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Full text of "A practical arithmetic for elementary schools"

QA IOZ C&4 uc ] , * t LIBRARY OF THE UNIVERSITY OF CALIFORN: Received <S//j , 1891 succession No. (o fa fo . Class No. L CONSTABLE'S SCIENTIFIC READING BOOK. By Prof* TYNDALL, KELLAND, BALFODR, ARCHER, &c. 2s. 6d. ENGLISH GRAMMAR. RUDIMENTARY ENGLISH GRAMMAR, By the Rei JAMES CTJRRIE, Principal of the Church of Scotland Trainir College, Edinburgh. Pp. 64, 6d. THE PRACTICAL SCHOOL GRAMMAR. By the sair Author. Pp. 128, Is. 6d. A GRAMMAR OF THE ENGLISH LANGUAGE, with Sketch of its History. By WILLIAM FRANCIS COLLIER, Esq. LL ] Cloth, Is. 6d. PRACTICAL TEXT-BOOK OF GRAMMATICAL ANAL^! SIS. By W. S. Boss, Author of "A System of Elocution." Cloth, 1 FRENCH GRAMMAR. VAN EXSY FRENCH GRAMMAR FOR BEGINNERS. I M. MICHEL, B.A. French Lecturer, Training College, Edinburg Cloth, Is. MODERN PRACTICAL FRENCH GRAMMAR. By tl same Author. 3s. 38 COCKBURN STREET, EDINBURGH. Prospectus and Specimen Pages on Application THOS. LAURIE, EDUCATIONAL PUBLISHER, ARITHMETIC. TtACTICAL ARITHMETIC FOR ELEMENTARY SCHOOLS. Price Is. 6d. ; or in two parts at 6d. and Is. each. ANSWERS, Is. By tiie Rev. JAMES CURRIE, A.M., Author of "Common School Education." FIRST STEPS IN ARITHMETIC. By the same Author, 6d. DAILY CLASS REGISTER OF ATTENDANCE AND SPELLING AND DICTATION. SPELLING AND DICTATION CLASS BOOK ; with Ety- mological Exercises. By an INSPECTOR OF SCHOOLS. Cloth, Is. 3d. ETYMOLOGICAL EXERCISES FOR ELEMENTARY CLASSES. 4d. ; cloth, 6d. ! \ ELOCUTION AND RECITATION. POETICAL READINGS AND RECITATIONS ; with Intro- ductory Exercises in_Elocution. By It. and T. ARMSTRONG. Pp. 160, cloth, Is. ELEMENTS OF ELOCUTION AND CORRECT READING. By Canon RICHSOK, of Manchester. Is. 6d. A SYSTEM OF ELOCUTION. By W. S. Ross, late of Clare College, Scorton. Pp. 480, 3s. COMPOSITION. FIRST STEPS IN ENGLISH COMPOSITION. Pp. 60, 6cl. PRACTICAL TEXT-BOOK OF ENGLISH COMPOSITION. By the same Author. Cloth, Is. ENGLISH LITERATURE. ^MPENDIUM OF ENGLISH LITERATURE. By THOMAS ARMSTRONG, Head Master, Heriot Schools, Broughton. 2s. GRAY'S ODES ; with Notes and Grammatical Analysis. By W. S. Ross. GERMAN. WERNER'S FIRST GERMAN COURSE. By the Author of " Henry's First History of England." Is. Third Edition. 38 COCKBURN STREET, EDINBURGH. Prospectus and Specimen Pages on Application. Constable's ^Educational Series. A PRACTICAL ARITHMETIC FOR ELEMENTARY SCHOOLS BY JAMES CURRIE, A.M. PRINCIPAL OF THE CHURCH OK SCOTLAND TRAINING-COLLEGE, EDINBURGH J AUTHOR OF " EARLY AND INFANT SCHOOL-EDUCATION," " COMMON SCHOOL EDUCATION," ETC. -11 /> A QV 4JFQtf && A NEW AND REVISED EDITION. THOMAS LAUKIE, EDUCATIONAL PUBLISHER, 38 COCKBURN STREET, EDINBURGH, AND 30 NEW BRIDGE STREET, BLACKFRIARS, LONDON. ALSO PUBLISHED. I. FIRST STEPS IN ARITHMETIC. Price 6d. Containing the Simple and Compound Rules, and formirg Pp. 1-G-i of the " PRACTICAL ARITHMETIC." II. ANSWERS to the EXERCISES in the PRACTICAL ARITHMETIC. Pries Is. PREFACE. THIS treatise of Arithmetic is designed to comprise all that is needed by the pupils of common schools, and by those of higher schools till they have completed their elementary education. It is not one of theory, since the instruction of pupils of their standing must be, in the main, practical ; nor, on the other hand, is it a mere collection of examples, since the only practical instruction worthy of the name is that which sets the processes before them in a rational way. It aims throughout at that just combination of theory with practice which is the greatest merit of an elementary text-book. The explanations are given concisely, and in the form in which they are likely to be soonest apprehended by the pupil ; whilst the exer- cises for practice will be found to be very numerous and carefully graduated. In particular, Notation and the four elementary operations, on a satisfactory knowledge of which the pupil's subsequent progress depends, are treated with great fulness. An introductory text-book of Arithmetic should not be a mere condensation of a higher one ; it should devote the space which it gains from the omission of certain of the more advanced rules to the ampler treatment of those which are fundamental. Where the arithmetic of a school is weak at all, it is in these rules that the weakness almost invariably lies ; and it is in these rules, according to the testimony of all competent authorities, that the most material improvement in the teaching of the subject is to be looked for. In the arrangement of the treatise the author has kept in view the requirements of the Privy- Council for Elementary Schools and Pupil- Teachers, although he has not limited himself by them. The Miscellaneous Exercises at the end have been taken chiefly from the papers of the Privy-Council and Dick Bequest Examinations. For the convenience of junior classes the early chapters, treating of the elementary operations with simple numbers and with money, and forming pp. 1-64 of the present work, are published separately under the title of " First Steps in Arithmetic." CONTENTS. PAOB TABLES OF VALUE, 5 NOTATION, 9 ADDITION, 17 SUBTRACTION, . . 25 MULTIPLICATION, 32 DIVISION, 39 MISCELLANEOUS EXERCISES, 48 COMPOUND ADDITION MONEY, 51 COMPOUND SUBTRACTION MONEY, ...... 54 COMPOUND MULTIPLICATION MONEY, ..... 55 COMPOUND DIVISION MONEY, 58 REDUCTION MONEY, 62 MISCELLANEOUS EXERCISES, 66 COMPOUND RULES WEIGHTS AND MEASURES, . . 70 MISCELLANEOUS EXERCISES, 79 BILLS OP PARCELS, 83 PRACTICE, .84 RULE OP THREE, 92 COMPOUND RULE OF THREE, 101 MEASURES AND MULTIPLES, 103 VULGAR FRACTIONS, 106 DECIMAL FRACTIONS, 114 SIMPLE INTEREST, 121 COMPOUND INTEREST, 124 DISCOUNT, ; 125 STOCKS, 127 BROKERAGE, 128 INSURANCE, . 129 PROFIT AND Loss, 130 SQUARE ROOT, 131 MENSURATION, 132 MISCELLANEOUS EXERCISES, 137 * v * y & . TABLES OF MONEY, WEIGHT, AND MEASURES. MONEY. I. Money of Account. 4 farthings, /. = 1 penny, d. 12 pence = 1 shilling, 5. 20 shillings = 1 pound, II. Coins In Circulation. BRONZE. 2 farthings = 1 halfpenny, \d. 2 halfpence = 1 penny. SILVER. 4 threepenny pieces = 1 shilling. 3 groats = 1 shilling. 2 sixpences = 1 shilling. 2 shillings = 1 florin,./! 2 shillings and sixp. = 1 half-crown. 5 shillings = 1 crown, cr. GOLD. 10 shillings -\ 5 So""" [=1 half-sovereign 2 crowns J 20 shillings "k 8 half-crowns f , , 10 florins f = l s vereign. 4 crowns J Paper money is also in use. One pound-note represents the value of 20s. , or one sovereign ; and there are also five-pound notes, ten-pound notes, twenty-pound notes, fifty-pound notes, and one-hundred-pound notes. The guinea, formerly a gold coin = 1, Is., is still recognised as a standard value, though the coin itself is not in use : so the half-guinea, or 10s. 6d. WEIGHT. m. Avoirdupois Weight is used for all common goods. 16 drams, dr. = 1 ounce, oz. 16 oz. = 1 pound, Ib. 28 Ib =1 quarter, qr. 4 qrs. or 112 Ib =1 hundred wt. cwt. 20 cwt. =1 ton, T. Also, 14 Ib =1 stone, st. IV. Troy Weight is used for weighing the precious inetals and jewellery. 24 grains, gr. 1 pennyweight, dwt. 20 dwt. = 1 ounce, oz. 12 oz. =1 pound, ft Note. The Ib Troy = 5760 gr. The Ib Avoir = 7000 gr. LENGTH. V. Lineal Measure Is used for measuring length, and is iience often called long measure. 1? inches, in. = 1 foot, ft. 3 reet = 1 yard, yd. 5 yards = 1 pole, po. 40 poles = 1 furlong, fur. 8 furlongs = 1 mile, ml. Tradesmen use what is called & foot- rule of three feet long for measuring with, on which the feet are divided into inches, and the inches into ei^ith parts, tenths, or sixteenths. For longer measurements, a tape or line of 22 yards, similarly divided, is com- monly used. Obsolete measures, but still used for special purposes, are the following : 1 line = Ath inch. 1 palm = 3 inches. 1 span = 9 inches. 1 cubit = 18 inches. 1 hand (for mea-) suring height of > = 4 inches. horses) ) 1 fathom (for mea- ) _ fi f . suring depth) } ~ 6 feet ' 1 geographical mile = 1 mile 266 yds. [nearly. 1 league = 3 geog. miles. 1 degree = 60 geog. miles. VI. Cloth Measure is used for measuring cloth. 2J inches = 1 nail, nl 4 nails = quarter, qr. 4 quarters = 1 yard, yd. Also, 5 quarters = 1 ell, E. The draper's rod, one yard long, is divided according to this measure; though in practice, fractions (six- teenths) of a yard are more commonly used. VTI. Land Measure is used for measuring land. Sur- veyors use a chain for this purpose, called Gunter's chain, 22 yards (or 4 poles) long, and divided into 100 parts or links. 22 yards = 1 chain of 100 Iks. 10 chains = 1 furlong. Note. The link = 75? inches. TABLES OF MONEY, WEIGHT, AND MEASURES. SURFACE. VIII. Square Measure, sometimes called superficial measure, is used for measuring surface or area. 144 sq. in. = 1 sq. ft. 9 sq. ft. = 1 sq yd. 30$ sq. yd. = 1 sq. po. (or perch, per.) 40 sq. po. = 1 rood, ro. 4 roods = 1 acre, ac. 640 acres = 1 sq. ml. Still used for special purposes are the following measures : 100 sq. feet = 1 square of flooring. *?** T }= 1 rod of brickwork. 36 sq. yd. = 1 rood of building. Land-surveyors, as stated above, use the chain of 100 links, though they express the result of their measure- ments in this table : 10,000 square links = 1 square chain. 10 square chains = 1 acre. SOLIDITY. IX. Cubic Measure Is used for measuring the contents of solid bodies, e.g., masses of stones or earth (hence often called solid mea- sure), or of bodies which have tli*i shape of solids, e.g., rooms, cis- terns, etc. 1728 cubic in. =1 cubic ft. 27 cubic ft =1 cubic yd. Shipping is measured by tonnage, timber by loads, and general goods sometimes by barrel-bulk, thus : 42 cub. ft. = 1 ton shipping, T. sh. 40 cub. ft. rough ) timber > = 1 load, lo. 60 do. hewn ) 5 cub. ft =1 barrel-bulk, B.B. CAPACITY. X. Measure of Capacity is used for the measurement of liquids, and also of dry goods, like grain, etc. 4 gills, gi. = 1 pint, pt. 2 pints = 1 quart, gt. 4 quarts = 1 gallon, gal 2 gallons = 1 peck, pk. 4 pecks = 1 bushel, bu. 8 bushels = 1 quarter, qr. The peck, bushel, and quarter are nsed for dry goods only. For wine and beer, casks of various sizes are used, of which the Eort common are FOR WINE. The puncheon = 84 gal. The pipe = 126 gal. The tun = 252 gaL FOR BEER. The kilderkin = 18 gal. The barrel = 36 gaL The hogshead, Jihd. = 54 gal. But these casks are not standard measxires, and vary in their capacity. The imperial gallon contains 277 '274 cubic inches. TIME. XI. Measure of Time. 60 seconds, sec. = 1 minute, min. 60 minutes = 1 hour, ho. 24 hours = 1 day, da. 7 days = 1 week, wk. 52 wks. 1 day, or ^ _ r 365 days f ~ Y ' V 366 days = 1 leap year. 100 years = 1 century. The year is divided into 12 calendar months : January 31 days February 28 March 31 April 30 May 31 June 30 July 31 days August 31 September 30 October 31 November 30 December 31 Every year (with very rare excep- tions) whose number is divisible by 4, is a leap year ; in which February has 29 days. Thirty days have September, April, June, and November : All the rest have thirty-one, Excepting February alone, Which has but twenty-eight days clear, And twenty-nine in each leap year. The lunar month = 29 da. 12 ho. 44 min. The solar year = 365 da. 5 ho. 48 min. 48 sec., i.e., nearly 365 days 6 hours (the Julian year). QUARTERLY TERMS. In England. Lady-Day, . March 25. Midsummer, . June 24. Michaelmas, . Sept. 29. Christmas, . Dec. 25. In Scotland. Candlemas, . Feb. 2. "Whitsunday, . May 15. Lammas, . Aug. 1. Martinmas, . Nov. 11. TABLES OF MONEY, WEIGHT, AND MEASURES. The centimes are reckoned, among Christian nations, in numerical order from the birth of our Lord (called the Christian era) : thus the years 1 to 99 are the first century, 100 to 199 the second, and so on. This is the nine- teenth century. Any particular year, e.gr., 1864, is denoted 1864 A.D., i.e., Anno Domini, in the year of our Lord. The years before the birth of our Lord are reckoned back in order from that event: thus 1460 A.C., means Ante Christum, or before Christ. INCLINATION. XII. Angular Measure is used for measuring the angle or inclination of one line to another. 60 seconds, " =1 minute, 60' = 1 degree, 90 = 1 right angle, L 860" = 1 circle, The following Tables are subjoined for reference : Paper Measure. 24 sheets = 1 quire, qu. 20 quires = 1 ream, re. 21^ quires = 1 perfect ream. Cloth Measure. 5 quarters = 1 English ell. 3 quarters =. 1 Flemish ell, FI. E. 6 quarters = 1 French ell, Fr. E. 37 inches = 1 Scotch ell, S. E. Apothecaries' Weight. OLD MEASURE. 20 grains, gr. = 1 scruple, j^ 3 scruples = 1 drachm, 3 8 drachms = 1 ounce Troy, 5 12 ounces =12) Troy. NEW MEASURE (1862). 437J grains = 1 ounce Avoir. Apothecaries' Fluid Measure. 60 minims, TT^ = 1 fluid drachm, /. 3 3 fl. drachms = 1 fluid ounce, /. ^ 16 ounces = 1 tb 20 ounces = 1 pint, O 8 pints = 1 gallon, C FOREIGN MONEY. United States. 10 cents = 1 dime. 10 dimes = 1 dollar, $ 1 dollar = 4s. 2d. France. 100 centimes = 1 franc. 1 franc = 9d. nearly. Canada. Accounts are kept in s. d. currency of which 1 = 16s. 8d. sterling. East Indies. 16 annas =. 1 rupee. 1 rupee = Is. lO^d, OLD SCOTCH MONEY AND MEASURES still recognised in Scotland for certain purposes. Money. 1 shilling Scots = Id. sterling. 1 Scots = Is. 8d. do. being one-twelfth of the same names sterling. 1 merk = Is. ld. Lineal Measure. 37 inches = 1 ell. 6 ells == 1 fall. 4 falls = 1 chain. 1 chain = 1J Imp. chain nearly. Square Measure. 36 sq. ells = 1 square falL 40 sq. falls = 1 rood. 4 roods = 1 acre. 1 acre = 1J Imp. acre nearly. Liquid Measure. 4 gills = 1 mutchkin. 2 mutchkins = 1 chopin. 2 chopins = 1 pint. 8 pints = 1 gallon. 1 gallon = 3 Imp. gallons nearly. Dry Measure. 4 pecks = 1 ftrlot. 4 flrlots = 1 boll. 10 bolls = 1 chalder. The Wheat Firlot was nearly equal to an Imp. bushel (= '998 bush.); the Barley Firlot nearly equal to 1J bush. (= 1-456 bush.) The Boll weighs 140 tt> Avoir. PROPOSED DECIMAL COINAGE. 1 mil = one thousandth part of 1, or = Jd. less j^d. 10 mils =1 cent, one-hundredth of L 10 cents = 1 florin, one-tenth of 1. 10 florins = 1. 1. 2. NUMERATION AND NOTATION. Numbers of One Place. One finger and one finger make two fingers. Two fingers and one finger make three fingers. Three fingers and one finger make four fingers. Four fingers and one finger make five fingers. Five fingers and one finger make six fingers. Six fingers and one finger make seven fingers. Seven fingers and one finger make eight fingers. Eight fingers and one finger make nine fingers. Bf. 1 One, two, three, four, five, six, seven, eight, nine, are the names of numbers used in counting. The naming of numbers is called Numeration. One, three, five, seven, nine, are called odd numbers. Two, four, six, eight, are called even numbers. These nine numbers mean so many ones, or units as they are called ; thus two means two ones or two units, three means three ones or three units, and so on. EXERCISE I. Bf. 1. Repeat the table of units, as given above. 2. Repeat it, using balls, marbles, boys, etc., instead of fingers. 3. Repeat it with the numbers alone, thus, " one and one are two." 4. Count from one up to nine, and from nine back to one. 6. Count the odd numbers from one to nine ; from nine to one. 6. Count the even numbers from two to eight ; from eight to two. 7. Name the two numbers next above five, eight, three, etc.* 8. Name the two numbers next below six, nine, four, etc. 9. Hold up three fingers, five, seven, etc. 10. How many wheels has a cart ? How many halfpence in a penny? How many pence in a threepenny-piece ? How many letters in the word " dog" ? How many legs has a cow ? etc. 11. If I have four pence and give one away, how many do I keep ? If I have six marbles, and get one from James, how many have I ? etc. The nine numbers are denoted by signs or figures, thus : one, two, three, four, five, six, seven, eight, nine, 12 3456789 The figuring of numbers is called their Notation. I Bf. means that the ball-frame may be used for illustration. > Etc. means that various other questions of the same kind may be given. 10 NUMERATION AND NOTATION. EXERCISE II. 1. Write down the figures (1.) even along ; (2.) up and down. 2. Name the numbers in Ex. iv. sect. 16. 3. Write down the figures for the same numbers.* 3. Numbers of Two Places. If I count nine on my fingers, I find one finger over. Nine fingers and one finger make ten fingers ; which is the whole number of them. If I wish to count beyond ten, I must begin again and go round a second time ; that will give me two-times ten or two tens. Three times round will give three-times ten or three tens ; and so on, up to nine-times round, which will give nine-times ten or nine lens. One ten is called Ten, denoted by 10. Two tens are Twenty, 20. Three tens Thirty, 30. Four tens Forty, 40. Five tens Fifty, 50. Six tens Sixty, CO. Seven tens Seventy, 70. Eight tens Eighty, 80. Nino tens Ninety, 90. The tens are numbers of two places. They are denoted by the figures for the units with a cipher on the right. The value of a figure is increased ten times by its being written in the second place from the right : thus 3 denotes three units, but 30 denotes three tens. Hence the notation we use is called the decimal 2 notation. The cipher is used to fill up the first or right-hand place, when that place contains no units or nothing ; hence it is commonly called nought or nothing. It is never used alone. EXERCISE III. 1. Repeat the table of tens ; backwards ; by odds ; by evens. 2. Count the tens. 3. Name the tens next above forty, sixty, etc. ; next below thirty, eighty, etc. 4. How many fingers have six boys ? eight boys ? etc. JBf. 5. How many boys together have thirty lingers ? seventy ? etc. Bf. 6. How many units in eight tens ? six tens ? etc. 7. How many tens in thirty units ? in seventy units ? etc. 1 Either from the copy or to dictation. /The teacher may vary the exercise by having the figures pointed out on the board from columns written by him- self. 2 From the Latin word decem, ten. NUMERATION AND NOTATION. 11 One ten and two units twelve, One ten and three units thirteen, One ten and four units fourteen, One ten and five units fifteen, One ten and six units sixteen, One ten and seven units seventeen, , One ten and eight units , eighteen, , One ten and nine units The tens-units are also num , nineteen, , bers of two places ' 8. If I have ninety marbles and give away ten, how many do I keep ? It 1 have seventy, and get ten more, and other ten, how many have 1 / etc. 9. Write down the figures for the tens below each other. 10. Name the numbers, Ex. vi. sect. 17, Nos. 1, 2. 11. Write down the figures for these numbers. One ten and one unit are called eleven, denoted by 11 12 13 14 15 16 17 18 19 the first being the units' place, the second the tens' place. The names of the numbers from 13 to 19 are formed by put- ting the number of the units before that of the tens ; thus thirteen is three and ten, fourteen is four and ten, etc. The names of all the other numbers of two places are formed by putting the number of the tens before that of the units j thus Two tens and one are called twenty-one, denoted by 21 Two tens and two twenty-tivo, . 22 Etc. etc. etc. Three tens and one thirty-one, . 31 Three tens and two thirty-two, . 32 Etc. etc. etc. When numbers of two places are written below each other, units are written below units, and tens below tens. EXERCISE IV. 1. Repeat the table of tens-units fron ten to twenty, from twenty to thirty, etc. 2. Count the tens-units from ten to 'twenty, from twenty to thirty, etc. 3. If one boy holds up the fingers of his right hand, and other three boys all their fingers, how many fingers are up ? how many if another boy holds up his ? if another? if one boy removes his ? etc., Ef. 4. If I hold up seven fingers, how many girls must hold up all their fingers to make twenty-seven ? to make thirty-seven ? etc., Bf. 5. Count by tens from thirty-one, from forty-two, etc. Count by tens back from ninety-eight, eighty-seven, etc. 6. How many are 1 ten and 4 ? 2 tens and 6 ? 4 tens and 7 \ etc. 7. What tens and units make up 18, 27, 33, 47 ? etc. 8. Figure from ten to twenty, twenty to thirty, etc. 12 NUMERATION AND NOTATION. 9. Figure 2 tens below 2 units, 3 tens below 3 units, etc. ; 9 ur.Hs below 9 tens, 8 units below 8 tens, etc, 10. Name the numbers in Ex. vi. sect. 17, No. 3-25. 11. Write down, or tell in order, the figures for these numbers. 5. Numbers of Three Places. Nine tens and one ten make ten tens. As we put ten units together, and call them one-ten, so we put the ten-tens together and call them one hundred. JBf. One hundred is denoted by . 100 Two hundreds . . 200 Three hundreds . . 300, and so on. The hundreds are numbers of three places. They are denoted by the figures for the units with two ciphers on the right. The value of a figure is increased a hundred times by its being written in the third place ; thus 3 denotes three units, tut 300 denotes three hundreds. The two ciphers are used to fill up the first and second places, when these places contain no units and no tens. EXERCISE V. 1. Count the hundreds, backwards, by odds, by evens. 2. Name the numbers in Ex. ix. sect. 19, Nos. 1, 2. 3. Tell in order the figures in these numbers. 4. How many tens in 100, 500, 800 ? etc. 5. How many hundreds in 10 tens, 70 tens ? etc. 6. Figure the hundreds in an up-and-down line. 7. Figure 1 hund. below 1 ten, 2 hund. below 2 tens, etc. 9 tens below 9 hund., 8 tens below 8 hund., etc. 8. Figure 1 h. below 1 1. below 1 u. 2 h. below 2 t. below 2 u. etc. 9 u. below 9 t. below 9 h. 8 u. below 8 t. below 8 h, etc. 9. Write down the figures for the numbers Quest. 2. ^- Numbers consisting of hundreds, tens, and units are also numbers of three places ; the first being the units' place, the second the tens' place, and the third the hundreds' place. Their names are formed by combining, in their order, the number of the hundreds, the number of the tens, and the number of the units. Thus 146 denotes 1 h. 4 t. 6 u., and is called one hundred and forty-six. 270 2 h. 7 t. u., two hundred and seventy. 804 8 h. t 4 u., eight hundred and four. Where there are no units, or no tens, these are omitted in the names, as in the last two numbers. When numbers are written in column, the same places must be kept below each other. NUMERATION AND NOTATION. 13 EXERCISE VI. 1. Count from one hundred to nine hundred and ninety by tens, and from nine hundred and ninety to one hundred by tens. 2. Count from two hundred and forty to two hundred and fifty. five hundred and sixty to five hundred and seventy, etc. 3. Name the numbers in Ex. ix. sect. 19, No. 8-25. 