^ ins. by 3 ins. and costing 6 pies each. 3. Which is the system of payment most advantageous for the student if the rate of interest obtainable is 6/ in the following case ? ** For students commencing the course the entrance fee is Rs. 30. The entrance fee is payable by all students at the commencement of the course or may be paid in three instalments of Ri2 each at the beginning of the first, secord and third years respectively." C, A. ?9 450 ARITHMETIC VI. UNIVERSITY OF PATNA. ENTRANCE PAPERS. 1918. COMPULSORY PAPER. 1. (a) Multiply 876095 by 567049- Or, Two numbers when divided by a certain divisor leave the remainders 4375 an d 2986 respectively ; but when the sum of the two numbers is divided by the same divisor, the remainder is 2361. Find the divisor. (b) Find the G. C. M. of 64176 and 119184. Or, What is the least number which, when divided by 6, 8, 12, 15, or 20, leaves a remainder of 5 ? 2. (a) Simplify 4? -2*. I 2+-. 5-i Or, Find the cost of 313 articles at 2. i>js. I id. each. (d) Multiply 3*25 by o'oi33, and divide the product by 3*64. Or, Find the value of ^~ -* of 3 guineas, and express the result as ito. 4^* decimal fraction of 5. 3. (a). What sum will amount to R587. 8a. in 3} years at 5 pet cent, per annum simple interest ? Or, In what time will .12345. 13 s. g\d. double itself at 4 per cent, pef annum simple interest ? (6} A does ff of a piece of work in 14 days ; he then calls in /?, and they finish the work in 2 days. How long would B take to do the work by himself ? 1018. ADDITIONAL PAPER. 1. Evaluate s/*67 - */*o7 to 6 places of decimals. 2. How many litres of water weigh 1000 Ibs., given that one cubic foot of water weighs looo ozs,, and one metre =39 '37 inches ? PATNA ENTRANCE PAPERS 45! 3. Find the value of the following series correct to four places of decimals : - 1 +_'_+_!_+_ L+._ M 3'i 3"2 3"3 3"4 4. What must be the least number of soldiers in a regimant, to admit of its being drawn up 5, 6, 7, 8, 9, or 10 deep, and also of its being formed into a solid square ? 1919. COMPULSORY PAPER. 1. (a) Multiply 79094451 by 7640950. Or, Find the greatest and least numbers of six digits which are exactly divisible by 789. (b) A heap of pebbles can be made up exactly into groups of 25 ; but when made up into groups of 18, 27 and 32, there is in each case a remainder of II : find the least number of pebbles such a heap can contain. Or, A grocer buys 10 cwt. 3 qrs. 21 Ibs. of sugar for ^30, and pays 12s. 6<f. for expenses ; at what rate must he sell it per pound to clear 6s. 3</. by his bargain ? 2. (a) Simplify Express of 12s. 6d. + '625 of 7*. 6d. - -565 of 165. 6d. as the 'decimal of i. (b) Find the cost of 9 yds. 2 ft. 10 in. at 5*. 7J^1 per yard. Or, What would be the cost of painting the four walls of a room whose length is 24 ft. 3 in., breadth 15 ft. 3 in., and height if ft. 6 in., at 4*. a square foot ? 3. (a) What sum will amount to ^"425. 19*. 4^. in 10 years at 3} per cent, simple interest ? Or, If the 6d. loaf weigh 4*35 Ibs* when wheat is 5*75*. per bushel, what ought to be paid for 493 Ibs. of bread when wheat is 9*7*. per bushel ? 452 ARITHMETIC (5) If 200 men can make an embankment 5 miles long in 25 days? how much overtime must 60 men work to finish an embankment 2 miles long in 32 days, 12 hours being a day's work ? Or, A man walks a certain distance, and rides back in 3 hrs. 45 min. ; he could ride both ways in 2$ hrs. How long would it take him to walk both ways ? 1919. ADDITIONAL PAPER 1. Find the square root of '00249976000576. Or, A square field contains 40 acres. Find the cost of running a fence round it at 2s. 6d. a yard. 2. Given that a metre contains 39*37 inches, express five miles ia kilometres and metres, correct to the nearest metre. Or, Find the value of !_ l +JL- I. +_. I __. 1-2 1-2-3 i'2'3'4 I '2 -3 '4 '5 correct to 4 places of decimals. 1920 COMPULSORY PAPER. 1. Multiply 915625 by 961024. 2. Prove that 95785 - 94340* = 16575*. 3. Reduce to its lowest terms Hsifl* 4. Simplify of-^})ofHLi v + T 9** lOflT. T *' 4tons3cwt. 5. Express J of 7*. 6d. 4-1-25 of 51. -0-545 of 9*. 2d. as a decimal of 6. Find the cost of 56375 articles at 2. i$s. yd. per hundred. 7. A tradesman who commenced business 5! years ago increased his capital at the rate of 1 5 per cent, per annum, simple interest, and it now amounts to 5960. What sum did he start with ? PATNA ENTRANCE PAPERS 453 1920. ADDITIONAL PAPER. r > Find the smallest number that must be added to 153*140025 to maki it a perfect square. CV-, A piece of silk cost ^84. os. 40?., and there were as many yauls in the piece as there were pence in the price of a yard. Find the length of the piece. 2. Either^ The Great Wall of China is said to be 2400 km. long and 7625 mm. thick at the bottom. Find, to the nearest square foot, the area of the ground it stands upon, (i metre = 39*37 inches.) Or, Employ the contracted method to divide 2*6289475 by 306*5 correct to the sixth decimal place. ANSWERS TO EXAMPLES. Examples. 1. 1. Ten ; sixteen ; forty-eight ; ninety-nine ; seventy-six ; fbrtythree ; fifty ; thirty-one ; sixty-two. 2. One hundred ; one hundred and eleven ; nine hundred and two ; six hundred and twenty ; three hundred ; one hundred and three ; two hundred and thirty-four ; one hundred and thirty. 3. Nine thousand, two hundred and sixteen ; five thousand, four hundred and nine ; five thousand and four ; one thousand and eleven ; one thousand, two hundred and ten ; nine thousand ; nine thousand, nine hundred and ninety-nine. 4. Twelve thousand, three hundred and forty-five ; twenty- thousand, one hundred and three ; forty thousand and forty ; fifty thousand and one ; ninety thousand, six hundred ; eighty-nine thousand, three hundred and forty-six. 5. Five hundred thousand ; seven hundred and eight thousand, nine hundred ; one hundred and two thousand and thirty ; three hundred and nine thousand, eight hundred and nine ; three hundred and seventy-nine thousand, five hundred and eighty-six. 6. Seven million, two hundred and thirty-four thousand, six- hundred and fifty-one ; seven million, ninety thousand, seren hundred and nine ; nine million ; seven million, eight hundred thousand and'forty ; three million, five hundred and sixty-seven thousand, eight hundred and ninety-one. 7. Thirty-two million, five hundred and sixty-seven thousand, eight hundred and ninety-two ; thirty-four million, eighy-three thousand and ninety two ; ninety million, nine thousand ; fifty-five million , five hundred thousand and fifty-five. 8. Seven hundred and eighty-nine million, three hundred and forty-five thousand, six hundred and twenty-one ; three hundred and ninety million, eighty-five thousand two hundred and twenty- two million. 9. Seven thousand and nine million, fifty-six thousand, seven hundred 5 three thousand two hundred and fifty-nine million, two hundred and eighty-seven thousand, eight hundred and ninety-one ; eight thousand and seventy million, eighty-eight thousand, two hundred. 10. Thirty-two thousand and five hundred million, ninety -four thousand and one ; three hundred and eight thousand five hundred and six million, eight thousand, two hundred and thirty ; one billion, three hundred and fifty-seven thousand nine hundred and ANSWERS TO EXAMPLES 455 eighty-six million, four hundred and twenty-eight thousand, one hundred and twenty-three. 11. 70, 2 ; 300, 50, 9 ; 4000, 200, 3 ; 70000, 800, 9 ; looooooooo, 300000000, 400000, 50000, 700, 80, 9 ; 3000000000000, 70000000000, 9000000000, 4000000, 70000, 8000, 20, 3. 12. Counting from left, the zeros respectively indicate the absence of thousands, tens ; tens of millions, hundreds of thou- sands, tens of thousands, hundreds, units ; tens of thousands of millions, thousands of millions, tens of millions, thousands, tens. 13. (10,000) ten thousand ; (9)999) nine thousandi nine hundred and ninety-nine. Examples. 2. I. 13 ; 17 ; 19; 12 ; II. 2. 23534540527. 3. 77 ; 9 ; 84 ; 63. 4. 342 ; 486 ; 504 ; 900. 6. 203 ; 430 ; 555 ; 400. 6. 892 ; 704 ; 640 ; 512. 7. 7)835 ; 9,028 ; 6,009 ; 4,000 ; 6,085. 8. 5,992 ; 8,074 5 2,003 5 4)040 ; 3>4O3- 9. 1,200 ; 80,008 ; 18,454 ; 36,012 ; 90,000. 1O. 20,070 ; 30,008 ; 54,400 ; 16,004. II. 405,000 ; 800,040 5 702,074. 12. 3,000,904 59,000,400 ; 15,000,050 ; 108,003,004 ; 4,005,000. 13. 5,000,700,028 ; 3i5)764>oo9,oo3. 14. 3,000,000,000,050 j 405,000,010,020,007 ; 1,000,001,001,000 ; 6 ,000,000,000,006 . 15. 512,255,762,713,473. 16. 12,000,000,000,012 ; 700,000,000,700,700 5 3,000,003,003,303. IV. 7,305,000,502,006,024 ; 47,000,047,047,047. 18. 1,000,000 ; 99)999- 19. The number expressed in figures is 7707 5 therefore (count- Ing r rom left), the first boy's mistake consisted in writing three ciphers unnecessarily to the right of the first 7, and two ciphers instead of one to *he ri^ht of the second 7 ; the second boy's mistake consisted in omitting to write a cipher to the right of the second 7. Examples. 3. L Three lacs, forty-five thousand, five hundred and forty three ; thirty lacs, twenty thousand and fifty ; seventy-nine -lacs, ninety 456 ARITHMETIC thousand, five hundred and seventy ; seventy lacs, fifty thousand, three hundred and four. 2. <5ne crore, twenty-three lacs, forty-five thousand, six hun- dred and seventy-eight ; thirty crores, fifty-seven lacs, fifty thousand and eighty ; four crores, fifty lacs. 3. Twenty-three crores, seventy-eight thousand and one ; seven hundred and eight crores, nine lacs, four thousand and eighty ; three hundred and seventy-nine crores, forty-eight lacs, fifty-seven thousand, six hundred and twelve. 4. Eight hundred and twenty-seven crores, forty lacs, fifty- seven thousand and nine ; three hundred and fifty crores, owe thousand, two hundred and thirty ; three hundred and ten crores, thirty-seven lacs, five thousand and forty. 6. One hundred and twenty-three crores, forty-five lacs, sixty- seven thousand, eight hundred and ninety ; six hundred crores, seven lacs, eighty-nine thousand ; five hundred and one crores, seven lacs, two thousand and nine. 6. 1,14,000 ; 78,00,000 ; 15,04,030 ; 7,00,007. 7. 1,00,00,500 ; 28,03,00,004 ; 20,00,00,000 ; 1,01,01,031. 8. 300,05,04,000 ; 101,01,00,101. 8. 328,17,45,71$. 10. 705,17,24,738. 11. One hundred thousand ; ten lacs ; ten million. 12. 103,028,401 = 10,30,28,401 which is read ten crores, thirty lacs, twenty-eight thousand, four hundred and one. 13. 103,07,00,704=1,030,700,704 which is read one thousand and thirty million, seven hundred thousand, seven hundred and four. Examples. 4. L 6. 2. 9. 3. 49. 4. 99. 5. 75. 0. 264. 7. 609. 8. 664. 9. 1990. 1O. 60010. 11. 2764. 12. XLIV. 13. LXVI. 14. LXX1X. 16. LXXXIII. ,16. CXLIX. 17. CDXXXX I. 18. CMXC. 19. MCCCLI. 20. "VDCLXX. 21. MMMCXLIX. 22. XLVCMLXXVIII. 23. M. Examples. 5. 1. 21. 2. 30. 3. 31. 4. 29. 6. 34. 6. 98. - 7. 99. 8. 77, 9. 140. 10. 163. \ ANSWERS TO EXAMPLES 457 11. 1323. 12. 1151. 13. 792- 14. 2727. 15. 2000. 16. 14129. 17. 9996. 18. 3674. 19. 5620. 20. 4696. '21. 14617$. 22. 59038. 23. 234671. 24. 379462. 25. 45271. 26. 2262514 27. 920114. 28. 982255. 29. 7474095. 30. 3967934L 31. 42450564. 32. 496651. '33. 92439. 34. 8082862. 35. 931979. 36. 531284. 37. 5694685. 38. 311989. 39. 9925098. 40. 984610763. 41. 74307. 42. 10246451. 43. 765168567. 44. 3129223218. 45, 46451330.46. 3936. 47. 1890. 48. 365. 49. 741. 5O. 2040. 51. 138187. 52. 42004 rupees, 53. 7193165 maunds. 54. 1468. 65. 163554. Examples. 6. 1. 43. 2. 52. 3. 222. 4. 543. 5. 4321. 6 - 25. 7. 49. 8. 8. 9. 9. 10. 33. II. 189. 12. 90. 13. 178. 14. 459. 15. 315. 16. 4641. 17. 47017- 18. 30532. 19. 27273. 20. 41976. 21. 2679. 22. 689357. 28. 687590. 24. 735347. 25. 6499247. 26. 5546. 27. 85416. 28. 707467. 29. 3562. 30. i. 3L 688881. 32. 390704. 33. 61059. 34. 999981 ; 999695 ; 990525 ; 900554 ; 956500. 35. 92964. 36. 99971. 37. 9998999. 38. 9921. 39. 83 years. 40. In 1642. 41. 923. 42. 117681 rupees. 43. 325 rupees. 44. 9460 rupees. 46. 16516. 46. 777101. 47. 6390. 48. 2000. 49. 35242 rupees. 50. 30000600. 51. 4503600. Examples. 7. 1. 458. 2. 62784. 3. 2740. 4. 288. 6. 19835. 6. 970. 7. 9960. 8. 14006. 9. 92788. 10. 99803. Examples. 8. 1. 46. 2. 96. 3. 84. 4. 195. 6. 282. 6. 522. 7. 784. 8. 684. 9. 765. 10. 987. 11. 2835. 12. 79"- 13. 19470. 14. 35445. 15. 73648. 16, 315824. 17. 623245. 18. 769527. 4$S ARITHMETIC 19. 68158 ; 102237 ; 136316 ; I739S 5 204474 ; 238553 ; 272632 & 306711. 20. 3625. Examples. 9. 1. 10770. 2. 281400. 3. 195250. 4. 421800. 6. 35100^ 6. 5760300. 7. 24040000. 8. 81036000. 9. 183018000=. 1O. 656550 ; 5836000 ; 51065000 ; 437700000 ; 3647500000. Examples. 1O. 1. 20250. 2. 88592. 3. 51060. 4. 17153400, 6. 7920848. 6, 7845984. 7. 501264. 8. 2877420,. 9. 41269151. 1O. 712823175. 11. 546962350. 12. 8741795904, 13. 60956040000. 14. 73866065616. 16. 4278833730. 16. 7716453390592. 17. 22237262250000.. 18. 38934178244-. 19. 2993392500000. 20. 8784920736579* 21. 2247882292480. 22. 27706959000. 23. 62834211900. 24. 581199247904. 25. 10612283522500. 26. 234916991513. 27. 83779349418000. 28. 47619. 29. 45708. 30. 93652. 31. 99148. 32. 73350. 33. 140624, 84. 230690. 35. 505260. 36. 82764 37. 711360. 88. 2170671. 39. 316875 rupees. 4O. 10727350* 41. 20692 maunds. 42. 33114. - 43. 3744- Examples. 11. 1. 432. 2. 4720645. 3. 16905000. 4. 1905700. 5. 1153800. 6, 44274384. 7. 1314. 8. 86400. 9. 3200. 10, 399735. 11. 9425. 12. 2208. Examples. 12. 1. See the Multiplication Tables. 2. 576. 3, 2500. 4. 4624. 5. 10000. 0. 12544. 7. 61504. 8. 531441. 9. 763876. 10. i ; 8 ; 27 ; 64 5 125 ; 216 ; 343 5 5* 2 ; 729 ; 1000 ; 1331 ; 1728 j, 2197 ; 2744 ; 3375 5 4096 ; 49*3 ; 5832 ; 6859 ; 8000. 11. 804357. 12. loooooo. 13. 679151439. 14. 170953875.' 15, 29503629. 16. 62913. ANSWERS TO EXAMPLES 459 1. 1 88. 4. 2333, rem. i, 7. 20511, rsrn. I, 1O. 2469. Examples. 13. 2. 4617. 3. 3542, rem. I. 6. 2675. 6. 30042. 8. 8203, rem. I. 9. 11419, rem. 2. 11. 20040. 12. 15555, rem. 2, 13. 15067, rem. i. 14. 14557, rem. 3. 15. 13155, rem. 4. 16. 541, rem 2. 17. 6569, rem. 3. 18. 4640. 19. 4809, rem. 2. 20. 4313, rem. 5. 21. 2005, rem. 2. 22. 8013, rem. 7. 23. loooo, rem. i. 24. 8666, rem. 6. 25. 3897, rem. 2. 26. 2456. 27. 3200. 28. 7070, rem, 7. 29. 2440, rem. 2. 30. 3004, rem. 8. 31. 1498, rem 8. 32. 1947, rem. 4. 33. 2002, rem. 4. 34. 169, rem. 29, 35. 11404, rem. 22. 36. 135, rem. 30. 37. 407, rem. So. 38. 521, rem. 89. 39. 87, rem. 300. 4O. 694, rem. 2. 41. 48, rem. 101. 42. 45, rem. 254. 43. 1 60, rem. 289. 44. 58, rem. 356. 45. 44, rem. 357. 46. 453, rem. 219. 47. 706, rem. 354. 48. 112, rem. 4543. 49. 234, rem. 641. 50. 3263, rem. 931. 51. 1017, rem. 2556 62. 381, rem. 1664. 53. 2559, rem. 2316. 54. 6652, rem. 5423. 65. 114285, rem. 3351. 66. 1250, rem. 539. 57. 15200^ rem. 10321. 58. 15005, rem. 54720. 59. 1338, rem. 110580. 6O. 423297, rem. 37606. 61. 240100, rem. 117400. 62. 420, rem. 114903. 63. 63261, rem. 6731383. 64. 8425323113, rem. 75. 65. 9886426883, rem. 672. 66. 507. 67. 36, 68. 528 times. 69. 13, 7O. 229 times. 71. 30115. 72. 7674^ 73. 375 rupees. 74. 256 days. 75. 22. Examples. 14. 1. 17280, rem. i. 2. 26310. 3. 20089, rem. 2. 4. 2558, rem. 2. 5. 3842, rem. 5. 6. 14057, rem. i. 7. 4320, rem. 7. 8. 2207, rem. 7. 9. 3456, rem. 7. 10, 52731, rem. 5. 11. 67253, rem. 4. 12. 10437, rem. 8. 13. 32198, rem. 10, 14. 49538, rem. 10. 15. 58491, rem. 6. <46o ARITHMETIC 16. 228850, rem. 7. 17. 455961, rem. 7. 18. 649772, rem. 10. 19. (i) 1728394, rem. i ; 1152263 ; 864197, rem. i ; 691357, rem. 4 ; 576131, rem. 3 ; 493827 ; 432098, rem. 5 ; 384087* rem. 6 ; 345678, rem, 9 ; 314253, rem. 6 ; 288065, rem. 9 ; 265906, rem. ii ; 24691 3, ( rem. 7 ; 230452, rem. 9 ; 216049, rem. 5 ; 203340, rem. 9 ; 192043, rem. 15 ; 181936, rem. 5 ; 172839, rem. 9. (ii) 40352015 ; 26901343, rem. i ; 20176007, rem. 2 ; 16140806 ; 13450671, rem. 4 ; 11529147, rem. I ; 10088003, rem. 6 ; 8967114, rem. 4 ; 8070403 ; 7336730 ; 6725335, rem. 10 ; 6208002, rem. 4 ; 5764573, rem. 8 ; 5380268, rem. 10 ; 5044001, rem. 14 ; 4747295) r em. 15 ; 4483557) rem. 4 ; 4247580, rem. 10 ; 4035201, rem. 10. v(iii) 493827160, rem. i ; 329218107 ; 246913580, rem. I ; 197530864, rem. I ; 164609053, rem. 3 ; 141093474, rem, 3 ; 123456790, rem. I ; 109739369 ; 98765432, rem. I ; 89786756, rem. 5 ; 82304526, rem. 9 ; 759734O9, rem. 4 ; 70546737, rem. 3 ; 65843621, rem. 6 ; 61728395, rem. i ; 58007313 ; 54869684, rem. 9 ; 51981806, rem. 7 ; 49382716, rem. I. Examples. 15. 1. 2io. 2. 465- 3. 1035. 4^ 2850. 6. 5050. 6. 12*54. 7. 33I5. 8/ 15150. ^ 9. 245. 10. 44818. ll' 4568. 12. 37951- 13. 4628 and 3899. 14. 5444 and 4556. Examples. 16. 1. 17472. 2. 337050. 3. 672840. 4. 132624. 5. 244160. 6! 94976. 7. 2599400. 8. 601425. 9. 1233282. 10. 143472. 11. 446048. 12. 3532008. 13. 295100780. 14. 1220242681. 15. 3625. 16. 1645- 17. 4060. 18. 2IOO. 19. 18225. 20. 2300. 21. 12250. 22. 15625. 23. 25875. 24. 11088. 26. 281718. 26. 2039796. 27. 4201 58. 28. 4182640, 29. 8267519. 30. 36950. 3L 5563- 32. 31220. ANSWERS TO EXAMPLES 461 33. 53I75- 34. 4560. 35. 59175. 36. 1225. 37. 3025. 38. 7396. 39. 9409. 40. 105625.. 41. 216225. 42. 606841. 43. 802816. Examples. 17. 1. 39. 2. 23. 3. 42. 4. 68. 5. 23. 6. 330, rem. 24, 7. 540, rem. 40. 8. 372, rem. 20. 9. 755, rem. 84. 10. 677, rem. 117. 11. 2935, rem. 168.. 12. 12882, rem. 58. 13. 359) rem. 319. 14. 2057, rem. 294. 15. 1422, rem. 138. 16. 389, rem. 4. 17. 34, rem. 56. 18. 89, rem. 345. 19. 827, rem. 46. 20. 89, rem. 346. 21. 12, rem. 3456. 22. 129, rem. 22. 23. 157, rem. 42. 24. 123, rem. 67. 25. 38, rem. 1368. 26. 46, rem. 894. 27. 783, rem. 10743. 28. 122, rem. 893. 29. 9733) rem. 176. 30. 2716, rem. 187. 31. 75, rem. 3. 32. 937) rem. 4. 33. 255, rem. I. 34. 313, rem. 20. 35. 3310, rem. 19. 36. 5515, rem. 17. 37. 670, rem. 14. 38. 1103, rem. 16. 39. 30, rem, 42. 40. 24, rem. 14. 41. 22, rem. 19. 42. 20, rem. 21. 43. 16, rem. 34. 44. 21, rem. 29, 45. 108, rem. 66. Examples. 18. 1. 2195. 2. 75582. 3. 871882. 4. 6. 85040. 7. 1595. 8. 8832. 9. 11. 49, rem. 74. 12. 118, rem. 53. 14. 2012, rem. 284. 15. 1064, rem. 3045. Examples. 19. 1. 2771928. 4. 94876320. 7. 1 53660000. 1. 14. 0. 4. 11. 4- 16. 14. 2. 7386918. 5. 627399162. 8. 3I3I99250. Examples. lOa. 2. 6. 3. 2. 4. 7. 31. 8. 2. 9. 12. 14. 13. o. 14. 17. 83. 18. 65. 19, 304166. 6. 18776. 92080. 10. 45I3S. 13, 113, rem. 79. 16. 866, rem. 2377. 3- 3747321. 6. 222013980. 9. 6783119796. 3. 5. 20. 2. 10. 28. io, 15. 450* 2oa 20. a . 4 62 1. 2548. 6. 9001. 11. 1477- ^15. 40023 tim 19. 1 50; 83. 23. 89. 27. 23 years. 31. 615. 35. 313288352. 36. 475 rupees. 37. A, 58 ; B, 34 ; C, 42. 38, A 9 40 rupees ; J3, 39 rupees ; C, 30 rupees. 39. 135 rupees. *4<J, 1 8 per rupee. 41. 60 seers ; 100 seers. ^42. 1800 rupees 48. 5 years. 44. 10 years ; 70 years. 45. 60. 46. 3 P. M. ARITHMETIC Miscellaneous Examples. 0. 2 . 2022. 3. 8611. 4. 621. 5. 788. 7 . 316. 8. ii. 9. 3791- 10. 17. L2 . 6354. 13. 33794 . 14. I. s, rem. 21. 16. 532. 17. 176. 18. 34. 20. 7 times. 21. 1545- 22. 1 59943- 24. 362. 25. 514590. 26. 99 and 106, 28. 176913. 29. 189461. 30. 71265. 32. 134807. 33. 545 pice. 34. 812168364. 1. 6240. 5. 1 2 la. Examples. 31. 2. 16640. 3. 1153280. 4. 591680. 6. 3720. 7. 6040. 8. 8300. 9. 59328/J. 10. 142080^. 11. 653184^. 12. 38700^. 13. 21624^. 14. I35324A 15. 5i87A 16. 7641^. 17. 13055^. 18. 194 pice ; 582^. 19. 501 pice ; 1503^. 20. 635 pice ; 1905^. 21. 7410. 22. 1632. 23. 631. 24. loo, 25. 3896. 26. 482. 27. 14400*. 2& 4800*. 29. 14180$. SO. 61005. 31. 405^. 82. 532J. 33. 6175. 34. 719*. 35. 84000*. 36. i6o8oo</. 37. 16848000*. 38. 109320*. 39. 121560: 40. 184200*. 41. 8700*. 42. 21700*. 43. 18830; 44. 960000?. 45. 293616?. 46. 7332?. 47. 3229?. 48. 6758?. 49. 2691?. 60. 37 crowns ; 370 sixpences ; 555 fourpences. 51. 42 crowns ; 420 sixpences ; 630 fourpences. 62. 63 crowns ; 630 sixpences ; 945 fourpences. 63. 19 half-crowns. 64. 255 threepences* 55. 36000?. 56. 28224 half-pence. 57. 100 oranges. 68* 2286 farthings. 59. 125 books. 60. 55 children. 61. 396 beggars. 1. 4. 7. ao. 13. 16. 19. J22. R52. i a. 4^. R20. 90. 13*. 3; 5*. . OJ. ?28. ^48. 31. Ris. ANSWERS TO EXAMPLES < Examples. &. 2. Ri6o. 60. ip. 3. R405. i a. $f / 5. R4o. 1 1 a. i ip. 6. R57. 130. ii 8. R247. 40. 2^. 9. R52. la. $p. 11. R59. 20. #. 12. R48. 20. 6/>. 14. R69. 130. 15. Rl20. 17. 29. 55. 3</. 18. ^37. 3*. 4</. 20. 10. 85. 6</. 2J. 3. 9J. 5t//. 23. 8. 55. i\d. 24. ^4. lu. 10 d 26. 49- 5* 27. ^28. 71. 29. 9. i8j. 30. 40- IQJ. 32. 4. 1 10. 33. 151. 463 Examples. 23. 1. Ri. na. 2 pice. 2. R2. 140. i pice. 3. RS. la. i pice. 4. R2. 90. 2 pice. 7. R3. oa. &. 110. R85. I2a. lop. 13. Ri888. 16. R4657. i a. 5^. '19. R238o$. 120. 7^ ^52. ,470. 19^. 25. 5746. .28. 5. 8. 11. 14. 17. 20. 23. 26. 29. R2. 9*. R2. 14^. 6^. R82. 90. RiSSo, na. 4/$. Ri7776. 6a. ic^. R2222I. 30. 6^. ,1010. 5J. 9^. 6. 9. 12. 15. 18. 21. 24. 27. R2. I $a. R52. 120. qp. R5i8. 20. Ri973. 140. 7/>. R23930. 100. ip* ^509- u. 5<i ; 10103. oj. i 17^. 1758. Examples. 4. 1. R6. 30. i pice. 2. Ri. 120. 3 pice. 3. R9. 100. 3 pice. 4. R3. no. < 7. 10. as. 16. 19. 5. R39- $ 9*- 7^- ^2. I2J. 7. i$s. ild. 8. 11. 14. 17. 20. R273. 3*. 6. R9. 80. , 9. 12. 15. 18. 21. Rio. 80. i ^20. IS*. /II. I2J. 32. ^809.61.9!^* 23. 467* 4' "*< 24. 464 ARITHMETIC / Examples. 25. L Rio. 100. I pice ; Ri7 ". 3 pice ; R24. 130. i pice* 2. R48. 140. 6p. ; P.68. 70. 6^. ; R88. 00. 6p. 3, R439- 4*- iA ; 519. 10. up. ; R638. 140. 8#. 4. ^89. i6j. 3<* ; ^209. IU. 30*. ; ^269.85.9^. 6. ^226. I2J. 4jp ^- ; ^302. 3s. 2d. ; ^491. OJ. l|0*. 6. 201. I9J. 4i< ^. 5 ^363. ioj. 10^0*. ; ^484. I4J, 60*, 7. R47. 140. 2 pice ; R73 ; R$7. 00. 2 pice. 8. R2228. 100. ; R3939- I4- 3^- 5 ^3979- Ha- 9. R6io6. 120. 4 #. ; R59H. 50. %p. ; R7O35. 1O. ,2819. 19J. 7 J0 1 . ; ^2228. 2s. 8^/. ; ^27851. 135. 4d. 11. ^4816. 13.?. 2\ W 5 ^353- oj. 6^ ; ^20434. 6j. $d. 12. Ri. 140. 13. Ri26. 14. ^10. 2j. 6d. 15. ^37. I4J. 2iiL 16. R5468. 120. 17. 266. 17^. 60*. 18. Rioo3i. 4a. Examples. ^0. 1. R75- 7' 2 pice ; Ri2i. 60. 2 pice. 2. R288. 70. $*. ; R366. 70. 3p. 3. Ri6i8. 30. 6f. ; R27O6,. 4. R6oi 5. 30. 9^, , ; R8490. 70. 6p. 5. ,2235. 125. 60*. ; ^490*. 6. 12763. ioj. ( <>d. ; ^4285. ly. 9i/aC 7. / 4934. ioj. <* I* ; ^5432. ioj. 9l^- a 7783. 185. ic 3^0*. ; ^8624. I3J. jojol 9. R2754* 9. 9# 10. Ri799. 120. 9A Examples. 27. L R3. 20. i pice. 2. R4- I3- 3 pice, 3. R7. 70. 7/. 4. Rio. 120. 4/. 5. Ri2, 130. ip. 7. RiS- 5* 8. Rio. 10. i ip. 1O. II 13. ^3- 7J- i 16. Rs6. 70. 19. R4i. 30. 22. ^55, I3J 25. Ri. 20. f 28. Rl2. 11. 14. 17. 20. 23. 26. 29. 55. I2J. 2. 7J. i a Ri45. 120. Ri38. 20. R3. 4. ^. 125. isj. fid. 6. 9. 12. 15. 18. 21. 24. 27. 30. R6. 150. BI43- ^420. 2J. R5. 120. - SI. 30. 9A 32. 10 annas. 33. 20. 8^. 34. IOV s>6d ANSWERS TO EXAMPLES 4&S Examples. &. 1. Ri3- 9*. # 2. R37. 9/z. 100. 3. R2. 120. 90. 4. Ri2. 7 a. 4$. 5. R40. ioa. iq^. 6. R6i. oo. i^. 7. $a. %f>. 8. R2. 2a. 20. 9. 43. i6j. %d. 10. 22. i$s. %d 11. ^5.25.2|^. 12. ^3. OJ. lid. Examples. ^9. 1. R5. la. ip. 2. R4. 1 5 a. 7/. or 8^. 3. Ri. loo. 6A 4. R3- 4"- 5A 6. R7. io. 2^. 6. R3. 150, 2p. 7. Rio. 130. iq p. 8. R9. 3. io/. 9. ^5. iu. 6|^. 10. 4. 5*. iQd. 11. 11. los. 3|^. 12. 4- 19^. 9^- 13, 2. i$s. i\d. 14. 2. i$ tj. 5i^ 15. R204. no., rem. 80. 16. Ri43. 80. 0; ^., rem. 38^. 17. R6$. 80. 3^., rem. 15^. 18. R98. I2a. 2p.> rem. 989^. 19. 14. ioj. 6</. , rem. 6</. 20. ^127. l6j. 2 \d*i rem. 230^. Examples. 3O. 1. 9. 2. 15. 3. 24. 4. 21. 6. 56. 8. 28, rem. R2. na. fy. 7. 21, rem. R3. 7 a. 4^. 8. 40, rem. R3. la. qp. 9. 32, rem. iS. 31. 3^. 10. 102, rem. 8. 3^. 4%d 11. 57. 12. 184. 13. 300. 14. 3426. 15. 7 days. 18. 100. Examples. 31. 1. 1 192320 gr. 2. 170880 gr. 3. 21927 gr. 4. 165000 gr. 6. 319896 gr. 6. 41865 gr. 7. i Ib. 4 oz. 6 dwt. 21 gr. 8. l Ib. 6 02. II dwt. 19 gr. 9. 10 Ib. o oz. 12 dwt. 4 gr. 1O. 17 Ib. 4 oz. 6 dwt. 16 gr, 11. 2 Ib. 3 oz. o dwt. 23 gr. 12. 3 Ib. o oz. 9 dwt. 9 gr. 13. 24 Ib. 6 oz. 8 dwt. 13 gr. 14. 2 oz. 16 dwt. 22 gr. 16. 2 Ib. 6 oz. 14 dwt. 8 gr. 18. i Ib. 4 oz. 8 dwt. 8 gr. ; 8 Ib. 9 oz. I dwt. 8 gr. ; 116 Ib. 9 oz. 19 dwt. 1 6 gr. 17. 8 oz. 6 dwt. 16 gr. ; 20. 18. 4 Ib. 9 oz, 19. 3 dwt. 1 8 gr. * 20. 34. C, A 30 466 ARITHMETIC Examples. 3$. L 4386816 dr. 2. 1218560 dr. 3. 2005392 dr. 4. 5361664 dr. 6. 1240064 dr. 6. 84156 dr. 7. I ton 14 cwt. 3 qr. 14 Ib. 3 oz. 15 dr. 8. 4 cwt. I qr. 6 Ib. 4 oz. 9. 12 Ib. 6000 gr. 10. 63775 tons 10 cwt. o qr. 22 Ib. 6000 gr. 11. 38 Ib. i oz. 6 dr. 12. 14 cwt. 3 qr. 26 Ib. 8 oz. 13. 1 1 tons 9 cwt. 3 qr. 4 Ib. 14. 3 Ib. 4 oz. 6 dr. 15. 6 tons 8 cwt. 2 qr. 18 Ib. 16. 2 tons 15 cwt. o qr. 3 Ib. 15 oz. 14 dr. ; 34 tons II cwt. 3 qr. 14 Ib. 3 oz. ; 129 tons 6 cwt. 2 qr. 19 Ib. 10 oz. 2 dr. 17. I cwt. 2 qr. 27 Ib. 5 oz. ; 500. 18. 2 tons i cwt. 3 qr. ii Ib. 8 oz. 19. 2 cwt. 2 qr. 2 Ib. 2O. 768. 21. A pound of feathers is heavier by 1240 grains. 22. 175 Ib. Troy. Examples. 33. 1. 8140 kanchas ; 10175 tolas. 2. 6448 kanchas ; 8060 tolas. 3. 4796 kanchas ; 5995 tolas. 4. 6176 kanchas ; 7720 tolas. 5. 2288 kanchas ; 2860 tolas. 6. 7040 kanchas ; 8800 tolas. 7. i md. 32 seers 14 ch. 8. i md. 12 seers I ch. i kancha, 9. 12 md. 1 8 seers 3 ch. 10. 31 md. 10 seers. 11. 31 md. 13 seers 13 ch. 12. 41 md. 13 seers 7 ch. 13. 81 md. 12 seers I ch. i kancha. 14. 4 md. 27 seers 13 ch. 16. 7 md. 31 seers 10 ch. 2 kanchas. 16. i md. ii seers o ch. 3 kanchas ; 5 md. 38 seers 3 ch. 2 kanchas ; 30 5 md. ii seers 8 ch. 3 kanchas. 17. 39 *eers l ch - > 2 5- 18. 595 md. 2 seers 3 ch. 19. i seer 2 kanchas. 20. 640. 21. 18900. 22. 75. Examples. 34. 1. 20 tolas. 2. 2280 tolas. 3. 3816 tolas. 4. 6792 tolas. 5. 45 120 tolas. 6. 72600 tolas. 7. 5 can. 7 md. i seer. 8. 16 md. i viss 2 seers 6 polf. 9. 3 can. 12 md. 7 viss i seer 5 poll, i tola. ANSWERS TO EXAMPLES 467 10. 4 can. 16 md. 3 viss 2 seers 2 poll. 2 tolas. 11. 2 viss 2 seers 4 poll. 12. i can. 8 md. 7 viss. 13. 86 can. 5 md. 14. 4 md. 3 viss 3 seers 6 poll. 16. ii can. 14 md. I viss I seer 6 poll. 16. i can. 3 md. 2 viss 2 seers 6 poll. ; 1 1 can. 19 md. 6 viss 4 seers ; 38 can. 9 md. 4 viss 6 poll. 17. 12 md. 4 viss ; 40. 18. 15 can. 13 md. i viss 24 poll. 19. i md. I viss I seer I poll. 20. 960. 21. 4375. Examples. 35. 1. 73728000 dhans. 2. 801792 dhans. 3. 756608 dhans. 4. 23224320 dhans. 5. 31488 dhans. 6. 1257984 dhans. 7. i can. 33 seers 24 tanks. 8. i can. 7 md. 12 seers i tank. 9. 1 8 md. 39 seers 39 tanks 2 mash as. 10. 135633 can. 13 md. 24 seers 32 tanks. 11. 2 md. 3 seers 22 tanks 2 mashas. 12. 2 can. 5 md. 37 seers 1 1 tanks. 13. 12 can. 3 md. 14 seers 36 tanks. 14. 3 can. 3 md. 32 seers 59 tanks. 16. 7 can. 8 md. 10 seers 3 tanks. 16. 16 md. 36 seers 53 tanks ; 6 can. I md. 32 seers 36 tanks ; 39 can. i md. 25 seers 15 tanks. 17. 3 md. 32 seers 56 tanks ; 400. 19. i md. i seer i tank. 18. 20. 36. 1 8 can. 8 md. 9 seers. 6400. Examples. 1. 4500 in. 2. 39600 in. 3. 6. 182556 in. 6. 209880 in. 7. 9. 1 1 10 in. 10. 1467 in. 11. 13. 28 po, 2 yd. 14. 36 po. 4 yd. 16. 35 po. 3 yd. I ft. 6 in. 17. 18. I mi. 36 po. 5 yd. I ft. 19. "20. i mi. 2 fur. 4 po. 2 ft. 5 in. 21. "22. I mi. 7 fur. 6 po. i ft. 23. 3 mi. 5 fur. 24 po. 3 yd. 2 ft. 3 in. 24. 15 mi. 4 fur. 28 po. 2 ft. 6 in. 26. 504 in. 26. 6310. 37. 126 in. 28. 100 nails. 29. 44 nails. 30. 50 ells. 31. 8000, 190080 in. 4. 380160 in. 612018 in. 8. 762 in. 184878 in. 12. 431766 in, 16. 19 po. 2 yd. I ft. 6 in. 6 po. I yd. 10 in. I mi. i fur. 9 po. 4 yd. 6 in. 5 po. 10 in. 468 ARITHMETIC 1. 4. 7. 10. 13. 16, 17. 18, 19. 20. 21. 23. 25. L 5, 8. 10. Examples. 37. 2. 4704480 sq. in. 3. 47358432 sq. in. 6. 127692 sq. in. 9. 17546220 sq. in. 12. 24 sq. po. 14 yd. 15. 29808 sq. in. 8028979200 sq. in. 5. 7880004 sq. in. 8. 300384 sq. in. 11. 12 sq. po. 2 yd. 14. 