4. Tell in their order the figures in these numbers. 5. Figure below each other two hundred and twenty-two, two hundred and two, two hundred and twenty, two hundred, twenty, two : etc. Repeat the same, beginning with the units. 6. Figure the numbers in Quest. 3. 7. Numbers of One Period. All numbers of one, two, or three places that is, all num- bers from 1 to 999 are numbers of one period. Numbers of one place may be written with their period completed by putting two ciphers to the left hand. Thus, since 6 units is the same as hundreds tens 6 units, the number 6 may be written 006, and read no hundred and six. Numbers of two places may be written with their period completed by putting one cipher to the left hand. Thus, since 6 tens 5 units is the same as hundreds 6 tens 5 units, the number 65 may be written 065, and read no hundred and sixty-five. A cipher placed to the left hand of any figure does not alter its place, nor, consequently, its value. EXERCISE VII. 1. What are the numbers whose figures in order are three, two, one ; four, nothing, six ; six, four ; seven, two, nothing, ? etc. 2. What figures in order denote two hundred, two hundred and six, five hundred and thirty-two ? etc. 3. What are these numbers made up of ? Ex. ix. sect. 19. 4. Figure their several parts in order below each other ? 5. Point out the tens' place in them ? units' place ? hundreds' ? 6. What numbers are made up of these parts, 3 h. 2 1. 6 u. ? 4 h. Ot. 7n.l 7h.4t.0u.? 8h. 4u.? etc. 7. Read these numbers, 7, 17, 20, 34, etc. (1.) as they stand ; (2.) with their periods filled up ? 8. Read these numbers, 00808080008888080, etc. 9. Take any number, as 5. What does it denote with one nought before it ? with two ? with one after it ? with two ? with one before and one after it I Which nought increases its value ten times ? which leaves it unaltered ? What two noughts increase its value one hundred times ? what two leave it unaltered ? What two increase its value ten times ? etc. 14 NUMERATION AND NOTATION. 10. Write the numbers, eight, ten, twenty-five, etc. (1.) as incom- plete periods ; (2.) as completed periods. 1 11. Write in figures : fifty-three, thirty-seven, ninety-four, one hundred and seventy, four hundred and sixty-nine, eight hundred and eight, seven hundred and fourteen, seventy-eight, two hundred and eighteen, five hundred and five, six hundred and sixty, three hundred and thirty-three, nine hundred and forty one, five hundred and sixteen, etc. %.* When the pupil has obtained perfect facility in reading and writing num- bers of one period, he may proceed with their addition, subtraction, and multi- plication, returning afterwards to the notation of larger numbers. 8, Numbers of Two Periods. Nine hundreds and one hundred make ten hundreds. As we put ten tens together and call them one hundred, so we put the ten hundreds together and call them one thousand. One thousand is denoted by . . . 1,000 Two thousands, .... 2,000 Ten thousands, .... 10,000 Eleven thousands, .... 11,000 One hundred thousand, . . . 100,000 Three hundred and forty-seven thousand, . 347,000 Any number of thousands is written as if it were units, with three ciphers on tlie right. If the number contain also hundreds-tens-units, these are written in place of the cyphers. Thus One thousand five hundred is denoted by . 1,500 Two thousand six hundred and thirty, . 2,630 Ten thousand four hundred and twenty-five, 10,425 Eleven thousand seven hundred and eight, . 11,708 One hundred thousand one hundred end thirty, 100,130 Three hundred and forty-seven thousand three hundred and forty-seven, . . . 347,347 Every number of thousands has from four to six places, forming two periods. The first period containing the hundreds -tens- units, if there are any ; the second the thousands. %* The two periods are often separated by a comma, as above, to prevent mistakes in reading numbers ; but by practice the pupil vill soon be able to do without it EXERCISE VIII. 1. Read the numbers, Ex. x. sect. 20. 2. Write to dictation the numbers in same Exercise. 3. In 501274 (or any of the numbers in same Exercise), how many thousands ? hundreds ? tens of thousands ? units ? hundred thousands ? tens? i Counters may be used to aid the pupil in writing numbers of one period ; see Note, section 9 NUMERATION AND NOTATION. 15 4. In 347029 (or any of the numbers in same Exercise), what does the 3 denote ? the 9 ? ? 7 ? 4 ? 2 ? 5. What figures in order denote six thousand three hundred? or any of the numbers in the same Exercise ? 6. What numbers are denoted by the following sets of figures in order, 4, 2, 4, 8 ? 8, 0, 7, 9, 2 ? 3, 6, 5, 2, 0, 1 ? etc. m Numbers of Three Periods. Nine hundred thousands and one hundred thousands make a thousand thousands, which we call one Million. One million is denoted by ... 1,000,000 Two millions are 2,000,000 Ten millions, 10,000,000 Eleven miUions, 11,000,000 One hundred millions, .... 100,000,000 Three hundred and forty-seven millions, . 347,000,000 Any number of millions is written as if it were units, with six ciphers to the right. If the number contain also thousands, hundreds, tens, and units, these are written in place of the ciphers, thus : One million five hundred thousand is denoted by 1,500,000 Two millions six hundred and thirty thousand, 2,630,000 Ten millions four hundred and twenty-five thousand, 10,425,000 Eleven millions seven hundred and eight thousand five hundred and ten, .... 11,708,510 One hundred millions one hundred thousand and one hundred, 100,100,100 Three hund. and forty-seven mills, three hund. and forty-seven thousand, three hun. and forty-seven, 347,347,347 Every number of millions has from seven to nine places, forming three periods; the first called the units 1 period, the second the thousands', and the third the millions'. EXERCISE IX. 1. Read the numbers, Ex. xii. sect. 21. 2. In 243,076,549 (or any of the above numbers), how many hun- dreds ? tens of thousands ? tens of millions ? units ? hundreds of thou- sands ? etc. 3. In 804395276 (or any of the above numbers), what does the 5 de- note? 4? 8? 0? 6? 7? etc. 4. What figures in order denote seven millions and thirty thousand, or any of the above numbers ? 5. What is denoted by the 1st place, 2d period ? 2d place, 1st period ? 1st place, 3d period ? 2d period ? 1st period ? 3d place, 1st period ? etc. 1 6. Write to dictation the numbers, Ex. xii. ect. 21. 1 This questioning may be continued with the help cf three periods of counters; thus 69G 999 999 These may be also advantageously used in the following exercises in dicta- 16 NUMERATION AND NOTATION. 10. Numbers of more than Three Periods. Nine hundred millions and one hundred millions make a thousand millions. One thousand millions are denoted by 1,000,000,000 Ten thousand millions, . . 10,000,000,000 A hundred thousand millions, . . 100,000,000,000 Thousands of millions are written as if they were thousands, and six ciphers are added. If there are also millions, thousands, and units, these are written in place of the ciphers, thus : One thousand two hundred and thirty millions is 1,230,000,000 Ten thousand five hundred and sixteen millions, five hundred and sixteen thousand, . 10,516,516,000 One hund. and thirty-seven thous.,one hund. and thirty-seven mills., one hund. and thirty-seven thousand one hundred and thirty-seven, 137,137,137,137 Every number of thousands of millions contains from ten to twelve places, forming four periods ; which may be separated by commas, as above. Still larger numbers may be expressed by a fifth period, com- mencing at a million of millions, or, as it is called, a Billion ; or even a sixth period for thousands of billions, thus : B. M. U. 137,137,137,137,137,137 But numbers of more than three periods rarely occur. J]_ > Appendix on the Roman Notation. Numbers are sometimes denoted by another set of characters, called Roman.* These are seven in number, thus : 1 is denoted by the letter I, 5 by V, 10 by X, 50 by L, 100 by 0, 500 by D, and 1000 by M. EXERCISE X. 1. Name the letters, with the numbers they denote. 2. Write down the letters, with the numbers they denote. tlon. Tims the pnpil may be asked to read 28 14 7, or to write numbers In that way in the first instance, and then to supply the necessary ciphers. i So called from having been used In the ancient Roman notation. The ordinary characters are often spoken of as the Arabic, from having come to us through the Arabs. ADDITION. 17 12. To denote other numbers, these seven characters are combined in two ways First, a character following another of greater or equal value adds thereto its own value ; thus VI denotes 5 + 1, or 6. Second, a character preceding another of greater value subtracts therefrom its own value ; thus IV denotes 5 1, or 4. The only numbers which are denoted by subtraction are the units next under V and X, and the tens next under L and C ; thus 4 is de- noted by IV, 9 by IX, 40 by XL, and 90 by XC. All the rest are denoted by addition. I 1 II 2 III 3 IV 4 V 5 VI 6 VII 7 VIII 8 IX 9 X 10 XX 20 XXX 30 XL 40 L 50 LX 60 LXX 70 LXXX 80 XC 90 XI 11 XII 12 XIII 13 XIV 14 XV 15 XLI 41 XLII 42 XLIII 43 etc. C 100 CO 200 CCC 300 CCCC 400 D 500 DC 600 DCC 700 DCCC 800 DCCCC 900 CX 110 CXX 120 CXXIV 124 CXLIX 149 CCXXX 230 CCCLXI 361 DXC 590 DCCIII 703 etc. M 1000 MC 1100 MCC 1200 MD 1500 MDLXIV 1564 MDCX 1610 MDCXCII1692 MDCCC 1800 MM 2000 The Roman characters are now used only to denote numbers, e.g., the chapters of a book, the hours on the clock, the houses in a street, and the years ; never to calculate with. EXERCISE XL 1. What numbers are denoted by V, X, IV, XX, XXII, XL, etc. ? 2. Name, or write down, letters for the numbers, Ex. iv. sect. 16. 3. Name, or write down, letters for the numbers, Ex. vi. sect. 17. 4. Name, or write down, letters for the numbers, Ex. ix. sect. 19. 5. Do. do. 1250, 1365, 1473, 1582, 1624, 1738, 1806, 1835, 1864. 13. ADDITION. Ex. Of four flocks of sheep, one contained 35, the second 29, the third 50, and the fourth 47. They were put into one field ; how many sheep were there in all ? Here we have to find one number as large as four given numbers together. The number to be found is called the sum. The sum is got by adding the four given numbers together. The process of adding is called addition; and when the things to be added are of one kind, as here simple addition. The sign of addition is + (plus) : thus 1 + 1 are 2. We cannot find the sum of the above four numbers at once ; they are too large. We must therefore add them in parts ; for which purpose we must leajn the addition of the first nine numbers. 18 ADDITION. 14. Addition Table. %* This Table should be learnt first in lines even along ; thus, 1 and 1 are 2 ; 2 and 1 are 3, etc. ; afterwards in lines up and down. Lf. 1 and |2 and |3and |4and 5 and 5 and 7 and |8 and 9 and 1 are 2,1 are 3.1 are 41 are 5 lare 6 lare 7 1 are 8 1 are 9 lare 10 2... 3,2... 42... 52... 62... 7 2... 8,2... 92... 10 2 ... 11 3... 43... 5;3... 63... 73... 8 3 ... 93 ... 10 3 ... 11 3 ... 12 4... 5 4... 6.4... 714... 84... 9 4 ... 10 4... 114 ... 12 4 ... 13 5... 6 5... 75... 85... 95... 10 5 ... 11 5 ... 125 ... 13 5 ... 14 6... 7 6 ... 8:6 ... 96 ... 106 ... 11 6 ...126 ... 136 ... 14 6 ... 15 7... 8 7 ... 9.7 ... 107 ... 11-7 ... 12 7 ...137 ... 14 7 ... 15 7 ... 16 8... 9 8 ... 108 ... 118 ... 128 ... 13 8 ... 148 ... 15 8 ... 16 8 ... 17 9... 109... 11J9... 129... 139... 14 9... 159... 16 9 ... 17 9 ... 18 EXERCISE I. Bf. 1. Repeat the several lines of the table even along ; backwards ; by odds and evens. 2. Repeat the several lines up and down in the same orders. 3. 5 and 6 are ? 8 and 3 are ? 4 and 9 are ? etc. 4. 2 + 3 + 5 are ? 6 -f 3 + 8 are ? etc. 1 5. 2 + 4 + 3 + 7 are ? 5 + 2 + 2 + 6 are ? etc.* 6. 2 books and 3 books are ? I have 5d. and John 4d., how much have we both ? John had 3 marbles ; if he bought 6 and gained 7, how many has he now ? etc. 7. Write down the columns of the table in order. . If one of the numbers to be added contains tens and units, add the units as if they were alone, and prefix the number of tens. Thus 11 and 1 are 12 ; 12 and 1 are 13 ; 13 and 1 are 14. 11 and 2 are 13 ; 12 and 2 are 14 ; 13 and 2 are 15. Etc. etc. etc. EXERCISE II. 1. Repeat the several lines of this table from 11 to 19, (1.) even along, (2.) up and down. 2. Repeat a similar table for 21-29, 31-39, etc. 3. 11 and 4 are ? 17 and 8 are ? etc. 4. 5 + 19 + 4 are ? 17 + 6 + 5 are ? etc. 5. 16 + 7 + 2 + 4 are ? 13 i + 4 + 9 + 6 are ? etc. 6. Write down any line of this Table in order. EXERCISE III. Count forward from 1, 2, 3, 4, 5, 6, 7, 8, 9 by twos, then by threes, fours, etc., up to nines. i In Ques. 4, the sum of the first two numbers, and in Ques. 5, the sum of the first three, should not exceed nine. ADDITION. 19 16. Addition of Numbers of One Place. Ex. John had 8 marbles, James had 4, William had 7, and Henry 5 ; how many had they amongst them ] 8 We can find the sum of these small numbers 4 without writing ; but if we wish to write down 7 the process, we set the numbers below each other, 5 and add step by step, thus (5 and 7 are) 12 ; (and 4 are) 16 (and 8 are) 24 24 which is the sum required. %* The words within parentheses may be used for some time by the pupil, but should be omitted at the earliest moment he can do without them. The addition may be proved to be correct by adding the column downwards from the top. The sum of any series of numbers is the same in whatever order they are added. EXERCISE IV. (l) (2) (3) (4) (5) (C) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (IS) (1. 8 9 2 6 8 3 5 6 7 2 1 4 5 6 7 8 9 2 (2.) 7 4 5 1 6 2 9 7 2 1 3 6 5 9 4 (3.) 5 7 7 5 4 1 8 7 5 6 9 2 5 4 1 3 2 (4. 6 6 8 4 8 6 4 9 4 3 2 1 8 6 9 3 1 (5. 4 9 3 5 3 8 7 2 1 5 6 8 5 6 9 8 (6-, 3 8 1 2 4 8 3 6 9 4 5 9 1 7 3 4 CM 2 5 7 2 9 5 8 6 5 3 1 2 3 4 1 (8.) 9 4 6 3 7 6 2 4 9 3 4 1 5 7 3 (9. 1 5 9 5 4 3 2 1 1 2 3 4 5 6 7 8 (10. 5 1 4 4 4 5 6 7 8 9 8 7 6 5 4 3 2 1 (11. 4 5 3 5 1 2 3 4 5 6 7 8 9 4 3 8 (12.) 8 9 2 2 6 1 5 3 4 7 4 3 8 5 2 2 9 (13.) 7 6 5 1 7 5 6 4 3 9 2 1 8 4 6 3 8 (14. 3 2 7 2 6 4 5 4 9 1 2 3 5 8 2 9 (15. 5 6 4 1 7 3 6 5 4 2 9 2 1 1 7 (16. 8 7 7 9 1 4 7 8 2 1 5 3 9 4 3 6 6 4 (17. 2 6 8 3 6 7 5 4 8 6 2 1 4 7 2 5 (18. ) 9 5 3 4 8 2 4 3 2 6 9 5 4 3 2 1 (19. 1 5 1 5 6 1 2 4 6 8 2 4 6 8 2 4 6 (20. ) 7 8 4 7 1 3 5 7 9 1 3 5 7 9 1 3 5 7 (21. ) 4 2 2 9 4 2 3 6 2 4 7 3 5 8 4 6 9 5 (22.) 4 3 5 4 3 4 7 8 9 5 6 8 2 1 (23. 6 6 1 1 3 4 9 6 8 2 1 4 3 7 7 6 4 (24.) 8 9 7 2 5 6 2 1 4 9 3 2 8 6 4 3 2 (25.) 1 8 8 3 4 5 8 9 1 4 7 6 8 5 1 2 3 %* These numbers may be added in parts of columns, or in whole columns, up down from left to right from right to left. And the pupil sbould work at them a little every day till he attains expertness in adding. 20 ADDITION. 17. Addition of Numbers of Two Places. The Table given, sect. 14, serves also for the addition of tens, thus : If 1 and 1 are 2, 1 ten and 1 ten are 2 tens, or 10 and 10 are 20. 2 and 1 are 3, 2 tens and 1 ten are 3 tens, or 20 and 10 are 30. Etc. etc. etc. EXERCISE V. Perform Ex. i. Quests. 1-5, with tens. Ex. Of four flocks of sheep one contained 35, the second 20, the third 50, and the fourth 47. They were put into one field : how many sheep were there in all ? Set the numbers below each other in their places. 35 Then in the units' column : (7 and 9 are) 16, (and 5 29 are) 21 (units ; set down) 1 (in the units' place), and 50 carry 2 (tens to the tens' column). Next, in the tens 47 column : (2 and 4 are) 6, (and 5 are) 11, (and 2 are) 13, (and 3 are) 16 (tens. Set down the) 6 (in the) 161 tens' (column), and (the ten tens as) 1 hundred (in the hundreds' column). EXERCISE VI. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. . 11. 12. 13. 14. 15, 20 70 46 23 14 22 34 54 72 29 13 27 41 64 39 30 40 50 13 43 19 96 34 49 64 70 91 27 36 44 40 7 64 14 50 47 94 18 81 49 17 9 4 51 63 50 20 36 29 69 90 25 60 70 80 90 40 56 4 20 60 60 45 56 24 47 18 26 43 31 83 7 91 54 7 40 9 69 73 33 53 37 43 62 15 24 19 9 48 17 70 80 92 20 41 64 74 51 64 82 39 24 47 64 8 80 10 87 34 76 92 82 27 39 51 63 75 87 99 9 90 5 71 47 92 10 45 14 17 20 23 6 9 2 49 30 50 25 56 85 86 37 35 38 41 44 47 50 53 80 40 30 34 81 24 48 29 94 91 87 84 62 59 72 27 50 40 23 73 37 35 15 62 59 18 60 53 27 9 93 16. 17. 22 + 80+6 + 12 + 15 + 93 + 27 + 36 + 48 + 51 + 70+10 + 29 + 8. 18. 37 + 45 + 15 + 7 + 1 + 27 + 39 + 82+99 + 4+54 + 37 + 10 + 29. 19. 28 + 57 + 3 f 30 + 17 + 37 + 90+25 + 41 + 8 + 59 + 32+87 + 40. 20. 29 + 5 + 16 + 34 + 64 + 72 + 19+7 + 38 + 64+28 + 11 + 58 + 38. 21. 18 + 90+21 + 7 + 9 + 8+15 + 27+47 + 50 + 62+71 + 89 + 69. 22. 30+54+4+23 + 93 + 47 + 50+41 + 39+8 + 17 + 28 + 60. 23. 16 + 84+17 + 30 + 85 + 74 + 32+91 + 11 + 22 + 50+5 + 15 + 66. 24. 93 + 9 + 8 + 17+29 + 40+57 + 85 + 36 + 21 + 73 + 17 + 76 + 82. 25. 87 + 53 + 20 + 6 + 9 + 14 + 65 + 89 + 53 + 28 + 70 + 38 + 67 + 2. ADDITION. 21 EXERCISE VII. 1. 10 + 11 are ? 10 + 12 are ? 10 + 13 are ? 10 + 21 are ? etc. 2. 20 + llare ? 20 + 12are ? 20 + 13are ? 20 + 21 are ? etc. 3. 30 + llare ? 30 + 12are ? 30 + 13are ? 30 + 21 are ? etc. 4. 40 + llare ? 40 + 12 are ? 40 + 13 are I 40 + 21 are ? etc. 5. Add the remaining tens in a similar way. 6.50 + 25are ? 20 + 18are ? 40 + 29 are ? etc. 7.22 + 15are ? 34 + 18are ? 75+24 are ? etc. V* In this last question, it is easier to add the tens first; thus : 34+18 are 4 tens and 12, that is 52. Addition of Numbers of One or more Periods. The table given, section 14, serves also for the addition of hundreds, thousands, etc. ; thus, If 1 and 1 are 2, 1 h. and 1 h. are 2 hs., or 100 and 100 are 200. 2 and 1 are 3, 2 h. and 1 h. are 3 hs., or 200 and 100 are 300. Etc. etc. etc. EXERCISE VIII. Perform Ex. i. Questions 1-5, with hundreds. Ex. Four heaps of bricks were lying in a field. The first con- tained 208 bricks, the second 349, the third 160, and the fourth 87 ; how many bricks were there in all ? Set the numbers below each other in their places. In the units* column (7 and 9 are) 16, (and 8 are) 24 (units ; set down) 4 (in the units' 208 place), and carry 2 (tens). 349 In the tens' column (2 and 8 are) 10, (and 6 160 are) 16, (and 4 are) 20 ; (set down) (in the 87 tens' place) and carry 2 (hundreds). In the hundreds' column (2 and 1 are) 3, (and 804 3 are) 6, (and 2 are) 8, (set down 8 in the hundreds' place). Sum, 804. %* After some practice in adding, the words within parentheses should be omitted. Rule. Set the numbers below each other in their places ; and add the columns in their order from the units, carrying the tens. 19. 4 1. 100 300 500 700 900 400 600 800 2. 200 500 900 100 300 800 600 700 3. 418 296 306 851 628 435 200 753 4. 524 615 500 924 705 396 527 713 ADDITION. EXERCISE IX. 5. 6. 7. 8. 638 793 814 701 800 215 427 593 524 300 324 414 357 618 650 710 184 509 379 327 225 493 800 967 604 215 930 413 593 336 247 258 9. 649 524 700 810 81 47 913 27 10. 547 64 147 291 17 364 84 913 11. 890 47 562 50 900 73 654 209 12. 736 624 93 14 257 39 572 809 13. 365+210 + 93 + 27+110+345+563 + 207+824+85+127. 14. 241 + 56 + 37+26^+357+842 + 506+37+81 + 190+429. 15. 306 + 194+516 + 70+7 + 829+593+601 + 72+720+18. 16. 501+600+60+372+144+11 + 111+29 + 360+306+71. 17. 76+706 + 760+370 + 307 + 37+377+84 + 804+840+9. 18. 275 + 360+910 + 989+724+57+507+37+7+190+273. 19. 188+560+108+506+56 + 15 + 7+180+18+56 + 566. 20. 673 + 840+737+928+517+349 + 210+500+618+819. 21. 307 + 509+910+117+250+638 + 356+951+117+89 22. 15 + 27+119 + 94+101 + 709 + 364 + 87+2 + 370+241. 23. 293 + 18+573+194+346 + 504+673+936 + 19 + 207. 24. 64 + 604 + 406 + 600 + 640 + 460 + 46 + 83 + 803 + 830. 25. 199 + 96 + 737+307+516+93+7+16 + 738+259+59. EXERCISE X. 1. 2. 3. 4. 5. 6. 7. 8. 9. 1,000 1000 5000 7000 1896 4567 8456 2408 9406 2,000 1100 500 700 1304 8432 7349 5493 1250 4,000 1200 4000 70 1940 9064 9118 9621 6430 6,000 1300 40 7 1284 2345 2565 8504 8094 8,000 1400 800 600 1700 7298 3894 7632 5432 9,000 1500 9000 4000 1676 5934 5248 4562 8006 7,000 1600 5000 900 1864 6309 7348 3901 9210 5,000 1700 600 6000 1547 7124 9176 2008 5090 10. 11. 12. 13. 14. 15. 16. 17. 3476 2930 8046 10,000 30,000 70000 80000 27,300 593 456 810 30,000 40,000 30000 500 34,000 24 3948 9 50,000 70,000 6000 60 26,900 896 27 9421 90,000 80,000 200 50000 84,200 7208 639 39 80,000 10,000 8000 9000 53,700 5009 7204 840 40,000 30,000 90000 40 85,600 648 408 7240 20,000 60,000 600 30000 28,400 8 3072 384 50,000 50,000 50000 700 61,060 ADDITION. 2; 18. 19. 20. 21. 22. 23. 24. 43,214 73059 83426 29070 45623 82472 19465 28,970 84320 34924 50846 72020 846 3947 36,429 92000 85241 63147 93647 9701 64 82,456 84372 12345 94621 804 35624 94702 93,484 50028 66666 80403 9562 256 876 21,086 90200 93002 70002 93 7 5724 73,481 89301 47020 70020 84756 9470 12730 18,498 56238 13076 70200 7250 85064 9400 25. 26. 27. 28. 29. 30. 100,000 300,000 400000 648,724 910,317 542300 300,000 200,000 8000 720,720 843,256 272484 700,000 700,000 90 843,843 123,000 364862 800,000 60,000 900 920,000 456,700 127859 400,000 50,000 9000 647,000 506,840 730640 900,000 500 80000 564,300 920,100 827938 500,000 800,000 800000 734,310 800,701 910400 600,000 500,000 60000 173,094 308,452 478915 81. 843 + 2465 + 724+17+10934+59470 + 107+20094 + 800. 32. 927 + 250 + 3070 + 601 + 38 + 731 + 1 456 + 1 001 + 27 + 374. 33. 493 + 913 + 67 + 500+610+1100 + 1420 + 3706 + 3076+3760. 34. 39 + 280 + 563 + 730+525 + 3482 + 79 + 2496 + 7314+326 + 89. 35. 470+1493 + 293 + 674+825 + 300 + 93 + 1910+2564+836 + 932 36. 9246 + 29805 + 367934 + 39 + 493 + 9 + 90 + 49321 + 7007. 37. 8439 + 7246 + 297 + 800 + 2094 + 73825 + 493 + 12345 + 936. 38. 4731 + 8472+938+76+3938 + 425 + 18 + 967+2005+6790. 39.4901 + 829 + 736 + 90 + 894 + 3247 + 9694+8482+386. 40. 7000 + 770 + 9382 + 54 + 504 + 5004 + 5040 + 5400 + 7054. 41.348 + 7 + 77+777 + 7777 + 77777+9 + 49+17248 + 34. 42. 2693 + 301 + 4 + 404 + 39456 + 327 + 999 + 45602 + 18. 43. 24962 + 376 42 + 4936 + 2754 + 930 + 18500 + 2590 + 196. 44. 93642 + 80010 + 930 + 18275 + 60600 + 66000 + 60060. 45. 7285 + 93271 + 893 + 7249 + 90000 + 18506 + 375 + 9640. 46. 8546 + 2764 + 94681 + 27600 + 9300 + 71486 + 8206 + 9. 47. 45894 + 318 + 7462 + 80001 + 90309 + 7402 + 70906. 48. 437 + 938 + 94 + 7300 + 1805 + 72468 + 79005 + 9406 + 50. 49. 6293 + 946 + 8001 + 92465 + 716 + 24070 + 807 + 5005 + 397. 50. 5484 + 29367 + 937056 + 720000 + 804906 + 100000 + 9040. 51 . 249356 + 730854 + 272494 + 800800 + 549304 + 20400 + 701. 52. 42836 + 90045 + 89362 + 5279 + 7264 + 7649 + 1200 + 937. 53. 5000 + 50000 + 50 + 505 + 5050 + 5 + 555 + 55555 + 550. EXERCISE XI. Below the sum of the following numbers, write the uppermost, and add again ; below that sum write the second from the top, and add again ; continue the addition in this way till all the numbers are taken in, and find the sum. 24 ADDITION. 1. 235 + 196 + 450 + 600 + 801. 2. 342 + 94 + 502 + 86 + 300. 3. 279 + 50 + 116 + 270 + 207. 4. 100 + 50 + 322 + 901+626. 5. 736 + 941 + 257 + 509 + 316. 6. 241+80 + 173 + 428 + 299. 7. 864 + 731 + 279 + 333 + 67. 8. 420 + 204 + 176 + 815 + 700. 9. 304 + 430 + 82 + 73 + 371. 10. 536 + 801 + 78 + 306 + 420. 11. 216 + 39 + 500 + 493 + 811. 12. 340 + 610 + 93 + 217 + 536. 13. 117 + 711+270 + 207 + 453. 14. 820 + 304 + 916 + 732 + 564. 15. 936 + 576 + 429 + 827 + 517. 16. 320 + 600 + 66 + 308 + 201. 17. 524 + 47 + 39 + 809 + 468. 18. 279 + 320 + 809+543 + 397. EXERCISE XII. 1. 2. 3. 4. 5. 238946 900500 1,000,000 8000000 3,564,236 72400 2736 3,000,000 800000 2,564,304 930 93 8,000,000 80000 2,197,629 645046 84293 4,000,000 8000 8,46D,038 8434 701 856 6,000,000 800 7,382,0<J3 67 73900 7,000,000 80 2,946,904 93248 2784 9,000,000 90000 3,842,460 100484 932043 2,000,000 7000000 8,080,803 6. 7. 8. 9. 10. 3456729 9203564 37,240,000 72,483,624 193,700,070 3040506 964383 93,280,000 8,734,724 270,937,000 3004005 728 87,200,400 9,328 384,256,070 3000400 92100 93,400,860 904,374 930,184,293 2790364 8056720 85,085,023 87,208,936 127,249,130 8710800 5296 62,473,903 97,318 147,234,876 56231*33 931724 24,084,573 9,433,729 310,249,364 7703804 8403203 16,946,004 47,082,970 172,849,564 11. 1234567 + 7238049 + 3947246 + 8420800 + 9220000. 12. 8004930 + 12340 + 7248436 + 9436 + 87 + 72456 + 9384567. 13. 72483624 + 8734724 + 9328 + 904374 + 87208936. 1 4. 27007<~70 + 2700707 + 94302 + 734 + 85693 + 9438729. 15. 37248734 + 946432 + 87324 + 9256491 + 80724300. 16. 1 25000890 + 700700700 + 193299870 + 240019000. 1 7. 738456938 + 248724807 + 301234563 + 384965724. 18. 2000000 + 7304524 + 5428946 + 7289476 + 1S0050 + 72004. 19. 47849562 + 93859627 + 2507923 + 804974 + 2904 + 93006. 20. 192196924 + 534920815 + 8256293 + 79000600 + 180000018. EXERCISE XIII. . 1. John has 38 marbles ; he buys 20 more, wins 17, and gets 11 from a friend. How many has he now ? 2. In a school, the first class has 15 scholars, the second 24, the third 27, the fourth 30, and the fifth 31. How many scholars are in the school ? 3. If I pay 8 shillings for bread, 14 shillings for tea, 7 shillings for sugar, and 11 shillings for butter and cheese ; how many shillings do I pay? SUBTRACTION. 25 4. In a wood there are 41 oak-trees, 18 firs, 63 beeches, and 9 elms. How many trees in all ? 5. A traveller went 110 miles by train, 62 miles by steamer, 17 miles by coach, and then he had to walk 2 miles. What was the length of his journey? 6. England has 52 counties, Scotland 33, and Ireland 32. How many counties in the whole ? 7. A class of 26 pupils receives 14 new ones. How many pupils has it now ? .8. Three apple-trees in a garden were shaken for fruit : if one gave 516 apples, and the other two 620 each, how many apples did they give in all ? 9. Three omnibuses started on a pleasure-trip : one carried 23 per- sons, the second 32, and the third 26. If 4 were taken up by the way, how many persons were there in the party ? 10. A grocer pays 140 for shop rent, 37 for taxes, 11 for rent of cellars, and he spends 75 on repairs. What is the whole expense ? 11. In a railway train there were 79 first-class passengers, 101 second-class, and 249 third-class. How many passengers in all ? 12. When will a boy born in 1855 be 69 years old ? 13. From Glasgow to Stirling is 30 miles, from Stirling to Perth 31, from Perth to Aberdeen 90. How far from Glasgow to Aberdeen ? 14. A merchant owes to one creditor 4275, to a second 531, to a third 300, and to a fourth 3005. How much does he owe ? 15. A basket of eggs contains 232, another contains 35 more than the first, and a third 101 more than the second. How many eggs in all? %* Only a few problems of the very simplest kind are presented at this stage : the pupil will be able to continue them to more advantage when he has learnt the four elementary rules. See Ex. 55. 23. SUBTRACTION. Ex. Of 689 trees in a park, 327 were cut down. How many remained standing ? Here we have to find the difference between two given num- bers, or what remains when the less is taken from the greater. The greater of the two numbers is called the Minuend, which means the number to be diminished ; the less is called the Sub- trahend, which means the number to be taken away. The number which remains is called the Difference or Re- mainder. The process of finding it is Subtraction ; called, when the things are of one kind, as here, Simple Subtraction. The sign of Subtraction is (minus) ; thus 2 1 is 1. We cannot find the difference between 689 and 327 at once ; the numbers are too large. We must, therefore, subtract them in parts ; for which purpose we must learn the subtraction of the first nine numbers. 26 SUBTRACTION. Subtraction Table. 1 from 2 from 3 from 4 from 5 from 6 from 7 from 8 from 9 from 2 is 1 3 is 1 4 is Ij 5 is 1 6 is 1 7 is 1 8 is 1 9 is 1 10 is 1 3... 2 4... 2 5 ... 2 6 ... 2 7 ... 2 8... 2 9 ... 210 ... 2 11 ... 2 4... 3 5 ... 3 6... 3 7 ... 3 8... 3 9 ... 310... Sill ... 3 12... 3 5... 4 6 ... 4 7... 4 8... 4 9 ... 4 10 ... 4 11 ... 412 .. 4 13... 4 6... 5 7 ... 5 8 ... 5 9 ... 5 10 ... 5 11 ... 512 ... 5J13 ... 5 14 ... 5 7 ... 6 8 ... 6 9 ... 6 10 ... 6 11 ... 6 12... 613 ... 614... 615 ... 6 8... 7 9 ... 710... 7|11 ... 712... 7 13... 714... 715... 716 ... 7 9... 810... 8|11 ... 8I12... 813... 8 14 ... 8 15 ... 816 ... 8 17 ... 8 10 ... 911 ... 9jl2 ... 9,13 ... 914 ... 915 ... 9 ( 16 ... 9J17 ... 9J18 ... 9 EXERCISE I. 1. Repeat the several columns backwards by odds by evens. 2. Subtract the units in each column from its highest number. 3. 3 from 8 leaves ? 4 from 13 leaves ? etc. 4. 9 less 2 less 3 is ? 17-8-4 is ? etc. 5. To 7 add 3 and take away 4? 9 + 8-2-2 is ? 6. From 5 books take 2, and how many remain ? John had 6 marbles ; if he lost 3 and then 1, how many had he ? Jane has 7 pence ; if she gets 6 pence more and gives away fourpence, what has she now ? etc. 7. Write down the columns of the Table in order. 24. Subtraction of Numbers of Two Places. The Table given above serves also for the subtraction of tens ; thus : If 1 from 2 is 1, 1 ten from 2 tens is 1 ten, or 10 from 20 is 10. If 1 from 3 is 2, 1 ten from 3 tens, is 2 tens, or 10 from 30 is 20. Etc. etc. etc. If 2 from 3 is 1, 2 tens from 3 tens is 1 ten, or 20 from 30 is 10. If 2 from 4 is 2, 2 tens from 4 tens is 2 tens, or 20 from 40 is 20. Etc. etc. etc. EXERCISE II. Perform Ex. i. with tens instead of units. Ex. A woman had 76 eggs in a basket ; if she sold 34, how many had she remaining ? Set down the subtrahend below the minuend in 76 its place ; then, subtract the places in their order. 34 4 from 6 is 2 units ; set down the 2 in its place. 3 from 7 is 4 tens ; set down the 4 in its place. 42 Total difference, 42. SUBTRACTION. 27 To prove the result, add together the subtrahend and the difference ; the sum should be the minuend, since what is taken away from a number and what is left of it make up between them the whole number. . EXEKCISE III. (1.) (2.) (3.) (4.) (5.) (6.) (7.) (8.) (9.) (10.) (11.) (12.) 84 56 76 48 59 37 29 70 86 91 64 73 32 24 36 25 32 21 19 30 20 31 20 52 13.47-24" 16.39-19 19.81-41 22.85-42 14. 78-51 17. 40-20 20. 56-36 23. 71-31 15. 63-30 18. 93-63 21. 78-47 24. 99-57 25. Though the minuend must always be greater than the sub- trahend, any place of the minuend except the highest may be less than the place below it of the subtrahend. Ex. A teacher has 45 steel pens ; if he distributes 29 to his class, how many are over ? 9 from 5 cannot be taken ; change one of the tens into units, making 15 units in all ; 9 from 15 is 6 45 units, set down the 6 in its place. 29 2 from 3 (the 3 tens remaining) is 1 ten ; set down the 1 in its place. 16 Total difference, 16. Rule. Write the less number under the greater in its place ; subtract the columns in their order beginning with the units' ; change one of the next highest name when necessary. Or thus, 1 9 from 5 cannot be taken ; add 10 units to the 5, 45 making 15 in all ; 9 from 15 is 6 units. 29 Add 1 ten to the 2 tens ; 3 from 4 is 1 ten. Total difference, as before, 16. 16 In adding 10 units to the minuend and 1 ten to the subtra- hend, we have added the same number to both. This does not alter their difference ; but makes it easier to find, by keeping each place of the minuend greater than the place below it of the subtrahend. Rule. Write the less number under the greater in its place ; subtract the columns in their order beginning with the units' ; add ten to any place of the minuend which is less than the place below it of the subtrahend, and one to the next place of the subtrahend. i Both methods of subtraction are given ; the teacher may choose either. 28 SUBTRACTION. EXERCISE IV. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. a. 35 47 53 64 71 60 82 91 47 24 63 30 44 28 34 41 17 39 27 35 49 29 35 53 19 17 45 21 27 9 16 27 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 6. 44 21 43 94 42 76 48 32 51 36 22 74 52 81 34 45 18 12 24 47 25 39 29 17 37 17 13 49 26 39 27 19 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. c. 53 70 42 80 43 30 52 63 74 85 96 97 50 32 43 77 18 43 19 37 19 12 24 34 45 67 29 38 26 27 17 58 49. 66-17 57. 21-13 65. 74-16 73. 44-27 50. 47-23 58. 38-19 66. 81-25 74. 58-39 51. 23-14 59. 72-43 67. 62-37 75. 86-48 52. 55-27 60. 83-54 68. 50-23 76. 90-54 53. 70-34 61. 51-26 69. 27-9 77. 93-65 54. 84-27 62. 66-37 70. 34-15 78. 45-29 55. 95-46 63. 80-43 71. 53-27 79. 74-36 5ti. 60-24 64. 91-54 72. 67-39 80. 82-43 EXERCISE V. Perform tlie above exercise mentally. * # * In doing so, it is more convenient to subtract the tens first, and then the units ; thus in 35-17, 10 from 35 leaves 25, and 7 from 25 leaves 18. 26. Subtraction of Numbers of One or more Periods. The Table given, sect. 23, serves also for the subtraction of hundreds, thousands, &c. ; thus : If 1 from 2 is 1, 1 liund. from 2 hund. is 1 himd., or 100 from 200 is 100. If 1 from 3 is 2, 1 Lund, from 3 hund. is 2 hund., or 100 from 300 is 200. Etc. etc. etc. If 2 from 3 is 1, 2 hund. from 3 hund. is 1 hund., or 200 from 300 is 100. If 2 from 4 is 2, 2 hund. from 4 hund. is 2 hund., or 200 from 400 is 200. Etc. etc. etc. EXERCISE VI. Perform Ex. i. with hundreds instead of units. Ex. 1. Of 689 trees in a park, 327 were cut down : how many remained standing? 7 from 9 is 2 units ; set down the 2 in its place. 2 from 8 is 6 tens ; set down the 6 in its place. 327 3 from 6 is 3 hund. ; set down the 3 in its place. Total difference, 362. 362 SUBTRACTION. 29 Ex. 2. How much greater is 6073 than 484 ? In this example, there is a cipher in the minuend, and the highest place of the minuend has no place below it in the subtrahend. 4 from 13 is 9 for the units' place. 6073 8 from 16 (changing one of the next highest name, 484 which is thousands) is 8 for the tens' place. 4 from 9 (the 9 hundreds remaining when the one 5589 thousand was changed) is 5 for the hundreds' place. from 5 is 5 for the thousands' place. Or thus : 4 from 13 is 9 for the units' place. 9 from 17 is 8 for the tens' place. 5 from 10 is 5 for the hundreds' place. 1 from 6 is 5 for the thousands' place. EXERCISE VII. 1. 2. 3. 4. 5. 6. 7. 8. 9 796 805 909 483 857 564 769 960 637 454 403 100 150 724 203 456 500 415 10. 758-342 13. 576-420 16. 7345-5135 11. 975-600 14. 874-574 17. 8500-7000 12. 856-326 15. 716-516 18. 2021-1020 EXERCISE VIII. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 547 635 248 429 511 924 700 801 540 707 800 600 219 427 154 274 364 519 451 605 229 593 209 405 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 713 391 420 706 300 401 535 297 316 02 732 194 256 98 301 279 107 208 328 198 49 541 342 94 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 3429 5642 7396 4524 8527 5418 4000 5040 6080 9004 2763 3804 5409 2790 6050 2980 2534 3956 2500 5084 5. 36. 37. 33. 39. 40. 41. 42. 43. 44. 7320 8074 2094 7000 5484 9302 7549 1368- 2008 7309 2496 1943 859 3456 2390 1903 5840^,66^.1123 978 Ip if iv ii7 V /> , oar 30 SUBTRACTION. 45. 407-298 46. 630-450 47. 275-87 48. 116-58 49. 730-5G3 50. 805-496 51. 357-192 52. 207-84 53. 476-189 54. 520-218 55. 600-315 56. 2809-939 57. 7340-2093 58. 9008-572 59. 1009-450 60. 7084-3921 61. 8000-1090 62. 5009-3094 63. 9101-9011 64. 7308-5904 65. 8234-4731 66. 2890-1936 EXERCISE IX. \* In the following, find the first remainder less than the subtrahend. 28. 1. 402-86 2. 530-105 3. 736-209 4. 900-121 5. 437-99 6. 215-67 7. 600-143 8. 816-197 9. 701-156 10. 2760-672 11. 8207-1938 12. 6094-856 13. 9400-2763 14. 8405-1504 15. 3091-750 16. 7463-1976 17. 5000-987 18. 5185-1978 19. 7320-2094 20. 9017-1853 1. 45060 29360 2. 38905 19450 EXERCISE X. 3. 4. 5. 27936 84571 73021 10007 25038 49950 6. 7. 45239 84901 29308 56402 378923 194033 14. 2567283 730946 934856 256094 15. 45070134 29098040 10. 734085 508506 11. 400000 40401 12. 501020 392406 23900140 4015002 17. 50000014 6010305 13. 276408 120394 18. 100200300 100199025 19. 25678-19341 20. 38056-9456 21. 45804-993 22. 50600-5600 23. 89476-4890 24.793246-45600 25.840300-524080 26. 60S409- 27. 900000- 28. 257931- 29. 456890- 30. 8409302- 81. 10000000- 32.57340506- 93560 33. 90909 34. 80002 35. 193456 36. 908567 37. 1001001 38. 8530205 73894219-25934764 170170170-7107100 59340947-20560724 123456789-98764532 10000000-100000 500500500-650650 %* In the following, find the first remainder less than the subtrahend. 39. 56030-9807 43. 60930-9493 47. 730294-165085 40. 10101-3427 44. 127936-29647 48. 100901-10192 , 41. 27092-5083 45. 982401-109472 49. 605090-92071 42. 47138-7509 46. 273408-84279 50. 400000-101010 29. EXERCISE XI. 1. Count back by twos from 100, from 101. 2. Count back by threes from 102, from 101, from 100. 3. Count back by fours from 100, from 101, from 102, from 103. 4. Count back by fives from 100, 101, 102, 103, 104. 5. Count back by sixes from 102, 101, 100, 103, 104, 105. 6. Count back by sevens from 105, 104, 103, 102, 101, 100, 106. 7. Count back by eights from 100, 101, 102, 103, 104, 105, 106, 107. 8. Count back by nines from 108, 107, 106, 105, 104, 103, 102, 10L ** This and the following Ex. should be practised along with the foregoing. SUBTRACTION. 31 EXERCISE XII. 1. 8+2+9-5-4+1-3 + 8+2-7-1 + 3 + 6-4- 2. 7 + 4-3+5+7-5 + 9 + 6-7-3+9 + 4-9-1 + 8 + 5-7-3. 3. 15-8 + 9-4+9+5-3-7 + 4-5 + 10 + 20-11-7 + 4 + 8-5-10. 4. 22 + 8-11-4+8 + 4-7-1 + 9 + 4-3 + 2-7+9-4-3 + 6 + 7-5. 5. 40+3-7-4+20-7-9-4 + 8-7 + 6-8 + 10-5-7-2 + 8 + 10. 6. 14+3-9+10-6-2 + 20-5 + 9-7-10 + 11 + 11-8 + 13-4-12. 7. 36 + 9-4-8 + 2+5 + 9-12-10 + 7 + 9-4-8 + 5 + 4-6-8 + 20. 8. 19 + 9-5-8 + 7 + 10-5-11 + 20-11 + 9+4-5+5-4 + 9-7-4. 9. 28 + 10+7-12-10 + 6-7-4+3 + 1-9-8 + 11 + 5-7-9 + 8 + 4. 10. 50 + 10-20+30+16-10-10 + 20+7-10 + 20 + 50-30-7-10. 11. 49-5 + 12-9 + 16-10 + 8 + 13-5-8-7 + 11 + 7 + 15-30-10 + 9. 12.- 53 + 8-11+5 + 9-14-15 + 9 + 30-4-9 + 12 +20-5 + 16-13- 9. Etc. etc. etc. EXERCISE XIII. How many are 37 - 29 + 48 - 33 + 79 - 15 ? Here, instead of subtracting 29 from 37, then adding 48, and so on, it is shorter to add together the numbers which are +, then add to- gether the numbers which are , and find the difference of the two Bums, thus : 37 -29 For it is the same thing whether, in find- + 48 - 33 ing 9 - 2 - 2, we say 2 from 9 is 7, 2 from + 79 - 15 7 is 5 ; or 2 and 2 are 4, 4 from 9 is 5. 164 -77 is 87 j.. 125+37-84-10+76 + 53-101+56+279-184-45+293. 2. 74-40 + 51-9 + 29 + 16-19-5 + 36-27 + 40 + 11. 3. 18+15-10 + 40 + 36-19-14 + 23-39 + 20 + 16-19. 4. 56 + 20-43-27 + 39 + 24-31 + 64-45 + 21 + 10-34. 5. 90 + 45 + 16-49-51 + 6-15 + 39-60 + 49 + 53-19. 6. 36-19+53-29 + 36-24-11-B 4 64 + 17-24-9 + 14. 7. 49+36-29-14+20+36-18-y +25 + 84-59-27+40. 8. 74 + 52-63-10 + 29 + 37-45-37+22-51 + 69-19 + 26. 9. 192-56-14+58 + 213-191 + 64-49+346-154-48 + 90. 10. 724-593 + 824-48+93 + 702-500+293-59-73 + 256-100. 11. 50004 - 8456 - 401 + 4592 + 9400 - 10100 + 734 - 809. 12. 29340 - 4560 - 9390 + 7248 - 15600 + 93402 - 56840. 30. EXERCISE XIV. 1. A woman went to market with a basket of eggs containing 342 : if she sold 192, how many did she bring back ? 2. John has 95 nuts, but gives 37 to William. How many does he keep? 3. A teacher gives out pens to a class of 60 scholars, but the box lias only 37. How many does he want ? 4. A cheese weighs 78 pounds. How much heavier is it than an- other which weighs only 47 pounds ? 5. A tradesman owes 260, but he has only 137. How much does he require to pay his debts '/ 32 SUBTRACTION. 6. A cask of sugar contains 539 pounds' weight. How much must be sold to leave 257 pounds ? 7. James has 24 marbles, and his brother gives him 37. How many must he buy to make up 100 ? 8. If a school has 374 scholars, of whom 27 are in the first class, and 32 in the second ; how many are in the other classes together? 9. A green-grocer received a basket of apples and pears, 264 in all : 157 were apples ; how many were pears ? 10. A house is worth 520, but it will cost 84 to repair it. How much should it be sold for? 11. Edinburgh to Dunbar is 29 miles, and Edinburgh to Berwick is 57 miles. How far from P-unbar to Berwick ? 12. A tradesman earns 16s. a week, and spends 13s. How much does he save in four weeks ? 13. A farmer had in his yard 31 fowls, 17 geese, 24 turkeys, and his ducks made up the entire number of his poultry to 87. How many ducks had he ? 14. How much of 385 yards remains if 93 yards be cut away from the piece ? How often may 93 yards be cut away, and what will remain ? 15. A train started with 374 passengers. At the first station 16 went out and 9 came in ; at the second, 11 went out and 25 came in ; at the third, 3 went out. How many passengers left the train at the terminus ? See Ex. 55. 31. MULTIPLICATION. Ex. Five boxes of oranges contained 125 each, how many oranges were there in all ? Here we have to find a number equal to 125 repeated 5 times. We could find that by adding 125 to itself 5 times ; but a shorter way is to multiply 125 by 5. The number to be repeated is called the multiplicand. The number of times it is to be repeated, multiplier. Both are sometimes called the . . factors. The result is called the ... product. The process is called multiplication ; and, when the multipli- cand is of one kind as here, simple multiplication. The sign of multiplication is X (multiplied by) ; thus 2X2 are 4. We cannot find how much 5 times 125 is "by one step ; the multiplicand is too large. We must therefore do it in parts ; for which purpose we must learn the multiplication of the first nine numbers. 32. MULTIPLICATION. Multiplication Table. 33 2 times 1 are 2 2 ... 4 3 ... 6 4 ... 8 3 times 1 are 3 2 ... 6 3 ... 9 4 12 4 times 1 are 4 2 ... 8 3 ... 12 4 ... 16 5 times 1 are 5 2 ... 10 3 ... 15 4 ... 20 6 times 1 are 6 2 ... 12 3 ... 18 4 ... 24 7 times 1 are 7 2 ... 14 3 ... 21 4 ... 28 5 ... 10 6 ... 12 7 ... 14 5 ... 15! 5 ... 20 6 ... 18! 6 ... 24 7 ... 21 7 ... 28 5 ... 25 6 ... 30 7 ... 35 5 ... 30 6 ... 36 7 ... 42 5 ... 35 6 ... 42 7 ... 49 8 ... 16 9 ... 18 10 ... 20 11 ... 22 12 ... 24 8 ... 24 8 ... 32' 8 ... 40 9 ... 27 9 ... 36 9 ... 45 10 ... 3010 ... 4010 ... 50 11 ... 3311 ... 4411 ... 55 12 ... 3612 ... 4812 ... 60 8 ... 48 9 ... 54 10 ... 60 11 ... 66 12 ... 72 8 ... 56 9 ... 63 10 ... 70 11 ... 77 12 ... 84 8 times 1 are 8 2 ... 16 3 ... 24 4 ... 32 9 times 1 are 9 2 ... 18 3 ... 27 4 ... 36 10 times 1 are 10 2 ... 20 3 ... 30 4 ... 40 11 times 1 are 11 2 ... 22 3 ... 33 4 ... 44 12 times 1 are 12 2 ... 24 3 ... 36 4 ... 48 5 ... 40 6 ... 48 7 ... 56 8 ... 64 9 ... 72 10 ... 80 11 ... 88 12 ... 96 5 ... 6 ... 7 ... 8 ... 9 ... 10 ... 11 ... 12 ... 45 54 63 72 81 90 99 108 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . .. 50 .. 60 .. 70 .. 80 .. 90 .. 100 .. 110 .. 120 5 6 7 8 9 10 11 12 ... 55 ... 66 ... 77 ... 88 ... 99 ... 110 ... 121 ... 132 5 ... 60 6 ... 72 7 ... 84 8 ... 96 9 ... 108 10 ... 120 11 ... 132 12 ... 144 *** This Table should be learnt first in lines even along, then in lines up and down. The pupil should practise it daily till he has it thoroughly at command. EXERCISE. I. Bf. 1. Repeat the several lines even along ; backwards j by odds ; by evens. 2. Repeat the lines up and down ; backwards ; by odds ; by evens. 3. 4 times 5 are ? 6 times 9 are ? 8 times 7 are ? etc. 5 times 4 are ? 9 times 6 are ? 7 times 8 are ? etc. 4. How many fingers have 8 boys? How many wheels have 9 carts ? How many days have seven weeks ? How many farthings have four pence ? How many units in 5 tens ? How many marbles have 9 boys with 11 each? What cost 6 oranges at 2 pence each? 7 fowls at 3 shillings each ? etc. 5. Name two factors of 18, 24, 96, etc. 6. How many times 7 is 63 ? 21 ? 70 ? etc. 7. 36 is 9 times ? 72 is 6 times ? etc. 8. 8 times 6 + 2 are ? 5 times 8 with 9 added are ? ete. 4 times 12 less 9 are ? 7 times 5 - 6 are ? etc. 9. 2 times 4 and 3 times that are ? etc. 6 multiplied twice by 2 are ? etc. 10. Write down the several columns of the Tab! 3. 34 MULTIPLICATION. OO. The Table given above serves also for the multiplication of tens, hundreds, etc. Thus If 2 times 1 are 2, 2 times 1 ten are 2 tens, or 2 times 10 are 20. If 2 times 2 are 4, 2 times 2 tens are 4 tens, or 2 times 20 are 40. Etc. etc. etc. If 3 times 3 are 9, 3 times 3 tens are 9 tens, or 3 times 30 are 90. Etc. etc. etc. EXERCISE II. Perform Ex. i. with tens in the multiplicand. If 2 times 1 are 2, 2 times 1 h. are 2 h., or 2 times 100 are 200. If 2 times 2 are 4, 2 times 2 h. are 4 h., or 2 times 200 are 400. Etc. etc. etc. EXERCISE III. Perform Ex. i with hundreds in the multiplicand. Multiplication by Units. Ex. Five boxes of oranges contained 125 each, how many oranges were there in all ? Set the multiplier below the multiplicand in its place ; then, multiplying each place in its order, 5 times 5 are 25 units j set down 5 units and carry 2 tens. 125 5 times 2 are 10, and 2 are 12 tens ; set down 5 2 tens and carry 1 hundred. - 5 times 1 are 5, and 1 are 6 hundreds. 