33 sq. po. I yd. 6 ft. 108 in. 1 ac. 2 ro. 1 8 po. 19 yd. 4 ft. 72 in. 7 ac. 3 ro. 10 po. 8 yd. 4 ft. 72 in. 2 ac. 23 po. 8 yd. 2 ft. 36 in. 2 ac. 2 po. 25 yd. 3 ft. 72 in. 5 sq. yd. 5 ft. 34 in. 22. 25 sq. po. 5 yd. 7 ft. 62 in. 24. 4390848 sq. in. 26. 752716800 sq. in, 80760240 sq. in. 200196 sq. in. 22632732 sq. in. 32 sq. po. 3 yd. 2 sq. po. 3 ft. 94 in. i ac. 2ro. ii po. 28yd. 51 hi. 48400 sq. yd. Examples. 38. 23280 ga. 2. 4025 ga. 6399 ga. 6. lonooga. 2 cot. 4 ch. 8 ga. i bi. 1 1 cot. 4 ch. 3. 42140 ga. 4. 124000 ga* 7. I bi. 6 cot. 15 ch. 9. i bi. 4 cot. 10 ch. 12 ga. Examples. 39. 1. 139968 cu. in. ; 326592 cu. in. ; 559872 cu. in. ; 746496 cu. in. ; 933120 cu. in. ; 1819584 cu. in. 2. 2 cu. yd. 17 ft. 768 in. ; 21 cu. yd. 4 ft. 966 in. Examples. 40. 1. 404 gills. 2. 2816 gills. 3. 1504 gills. 4. 1696 gills, 5. 9344 gills. 6. 18176 gills. 7. 159744 gills. 8. 50432 gills. 9. 428032 gills. 10. 31 gall, i qt. 11. i barrel 28 gall. 3 qt. i gill. 12. 2 barrels 34 gall, i qt. 13. 6 barrels 9 gall. 3 qt. i gill. 14. i qr. 3 bus. 2 pk. i gall. 3 qt 15. 5 bus. 3 pk. 3 qt. I pt. 16. i last 2 qr. I bus. 2 pk. I gall, i qt. 17. 4 lasts i Id. 3 qr. I bus. 3 pk. i qt. I pt. i gill. 18. 25 Ib. avoir. 19. 3500 Ib. avoir. 20. 64 ; 32, ANSWERS TO EXAMPLES 469 Examples. 41. 1. 25923 sec. 2. 637800 sec. 3. 1512000 sec. 4. i hr. 23 min. 20 sec. 5. i da. 3 hr. 26 min. 5 sec. 6. i da. 3 hr. 46 min. 40 sec. 7. I wk. 4 da. 13 hr. 46 min. 40 sec. 8. 94. 9. 121. 10. 244. 11. 577- 12. 289. 13. 821. 14. Thursday. 15. Wednesday. Examples. 4. 1. 26247". 2. 865535". 3. 1296000". 4. i. 6'. 40". 6. io*. 32'. 36". 6. i rt. gle. 26. 40'. 7. i rt. gle. 47*. 36'. 8. 3 rt. gle. 4*. 20'. 54". Examples. 43. 1. 24000. 2. 104 reams 3 quires 8 sheets. 3. 432. Examples. 44. 1. H2ogr. 2. 1632 gr. 3. 24960 m. 4. 192000 m. 6. 612309 m. Miscellaneous Examples. 45. 1. 61200. 2. Ri9. I3<*. 6A 3- 569. u. f\d. 4. 479 mi. 2 fur. 6. Ri3. 30. 6. 2028. 7. ia. 4p. 8. is. 9j< 9. 16384. 10. 105 parcels, 30 seers rem. 11. 96. 12. 1920. 13. n. 14. Ri88. ua. <$. 15. Bi2. 150. 6. 16. R4& H*. 9A 5 ^343- 6a. $p. 17. R2. ioa. #. 18. 8500. 130. o/. 19. i. is. nrf. 20. ES. ia. 21. B3754. 9. 9/. 22. 6^.3^. 23. 56 yr. 3 mo. 7 da. 24. 160. 25. 5 sec. 26. 3960. 27. 2 ft. 7 in. 28. 4196. 29. 83. I2a. 30. 32. 110. #. 31. 66. 12*. 6^. 32. 17. 33. R687. loa. 34. 30. 5*. i\d. 35. 66. 13*. 4^. 36. 104. 37. 53. 38. 130 Ib. 39. 1 6 yr. 4 mo. 2 da. 40. 4*. id. 41. 2s. 6d. 42. 62. 43. 12 seers. 44. 5 md. 45. 8 min. 18 sec. 46. 5 ft. 4 in. 47. i6th September. 48. Friday the 8th of May. 49. 53 hours. 50. 192000 miles per sec. 51. 68. 52. 19. 47O ARITHMETIC 63. 3 yd. 54. R2. 3*. 56. 11088. 66. 4497 times. 67. 18000. 68. 82745. 69. 41 yd. 4 in. 00. 28 yr. 13 wk. 4 da Examples. 46. 1. 84. 2. 44- 3. 50. 4. Receives ^13. 135. *)d. 5. Ri. 70. &. Examples. 47. 1. Gains 82. Sa. 2. 821. i. 6p. 3. 830. 4. B;. 120. 6. 30. 7<?. 60. 6. Ri. loa. ?>p. 7. #. 8. 4^. 9. 1. is. 10. 24 qr. 11. 8*. $d. per yard. 12. Ri. 5*. per Ib. 13. Gain 12 j. 6</. 14. 4^ 15. (i) RE. 2a. ; (ii) Ri. 30. Examples 48. 1. 4<*. 2p. 2. 1. 4J. 3, 15^. 4. R9. 6^f. 5. 2^.3^ 0. 2j. 3^. 7. 2tf. . 8. 6 seers. 9. 9 Ib. 10. 2J. 6//, Examples. 49. 1. -<4, 823. 6a. ; ^, Ri6. i. 9^. 2. ^4, ^12. 6s. -]\d. ; ^?, ^16. os. io\d. 3. The two get 834. 3a. i^. each ; the rest R22. 4. 4^. each. 4. Each man, 20. 40. 6/>. ; each woman, R26. 4a. 6p. 6. A, Ri6. 6a. io^. ; #, 813. 6^>. io/>. ; C, R9. 6. io/. 0. A, Rii3. 13^. 3^. ; B, Rio6. 13^. 3^. ; C, Rio8. 13^. 3^. 7. ^40. Examples. 50. 1. Boy, Rio. 6a. qp. ; girl, R$. 3a. 2p. 2. -4's share-RiS. ga. 6p. ; ^ ; sRio. 6a. tf. ; C's-Rs. 30. 2/ 3. Each man, Ri2. 8<z. ; each woman, R6. 40. ; each boy, 83. 20. 4. ^, ^6. I4J. 6d. ; ^, ^3. 7^ 3</. ; C, ^l, 13^. *]\d. 5. One gets ^5. 3-r. gd. ; and the other two, 2. us. ioj</. each, 8. -4, R26. 15^1. #. ; -ff, Ri2. 8^. 6/. Examples. 51. 1. 12 2. 10. 3. 12. 4. 16, 6. ii rupees, 22 half-rupees, 44 quarter-rupees. 0. 32. ANSWERS TO EXAMPLES Examples. 52. 471 1. 3. la. 9/. 2. Rio. 20, 3. The price of a horse is 875. 8a., of a cow, 825. 8a. and of a sheep, R$. 80. 4. A markn|^/. ;a gulden u. n</. ; arouble=3J. \\d. 5. R38. 4. 6^. Examples. 53. 1. 2,3- 2. 3) 5) 9. 3. 2, 3) 4) 9- 4. 2, 3)4) 5 1) 10. 5. 2) 3) 4) II. 6. 2, n. 7. 2) 3) 5) 10. 8. 2, 4- 9. None. 10 5- 11. 2) 3) 4) 8, ii. 12. 2, 3) 4) 8, 9) " 13. 3)5- 14 5- 15. 2, 4) 5) 8,10. 16. 2, 4, 5, 8, 10. 17. 3>9- 18. 3, ii . 19. 2)3- 20. 2, 3) 5) 9) 10. 21. 7- 22 . ii. 23. 13. 24. 7, ii)i3 25. n. 26 . 7, 13. 27. None. 28. 7) ii) 13 29. 6, 12 30, , 6, 12 . 31. 6, 12, 30. 32. None. 33. 2 ; i. 34. i ; 7 ; 2. 35. 2717. Examples. 54. 1. 2*. 2. 2.3. 3. 2.3*. 4. 2 8 .3. 5. 3*. 6. 2 6 . 7. : 2*.3. 8. 2.5'. 9. 3'-7. 10. 2". 11. 2*. 5. 12. 2 S .II. 13. 3 8 .ii. 14. 2*. 5 s . 16. 2 a .3*. 16. 2 4 .II. 17. 38.13. 18. 2 5 .3 a . 19. 3 2 .5.n. 20. 5*. 21. 3 8 .37- 22. 2.3.5 2 -7. 23. 2 4 .3*. 24. 2*.5.II. 25. 2 4 .5*. 26. 2-5 2 .73. 27. 2 ; 7 .3 3 .5. 28. 3 8 .7.] [3. 29. 2 9 .3 8 . 30. 2 8 .3.5 8 .23.29. 31. prime. 32. prime. 33. 3*. 34. prime. 35. prime. 36. prime. 37. prime. 38. 3*.23. 39. prime. 40. prime. 41. iiVji. 42. 3-I3 8 - 43. 17.269. 44. prime. 45. 23-31. 46. prime. 47. 13-503. 48. 11.163, 49. prime. 50. 29.47. 51. 10. 52. ii. 53. ii. 54. 5i7. 55. 5)7. 56. 6, 8, 12, 24. Examples. 55. L 3- 2. 4. 3. 5- 4. 18. 5. 5- 6. 12. 7. 75- 8. 4. 9. 24. 10. 5- 11. 4- 12. No common factor. 13. 56. 14 . 25. 15. 28. 47* ARITHMETIC Examples. 56, L 48. 2. 2. 3. 4. 4. 12, 5. 29. 6. 124. 7. 101. 8. 143. 9. 377. 10. 7. 11. 133. 12. 25. 13. 19, 14. 15. 15. 53. 16. 28. 17. 39. IB. 113- 19. 173. 20. 147. 21. 221. 22. 3. 23. 57. 24. 287. 26. 213. 26. 221. 27. 15. 28. 1536. 29. 257. 30. 6. 31. No. 32. Yes. 33, No. 34. Yes. 35. No. 36. No. 37. Yes. 38. Yes. 39. No. 4O. 37. 41. 37. 42. 23. 43. 17. 44. 3. 45. 5. 46. 3. 47. 63. 48. 17. 49. 57. 60. 2. 51. 2. 52. Rx.4*. 53, 3^. 54. 16. 55. 32. 56. No. 57. i So gall. 68. i tola. Examples. 57. 1. 96. 2, 3724. 3. 891. 4. 3520. 5. 7488. '6. 259488. 7. 672. 8. 23374- 9. 87087. 10. 759655. 11. 49077. 12. 734877. 13- 96672. 14. 159137. 16. 183645. 16. 2672700. 17. 2310. 18. 2376. 19. 115256. I2a. 2O. 64. 21. 390. Examples. 58. 1. 48. 2. 48. 3. 720. 4. 36. 6. 2520. 6. 1680. 7, 28050. 8. 360. 9. 1890. 10. 7560. 11. 7200. 12. 144. 13. 8415. 14. 7920. 15. 792. 16. 3570. 17. 228150. 18. 98280. 19. 49140. 2O. 5481. 21. 237510. 22. 2520. 23. 1680. 24. 10800. 35, 98280, 26. 189. 27. 389. 28. 141. 29. 1296 sq. in. 30. 189. 31. 14 min. 32. 90 miles. 33. 131 yd. 9 in. 34. 677. 35. 232792560. 36; 75 yards. Examples. 59. L. 40. 2. AS. 3. iq. 4. i seer. 5. 5<*. 6. 9* 7. 7 in. 8. 5^. 9. 10 in. 10. 4d. 11. 3 pice. 12. 3 cwt. 13. 160 yd. 14. 6 ch. 15. 9 sq. in. 16, 7 Ib. 17. 60. 18. 9*. 19. I ft. 20. 4* 21. l$mio. ANSWERS TO EXAMPLES 473 Examples. CO. 1. f. 2. J. 3. iV 4. A- 5. 7. M- 8. I T - 9. J. 10. 1 11. 13. & 14. A- 15. If- 16. tt$- 17. 19. f- 20. tt- 21. ?? 22. 1- 23. :25. 26. !? 27. i 28. W- 29. 3. Examples. 61. 1. |. 2. f 3. i 4. f. 5. f. 6. f. 7. f. 8. 5. 9. f. 10. J. 11. f 12. 5. 13. f. 14. |. 15. f. 16. J. 17. J. 18. j. 19. A- 20. f Examples. Ola. i 6. |. J. 12. |. m- is. H- &. 24. H. If- 30. TJJS ^1. . 32. ?. 33. Jf- 34. f 35. & Examples. 61b. 1. J. 2. f. 3. f. 4. JJ. 5. A. 6. A. 7. A- 8 - T 6 A- O- S- 10- 11- iV !2- * Examples. 6. 1- - 2. *. 3. ^. 4. fi. 5. V. 8 - IW- 7 - W- 8 - W- e J JJ 1 - 1O 11 -WSJ* 12. ^88^. 13. $*ft. 14. ^2Aii 4 15. 17. ISM- 18. W^- 19- WA 20. Examples. 63. 1. 3j. 2. 2j. 3. 4f. 4, 5t- 5- 3f 8. 3J. 7. 9. 8. 6|. 9. 9A- 1O. 5 11. 2}}. 12. 4. 13. 3]. 14. 4tf* 15. 3} 16. 4f. 17. 2^. 18. y^j. 19. 56}. 20. 7. 21. 28)}. 22. 329. 23* icij. 24. 10. 25. 4*1 86. 2}. 27. nJJi. 28. KH^. 29. n. 30. 474 ARITHMETIC Examples. 64. i. A. It- 2. *,& 3. if if. 4. AtAi l,tftt- e. ttfcttftti. 7- H.W.M- s. M,ftf, 8. MitfbM- 10. a&.i&b.sg,,. 11. ffi,M 12. Mtf4fo 13- 48.fi.lt-' I*- H.A.A- 20. ttH,fHt,iHf>mt>mf. 21. iff, m, m, m 22. Mf,!|J>&WA,iA. 23. An m> vtfv ilfa 24. H. W, *<>**> it- 25. H, MSn rifc. AW 20. H.W'.M.H.IJ. 27. Examples. 6ft. L I- 2. A. 3. ft 4. if. 5. t 6 I?. 7. $$ greatest, J$ least. ' 8. JJ greatest, J least. 0. |? greatest, J least. 1O. $f greatest, | least. IL ^, greatest, T % least. 12. ,% greatest, least. 18. |,A,f 14. JI.A./i- 16. H.5.I- 16. V,3fiV 17- Uift>ft. 18. Uf,Hfl> 19- fifcttif 20. Uf,fi,f,^. 21. H.ttt* Examples. 66. 1. zj. 2. i?. 3. f. 4. i^. 6. 2 T V 6. i|f. 7. i. 8. 7&. 9. if- 10. f. 11. i. 12. i. 13. ijft- I*- ii*- 16- ilM- 16- iH- 17. tf. 18. I^,. 19. T ^. 20. 4^. 21. 2ii. 22. *ff- 28. 2^,. 24. 4A- 25. 2jjf 26. ifJJ. 27. 3*8- 28. 2. 29. sHIi 3O. , Examples. 67. 1. #. 2. I4i- 3. I2&. 4. isiS- 5. 23f. 6. 29j|. 7. sf. 8. 4 ift. 9. lo^j. 10. u, IL I4H- 12- "i 13. i6of*. 14. 34H- 15. 13$. 16. 3iA- 17. 976.V 18- JS4>. 19. I7|J. 20. 6,%. 2L 29.9.s}|#. 22. 7.l7 ANSWERS TO EXAMPLES 475 23. 15 yd. 3 ft. 6{ in. 24. Ib. I oz. 3& dr - 25. 21 oz. o dwt. igj gr. 26. 20 hr. 34 min. 33}! sec. Examples. 68. 1. t 2. 6}. 3. ft. 4. A- 8- tt- A- 7. H. 8. . 9. # s . 10. A. 11. ^. 12. m 13. li*. 14. 5. 16. &. 16- A- 17. J. 18. . 19. . 20. ffl. 21. &. 22. &. 23. A- 24. M. Examples. 69. 1. 3J. 2. *&. 3. 3*. 4. 5f. 5. Sit- 5ft- 7. 6J. 8. 8 T ^. 9. 2f. 10. 3|&. 11. Jf. 12. s. 13. i||. 14. 8J8. 15. 9lf 16. 9lf 17- i&- 18. 6^. 19. 6 3 7 ,. 20. 9.. 21. 2|. 22. 6f. 23. 8 T V 24. 9^- 26. 8f 26. I2,\. 27. I3tf. 28. 10^. 29. I T V 30. 6/ T . 31. 6JJ. 32. ;A- 33- 3- 34. 2-ft. 35. 9*. 36. i|. 37. I2|?. 38. I4A- 39. f. 40. |. 41. Rio. 12. ij. 42. Bs. 12. 9^. 43. &4. 4. 3}. 44. jio. 9. 5^'. 46. ^5. 12. io&. 46. 6 yd. sf in.. Examples. 7O. 5. si. 10. loj. 16. 195. 2O. xooft, 21. 33^- 22. 62}. 23. 328^. 24. 198?. 26. 213. 26. J224J. 27. 487?- 28. 177*. 29. 28^,. SO. 38^- 31. i8$ft. 32. 44^1- 33. 899i%- 84. 386^. 36. 22 9 99i%fe. 36- 3i9o. 87. 209^,. 38. 6399ft- 39. i . 18 . 1 ft 40. ^4.9-7A- 4L 850. 7. 2 J. 42. R49.4.5J- 48. 2.4.5- 44. 36.7-2*. Examples. 71. 1. *. 2. ft. 8. ,V 4. f 6. A- <* rf* 7- ifr 8. T J B . 8. A. 10. 1. 4i 2. 7- 3. i&- 4. 3?- 6. I9i 7. ioj. 8. 300. 0. 2i 11. 22i 12. 23i 13. I8|f. 14. 60?. 16. 6&- 17. I3i 18. 47f. 19. 66f. 47$ ARITHMETIC 11. 1 ^ . 12. jh. 13. sfc. 14. ,1,. 15. 16- !,&*. 17. i*. 18. A- ! A- 20. |f. 21. if 22. ^. 23. $\. 24. A- 25. 42*- -26. I2&. 27. 178}. 28. 6H- 20. 15!}$. 30. 31. 62,V 32. 3&H1- 33. 2if. 34. i| 36. Ri . 5 6. 36. R2 . 8 . 7&. 37. i . 17 38. 7 13 io$Jr. 39. 80. 40. 192$. 41. 42. IJ2&. 43. 4^. 44. 26$. 45. 217$?. 46. I46}fg. 47. R2 . 9 . of. 48. Ri . 4 o T V 49. 832 . 5 6^. 60. R6.4-2}!- W- ^3.".6i 52. ^3 19 ** Examples. 7^. 1. f. 2. H- 3. |. 4. ,ft. 5. 4*. 6. rffc. 7. f 8. A- 9. f 10. 3J. 11. if 12. 2f. 13. 3A- 14. 25*- 15- 3- 16. I4i 17. 10. 18. 8. 10. 28^. 20. I6&. 21. 9J- 22. 8. 23. 3f. 24. 35- 25. 8. 26. i$fj. 27. ia|. ^8. 40^. 29. a86|f. 30. f 31. 2^. 32. 3ii. ^8. if. 34. A. 36. A- 36. 28. 37. 294. Examples. 73. 1. 1386 in. 2. 2574 in. 3. 5742 in. 4. 7722 in. 6. 9702 in. 6. 39582 in. 7. 673308 in. 8. 274428 sq. in. 0. 509652 sq. in. 1O. 1136916 sq. in. 11. 1528956 sq. in. 12. 1920996 sq. in. 13. 59864508 sq. in. 14. 4033699560 sq. in. Examples. 74. 1. i$. 2. ft. 3. i. 4. i*V. 5. ij. 6. 14* 7. 3- 8- 2|. 9. i8f. 10. if 11. f 12. . 13. I if. 14. 6f. 15. I7&. 16. 9&- 17. 2ftf 18. if 19. if 20. 2f 21. & 22. if 23. 6A- 24. 3*%- 25. 16. *26. ff. 27. lA' 28. ,V 20. 93J. 30. The former. Examplea 75. 1- &;x. a- A;2f s. T ir;*f 4. ^^ 7. A;409i. s. ; 42. ANSWERS TO EXAMPLES 477 9. i;i57*. 10. )*;63. 11. ij ; 8. 12. A 8 * 13. 3 in. 14. 2f, 15. i tnin. 45 sec. Miscellaneous Examples. 1. 6j. 2. ij. 3. si 4. A. 6. A. 6. 5. 7. i6,V 8. j. 9. 4 . 19 . 5$. 1O. 840 . 6 . 10$. 11. 950 Ib. 12. Ri$2o. 13. 50. 14. 4i*3i- 15. . 16. |. 17. ,V 19. |. 20. iV 21. J. 22. ,%, 23. |j. 24. dfc. 25. A- 2 6. 3- 27 - R6 - 28. ^7 20 * 29. 155. 30. 22 miles. 31. 400 inches. 32. 8, 6, 3, 2 ; 24 kings in all. 33. 34. 34. i j}. 35. }f 36. $ times. 37. &. 38. 27 hours. 39. 3$. 40. 310. 41. 13 ; 17. 42. 36. Examples 77. 1. f 2. i&. 3. 3 |. 4. 12. 6. A- 6. 2&, 7. !!i- 8. 4l V 9. ,V 10 - 5- 11- 3. 12. 3- 13. 5iV 14. 96 5 V 15. 17. 16. Mf- 17. 4- 18- I- 19. ii^. 20. A. 21. IS. 22. 8JgJ. 23. f. 24. i& Examples. 78. 1. i. 2. if 3. ft. 4. 2f?f *6. 2 5 f 6. i s 7- iAV 8. 6^W 9. 4 A. 10. *. 11. ?2- 12. t Examples. 79. 1. AV 2. i 3. Jfc. 4. 6. 5. i. 6. ! 7. ij. 8. i. 9. i*. 10. A- 11- } I 2 - iV 13. 3&. 14. i. 15. 2. 18. AV 17. 22j. 18. i 19. 2. 20. AV 21, 22|. 22. ff. 23. 2ji. 24. fj. Examples. 80. 1. 3- 2. 3 i 3. T&V 4. 3i 9i5- 6. z|. 7. 12. 8. 7&- 0- 7aV 10. ii 11. 4*- 12- 13. 4& 14. i. 16. #J. 16. I2j. 17. 4||{. 18. 478 ARITHMETIC Examples. 81. L I. 2. ij. 8. 7&- 4. 6&. 6. AV 6. 5AV 7. 1 l . 8. 3 Jft. * i* 10. g}f. U. 7l 12. 4i- 13. 4*. 14. 5f- 16. 8- >16. I2&. 17. i. 18. 10. 19. i t V 20. |J. 21. i. 22. 2$. Examples. 8. 'L A. 2. 2. 3. 2. 4, i#M. 6- 3i- 6. 2fh. 7. i. 8. I4&. 9. 1 T . 10. ij. 1L ?Wj- 12. f?- 13. 3. 14. 5AV 15- 16. 49- 17. If 18. A- 19. Si- 20 - 21. iA- 22 - f 23 * 24 ' 4> 26i 26. i. 27. A. 28. ,&& 29. 84$?. 30. 31. !%. 32. 22}. 33. J||. 34. 3 iV 35. Examples. 83. 1. R3 . 10 . 4 2. Bi . 10 . 8. 3. Ri. I4. 4. R8 . 8 . 8. 6. Ri . 3 . 6. 6, 7<z. ty. 7. 33 . 16 . 4- '8. ^58. ioj. 9. 29. 14.?. 10. R70 .9.4. U. Ri . 12 . 8. 12. Ri . 2 . 8. 13. 11.5- 955- 14. 38 . 8*. 15. 6s. 3* 16. R$2 . 6 . 10}. 17. Ri9 9 . 9i 18. I9J. 6^- 19. 15. 10 .2}. 20. R284.2.6 54 V 21. ^22.14.3}!}- 22. 4 cwt. 2 qr. 24 Ib. 12 oz. 23. 343 yd. I ft. lof in, *24. 25 min. 25^ sec. 25. 2 pk. i$} gall. 26. Ri46 . 11 . 11. 27. Ri . o . sf 28. Ri22 .3.8. 29. ^7 . 19 . ioj, 30. 22.1.9}!- 31. R3i-8.6 3 V 32. RS . 10 . 7. 33. ioj. iifj^. 34. i2a.9iS&>. 36. 2 . 8 . ;i '86. 16*. ioj^. 37. R8 . 5 . i. 38. Ri4 . 6 . 89. 3-i8.5i}tt. 40. }iofR6. iia,,fofR7,R5. 4L 14- IS- 2 - 42. R8.9.4*. 43. R6 . 5 . 9 |. R2i7.i?.6 45. 18 ANSWERS TO EXAMPLES 479 Examples. 84. - 3*- 2. 9i 3. <&. 4. 7i- 6. 723- '. 7i- 8. 3*. 9. ttf. 10. |. 11. }. 12. I. A- 14. ?$&. 16. #fo. 16. &. 17. J3J. 18. 22. &. 23. 34. I. 39. 44. 49. 7 18. 19. f}|. 20. 24. 