625 Product, 625. Hule. To multiply by units, multiply each place of the multiplicand in order, carrying tens. The answer may be proved by adding the multiplicand to itself 5 times ; the sum should be the same as the product. Or we may multiply by 4, the number next below the multiplier, and add the multiplicand to the product. EXERCISE IV. 1. Multiply the following numbers by 2, 3, etc., to 12, in order : 13 21 31 41 51 61 71 81 91 14 22 32 42 52 62 72 82 92 15 23 33 43 53 63 73 83 93 16 24 34 44 54 64 74 84 94 17 25 35 45 55 65 75 85 95 18 26 36 46 56 66 76 86 96 19 27 37 47 57 67 77 87 97 20 28 38 48 58 68 78 88 98 29 39 49 59 69 79 89 99 80 40 50 60 70 80 90 100 MULTIPLICATION. 3 5 8. Multiply the several columns mentally. 8. 2 times 27 are ? 3 times 32 are ? 4 times 48 are ? etc. 4. Multiply the following numbers by 2, 3, etc., to 12, in order : 1. Ill 11. 893 21. 2461 31. 24682 41. 34194 2. 222 12. 248 22. 5382 32. 74394 42. 21384 3. 333 13. 604 23. 2081 33. 31208 43. 75689 4. 444 14. 573 24. 4095 34. 24295 44. 38472 5. 555 15. 421 25. 2496 35. 19064 45. 29319 6. 666 16. 298 26. 5162 36. 70538 46. 82964 7. 777 17. 157 27. 7349 37. 25819 47. 70109 8. 888 18. 820 28. 8210 38. 39147 48. 10840 9. 999 19. 659 29. 9347 39. 16731 49. 30028 10. 427 20. 416 30. 1924 40. 42858 50. 90084 %* This exercise Is designed to be performed orally from the book as well as on slate. Multiplication by Factors. .Ek Multiply 248 by 24. Since 24 is 6 times 4, we multiply by 24, if 248 we multiply first by 6, and then that product by g 4 ; thus : r The result may be proved by multiplying by /? 3 and 8, or by 2 and 12 ; which are also factors _ of 24, and which should therefore give the same 5952 product. A number like 24 which is made up of factors (other than 1) is called a composite number. A number like 7, 11, or 23, which is not made up of factors, is called a prime number. Multiplication by two factors may be used in the case of all composite multipliers between 12 and 144. Practice in multiplying will show the pupil that three factors may often be used for a multiplier with advantage ; thus, 252=4X7X9. EXERCISE V. Multiply, using factors : 1. 536x14, 15, 21, 22. 6. 4732x77, 81. 84. 2. 270x25, 27, 28, 32. 7. 2096x88, 96, 99. 3. 905x33, 42, 44, 45. 8. 8405x108, 121, 132, 4. 827 x 54, 55, 56. 9. 7289 x 144, 160, 270. 5. 638x63, 66, 72. 10. 8175x420, 840. 11. 3497 x 16, 18, 48, 72, in two ways. 12. 7302 x 24, 36, in three ways. " 36 MULTIPLICATION. 36. Multiplication by more than One Place. A cipher annexed to the right of a figure increases its value 10 times, that is, multiplies it by 10. Therefore, to multiply by 2 tens or 20, multiply by 2, and annex the cipher ; to mul- tiply by 30, multiply by 3, and annex the cipher ; and so on. Similarly to multiply by 200, multiply by 2, and annex two ciphers ; to multiply by 300, multiply by 3, and annex two ciphers ; and so on. Rule. To multiply by tens, hundreds, etc., multiply by the left-hand figure, and annex the ciphers. EXERCISE VI. 1. Multiply the columns in Ex. iv. by 20, 40, 50, 90. 2. Multiply tlie same columns by 300, 600, 700, 800. O 7 Ex* A book contains 356 pages, and each page 237 words : how many words are in the book ? Set the multiplier below the multiplicand in its place ; then multiplying by the 7 units, 237 we have ..... 2492 Multiplying by the 3 tens, we have . . 10680 Multiplying by the 2 hundreds, we have . 71200 Product by whole mutiplier is . . 84372 The result may be proved by interchanging . the multiplier and multiplicand, that is, multiplying 237 by 356 ; which will give the same product. Rule. To multiply by a number of several places, multiply by each place in order from the units, and add the several products. V* The pupil may by and by omit the ciphers, denoting the tens and hundreds in the second and third lines of multiplication ; being careful to place the right-hand figure of each line exactly under that place of the multiplier which gives it. Should there be a cipher in the tens or some higher place of the multi- plier, it is simply passed over in multiplying. EXERCISE VII. I 2364 x 29, 37, 43. 5. 8256 x 17, 93, 49. 9. 40001 x 81, 28, 34. i 4328 x 39, 51, 86. 6. 6439 x 38, 57, 61. 10. 73000 x 47, 59, 92. 3. 5936 x 28, 46, 59. 7. 20480 x 71, 43, 53. 11. 90000 x 27, 64, 79. 4. 9320 x 19, 73, 31. 8. 30093 x 98, 83, 78. 12. 70091 x 75, 88, 99 MULTIPLICATION. 37 EXERCISE VIII. 1. 85627x183, 297,403. 2. 47231x245,318,721. 3. 93086x240, 825, 649. 4. 23456x409,207,308. 5. 73610x930, 470,290. 6. 85093x418,738,562. 7. 72170x936,259, 816. 8. 37293x904, 506,801. 9. 80050x629,350,680. 10. 90000x456,789,910. 11. 70700x843, 529,365. 12. 90280x706, 504,209. 13. 456789x297,399, 536. 14. 724936x840,908,273. 15. 459630x364, 814, 518. 16. 536298x230, 563, 720. 17. 210830x821, 913, 713. 18. 914567x439, 546,208., EXERCISE IX. 1. 500606x5423, 6106. 2. 730000x2936, 8492. 3. 700000x4028, 5003. 4. 830830x6300, 7240. 5. 308070x8740, 5007. 6. 934764x23418, 93125. 7. 621930x19728, 73465. 8. 493628x27368, 93480. 9. 840300x19030, 80807. 10. 621934x70029, 54309. 11. 493002x56721, 12765. 12. 2389745x4567, 7394, 6270. 13. 6348576x7321,8492,1029. 14. 2930840x6080, 5090, 7200. 15. 7394900x8936, 2009, 5900. 16. 8002006x7290, 5718, 3290. 17. 7802058x35467,29631. 18. 4932096x84932, 94629. 19. 7007007x93021,80709. 20. 3489493x29100,28101. 21. 9000000x73500,82090. 22. 4290000x80972,50608. EXERCISE X. 1. 25473809x258956,817456. 2. 73890496x483921,293185. 3. 90900900x259671,798491. 4. 25608709x408506,930850. 5. 70409360x273093,129608. 6. 49328914x506090,709080. 7. 82483949x210000,930039. 8. 72340090x724801,520936. 9. 53042485x493094,891172. 10. 73249000x938950,249056. 11. 490562001x362987, 450893. 12. 293904510x450813,920854. 13. 710842930x293050, 493096. 14. 256849361x259928,936190. 15. 209209209x123456,789012. 16. 600040068x900405,908550. 17. 394620100x736493,856190. 18. 824904561 x 437285, 737292. 19. 296382173x555555,505050. 20. 493084095x828561,400800. 38. Squares and Cubes. A figure like this, which has 4 rows of counters, .... each containing 4, is called a square. The .... number of counters we see by counting to be 16 ; that is, the number even along (4) multi- plied by the number up and down (4). Bf. Similarly 7 rows of trees with 7 trees in each would be a square of 49 ; 10 lines of soldiers with 10 soldiers in each line would be a square of 100. When any number is multiplied by itself, the product is called the square or second power of that number. The square of 4 is denoted 4 a . 39. 40. 38 MULTIPLICATION. EXERCISE XI. 1. Repeat the squares of 1, 2, 3, 4, etc., up to 12. 2. Find the squares of 13, 14, 15, 16, 17, 18, 19, 20. 8. Find the squares of these numbers : 1. 784 5. 3456 9. 23456 13. 75423 17. 50005 2. 937 6. 2930 10. 90307 14. 20056 18. 728946 3. 508 7. 4500 11. 58126 15. 90030 19. 809407 4. 610 8. 7000 12. 37000 16. 80705 20. 916738 "When a number is multiplied twice by itself, the product is called the cube or third power of that number ; thus 4X4X4 64. The cube of 4 is denoted 4 3 . > %* This may be illustrated by a small cube of wood, or, better still, by a box of such cubes. EXERCISE XII. 1. What are the cubes of 1, 2, 3, etc., up to 10? 2. Find the cubes of these numbers : 1. 789 4. 4506 7. 12000 10. 67809 2. 405 5. 5730 8. 37100 11. 40506 3. 623 6. 9825 9. 24089 12. 12345 EXERCISE XIII. 1. How many eggs in 16 boxes, each having 96 ? 2. How many pupils in a school which has 7 classes of 23 each ? 3. How many hours in 36 days ? 4. How many pence in 47 half-crowns ? 5. How many oranges, at 15 for a shilling, will 25s. buy ? 6. How long a journey shall I make in 27 days, at 18 miles a day ? 7. How many yards of linen in 387 pieces, each 35 yards ? 8. How many bottles in 45 dozen and 5 ? 9. How many pages in a yearly volume, of which a monthly part has 96 ? 10. What cost a railway 49 miles long, at 4500 a mile ? 11. A postman delivers 29 letters each morning and evening for a week ; how many did he deliver in all ? 12. A pipe pours into a cistern daily 13410 gallons water ; how many gallons will it pour in during November ? 13. A house of five storeys has seven windows in each, and twelve panes of glass in each window ; how many panes of glass are there in all? 14. Three men, in business together, receive 672 each of the profits at the end of the first year ; what were the whole profits ? 15. If a baker reckons 13 to a dozen, how many biscuits does he count to 136 dozen ? 16. A merchant's office occupies 43 clerks at 2 a week each, and 24 at 3 ; what sum is required in a year for their wages ? 17. There are 29 trees in the side of a square plantation ; how many trees has the plantation I 41. DIVISION. DIVISION. 39 Ex. A box of eggs, containing 852, is to be divided amongst a number of families, each getting 6 ; how many families will be served ? Here we have to find how often 6 is contained in 852. We could find that by subtracting 6 from 852 successively till nothing remains, and then counting the number of 6's we have got, but a shorter way is to divide, 852 by 6. The number to be divided is called the dividend. The dividing number is called the divisor. The number of times the divisor is contained in the dividend is called the quotient. The process of dividing is called division ; and, where the dividend is of one kind as here, simple division. The sign of division is -j- (divided by) ; thus, 4 -7- 2 is 2. We cannot find how often 9 is contained in 243 by one step ; the dividend is too large for that. We must therefore do it in parts, for which purpose we must learn the division of the first nine numbers. 42. Division Table. 2 i n 3 i a 4 ] n 5 ii i 6 i: a 7 i n 2 i s 1 3 3 1 L 4 ] LS 1 5 is \ 1 6 i I 1 7 i s 1 4 |% 2 6 < I 8 2 10 2 12 2 14 2 6 3 9 \ \ 12 3 15 3 18 3 21 3 8 >e 4 1 2 i [ 16 4 20 4 24 4 28 4 10 5 1 5 \ 5 20 5 25 5 30 5 35 5 12 6 1 8 ( 5 24 6 30 6 36 6 42 6 14 . 7 2 1 \ r 28 7 35 7 42 7 49 7 16 8 2 4 \ J 32 8 40 8 48 8 56 8 18 9 2 7 < ) 36 9 45 9 54 9 63 9 20 10 3 1( ) 40 10 50 10 60 ] LO 70 10 22 11 3 3 1] L 44 11 55 11 66 1 11 77 11 24 - 12 3 6 IS I 48 12 60 12 72 ] 12 84 12 8 in 9 ID 10 i n 11 ii a 12 ii i 8 is 1 9 is 1 10 i 3 1 11 i 1 1 12 is 1 16 " 2 18 2 20 2 22 2 24 2 24 3 27 3 30 3 33 3 36 3 32 4 36 4 40 4 44 4 48 4 40 45 5 50 5 55 5 60 5 48 e 54 6 60 6 66 6 72 6 56 \\ 7 63 7 70 7 77 7 84 7 64 8 72 8 80 8 88 8 96 8 72 mt 81 9 90 9 99 9 1 08 9 80 M 1C 90 10 100 10 110 1 1 20 . 10 88 11 99 11 110 11 121 1 1 1 32 . . 11 96 IS 108 12 120 12 132 1 2 1 44 . . 12 40 DIVISION. EXERCISE I. Bf. 1. Repeat the lines of this Table up-and-down ; backwards ; by odds ; by evens. 2. Repeat the lines even along in the same way. 3. 2 in 8 is ? 5 in 35 is ? 9 in 72 is - ? etc. 4 in 8 is ? 7 in 35 is ? 8 in 72 is ? etc. 4. How many pence in 8 farthings ? Divide 15 shillings among 5 persons. Divide 40 marbles among 8 boys. How many oranges at 2d. each can I buy with 16 pence ? etc. 5. Write down the several columns of the Table. 4:0. This Table serves also for the division of tens, hundreds, etc. Thus If 2 in 2 is 1, 2 in 2 tens is 1 ten, or 2 in 20 is 10. If 2 in 4 is 2, 2 in 4 tens is 2 tens, or 2 in 40 is 20. Etc. etc. etc. If 3 in 3 is 1, 3 in 3 tens is 1 ten, or 3 in 30 is 10. Etc. etc. etc. EXERCISE II. Perform Ex. i., Nos. 1, 2, 3, with tens in the dividend. If 2 in 2 is 1, 2 in 2 hunds. is 1 hund., or 2 in 200 is 100. If 2 in 4 is 2, 2 in 4 hunds. is 2 hund., or 2 in 400 is 200. Etc. etc. etc. EXERCISE III. Perform Ex. i., Nos. 1, 2, 3, with hundreds in the dividend. 44. Division by Numbers of One Place. Ex. How often is 3 contained in 963 ? Place the divisor to the left of the dividend. 3 in 9 hundreds is 3 hundreds. 3 ]_963 3 in 6 tens is 2 tens. 3~2i 3 in 3 units is 1 unit. Quotient, 321. EXERCISE IV. Divide 1. By 2 : 86, 128, 420, 642, 864, 4806, 6428. 2. By 3 : 63, 96, 123, 249, 630, 963, 6093. 3. By 4 : 84, 168, 244, 488, 804, 884, 4084. 4. By 5 : 105, 155, 250, 355, 505, 4550, 5035. 6. By 6 : 126, 246, 306, 426, 5460, 6048, 12660. 6. By 7 : 147, 217, 357, 714, 6377, 7063. 7. By 8 : 168, 248, 320, 880, 1608, 5680. 8. By 9 : 189, 279, 540, 3609, 4599, 8190. DIVISION. 41 . The places of the dividend do not often contain the divisor evenly ; there is generally a remainder. 2 in 3 is 1 and 1 over 3 in 4 is 1 and 1 over 4 in 5 is 1 and 1 over 5 in 6 is 1 and 1 over EXERCISE V. in 5 is ? in 7 is ? etc. in 5 is ? in 7 is ? etc. in 6 is ? in 7 is ? etc. in 7 is ? in 8 is ? etc. *** The exercise should be continued up to 12 as divisor. 4:6. Ex. 2. A box of eggs, containing 852, is to be divided amongst a number of families, each getting 6 ; how many families will be served ? Set the divisor to the left of the dividend. Then 6)852 6 in 8 hundreds is 1 hundred and 2 hundreds over ; T42 set down the 1 in its place, and change the 2 hun- dreds into tens, making 25 in all. 6 in 25 tens is 4 tens and 1 ten over ; set down the 4 in its place, and change the 1 ten into units, making 12 in all. 6 in 12 units is 2 units. Quotient, 142. Rule. To divide by a number of one place, divide the places of the dividend in order from the highest, carrying the tens. The result may be proved by multiplying the quotient by the divisor ; the product should be the dividend. EXERCISE VI. Divide 1. By 2 : 98, 258, 374, 454, 526, 598, 638, 694, 738, 876, 938, 972. 2. By 3 : 87, 378, 465, 471, 513, 582, 648, 657, 726, 735, 879, 978. 3. By 4 : 96, 492, 536, 548, 620, 676, 768, 792, 860, 892, 948, 956. 4. By 5 : 565, 590, 675, 680, 745, 775, 865, 880, 930, 975, 7345. 5. By 6 : 150, 672, 726, 744, 804, 852, 918, 990, 6834, 8526, 8730. 6. By 7 : 161, 798, 805, 875, 910, 987, 7847, 7952, 8596, 8764, 9233. 7. By 8 : 256, 896, 960, 992, 8976, 9544, 1896, 1944, 2888, 3976. 8. By 9 : 144, 252, 423, 603, 828, 1026, 2160, 3267, 5040, 6543, 7038. 9. By 10 : 730, 840, 9320, 4500, 7310, 2030. 10. By 11 : 748, 396, 594, 286, 7942, 8503, 25894, 92477 56089. 11. By 12 : 348, 564, 936, 3888, 57372, 20928, 3708, 94020, 67308. 47. Ex. How often is 6 contained in 24295 ? Dividing as before, there is a remainder of one 6)24295 after dividing the units. This is annexed to the 4049J quotient with the divisor below in the form , which denotes one-sixth t or the sixth part of one. 42 DIVISION. In multiplying the quotient in this case by the divisor to prove the result, the remainder must be added to the product ; thus, 4049 X 6+ 1= 24295. EXERCISE VII. Divide 1. 2. 3. 4. 5. 6. 1. By 2, 345 467 931 857 1129 2525 2. By 3, 472 305 721 922 2684 7055 3. By 4, 105 653 437 829 5634 8631 4. By 5, 732 482 911 573 8421 7018 5. By 6, 515 833 791 273 5927 6381 6. By 7, 452 635 134 608 3210 7962 7. By 8, 123 537 817 909 4561 8347 8. By 9, 258 316 501 823 7082 1293 9. By 10, 137 259 533 471 2563 9327 10. By 11, 564 800 601 942 3874 6088 11. By 12, 373 529 705 637 1949 2009 48. Division by Factors. In dividing by any composite number up to 144, we may get the quotient by dividing by its two factors successively. E.g., in dividing an apple into 4 parts, we first divide it into 2 parts, then each of these again into 2 parts. Ex. Divide 3568 marbles into parcels of 24. The factors of 24 are 6 and 4. 6 13568 Dividing first by 6, we have for quotient 4 594 4 \ 594 (parcels of 6), and 4 (marbles) over. -j.48 2 1 ^ Dividing next by 4, we have for quotient 148 (parcels of 4 sixes or 24's) and 2 (parcels of 6) over. Adding now the second remainder (2 parcels of 6, or 12 maibles) to the first (4 marbles), we have for total re- mainder 16 marbles : 6X2-f4 = 16. Hence, to get the real remainder, multiply the first divisor by the second remainder, and add the first remainder to the product. If there be no second remainder, the first is the real one. EXERCISE VIII. 1. 234564-14, 15, 21, 22 6. 905036-7-84, 88, 96 2. 37095-25, 27, 28, 32 7. 249076^99, 108 3. 9085133, 42, 44, 45 8. 593250-7-120, 132, 144 4. 8437954, 55, 56, 63 9. 731105-M6, 18, 48, 72, in two ways. 5. 6592766, 77, 81 10. 847644-r-24, 36, in three ways. 49. Division by more than one Place. As a cipher annexed to the right of a figure multiplies it by 10, so a cipher removed from the right of a figure divides the number by 10 : thus, 20 -j- 10 = 2. DIVISION. 43 If the dividend do not end in a cipher, then the figure in the nnits' place is removed for a remainder : thus, 21-7-10 = 2-^. If the divisor contain more tens than one, as 30, divide nrst by 10 as one factor, and then by the other factor, 3 ; that is, remove the units' place of the dividend for the remainder, and divide by the second factor, carrying what is over in this divi- sion to the remainder. Thus, 63 -*- 20 = 3^y ; 73 + 20 = 3 J. To divide by a number of hundreds, remove the two last ciphers of the dividend, or the two last figures of it, for re- mainder, in a similar way. Thus, 200 -i- 100 = 2 ; 564 -i- 200 ^"JTJO* EXERCISE IX. Divide by 10, 30, 50, 70, 90 1. 370 7. 1200 13. 2474 19. 32814 2. 290 8. 6600 14. 3935 20. 56732 3. 835 9. 8800 15. 5066 21. 83940 4. 672 10. 7000 16. 7317 22. 50761 6. 425 11. 4800 17. 8058 23. 69005 6. 901 12. 6300 18. 9720 24. 85436 EXERCISE X. Divide by 200, 400, 600, 800, examples 7-24 in last Exercise. , Ex. How often is 234 contained in 849726 ? 234 in 8 or in 84 cannot be taken, but 234) 849726(363 !,& in 849 (thousands) is 3 (thousands), 702 and 147 (thousands) over. Set down the 3 in the thousands' place of the quotient, and carry the 147 to the hundreds' place, making the next part of the dividend 1477 (hun- dreds) in all. 234 in 1477 (hunds.) is 6 (hundred), and 73 (hunds.) over. Set down 306 the 6 (hunds.) in its place in the 234 quotient, and carry the 73 (hunds.) to the tens' place, making the next 72 part of the dividend 732 (tens) in all. 234 in 732 (tens) is 3 (tens), and 30 (tens) over. Set down the 3 (tens) in its place in the quotient, and carry the 30 (tens) to the units' place, making the next part of the divi- dend 306 (units) in all. 234 in 306 (units) is 1 (unit), and 72 (units) over. Set the 1 (unit) in its place in the quotient. The 72 units are remainder. 44 DIVISION. This form of division, which is required when the divisor contains more than one place, is known as Long Division. EXERCISE XL 1. 370374-25, 37, 43. 2. 298354-34, 49, 51. 3. 73632-1-47, 93, 39. 4. 802944-19, 26, 41. 6. 900004-73, 61, 17. 6. 50032-29, 53, 98. 7. 17918-13, 34, 82. 8. 47320-38, 91, 47. 9. 20971-67, 82, 93. 10. 5428023, 46, 85. EXERCISE XII. 1. 456824-251, 183, 342. 2. 409364-301, 457, 631. 3. 238434-113, 911, 564. 4. 890404-824, 159, 296. 5. 900004-457, 734, 825. 6. 123844-391, 516, 364. 7. 730274-801, 709, 208. 8. 290414-257, 314, 846. 9. 928814-934, 652, 293. 10. 799484-418, 506, 853. 11. 5608024-293, 791, 846. 12. 2935444-151, 258, 174. 13. 8587414-325, 291, 397. 14. 4853614-851, 702, 813. 15. 9341104-561, 582, 738. 16. 5006364-921, 309, 257. 17. 7000004-416, 526, 736. 18. 205428^-901, 754, 815. 19. 9340654-297, 358, 492. 20. 7144084-824, 964, 708. EXERCISE XIII. 1. 74893184-37, 74, 89. 2. 2934821-7-41, 73, 97. 3. 73486404-594, 416, 607. 4. 2684816-5-208, 541, 732. 5. 60845164-2342, 5684. 6. 54031444-9348, 2571. 7. 72561544-3040, 8009. 8. 91446089401, 5008. 9. 82717504-3075, 4908. 10. 91939324-5671, 2943. 11. 573380644-5473, 3024, 9902. 12. 630927064-2931, 4708, 5004. 13. 72491840-7-3040, 8009, 5231. 14. 20018414^-7298, 6804, 77?4. 15. 921006254-5136, 1984, 2875. 16. 800000004-8345, 6205, 7095. 17. 538054484-4001, 8936, 9027. 18. 7300692-7506, 9324. 19. 90000008931, 7295. 20. 82035704583, 9308. 21. 2568903684-28, 79, 35. 22. 9314562044-17, 47, 82. 23. 2490860224-457, 329, 704. 24. 3036067964-293, 718, 274. 25. 7240880434-8561, 2793. 26. 3659057804-5006, 2918. 27. 8543724004-9300, 8540. 28. 2936001704-2005, 7009. 29. 8759127804-3054, 7090. 30. 2934000004-7200, 5090. , To find an Average. Ex. A boy gets 23 marks on Monday, 17 on Tuesday, 28 on Wednesday, 31 on Thursday, 25 on Friday, and 14 on Saturday : what is his average number of marks daily for the week? Here the sum of his marks for the whole week is 138. There is a certain number of marks, which had he got every day of the week, the sum of his marks at the end of the week would DIVISION. 45 have been the same as it is now. That is the number we wish to find. The average of a series of numbers is that number which, if repeated as often as there are numbers, will amount to their sum. It is found by dividing the sum of the numbers by their number ; thus 138-J-6 = 23. EXERCISE XIV. Find the average of the following numbers : 1. 27, 37, 42, 50, 22, 24. 6. 2738, 3624, 3001, 2. 13, 49, 35, 64, 53, 42. 7. 937, 1001, 1100, 1010, 1110. 3. 93, 87, 59, 67, 73. 8. 856, 1533, 930, 1399. 4. 29, 30, 37, 32, 33. 9. 8973, 10704, 9320, 14976, 9999. 5. 125, 250, 315, 193. 10. 27345, 73421, 85648, 79286. - Fractional Multipliers and Divisors. Ex. A train runs 27 miles an hour for 14| hours ; what distance will it go in the time ? 27 The distance is 27 miles repeated 14 times -a and | a time ; which is got by multiplying 27 by 14}. To multiply by j, multiply by 3 and divide 20 the product by 4. Then in multiplying by 14, the right-hand figure of the first line, being units, 27 _ is set in the units' place. 398^ The number f , which is less than 1 is called a fraction. If one is divided into 2 equal parts, each is called a half ; if into 3, each is called a third ; if into 4, a fourth ; and so on. A fraction is denoted by two numbers, the one written below the other ; thus one-half is written J, one-third , one-fourth | ; if more than one part be taken, the upper figure denotes how many, thus three-fourths is written j. The number 14f , which consists of a whole number and a fraction, is called a mixed number. EXERCISE XV. 1. Find one-half of 38, 57, 108, 265, 798, G357. 2. One-third of 51, 252, 254, 768, 784, 8472. 3. One-fourth of 56, 92, 94, 397, 3828, 8927. 4. Multiply by f : 85, 101, 357, 456, 2456, 7530. 5. Multiply by | : 84, 356, 537, 933, 1272, 7000. 6. 8456x41, 6}, 15J, 27$, 139|, 308. 7. 93582 xlOJ, 200*, 750, 30|, 5$. 46 DIVISION. Ex. How often is 29 J contained in 9384 ? The numbers cannot conveniently be 29J 9384 used for divisor and dividend as they 4 4 stand. 117 )37536(320i*A Multiply both by 4, the fraction in the 351 divisor being fourtlis. This will give a ~243 new divisor and dividend four times greater . than those given ; but which will be free from fractions, and will give the same quotient. EXERCISE XVI. 1. 3482-^-3$, 6, S. 6. 900536-^-12*, 74$, 256$. 2. 8506-Hi|, 5 94. 7. 852079-f-5A, 301, 3651. 3. 72584-7-27$, 54f, 79. 8. 205930-^-152, 85$ , 365J. 4. 59321 H-19J, 68^, 128$. 9. 730526-~29i, 217$, 8342^. 5. 80999-^-15$, 265, 94g. 10. 