8}iJ. 25. 4 2&. 29. flft. 30. A- 35. ,%. 40. - & 21. 26. 31. //*. 36. lift. 41. 3- 45. 2& 46. 60 - i%. ! 27. 32. 37. 47. A. 48. 28. 33. J. 38. , . 43. 9- 62. 53. Miscellaneous Examples. 85. 4. ^7.2. i. 7. ^i . 13 7 10. j2.gj. 16. 1^5. 3. B8. 6. E3. 8- 6 T 6 s ft. 11. Ri. 6a. 12. {. 16. -ff. 17. # . ; 812. 8a. ; Bi2. Sa. 6. 191. ii\d. 9. Bi22 . 13 . 13. i 14. 18. R;85862. 7. 12*04006. 8. '013005. 9. "oooioooi. 19. RS . 4 . 6 . 20. sV 21. 72 oz. 22. 12 Ib. avoir. 23. ^,. Examples. 86. 1. '3. 2. 2-oi. 3. -07. 4. '104. 6. '0008. 6. -000009. 1O. 100*502. 12. 290, 2-9 ; 29000, -029. 14. '2, -002 ; 20, -00002. 16. 70-3, -703 ; 7030, -00703. 18. '07, '0007 ; 7, -000007. 20. 234-5, 2-345 ; 23450, -02345, 22. 1232, 12-32 ; 123200, '1232. 11. 7> 7 ; 7000, '007. 13. 2, *02 ; 200, '0002. 15. 34 "34 ; 3400, '0034. 17. 10-03, -1003 ; 1003, -001003. 19. 39^1 3*9* 5 392oo, -0392. 21. 30000, 300 ; 3000000, 3. 38. 'I. 24. -oi. 26. 35 ; 70-5 ; 40. 26. -25 ; -06 ; 1. f e. A- *! . Examples. 87. 2. fa. 3. A- 4. f. 7. ^ 5 . 8. A. 9. }|. 12. W* I 8 ' i 14. iV 6. fa. 10, . 4 8o ARITHMETIC 17 21. 2. 22. 26. 3&. 27. 31, I2& 32. ii 30. ndfr.87. 7. 23. 8J. 28. 33. 38. -9. 10 - 24. if. 29. 34. 39. 12. 25. 30. 35. 40. 2*4. 41. *003. 42. '0725. 43. '0329. 44. '09. 46. T2345. 46. *oo2. 47. 200*03. 48. *oi. 49. '0125. 50. '00079. Examples. 88. 1. 20-163. 2. 37*479. 3. 43*31. 4. 80*33. 6. 10*36411. 6. I. 7. 10. 8. 909*9099, 9. 14*53302. 10. 8. 11. looo. 12. 417*11157. 13. 669*2981. 14. 657*2236. 15, 732*131. 16. 8347*23478. 17. ^747*0199' 18. 41*4819 min. 19. 332*475 ft - 2 0. 41*307 in- Examples. 89. 1. 7-084. 2. 1*9711. 3. 1*09922. 5. 62*65. 6. 204-103. 7. "000275. 9. 7*5554623. 10. 342*817. 11. 7- jooi. 14. ^-99949. 16. 696-162. 17. 83*9583. 19. 128-471, 20. 673-14159. 4. 19970334. 8. '0118766. 12. 2-063. 15. 9*88309. 18. 1999*25218*. 21. By 27183*. 1. 74*52. 2. 6. -001024. 6. 9. 40*804. 10. 13. 4264014. 14. 18. -589*12. 19. 22. -819. 23. 26. 30917497, 27. 30. 48*6328,503. 31. 34. 2-16. 35. 38. 2401. 39. 42. 2*607255. 43. Examples. 9O. 36*2. 3. '13446. 4i 000324, 7. 28*00028. 8. 30228. 11. 1-62023. 12. 8. 15. -58. 16. 8. 17. 00008. 20. '0000423. 21. oooi. 24. '82008. 25. 1209*11. 28. '096. 29. 15*625. 32. '015625. 3*. 1*331. 36. I. 37. 00081. 40. 27-5. 4L 7*5667. 44. 900-0025. 45. 6006. 2456*8884. 0003125. 216*32. 00003738028*. 3'5- 1344620025. '00008. 000000125. 38*9375. 421*36875. ANSWERS TO EXAMPLES 481 Examples. 91. 1. 1*27. 2. 1-372. 3. 1-2. 4, -00043, 6. 1-99. 6. '0000479. 7. '00002637 $ 8. 10*3. 9. -000002. 1C. 17-125. 11. -0000000212.12. '0528. 13. 1-84782... 14. '00009... 16. 2-49367... 16. -00040... 17. -00002...* 18. 371428... 19. 1*30586... 20. -01900... 21. -00003... 22. 2*0625. 23. '46625. 24. '004857... 25. -236. 20. !2-i8i8i8...27. 2*29375. 28. -000540... 29. '659. 30. -001666... 31. 31*25. 82. 352*25. 33. '24. 34. 2532. 35. 1200. 36. 640. 37. -002. 38. '374. 39. 20. 40. 2040000. 41. 22500. 42. 58070. 43. 3596. 44. 12132. 45. 17500. 46. 1*4. 47. 750000. 48. '007853. 49. 128-18518... 50. 5*20833... 51. 33'33333 52. '08366... 53. '02320... 54. '00650... % 55. 33057851-23966... ^6. 83-33325. 57. 9-58904... 58. -01216... 69. 350. 6O. 752. 61. 2-533333... 62. 6-3125. 63. '000092... 64. 32714-285714... 66. 5628-571428... 66. 119175. 67. 1145*833333... 68. -018181... 69. '021428... 70. 377777777.- VI. '9- 72. 8. 73. '27. 74. -5. 75. '25. 76. 75- 77. '125. 78. -375. 79. I-4375- 80. 3*09375- 81- 9*375. 82. 3-28. 83. 2'68. 84. -33333 85. '16666... 86. '28571... 87. '27272... 88. '69230... 89. 1*44444... 90. 7*18181... 91. 8-33333... 92. 10-34482... 93. 58-41666.., 94. -8, 75, '6666... 95. -5, '4166..,, -2727... 96. '55> *5333-> '525. 97. '375> '3i25 '2187... 98. -44* '4333-j '35- 99. 7777-..> 7142..., '6. 1OO. -0216. 101. -1125. 102. 3*135., 103. -2. Examples. OS. L '25 ; 10875. 2. '03 ; 72-12. 3. '004 ; -4. 4. '24 ; 6. 6. '005 ; I'd. 6. -12 ; 7*2. 7. 'oooi ; '08. 8. '06 ; 11754-6, 9. "03 ; I'S. 10. '06 ; 180. 11. '05 ; 140. 12. '025 ; 1-5. C. A. II 482 ARITHMETIC Examples. 03. 1. Non-terminating. 2. Terminating. 3. N.-T. 4. T. 6. N.-T. 6. N.-T. 7. N.-T. 8. N.-T. 9. N.-T. 10. N.-T. 11. T. 12. T. 13. T. 14. T. 16. N.-T. 16. 3, 6, 7, 9, n, 12, 13, 14, 15, 17, 18, 19. Examples. 94. 1. -3. 2. '2. 3. '714285. 4. n6. 6. 1-18. 6. 1-538461. 7. '46. 8. 1-009. 9. -27. 10. 3-230769. 11. 11-904761. 12. -o4. 13. 3780003. 14. -2083. 15. 3-8846153. 16. 7-481. 17. 5*285714. 18. 10-676923. 19. 7-13. 20. 9-6428571. 21. 1*06198. 22. 13-94230769. 23. 4-80357142$. 24. 3'456o97. 25. 5*12. 28. '(3. 27. 6-571428. 28. 1-772. 29. '126984. 30. 4-8. 31. -16. 32. '015. 33, -o6i. 34. '0601$. 35. '060015. 36. 8-106. 37. 3-13714235. 38. -6588235294117647- 39. 2-105263157894736842. 40. -0869565217391304347826. 41. 10*96. 42. -699906. 43. ,2*307692. 44. 2-857142. 45. 27-27. 46. 2*27. 47. 7-8695652173913043478266. 48. 1671428$. 49. 6-676923. 50. 642*857142. 51. '82. 62. 00072. Examples. 95. 1. '234534. 2. '347<>7. 3. '67670". 4. '234^. H, -001231. 6. '1234523. 7. -1234123. 8. '12345623. 9. '3444444, '2*42424, '2678678. 10. -1620202020202, '1234234234234, 'J76537653765. 11. '233,7*7. 12. -34, 7<57, 722. 13. '3077)767<J. 14. '076767, 777777i '006123. 15. '23888$, '123412, -023232. 16. *3333> '767^ '7230. 17. 77777777, -12424242, '24723723. 18. 3-4444444, '2686868, '123123!. 19. 3-4022*, 7823>'3iH. 20. '4232323, 7^727271 '1203203. Examples. 90. 1. I- 2. A. 3. . 4. . 5. ft. 6. 1 %. 7. f{. 8. AV iMlB 10- rm* 11- iirmiF. 12- i?^ ANSWERS TO EXAMPLES 4&3 13. 3 *W- 14. 3{ft 15. 18. A- 19. ajf- 20. loHft. 21. uftfo. 22. 23. rffc., 24. tfg. 25. T }j. 26. ,fc. 27. 28. JJ&. 29. 3*W 30. Jfljfo. 31. &fa. 32. 83. #. 34. H. 35. W. 36. |J. 37. 38. SW- 39. -Wo 1 - 40. *flW- 4L . %Wb 42. 43. %W- 44. 1W1. 45. \V. 46. 47. %yyyt. 48. ajjji. 53. -i. 54. -368. 55. 17. 56. -ooi. 57. -3. 58. 4. 59. 4. 60. lo. Examples. 07. L 378. 2. 793! 3. 11-095. 4. 6-48453. 5. 4*82^7. 6. I'03i3o829oi. 7. 2-8579. 8. 8-98. 9. 10-345. 1O. 8-002. 11. -10291837. 12. 5*348655. 13. 19-17230127. 14. -00936663. 15. 11-17997. 16. 1723082719. 17. 9. 18. 5-87263. 19. 75*oi35464357246$. 20. 4. 21. 11-5977942- 22. 2-6542987441. 23. 92*4687 545 56 556734. 24. 37593- 25. 3'O777<M9- 26. -39489560667778. 27. '911001. 28. 3*3876$. 29. 2-472876- 30. 676J2J, 31. -8916. 32. 6*3458. 33. 2-4644933412260!. 34. '4312. 35. 3-89386295. 36. 7161605349724. 37. 3-6442255331. 38. -1236786. 39. 771-0735127582. 40. 29*6236196$. Examples. 98. 1. -ool 2. ri8. 3. 1-338842... 4. -iS. 5. -1686419753. 6. 51-962. 7. -5. 8. 106-5625, 9. 2335-882352... 10, 1-518141... 11, 2794932.., 12. 7857U2. 13. -236232... 14. -08281853. 16. 69-3957. Examples. 99. 1. 120-42857 L 2. 13316-875. 3. '07$. 4. 5. 5. #$& or -5048... 6. 350. 7. T2. 8. '03483. 9. 20. 10. '380952. 11. '125. 12. 11344-6. 18. 8. 14 sW& r '22269... 15. 998-001. 16, 44 ARITHMETIC 1. 6. 10. 14. 17, 23. 26. 29. 32. 34. 36. 40. 43. 46. 49. 60. 62. 66. 1. 4. 8. 12. 16. 20. 24. 26. 82. 87. 1. 2. Examples. 100. 3. 32^0". 4. y6g. 5. so/. X58o-8/. 8. 93*30*. 9. 1603*8402. 7. 50. 2-4^. 12. 3. 7s. 13. 2. 00. 3*84^. 16. 2. i$s. 2*40*. 16. 120. 11*52^. I ft. 1*824 in. 19. R6. 120. 9A 22. 45. 10. 6>>. 25. 28. 81. 18. 21. 24. 27. 30. 4 cwt. 2 qr. 20*16 Ib. Rl2. 50. 1*2^. R2. I2a. 10*464^. u. 137*'^. a- 302-4?. 7. 789-0310. 11. 2. 60. 7*#. 34. 4*. 3'8#- 120. 8*#. 4. 90. i*2/. l6j. 6*9120*. R2. 80. 6*7/. 10 md. 13 seers 4*84 ch. 33. a po. 2 yd. i ft. 3*9375 in. 35. 7. 120. 87. 2s. 3*0450*. 38. ;l68. 7*. 5*690". 41. 68. 30. rip. 42. 815. 20. R3. 140. 44. 17. 10. SA 45. 4. 150. jl. 3*. OJ0". 47. 12s. lid. 48. ^34. 145.6791^. H of 3. 90.) -025 of Rioo. 100., '32 of 5. 80. 3J of 10*., -256 of I*., -0034 of i. 51. 4. 120. 2*6/. 2-592^. 63. 9iff^. 54. i6s. 55. 68. 20. 5' I ton 17 cwt. 2 qr. 4 Ib. 57. 6 md. 58. Jo?. j. 7*1250". I ton 8 cwt. I qr. 8 Ib. 22 hr. 19 min. 4*275 sec. 113.70. 89. 7. iy Examples. Ri7'359375- i '42045 mi. 3-640625. 7-3I875. 7-2395^- 751875. 578481... 4780219. 36. 83. 35. 38. 2. 5. 101. 3. 4*4642857! tons. 9. 13. 17. 21. 25. 29. 771 5974 da. 6. ^40-95- 8*5. 3'9S- 620543.,. 1-045138. 009142857. 1*375. 14. 1*0042011... 18. 8296. 22. 1*06875. 26, I5'054375. 30. 84. 75595238& 85. *oi. 0102339... 39. J -0384615. Miscellaneous Examples. 102. The value of 2 is x&o ; of 7> lo&ro 5 of 3, imfinro 0076 | f^ff. 3. 72 ; 3$i$. 4. '000282. 7. 775. 11. 1-183. 15. 572. 19. -659375. 28. '481283.^ 27. 1*045918*.. 31. '26041(5. 86. '171296. 40. -328. -362, ANSWERS TO EXAMPLES 48$ 6. R225. ii*. #. 7. i ton 19 cwt. 3 qr. 3 Ib. 8. "50& 9. 89000. 10. '6962. 11. 64-09, 49-3, 1-3. 12. 1510640. 13. 8000 times. 14. 29 times ; 1-576 gallons over. 16. 21 times ; rem. 2-02. 16. '$. 17. 1 50 A** 18. 7'o59toas, 19. 8-571875 Ib. 20. ^33. is. i\d. 21. OSS- 22. -oos84...b, 28. 45 yd. 2*1812 ft. 24. 1142 ; '054 in. 25. '8095. 26. 81-649296. 27. 448-52990016. 28. 8. 29. 8000. 30. -15. 31. 82.9*. 8/fr. 82. RSiooo. 33. 9-5087... 84. 4*5 Ib. greater. 35. I5'i years. 36. 36 min. 24 sec. 37. 2s. 6d. 38. 820, 830. 39. A, 36 ; , 12 ; C> 4- 40. i I. 5. 9. 18. 15b. 16. 17. 19. Examples. 103. 2-1053. 2. -05882. 3. 1*0313. 3949. 6. PI i. 7. 2-00. P33. 10. 1-250. 11. 1*167. 1*41069. 14. -28768. 15. -20273. 632. 15c. -182. (i) 378400 ; (ii) 736000 ; (iii) -5207 ; (v) 2-010 ; (vi) 2*000 ; (vii) '03407 ; 3456800 ; 80057000. 18. (i) 4 ; (2) 3*9 J 143. 20. 3*14159* 4. 75*014. 8. 1*50. 12. -26667. 15a. -909. (iv) 7*385 ; (viii) -009063. (3) 3'93. Examples. 103 (1). 1. 1*14286. 2. 1-02041. 3. '85714. 4. -95238- Examples. 1. 7-306. 2^ 4*233- 3. 6. 18979409- & 64-20153. 7. 8a. '33799- 8b. 23-91753. 9. lOa. -8499*. 10b. -0415^. lOc. lOe. 25978. lOf. 231. lOg. 12. -344- 18. 1-229. 14- 16. 1937204, 17. 530'I3237 18. 20. 1084101*7079601. 20a. 21. -281. 22. 23*207065. 23. 103a. 0076. 4. 1180*5103. 7704746. 8. -392754- 66*939. 1. '143292. 10-367. lOd. 1*113. 28,632,0001000. 11. 1*617. 12-310. 15. -1178. 8231*60553* W. 1072-476227. 0065. 20b. -00785. 91330- 24, *37i* 436 ARITHMETIC 1. 6. 8. 11. 14. 17. 20. 23. 26. 29. 32. 35, 88. 40. 42. 45. 1. 4. 7. 10. 13. 10. 19. 21. 23. 25. 37. 30. S3. Examples. 103b. '062. 2. 1-892. 3. '140. ^ 6. roil. 6. Examples. 104. 2. ^843. i$s. 3. &49- ^1675. i6s. 2523. ga. ^42. 155. B453- H*. 6 ^1730. I5J. 1^4894. 20. 8/. 20-888. i'525. Ri3oo. 6. 130. #. 542. 5*. 400. 12*. fy. 6. 9. 12. 16. 8747. 5*. #. 18. ^8002. 7J. 4^T. 21. 87033. 7a. &. 24. ^280508. i$s.\7%d. 27. Ri$o6o. ^191898. 12*. 30. 2771. I9J. ^39247. 3003. 36. 243. 15^. 18. 10. 22. . 25. 28. 31. 33. 644434. 110. 4^. ^33673. 95. ^5027. lu. 1073. i$a. ofy. 7. 8327. I2a. 10. 4. lu. 8rf. 13. 226. ga. j8, us. $d. 2830. I2. 6^. 251. 151. 6 1</. &38397- loa - 6 A ^11714. 18^. nj 49514. 3 9fA 34. 2?8979- 3^- 4 20994. 8d5. ioj^. 34075- I4- i^- 7661. 9. oj^. 43. 72. 6a. Examples. 2. 44. 00. 37. 39. 41. 44. 46. 105. 5. 8. 11. 14. 17. 3. 6. 38. 20. io|^. 9. 27. 00. 2fc*. 12. ^150. i7J.6jfj^.l5. 25. loa. 6J ,68.14^9^. ^57. 8j. 67. 7*. 2/. 2. 6s. i$d. 1835. na. 4067. 2a. i last o Id. 4 qr. 7 bus. 11. i$J. 7t^ 24. 2529 md. 7 seers 8 ch. 26. 265.9^.5^. 28. 14, 15* 5fc ^239. 7* 9i^ 31. 92. la. $&* 4. 4664. 31. 9^- i|4 ^93. os. s 108. 15^. 100. ja. 180. 20. pk. 18. 109. 17^. 3</. 20. ^4279. 6s. 7\d. 22. 19 cwt. 3 qr. 9^ ^ 30 tons 6 cwt. I qr. 14 Ib. 26. 15*. ioj*/, 29. 45.4^1.6^. ' 32. 959. 7- IP* 35. 7999. ANSWERS TO EXAMPLES Examples. 106. 3. 1 8. 4. 24. 5. 36. 8. 84. 9. 105. 10. 231. 13. 504. 14. 6006. 15. 66990. 18. 2. 19. 3600. 20. 900. 1. 21. 2. 24. 3. 27. 4. 31. 5. 32. 6. 81. 7. 75. 8. 96. 9. 165. 10.^234. 11. 222. 12. 135. 13. 345. 14. 440. 16. 804. 16. 847. 17. 2222. 18. 1679. 19. i ooi. 20. 1234. 21. 9070. 22. 7906. 23. 9876. 24. 4607. 25. 56804. 26. 80047. 27. 15367. 28. 600098. 29. 543200. 30. 123456789. 3L 41. 32. 80. 33. 76. 34. 105. 35. 252. 36. 5. Examples. 107. 1. 30. 2. 40. 6. 64. 7. 42. 11. 315- 12. 756. 16. 2. 17. 15. Examples. 1O8. 1. 3*4. 2. 2-17. 3. 6*25. 4. 9*08. 6. -08. 6. -073. 7. 32-9. 8. 2-403. 9. -0231. 10. '0045. 11. 15*367. 12. '897. 13. '001849. 14. i 'ooi. 16. 968-8669. 16. 27-6025... 17. 1-3038... 18. 15-4147... 19. 2-2360... 20. 29-6063... 21. '3162... 22. 7071... 23. 4*8062... 24. -9486... 25. 4*4721... 26. '1264... 27. '0252... 28. 2-6457... 29. 8-1240... 3O. 3-6055... Examples. 109. 1. *$. 2. 74*. 3. sf 4. io&. 6. ij. 6. l'<5. 7. 5-3. 8. 1-83. 9. 2-83. 10. -20". 11. 1-322... 12.. '845... 13. *8i6... 14. 790 15. 763... 16. '577 17. '645... 18. 1-568... 19. -632... 20. 20-493... 21. 7?. Examples. I1O, 1. 2-236067... 2. 4-123105... 3. 27-602536... 4. -019598... 6. 774596... 6. 1723050... 7. '264575.M 8 * '921954- 9. 87-286883.., 1O. -612372... U. 15-414765... 12. 1-303840... 13. '845154... 14. 4-882304... 16. '030708... 16. 3'i62277M 488 1. 4. 7. 11. 14. 18. 21. 24. 28. L 5. 10. ARITHMETIC Examples. 111. 3. 36. 4. 48. 9. 89. 14. 956. L II. 2. 25. 3. 36. 4. 48. 5. 49. 6. 72. 7. 13. 8. 57. 9. 89. 10. 97. 11. 247. 12. 473. 13. 945. 14. 956. 16. 6031. 16. 551. 17. 9009. 18. 2222. 19. 45333. 20. IIIHIIII. Examples. 11. 1. 2*6. 2. 5-1. 3. '79. 4. 40-1. 5. 2-65. 6. '197. 7. -957. 8. -ioi. 9. f 10. &. 11. 3f 12. i9i 13. -3. 14. 11-6. 15. !$'&. 16. 3f. 17. 2j. 18. rj. 19. i '523... 20, 2-223... 21. 2-884... 22. i'959... 23. '928... 24. -646... 25. -464... 26. -584.., 27. '167... 28. 1759.^ Examples. 113. 1. 1*523913... 2. 2-884499... 3. i*959i7**~ 4. '125992... 5. '144224... 6. 2-648751*^ Examples. 114. 1. 4. 2. 22. 3. 36. 4, 6-3. 6. 9. 6, 2*6. 7. 54. 8. 4. 9. 5. 10. 2-434^. Examples. 115. 1 80 sq.ft. 2. 320 sq.ft. 3. 117 sq. ft. 64 sq. ft. 106 in. 5. 78 sq. ft. 51} in. 6. 70 sq. yd. 8 ft. lift. 8. 2ft. 4 in. 9. 99yd. 10. 8 ft. 9 in. 1067 sq. ft. 16 in. 12. 14 sq. yd. 81 in. 13. 392. 18. 15. Ri36. Sa. 16. 9. 15*. 17. 128 sq. ft. 556sq. yd. 19. 15888. 20. Ri6o. 15*. 78 j sq. yd. ; i. 6s. $d. 22. 4800 sq. ft. 23. 15 ft. 2l$f sq. ft. 26. if in. 26. 27? in. 27. Rni2. 80. 26 yd. 2 ft. 29. 