45067824-i-14|, 58$, lOOfc. Multiplication and Division Combined. Ex. What number results from multiplying 57 by 16, and dividing by 24 ? To multiply by 16 is the same as to multiply by 2 and then by 8 ; and to divide by 24 is the same as to divide by 3 and and then by 8. We may strike out the 8 from both terms ; since to multiply a number by 8 and then to divide it by 8 leaves it unaltered. So that 57 x 16 24 The striking out of a factor common to a multiplier and a divisor is called cancelling. Cancelling may sometimes be performed more than once in the same exercise ; thus EXERCISE XVII. Perform the following operations, cancelling where possible, 1. 9x7 8x15 24x12 16x6 33x14 48x24. 2. 8. 3 45x36 5 84x48 18 105x21 8 35 117x48 57x25 72 81 89x32 84 157x81 49 238x63 108 181x36 40 66x45 44 124 x If 108 119 54 99 00 DIVISION. 47 *. 85x9x12 45x16x18 24x15x21 30x14x24 42x16x32 4x18 36x45 40x35 20x28 48x35 5. 59x10x33 63x8x25 18x14x28 50x34x21 9x8x6 11x60 35x32 9x36 14x25 3x4 6. 147x24x18 240x65x8 306x28x63 564x84x33 72x45 16x30 35x102 88x144 Any number is divisible exactly 1. By 2, when its last place is divisible by 2. 2. By 4, when its last two places are divisible by 4. 3. By 8, when its last three places are divisible by 8. 5* By 9^ } waen tne sum f its places is divisible by 3 or 9. 6. By 5, when its last place is 5 or 0. 7. By 10, when its last place is 0. EXERCISE XVIII. 1. 243x316 79x104 348x252 219x573 391x215 893x4128 228 432 384 693 300 376 2. 256x216 750x375 358 x 516 250x700 295x415 312x462 8. 584x2928 73x321 92x840 300x200 843x356x296 3024 412 342 6000 296x560 54. EXERCISE XIX. 1. How many scores in 340 ? 2. How many one-dozen baskets may be filled out of 468 bottles ? 3. How many pieces, each 25 yards, may be got from 6425 yards. 4. How many forms, of 15 each, will hold 675 scholars ? 5. Into how many parcels of 16 may 432 marbles be divided ? 6. How often can I subtract 64 from 2304 ? 7. What must 73 be multiplied by to give 22995 ? 8. How many regiments, each 829, are in an army of 38963 men ? 9. If 2664 be dividend, and 36 be quotient, find the divisor. 10. How many boxes will hold 7000 oranges, if each hold 125 ? 11. If a man divides 728 equally among his 4 children, what is the share of each ? 12. How many years' rent of a house at 6 is 792 ? 13. If the journey from London to Edinburgh, which is 385 miles, be made in 11 hours, what rate is that per hour ? 14. What multiplier of 346 gives 81964 as product ? 15. If a tradesman saves 5 shillings a week, in how many weeks will he save 850 shillings. 16. What is the nearest number to 850 which can be divided evenly by 27 ? and the next nearest ? 17. The year 1864 began on a Friday, how many Fridays had it I and how many Sundays ? 48 DIVISION. 18. In a certain city there died in the month of April 23790 persons, what was the daily number of deaths on an average ? 19. A banker has a box with 7460 shillings, 24 five-shilling pieces, and 50 florins, how often can he change a pound ? 20. Five trains left London Bridge for the Crystal Palace, the first with 379 passengers, the second with 250, the third with 483, the fourth with 579, and the fifth with 294 : what was the average number in each train ? 21. A regiment of 1170 men had one man killed or wounded in battle for every 18 men in it : how many remained fit for service ? 22. A cargo of tea, 435 chests, each 180 pounds' weight, is to be packed in boxes, each containing 54 pounds : how many of these must be ordered ? 23. What must I add to the square of 154 to contain exactly the square of 27 ? . MISCELLANEOUS EXERCISE ON THE FOUR RULES. I. 1. Printing was invented 1440 A.D., and the first book was printed in England 34 years thereafter : what was its date ? 2. If a farmer sells 35 oxen for 12 each, 253 sheep for 2 each, and 159 lambs at 1 each, what does he receive for all ? 3. The circumference of the earth is 24900 miles, in how many days could a ship sail round it at 9 k miles an hour? 4. How much higher is Mont Blanc, the highest mountain in Europe, which is 15,680 feet high, than Ben Nevis, the highest in Britain, which is 4368 feet high ? 5. To half the sum of 85 and 57 add half their difference. 6. A clerk, engaged for five years, receives 80 salary the first year, and an advance of 15 each year : what is his average yearly salary ? 7. The six largest cities in England are London with 2,362,236 in- habitants, Liverpool with 375,955, Manchester with 316,213, Bir- mingham with 232,841, Leeds with 172,000, and Bristol with 137,000 : what is the population of these cities together ? 8. Sir Isaac Newton was born in 1642 and died in 1729 : how old was he at his death? 9. Three apples were given to each of 178 pupils of a school, but 672 apples were provided in all : how many more pupils could have been served ? 10. I met 7 flocks of sheep, of one score each, .on their way to market, 5 of twoscore and nine each, 6 of threescore and ten each, and then one of 19 : how many sheep did I pass ? 11. From London to Peterborough is 76 miles, from Peterborough to York 115 miles, from York to Newcastle 72 miles, from Newcastle to Berwick 65 miles, from Berwick to Edinburgh 57 miles : what is the distance from London to Edinburgh ? 12. What number added to 7803 will make up the third part of 87003? - 13. To 7 times the sum of 909 and 98, add 7 times their difference. 14. A train contains 1097 passengers ; of these, 286 are first-class, and half as many more second-class : how many third-class are there? 15. What divisor of 44934 gives 348 as quotient, and 42 over? MISCELLANEOUS EXERCISES 49 16. Find the number of days in a leap year. 17. A teacher buys 100 boxes steel-pens, containing one gross each. He has 563 pupils in school : after serving them with pens 7 times, how many remain ? 18. The ship " Graceful," from Charente to Leith, discharged 2552 one-dozen cases brandy, 122 two-dozen cases, and 16 three-dozen cases : how many gross of bottles were in her cargo ? If 6 bottles go to a gallon, how many gallons of brandy ? 19. A shelf in a library contained History of England, 10 volumes ; British Poets, 75 volumes ; Goldsmith's Works, 4 volumes ; Waver- ley Novels, 25 volumes ; British Essayists, 45 volumes ; and the shelf below contained exactly the same number : how many volumes were on both ? 20. What must be added to the third part of 1395 to bring it up to the fifth part of 3790? 21. Find the product of three numbers, of which the first, 374, ex- ceeds the second by 93, and the third by twice as much. 22. In what time will 3 pipes empty a tank of 429165 gallons, if they run off respectively 450, 500, and 535 gallons per hour ? 23. If a stage-coach travel 5 miles an hour, how far will it go in two days of 9 hours each ? 24. An army of 69776 men was drawn up in squares of 28 in a side ; how many squares were there ? 25. Find the difference between the square of 9009, and the cube of 909. MISCELLANEOUS EXERCISES-^mftVwed. II. 1. Julius Caesar invaded Britain 55 B.O. : how long was that before the union of England and Scotland in 1700 ? 2. How often does a clock strike in a year ? 3. A boy, working 8 hours a day, can point in a year 33979280 pins : how many can he point in an hour ? 4. A travels 3 miles an hour, B 4 : when B has gone 45 miles, how far has A gone ? 5. Great Britain and Ireland contain 121385 square miles ; the British possessions in Europe, 145 ; in Asia, 928610 ; in North America, 768577 ; in South America, 89000 ; in Africa, 201403 ; in the West Indies, 73384 ; in Australasia, 560000. What is the whole area of the British Empire ? 6. Michaelmas is 86 clear days before Christmas : what is the date of it? 7. January 4, paid into savings' -bank, 14 shillings ; February 1, paid in 13 shillings ; February 28, drew out 11 shillings ; March 14, paid in 19 shillings ; March 31, drew out 25 shillings ; April 24, paid in 17 shillings ; May 3, paid in 9 shillings ; May 25, drew out 15 shil- lings ; June 1, paid in 16 shillings. My account was then balanced : how much had I at my credit ? 8. Adam lived 930 years ; Seth, his son, was born when he was 130 years old, and lived 912 years : how long did they live together ? 9. A bag of nuts, containing 3000, was divided among a school ; the pupils above 9 years got 35 each, and those below 9 (who were exactly the same number) got 25 each : how many pupils were in the school? 10. A railway guard makes two journeys every lawful day from D 50 MISCELLANEOUS EXERCISES. Edinburgh to Glasgow and back ; if these towns are 47 miles apart, what distance has he travelled, after being in his situation five years ? 11. Three regiments form squares, the side of the tirst being 33 men, of the second 29, and of the third 27 : how much stronger is the first regiment than the second, and the second than the third ? 12. How often will a cart-wheel, 16 feet round, revolve in going a mile, which has 5280 feet ? 13. A railway 273 miles long has a station every 10 miles on the average : how many stations has it ? And what is the length of a railway which has 18 stations, distant on the average 7 miles from each other ? 14. George I. of England began to reign 1714 A.D., and reigned 13 years ; George n. reigned 33 years, George in. 60, George iv. 10, and William iv. 7 years. Queen Victoria succeeded William ; in what year did she begin to reign ? 15. The sea route from London to Hamburgh is 482 miles. When the London steamer is 130 miles on its way, and the Hamburgh steamer 210 miles on its, how far are they apart ? 16. If Scotland produced in 1864, 23000 tons pig-iron weekly, what was the produce for the year ? and at 3 a ton, how much did it add to the wealth of the country during the year ? 17. A farm has 5 fields, the first containing 89 acres, the second 101, the third 174, the fourth 92, and the fifth the average of the other four. It is to be divided into as many fields of equal size : how many acres will each contain ? 18. (a) A legacy of 1595 is left to two charities, of which the one receives half as much again as the other : what was the share of each ? (6) Out of a legacy of 8578, 730 were devoted to charitable purposes ; the rest was to be divided into 9 shares, of which the eldest son was to get four, the second three, and the youngest two : how much did each get ? 19. If a candidate at an election is returned by a majority of 291 votes out of 3579, how many voted for the unsuccessful candidate ? 20. If I bought 79 shares in the Great Western Railway at 64 each, and sold out at 69, what did I pay for them, and what did I gain? 21. The exports from Liverpool to the United States in 1861 were 8223587 ; in 1862, 11986233 ; and in 1863, 13765217. How much did the increase in 1862 exceed that in 1863 ? 22. In a journey of 37 hours, I travelled one-third of the time at 24 miles an hour, and two-thirds at 27 miles : what was the length of the journey ? 23. Divide 318 apples among 18 boys and 8 girls, giving each boy twice and a half as much as a girl. 24. Handel, the great musical composer, died in 1759, aged 75 ; and Haydn was born when Handel was in his forty-seventh year; what year was that ? 25. A tank of water contained 75000 gallons. A supply was drawn off by 3 pipes, which ran for 10 hours at the rate of 255 gallons each Eer hour ; but during that time two pipes ran into the tank 335 gal- >ns each per hour : how much water was left ? 26. Find the difference between the square of the sum of 28 and 39, and the sum of their squares. MONEY. 51 27. A sum of money was divided between A, B, c, and D, so that A got 260, B 375, c the excess of B'S share above A'S, whilst D was to receive 25 from A, 48 from B, and 17 from c, as his share. What were the shares of all four ? 28. The sum of 2 numbers is 428, and their difference 194 ; find the numbers. %* Half the smn-fhalf the difference gives the greater. Half the sum -half the difference gives the less. 29. A tradesman, out of his weekly savings for a year, bought a table that cost 22s., 6 chairs that cost 7s. each, a carpet of 20 square yards in size at 3s. a yard ; he had besides 32s. over : how much had he saved every week ? 30. A farmer paid 780 for cows and sheep. Of t]iis sum he paid 350 for 25 cows ; if a cow cost 7 times as much as a sheep, how many sheep did he buy with the rest of his money ? 56. MONEY. MONEY OF ACCOUNT TABLE I. Accounts are kept in pounds, shillings, pence, and farthings sterling. Pounds are denoted by the letter , thus 40. Shillings by the letter s., or by a line ; thus, 3s. or 3/. Pence by the letter d. 9 thus 9d.* Farthings, which mean fourths of a penny, are denoted by fractional numbers ; thus, one farthing by ; two farthings, or one halfpenny, by Jd. ; and three farthings by f d. Pounds, shillings, and pence, when written in columns, are denoted by s. d. placed over the column. EXERCISE. Read off the sums in Ex. i. sect. 58. t COMPOUND ADDITION. Ex. If I have paid into the bank in January, ,27, lls. 3jd. ; in February, 23, 14s. 8^d. ; in March, .13, 19s. 9|d. ; in April, 7, Os. 2jd. ; in May, 2, 7s. lid. ; and in June, 17s. 3jd. : how much have I put in during the six months ? We have here to find the sum of the six payments ; which we do by addition. Write the numbers below each other, pounds below pounds, shillings below shillings, and pence below pence. i is the first letter of libra, a Roman weight ; *. and d. the first letter* of iolidus and dtnariu-s, Roman coins. 52 MONEY. Then, adding the farthings' column, we have 2-3-6-8-9 farth- ings, which is 2|d. ; set down the Jd. and carry the 2d. Adding the pence column, we have, by simple addition, 29d., that is 2/5 ; set down the 5d. and carry the 2/. Adding the shillings' column, we have, by simple addition, 70/, that is 3, 10s. ; set down the 10s., and carry the 3. Adding the pounds' column, we have, by simple addition, 15. Sum, 75, 10s. 5|d. Rule. Write the numbers below each other so that each column may be of the same name ; add each column in its order, carrying as many of the next highest name as are con- tained in its sum. The result may be proved, as in simple addition, by adding the columns from the top downward. The addition of quantities of different names, as here, is called compound addition. 75 10 5 58. EXERCISE I. 1. 2. 3. 4. 5. s. d s. d. s. d. s. d. 2 s. d. a. 3 6 10 I 2 10 9^ t 11 9 10s [ 7 9 4* 5 10 9; b. 4 16 8. 384. 543 11 9 2; 7 9 6. c. 930 \ 9 5 6j 729; 507; 1 14 10; d. 5 14 10; 7 7 103 800 3 17 2, 3 9 e; e. 10 9 4 15 11' 7 2 11- 11 9 5; 17 10 /. 6 18 6, 729 8 2} 8 2 4j 2 17 93 a. 19 3; 12 5* 9 11 12 Oi 8 5 101 *. 11 5 8J 6 16 10; 7 10 9 976^ 11 11 4a . Ill; 8 2 3^ 11 15 18 91 f 879^ j. 2 3 9} 756; 4 10 11 10 9 4, 3 11 5J A. 12 2 10 10 11 11. 12 12 9} 500' 276 & 9 6 5J 2 19 6 1 17 1* 7 1<H L 994 m. 8 10 11; 3 15 4 248; 549i 12 4 10, n. 5 14 3^ 4 17 2J 5 15 71 6 10 10; 1 19 11 o. 10 2 5 10 H 10 18 2^ 696 p. 1 17 9J 11 9 4 909 11 9 2; 11 4 4 EXERCISE II. 1. Count from Id., 2d., 3d., etc., by lid., 2d., 2fd., 4|d., etc. 2. Count from 1 sh., 2 sh., 3 sh., etc., by 1/3, 2/4, 1/3 J, etc. 8. Count from 3, 4, 5, etc., by 7/8, 13/4, 12/6, etc. V Ex - J - and 11 tor oral practice, whilst the pupil is working the following Exercise. The same remark applies to the subsequent rulefc, COMPOUND ADDITION. 53 59. EXERCISE III. 1. 2. 3. 4. 5. 8. d. s. d. s. d. s. d. 5. d 73 10 91 14 10 8 * 47 17 10 63 14 5f 72 8 41 47 5 ol 92 4 39 14 8i 28 10 10 17 8 9j 3 7 91 37 16 4 7 19 10; 37 15 71 93 15 2j 13 17 4 29 18 llf 72 12 6l 9 9 9} 82 10 4f 28 9 3| 15 14 oj 84 11] 8 5 29 79 1 11 80 5 11 34 8 5 59 10 0; 92 10 10j 9 11 u 6. 7. 8. 9. 10. 9. d. s. d s. d. s. d *. d. 42 8 28 10 59 OOOO 147 82 7 2 5 17 CO il O 293 118 500 14 10; 10 0; 8 \ 673 : 200 [ - 74 10 18 *** COOrH l-H 534 1 10 9 6 5 Of 72 10J 973 rj 94 3 6] * 9 10 4f 904 15 2 5 18 38 14 1 11 459 226 19 4 ill ol 7 192 18 2 17 5; 28 t 990 14 19 51 o] 673 49 17 0* 15 8J 19 3J 305 2 11 201 7 , 309 17 6 200 18 2 17 8 el 38 18 6J 802 14 25 8 55 16 11. 934, 18, 6 + 84, 0, 9 + 702, 15, 21 + 39. 4. + 740, 0, + 85, 16, 2 + 156, 18, 6| + 529, 5, 1J. 12. 617, 10, 11| + 290, 0, 10j + 38, 5, 6 +93, 0, OJ + ^549, 7,2| + 29, 10, + 709, 18, 4f +8l, 16, 1. 13. 127, 14, 81 + 29 3> 11, 5l + 3Q, 10, 101 + ^458, 10, 9 + 500, 17, 7f + 110, 19, 2 + 301, 1, 11$ + 824, 0, 0^ + 629, 5, 5. 14. 543, 10, + H 17 +7, 10, 6f + 829, 7 +471, 10, 28, 15, + 728, 16, 10| + 840, 0. 11. 15. 293, 18 + 72, 19, l + 9, 10, 5? + 820, 15 + 94, 18, 65 + 571, 15, 4| + 629, 18, 4 +930, 15, 10 J. 16. 2005, 7, 6 + 943, 18, lJ + 564, 9 + 7248, 0, 9^ + 1508, 10, 8| + 592, 8, 0^ + 9408, 2, 10 + 93, 0, 11|. 17. 329, 14, 4^ + 73, 18, 5i + 493, 9, 4^ + 701, 1, 7i + 592, 10, 11 + 17, 4, 8 + 9, 7, 6^ + 341,19, 8| + 700, 1, 11. 18. 112, 9, 4 + 257, 3, 0^ + ^62, 11, 7| + 79, 19, 91 + .790, 8, 2 + 173, 13, llf + 459, 12, 10 + 614, 14, 11^ + 998, 19, 5|. 19. 72, 7, 10 + 394, 6, 4| + 593, 0, 8i + 360, 0, llf + 94, 15, 81 + 250, 11 + 37, 18, 0| + 84, 15, 6f + 4^20, 18, 6 + 13, 2, 1|. 20. 640, 10, 11 + 93, 4, 7^ + ^870, 19 + 250, 0, 9f + 550, 9, 1 + 709, 13, 6f + 1, 2, 3^ + ^85, 16, 6 + 924, 15, l + 9, 2, 8f. 21. 279, 18, 6 + 90, 17, 3| + 250, 4, 10 + 79, 18, If + 100, 15 + 25, 0, 6^ + 365, 19, 1 + 209, 14, 7| + 99, 18, 4 +805, 7, 6J. 22. 8408, 14, 10 + 2930, 10, 44 + 6009, ig ? Of +509, 7, 111 + 93, 10, 6^ + ^793, 10, 0^ + ^209, 18, 1 + 3085, 2, Of + 94, 18, 2J. 23. 2563, 14, l + 846, 10, 01 + 2 564, 0, lOi + 865, 17, llf -f 590, 0, 6 + 859, 2, l+9337, 19, 0^ + 820, 7, 6 + 94, 17, 6f. EXERCISE IV. Work the auestions Ex. iii. as directed Ex. xi. p. 23. 54 MONEY. 60. COMPOUND SUBTRACTION. Ex. It I pay a debt of -28, 18s. 5^d. out of a sum of 63, 13s. 4jd., how much have I over? We have here to find the difference of these two sums of money ; which we do by subtraction. Write the subtrahend below the minuend in its place. 2 f. from 1 f. cannot be taken ; change one of the pence, making 5 f. in all ; 2 f. from 5 f. leaves 3 f. 5d. from 3d. cannot be taken ; change one of the shillings, making 15d. in all ; 5d. 34 14 lOf from 15d. leaves lOd. 18s from 12s. cannot be taken ; change one of the pounds, making 32s. in all ; 18s. from 32s. leaves 14s. 28 pounds from 62 pounds leaves 34. Rule. Write the subtrahend below the minuend so that each column shall be of the same name ; subtract each column in its order, changing one of the next highest name when necessary. The result may be proved, as in simple subtraction, by add- ing together the subtrahend and the difference. The subtraction of quantities of different names, as here, is called compound subtraction. Or thus i 1 Then, beginning with the lowest name, 2 from 1 cannot be taken ; add Id. or 4 farthings, making 5f. in the minuend ; 2f. from 5f. is 3f. Then 6d. from 4d. cannot be taken ; add Is. or 12d., making 16d. in the minuend ; 6d. from 34 14 10| 16d. is lOd. Then 19s. from 13s. cannot be taken ; add 1 or 20s., making 33s. in the minuend ; 19s. from 33s. is 14s. Then 9 from 3 cannot be taken, but 9 from 13 is 4 ; and 3 from 6 is 3 for the tens' place ; making 34. Rule. Write the subtrahend below the minuend so that each column shall be of the same name ; subtract each column in its order, beginning with that of lowest name, and carrying as in compound addition ; if any name in the minuend is less than the same name in the subtrahend, add to it one of the next highest name changed to its own, and add one to the next name in the subtrahend. 1 Both methods of subtraction are given as in simple subtraction, sect. 26 ; the teacher may choose either. COMPOUND SUBTRACTION. EXERCISE I. 1. 7J-2A, 5A-31, 8f-5f, 10^-7, 9A-1A, etc. 2. 5j-3|, 71-6?, 71-5$, 9-7, 111-8J, etc. 3. 6/5-3/2, 8/11-5/6, 7/9-1/9, 3/6-2/4A, 14/10^-7/4, etc, 4. 4/3-2/6, 7/2-3/8, 8/4-4/7, 8/41 - 6/5, 13/2J - 8/8, etc. EXERCISE II. 1. Count back from I/, 2/, etc., by 2d., 3|d., 43d., etc. 2. Count back from 20/, 19/, etc., by 1/3, 1/4, 2/2, l/7, etc. 3. Count back from 5, etc., by 10/6, 12/8, 13/4$. EXERCISE III. 55 1. 2. 37 8 4A 93 10 3| 19 5 10J 39 6 9 3. 84 7 101 53 17 I} 47 29 4. 5. 17 8} 205 2 9 8 10$ 126 12 8 1 6. 7. 8. 9. 10. 730 2 6.1 704 14 91 294 9* 360 10 6^ 545 12 428 17 8j 396 a 89 10 9 ^ 219 19 0} 293 18 OJ 11. 848 Oj ?- 274 10 1 19. 8000 1 - 1793 10 0* 12. 763 10 11 [- 294 18 2| 20. 3030 13 - 2594 71 13. 540 14. 643 15 8 J- 290 i- 19 15 9 21. 22. 2000 903 01- 61- 17 50 9 7 15. 1938 17 6 I- 209 19 8 23. 1000 - 295 T$ 16. 2467 14 8 - 938 15 6 24. 3724 6 104- 1936 2 11 17. 3091 10 11 - 1857 16 111 25. 5704 13 8 - 2945 2 105 18. 4000 - 993 1 u 26. 8407 7J- 899 19 91 EXERCISE IV. Find the first remainder less than the subtrahend in- 1. 2761 13 4i 564 17 63 10. 8473 16 4 - 1005 5 10A 2. 4095 14 of- 709 19 1 11. 10000 - 2946 5j 3. 8740 ?2 ~ 1096 10 91 12. 7338 2 11; - 943 4 9: 4. 5436 10 81- 854 12 0? 13. 7009 9 - 856 4^ 5. 9425 16 2 - 1906 17 2} 14. 1946 10 10 - 405 16 11. 6. 7464 13 11 - 948 17 6| 15. 6429 14 6. J - 842 15 10; 7. 4763 01- 742 11 4 16. 8754 12 3^ - 947 13 6 8. 6000 - 823 Of 17. 5431 18 l' - 739 11 41 9. 2346 2 10 - 473 0J 18. 9402 14 ; : - 1246 16 8 f COMPOUND 1MULTIPLICATIOK Ex. What cost 9 chests of tea at 24, 14s. 7Jd. per chest ? We have here to find 9 times the price of one chest ; which we do by multiplication. 56 MONEY. Write the multiplier under the pence column s. d. of the multiplicand. 24 14 7J Then, beginning with the lowest name, 9 __ 9_ times If. are 9f., which is 2d ; set down If., 222 11 5 and carry 2d. 9 times 7d. are 63, and 2 are 65d., which is 5/5 ; set down Bd., and carry 5/. 9 times 14s. are 126s., and 5 are 131s., which are 6, lls. ; set down lls., and carry 6. 9 times 24 are 216, and 6 are 222. Total product, 222, lls. 5^d. The result may be proved by dividing the product by the multiplier (see sect. 65), which will give the multiplicand. Rule. When the multiplier is not above 12, multiply each name in the multiplicand by it in order, beginning with the lowest, and carry as in compound addition. When the multi- plier is not greater than 144, and has two factors, neither above 12, multiply by each factor in succession. The multiplication of quantities of different names, as here, is called Compound Multiplication. EXERCISE I. upy e oowng y , , , ec., up o successvey : ., 3id., 3|d., 4Ad., 51d., 5|d., 6 Ad., 63d.. 71d., 7id., 81d., 8jd., 9Jd., 9jd., lO Multiply the following by 2, 3, 4, etc., up to 12 successively : 2M., i 9j EXERCISE II. Multiply by 2, 3, 4, etc., to 12 successively : 1. 6d., 8d., 10d., 1/1, 1/4, 1/8, 2/1, 2/7, 3/4, 3/6, 4/2, 4/10, 5/6, etc. 2. 10A, 10/3, 10/9, ll/, 12/2, 12/8, 13/3, 13/10, 14/4, 15/1, 15/11, etc. EXERCISE III. x 2. x 3. x 4. x 5. x 6. x 7. x 8. x 9. xll. x!2. EXERCISE IV. 1. 7, 8, 4i x 2, 4, 7, 8, 9. 4. 21, 4. 8} x 3, 8, 2, 7, 5. 2. 10, 9, 4x3, 6, 8, 10, 11. 5. 34, 17, Iljx7, 11, 9, 12, 3. 8. 16, 5| x 4, 5, 7, 9, 12. 6. 43, 10, 10* x 5, 8, 4, 10, 7. COMPOUND MULTIPLICATION. 57 7. 87, 9, Of x!4, 15, 21, 22. 8. 92, If, 4* x25, 27, 28, 32. 9. 127, fc, 6| x35, 42, 44. iO. 209,15,7|x45,48,54. , 11. 543, 18,21 x56, 60, 63. 12. 708, 13, l| x 64, 72, 77. 13. 900, 0, 91 x 84, 99, 108. 14. 1256, 10, Of x 121, 132, 144. \* Multiply by three factors. 15. 18, 9, 41 xl!2, 125. 16. 37, 0, 9| x 105, 126. 17. 85, 17,2^x192,216. 18. 90, 14, 83 x 128, 135. 19. 74, 8, 111 x 147, 162. 20. 105, 15,0^x189, 210 197 16 10 price of 8 chests. 64 chests. 4 chests. 68 chests. 63. Multipliers of Two Places. Ex. Find the price of 68 chests tea at 24, 14s. 7Jd. per chest. The number 68 cannot be re- solved into two factors under 12. . d. Take the next less which can, that 24 14 7^X4 is, 64. Find the price of 64 chests 8 (8X8), and add the price of 4 chests ; for 68 = 8 X 8+4. The price of 64 chests is found -J582 14 g as above : the price of 4, by mul- 93 18 5 tiplying the price of one (first line) ,, r by 4 ; the price of 68 by adding the price of 64 and the price of 4 together. Other factors which might be used are 9X7 + 5 and 10 X 6 + 8, either of which pairs may be taken to prove the result, Rule. When the multiplier is not above 144, and cannot be resolved into two factors under 12, multiply by the factors of the next less number which has them, and add the product of the multiplicand by the difference between that number and the multiplier. It is advisable to take factors for the number next above the multiplier, when that number exceeds it only by 1, and then subtract the excess; thus, 39 = 10X4 1. In the present case we might have taken 68 = 10x7 2. EXERCISE V. 1. 2, 14, 2S 2. 7, 10, 9] x 13, 17, 19, 24, 29, 31. x 34, 38, 43, 51, 58, 61. 3. 13, 8, 5 4. 34, 3, 2^ 6. 60, 0, 9; fe x 62, 69, 74, 78, 82, 87. ; x 91, 94, 101, 106, 117, 123. x 129, 135, 142, 145. U06. 58 MOXEY. . Multipliers of Three Places. Ex. Find the price of 457 chests at .24, 14s. 7|d. pel chest. 24 14 7 X 7 = X173 2 2| price of. 7 chests. _ 10 247 6 Oi X 5 = 1236 10 2| 50 10 2473 5 X4= 9892 18 ,,400 Total product, 11301 14 1J 4^7 Rule. Multiply by factors for 100 (10 X 10). Then multi- ply the multiplicand by the number of units in the multiplier, ten times the multiplicand by the number of tens in it, and a hundred times the multiplier by the number of hundreds in it . add these three products for the total product. EXERCISE VI. 1. 9, 13, U x 257, 381, 473. 7. 59, 7, 31 x 915, 638, 187. 2. 13, 10, g^x 319, 459, 542. 8. 73, 8, 10x562, 784, 268. 3. 19, 8, 5J x417, 534, 629. 9. 83, 15, 71 x 400, 701, 511. 4. 23, 10, Of x 566, 671, 713. 10. 89, 0, 5 x208, 962, 609. 6. 31, 19, 41x647, 738, 825. 11. 93, 14, 2^x354, 849, 276. 6. 43, 1, 11 J x 724, 850, 993. 12. 109, 7, 9 x 417, 651, 767. Multipliers of Four Places. The same method is used for multiplying by thousands. Rule. Multiply by factors for 1000 (10X10x10). Then multiply the multiplicand and the successive products by the several places of the multiplier in order, beginning with the units' place ; add these products for the total product. EXERCISE VII. 1. 13, 18, 5^x1924, 2438. 4. 57, 10, 7| x6234, 7941. 2. 19, 5, 104 x 2741, 3925. 5. 69, 5, 8^ x 8301, 9042. 8. 27, 3, 4 X4837, 5529. 6. 124, 15/6 jx 4520, 6009. V* These products are obtained more easily by practice. B COMPOUND DIVISION. Ex. 1. Divide .93, 15s. 9|d. equally among 7 persons : what is the share of each ? Write the divisor and dividend as in simple division. COMPOUND DIVISION. 59 Then 7 in 93 is X13 and 2 over ; set down the 13, and carry 7 ) 93 15 9| the 2 to the shillings, making "13 7 ll f 55s. in all. 7 in 55 is 7s. and 6s. over ; set down the 7s. and carry the 6s. to the pence, making 81d. in all. 7 in 81 is lid. and 4d. over ; set down the lid. and carry the 4d. to tJie farthings, making 19 farthings in all. 7 in 19 is 2 farth. and 5 farth. over ; set down the 2 farth. and, as the division is now finished, there is a remainder of 5 farthings, divided thus, f . Quotient, 13, 7s. lljf. The result may be proved by multiplying the quotient by the divisor, and adding the remainder, which will give the dividend. Ex. 2. Divide the same sum ^ ~ equally among 28 persons. Resolve the divisor into its two 13 7 11H-5 ( 19f. factors (7X4), and divide by each 3 611|-f2) in succession. Quotient, 3, 6s. llj Jf. The result may be proved by reversing the order of factors in dividing, or by multiplying the product by the divisor. Rule. When the divisor is not above 12, divide each name by it in order, beginning at the highest, and carry the remainder to the next lower. When the divisor is not above 144, and has two factors neither above 12, divide in the same way by each factor in succession. The division of a quantity of several names, as here, is called compound division. 66. EXERCISE I. 1. 2d. 3d. 5d. 6d. 7d. etc. 4-2, 4. 10. 1/3,1/6, 1/9, 2/, 2/3, etc. 4-6/12. 2. lid. 3d. 4d. 6d. 7d. etc. 4-3, 6. 11. 1/OJ, 1/2, l/3, 1/5 A, etc. 4-7. 3. lid. 2Ad. 3M. 5d. 6|d. etc. 4-5. 12. 1/1*, 1/4$, l/7i, 1/10, etc. 4-11. 4. ifd. 3|d. 5jd. 7d. etc. 4-7. 13. 1, 1, 4, 1, 8, etc. 4-2, 4, 8. 5. 2d. 4d. 6d. 8d. etc. 4-8. 14. l,2/6,l,5/6,l,8/6,etc. 4-3,9. 6. 2}d. 4$d. 6|d. 9d. etc.4-9d. 15. 1, 1, 5, 1, 10, etc. 4-5, 10. 7. l/,l/2, 1/4, 1/8, 1/10, 2/, etc. 4-2,4. 16. 1,4, 1,10, 1,16, etc. 4-6, 12. 8. lli,l/l$,l/3|,l/6,l/8|,etc. 4-3,9. 17. 1, 1, 1, 4/6, 1, 8, etc. 4-7. 9. 1/0$, 1/3, 1/5$, 1/8, 1/104, etc. 4-5. 18. 1, 2, 1, 7/6, 1,13, etc. 4-11. 60 MONEY. EXERCISE II. 1. 8 19 72 -2, 3, 4, 5. 13. 89 14 102-14, 15, 21. 2. 7 5i 3. 4, 5, 6. 14. 91 2 81- -24, 27, 22. 3. 19 10 3; -4, 5, 6, 7. 15. 156 17 3j- -25; 28 100. 4. 27 15 6; -5, 6, 7, 8. 16. 193 5 - -30, 32, 108. 5. 79 1 111 -6. 7, 8, 9. 17. 279 6 104- -84, 96, 99. 6. 54 -7, 8, 9, 10. 18. 309 1 4|- -80, 81, 35. 7. 60 5 7l -8, 9, 10, 11. 19. 600 10 101- -77, 72, 121. 8. 86 14 9' -9, 10, 11, 12. 20. 793 15 6 i- -70; 64, 18. 9. 43 6 11; -10, 11, 12, 7. 21. 72 5 6}- -56, 63, 16. 10. 37 18 1; -11, 12, 5, 9. 22. 68 7 84- -48, 50, 144. 11. 5 17 5 -12, 6, 7, 10. 23. 81 19 02- -42, 44, 132. 12. 3 12 9* -7, 9, 4, 5. 24. 69 2 7g- -36, 40; 33. 67. Divisors of Two or more Places. -Ex. Divide ,93, 15s. 9|d. among 43 persons. Rule. Divide each name m order by the divisor, be- ginning at the highest ; and carry each remainder to the next lower name. 43)93 15 _86 y 20 )155 s. 129 9|(2 3 ** The 40 farthings over are written in the quotient with the divisor below tham, as & EXERCISE III. ^29, 37, 53, 71, 83. ^19, 41, 67, 86, 91. f-52, 23, 47, 95, 13. f-124, 213, 352, 793, 61. ^225, 538, 401, 191, 17. M15, 116, 237, 73, 85. -372, 416, 509, 1000, 1937. f-562, 57, 829, 900, 2340. ^1256, 4073, 236, 800, 158. ^721, 1356, 2943, 673, 78. 7-2905, 7238, 825, 34, 304. T-59, 97, 652, 8905, 4005. 1. 567 10 3i 2. 734 18 5 3. 392 15 4J 4. 78 2 11- 6. 27 18 o; 6. 115 10J 7. 1897 14 3! 8. 2700 18 0^ 9. 8035 17 5; 10. 5682 11 3, 11. 73582 14 7 12. 290732 9 li COMPOUND DIVISION. (J3. Fractional Multipliers. Ex. What cost; 8} packages if 1 package cost 5, 17s. 9|d. ? Multiply first by the fraction (}), then by the whole number (8). Add the products. ^ V %01 5 17 91 _ _ 4)17 13 3| 3| 2 4 47 EXERCISE IV. 51 10 1. 7, 10, 3 x71,9J, 11J. 2. 14, 15, 7|x4f, 6|, 8. 3. 24, 19, 3 x 15i, 27|, 36? 4. 71, 5, Ilx49f, 84 j, 10( 5. 91, 15, 6 x 73J, 591, 91 j 6. 256, 14, 10x291 13 J, 681 7. 509, 8, 3} x 231 J, 4501, 6713 . 8. 891, 11, 1^x3071,5931, 713J. Fractional Divisors. Ex. It 17} yards cost 9, 18s. lOjd., what is that per yard? We have to divide the 17} 9 18 10 J whole price by the num- 4 4 ber of yards to get the 71 price of one yard. Multiply both divisor and dividend by 4 to re- move the fraction from the divisor. )39 15 20 795 71 85 71 14 12 )174 142 32 4 )128 71 57f. EXERCISE V. 1. 7, 10, 11J^5.J, 6 2. 11, 14, 5JH-81, 1 3. 29, 5, Oi-r-18i, 2 4. 36, 7, 2|-^2l|, 87J, 52J. 8. 643, 0, 5^83^ 173J, 824J. Money Divisors. " Ex. How Dften is 5, 13s. 6jd. contained in 39, 14s. 7|d. ? 62 MONEY. Rule. Reduce divisor and dividend to the same name, and proceed as in simple division 39, 14, 7|-~5, 13, 6|=38143f.-j-5449=7. EXERCISE VI. %* To be performed after reduction has been learnt. 1. 27, 17, 3^-6, 3, 10. 6. 63, 8, Of -f-21, 2, 8|. 2. 137, 8, 9-i-8, 19, 4j. 7. 671, 10, 1 47, 19, 3.i 3. 361, 2, 9f-7-72, 4, 6f . 8. 268, 10, 3-4-100, 9, lOj. 4. 2090, 0, 7j-r-81, 0, 91. 9. 675, 19, 3-f-75, 2, U 5. 459, 18, 2^-24, 17, 8J. 10. 870, 0, 5-r-39, 18, 5J. 70. REDUCTION. MONEY OF ACCOUNT TABLE I. From a Higher to a Lower Name. Ex. In 7, 13s. 3|d., how many farthings ? We cannot change this sum to farthings by one step, as it is too large ; we must therefore do it in parts, changing first the pounds to shillings, then the shillings to pence, then the pence to farthings. Thus, to change the pounds to shillings, since there are 20/ for s. d. every pound, there will be 20 7 13 3j times as many shillings as 20 pounds ; multiply 7 by 20, and "153" 3| 8 h. in the sum. add the 13/ already in the sum, 12 making 153 sh jggg- j penee ln the aum . To change the shillings to 4 pence, since there are 12d in >.,, every shilling, there will be 12 7359 times as many pence as shil- lings ; multiply 153 by 12, and add the 3d. already in the sum, making 1839d. To change the pence to farthings, since there are 4 farth. in every penny, there will be 4 times as many farthings as pence ; multiply 1839 by 4, and add the 3 farth. already in the sum, making 7359f. in all. Rule. Multiply each name, in order from the highest, by the number of the next lower which it contains, adding to each product the number of the lower in the given sum. REDUCTION. 63 The process of changing from one name to another is called Reduction. The result may be proved by changing back the farthings to pounds ; dividing by the same numbers by which we have multi- plied. If ,7, 13s. 3jd., when changed to farthings gives 7359f., 7359 farthings, when changed to pounds, must give 7, 13s. 3|d. (See sect. 71.) EXERCISE I. 1. How many farthings in l|d., l*d., Ifd., 2d., 2}d., etc., to 12d. ? 2. How many pence in 1/1, 1/2, etc., 2/1, 2/2, etc., to 20 ? 3. How many shillings in 1, Is. ; 1, 5s., etc. ; 2 ; 2, 7s ; 10 ? EXERCISE II. (1.) To pence 75; 352; 1001; 2450; 23, 10s; 179, 17s. ; 305, 19s. ; 5024, 15s. ; 734, 17s. 4d. ; 809, 10s. 8d. ; 2702, Os. lid. ; 6304, Is. 7d. (2.) To halfpence 5/, 7/, 13/, 8/2, 18/3, 14/7^, 53/8^, 15, 23, 17s., 27, 9s. 10d., 150, Os. 7d., 207, 19s. 0*d. (3.) To farthings-4/, 9/, 24/, 37/, 3/4*, 11/9J, 19AJ, 15/Ofc 29/10J, 72/8, 13/9|, 194/0*. (4.) To farthings 1. 93. 5. 39, 17. 9. 4, 17, 10. 13. 922, 10, OA. 2. 201. 6. 125, 8. 10. 172, 0, QL 14. 507, 19, 11. 3. 485. 7. 709, 10. 11. 250, 0, Oj. 15. 1854, 0, 3. 4. 7392. 8. 4890, 19. 12. 793, 15, 11J. 16. 3000, 10, 10J. 71. From a Lower to a Higher Name. Ex. To what sum of money are 37227 farthings equivalent ? Here we have to change the farthings to the highest name. We cannot do this at one step, as the number is too large ; we must therefore do it by several steps, first changing thj farthings to pence, then the pence to shillings, then the shil- lings to pounds, thus : To change for the far- 4 | 37227 9306 j = pence in the sum. 77(5 6 j = shillings, etc. in sum. things to pence, since it takes 4 farthings to make 1 penny, there be only one-fourth as many pence ^ 38 15 6 i=Punds, etc. in sum. as farthings ; which is got by dividing the number of pence by 4, giving 9306|d. To change the pence to shillings, since it takes 12 pence to make 1 shilling, there will be only one-twelfth as many shil- lings as pence ; which is got by dividing by 12, giving 775s. 64 MONEY. To change the shillings to pounds, since it takes 20 shillings to make 1 pound, there will be only one-twentieth as many shillings as pounds ; which is got by dividing by 20, giving ^38, 15s. 6|d. Rule. To change a sum of money from a lower to a higher name : Divide by the number of the lower contained in the next higher, and so on till the required name be reached. The result may be proved by changing back the pounds, shillings, and pence to farthings. If 37227f., when changed, give 38, 15s. 6|d., so must ,38, 15s. 6|d., when changed back again to farthings, give 37227f. EXERCISE III. 1. How many pence in 4 f. 5, 6, 7, etc., to 48 f. 2. How many shillings in 12d., 13d., etc., 24d., 25d., etc., to 240d. 8. How many in 20/, 40/, etc., 21/, 22/, etc., 30/, 31/, etc., to 200/. EXERCISE IV. 1. To shillings from farthings 912, 1344, 1680, 2352, 737, 501, 1079, 1893, 600, 903, 1807, 2356. 2. To shillings from halfpence 360, 432, 552, 768, 247, 301, 423, 593, 827, 1327, 1613, 2597. 3. To pounds from pence 6480, 2376, 4800, 11040, 35721, 60089, 23459, 45930, 49087, 780923, 56421, 93000. 4. To pounds from farthings 23496, 39408, 45082, 69857, 289508, 543306, 60085, 932092, 1000000, 2456793, 4560000, 5369480. COINS IN CIKCULATION TABLE II. From a Higher to a Lower Name. Ex. 1. In 7, how many half-crowns ? Since there are 8 half-crowns in l, 7 will have 8 times as many half-crowns, that is 7x8 or 56 half-crowns. Ex. 2. In 8 florins, how many groats ? Since there are 6 groats in 1 florin, 8 florins will contain six times as many groats, that is, 8X6, or 48 groats. EXERCISE V. 1 to cr. and fl. 75 ; 37, 15 ; 114, 10 ; 204, 5 ; 493, 10 ; 500. 2 to halfcr. and sixp. 83, 7/6 ; 52, 2/6 ; 94, 15 ; 173, 12/6 ; 79. 8 to gro. & threep. 13 ; 28, 10 ; 47, 18/6 ; 52, 10 ; 73, 8, 8. i. Half-sovereigns to halfcr. and shillings 59, 107, 293, 408, 96, 315. 73. REDUCTION. 65 EXERCISE VI. 1. Crowns to half-crowns and sixp. 345, 201, 793, 1248, 930, 300. 2. Crowns to shillings and groats 410, 293, 548, 702, 1564, 2738. 3. Half-crowns to shill. and threep. 450, 379, 901, 763, 1001, 2100. 4. Half-crowns to sixpences and pence 93, 58, 176, 290, 315, 728. EXERCISE VII. 1. Florins to shillings and groats 345, 290, 1000, 1293, 5681, 1807. 2. Shillings to sixpences and threep. 195/6, 37/6, 87/6, 27/6, 45/, 105/. 3. Shillings to groats 19/4, 25/8, 37/4, 56/4, 93/8, 82/4. 4. Shillings to threepences -63/9, 70/6, 82/3, 29/9, 71/6, 42/3. EXERCISE VIII. 1. Sixpences to pence and halfpence 378, 290, 573, 900, 1856, 2073. 2. Groats to halfp. and farthings 250, 316, 843, 569, 1789, 3476. 3. Threepences to pence and farthings 73, 101, 1236, 578, 1936, 2001. From a Lower to a Higher Name. Ex. 1. In 375 florins, how many pounds ? Divide the number of florins by 10, since there will be only one-tenth as many florins as pounds ; giving ,37, 10s. Ex. 2. In 720 pence how many crowns ? Divide the number of pence by 12 to bring it to shillings, and the shillings by 5, since there will be only one-fifth as many crowns as shillings ; giving 12 crowns. EXERCISE IX. 1. Halfpence to groats and shill. : 496, 728, 916, 236, 1020, 2000. 2. Pence to threep. and sixp. : 240, 324, 825, 113, 1562, 8249. 3. Farthings to groats : 960, 376, 420, 810, 1256, 9000. 4. Threepences to shillings : 240, 813, 190, 1000, 2483, 9267. EXERCISE X. 1. Shillings to florins and crowns 324, 290, 732, 1000, 2736, 5028. 2. Groats to shill. and half-crowns 298, 728, 1000, 2564, 4916, 952. 3. Pence to florins and crowns 934, 960, 2562, 8426, 3560, 5240. 4. Farthings to sixp. and florins-2456, 8400, 3000, 5249, 738, 7004. EXERCISE XI. 1. Florins to crowns and half-sovs. 1248, 4000, 1214, 793, 501, 910. 2. Sixp. to half-crowns and sovs. 317, 819, 1584, 4008, 704, 3048. 3. Halfpence to florins and sovs. 726, 8400, 906, 834, 2894, 5000. 4. Threepences to half-guineas- 493, 724, 840, 1000, 4934, 1960. 6. Farth. to groats and guineas- 8400, 9346, 7245, 2309, 6451, 8243. 74. 75. 66 MONEY. Coins not contained in each other." Ex. 1. In 36 crowns how many florins ? Change the crowns to shillings, and then the shillings to florins, giving 36 * 5 > or 90 florins. 2i Ex. 2. In 9 groats, how many threepenny-pieces ? Change the groats to pence, and the pence to threepenny- 9x4 pieces, giving '- or 12 threepenny-pieces. 3 Rule. Change the given name first to some lower name, which is contained evenly in the name required. EXERCISE XII. 1. Groats to threepenny-pieces 192, 252, 972, 396, 468. 2076. 2. Threepences to groats 708, 324, 96, 4782, 725, 589. 3. Crowns to florins 200, 370, 630, 1000, 484, 1297. 4. Florins to crowns 450, 995, 857, 500, 21170, 5000. 5. Half-crowns to florins 120, 840, 1000, 380, 296, 2483. 6. Florins to half-crowns 660, 3000, 1750, 475, 793, 215. 7. Pounds to guineas- 621, 793, 800, 1750, 2000, 576. 8. Guineas to pounds 347, 725, 240, 2000, 152, 937. EXERCISE XIII. MISCELLANEOUS. 1. A owes me 72, 19s. 3d. ; B 192, 16s. 9fd. ; c 258, 10s. Oid. I have goods worth 174, 16s, 4d., and in the bank, 62, 18s. 7d. : what am I worth in all ? 2. My butcher's bill is 7, 19s. 6d. ; my baker's, 9, lls. 5d. ; my grocer's, 11, 15s. Ofd. ; my green grocer's, 1, 17s. 6d. ; my shoemaker's, 1, 5s. 6d. ; and my tailor's, 2, Os. ll^d. : what sum will pay the whole ? 3. A house has three storeys, of which the rent of the first is 60, 10s. ; of the second, 42, 7s. 6d. ; and of the third, 25, 5s. : what is the entire rent ? 4. What was lost on a cargo which cost 1749, 14s. 6d., and sold for 1393, 2s. lOf d. ? 5. A workman's weekly wage is 33/, what must he spend to save 4/9 a week ? 6. If I send my servant to pay an account of 17/6 for bread, 9/8 for butter, 2/4 for eggs, and 3/9| for vegetables, and give him a five-pound note, what should he bring back ? 7. What cost 27 yards silk at 1, 13s. 5d. per yard ? 8. What cost 36 tb tea at 3/9^ per Ib ? 9. What cost 11 sheep at 1, 18s. 5d. each ? 10. What cost 16 stones sugar at 7/10% per stone ? 11. What cost 21 tons iron at 3, 7s. 3|d. per ton ? 12. What cost 15 oz. silver at 5/3 1 per ounce ? 13. If 39 yards cloth cost 1, 17s. 4*>d., what is that per yard ? 14. If 26 Ib tea cost 4, 16s 5d. what is that per ft } MISCELLANEOUS EXERCISES. 67 15. If 1 cwt. of sugar (112 Ib) cost 2, 13s. 8<L, what is that per tb? 16. If a quarter of wheat (32 pecks) cost 52/, what is that per peck? 17. If a cask of wine (140 gallons) cost 116, 13s. 4d., what is that per gallon ? 18. If 7 doz. sherry cost 11, 4s., what is that per bottle ? 19. A gentleman gave 6d. each to a number of poor persons : how many would he relieve with 100 ? 20. If I hold in my hand one of all the coins in use, and add a guinea to them, how much have I ? 21. What is the annual income of an art-union which has 963 guinea subscribers ? 22. A ploughman's wages are 5 guineas a quarter ; he receives also a yearly allowance of 6, Os. 3d. for oatmeal, and 40/ for potatoes : what are his yearly wages ? 23. A family finds its monthly account with the grocer as follows : 1 R> tea at 4/ ; 6 lb sugar at 5d. per lb ; 3^ lb soap at 4d. per ib ; 1 lb soda at Id. ; 4 lb butter at Is. 3d. per lb ; 4 lb cheese at lid. per ft) ; 1 lb currants at 6d. ; 1 Ib raisins at 8d. ; lb almonds at 4s. per lb : what is it in all ? 24. A draper has in bank 39, 14s. 6d. ; goods to the value of 136, 15s. O^d. ; and credits for 53, Os. ll|d. ; but he owes 47, 16s. 2d., and his bad debts amount to 11, 4s. 3$d. : what is he worth? 25. A merchant borrowed 700 ; he has paid three instalments of 150 : what does he still owe, allowing 33, 5s. for interest ? 26. What cost 25 hogsheads beer at 4, 2s. 8d. each ? 27. What cost 33 bales cotton at 7, 15s. 9d. each ? 28. What cost 58 yards cloth at 18/9 per yard ? 29. If a year's wage is 114, what is that per week and per day? 30. If 2 chests tea, each 40 lb weight, cost 16, 3s. 4d., what is that per lb ? 31. If 19000 cubic feet gas cost 5, Is. 4d., what is that per 1000 cubic feet ? 32. How much has a tradesman drawn during the day, who finds in his drawer at night 1 sovereign, 3 half-sovereigns, 7 crowns, 9 half- crowns, 3 florins, 38 shillings, 14 sixpences, 9 groats, 6 threepennies, with 66 id. in copper ? 33. A Frenchman, about to travel in England, changes 7000 francs into English sovereigns : how many does he receive at 25 francs for one sovereign ? 34. The money raised in a penny subscription, which had 12936 names, was divided into three equal shares : what was the amount of each? 35. A tradesman draws on Monday, 2, 13s. 5d. ; on Tuesday, 1, 19s. 7|d. ; on Wednesday, 2, Os. 9^d. ; on Thursday, 1, 15s. 3d.; on Friday as much as on Wednesday and Thursday together ; and on Saturday twice as much as on Wednesday : what were his week's drawings ? 36. A merchant bought tea for 259, 19s. 3d., sugar for 192, Os. 5|d., and coffee for 207, Us. 6d. : what must he sell the whole for to gain one-fourth of what he paid for them ? 37. I have 65 guineas, and my friend 60 and 60 crowns : what ii the difference between us ? 68 MONEY. 38. A sum_of 2765, 10s. is to be divided between A, B, and c ; A gets 459, 15s., B twice as much : what remains forc.'s share ? 39. I exchange 82 cwts. sugar at 5, 2s. llgd. per cwt. for 19 chests tea at 23, 15s. 6^d. : how much should I pay besides ? 40. If 1 have put 9 into the post-office savings'-bank during a half- year, what have I saved per week ? 41. A house rented at 62, 10s. sold for 1125 : how many years' purchase was that ? 42. A farm of 73| acres is rented at 138, 5s. 7 Ad. : what is that per acre ? 43. The old pound Scots was 1/8 : how many in 250 ? 44. I paid an account with 25 half-guineas, 25 half-sovereigns, 25 half-crowns, and 25 sixpences : what was its amount? 45. The visitors to a menagerie were 153 at 2/6, 439 at I/ with 52 at half-price, 736 at 3d., and 237 scholars at 2d. each : how much money did it draw in all ? 76. EXERCISE XIV. 1. The receipts of a railway for the first week of February were 2075, 16s. 2d. ; for the second, 2192, 19s. 8id. ; for the third. 1989, Os. P|d. ; and for the fourth, 2530, 17s. : what were its ave- rage weekly receipts for the month ? 2. A bankrupt paid 5/3.^ per pound on a debt of 425, what was his estate ? and how much does a creditor lose on a claim of 37 ? 3. Bought 45 railway shares at 23, 10s., and sold out at 25, 16s. 6d : what did I gain? 4. At a collection s* a church door there were in the plate 375d. 749 halfpence, 45 groats, 28 threepences, 7 sixpences, and 3 shillings : how much in all ? 5. I bought a book in 3 volumes at half-a-guinea a volume ; dis- counting of the price, what did I pay ? 6. A dinner-bill for 23 persons came to 8, 12s. 6d. ; if five were guests, what had each of the others to pay ? 7. A draper bought 37 yards cloth at 7/9^ per yard ; if he gained 30/, what did he sell it at per yard? 8. Bought oxen and lambs for 193, 17s. 6d. ; if the oxen cost double of the lambs, what cost each? 9. A factory consumes 11 tons coal per week at 9/7^ per ton, what is its annual outlay for coal ? 