1024 sq. ft. 30. 300. 31. R666. iia. Examples. 110. 220 yd. 2. 22 ft. 5 in. 8. 280 yd. 5*656. ..yd. 6. 42-42, ..ft. 7. i8ft. 8. 48yd. 77 yd. 2 ft. ii in. 4. 9. 50 yd. 34yd. ANSWERS TO EXAMPLES 4*9 Examples. m. 1. 60 yd. 2. 37 yd. 1} in. 3. 60 yd. if in. 4. 44. ja. i}p. 5. ^23. is. $d. 6. 648 sq.ft. '7. 495 sq. ft. 8. 88 sq. yd. 6 ft. 9. 288yd. 10. 96 yd. 11. 211 yd. 12, 176 yd. 2 ft. i| In. 13. 46. 4<7. 14. i7. 16. $. os. 4$df. 16. i57iyd. 17. Ri. loa. 7^ 18. 4s. 8iVr< 18. *i yd. ao. i6 in. 21. R3499- 3*- ^ 5.22. Rii4. I2a. 23. 5} ft. ^4. R83. 14*. io| 25. Ii9. 14*. 26. 5}f. ^7. Width, i8J ft. ; height, 14$ ft. 28. Rl3.6a. Examplea 118. 1. 12 bi. 2. 52 bi. 10 cot. 3. 108 bi. 7 cot. 8 ch. 4. 207 bi. 7 cot. 3 ch. 4 ga. 5. 357 bi. 9 cot. 3 ch, , 4 ga. 6. 7427 bi. 8 cot. 7. 4992 bi. 10 cot. 16 ga. 8. 12188 bi. 19 cot. 14 ch. 8 ga. 9. 27 bi. 12 cot. 8ch, 10. 8 bi. i cot. 4 ch. 41. 6 bi. 9 cot. 2 ch. 8 ga. 12. 19 bi. 12 cot. II ch. 4 ga. Examples. 119. 1. 400 cu. ft. 2. 183} cu. ft. 3. I57$cu. ft. 4. 8 j cu. ft. 5. 4952$} cu. ft. 6. 42! cu. ft. 7. 843! Ib. 8. 10080. 9. 3750 times. 10. 48 min. 11. 24. 12. i ton 16 cwt. 13. 2800 times. 14. '027. 15. 62$. 16. 4i. 17. 16 ft. 9 in. 18. 2 ft. 19. 1466. 100. 8 p. 20. i6407}$l tons. 21. Ri70. 32. I33i 23. 4 in. 24. 3yd. 25. 2564 Ib. 26. 675 Ib. 37. 60. 28. 15*404! ft. 29. R5520. 30. R2;6. 50. #. ; 31440, Examples. 1*0. 1. 4 yd. 7 A in. 2. 6 yd. 2 ft. 8 T V in. 3. I sq. yd. 4 ft. in in. 4. 2 sq. yd. 4 ft. 40} in. <5. 4 sq. yd. 4 ft- 12} in. 6. 2 sq. ft. 26}$ in. , 7. I cu. yd. 3 ft. 480 in. 8. 2 cu. yd. 20 ft. 1048 in. e. 10 cu. ft. 300} in. 10. 3 cu. ft. 47ijf in* 493 ARITHMETIC 11. 8 ft. 7'. 12. 34 ft. 7'- 6". 13. 8ft. ii'. 6". 8'". 14. 10 ft. 9'. 10". 6"'. 15. 56 sq. ft. 5'. II". 6'". 16. 70 sq. ft. $'. o". 4'"- 6". 17. 62 cu. ft. i'. o". 6'". 8 lf . 18. 28 cu. ft. i 1 . 8". o'". 5 lf . 4*'' Examples. 191, 1. 7 sq. ft. 72 in. 2. 67 sq. ft. 12 in. 3. 132 sq. ft. 117 in. 4. 217 sq. ft. 14 in. 6. 316 sq. ft. 36 in. 6. 129 sq. ft. 54 in. 7. 98 sq. ft. 8oJi in. 8. 130 sq. ft. 140 in. 9. 228 sq. ft. 83JJ in. 10. 2459 sq. ft. 107^ in. 1L 38 cu. ft. 1161 in. 12. 127 cu. ft. 304 in. 13. 874 cu. ft. 1510^ in. 14. 471 cu- ft- 5 8 5ll in - 15. 339 cu. ft. 453H in. Examples. L 6a. 2. R2. 8a. 3. 40. 4. 2 md. 20 seers. 6. 2 ft- 7 - 5^- 8 - R 35 I2a - 9 - 5 J - I0 ^- 10 * 36 ^ i- 12. ^2. I2J. 6dT. 13. S- I 4 - R2 Examples. 1. 30 da. 2. 60. 3. 270 da. 4. 700 mi. 5. 91* 6. 4$ da. 7. 7- 8. 4$ da. 9. II D , 10. 4md. 11. 270. 12. 270. 13. 2* Examples. 124. 1. R2079. 2. R2o. 3. Ri$. iia. 4. R6so, 6. 10. icw. 6. R48. 7<i. 7. 240. 8. 48. 9. 12. 135. 10. 36 Ib. 11. R8. 12*. 12. R9- no. t&p* 13, 20. 14. 8J^. 15. ^2.6^.8^. 16. 7^.6^. 17. R3937. 8. 18. ^816. i6s. 10. Ri764o. 2O. R240. 21. R472. i3. 7-^. 22. 7* da. 23. R3i.M*.24. ^i. 8*. 20. Ri68. 26. iw. $d. 27. ^3- 125. 28. 140.8^ 1*. 94f. 30. 2iimd. 31, R937 8. 32. 17^ days. 83. 16^. 34. 4618. 35. 117^. 36. 391$ yd. 87. 40}?. 88. I2&. 38. 433i- , ANSWERS TO EXAMPLES 4C1 41. iQof. 42. R7.6.6#. 48. i$. 44. 12. 45. R6o.. 46. 100 grains, 47. 8&. 48. 390. 49. Ri. 50. i Ib. 8 02. Examples. 125. 1. 6. 2. 6. 3. 8. 4. 15. 6. 10. 6, 1 1 md. 8 seers. 7. 4. 8. 2 hr. 40 min. 0. 12 oz. 10. 9J. 11. 48. 12. 1 80 days. 13. 46! days. 14. 4ijdays. 15. 4. 16- 6 months. 17. 35iV Examples. 186. I. 2. 2. $. 3. 3- 4. 7- 5. 50. 6. 6;i 7. 22?. 8. 32. 9. ioi 10. 50. 11. 8J. 12. 534- 13. 75. 14. 84. 15. 23$. 16. 60 yd. 17. 7* ^ 18. 2*. 4*/. 19. 8. 20. 100. 21. icf. 22. 15- Examples. 1&7. 1. 6. 2. 3i 3. nt. 4. 3&- 5 - 34- 6. 3. 7. 16. 8. 33if*- 9- 26 iV 10 - I0 * II. Ri2. 3. 12. R8o. 13. 16 days. 14. Rn8. 120, Examples. 1. 93. 120. 2. 471. ". 3- Ri7i- 14*. 4=. 6. la. %p. 6. 3</. 7. 2967. 3*. 8. 4000. 9. Ri92o. 10. ^396. I2J. 11. R288o. 12. /i8o. 13. 722.13.4. 14. # 15. A. 16. 3200. 17. 3000, Examples. ISO. 1. 4Jhr. 2. iJHa. 3. i^fe hr. 4. 4 da. ; A |, ^i, C T V 5. 12 da. 6. i hr. 7. 7iV min - s - 4i te 9. ^, 20^ da. ; ^, 8Ji J C, 7JS. 10. 2^ da. 11. 18 da. 12. I3jda- 13. 120 da. 14. 4$ da. 15. Each in 60 da. 16. 7 jf. 17. SiM to I- " hr. 19. 16. 20. 6i. 21. At 10. 22. 32- 23. 25 da. 24. 76. 25. lafmin. 26. 4 hr. 27. $6j da.. 492 ARITHMETIC Examples. 180. 1. 2 h. 39$J m. P. M. 2. 2 h. 48^ m. P. M. 3. 9 P. M., Friday. 4. After 112 da. 12 hr. (true time); first, 7 h. 48} m. P.M.; second, 8 h. i8f m. P. M. 6. 8 h. 47 tW m - A M - 8. The slower must be put on 13^4 min. ; or the faster put back I3JJ min. 7. 3 P. M., Dec. 3. 8. 9 min. 9. J min. 10. 4 P. M. 11. Tuesday, 4 P. if. 12. ,V min. past 9. 13. Tuesday next, 4 h. 54} m. P. M. and 4 h. 32$ m. P. M. 14. ioj$ min. past 6. 15. $ sec. 16. i h. 50,^ m. P. M. 17. On March 13, at the same hour at which it was put right. 18. 5 <& a > at th* same hour ; after 235 da. at the same hour. 19. 2sAr min. Examples. 131. L (i) io}$ min. past 2 ; (ii) 27^ min. ; (iii) 43^, min. ; (iv) 24 min. ; (v) 34} J min., and $2/ T min. 2. (i) i6j\ min. past 3 ; (ii) 32^ min. ; (iii) 49^ min. ; (iv) 3& min., and 29^ min. ; (v) 40^ min., and 57& min. $ (0 3 2 A min. past 6 ; (ii) 16^ min., and 49 T \ min. ; (iii) no time ; (iv) I9/ T min., and 45^ min. ; (v) S^j min., and 56^ min. 4. (i) no time ; (ii) 16^ min., and 49^ min. past 12 ; (iii) 32^ min. ; (iv) i3 T \ min., and 52^ min. ; (v) 24 min., and 41^ min. (i) 3 8 A * P ast 7 5 (") 2 iA min., and $4i\ min - 5 (iii) 5 A min. ; (iv) 25^ min., and 51^ min. ; (v) 14^ min. * (>) 54A mm - P* 8 * I0 (") 5A min - and 3 8 A min - 5 (iii) 2i& min. ; (iv) 2^ min., and 4i& min. ; (v) 13^ min., and 30^ min. 7. 32jf min. past 2. 8. 27$ff min. past 5. 9. 4ijfi min. past 5. 10. 4i\ min - P* 8 * I2 - 41. i min.-div. put back. 12. Gains 5^A min. Examples. 1S. 1. In 45 sec. 2. 417 mi. 8. At 7-30 p. M. ; 300 mi. from Cal. 4. At 5 h. 34| m. A. M. ; 257$ mi. from Cal 5. 4$ sec. ANSWERS TO EXAMPLES 493 6. 36 sec. 7. 3$ and i$ mi. per hr. 8, , i hr. 26^ min. 9. 1 50 yd. 10. n h. 38$ m. A, M. 11. ii9imi. 12. 12 mi. from Cal. 13. 7 miles. L*. 5 min. 24)) sec. after B starts. 15. 9 h. 9& m. A. M. 16. 240 mi. 17. 6 mi. and 5 mi. per hr. 18, 7 mi. 18a. iiiml, 19. 9 hr. 37$ min. 20. 10 hr. 46 min. 21. 46. 22. 16 min. 42 sec. 23. 3 hr. 55 min. 24. 28 min. Examples. 133. 1. (i) 10 hr. ; (ii) if hr. 2. (i) 74 hr. ; (ii) ijhr. 3. 3ii da. 4. 300 da. ; 300 da. 5. 3 hr. ; 6 hr. Examples. 134 1. 5^ min. 2. 79A X d - 8 4. 9 min. 36 sec. 6. C can give B 5 points, 6. B wins by 126 yd. 2 ft. and by i min. 16 sec. 7. 5. 8. C wins by 6of yd*. 9. A> i min. isjff sec. ; B, i min. 2o| sec. ; C, i min. 23 sec. 10. A wins by 68ff yd. 11. 9. 12. A in i6$& sec. ; #, 17! sec. ; C, i8f sec. 13. 176 yd. 14. 5. 15. Ami$ min. 50 sec. ; B in i6min. 20 sec. ; in 16 min. 40 sec*. 16. Cwinsby Examples. 135. 1. i8&. 2. 60. iojf 3. ico. 4. 82. 40. 6fy. 5. 6. 1885. 7. ioda. 8. 3fd a - 3. 10. 100. Examples. 136. 1. 10. 2. 45- 3. 264. 4. 75. 5. 8. 6. lotf 7. B37- 8*. 8. 30. 9. 24. 40. io&. 10. 21 mo. 11. 8, 12. 6. 13. 43i da. 14. 120. 15. 6 ft oz. 16. ii. 4^. 17. loj. 8<* 18. 8*- 19. 27. 2O. 9- 21. 25. * 22. 10. 23. 13*. 24. 4t 25, 6f ox. 26. 49&S' 494 ARITHMETIC 27. 8. 28. 4. 29. 7. 30. 4. 31. 8. 32. so|. 33. R6o. 7*. g^lf. 34. 75 ac. 35. I9joz. 36. 20. 37. 3- Examples. 137. L R20. 2. R3SR4- 3- 180 gr. ; 87 & gr. 4. Ri3- 6. RS ; R20. 0. 48 da. 7. 28 da. 8. 54^ da, 9. 4 da. 10. A man in 7$ hr. ; a boy, 18 hr. ; a man and a boy in 5} ht. 11. 6. 12. 10 hr. Examples. 138. 0. f 2. |. 3. A- 4. |. 5. if. 6. jjf. 7. |. 8. |. 9. J. 10. 5 : 4. 1L lU. 12. I 1 1. 13. I : 4. 14. 7 I 8 is greater. IS. 18 I 29 is greater 16. 4:5 greatest, 2 : 3 least. 17. 7 I " greatest, 3 : 7 least. 18. Yes. 19. No. 20. Yes. 21. lof 22. $! 23. '0002. 24. 1 8 Ib. 25. i. 6s. 8<f. 26. 45 men. 27. 2. $j. 28. 30 hr. 29. 7*. 30. 14. 31. 39, 32. 7280. 33. yV 34. 3|. 36. '06. 36. 25. 37- 4H- 38 - "a- $A 39. 17 : 10. 40. 27 I 64, 4i. 2 : i. 42. 192 : 240 : 280 : 315. 43. 2. 5^. %)<t. 44. 18500 oz. 45. 33ft. 46. 15 : 16. 47. 32. 48 30 gall., 20 gall. 49. 40 gall. 50. 16:15. Miscellaneous Examples. 130. 1. I7 . 2. 8204. 3. 3 8 .5'7 8 -" 2 .i3 8 ; 5- 4. JfJ- 5. R369. 20. #. 6. 1 8. 7. 9996 and 1020. 8. R65. 15.6A 9. 8. 10. 25. 11. ^269. u. 9K 12. 158415. 13. 3020 men ; 2700 women. 14. RISI. 2a. 16. 63 times. 16. 3& 17. 123. ? 18. i. los. 19. 84. ISO. R8. 2a. 6/. to each of $ ; R4* ia. 3^. to each of the others. 21, 13. 22. '0203125. 23. '016. 24. I4J. 6 6. 26. 720. 27. 162 dollars. 28. 13^ gall. 419, 112 sq. yd. 7 ft. 3O. 4} f hr. 31. 50 years. 82. 10 seers. 34. *o8j. 35. RIIO. *a. ; i ft. 36. $0. 87, f}^. 38. The first person gains Ri. no. 6>. more. ANSWERS TO EXAMPLES 495 39. 43. 47. 50. 63. 66. 69. 61. 64. 67. 70. 73. 76. 78. 81. 84. 88. '93. 97. 98. 101. 104. IO7. 110. 113. L16. 118. 455. 14. 9600. 40. 44. 48. ; T 4L 4. 45. 82790. loa. 22. i$s. ; 7. 12s. 8</. 4 sq. ft. 18 in. i. 7*. id. and $d. Ri. 100. fy. ; Ri. ga. 4 gall. 51. 64. 57. . 60. 62. if ft. 42. 85888. 40 grains. 46. '6552. &. 49. 14. 42 boys ; 20 fruits. 52. . I P. M. 13*. lo 2. oj. 14 ; 28 120 mi. from Cal. 65. ; fj. 68 71. 42. Monday, 12 h. 8 m. P. M 82560. 79. 59^3. 1-15 o'clock. 82. 8 mi. per hr. 85. 72. 89. 45. 90, 13} da. 55 min. 9} vyeeks 3f hr. 172800. After I2 2mi. 42 ft. ; n h. 56 m. A. M 80. 2250. 16 Ib. 6 : 5 74. 65. 83600. 68. 27-1. ; 341. 5*. 63. n P. M. 66. 39. min - 69 - B 2 * 20 - 72. 128. 75. 14! da. 77. 66 yd. 14 yd. ; 7 yd. ; 2 yd. 2ft. 83. i^ mi. ; 2 hr. 86. 2} hr. 87. look, 91. . 92. 5 4 da. yd. 55$ sec. 94. 20}? yd. 95. 10. 96. 29 of wine to 41 of water, A, 85. 4*. ; 5, 817. I2a. ; C, 824. 4^ and 16}? min. past 2. 99. 3oJ| sec. 100. 18. A cow, ji ; a sheep, $s. 102. 7 I 17. 103. J. 7j. 105. 4 mi. per hr. 106. 108. 2 oz. 109. ni - 55 min 112 ' 1 52 da. 114. 4 gall. 115. A in 36 days ; B> 48 ; C 9 28|. 117. 360 sec. 119. 15. 12O. Examples. 14O. 100. 103. B wins by A 2 gall. 5 min ' X 5 sec ' 491. 8s. 20 mi. per hr. 2 I i. L Ri. 9*., 83. 2*., 84. -2. 8. 2j., /;6. 15^., 2. 1 4. 75, 100, 112$, 120, 125. 7. 66 ; 71. 10*. 8. 11. 840,830,820. 12. 14. 818,86,88. 15. R6. 4^. ^.1 18^. 3. 7i 4i> 6|, 6. 3j i 17 6. 6. Rio6. 100$ Ib. 9. 250 Ib. 10. 50^)00. Ri2,Ri6, R8. 13. 8240,880,840. 8, 6, 16. 12,10,8. 496 ARITHMETIC 17. R6, Rio, R$. 18. 5* 7$<*., 7 19. Each man 5;., each woman 31., each boy 25. 20. R2. 21. Men 27*., women 274., children us. 3</. 22. ^iS, ^12, 23. }$ cwt. 24. 20, 30, 40, 50. 25. 50. 20. 40 rupees, 48 eight-anna pieces, 64 four-anna pieces. 27. Each man R2. 80., each woman Ri, each child Rf . 28. f,f,i. 29. 80. The radii are -- and - ft. 31. 180 gr. v3 v3 82. R25000. 33. 57. Examples. 141. L R70, Rico, Riso. 2. R78o, R52O. 3. 1200. 4. R4500, R3000, RSOOO. 6. R3372. 8a. 6. 480, 360, 7. ^17. ioj., 15, ^12. 8. R7, R6, R4. 8a. 9. ^286, ^163. 161. 10. B483Jfti &498i!i ft2z8U. 11. 100. 15. 366. 13. Ri68. 120. 14. 30. Examples. 142. L In the ratio of 3 to i. 2. 8 : 5. 3. In the ratio of 9 to n 4. 197 : 1 80. 5. In the ratio of 33 *. 2. 6. I I 4 7. 8$ Ib. of each. 8. 25 md. at RS, 35 md. at Ra. 4*. 9. 4! gall. 10. 20 I 7 ; 51. i$//. 11. In proportion of 3, 3, 2, V 12. In proportion of i, i, 5. 13. 10 gall 14. In proportion of 4, 6, 9. 15. In proportion of 52, 78, 51, 6& Examples. 143. L 3. 2.. I3f. 3. 7f 4. 4'34- 6. n|. 6. R4. 80. 7. 125. 8. 2 . 19 . 4t 9. 10 st. 10. R4 . 8 . 9}^ U. 8|mi. 12. lojst. 13. 14 yr. 14. 43 yr. 15. 8$ st. 16, ii yr. 17. RS. iia. 18. 7. *19. 63, 75*. Examples. 144. L |, 2. i. 3. xta ^ db- & i< 6. 7. jlo, loj. 8. 3^. 9. 1218. 10. Y % sq. in. ANSWERS TO EXAMPLES 497 1L 4 cwt. I qr. 14. 600. 12. R750. 15. RSI . 15 . 7j. 13. 35929. 16. 450. Examples. 145. I. 2S p. C. 2. 16} p. c. 3. 3J p. c. 4. 40 p. t. 6. 42? P- c. 6. 35 P- c. 7. 88| p. c. 8. ioJS p, c 9. 468} p. c. 10. 138 p. c. 11. s P * I 2 - 20 p. C. 13. 20 p. C, 14. 57} p. c. 15. 210 p. c. 16. 50 p. c. 17. 8?** p. c. 18. 24 p. c. 19. I2i p. CS 20. Nitre 75 p. c.) sulphur 10) and charcoal i [5. 21. 8ip.c. Examples. 146. 1. 220. 2. 1200. 3. 25. 4. 10800. 5. 100. 6. 1296/7. 7. R4875. 8. RSOOO. 9. 13000. 10. R78. 20, Miscellaneous Examples. 147, L 100. 2. RSooo. 3. R4545&- 4- 128. 5. Ri53ilJ, 6. 35 p. c. 7. 54? p. c. 8. 2 Sir P* c. decrease. 9. 50 Ib. 10. 9AP-C, 11. 1 8^ tp. c. 12. 9A P- c. Examples. 148. 1. Ri75. 2. 245. 3. R75f 4. R7003. . R28000. 6. ^914?. 7. R3ooo. 8. ^101 . 10 . 9. Rioooo. 10. 260. 11. Examples. 149. L 6. 8. 12. 16. 20. 25 p. c. 2. 25 p. c. 3. 4 p. c. loss. 6. 71$ J p. < R8o ; lo. io|^, 9. is. 5^. 2s. i${d. 13. 12$ p. c. 8md. , 17. !43^rRi2. ?j. 2L R2.o.4$. \ p. c. gain. 25. 50 p. c. 17 p. c. 29. 26AP- c. 25 p. c. 4. :. gain. 7. 10. 12. 11. 14. 2&a. 15. 18. R2320&19. 22. 8. 23. 26. 2ffrf. 27. 30. i6| p. c. 3L C. 33 J P- c. 33i P. c. 95*. 4}< R320. 6 p. c. gain. Loses 16 p. c. Ri5o. A. 32 4<j8 ARITHMETIC 32. R22J. 33. 25 yd. 34. Gains 30^ p. c. 36. 4 for 3. ; 512. 36. I Ib. to 2 Ib. 37. 2. #. 38. I7i p. c. ; 2 I I. 39. R23 . 5 . 4. 40. 19 I 12. 4L 1:2. 42. 21 p. c. 43. R46o. 44. 33$ p. c. Examples. 15O. L R7. 4<*. 2. R21. 6a. 3. R45- 4. R263 . 10 . 9. 5. Rn . 12 . 6. 6. R27O. Examples. 151. 1. R24. 2. 60. 3. R3I5- 4. 57. 12*. 6. R222. 12*. 6. 112. 7. R40.i3.8}J;R536.i-4|. 8. 32. 10. 6; ^357- 15-6. 9. Rio8 . 5 . 7&h R334 . I- 4J1- 10. R285. 11. ^372. Ss. 12. R440.8.4J. 13. 763 13 *V 14. 406 . 4 iJtJ- 15. 226 .i.ii. Examples. 15$. L R33.5-4- 2. 100. 3. 157. w, 4. R5.I2.6. 6. R2.0.3. 6. R3.I4.7- Examples. 153. L 2. &s. 2. R2o. 4a. 3. R4 . 13 . ifjf. Examples. 154. 1. 2j. 2. sJ. 3. 3j- 4. 5. $! 6. 3i. 7. 2j. 8. ty. Examples. 155. i L 3yr. 2. 3jyr. 3. 3^ yr. 4. 4yr. 9mo. 5. 2 yr. 3 mo. 24 da, 6. 97 days. 7. 64 yr. 8. 3 yr. 9. 5 yr. 10. I5th April. 11. 16 mo. Examples, a 150. * L R750, 2. R4266 . 10 . 8. 3. 170 .6.3. 4. 1050. 6. R400. 6. R73o. 7. R8oo. 8. RlSo. 9. 265. . 10. 33 .13.4. 11. R672 . 4 . 4. 12. ^1022 , 14 . 7. ANSWERS TO EXAMPLES 499 Miscellaneous Examples. 15T. L 6J. 2. RSOO. 3. 570. 4. 3yr. 5. loyr, 6, 6 p. c. 7. R9733 -5-4. 8. R4oo ; 7$. 9. 8& 7*. 10. 