10. In 93 American dollars ($1 = 4/2), how much sterling money? 11. A public work employs 25 labourers at 13/6 a week, and 15 at 15/9 : what sum is expended annually in wages ? 12. If an apprentice's wages are 4/6 a week the first year, and are advanced 1/6 a week each of his five years' service : how much does he receive in all ? 13. What is my clear income, if I am assessed 3/4A per pound on 375? 14. If a tradesman's wages are 95 per annum, what should be his daily expenditure to save 10 a year ? 15. A farmer's profits for 1860 were 407, 11s. 6d. ; for 1861, 493, 2s. 8d. ; and for 1862, 430, 3s. lOd. : how much does the increase for 1861 exceed that for 1862 ? MISCELLANEOUS EXERCISES. 69 ' 16. Bought 38f yards at 17/6J per yard ; retaining 5J yards, I sold the rest so as neither to gain nor lose ; how did I sell it ? 17. The stipend of 153 clergymen is 150 each, but there is a fund of 4082, 15s. available for equal distribution among them : to what does that bring up the stipends ? 18. If the amount of deposits in a savings'-bank is 15645, 14s. 3d., and the number of open accounts 935 : find the average amount of each. 19. Divide a legacy of 3000 among 3 sons and 4 daughters, so that each son shall receive twice as much as each daughter. 20. If 2500 persons cross Waterloo Bridge daily, paying a toll of id. : how much is raised yearly ? 21. My bank-book for April shows these entries April 3, paid in 13/6 ; April 9, paid in 7/10 ; April 16, received 7/3 ; April 23, paid in 5/6 ; April 30, received 14/9 : find the increase to my credit forthe month. 22. What is the annual saving to the owners of a factory employ- ing 550 hands, if wages fall three-halfpence a day ? , 23. My gas account for the last quarter was 7500 cub. feet at 4/7J per 1000 : what had I to pay ? and what will I save next year, if I burn 1500 cub. feet less each quarter? . 24. A grocer mixes 12 gallons whisky at 18/6 with 18 at 16/6, and 15 at 14/6 : find the value of the mixture per gallon. 25. I hold a sum of money, consisting of 5 five-pound notes, 25 sovereigns, 25 half-sovereigns, and 16 half-crowns, of which three- fifths belong to a friend ; how much is mine ? 26. A merchant who began business with a clear capital of 2396, 15s. 6d., increased it by one-third for three successive years : what was it at the end of that time ? 27. A, B, and c subscribe to a venture of 7260 in 10 shares. A has 2, B 3, c 5 shares : what did each subscribe ? 28. A merchant bought 7 chests tea, each 48 Ibs., for 73, 8s. 3d. Three of the chests he sells at 4/6 per Ib : what must he sell the rest at to gain 4, 18s. on the whole? 29. If my income is 500 gs. a year, what income-tax do I pay at 7d. per pound, and how much more would I pay at 7d. per pound ? 30. An hospital contains 125 boys. At the beginning of the year 16 leave, and 11 new boys are admitted : what will be its expenditure for the year, if each boy costs 11, 17s. 4d. 31. Farmer A drove to market 8 oxen, which he sold at 15, 10s. a head, and two score of sheep which he sold at 1, 18s. a head ; but he bought a horse for 27 gs., and a gig with harness for 11 gs. : what money did he bring home ? 32. Fanner B bought 23 oxen at 9, 10s. each. One of them died ; but he fattened the rest at an expense of 25/ each, and then sold the lot for 300 guineas : what did he gain on each ? 33. A merchant paid a bill of 257, 10s. Of that 94, 15s. was for sugar at 5d. per lb, and the rest for tea at 5/6 per Ib : how much of each did he buy ? 34. In a school of 350 children, the quarterly fee is 5/6 for the first division, which contains 108, 4/9 for the second, which contains 236, and 3/ for the third, which contains the rest : how much is drawn in fees for the quarter, and what would be gained if they were raised 6d., 9d., and I/ for the first, second, and third divisions respectively ? 35. A cheque for 63, 15s. is paid in an equal number of crowns, half-crowns, and shillings : how many of each ? 70 WEIGHTS AND MEASURES. WEIGHTS AND MEASURES. General Rule. For adding, subtracting, multiplying, dividing, and reducing weights and measures, the rules are the same as for performing these operations with money. Avoirdupois Weight Table III. EXERCISE I. 1. 2 i. 3. 4. 5. 6. T. cwt. qr. cwt, qr. Ib qr. Ib oz. T. cwt. qr. cwt. qr. Tb Ib oz. dr. 27 15 2 13 1 18 2 25 14 84 13 2 14 2 15 26 13 12 45 17 1 17 3 15 1 20 11 93 17 1 16 3 12 20 11 10 83 9 3 19 2 16 3 9 10 60 10 3 908 15 9 1 56 3 8 3 7 2 18 6 74 19 5 1 13 17 15 9 70 8 2 11 11 13 13 26 8 3 17 2 11 8 10 4 92 19 1 14 1 27 3 21 15 45 15 2 19 1 9 14 7 13 7. 8. 9. 10. 11. 12. T. cwt. qr. cwt. qr. Tb qr. Tb oz. T. cwt. qr. cwt. qr. Ib qr. Ib oz. dr. 73 19 1 14 2 19 3 17 11 115 10 3 10 3 12 2 20 7 9 29 7 2 9 3 17 1 23 12 79 15 2 7 3 15 1 23 13 14 13. 36 tons 14 cwt. 3 qr. +17, 18, l, + 94, 10, 3 + 7, 2, 1. 14. 15 cwt. 1 qr. 27 lb + 16, 0, 20 + 8, 2, 19 + 19, 3, 23. 15. 1 qr. 20 Ib 12 oz. + 3, 14, 14 + 2, 19, 7 + 3, 24, 13. 16. 275 tons 1 cwt. 2 qr. + 193, 14, 2 + 400, 15, 2 + 640, 18, 1. 17. 13 cwt. 2 qr. 19 lb + 8, 3, 14 + 19, 2, 11 + 11, 1, 15. 18. 2 qr. 24 Ib 7 oz. 8 dr. + l, 20, 14, 12 + 3, 10, 11, 4 + 1, 8, 7, 11. 19. 157 tons 10 cwt. 2 qr.- 59, 15, 3. 22. 81 tons 11 cwt. Iqr.- 37, 14, 3. 20. 13 cwt 2 qr. 20 Ib - 9, 3, 21. 23. 7 cwt. 3 qr. 14 Ib - 4, 3, 19. 21. 3 qr. 15 Ib 12 oz. - 1, 17, 9. 24. 113 Ib 14 oz. 7 dr. - 75, 15, 10. Note on Subtraction Instead of saying 15 dr. from 9 dr. Tb oz. dr. cannot be taken, change one of the oz., making 25 dr. in all ; 18 11 9 15 from 25 leaves 10 dr., and so on ; it is simpler, in this and 11 14 15 some of the following tables, to say, 15 dr. from 16 leaves 1, ~~Q 12 10 1 and 9 are 10 ; 14 oz. from 16 oz. leaves 2 oz. which with 10 oz. makes 12 oz., etc. That is to say, instead of adding one of the higher name to the lower befo^t subtracting, we may subtract at once from one of the higher name, and add the difference to the number of the lower in the minuend. EXERCISE II. 1. 13 tons 14 cwt. 2qr. x 6, 15, 32i, 63. 2. 5 cwt. 3 qr. 16 Ib x 5, 21, 36|, 79. 3. 3 qr. 9 Ib 14 oz. x 11, 72, 87, 108. 4. 193 tons 19 cwt. 3 qr. 91b x 144, 172, 360. 6. 15 cwt. 1 qr. 24 Ib 7 oz. x 96, 473, 840. 6. 1 qr. 17 Ib 11 02. 15 dr. x 120, 285, 793. WEIGHTS AND MEASURES. 71 7. 843 tons 15 cwt. 8. 25 tons 1 cwt. 1 or. 20 Ib 8, 36, 32j, 193. 9. 61 cwt. 3 qr. 14 ft 10. 173 tons 5 cwt. 11. 83 tons 15 cwt. 3 qr. 3 Ib 8 oz. 12. 10 cwt. 3 qr. 19 Ib 4 oz. 7 dr. 4, 15, 62i, 279. 12, 99, 18$, 370. 10, 21, 132, 562. 84, 390, 821. -49, 913, 770. 78 EXERCISE III. 1. 79 tons to Ib 10. 27645780 oz. to cwt. 2. 25 tons 7 cwt. 18 Ib to oz. 11. 72480 oz. to dr. 3. 19 cwt 1 qr. 21 Ib 9 oz. to oz. 12. 17250 Ib to T. 4. 2 qr. 15 Ib to dr. 13. 9456 dr. to oz. 5. 27 ft) 15 oz. 12 dr. to dr. 14. 694721 oz. to T. 6. 10 cwt. to stones. 15. 932450 dr. to cwt. |7. 127 tons 15 cwt. 1 qr. to ft 16. 57289 oz. to qr. 8. 57 Ib to dr. 17. 123456 oz. to cwt. 9. 15 cwt. 7 Ib to oz. 18. 93000 Ib to T. Troy Weight Table IV. EXERCISE IV. 1. 2. 3. 4. 5. Ib oz. dwt. oz. dwt. g r ft oz. dwt. gr. Ib oz. dwt. oz. dwt. gr. 125 9 15 27 15 21 29 8 14 22 193 8 13 85 14 19 27 11 19 36 11 20 36 4 12 15 59 11 15 19 15 20 98 5 10 9 14 11 8 2 10 6. 7. 193 10 230 13 11 50 30 3 10 8 50 79 7 11 7 18 8 Ib oz. dwt. gr. 193 7 13 15 ft. 96 oz. dwt. gr. 10 13 20 79 7 11 14 5 10 13 23 85 11 9 19 37 11 7 21 8. 731blOoz. 14 dwt. +94, 9, 9 + 150, 2, 11 + 8i 11, 15. 9. 11 oz. 14 dwt. 20 gr. + 19, 12, 16 + 24, 11, 22 + 30, 7, 18. 10. 453 ib 11 oz. 17 dwt. -87 Ib 11 oz. 19 dwt. 11. 285 oz. 11 dwt. 17 gr. -97 oz. 12 dwt. 23 gr. 12. 64 Ib 7 oz. 13 dwt. 17 gr. -28 Ib 10 oz. 15 dwt. 20 gr. EXERCISE V. 1. 23 ft> 5 oz. 11 dwt. x 7, 56, 130, 257. 2. 54. oz. 13 dwt. 20 gr. x 9, 24, 560, 365. 3. 18 Ib 10 oz. 14 dwt. 22 gr. x 11, 150, 4/9. 4. 136 Ib 10 oz. 14 dwt. -f- 8, 42, 70, 192. 5. 172 oz. 10 dwt. 15 gr. 3, 81, 92, 268. 6. 93 Ib 2 oz. 17 gr. -f- 108, 236, 807, EXERCISE VI. 1. 10 oz. 15 dwt. to gr. 7. 5932 gr. to oz. 2. 7 Ib 8 oz. to dwt. 8. 2400 dwt. to Ib 3. 21 Ib 9 oz. 18 dwt. to gr. 9. 29324 gr. to Ib 4. 14 Ib to gr. 10. 7256 gr. to dwt. 5. 8 oz. 15 dwt. to dwt. 11. 10000 dwt. to Ib 6. 5 Ib 11 oz. to gr. 12. 9000 gr. to oz. , 72 79. 1. yds. ft in. 118 2 7 72 1 8 92 6 240 1 10 74 2 8 154 1 11 6. 237 1 8 194 2 9 WEIGHTS AND MEASURES. Lineal Measure Table V. EXERCISE VII. 2. 3. 4. ml. 93 fur. 5 po. 22 po. yds. ft. 87 4 2 fur. 82 po. yds. 24 2 118 2 15 36 2 1 25 20 4 70 10 59 5 2 90 1 4 120 7 11 89 1 61 17 3 81 6 4 94 3 2 35 10 1 59 4 17 16 4 1 70 6 2 7. 8. 9. 56 5 20 37 2 1 85 21 5 28 6 18 18 3 2 25 22 5 ;o. yds. ft. in. 7 3 2 11 27 4 2 10 8518 36 2 5 60 1 2 9 1427 10. 17 3 2 10 8 3 2 11 11. 27 yds. 1 ft. 3 in. + 58, 2, 8 + 37, 2, 5 + 84, 1, 11. 12. 84 ml. 3 fur. 20 po. + 17, 7, 8 + 29, 5, 15 + 47, 6, 17. 13. 28 po. 3 yds. 1 ft. +54, 2,.2 + 30, 5, 2 + 25, 1, 2. 14. 19 fur. 20 po. 2 yds. +17, 10, 1+24, 15, 3 + 49, 10, 2. 15. 70 yds. 2 ft. 3 in. + 39, 1, 10 + 25, 2, 8 + 40, 1, 7. 16. 418 yds. 2 ft. 7 in. -250, 2, 11. 17. 73 ml. 2 fur. 15 po. -38, 7, 10. 18. 47 po. 3 yds. 1 ft. - 38, 1, 2. 19. 56 fur. 3 po. 2 yds. -27, 15, 3. 20. 18 po. 2 yds. 1 ft. -7, 3, 2. 21. 290 yds. 1 ft. 11 in. - 49, 2, 11. EXERCISE VIII. 1. 37 yds. 2 ft. 8 in. 2. 90 ml. 7 fur. 24 po. 3. 82 po. 5 yds. 1 ft. 4. 25 fur. 20 po. 4 yds. 1 ft. 5. 13 po. 2 yds. 1 ft. 2 in. 6. 64 yds. 2 ft. 11 in. 7. 160 yds. 2 ft. 2 in. 8. 129 ml. 7 fur. 20 po. 9. 137 po. 4 yds. 2 ft. 6 in. 10. 68 fur. 29 po. yd. 1 ft. 6 in. 11. 1 fur. 35 po. 12. 292 yds. 1 ft. 6 in. x 6, 49, 315. x 9, 24, 482. x 56, 800, 493. x 81, 720, 848. x 18, 350, 925. x 36, 450, 637. -+2, 22, 118. H-4, 15, 110. ~6, 10.4, 236. -M3.4, 210, 375. .5-22$, 150, 561. 4-204J, 405, 914. 1. 14 ml. 5 fur. to yds. 2. 7 fur. 25 po. to ft. 3. 4 yds. 2 ft. to in. 4. 29 po. 3 yds. to in. 5. 30 ml. 25 po. to ft. 6. 17 po. 4 yds. 2 ft. to ft. 7. 5 fur. 39 po. 2 yds. to yds. 8. 58 fath. 3 ft. to ft. 9. 13 hands 2 in. to in. EXERCISE IX. 10. 11. 12. 13. 14. 15. 16. 17. 18. 2597 yds. to fur. 9256 po. to ml. 29738 ft. to ml. 2500 ft. to fath. 375960 yds. to ml. 593 in. to yds. 63 in. to hands. 2570085 ft. to fur. 47268 in. to po. COMPOUND RULES. 73 80. Cloth Measure Table VI. EXERCISE X. 1. 2. 3. 4. 6. yds. qr. nl. ells qr. n: yds. qr. in. ells qr. nl. yds. qr. nL 256 2 3 73 4 2 64 2 7 192 4 2 74 2 1 93 1 2 19 2 1 29 8* 96 4 3 47 3 2 80 3 1 156 1 3 158 3 90 2 72 1 51 yds. qr. in. 100 22 118 4 1 28 3 6i (5.) 315 2 1 (7.) 85 3 7A 56 1 3 54 2 2 58 7 196 43 47 3 8j 8. 384 yds. 2 qr. 1 nl. +79, 1, 3 + 74, 3, 2 + 17, 1, 2. 9. 97 ells, 4 qr. 3 nl.+17, 3, 2 + 39, 1, 1 + 14, 3, 2. 10. 114 yds. 3 qr. 7 in. +200, 1, 2^ + 74, 3, 4 + 92, 1, 6J. 11. (1.) 94 ells 3 qr. 3 nl. -47, 4, 1. (2.) 173, 4, 1-85, 4, 3. 12. (1.) 83 yds. 2 qr. 2 nl. - 29, 3, 1. (2.) 92, 3, 6^ in. - 56, 2, 7J in. EXERCISE XI. 1. 16 yds. 3 qr. 3 nl. x 6, 15, 87. 4. 17 yds. 2 qr. 2 nl.^-6, 47, 71. 2. 38 ells 4 qr. 2 nl. x 9, 24, 123. 5. 141 ellsS qr. 3 nl.^-7, 81, 156. 8. 73 yds. 2 qr. 8 in. x 12, 63, 274. 6. 74 yds. 2 qr. 6 in. 4-8, 84, 121. EXERCISE XII. 1. 7 yds. 3 qr. to nl. 7. 756 nl. to yds. 2. 56 ells to nl. 8. 1000 nl. to ells. 3. 3 qr. 2 nl. 1 in. to in. 9. 250 in. to yds. 4. 240 ells to yds. 10. 680 yds. to ells. 5. 37 ells 3 qr. to yds. 11. 2764 nl. to ells. 6. 18 yds. 2 qr. 7 in. to in. 12. 1296 in. to yds. 81. Land Measure Table VII. EXERCISE XIII. 1. 2. 3. 4. ml. fur. eh. Ik. ml. far. ch. fur. eh. Ik. ml. fur. ch. Ik. 3 7 8 50 13 6 7 6 8 70 63 7 8 70 19 5 9 64 49 2 8 5 9 60 25 7 9 83 72 1 6 36 25 3 7 3 7 54 25 6 7 90 19 7 9 7 6 26 5. 11 2 5 46 10 5 4 9 30 171 6 9 45 7 3 8 54 766 1 5 20 83 7 8 93 6. 8 fur. 7 ch. 60 Ik. +7, 8, 45 + 5, 3, 28 + 2, 5, 73. 7. 7 ml. 2 fur. 8 ch. 50 Ik. + 19, 7, 5, 60 + 25, 3, 9, 80. 8. 34 ml. 7 fur. 6 ch. 40 Ik. -15 ml. 7 fur. 8 ch. 90 Ik. 9. 154 ml. 2 fur. 5 ch. 85 Ik. - 76 ml. 6 fur. 5 ch. 86 Ik. WEIGHTS AND MEASURES. EXERCISE XIV. ml. fur. ch. Ik. nil. fur. ch. Ik. 1. 7 8 45x5,36,120. 4. 8 64-4,21,56. 2. 3 2 9 60x7,96,213. 5. 15 3 9 848,63,123. 3. 7 6 8 56x9,132,260. 6. 16 5 9 50-10,19,235. EXERCISE XV. 1. 150 ml. 47 ch. to Ik. 5. 2596 Ik to fur. 2. 7 fur. 3 ch. to Ik. 3. 25 ml. 70 ch. to ch. 4. 347 ml. to Ik. 6. 9000 Ik. to ml. 7. 586 ch. to ml. 8. 8256 Ik. to ch. 82. Square Measure Table VIII. In the surface A B c D, let its length A D be 12 inches, and its breadth AB be 12 inches; then the surface contains 144 parts, 1-inch long and 1-inch broad, or, as they are called, 144 square inches. The area is found by multiplying the length by the breadth. The figure A B c D, being one foot long and one broad, is one square foot, which measure there- fore contains 144 sq. inches. Any area containing 144 square inches is regarded as one square foot ; e.g.,, a figure 18 inches long by 8 inches broad. A D F Observe that, whilst one square foot means one foot measured every way, or one foot square, any other number of square feet does not mean the same number of feet square. Thus, in Fig 2, if A B c D is one square foot, and D c E F is another part equal to it, then the whole A B E F is two square feet. But in Fig. 3, if the length A F is two feet, and the breadth AG also two feet, the figure is two feet each way, or two feet square, which, as we see, contains four square feet. COMPOUND RULES. 75 EXERCISE XVI. 1. IMS. ro. po. 39 3 29 57 2 20 93 15 64 2 39 12 1 12 27 3 17 2. ac. ro. po. yds. 172 2 34 24 85 27 20 276 1 11 15 93 3 7 8 57 2 18 17 190 1 22 10 3. ro. po. yds. ft. 3 25 8 8 2 17 21 5 3 3 7 7.4 1 28 27 6j 2 30 19 4| 14 22 3j 4. po. yds. ft. in. 24 20 8 47 20 18 1 98 17 5 3 87 36 27 6 79 8 9 2 24 24 25 4 110 5. yds. ft. in. 25 7 130 18 8 94 7 6 56 30 8 104 15 5 85 9 2 118 6. ac. ro. po. 93 2 18 47 3 20 7. ac. ro. po. yds. 256 3 36 25 98 3 38 29 8. ro. po. yds. ft. 3 27 21 7 1 19 26 8 9. po. yds. ft. in. 36 25 1 93 19 26 100 10. yds. ft. in. 25 7 110 18 8 129 11. 24 ac. 2 ro. 27 p. + 194, 3, 30 + 98, 1, 25+100, 1, 36. 12. 25 yds. 8 ft. 100 in. +11, 7, 94 + 56, 2, 120 + 62, 5, 85. 13. 2ro. 25 po. 26 yds. +1, 19, 13 + 3, 11, 14 + 1, 39, 29. 14. 36 po. 25 yds. 3 ft. +17, 13, 4 + 28, 30, 6 + 19, 26, 4. 15. 156 ac. 3 ro. 26 p. 7 yds. -98 ac. 2 ro. 27 po. 18 yds. 16. 258 ac. 2 ro. po. 15 yds. - 89 ac. 1 ro. 5 po. 26 yds. 17. 125yds. 8 ft. 56 in. - 16 yds. 8 ft. 100 in. 18. 25 po. 30 yds. 6 ft. -16 po. 30 yds. 7 ft. EXERCISE XVII. 1. 136 ac. 3 ro. 27 po. x 7, 24, 73. 2. 16 ac. 2 ro. 3p. 16 yds. x8, 91.4, 540. 3. 2 ro. 14 po. 25 yds. 6 ft. x 11, 84, 837. 4. 25 po. 18 yds. 3 ft. 110 in. x 35, 270, 492. 5. 7 yds. ft. 93 in. x 16, 105, 308. 6. 7 ac. 3 ro. 35 po. 3, 25, 85. 7. 8 ac. 1 ro. 11 po. 21 yds. 2 ft. 36 in. 4, 93 j, 256. 8. 1 ro. 6 po. 1 yd. 7 ft. 72 in. 12, 33, 324. 9. 22 yds. 2 ft. 6 in. 96, 300, 849. 10. 205 yds. 7 ft. 72 in. -72, 185|, 620. EXERCISE XVIII. 1. 27 sq. yds. 8 sq. ft. 90 sq. in. to sq. in. 10. 2. 191 ac. 3 ro. 31 po. to po. 11. 3. 25 ac. 1 ro. to sq. yds. 12. 4. 75 sq. ml. to ac. 13. 5. 84 ac. to sq. Ik. 14. 6. 19 p. 25 yds. 8 ft. to sq. in. 15. 7. 101 ac. 27 per. to sq. yds. 16. 8. 93 ac. 1 ro. 21 per. 9 yds. to sq. yds. 17. 9. 7 sq. yds. 8 sq. ft. 120 sq. in. to sq. in. 18. 50000 sq. in to sq. yds. 97326 sq. yds. to ro. 858 po. to ac. 29682 sq. ft. to ro. 8256000 sq Ik. to ac. 84720 sq. in. to po. 27384 sq. yds. to per. 139286 sq.in. tosq.yds. 76536 ac. to sq. ml. Cubic or Solid Measure Table IX. A piece of wood or brick 1 inch long, 1 inch broad, and 1 inch thick (or deep), that is, 1 inch every way, is called 1 cubic TO WEIGHTS AXD MEASURES. inth. If in Fig. 1, sect. 82, each square inch had 1 cubic incK placed upon it, the square foot would contain 144 such inches ; another layer would make 2 times 144, a third 3 times 144, and so on, till 12 layers would give 12 times 144 such inches, or 1728. But 12 such layers would reach 12 inches, or 1 foot high, and the figure would now be 1 foot every way, or 1 cubic foot, which therefore contains 1728 cubic inches. Similarly, u 1 yard contain 3 feet, 1 cubic yard will contain 3 feet every way ; that is, 3x3x3, or 27 cubic feet. EXERCISE XIX. 1. 2. 3. 4. 5. c. yd. c. f. c. in. yds. ft. in. yds. ft. in. yds. ft. in. yds. ft. in. 7 13 356 136 16 460 36 9 300 173 20 892 250 17 800 8 26 938 282 9 200 42 8 154 49 24 900 92 18 948 5 15 701 325 8 154 29 18 408 6 10 1263 482 21 938 54 24 293 6. 7. 9 7 1564 254 17 1628 18 17 567 538 25 1130 814 19 710 4 22 842 123 4 801 62 6 482 299 26 1628 298 20 1260 8. 248 c. yd. 14c. ft. 309 c. in. + 159, 24, 560 + 78, 15, 914, + 82, 3, 284. 9. 19 yd. 16 ft. 847 in. +34, 19, 936 + 22, 23, 1000 + 36, 19, 100. 10. 247 yd. 19 ft. 560 in. - 198, 24, 700. 11. 72, 25, 1608 - 27, 26, 1700. EXERCISE XX. c. yd. ft. in. c. yd. ft in. 1. 17 21 1500x6, 56, 13i, 156. 4. 26 2 1488-3, 7i, 87, 420. 2. 23 24 900x9, 84, 26.J, 632. 5. 160 2 1584-5, l2i, 36, 191. 3. 79 11 372 x 132, 720, 365, 800. 6. 92 24 576-56, 120, 43, 321. EXERCISE XXI. 1. 69 cub. ft. to cub. in. 9. 2576 cub. ft. to cub. yd. 2. 75 cub. yd. to cub. ft. 10. 14850 cub. in. to cub. ft. 3. 291 c. yd. 19 c. ft. to c. in. 11. 235 cub. ft. to B.b. 4. 17 lo. rough to cub. ft. 12. 1250 cub. ft. to lo. rough. 5. 189 T. sh. to cub. ft. 13. 6728 cub. ft. to T. sh. 6. 34 lo. hewn to cub. ft. 14. 9362 cub. ft. to lo. hewn. 7. 129 B.b. to cub. ft. 15. 795 cub. ft. to B.b. 8. 457 T. sh. to cub. ft. 16. 15728 cub. in. to cub. ft. 84. Measure of Capacity Table X. 5. qrs. Ira. pk. gal 114 6 3 1 84 4 2 74 7 1 56 5 3 1 90 1 2 28 6 3 1 EXERCISE XXII. 1. 2. 3. 4. qts. pt. gj. galls. qt. pt. bu. pk. gall. qrs. 1m. pt 17 1 3 48 3 1 120 3 1 78 7 3 6 2 27 2 1 59 2 84 5 1 9 1 3 36 1 78 3 1 93 6 2 7 1 1 42 3 24 1 1 27 I 2 14 2 20 2 1 143 3 1 56 4 3 13 1 3 7 3 84 2 49 2 3 C'UMrOUiN'D itULES. 7 i 6. 7. 8. 9. 10. qts. pts. gt. galls, qts. pts. bu. pk. gal. qrs. bu. pk. qrs. bu. pk. gal. 42 02 161 3 130 2 1 275 62 314 5 2 1 27 1 3 89 3 1 65 3 1 118 7 3 283 6 3 1 11. 94 galls. 3 qr. 1 pt. 2 gi. + 47, 2, 0, 3 + 84, 3, 1, 2. 12. 82 bu. 2 pk. 1 gall. + 56, 1, 1 + 70, 3, 1 + 62, 2, 0. 13. 156 qrs. 7 bu. 2 pk. 1 gall. +273, 6/1, + 193, 4, 3, 1. 14. 83 gal. 1 qt. pt 1 gi. -65, 1, 1, 3. 16. 72 galls. 3 qt. -25, 3, 1. 15. 64 bu. 1 pk. 1 gall. - 38, 2, 1. 17. 125 qrs. 6 bu. 2 pk. 1 ga. - 84,7,3. EXERCISE XXIII. 1. 25 galls. 2 qt. 1 pt. x5, 14J, 36, 93. 2. 64 bu. 3 pk. 1 gall. x 7, 24|, 63, 124. 3. 5 qr. 4 bu. 2 pk. x 9, 42, 85, 250. 4. 9 qts. 1 pt. 3 gi. x!2, 70, 101, 339. 5. 156 qrs. 4 bu. 3 pk. 1 gall, x 11, 108, 700, 413. 6. 52 galls. 2 qt. 4-6, 15, 84, 91. 7. 229 bu. 2 pk. -$-4, 27, 17, 130. 8. 508 qrs. 6 bu. -f-8, 55, 37, 185. 9. 2569 qts. 1 pt. 3 gi. 4-3, 77, 89, 400. 10. 1855 qrs. 7 bu. 3 pk. 1 gall.-f-7, 71, 239, 372. EXERCISE XXIV. 1. 5 galls. 3 qts. to pts. 10. 2572 gi. to qts. 2. 2 qts. 1 pt. 3 gi. to gi. 11. 593 pts. to galls. 3. 14 pk. 1 gdl. to pts. 12. 1876 qts. to pks. 4. 7 bu. to galls. 13. 705 galls, to bu. 5. 12 qrs. 6 bu. to pks. 14. 193 pks. to qrs. 6. 13 galls. 1 pt. 2 gi. to gi. 15. 628 pts. to pks. 7. 9 pks. 1 gall. 3 qts. to qts. 16. 3000 gi. to galls. 8. 5 bu. 3 pks. to galls. 17. 484 pks. to bu. 9. 284 qrs. 5 bu. 3 pks. to galls. 18. 1608 galls, to qrs. 35. Measurement of Time Table XI. EXERCISE XXV. 1. 2. 3. 4. 5. ho. min. sec. da. ho. min. wk. da. ho. yrs. \vk. da. ho. min. sec. 19 41 50 84 10 30 41 6 20 28 10 4 18 48 42 3 25 45 150 14 24 25 4 10 94 30 6 27 13 21 17 9 11 79 20 18 43 1 9 15 45 3 9 35 17 10 30 8 148 13 49 9 3 18 8 24 1 24 8 9 9 6 29 64 8 27 17 5 15 49 36 5 19 51 37 12 58 37 293 15 11 50 4 16 11 9 4 36 29 1 6. 7. 8. 8. 10. ho. min. sec. da. ho. min. -wk. da. ho. yrs. wk. da. ho. min. sec. 36 41 29 93 12 54 36 4 12 152 41 4 27 32 18 19 50 30 29 17 58 18 6 19 76 49 6 8 45 30 78 WEIGHTS AND MEASURES. 11. 11 yr. 27 wk. 6 da. +25, 36, 4 + 30, 40, 3 + 15, 26, 1. 12. 5 ho. 40 min. 36 sec. +20, 36, 51 + 11, 25, + 17, 0, 54. 13. 27 da. 14 ho. 46 min. +93, 10, 0, 31 + 87, 0, 0, 47 + 59, 0, 10. 14. 21 ho. 36 min. 14 sec. -19 ho. 45 min. 20 sec. 15. 93 yr. 28 wk. 4 da. - 57 yr. 36 wk. 6 da. 16. 13 ho. 46 sec. - 9 ho. 11 min. 17. 17 yr. - 8 yr. 10 wk. 3 da. 1* sec. EXERCISE XXVI. 1. 3 ho. 42 min. 7 sec. 2. 17 wk. 3 da. 11 ho. 3. 64 yr. 27. wk. 1 da. 4. 24 da. 20 ho. 16 min. 5. 13 wk. 1 da. 19 ho. x7, 8.i, 19, 73. x3, ISA, 47, 132. x 10, 132, 400, 89. x8, 63, 120, 247. x 11, 42, 160, 303. 6. 75 yr. 8 wk. 4 da. 7. 15 ho. 36 min. 9 sec. 8. 36 wk. 5 da. 13. ho. 9. 250 da. 14 ho. 58 min. 10. 293 yr. 5, 24, 19, 31. 9, 42, 220, 93. 6, 7i, 84, 140. 11, i, 34, 215. 12, 108, 71, 168. 1. 29 wk. 6 da. to ho. 2. 25 yr. 79 da. to da. 3. 3 wk. 5 da. 19 ho. to min. 4. 23 ho. 48 min. to sec. EXERCISE XXVII. 9. 10. 11. 12. 5. 39 wk. 5 da. to da. 13. 6. 115 yr. 140 da. to ho. 14. 7. 14 ho. 36 min. 50 sec. to sec. 15. 8. 7 yr. 10 wk. to wk. 16. 29650 sec. to ho. 78928 min to da. 5620 ho. to wk. 1795 da. to yr. 805 wk. to yr. 290786 ho. to yr. 8349250 sec. to da. 93078 min. to wk. 86. Angular Measure Table XII. In the circle (see Fig. A B c D), the whole circumference is supposed to be divided into 360 equal parts, called degrees (), each of which is divided into 60 minutes ('), and each minute into 60 seconds ("). Each quarter A B, B c, CD, and D A, contains therefore 90 degrees. The angle BOG has the same mea- surement as the quarter of the circle (B c) opposite to it ; it is 90, and is called a right angle. It is the angle at which a wall stands to the ground. If the angle E o c be half of B o c, it is 45 ; as is the arc E c. A familiar example of measurement on the circle is the reckoning of the position of places on the earth's surface by latitude and longitude. MISCELLANEOUS EXERCISES. 79 EXERCISE XXVIII. 1. 2. 3. 42<> 49' 57" 36 28' 29" 17 19' 36" 28 15' 25" 93 50' 10" 57 260 18 39 19 29' 13' 47' 21' 15' 45" 29" 50" 36" 42" 72 48 28' 39' 46" 49" 37 19 19' 20' 25" 36" 41 9 5. 26' 38' 37" 42" 28 18 6. 0' 7' 17" 29" 87. 7. 24 36' 49" + ll 25' 43" + 29 37' 59"+45 28' ll" + 37 41' 5" 8. 36 18' 24" + 56 49' 10" + 1 18' 11" 4- 40 5' 17" + 71 53' 19" 9. 72 45' 54" - 38 53' 59" (10.) 51 24' 27" - 27 36' 48" EXERCISE XXIX. 1. 36 27' 36" x 6, 21, 29, 97. 4. 186 24' 15"^5, 33, 12|, 83. 2. 24 38' 42" x 8, 45, 82.i, 130. 5. 829 30'^7, 108, 200, 158. 3. 55 10' 28" x 11, 37, 59, 242. 6. 732 35' ^4, 150, 149, 236. EXERCISE XXX. 1. 37 to " 7. 56280" to 2. 48 50' to " 8. 2794' to 3. 910 42' 25" to " 9. 324000" to L's 4. 1 to ; 10. 6896' to o 5. 73 to ' . 11. 4287' to 6. 2 L'S to ' 12. 98756" to EXERCISE XXXI. MISCELLANEOUS. 1. A ship delivered a cargo of 18 qr. 7 bu. 3 pk. barley ; half as much again of flour ; and as much wheat as both barley and flour together : how many quarters were there in all ? 