8533 .5.4. 11. ;i9' 12. 30000. 13. 819200. 14. 40 yr, Examples. 158. 1. 841. 2. 842. 6. ii. 3. 838.6.6. 4. 8141.2.8. 5. 731 3 3- 6. 343 .4.5-7. 6 4i -6.3. 8. 260 .9.1. 9. 814 . 2. 2f{. 10. 31 . 18 . 9 to the nearest penny. Examples. 159. 1. 81102. 8a. 2. 8327.13.1. 3. 8772.4.2. 4. 8855. 14*. 5. 82184 . 13 . 4. 6. 84328 .7.7. 7, 81.0.10. 8. 811.1.7. 9. 83278.2.11. 10. 8375 . 3/11. 11. ^90. 14 . i to the nearest penny, i2. 120. 13. 250. 14. 3125. 15. 815 . 3 . 3 to the nearest penny. 10. I5J. to the nearest penny. Miscellaneous Examples. 160. 1. 82-432. 4. 8625. 5. 83310. la. 6. 85184. 7. 810000. 8. 85000. Examples. 101. 1. 8170. 2. 81250. 3. 83562. 8*. 4. 1337. KW, 6. 1416 . 13 . 4, 6. 1005 .6.8. 7. 81600. 6. 8182. 8a. 9. 820000. 10. ;iooo. Examples. 162. 1. 85. 4. 2. 880.3.4. 3. 8151.140. 4. 8105 .6.8. 5. ,20 . 4 . 8}. 6. ,17 . 8 . a|f, 7. 4-2.4. 8 - * l $ s - 9 - R 7o8. ia. 10. 8482 . 14 . 8. 11. 81077 .8.6. 12. ^38 . 3 . g. 500 ARITHMETIC Examples. 103. 1. 2 years hence. 2. 3$} yr. 3. 3$ yr. 4. 9 mo. 6. 2\ yr. 6. 4j yr. 7. 3 mo. Examples. 164. 1. 20 p. C. 2. 2$p.c. 3. sJp.c. 4. 2jp,c. 5. 3 p. c. 6. 5 p. c. 7. 3iP- c. Examples. 165. 1, R8I34. 8a. 2. R53903 . 10 . 8. 3. 574.3.4 4. 4 yr. 5. 19 mo. 6. 3$ p. c. 7. R6oo. 8. 2800. 9. R4$o;6ip. c. 10. 200 ; 5 yr. 11. fti34|i. 12. 858.6.8. 13. &s offer. 14. R6of^. 15. 50 : 51 ; R49A- 16. 20 p. c. 17. R9i|. 18. 17!?- 19. 188 . 13 . 5J. , 20. 124. 21. ^375- "w. 22. R7I28 . ii . 10. to the nearest pie. Examples. 166. 1. R2.8.IO. 2. 247. loj. 3. 2. 8j. 4. R88. 130. 6. ilf s a. 6. 1 J|^j. 7. R9504- 8. R337.8*. 9. 20 p. c. 10. i6| p. c. 11. 12$ p. c. 12. 334 p. c. Examples. 167. L 7 mo. 2^ 2, 9 j mo. 3. 8 mo. 4. 6 mo. 6. 9th Juries Examples. 168. 1. Ri900. 2. 242 . 16 . 3. 3. R5034- 6a. 4. 935. 5. 106$. 6. Ri$oo. 7. R4$oo. 8. 7440. 9. R;o. 10. 22. ioj. 1L 1248. 12. si77i- 13. 530* Examples. 169. L R7a 2. Ri64i .5.3. 8. 40,000. 4. R27O. 6. 921. 41. 6. R779.2.8. 7. R6. 40. 8. 17. 9. 105. 10. 20 increase. 11. R375<> stock ; Rn. 40. increase*. 12. R34 decrease. 13. R2o gain. 14. No alteration.. ANSWERS TO EXAMPLES $OS 15. ^3o>5oo. 16. 22,500. 17. 7200. 18. 93t 19. I2 9 J. 20. Examples. HO. 1. 4 P. c. 2. 4iWP- 9 3. siP-c. 4. 3j& 5. 6. 74j. 7. 99. 8. 86 j. 9. 4^ p. c. 10. The latter. 11. The former. 12. j\ p. c. 13. 87040, 14. Miscellaneous Examples. 171. 1. iW P c. 2, 2\ p. c. 3. The former. 4. 32. 5*. 5. 77$. 6. 190. 7. ;i8oo ; 2 years sooner, 8. R9o,6oo. 9. Ri824. 10. 91. 11. 82$. 12. R840. 13. 108. 14. ^9880. 15. 830,000. 16. 4. 16$. ; 35 I 34. 17. 2261 ! 2260. 18. R2o,8oo, 19. 10. 20. Riooo and R2ooo. 21. 400, ^1200. 22. 83200. 23. 3! J p. c. 24. Rioo. 25. R27OO. 26. /2429i4|Jf 27. /75,ooo, 28, Examples. 17$. 1. 275 .15.5. 2. R37Q5 .7.6. 3. 360. 4. 4 . 17 . 4- 5. &2 . 13 . 4 per dollar. 6. no. 7. Rij. 8. 14. 9. R25. 15^. 10. Advantageous through London. U. .12 . 18 . 7jf. 12. I lose 10 p. c. 13. 8j. 2d. 14. ^83.6.8. 15. 56. $s. 16. Ri-u. 8/* 17. /8o. 18. ^4687. io.f. 19. Gains 11. 5^. 20. is. 4^. per rupee. 21. i Gold Mohur 71. ..eagle. 22. i Napo.8*55 rupees. 23. Ri. 8<z. 24. 2s. id. 25. One of the former -2 of the latter., Examples. 17a. I. 2-305 Km. 2. 3 Km. 4 Dm. 7 cm. 3. 120 Dm. 3 m. 2 dm. 7 cm. 4. 75073050 mm. 6. 30 Km. 7 Hm. 5 m. 8 cm. 6 mm. 6. 23000807 sq. m. 7. 5oo6oo g 04 sq. Dm. 8. 4 ha. 7 a. 40 ca. 9. 80700 ca. 10. 36 ha. 30 a. 70 ca. 11. 3 cu. m. 12 cu. dm. 35 cu. cm. 12. 5027004000 cu, mm. 13. 40 Kl. 7 HI. 3 dl. 2 ml. 14. 3 Mg. 4 Hg. 6 gr. 15. 13 fr. 7 dec. 5 cent. 16. ri m. 502 ARITHMETIC 17. 4125 times. 18. 5 days. 19. 8 Kg. 5 Hg. 20. 3 fr. 75 c. 21. 3 a. 5 ca. 22. 200 hectolitres. 23. '914.. .metre. 24. '621. ..mile. 25. 29-921808 inches, 28. 453-6. ..grams. 27. 1*2255. ..grams. 28. 4545*45 cu. cm. 29. 8 tonneaux 825 kilo. 30. 1056-8.. .grams. 31. 7. 6s. iod. 32. 2-20 Ib. 33. 10 Ib. nearly. 34. 5-25 m. 35. 1050 cm. 88. 5 francs. ,37. (i) 1000 ; (2) 1000000. 38. 37500 cu. cm. 89. 4 m. 40. 1000 grams. 41. 28*41. 42. 13*6 ; *S. 43, 1*5 cm. 44. 13 times ; '61 litres left. 45. 35*2. 48. 5 yd. 2 ft. 1-9051 in. 47. 121*8 ares. 48. 1*234 metres* *& 453 grams. 6O. 637^5 kilo. 51. 1*5 metres. 62. r8... francs. 63, ^44. 54. 19375 sq. yd. 55. 5*2 sq. metres* 68. (i) 2*54 cm. ; (ii) 1550 sq. in. ; (iii) 61 cu. in. ; (iv) 28 litres. 67. 16 grains. 68. 933*25 grams. 69. 1360000 grams. 82. i hr, 25 min. 20 sec. 63. 07716. 64. 3727 litres. Examples. 173. 1. 30. 2. 894. 3. 870. 4. 3. 5. 3j. mi. 8. Ri8. 7. $s. lod. 8. Tea 2j., coffee is. per Ib. 9, Tea 2*., sugar (>d. per Ib. 10. 2 and 5. 11. ,900 and yx>* 12. 25, 30 and 35 years. 13. 20, 10 and 15 years. 14. y4R54, #Ri8, CR8. 15. Riso. 16. 8342?. 17. 95, 60. 18. 40, 60. 19. 50, 300. 20. R6. 44. 21. 50. 22. I md. ; 5 md., 3 md. 23. 40^ mi. per hr. 24. 24JH$mi. 26. 1122 ft. 26. 15^% min. 27. 9|J min. 28. 40. 29. 20. 30. 70 oz. 31. 12 gr. 32. ii oxen, 24 sheep. 33. 8750. 34. 20 years'. 85. 3 p. c. 36. 3j weeks. 37. 19. 38. 15 Ib. 10 oz, 3d. 44 days ; 2 I i. 40. 200 cu. ft. 41. 3 hours. 42. 3 hours. 43. 65 gallons ; 13 hours. Examples for Exercise. 174a. 1. Ten billion, thirty thousand two hundred million, seven hundred and twenty thousand, and twenty-one. 2. 48910. 3. 47337?. ** 5'- f . *7. 6. & 8. 23-0424 j 22-9596. 7. B4.7.* ANSWERS TO EXAMPLES 5-3 8. 9. 13. 16. 20. 23. 27. 32. 37. 41. 45. 49. 54. 67. 60. 63. 67. 7L 74. 79. 83. 88. 91. 96. 98. 10L 106. 108. 112. 116. 120. 124. 128. 132. Three hundred and twenty crores, one lac, three thousand, ope hundred and two. 10091401. 10. R2 . 7 . 3. 11. 37- 12. li 0001596 ; "0051472. 14. 49110419796. 7045. 33211521848. 026. 28. 15. 33. |g. 921. 765. 212. R8. 30. *.* 7. 50. i'2375. 124727. 324. 17. 21. 24. 29. 34. 38. ^|- &3. I2fl. 46. ij 61. 55. 58. 25. 30. 36. 39. 43. 47. 52. 16. 18508984. 18. 48345. 19- 22. CMXLIV ; 499. i$. 26. I53-4II34. 27. 31. 32953856 dr. 3 po. 4 yd. 2 ft. 3 in. 14. 22. 5456. A- Ri. 4536360. 80. 85. 89. 92. 76. 42-6. Saturday. 729. 86. 120712. 90. 3. 8a. " 93. 96, '3125. 64. 68. 72. 75. I 4A * 84. 43-3- 9405- 934-12 sq. yd. 275 times ; rem. '003 9, 6 and 4 times. 99. i&V i2- li- 103 - 21 yd. 2 ft. 2j in. 106. I2a, 109. '000000142857. 110. S V 113. 1296. 114. f. 116. 3^ 117. 8. 118. 48. 12L 2s. 8i^ 122. *33. 125. 3*461538. 126. Wednesday. 129. 53. 130. 0432. 133. 3840. 134. 4O. 3-0688259.., 2. 44. 1421144. 4. 48. -og. 11. 62. 3f. 53. 700310. I25'56875</. 66. I min. 3osec. Ri6. 130. 3^. 69. 1 \. 61. 9 ; 7. 62. 424-8936. 8110328. la. 6/>, 4828-04... 73. 8466. 9*. 77. 769. 78. 137. 81. A/&. 82. }. ^125. 5*. 87. i 7702^ in. 5 and 7. 94. 2f. 97. 29400000, 65. 52084. 66. J. 69. 3. 70. 340 po. 5 yd. i in. 326764. 4461535. 1753. 00759... 100. 4J. 104. 112*4, 107. ^, 11L '8. 1386 sq. yd. 3 ft. 96 in. Rx. 8a. 8^. 119. 220. i|J. 123. 13. 182. 75. 2d. 127. 13. 4^. 131. 20. ^.3.5.7.673 ; 37.19.101 j G. C. M. 21 ; L. C. M. 2'.3,5.7.i9.xoi,673. $04 ARITHMETIC 135. 26. 136. I. 187. -657526. 138. *J*. 139. 4288*179204. 140. 250 times. Examples for Exercise. H4b. 1. 3210 ; 1023. 2. 12. 3. 3. 4. i6f min. 6. 46|. 6. J. 7. 5 p. c. 8. 4,7. 9. 4725. 10. 1050 sq. yd. 11. 6 h. 27^ m. P. M. 12. 46. 4a. 13. 3*2804. 14. 4. 16. 137- 16. 1250 ; '0125 ; '0000000125. 17. &5. loa. 18. Monday 8 P. M. ; & min. to 6. 19. IQJ. ; 6s. %d. ; 2d. 20. !% 21. 17*. 6</. 22. 1855. 23. 8^A- 24. 300 sq. yd. 26. 8 hr. 26. 22. 8j. 27. 169 I 191. 28. 9 T \ p. c. 29. 999976 ; 100141. 30. 172. 81. 19251, 18261, 17271, 16281, 15291, 15201, 14211, 13221, 12231, 11241,10251. 32. 3ihr. 33. 9963. 34. 11:9. 35. 33$. 36. 5. 37. 14. 38. 8750. 39. 7h. 34m. P.M. 40. 419* X9 -3- 41. 401:544.42. 4yr. 43. 150. 44. &. 46. 1015. 46. 3J days. 47. 9 days, 48. 16 ! 65. 49. 264. 6j. 8</. 60. 14. 61. 80. 62. 8156. 63. I hr. 64. 70. 65. 83 \ 92 ; 92 I 153. 66. ,4800. 57. 429. 58. *oi. 59. iij gallons. 60. n P. M. 61. 12 da. 62. In the first vessel ratio of wine to water is 1729 I 271 ; in the second 271 : 1729. 63. ,4840, 4400, 4000. 64. 20. 65. 7*875. 66. 453750 tons. 67. 45 days. 68. 440 ml. 69. 7 I i. 70. 53J. 71. 200. 72. 120. 73. 26. 74.' 17$ mi. and 9$ mi. per hour. 75. is. ioj*/. 76. Each man 3. 15*. ; each woman 2. ioj. ; each child ^i. 5* 77. 4 mo. hence. 78. 250. 79, 388 ; 11-32 gr. 80. Ri9. 80. 81. Loses }J|f min. 82. 20 hr. 16 min. 83. 1200. 84. 276 .6.1. 86. 8184 or 7434. 86. 10. 8j. 87. 126. 88. 12 hr. 89. 18/3 days ; on the supposition that they work 13 hours a day. 90. A 540, #360, 240. 91. R62I&. 92. 8500- 93. 61000. 94. 24 yd. per min, 95. 9 hr. 96. H3*}*gr. 97. R2. 130., R4- 80. 98. 10 for a rupee* 99. 1033. 100. 128-5016... 101. i in. ANSWERS TO EXAMPLES $O 3.02. The clock ought to have been set at 5 h. 3of }Jf m. P. M. 103. 1 50 mi. 104. A, 48 ; B> ^40 ; C, R35. 106. R26. 106. 63. 107. Iff. 108. 16 ft. 109. 12$ hr. ; A, 4$ J * 5}, 110. Ri. 8#. 111. 4*., 8a., Ri. 80., R4. 80., Rij. 80. 112. R24 1 T TJ . 113. R66o. 114. R24ooo, 115, 73 times. 116. $1 miles from P. 117. roa. 118. A's i| oz., #'s 2 oz, 119. Rio. 120. 280. 121.' -0218... 122. 2ft. 123. 7& yd. 124. Rg. 70. 3^. 125. 40. 126. R3. *<*. 127. 46. 128. '575. 129, 12. los. 130. 5^ days. 131. 4*ft. 132. 8ft. 133. Will lose 7 P- c. 134. 120. 135. 4$. 136. 15 yd. 137. iff hr. 138. 48. 155. 139. 35, 15, 10, 25. 140. 47^ p. c. 141. RS. 142. 576*0297502224. 143. 50 times, 144. They will run a dead heat. 145. 25. 146. 9. 147. 10. 148. 3 gallops. 149. 30 . 14 . 8|. 150. 3 ft. 151. 23^ days. 152. 43 wk. I da. 2 hr. 163. 6 ft., 8 ft. 154. Loses 53^ p. c. 155. 78. 156, 8. 6s. 157. 121. 168. 21? min. 159. Riosooo. 160. 6 ^2 in., 8 */2 in. 161. 12$. 162. 42 gallons. 163. 279 ; f . 164. Breadth, 6 yd. ; height, 5 yd. 165. 254$ min. 166. R67. 80. 167. 224, 336, 420. 168. 54!?. 169. 72. 170. t \V 171. 4 hr. 172. 2i| hr; 173. 66 min. 174. A must pay is. $d. and C is. 6ef. to B. 175. 40. 176. ii. 177. ^2359.15^.2^^. 178. , 1200. 179. 36 mi. and 4 mi. per hour. 180. 2333283$ francs. 181. 1327. 105. 182. 12. 183. 2313^. 184. '1115718. 185. 2i74ft. -,242 times. 186. n}. 187. 3. 188. 75. 189. The former ; customer loses 2*05 oz. in I Ib. 190. 58 miles. 191. 79 wk. I da. 22-83 hr. 192. 263^. 193. 3|Jdays. 194. jio. 196. RSOO. 196.6800:7221. 197. 2oth Oct. 1855. 198. 780 ac., 468 ac., 520 ac. 199. 3 times. 200. 3426 yd. 201. (i) 40 ; (ii) 60 ; (iii) 80. 202. A, R2476A J *> &W& 203. % 99^ ; 176^. 204. lid. 205. '125. 206. 3I7S- 207. C wins by ift% yd. 208. 19 ac. 209. R345. 210. R54. 14*. <#. ; 3^ P c. 211. 14^. l\d. ; gd. 212. '346574. 213. I min, 514 sec. 506 ARITHMETIC 214. 60 days. 215. ,606. 216. After 6 months,. 217. 15400. 218. 2j. 2&/. 219. i^V 220. 5000 sq. ft. 221. 322$ yards. 222. 29040 ft. 223. 76. 224. Gains 825!$}. 225. 550. 13*. 4A 226. -4, i^ of a chest ; -ff, & ; C, jfo. 227. 17 in. 228. 22 yd. 229. 43}^. 230. A, 876 ; B, 876 ; C, 840* 231. 770 ; i. 232. 10. 233. ^860.3^.11^. 234. 6 yd., 6 yd., 3 yd. 235. After 9 min. 236. 10. 237. lib. to 2 Ib. 238. 12 \ 1460. 239. 8411.126. 240. 3*. S^Vfc*/. 241. 7 in each way ; 7776. 242. 2 min. 27^ sec. ; 1080 yd. 243. 10. 244. Better 20 Ib., worse 40 Ib. 245. 500. 246. 1152. 247. 2364. I2J. 4i< 248. 2 ft, 249. B wins by 88 yd*. 250. &i8. 251. 12 bus., 12 bus., 36 bus. 252. RsJIS decrease. 253. 84. 30. i&). 254. lojf 255. 250 Ib, 257. 13^ days. 258. 3 : 2, (by volume). 259. 830780* 26O. 8276. la. 6p. 261. 50. 7J^. ; i>5498. ja. 262. 72 yd. 263. I min. 264. 43$}. 265. 80 Ib. 266. 81726 . 10 . 8. 267. 40. 3A gain. 268. ^1123 . 15 . 2. 269. 59 sq. ft. 21 in. 270. 39 yd. 271. ioj da. ; 4/& en. ft. 272. 65. 273. 895197. 2*. i$fc>. 274. 2s. tf. 275. 60. 276. 12 yd. 277. 3 da. 278. 27 da. 279. 2 st. 7 Ib. 280. 816500, 281. 3^ mi. 282. 64. 283. 9 cu. ft. l; 97 in. 284. ij hr. 285. 27. 286. 40 yr. 287. )2. 288. 60. 289. /i 508. 15*. 7}i&/. 290. 2399 Ib. 7A\ oz. 291. 160 yd* 292. 4 1 $ 7 #. 293. 1000 yd. 394. 17000 I 18067. 295. 3j pice. 296. 1668. 7*. i}ffc/. U97. 82 . 9 . 8. 298. 5$ da. 299. 49. 300. 26$. 301. 89.8.9. 802. 9. 303. 8370. 304. 161 sq. ft. 2 if in. 305. 25 mk 806. 2176. 807. 81500. 308. 1350. 309. 82 . 15 . 7^ 310. I4'5. 311. 2 in. 312. 5 min. ; J mi. 313. 68. 814. io&'jpi P* c - ^crease. 315. 12 p. c. 316. 4 yds 317. 933i Ib. 818. 49i min. 319. 18 da. 320. 33k 821* 844000 decrease. 822. 81705}^ ; i73^h- 323. i. 834% V3, Ja, |. 826, Faster 99 yd. ; slower 77 yd. ANSWERS TO EXAMPLES 326. i . 18 . 4. 827. Just passes. 328. R6 . 8. 329. 4i$. 330. R2. 30. 331. 900. 332- 333. 2,&. 334. 72 gall. 335. 4$ P- c. 336. w. 8< 337. ga. &. 338. 144 ; la. 339. 22 mi. 340. 4$. 341. R923oJ. 342. ,7995, 343. is. 9^^. 344. $ a > 4A 345. ^150.15^.346. 80 min. 347. 2601. 348. Ri925Wil 849. 1073. 4s. 0-6560736^. 350. 30. Problems. 1T5. 1. 942. 2. lod. 3. iij\ in. 4. 1083. 6. 80 guineas, 128 half-crowns. 6. ^. 7. 132. 8. ^275, 9. 6J ; 156^. 10. 223-358... ; 2O'o57...oz. 11. 34!. 12. The latter. 13. 3s. ii^d. 14. 15*. uj</., 155. io</., 15$. gd, 16. 3456, 2304. 16. 126 qt. 18. R5, R3, R2. 19. 2632. 2O. 3. 2L 36. 22. 424. 23. 60. 24. i^% 02. 25. 120000. v 26. 11960 sq. yd. 4 ft. 20*41 in. 27. 10 ft. 28. 100. ty. 29. 1319*472 ft. 30. 33^ Ib. 31. 8s. 32. Rro25... 33. 395. 34. 46^ hr. 35. Rio26. - 36. 6 hr. 59 min. 15 sec. 37. 54 tin<es. 38. 1 1 days. 39. B ; B } ff . 40. 13. 41. 50. 42. A mi. 43. i mile 980 yards ; I3f| miles. 44. 2\\% hr. 45. 20. 46. 36$ mi. per hr. ; 8 h. 37 m. A. M. 47. 29^ mi., 15/5 mu 48. 9r^ mi. per hr. 49. lof mi. 61. 115 min. 52. 167 min. * 53. 25 mi. 64. 11-30 A.M. 65. In 10 min., more. 56. A ^162, B ;n8, C^IO4. 67. A 1296, B 1872, C 1044, 58. 30. 69. 3. 60. R720, Ri28o. 61. f{. 62. n, 22 and 33 days 63. Tea is. 5j^., coffee 5*. lod. 64. 30 and 18. 65. 8 and 12. 66. 2'2o Ib. 67. 10 gall. 68. Man R25o, each woman R62. 8a., each child Ri5. loa. 69. R24, Ri5, Ri. 70. 30 yr. and 25 yr. 71. 10 p. c.. 72. I02i</. 73. RS . 7 . i^. 74. 30 times. 75. 123. 76. ,5000. 77. 4$ mi. per hr. 78, 79. 23 carats fine. 80. 4$ mi. per hr. 81. 82. 9 gall. 83. 2 I i. 84. 12 gall. 86. 5$ gall. 08 ARITHMETIC 88. I I I. 87. 3145 I 6424 I 1431. 88. 2s. 4<f. per stone. 89. Ri6o6o. 90. R2. fa ; la. ty>. 91. 87678. la. ; loa. 2'8#. 1 92. 7. 15*. 7i\%dT. 93. 10, 25, 50, 75. 94. i8j. ' 95. -4 2400, B R9oo, C R240, > R6o. 96. 28800 ft. '97. 15 rich, 85 poor. 98. 27} f $ cu. in. 99. 3923^. 100. R820. 10L 133. 102. 7&54A- 103. 818.8*. 104. 12960, R 1 1220. 105. .48000. 106. 6{p. c. 107. 48 mi. 108. jio. 109. 5j. 110. ^10538 . 12 . 6. 111. Ri45o8, Ri2o9o, R 12896, 9672. 112. 19$. . 113. 4942^. 114. 45 mi. per hr. 115. The steamer ; 16 hr. 116. 25. 117. 76. 118. 35 measures. 119. 30 seers. 120. 690. 121. 52. 122. R9i8o. 123. 1050. 124. I5;*iihjcu. in. 125. ^5.14^ 126. 8400, 127. 144. 128. RSOOO. 129. 25. 130. 3$ md. 131. 2f p. c. 132. 2d, 133. Ri. ga. 134. 450. 135. The second is R2o less. 136. 7. 137. 20 da. 138. R7. 8a., Rio. 139. R7- 8a., R9- 140. 30. 141. R2. 142. 7 and I. 143. R3- I2a. 144. By 3^. 145. 56306* ; 12577 57 IIT&' 148. n66|, 1169, 1000, 1002. 147. 48 centres, 31 outers. 148. 4. 4*- ^3> i. i6j. 149. R8. 150. R45OO. 151. R49. 152. 89. 153. 11. 154. JJ in. 165. Each man, R2 ; woman, R 2 ; boy, I2a. ; girl, 8. 156. 7 I 40. 167. 10, 15, 20. 158. 75 p. c. and 25 p. c. 159. 6| cwt. alloy, 2$ cwt. lead, J cwt. tin. 160. 80., 60., 40. 161, I md. 162. R2. 163. 6a. 164. 15 hr. 166. $*y hr. 166. 4 hr. 20 min., 7 hr. 35 min. 167. R46 . 10 . 8. 168. 3^ mi. 169. 4-25 p. M. 170. 18 mi. per hr. 171. 2$ mi. 172. R46. fa 173. R37350. 174. 120. 175. 7iVir- 176. Rso6 5}} decrease. 177. 140,168, 1 60; 840. 178. Ri5. 179. 20. 180. R4OO. 181. 15$. 182. 412.10*. 183. English navvies ; 4000. 184. 1050. 185. j34 8 . ii^/j. 186. 1199-36 $23437 S sq. yd. 187. 18^. 188. I23f 189. 2s. *d. 190. 33^. 191. 12. 192. 48 of each kind. 193. 90 mi. 194. 60 p. c. 195. 31. 196. 21420. 197. Rioo22. 40. 6#. 198. 1*39. 13^. ANSWERS TO CALCUTTA ENTRANCE PAPERS $09* 353. us. 7/ T </. 200. 3* i\d. 201. ^2000. 202. i.7A<* 203. 78 p. c. 204. ;4654&, ji35li> ;&! 205. 320. 206. 3 . 17 . ioj ; 5;. i$</. 207. noo ft. per sec. 208. i& mi. and f mi. per hr. 209. 2} days after 2nd starts. 210. ,13116.6.8. 211. 250. 212. 8 min. 4 sec. ; 8 min. 15 sec. ; 8 min. 26 sec. 218. 14 min.. 214. R22}. 215. 9)?} min. 216. R2oo. 217. 15 : 9 ' 5- 218. 75 sec. 219. 29 1 f{| j mi. per hr. 220. ,7.11.3.. ANSWERS TO CALCUTTA ENTRANCE PAPERS. 1858. L 33i- 2. AWw 3. 17320508... ; -5477225... 4. i&% oz. 1859, A. L 5 : 22. 2. 407 yd. 8. The former ; '2236. 4 ; ,857} ac. ; J&, 5. 2400. 155. oj}</. 6. 13-427 poles ; 'I735 * 7. ^1350. 1859, B. 1* 8333 hr. 20 min. 2. R6. 8. I j ; 0079. 4. io^|. 5. '00064 ; '009 and 400000. i860. 1. R9963. 2, 7564 ; -7071... 8. 29 ; 2. 4. 6, I86l. L 2243*18. 2. -035 ; ,#*. 8. ^2142. 5* 4 j^. * iio3|gjac. 5. -0316. 1862. L '54. 2. v ^ v 5 is. gd. 8. 4,1,0. 4. In 25}! min. 5. '03162.. 1863. 1. nj$ ; 11*2388... 2. 3&%%. 3. 143- 7'- 4. Mlldays. 6. 31*052. 6. ^529. 4*. 510 ARITHMETIC 1864. ! 54. 2. i ; 2. 3. /77. I4J. 4. 4j</. 5 '5885416*. 6. '014 ; -oooi, 6. 6800 I 7221. 1865 1. 79$!! ; 79'448 ; '3415- 2. -001764 ; xo. & 329^ yd. ; 1023. go. J\p. 4. 45 men. 5. RS4, ia. ic^. ; Ri6. 8a. 1866, A. 1. 2-183125 ; 120? ; 13316-875. 2. 96. i6s. 3. 39*05 ; 12-348... ; $d. 4. 12 days. 5. 2. i6s. 0*478447265625^, 1866, B. 1. '10444637 5 x. 2. 21. 31. 6|!</. 3. '00041616 ; 9*042 ; 21*7272... 4. 256*256 ; '0256256. 5. R2io. ** e. 1867. L 19 mi. 836 yd. 2 ft. 2. 102960 ; 320*87. 3. Loses ji, 3^. 4</, 4. 2;9|?|<* 6. 4T /bW '001275 U B 4 AW '001699.... 6. 9-f- 1868. L lu. 3^ ; 5. 2. 12-375 ; 1*816... 3. 440 miles. 4. 401 I 544* 6. 12. i&r. iof|4 6. 58! yd. 1869. L 4 i '02392609126984 2. 10. ioj. ; if. 8. '02 ; -0000002 ; -1414.^ ; '0004... 4. ^14. 7^. n</. 5. 16 years. 1870. L Ri5. n}Ja. ; 8091 cu. ft. 2. 9939992 JJ J ; (i) '001353 ; (a) 290 ; 2*52*7. 3. 140* ; 2*0025... 4. xof days. 5. Second. 6. 2070^. ANSWERS TO CALCUTTA ENTRANCE PAPERS $11 1871. L R2732. 13*. 2. A greatest, j% least ; 7. <w. 3J3^ ; 3. -001875 ; 67952-25 ; R68. 3*. ifc. ; 'i54 4. 55 miles. 5. 83250. IS/2. 1. Ri597- 100. 3^. 2. f ; 3. 5050 ; (i) '075758 ; (ii) '677166 ; 30*84. 4. 8197. na. 7HJA " 6 - R262. 1873. 1. (i) | ; (ii) R2569. 7*. 7A J R48. 2. '6033 ; u\j J -5048.- 3. R2o. i ia. 2|$/*. 4. 19 yr. 6. 5*1. yfe*. ; R5498. 7*. 1874. <! I? ; i I 161 ; 3328 226l28. M ; ^30769. 2. 63 days. 3. 3 T *& cu. ft. ; I5$J{ cu. ft. 4. 120000. ^5. R66666. 10^7. 80. ; Rio8. 1875. 1. 2 ; RSO ; '2213... 2. ^f. 3. R35. la. 4^. 4. 816540. 5. 858. 2*. ; 3|. 1876. ! ijftft 5 Rl 3- 13^- 6i%#. ; '4441... 2. 9 ; 23-04484... 3. 12^ yd. ; Ri. I2a. ;^2is. 161. Sjrt'. 4. 2co da. 6. 4$, 1877. 1. t 3. 9* 2. 89105. iff. 6fi. 3. 78. 15^. 4. 125. 5. 39 days. 6. 3312 ; 1878. i, 2062*649.^ 2. 1*00015... 3 f '375. 4. 824. 140. 6^V^. 5. -0099454 $65079. * 45 n. $12 ARITHMETIC 1879. 1. 400, So, 6, A, jfo, 11 j\ ni . 2. 104. 3. (c) '02704$ ; (d) *ooi. 4. 18 times. 5. 68 men. 6. Decrease 11. 4*. $d> 7. 18. 1880. 1. loo, 20, 3, A, jfo, 1T fo, ; J. 2. (a) /& ; () 40 ; (<:) 2-65. 8. Each boy, 4. 1 1 j.; each woman, 13. 13^.5 each man, 27. 61. 4. 65 gallons ; 13 hr. 5. C wins by ffi& yd. 6. I88l. 2- 4i% 5 3- 3. -78125 ; R38o. 6a. 4. -YiW ; 1*8549. 6. 15400. 1882. L 4321. 2. 5- 18^. 9^. ; *57. 8. 30030 sec. ; 15016, 10011, 6007, 2003, 1431, 463, 391 times respectively. 4. (i) 1600, (ii) 27*96424... 5. 18 da. 6. ^35000^ 1883. 1. i. 2. 30 ; 75- 3. 00694 ; R6 ; 10 . los. iod 4. 21. n *. 2jJ</. 6. 96f 6. 28$ years ; 8562. 8a. ,- 75 p.c. 1885. * ! 5 AV a - * 12 5 ' 2 5 "30472... 3, 3'46i 53 ; 1. ioj. 4. 513. 61. 6i</, ; 3*1224..., '2828... 6. 18 ; 8^ per cent 6. The first investment is belter ; 1342. ioj. ; 3^ per cent. 1886. L IH- 2. Viftfc- 3. HfjfU- 4. 5;-oon#. 5. 36. 171. 6^ 6. 28659. 60. 7. Ri2. 12*. 9^. ; gain Ri33f 1887. t (a) i ; (^) 350. ' 2. '0203125. 3. (a) 17. 12* 2^ 5 (^) R20oo. 4. xo. 6. RSIO. 6. 13*31 ; '471... ANSWERS TO CALCUTTA ENTRANCE PAPERS $13 1888. 1. A. 2. 1 1200 ; 37-96. 3. 1 38'4497 i ; 20. i6s. ?&d. 4* ^1034. 14*. 4j#< 6. isf days. 6. 6J ; lQQ. 1889. ! 5 I *59 I 394i2. 2. 8*62126... 8. ,5247. 2J. 4. 1*000127... 5. ,6705. 14^. 7*/. 1890. 1. 3 ; R2393I. 7*. 7^. 2. 73O5*4*o$ ; 8. 1771. 4. 60 days. 5. 104. 4*. 1891. 1- (*) if J (<5) I 2. 2-202642. 3. 408, 3 4. 9 hr. 41/5 min. 5. 820800. 6. 8J yd. 1892. 1. jgg. 2. 26219. 3. '312 ; -098 ; -998. 4. RI232. 140. ofj^. 6, ^2500. 1893. 1. (0 5i% 5 (2) 3- 2. -0789 ; ffi ; || ; ri. 3. ^ 3 45- 7^. fid 4. R238. 3*. 2Hf#- 0. R9o,ooo in the 4 per cent, stock and 73,000 in the 5 per cent Municipal debenture stock. 1894. 1- 37- ox. 8J 2. 491- &f. 3. i6s. o'tf&ijd 4* "9998. 5. 6 Rupees per head. 1895. 1. 1*00001. 2. 12345. 3. 3 francs 84 centimes. 4. r ft. Increase of 47 ; '6832876712. 1896. L Greatest number 23704 543, and least number 8 1 43. 2. (i) A ; (2) -075088. 3. 2-2677... *, &53I- 3. iQif A 5. if per cent. loss. 6. 21735. C. A. 33 $14 ARITHMETIC 1897. 1. -072 5. (a) A- 2 - Yes, 320th part ; 832. 9*. xfc. 8. 20. 4. 33iyr. 6. R6 per share. 6. 17724... 1898. 1. 20150. 2. ft ; '083. 3. 234 ; 8-0600. 4. 250. 5. i per cent, profit. 6. 23400. 1899. L 25. 2. ittJfNNf- 3. R6o6.iia.9i0. 4. 226 ; -226. 5. 3jJ per cent. 6. RiS. 1900. 1. 2520 sees. 2. 6 ; 5 \. 3. 8. 4. ,$$ ios. 3}< 5. 125. 6. Gains Rioo. 1901. 1. (a) 1-416 ; (J) -565- 2. (a) Yes ; () 68. 15*. 9^. 3. 4 Ib. S oz. 4. 4 per cent. 5. 86*42. 6. Ri ; 22 ; 169. 1902. 1. (a) Terminating ; (b) '848... 2. Rl5326. loa. %p. ; R734O. 8. 35 boys. 4. 2$ ; 8729. 5. ,100. 6, 4 per cent, R6a 1903. 1. (a) i ; (b) -0005681. 2. Yes ; (fi) 170. igs. 4^4 8. i|mm. 4. (6) 3^ ; 1-5118... 6, if gallon. 6. (t) 50. 1904. 1. 4. (a) 9979^0 39*6 poles, 2. 6. w*. 700. 3. 6. 22$ days. Ri5i7ia 1905. L 4. 165. 2. Rno i. 7 a. 2j#. ; 3 p. c. 3. 6. 4*467 ; 79*. ^3xx 1906. L 4. (2) 99679. 1667 ; 7746. 2. 5. (2)(a)l;(*)s. 17,?. 6<f. 8. Ri7. **. 6. 11} p. o ANSWERS TO CALCUTTA MATRICULATION PAPERS 51$ 1907. L 37128. 2. (i) $;(2)&. 3- 4. 110400. 5. R;6o. 8a. 6. 1908. 1. (i) Non-terminating ; (2) Ri. 80. 2. 934 . 18 . 2, 3. (a) Loses H^f min. ; () D wins by ;}f J yd. 4. (a) 2^3 ; () -5345. 5. Riooo, 6. Incr. R42$ Alter : 2. 2038^^ gall. 4, 5746. 8*. ; 6 p. c. 1909. 2. (1)75(2)65. 3. ^939. 13. 6; 371*173. 4. Ri228. 2#. ; ^588 . 10 . 10. 6. 3 ii 59> 33) I77 649, 1947 ; 20 p. c. ANSWERS TO CALCUTTA MATRICULATION PAPERS. 1910. Compulsory Paper. 1. (d) 504 ; 17280. 2. (I) 167^ 5 (2) '009: (3) 40. 3. (i) 83816. ioa. 8A ; (2) i6fyr. (3) 2'ii5...cu. in. Additional Paper. L 2501317. (<$) 15 ft. 2. (i) See p. 141 ; (2) 248ss'296...miles. 1911. Compulsory Paper. 1. () B6io. 2. (i) i ; (2) -0052083, () tx27. 8, (i) RSI. 150. &. ; (2) 227.12*. (6) iSomen. Additional Paper. 1. 469246 ; 54*0321 579 men. 2. (i) 3'I4I59 ; (2) *8. Compulsory Paper. 1. W 1%. 2, (i) A; (2) 12.13^.2^. W(i)3ip.c.; (2) R442. 7 a. 7fc*. 3. 25 men. () 2624 sq. ft. ; R95. lew, Additional Paper. L 37ri73. (W "7 ft. 2. 24855 miles, (3) -54931. $16 ARITHMETIC 1913. Compulsory Paper. 1. (2) 504 ; (*) 1890. 2. (i) J 5 (2) 0*2907. 0*) (0 0-0015625 ; (2) 82362. 8*. 3. (i) 3 P- c. ; (2) 28 yd. Additional Paper. 1* 5*403 ; () &36. loa. %. 2. 239*197... ; () See p. 79, 1914. Compulsory Paper. L 278523^. 2 - 247 ; (<*) 2160. 3. i*Htt- 0) 14650. 30. 6&. 4. 45*408^. W R564- 5- 4#. Additional Paper. L 2*646. 2. 20-8... I9IS- Compulsory Paper. 1- (0 75154060188. () 7908. (2) 504. () 28000, 2. (i) |. (2) 70-2702 ; 85-8. (3) (I) 11-938461$. (2) 85615. go. &. 3. (i) 10 p. c. (2) 83072. Additional Paper. L 13*057- 2. 3937X3937- W '41937. 1916. Compulsory Paper. L (2) 119 J W 2520. 2. (i) i ; (2) '0041(5. (b) (i) '1035546875 ; (2) i73- & 3. (i) I2j p. c. ; (2^ 60 men. Additional Paper. 1. -06435, 2. -097... ; (*) 1732. 1917. Compulsory Paper. 1. () 272428968896 ; 101793. () 756 ; 89 ft. 3 in. 2. (a) i ; 5* <*<* (*) 30 5 -583. 3. (a) Rio6. ; ; &37- 2a. 8/>. (^) 12^ p. c. ; 23 p. c. Additional Paper. 1. 7589 ; I4U mm, 2. 1-6487 ; 1-2501. ANSWERS TO MADRAS ENTRANCE PAPERS 517 Compulsory Paper. ! d) (&) 157. (2) 5910 cisterns, 33 gallons rem, 2. (I) 2|. (2) 2- 4 . 3. (i) 2. 2s. ; (J) 12. i5.nj. (2) 1350 ; W 20 days. Additional Paper. 1. (0 ^31623. (2) Length 1-875 ., breadth 0-625 m. 2. o'575i ; W 0-3770. 1919. Compulsory Paper. 1. (2) 391 5 (*) (2) 104329- 2. (i) R2. I5a. 4J?. (2) 0-00027. 3. (i) 43. 71. loj^/. (2) 4 years. Additional Paper. 1. S54'2coi ; (*) 1-224. 2. 0*2836 ; () 873. IQ2O. Compulsory Paper. a. (0 i ; (2) &--04I*. 3. (i) ^84.59^. 4K- ; (2) Additional Paper. -2, (a) 9 hours ; (V) R8oooo. - . .**.' ANSWERS TO MADRAS ENTRANCE PAPERS. ' 1880. 1 6$ ft. 2. 3 miles. 3. 'fci. 4. 3 annas, 6. Width, i8i ft. ; w height 14$ ft 6. 2469 ; '0788... 7. 7$ hours. 8. 3} per cent 9. Each childf 8960 ; each brother, 495. 518 ARITHMETIC 1881. 1. R24Z. 8a. 8/. ; 85267. iza. 7&. 2. R666. Z2a. 8. R37350. 4. 48* ; -2070... 6. & ; ^fo ; H 5 6. Z4 years. 7. Rzoo22. 40. 6f^. 8. 28500. &. Slower z 5/g miles ; faster 29}! miles. 1882. 1. '387. 2. 9. 8. Z3 ft. 4 in. 4. 294-151. 6. (*) 210 ; (b) Z79. 6. (a) Percentage obtained by A is 52, B 69, C* 64-2, D 687 and 7^49-2 ; (b) 64*3 in Arithmetic, 55*4 in Algebra, 46*2 in Euclid, 67*z in English, 65*2 in History, 62 i Geography, 41-9 in Handwriting ; (c) 60 per cent. 7. '428571. 8. 1711.12*. 9. 675 Ib. 1883. 1. 27 gallons. 2. 4^. 3. 27 days. 4. 4 hr. ;i345* i&f. 8^1 ; ^107. Z3^. 4</. ; R8 per cent. 6. Ay 4 hr. 20 min. ; B, 7 hr. 35 min. 7. 8. o. 9. zooo yd. 1884. 1. i. 2. RZ455, 4<*. tf. 8. 1*00904...; 4, 962* 3*. 5#. 6. The side of the cube is 7 in. ; 7776, 6. A 480, B 533. 5*. #., C 466. zoa. ty. ; z J per cent. 7. 3Z25. 8. 8. 9. 4j per cent. 10. 236. 90. 1885. 1. J, 2. !%, 64 Rupees. 3, z. 6f. o^. 4. z9. 3^, zoJJ 6. zz^. zoj^ 6. z92o. 7. 3Jp. 9^ 8. 4 years. 9. 5000, 10. -9196 ; 16. zoj, 11. 3500000 men. 1886. 1. z, 2. -9705. 3. 3. us. sld. .4, R39S5-3* "A 6. z-Z3 p. m. 2nd July. 0. 80 men. 7. 1000. 8. ;z8a 9. Ri7. 80. 10. 590344000 cub. ft. ; iff in. ANSWERS TO MADRAS ENTRANCE PAPERS 1888. 2. }. 3. 10. 4^. ; *i 14583. 4. 721. 155. 6^ 5, 8335000. 6. ,416. 13*. 4^. 7. 87. 20. 8. Increase of 8502. 80. 9. 8500000. 10. 500400. 1889. 2. }. 3. 08273029 ; 6s. 9j0\ 4. 81730. 130. 6# 5. 848. 20. 6. 1694. 13^. 90*. 7. 8380000. 8. 10 days. 9. 7500274. 10. 39. 3*-9^ 11. 2-08008. 1890. L 342 ac. 2 ro. 39 po. 2 sq. ft. 36 sq. in . ; 160 yd. 2. 1-5. 8. 8975358. 90. 2^0. 4. 30 weeks. 6. 86744273. 6. 4 months. 7. Increase of ,397. 8. 12 cwt. I qr. 19 Ib. 4 oz. ; 33. 2s. 6d. 9. 343 : 169. 10. 19487-171. 1891. 2. Ri, 110. 8^. 3. Ri. 100. 2<|^ \fi. 4. 9; 46-94718, 5. 12'. 6. 86. 60.40. ; 8158. 7. 291. 9,f. 5}^. nearly. 8. 20' after noon. 9. 100*. 10. 89180. 1892. 2. f. 8. 55. 30*. ; '0037 1 1 562 5. 4 867567. 90. 7$0. 6. j4i6. I3J. 40*. e. 3700965. 7. 3221625 tons. 8. 8355. 13*. 40- 9. 855. 80. 4$. 10. 3& per cent. 11. 25640000. 1894. 2. * 3. 5-r. 2\%%$d. 4. 81593.40.51^. 6. 893333. 5* 40- 6. ,976. us. & I 7. 823. 120. 40. 8. 4$ miles. 9. 81062. 10. 14*1625 per cent 1L 9-0073210. 1895. 2. IPO Q ^Bl * ** 044481. 4. 83359. 20. ioJ0. & 887885. 6. Between 7 ft. 6 in. and 7 ft. 7 in. 7. #25. 9. 827348. 120. 9. 878645. 130. 40. 10. 5 A per cent. 11. 10 ft 6 in. ARITHMETIC 1896. 2. fj. 3. 8123. 110. 3J^, 4. 825516373.90, 1 1-2] 6. 114 more men. 6. 81041. 100. 8/. 7. 8220000. 8. 83. 130. 6p. 9. 12*986. 10. 82,80. 11. 70*605009. 1897. 2. . 3. 83. 20. ip. 4. 833862. 140.3!^ 6. 8430. 130. 4^5. e. 84556. 40. 7. Increase 8535. 8. 87 ; 84. 60. ; 82 . 100. 9. 11*072. 10. 8152.90.6?*. 11. 2 fur. 3 po. 4 yd. 1898. 2. I A. 3. Ri. 20. ip. ; '0021875. 4. 8545. 70. n T V 6. Real time, midr light July 6 and 7 ; time indicated, 5 h 58 min. 7$ sec. 6. 86875. 7. 81200. 8. Br. 20. 9. 875306000. 10. 1st year 70848000 ; 2nd year 76161600, 1L 12*06 ; rem. "0501, 1899. 2. i^V 3. '0082175 ; 815. 150 6. 81940. 40. i/. 6. 868437. 80. 8, 37 m. nearly, 9. 1 1 per cent. 11. 8 ft. 9 in. , 6/. 4. 81159. IS** SiA 7. 7 miles. 10. 35640000. 1900. 2. }gj. 3. Ri. 4*- 3A 5 '00021859375. 4. 849770. 40. 6. 8773. 80. 6. 8504. 80. ii/. 7. 8531.80. 8. 11} per cent. 9. 4 per cent. 10. 1*25 per cent. 11. 68 yd. 2 ft. 10 in. 1901. 1. 86711070. 130. icp. ; 83777. 110, iqf. 2. $?. 8. Bi. 40. #. ; *o43$i&. 4. 84192. 150. 7^. 5. 837750- 6. 3i per cent. 7. 8171. 50. $0. loss. 8. * io T 2 T per cent gam* 9. 427 50000 or 427$ lakhs. 10. Ri. - is. 11. 907*00381 ; rtm. 008070038. ANSWERS TO BOMBAY ENTRANCE PAPERS 53 1 ANSWERS TO BOMBAY ENTRANCE PAPERS* >* 1865. 6. 40457.114.$*. 3. 52. 4. 405756 ; 12065. 120. S. 5i%. 6. -oj ; '632258064516129 ; -03115. 7. Increase by 3. 7^. 4JS4A 8. ^ R22222$ ; B R33333J ; C R44444* J ^ R46i53tt ; 9. 12 shares ; ^1460. 10. 53 hr. 11. 416*4 ; 12*3. 1866. 2. if ; (a) 1J. 3. R2000 ; 5000. 6. I49553$7Z4*& 6. (i) 44*1 531 57 J (ii) 1 1*569328 ; (iii) '4995O6 ; (iv) 50000 ; (v) 46-04. 7. R 1636363-^ ; increase of income Rioooo. 8. 515. i6j. 7i</. 9. 140. 3^. 10. 6^7 months. 1867. 2. 2, 2, 5, 3, 3, 7, 43- 3. 14!}. 4. 'S57H2 ; ifi|& 6. 788-423. 6. 4ja. ; R&. 8. 90 men. 9. 3 yr. 8 mo. 24 da. tO. no ; 90 ; 30 ; 10. 11. 78-0064... ; '0158.. . ; '3902... 12. R4i. io#. 8^. ; 8 cwt. I qr. 20^ Ib. 1868. 1. 52^5 yd. 2. RI4557- i- o^. 3. *&,> 4. 7^. 5. ^2 5 A%WA- 7. R76363'(53. 8. if J per cent. 9. 466. ; 1O. 9*. 4i 8 5 %</. 11. 23 men. 12. 30. = 13. 763... 1869. 1. 97 ; 1008. 2. I. 3. 2. 4. fo g^. 6. '6489583. 6. R393. 13^. ; R656. 5*. 8/. ; Rioso. 20. 8^. ; Ru8i. 7^. 7. 4 per cent. 8. ;i34^ w- io*95375^ . 9 days. (10. Increase R428. 1L 1769 ; 2O-8J. 1870. 2. iifll ; 11-8208. 3. tfe ; {ft 4. 81. 6. R256oo, 6. 401 : 544- 7. 5 dwt. sHJ ^- 5 3 <*wt* ISA ? r - 8. 35. 16*. io}l^ 9. 8900 ; R6oo ; 1L 533 ARITHMETIC 1871. L 192000 mi. 2. 1287 ; 9009. 8. Tjr Vo- 4. 6, 606. 6. 4jf 8. 83149. 9. R53j$f decrease. 1873- 5 I? 5 R$. 7. 915 ; 8*2956. 4*. 10. 1055 subscribers* L v 7 . 2. 13*. io* ; f . 3. ^ . 4. 1500. 5. 7^ hours* 6 2376. 5J. 7. 30780. 8. 76. 9. 55^. 10. 1234. 1874. 1. I. 2. 4-8. 3. j? 7 . 4. 213. 12J, 6. 814586. 6. 13*. 4<f. 7. R26. 8. 7 A P er cen t ; 1840 interest yearly ; 4^ L ^ per cent. 9. 36 miles and 24 miles per hour. 10. They are in order of magnitude. I87S. 2. 21 mi. 6 fur. 33 po. 3 yd. 2 ft. 7 in. 4. 528093440*, 6. II T &- 6 - J J A 5 vi- 7. 10*017, 8. . 8 s . -10j . nlTj 11. Ri875o ; 56. 4a. increase. 12. 3}f| s q, ft. 1877. 2. B is ^ of a mile in advance of A, 3. A- 4. Tea R2. 80,, sugar 20. %$. 6. 2174 ft ; 242 times. 8. ^4 8850, ^ 846 and C 1182. 7. 650. 8. 818. 8*. 9. -0061. 10. 9^0 miles. 1878. 1. 90, i8j. nfdl 2. i, 3. ^4. c i*. <)d. 4. 83 ft. 5 in. 6. 25 per cent 6. 7678. 20. ; 100. 2'8 #. 7. 1500. 8. ^215, 8j. VJfrd. 9. ^20. 10. ^2890, lo/*. 1879-80. L 48023601545943521. 3. ^5-oi.oJ d. 4. i;H. 6. Rxo. 6. 3i*M& pc ir cent ANSWERS TO BOMBAY ENTRANCE PAPERS I8808I. 1. 2>i 8 7 ; 4$ isJjf . 2. 3oJ| sec. 3, 26 coolies.. 4. 2 years, 6. 273. 8*. 9** 1881-82. 1. 1508. 15,7. 7jJi><* 2. I minute. 3. loj.iof**, 4. R2646, 6. 4 per cent. 6. '83 ; 75*h 1882-83. 1. us. u"2$d. ; -03671875. 2. 24 posts, 8. 9| weeks ; 341. 5-f. 4. 4328. is. 6d. 6. 77 yd. 2 ft. 1 1 in. 1883-84. 1. (a) 16,000,075,040,002 ; () I. ; (c) 24. igs. $\d. 2. 360 ; 2nd, 72 ; 3rd, 60 ; 4th, 45 ; 5th, 40 ; 6th, 36. 8. 22yd. 4. 86. us. 6. 13312 ; 93-05 ; 9*1*. 1884-85. l. ? ; 725 ; i. 13* 9/o^. ^ 2. 2. gs. oft*. 3. 15. 1 6s. zJ. 4. 7 increase. 5. 4 per cent, 1885-86. 1. "4857142. 2. 113 boys. 3. Si <F* *& 4 7**6** &<* 6. The latter investment more profitable ; ;457i2- 1886-87, 1. 5x7x11x13,22*9999891208453... 2. I92ift. 3. -k- 4. (i) 7 ft. 2 inches, (ii) 3*5752. 5. $io$. 6. 3$}? per cent*. 1887-88. 1. 6. 2. igs* $<L 3. 20 months*. 4. 20 ; 7 ; 55. i|<* 6. 5*. 4^. 1889^0. L 2. 2. 5{ ft long by $f ft. broad by 5f ft. deep. 3. 5-15 o'clock. 4, R32000. 5. 3 parts of the one to 13 parts* of the other. 5^4 ARITHMETIC 1891-92. * (0 J () '83- 2. Weight allowed is 100 Ib. ; they had 2 cwt. and 3 cwt. 3. They last agreed at 10 hr. 30 min. P. M. when they both indicated 10 hr. 30'. 50". 4. 8640, 1892-93- L '00502083 ; 15 annas ;jj pies ; J. 2. 10 days. 3. 3 hr. 30 min. 4. ^259. 3*. si^. 6. 401 : 544. 1893-94. in the Mofussil.) L 20577 ; 32490 ; *6j. 2. jn. us. t>\d. 3. 2 cwt. 2 qr, 20 Ib. 4. 83. 6^. $ 5. 12$ hr. 6. 4 Ib. of the inferior to 5 Ib. of the superior quality, 1893-94. (Set at Bombay.) i. (i) 24 ; (ii) AV 2. 32. 14*. 3^ 3, 3& months. 4. ^3 2j|j. 5. JJths. 6. At the same time on the after- noon of the 23rd August when the first clock will show 1-46* and the second 2-16'. 1894-95. L 146097 days. 2. 156. 3. 30. 4. \\\ days. 6. &2i6o. 6. 772 ; 15} annas. 1895-96. L '612345679. 2. Ri2. 3. 31 5 miles main line ; 1 89 branch line. 4. Ri25 ; I4| per cent. 6. ^looo stock. 1896-97- L 2757029. 2. 3762 ; 2280 ; ^6498. 3. XI J sec. 4. 4 per cent. 5. R6ooo. 1897-98. L |. 2. 55 men. 8. 9}? sec. 4. 3$ per cent 8. ^9880; ANSWERS TO PUNJAB ENTRANCE PAPERS $2 1899-1900. L 660539 5.854920, 2. 176 : 175, 8. s| miles ; 6 miles, 4- 1552.81.6^ 5. 124$. I9000I. 1. 1008. 2. -15795' 3. 27 days, 4. fti466 ; Ri2i6o, 6. '01041 ; 14*34 sec, ANSWERS TO PUNJAB ENTRANCE PAPERS. 1875. 1. 1,010,001 ; 766. 3. (i) 6J ; (2) |J$. 4. I ro. 14 sq. po. ; 72. 5. 999 ; 1772... 6. 1$, 1876. 1. 2881-161. ..revolutions. 2. 3281. la. 6p. 3. R8oo. i a. o/. 4. 22 f seers for a rupee. 6. 3992. i la, 81gjf#, 6. 220 days. 1877. L u. 1076^. 2. (a) 18 min, ; (b) n|Jf ; 11*8208, 3. 34-3I68 ; 5-858... 4. (a) 4838$! ; 0) S3i 5 R3il?- 1878. 1. 5. 100. 2/. 2. i& ; -197802. 3. '0063 ; '00296... 4. 338 sq. ft. 6. 5 per cent, per annum. 6. '316... ; 'oooi. 1879. 1. (b) and J| ; -28472. 2. 1). 3. 2'U5...cu. in. 4. () if* J W -0316. 5. 3i6'227...yd. I88l. 2. 9-45 ; 2*23?i" 3. *o3i68... 4. ^14, 13*. 8,V- I88 3 . 1. '6848. 2. 12. 3. ()}J}. 4. 3j hours, 5. 140 ; 170 ; 190, 6, 8507. 8a. decrease* $36 ARITHMETIC 1884. 2, '02688^ '002688 ; 25*6, 2*56. 8. I. 4. 1*0001. 6, R62. loo. 5ffe>. 6. 80. iiJifJA 7. The former greater. 1885. L I ; 123 times. 2. '08125 ; -0063 ; '038961. 3. R884. i$a. #. 4. ^9. 15*. 6. 2963520 ; 2420 sq yd. 6. The latter ; 40000, 1886. L '375 5 '612... 2. 7895- 3. J& 4. She loses. 6. RiQ5. 130. 0*96)*. ; Ri;2. 8a. 5AV 1887. L (c) 3-025. 2. 133. 6j. 8^ 3. 89. 60. 8J&*. per maund. 6. Rno29. 6<z. 7tS#- 6. 1*321... 1888. L iimi 2. *626893. 3. 16 days. 4. 17 per cent. 5. 32-907. 1889. 1, '322083 ; 799. 2. 2J miles. 3. 37651 ; 2 min. 6 sec. 4. Ri4.8a. 6fe*. 6. V3 greater ; the latter. 1890. 1. (a) I ;W 'oi 5789.^ 2. A; '39062 5. 3. 1324. 13* 9$$* 4. 8320 men, 6. R86& 6. q. 1891. 1. (i) i& ' W 2- '0064453125. 8. 10 years, * ^200 ; 5 years. 6. 45 gallons. 1892. 1* T)I > the ^^r comes nearest. 2. 5*90635. 3. 218972-16 gallons. 4. jioi66f ; 6ooa 6, 171. 3^ per g*L ANSWERS TO PUNJAB ENTRANCE PAPERS $37 1893. .1. Ricoo6oo. 2. 13713729903* 8, -41421. 4. Ri7. 5a. 9}^. 6. 32 miles. 1894. ' L '$71428 ; '428571. 2. Length 44 ft. f breadth 33 ft. 8. ^511. 4. Present value by common calculation! *.*., by deducting interest is 987. 80. ; present value by deducting discount is 8987. ioa. 5jJ#, 1895. 1. rcj(428$. 2. r25Mb. 3. 391* $29> *$** 4. 2*2360679. 6, Length 76*2 yd., breadth 38*1 yd, 1896. 1* /A 2. 2*21359. 3. 11400000 ; 17100000 ; 3800000. 4. 3930* 1897. 1. m. ^4. 4*. ; w. 3 ; c. ji, 161. 2. 2*4142136. 3. 12^ weeks. 4. 540. 1898. L R66. 100. $*. 2. I. 3. /66. 13^. 4^ 1899. 1. 18 ft 2. 6 yd. 3. 4 ; 6 ; 22. 4. si per cent. 1900. L 2000*301. 2. R7352. 150. neard-f. 8. loyd. ;?2 yd. 1901. L 5. or. 4< 2, 1*5789 I I orH I *6j. 3. 12 per cent 4. 79 I 49. 1902. L 3- 5-7- 1 1- 1 3-37- 2. 576a , ^4800. 6. 15 per cent, 5*8 ARITHMETIC 190J. L Prime ; 28721 x 373. H % a| hr, 3. Rass- 12*, 4. ^, RioS^A ; ^ Ri 3 i^ ; (7, Rio9fiJJ. { 5. * 190* L 2 10 .3.7* ; 2*.3.7 J 5.7.9. 2. -044; a R22 14. 3. 4* ft 2 56. 3. 2|. 6, gj,. 1905. L (l) 2*43*43 ; (2) 90017. 2. Man, Ri&7. la. ; woman, 807, 70. y. ; boy, B6^. ga. o/. ; girl, 841. 120. 3^. 3. RH4- 130. 6/. 4. R8i2. 8a. 6. (i) Sunday ; (ii)) 24 times. 1907. 1. 53- 2. ^S"5-2.4j. 3. Bi428sftf 4. 1^126720. 6. -4190242. 1908. L 1 92$ ft. 2. 00000292. . 3. 9iVPc. 4. 6*125 p. c.. 1909. 1- Ah 5 '04375 2. Ri. 80. x T V^. 3. i6 T W days.. 4. Rn6i. 60. ; R4Q2. loo. 1910. L 2525. 2. ?*& ; '0003125. 3. 1689. n*. 4. P. W.-^xooa. 15* 5|f^ 5 D^33. *s. 6&<t. 5. I9II. . 32'907- ; AV- 3. ^11870.3^.4-4925^ 4. 2 months hence. . 6 ' 5f days. 1912. L (*) 89. 2. i ; ^. 3. 8 days. 4. 2& p. c. ; 6* iStS: min. past 2. I9I3- L 109, 113 ; 544. 2. (ff) 1*4102* 8. R3 ; R.3. 100, ; Ri. 60, 4. 2269, ISA l#. 5 150-8, ANSWERS TO ALLAHABAD ENTRANCE PAPERS $29 ANSWERS TO ALLAHABAD ENTRANCE PAPERS. 1889. L i A ; 1-38461 . 2. (a) 3ft ; (*) -0063 ; -00196,.. 3. '0316... ; -ox, 4. 81300, 5. 12 min. 40}? sec. 6. 8 days. 7. 39! miles from their starting place, 1890, L '696294007 5 36*55086... 2. & 3. R86oo. 13*. lofeJ. 4. '99999... 5. 2'ii5,..cu. in. 6. xx& in, 1891. 2. izsUfttffU. 3. 351- * 5 per cent. 5. 9999 5 7- 1892, 2. () 12 ; (*) Wi & '84375- 8. $6| day. 4. ^3- ' 5. I'oooi... 1893. 2. 2 fur. 12 A po. 8. 3. 7}a* 4. 14* U* 3i< 6. ^350. iu. 8</. 6. 1869, 7, 79*032 ; 8 fg. 1894. 1. (a) 999 x 8o7-8o6i93- W - 1- 2. () 'oo6 j (*) S'OSQ... 8. 444 miles. 4. R$55. 1895. 1, (3) 4 feet square. 2. (a) ^\, ; (3) 17724... 8. R57oa 4. R7 .6. nj and 84. 15 . 3f. 5. 150 yards. 1896. 1. (a) I A $ W 25. 2. (a) '0203125 ; (^) aooioox. 8. 178 hr. 52 min, 30 sec. 4. 7} miles. 5. 82800, 1897. L 47 ; 127041- 2. '285. 8. "291, 17* Sftttf* 4. 315. 5, 2203*9062$ francs. 530 ARITHMETIC 1898. L 2* ; A 5 29- 2- () 3 J (*) -4t- 3. 3} ; 'ooi. 4. i. 9*. 4}< nearly. 6. 125. 1899. 1: 4} ; -63. 2. 42*68. 3. 435. is. 5< nearly. 4. 51 more days, 6. 2*2360(6) ; 7071. 6. 3ii$*/. 7. 25 per tent, loss ; 80. I90O. 1* ilHli 5 '067. 2. '29S67o ; A- 3- &*97. *. of. ; 3*165. 4. 3200 ; 83889, ga. 11*04^, 5. 289. 5^ nearly, 1901. L }}8 ft. 8. 3. 2. A ; '03391. 8, 16 yd. 4. R6- * 5. Ri68oo. 1902. 1. ^000279 ; 25234713* 2. 44- 8. 3285. 4. A, 48 ; B) 84. 6. 7s T u days. 6. 8J years. 1903. 1. ()34Jito. (*)'S. 2. (a) -0051472 5 QSTS- W'S- 8. 30 days. 4. 10-15 P.M. 5. 4} p. c. 1904. L (a) lljf 5 (J) -000125. 2. (a) 1714 15 3 5 W 3*1624. 3. /J5. 4. Ri 1 57-ioa, 5. ^19425. 1905. 1, (a) 8AV ; W '353^. 2. (<i) 4 cwt. I qr. 9*89 Ib. ; 10216... .5.4i. 8. I5lp-c. 4. 9iV*. 6. /2j loss, I, 42, 2. i6A* 8. 25 yr, 4. 3981, ANSWERS TO ALLAHABAD ENTRANCE PAPERS 531 1907. Yes. & 94*91. 4. 2*236... 1908. 1. 324513* 2, $. 8. See Art. 93*. 1909. ! 35543I* 2, 44p*c, 3, If, 4** 4. RS. 80. increase. 1910. 99iOoe>p99i099>o99 ; 845*9. * 3*5 J 873$. 3. 700, * I9II. 9f000jo89f009)0io ; I. 2. 1*609344 Km. 8. 54021, 1912. L Three billion) two hundred and three thousand and -Jx hundred million) and four hundred thousand. 2. d)i^ftr;(2) i.os.&i* 3. 1913. 2. II, , ...^'*- 8*880, 2. 6*01 c 532 ARITHMETIC 1917. 1- (0 if J ( 2 ) 0*027. 2. (i) 0-25 ; (2) 1-4142... 8. $ quart. 4. 781. 4$. 1918. 1. (a) 99*960 ; (6) R369H& 2. 288 ; 720, 3. Aug. 17, 2 h. 2J T S T m, A M. right time. 1919. 1. (0) II and 13, Wii\;0'2. 2. 8635, 4<*. 3. The former ANSWERS TO PATNA MATRICULATION PAPERS. 1918. COMPULSORY PAPER. 1. (a} 496788793655 5 5ooo. (b) 9168 ; I2. 2. () i ; 906' 7*. n< () -043225, -011875; i$s. 9<*'i575- ,8. (a) 8500 ; 25 years. . () 10 days. ADDITIONAL PAPER. COI333. 2. 453 litresjnearly. 4055. 4. 176400. 1919 COM ' ^"^ v PAPER. 1. (a) 604356745568450 (a) i ; -2o8j. oj. 8rf. ; 9 APPENDIX * 1. A. To prove that the multiplier and multiplicand may bfc interchanged without altering the value of the product. For example, to prove that 5x4=4x5. Place 5 dots in a line, and repeat this line 4 times, The number of dots in a row is 5, and there are 4 rows ; there- fore the number of dots altogether is 5 multiplied * ' * by 4. Again the number of dots in a column is 4, ' * ' * | and there are 5 columns ; therefore the number of dots ' altogether is 4 multiplied by 5. Hence 5X44X5. ' * ' ' \ B. The product of a recurring decimal by a whole number or py a terminating decimal may be obtained without converting them into vulgar fractions. It is evident that the product in such a case wjill be a recurring decimal, and that its period will contain as many digits as there are in the period of the multiplicand, Example i. Multiply 3-2456 by 7, 7*4 by 4 and 1*236 by n (i) 3*2456 (") 7H Oil) 1*236 7 4 " 227192 2-856 Ans. 1 3'596 $ i 3 227195 Ans. 13*599- 13'6* Here, we multiply in the usual way, and increase the la^ in the result by the figure (if any) carried from the ^^ '' column of the period of the m" u: ^ r 534 APPENDIX i. multiplicand. Thus we get (). We now add these lines in the usual way f out to do this correctly we extend each line (except the top line) as far as the right-hand figure of the top line) by repeating the digits of its period. The number of decimal places as far as the end of the first period in the result must be 3 + 2, /.*., 5. We therefore place the decimal point to the left of the 5th figure from the end ; and the required product is 51*43727. Example 3. i "32 56 x 1013*2 6. Example 4. "3256 x loo '326i x 100=32*562. Example 5. *x iooo*555x iooo 555*5. C. To divide a recurring decimal by a whole number, we >ceed as in ordinary division ; but instead of bringing down a to each time we br\ng down the digits of the period in rotation. /the divisor is a terminating decimal, we multiply it by that ^ower of 10 which will make it a whole number, and also multiply the dividend by the same power of 10 ; and proceed as in the case of division by a whole number. Example i. ^ Divide 32'624lby 5. 5 ) 32*6242424 6-5248484... ho. quotient is 6*5248. - 52, 202 Example 2 Divide 2723 by 53. Q uot y' 53") APPENDIX II. When an algebraical formula is applicable, it may be used with great advantage in simplifying fractions. r- ,, . rf '704 x '704-*296 Example. Simplify ~i2i-2. - z_ r * J 704- -296 Let 704 =*a } and '293= . Then the given fraction <?-& (a+Pta-b) , " "a^T a-b - ~<* + &==I '704 + '29:>-I.