2. A railway has three stations. The first is 3 ml. 5 fur. 21 po. 3 yds. from the terminus ; the second half as much from the first ; the third 4 ml. 20 po. 2 yds. from the second ; and the distance thence to the terminus is the average of the three distances mentioned. Find the length of the railway. 3. A footrule, 3 ft. long, was broken through at 17 in. 4 tenths : how long was the other part ? 4. A cart with coals weighed 1 ton 15 cwt. 3 qr. 25 lb, and the coals alone 19 cwt. 26 K> : what was the weight of the cart ? 5. Find the weight of 15 sugar loaves, each 25 lb 11 oz. ? 6. What distance is travelled in 37 da., each 8 ho., at 3 ml. 2 fur. per hour ? 7. What is the weight of 10000 sovereigns, each 5 dwt. 3 gr. ? 8. How many 6-lb packages may be made out of a hhci. sugar, weighing 5 cwt. 3 qr. 8 K> ; the tare, or weight of the cask, being 2 qr. 9. How many shirts may be made of 243 yds. cotton, each requiring 10. Twenty-five carts of coals weigh 23 tons 15 cwt. : find the aver- age weight of each. 80 WEIGHTS AND MEASURES. 11. How many sq. yds. in a court 138 ft. long by 64 broad? 12. How many chests of tea, 40 Ib each, were required to distribute 1 oz. to each of 2000000 poor people ? 13. Three lots silver plate were exposed at a sale ; one weighing 35 oz. 17 dwt. 9 gr., the second 19 oz. 16 dwt. 15 gr., and the third 14 oz. 6 dwt. : what cost the whole at 4/6 per oz. ? 14. How many hours had a boy, who was born Jan. 1, 1848, lived, when he was 10 yr. 7 wk. and 3 da. old ? 15. Out of 16 cwt. butter a grocer sells 27 Ib. daily for 5 days : how much has he still on hand? 16. 1000 Ib sugar are sold, each 1 oz. short in weight : find the real weight of the whole. 17. If the " lona" steamer sails 19 miles per hour, how far will she sail in 3 days, 6 hours ? 18. How much cloth will clothe a regiment of 560 men, if each suit takes 4 yds. 1 qr. 3 ill. ? 19. Twenty carts carried each 7 bars lead, and each bar weighed 2 cwt 25 Ib : find the total weight of lead. 20. A railway 47 ml. 4 fur. has 9 stations : what is their average distance ? 21. If 4 Ib gold are coined into 187 sovereigns, what is the weight of a sovereign ? 22. A square field is 93 yds. long : how many acres has it ? 23. If the circumference of the earth is 24899 ml., how many yds. of cotton-thread would reach round it ? 24. If I am 10 yr. 7 wk. and 3 da. old ; my elder brother 3 yr. 17 wk. 5 da. older than I am, and my younger brother 1 yr. 42 wk. 4 da. younger : what will be the average of our ages 3 yr. 5 wk. hence? 25. Two places on the same meridian are respectively 37 45' N. and 22 59' s. : find their difference in latitude. 26. What is the whole area of a farm, of which one field has 19 ac. 3 ro. 29 po. ; the second 27 ac. 36 po. ; the third 36 ac. 2 ro. 18 po. ; and the fourth 56 ac. 1 ro. 19. po. ? 27. A and B start from points 28 ml. 6 fur. 18 po. apart, to meet each other. When A has walked 4 ho. at 3^ ml. per ho., and B at 3J ml. per ho., how far are they still apart ? 28. A was born 28th Jan. 1844, and B 16th Nov. 1845, what is the difference between their ages ? If C was born 3 y. 45 da. later, what was his birthday ? 29. The daily supply of bread to an hospital is 47 Ib loaves : what weight of bread is sent per annum ? 30. Find the weight of 279 cub. ft. water, if 1 cub. ft. weighs 62 K> 7 oz. 4 dwt. Find also the weight of 1 cub. in. 31. The ship " Ino" landed a cargo of cotton in 960 bales, weighing in all 210 tons 15 cwt. : what was the weight of each bale ? 32. A line of 29 yds. 1 ft. is told off 18 times for sounding : what is the depth of the sea there in fathoms ? 33. How many miles of rails in a double line of railway 29 ml. 3 fur. 29 po. long ? 34. How many cub. ft. of air in a hall 60 ft. long, 21 broad, and 18 high? 35. How many chains would measure a road 17 ml. 4 fur. long? 36. If sound travels at the rate of 1116 ft. per second, in what time will the sound of a cannon-shot be heard 6 miles off? MISCELLANEOUS EXERCISES. 81 37. Divide a hhd. ale, containing 63| gall., into an equal number of one gallon, one quart, and one pint measures. 38. England is 50000 sq. miles in area : how many acres is that ? 39. A baker uses 6 qr. 5 bu. 3 pk. wheat weekly : how much is that in a year ? 40. Gold of the value of 500000 arrives from California : what is its weight avoirdupois, the price being 3, 18s. per oz. Troy ? 41. A ship sailing due north passed through 3 30' : how many nautical miles was that ? 42. A road 17 ml. 7 fur. 20 po. 3 yds. is repaired by 23 men : what share of the work falls to each ? 43. How often will the forewheel of a carriage, 5 ft. 8 in. round, revolve in a journey of 45 miles ? and how much oftener than the hind- wheel, which is 7 ft. 6 in. round ? 44. How long will it take to count a million of penny-pieces at 100 a minute ? 45. A watch gains 3' 25" daily ; if it starts from true time on Monday at 1 o'clock, what time will it show that day and hour three weeks ? 88. EXEECISE XXXII. MISCELLANEOUS continued. 1. Three parcels of paper contained respectively 3 re. 14 qu. 20 sh., 5 re. 16 qu. 10 sh., 7 re. 17 qu. 23 sh. : how much was there in all ? 2. What will remain of a piece of cloth 7 ells, 1 qr. 2 na., if 2 yds. 3 qr. are cut off for a coat ? 3. If a silver spoon weighs 3 oz. 4 dwt. 10 gr., what is the weight of 3^ dozen ? 4. What cost the gilding of a box, 6 in. long, 4 broad, and 4 deep, at 4^d. per sq. inch ? 5. The great bell at Moscow is said to weigh 443772 lb : how much does it exceed the weight of that of St. Paul's, which is 5 ton 2 cwt. Iqr. 22 lb? 6. If a puncheon of whisky contain 84 gall. 2 qt., how many dozen bottles will be required to draw it off, counting 6 bottles to the gallon ? 7. A journey of 87| ml. is performed in 2 ho. 12 min. 40 sec. : what is the rate per min. ? 8. Five stones butter are to be made up into parcels of lb, 1 lb, and 2 lb, the same number of each : what is that number ? 9. If a soldier takes 75 steps a minute of 2 ft. 8 in. each, in what time should a regiment march 7f ml. ? 10. To reach the bottom of a pit 12 ladders are required, each with 23 steps 1| ft. apart : how many fathoms deep is the pit ? 11. If the sovereign weighs 5 dwt. 3 gr., but becomes lightened by 4 gr., what is the weight of 100 sovereigns ? 12. If a class uses 36 sheets paper 5 days a week for writing, how much will it use in a year, allowing 6 weeks for vacation ? 13. Find the difference between 18 square feet and 18 feet square. 14. How many geographical miles between London, 51 30' N., and Saragossa, 41 46', which are nearly on the same meridian of longitude ? 15. If the weight of I/ be 3J T dwt., how many may be counted out of 3 bars silver, each 7 lb 9 oz. 14 dwt. ? 16. If 8 qr. 4 bu. 3 pk. weigh 18 cwt. 2 qr. 26 lb, what is the weight of a peck ? 82 WEIGHTS AND MEASURES. 17. A field 400 feet long by 175 broad is intersected along its length by a stream 21 links wide : what is the area of the field ? 18. From a cistern containing 2560 galls, are drawn on Monday 859 galls. 2 qts., on Tuesday 384 galls. 3 qts., on Wednesday the difference Between these quantities. How much must be drawn on each of the three remaining days of the week to empty the cistern ? 19. At 5/3 per ounce, what is the value of silver plate weighing 3 Ib 9 oz. 10 dwt. ? 20. A piece of cloth was 56 yards long : how much more would have given 7 yds. 2 qrs. 3 nls. to each of 8 persons ? 21. How much pure gold in spoons weighing in all 2 Ib 6 oz., 18 carats fine, i.e., 18 parts out of 24 being pure, and the rest alloy? 22. A journey of 127 miles 6 fur. is made by rail, at the rate of 26 miles 6 fur. per hour. That is only \ of the time a steamer would take, and f of the time a stage-coach would take. How much less time does the railway take than the steamer, and than the stage-coach ? 23. How many solid feet of earth ill a mass 15 yds. long by 11 broad and 9 thick * 24. Four hhd. sugar weighed 7 cwt. 2 qr. 1 Ib, tare 2 qr. 16 Ib ; 6 cwt. 1 qr. 18 Ib, tare 2 qr. 5 Ib ; 5 cwt. 3 qr. 20 Ib, tare 1 qr. 26 Ib ; 7 cwt. 3 qr. 24 Ib, tare 2 qr. 18 ft> : what is the net (or neat} weight ? 25. How many coils of rope, each 5^ ft., will stretch half a furlong? 26. What is the girth of a wheel which revolves 1365 times in 2 ml. 1 fur. 36 po. ? 27. Find the weight of water in a tank whose bottom is 10 ft. square, and depth 14 ft. (See No. 30, Ex. xxxi.) 28. If 1 acre yield 3 bu. 2 pk., what is the produce of 193 ac. 3 ro. ? 29. How many hurdles, each 4 ft. long, will enclose a park 9 chains long by 5.^ broad ? 30. If you waste 10 minutes daily, how much time do you lose in 3 years (counting 1 leap year) ? 31. If I walk 8 yds. 2 ft. 6 in. a minute faster than my neighbour, in what time will I be a mile ahead of him ? 32. A milkmaid, carrying a pitcher of milk containing 3 gallons, to be delivered equally to 16 families, loses 3 pints by leakage ; what must each family get ? 33. How many benches, 8 feet long, would seat a class of 42 pupils, allowing 1 ft. 4 in. to each ? 34. If the breadth of an oblong field is 17 ch. 56 Ik., and its length twice and a half as much, how many yards does a person walk who goes round the field three times ? 35. Light travels at 186,000 miles per second ; how long does the sun's light take to reach the earth, the distance being 95 millions of miles ? 36. The Jewish silver shekel weighed 9 dwt. 2^ gr., and its value was 2/3 : find the weight and value of the talent, which was 3000 shekels. Find also the weight and value of the silver vessels (" one silver charger of 130 shekels, and one silver bowl of 70 shekels") offered by the twelve princes of Israel, Num. vii. The present of each also included one golden spoon of ten shekels : what was the weight and value of the spoons, the shekel of gold being estimated at 1, 16s. 6d. ? BILLS OF PARCELS. 83 89. BILLS OF PARCELS. Ex. Mrs. Wilson bought of William Dixon, grocer : 4 ft> tea at 3/6 per lb ; 12 lb sugar at 5d. per lb ; 7 lb butter at 1/1 J per lb ; 10J lb rice at 3|d. per lb ; and 2 lb -urrants at 8d. per lb : what has she to pay ? The grocer presents her with an account drawn up in the following form. Sometimes, for ready money payment, a deduction or dis- count of 5 for every 100 of account or, as it is called, 5 per cent. is allowed. That is ^th part of the whole, and is 1 calculated with sufficient accuracy by allowing I/ for every pound, and 3d. for every 5/ besides. Mrs. WILSON, Bought of WILLIAM DIXON, Grocer. U March. 4 ft) tea @ 3/6, . . . . .0 14 12 lb sugar @ 5d., . . . 050 7 lb butter @ 1/1 1, . . . 07 lOi 10^ ft) rice @ 3^d., . . . 3 0| 2 ft) currants @ 8d., . . . 014 1 11 3i Discount, . . 016 1 9 9jr The tradesman discharges the account by writing below it, when he receives payment, "Received payment, William Dixon." These accounts are called " Bills of Parcels." 90. EXERCISE I. Make up the following accounts, allowing discount at 5 per cent. : 1. Mr. John Thomson bought of William ITendry, May 15, 1864 : 17 yds. lace at 6/6 per yd. ; 14 yds. cambric at 10/8 per yd. ; 58^ yds. calico at lOd. per yd. ; 36 yds. muslin at 7/4| per yd. ; 6f yds. linen at 3/8.}. 2. Mr. William Simson bought of Thomas Adams, June 14, 1864 : 7 \ lb beef at 8d. per lb ; 9 lb mutton at 7d. per lb ; 2 spring chickens at 2/9 each. July 3. 12 lb beef at 8^d. per ft ; 6^ lb pork at 6d. per lb ; 2 hares at 2/9 each ; 1 pair of pigeons at l/4d. 3. Mr. Alexander Cooper bought of Alfred Garland, Aug. 7, 1859 : 28 yds. Brussels carpet at 4/8^d. per yd. ; making do. 4/ ; 18 yds. stair carpet at 2/9 per yd. ; 3 hearth rugs at 17/8 each ; 2 pieces floor- cloth, 15| yds. each, at 2/10 per yd.; men's time laying down, 3/6. 4. Mr. "Henry Wallis bought of Robert Mapleton, January 1, 1563 * 84 BILLS OF PARCELS. 41 yds. super, black cloth at 16/6 per yd. ; 6 yds. tweed at 5/6 per yd. ; making and mounting 17/8 ; 2 black ties at 3/6 each ; 6 silk pocket handkerchiefs at 3/9 each ; 2 pair kid gloves at 3/. 5. Mr. Philip Chorley bought of Andrew Stewart & Cc. : 8 quires note-paper at 7d. per quire ; 5 reams foolscap at 22/6 per ream ; 12 boxes steel pens at 1/3 per box ; 3 bos. black ink at 2/6 per bo, ; 3 account books at 4/9 each. 6. W. H. Wilson, Esq., bought of Dalton & Tait : 4 dozen sherry at 35/; 6 dozen do. at 48/; 8 dozen port at 56/ ; 3 galls. Scotch whisky at 18/6 ; 6 dozen ale at 6/ ; 2 dozen claret at 63/ ; carriage paid, 11/4. 7. Mrs. Barton bought of Hobkirk & Son : 33 lb Cheshire cheese at 8|d. per lb ; 15 lb butter at 1/2 per lb ; 1 Belfast ham, 13i lb, at lid. per lb ; 3-J lb bacon at 9d. per lb ; 6 dozen eggs at 7d. per dozen ; 2 pks. fine flour at 1/2 per pk. 8. Mr. Robert Beaton bought of John Gardiner : Feb. 1, 1860, 5 lb tea at 4/4, 12 lb loaf-sugar at 5|d ; Feb. 28, 8 lb rice at 3^d. per lb, 3 It) currants at 5|d ; March 4, 4 fb coffee at 1/8, 6 lb loaf-sugar at 5^d., 3 lb table raisins at 1/2, 4 dozen oranges at 1/6 per dozen. 9. Wilson & Hill bought of Farmer Brothers, June 6, 1861 : 36 qrs. barley at 25/ ; 58 qrs. wheat at 52/ ; 16 qrs. of oats at 35/ ; 17 bushels pease at 3/8.] ; 19 bushels tares at 1/9|. 10. Mr. David Hodson bought of Thorn & Maclean : 4 chests tea, 180 lb each, at 3/8 ; 9 hhcls. sugar at 4, 19s. 3d. per hlid ; 7 bags of rice, each 1 cwt., at 1, 7s. 6d. per c\vt. ; 8 bags coffee, each 84 lb, at 1/5 per lb ; 64 sugar loaves, each 12^ lb, at 4|d. per lb. 11. Mr. John Smith bought of Thomas Rogerson, Fek 15 : 12 yds. flannel at 1/5 ; 30 yds. calico at 6^d. ; 18 yds. linen at 3/10 ; 6 pairs stockings at 1/2 ; 6 pairs blankets at 14/6 per pair. 12. Mr. R. Thompson bought of Hancock Brothers : 16 yds. broad eloth at 27/6 per yd. ; 18 seconds do. at 14/9 ; 13 yds. brown cloth at 11/10 ; 14J yds. scarlet at 24/4 ; 62 do. at 9/8. PRACTICE. Ex. 1. Find the price of 385 yards at 7^d. per yard. If we find the price at (3d. and then at l|d., and add the two prices, we shall have the price at 7Jd. Now, the price of 385 yards at Is. per yard being 385s., the price at 6d. will be one-half of that, or 192s. 6d. ; and the price a.t l^d. will be of the price at 6d., that is 48s. l^d. Adding the two together, we find the price at 7^d. to be 240s. 7|>d., that is, 1%, Os. 7jd. The working is written down thus : y:ls. s. d. Price of 385 at Is. 385 Price @ 6d. Price @ l|d. Therefore, Price @ 7jd. XT2~CT PRACTICE. 85 The parts of the whole price chosen are 6d. and 1 Jd. because these are even, or, as they are called, aliquot parts of the next highest name, that is, a shilling. The answer to this question might be got by compound multiplication ; but the process is longer. The method of find- ing prices by aliquot parts is therefore commonly practised ; hence it is called " Practice." Rule. Take aliquot parts of the next highest name, and find the prices at these ; add the several results, and reduce the sum to pounds. 92. Ex. 2. Find the price of 385 J yards at lOjd. per yard. Here the price at 1 s. is 385s. 6d. Then we might take as aliquot parts of a shilling, 6d., 3d., and l^d., and add the results ; but in cases like this, whei . the price differs from the next highest name by an aliquot part exactly, it is shorter to take that difference as an aliquot part, and subtract the result from the price at the next highest name. Thus : yds. s. d. Price of 385j- at Is. 385 6 Price @ l^d. Therefore, Price @ 10 Jd. EXERCISE I. ** The pupil should first be exercised in the aliquot parts to be taken in the following examples, till he can state them readily. 1. 73 yds. @ If, J, 2f, 3.J, 4J, 5J, 6J, 7J, 8Jd. 2. 294 @ 2i, 1J, 44, 3J, 4f, 5f, 6i, 7f, fyd. 3. 596$ @ 3$, lid, 2|, 6f, 4.J, 5|, 8, 7i, 9fd. 4. 7384 @ loi, Sf, 64, 9^, 11^ 11.J, f, 5"f, 9jd. 5. 8036J @ 7i, 2i, 9.i i, 11*, lO.i, 3f, 8^, 4Ad. 6. 5690 @ Hi, lOi, 9^ |, Il|, 8f, 7|, lOJ, fd. 7. 6853 83. -^ or w orkincr questions in which the price is shillings, the pupil must be familiar with the following tables : 1. ALIQUOT PARTS OF 1. 10s. Od. is A 4s. Od. is 2s. Od. is ^ Is. 3d. is ^ 6s. 8d. j 3s. 4d. Is. 8d. fa Is. Od. ^ 5s. Od. ^ 2s. 6d. i Is. 4d fr Os. 6d. ^ 86 PRACTICE. 2. ALIQUOT PARTS OF THESE PARTS. *. d. s. d. d. d. s. d. d. 2 6 is i of 10 10 is A of 10 7i is J of 1 3 4 2 6 1 8 j 5 - 10 10 10 \ 6 8 5 6" ^ 10 6 A 5 4 3 1 8 1 I 5 10 3 4 6 140 3 1 4 ; 4 10 \ 2 6 6 * 2 6 3 1 3 5 10 10 1 8 6 I 1 3 1 3 ; 5 8 -h , 10 5 A 5 1 3 .. 2 6 8 i 4 5 f 3 4 8 1 3 4 5 2 6 2^ 8 t 2 5 \ 1 8 8 i 1 4 4 A 4 l) 7 \ '* 5 4 A- 3 4 2 6 4 2 EXERCISE II. 1. Of one pound, what is , |, J, fc J, i, 2. Of one pound, what is |, f , f , f , |, &, 3. Of one pound, what is f , f , &, rf,, , 4. Of ten shillings, what is $, i, |, ^, ^, 5. Of five shillings, what is $, ^, i, J, |, 6. Of four shillings, what is \, \, \, J, \, 7. Of half-a-crown, what is \, |, ^, J, ^, 8. Of two shillings, what is \, \, J, |, 9. Of one and eightpence, what is }, \, ^ 10. Of one and threepence, what is -, , 6 A, A> A A, A, A* A ? , &, A, A. A A, etc. ? ^, A, ii, M> etc. ? , ^, f , T %, ^, etc. ? , ^, |, f , f , etc. ? &, X, ^, , f, f, etc. ? ^, ^, |, f , ^, etc. ? T \, f , |, f , etc. ? |, ^, ^ f , |, f, ^, etc.? , ^, ^, f, f, f, ^, etc. ? Q A Ex. 3. Find the price of 385 yards at 11/10^ per yard. " The price here being shillings, we take aliquot parts of the next highest name, the pound. If we find the price at 10/, 1/8, 2d., Jd., and add these seve- ral prices, we shall have the price at 11/10^. Thus : yds. s. d. s. d. Price of 385 @ 1 385 Price @ 10 Price @ 1 8 Price @ 2 Price @ 192 10 32 1 8 342 8 0^ Therefore, Price @ 11 10 J 228 3 10 J The aliquot parts should always be chosen so as to give the smallest number possible. Thus, if the price were 3/9, and if we took the largest aliquot part, viz., 3/4, we should still have PRACTICE. 87 two to take, viz., 4d. and Id. ; whereas, taking 2/6 and 1/3 aa the aliquot parts, we need only two. Ex. 4. Let the price of the 385 yds. be 17/6 per yard. yds. s. d. s. d. Price of 385 @ 100 385 48 2 6 Price Therefore, Price 2 6 | 17 6 336 17 6 EXERCISE III. 93 2/10, 5/8, 3/9, 13/4, 18/4, 7/9. 118 @ 4/6, 6/8, 6/3, 15/10, 19/4, 1/5. I/7L 4/3|, 6/21, 8/7|. 11/24, 14/32. 5/104, 17/8, r 3/72, 8/52, 1! 6/10, 16/11, 12/6'f , 15/i'4, 5/lll, 17/72. 2/1, 15/6, 13/41, 8/52, 3/81, 19/12. 9/54, 10/0 j, 15/6 J, 17/6, 14/2$, 4/T 548 805 724 327 2937 5608 7890 8000 }, 1, |, 8/7i, 9/2|, 15/6J, 7/ 95. ^ n ^ ne following examples, which involve fractional numbers of articles, it is better to work first for the value of the whole number, and then add the fraction of the price of one article. Ex. 5. What cost 397f at 10/6 ? yds. s. d. s. d. Price of 397 100 397 Price >j @ 10 I 198 10 Price 11 @ 6 ^T 9 18 6 Price of 1 @ 10 6 7 Price of 397 1. 236 ( 2. 693* ( 3. 600A( 4. 1594f ( 5. 7240|( 6. 94364 ( 10 6 EXERCISE IV. 208 15 6 2/3, 4/7*, 8/11, 13/4J, 18/2f, 7/6. 10/9, 14/8, 17/2J, 9/3^, 12/81, 1/7J. 17/8, 13/1J, 15/yj, 3/10.^, 4/8, 11/5J. 3/3, 7/9f^l3/2|, 5/10^ 18/6, 19/5._ i 14/1, 10/6, 10/8J, 15/7i 18/9, 2/5|. \ 17/4, 6/10J, 9/3, 18A1, 3/4J, 15/2. 9 D . When the price of each article is a number of shillings exact, the shortest way to find the price of the whole is by multiplica- tion. 88 PRACTICE. Ex. 6. What cost 278 yds. at 9 sh. and at 14 sh. ? sh. sh. 278 price at I/. 278 price at I/. 9 7 2(0 ) 250(2 price at 9/. 10) 194(6 price at 14/. 125, 2s. 194, 6s. What has been done in the latter case, in which the number of shillings is even, is to multiply by ^ instead of J ; which is shorter, and comes to the same thing. EXERCISE V. 1. 742 6/, 14/, 16/, 5/, 9/, 13/. 4. 1894 14/, 6/, 8/, 117, 13/. 2. 913 @ 3/, 7/, 8/, 12/, ll/, 19/. 5. 3565| @ 17/, 2/, 9/, 3/, 12. 3. 296i @ 16/f i 8/> 5^ 7/; 127, 157. 6. 7924 13/, 7/, 15/, ll/, I/. 97. Ex. 7. Find the price of 385| yds. at 3, 11s. 10|d. Here we find the price at 3 separately, and proceed for the rest as before. yds. s. d. s. d. Price of 385 @ 1 385 3 Price 3 1155 Price w 10 1 192 10 Price 99 @ 1 8 1 32 1 8 Price @ 2 JL 3 4 2 Price 01 | 8 oj Price of f <T 3 11 iol 2 13 10J 1 Price of 385 J@ 3 11 10 1385 17 8j f Rule. When the price consists of s. d. t multiply by the number of ; take aliquot parts for the rest, and add tho several results. EXERCISE VI. 5. d. s. d. s. d. s. d. s. d. 1. 235 @ 8 4 10 4 17 3 5 18 10 15 6 2i 11 7 11 2. 486 @ 9 2 3^ 7 10 6 24 7 2 30 18 9 18 2 11 3. 592 @ 1 5 6 9 15 10 20 10 1 25 1 7 3 16 lOa 4. 3560 @ 2 10 6 5 7 3 9 2 6} 7 2 3| 1 9 6 5. 5986 @ 7 13 4.i 11 10 6 23 10 9 14 13 0^ 2 18 10 6. 7852 @ 5 9 31 8 7 36 2 5J 9 17 3 5 2 11J 7. 8194 @ 12 10 9 20 15 7 41 8 7 28 5 6i 17 13 8 8.2936 @ ]4 5 3$ 2 19 9 6 14 2i 34 17 4f 100 19 4 9. 23%1 @ 50 7 6 43 1 9. 1 , 11 5 8} 17 9 3 7 2 11 10. 7852^ 120 10 9A 84 2 I" 70 16 2i 24 11 5 1 9 6J 11. 9324,} (& 55 1 11" 72 18 2i 24 5 01 36 2 0} 3 104 12. 56498 OZJ 27 17 39 18 46 8 OA 10 9i 92 14 6j PRACTICE. 89 98. E'j. 8. Find the value of 9 cwt. 3 qr. 7 tb at .4, 13s. 6d. per cwt. Here the quantity is compound. It comes to the same thing whether we multiply the quantity by the price of one, or the price of one by the quantity ; and we choose the latter as the simpler in this case. Thus we multiply the price of 1 cwt. by 9, which gives the price of 9 cwt., and take aliquot parts of a cwt. for the rest of the quantity. cwt. Price of 1 qr. ft 4 s. 13 d. 6 9 Price of > 9 2 1 7 1 42 2 1 1 6 3 5 6 9 4* ion Price of 9 3 7 ^45 17 5j Rule. When the quantity is compound, multiply the price by the highest name of the quantity, and take aliquot parts for the remainder of the quantity. Note. In the above example, the price of one of the highest name is given. If the price of one of the lowest be given, reduce the quantity to that lowest name, and find the value by a previous rule. EXERCISE VII. %* In this and the following Exercise, the exact calculation of the fractions of pence may be left till the pupil studies the chapter on that subject. 1. 5 cwt. 2 qr. 9 ft ( 2. 9 tons 16 cwt. 1 qr. ( 3. 25 ft 12 oz. 7 dr. ( 4. 27 ft 8 oz. 9 dwt. ( 5. 64 yds. 3 qr. 2 nl. ( 6. 144 yds. 2 qr. 1 nl. ( 7. 15 qrs. 5 bu. 3 pks. < 8. 36 bu. 2 pks. 1 gall. ( 9. 7 pks. 1 gall. 3 qts. ( 10. 7 qts. 1 pt. 1 gi. ( 11. 18 qrs. 7 bu, 2 pks. ( 12. 96 ac. 1 ro. 20 po. ( 13. 144 ac. 2 ro. 16 po. ( 14. 27 po. 20 sq.yd. 8 sq.ft. ( 15. 49s.y. 5s.f. 100 sq. in. ( 16. 10 rea. 7 qu. 6 sh. < 17. 42 yds. 2 ft. 9 in. ( 18. 63 yds. 1 ft. 7 in. ( 19. 25c.y. 18c.ft.144c.in. < s. d. s. d. s. d. s. d. 217 6 415 3 911 3 15 14 10A 314 5 7 9 10J 12 8 5] 13 17 llj 1 15 6 2 7 84 5 8 fa * 9 1J 711 5 914 7j 8 2 3J u 7 ;r 1 3 6 2 5 4* 873^ 409 2 3 71 3 7 8j 4 911; 562 317 6 414 6j 513 9 ; 7 6 OA 1 15 8^ 2 310 5 6 45 6 211 105 110 11A 276^ 2 8 OJ 2 7 81 1 9 5;r 3 7 J 1 17 8j 318 9j 624. i. 2 7 3^ 1 19 Ilk 24 7 6 15 8 2.- 36 18 OJ 12 10 10" 8 5111 978; 71311 10 5 61 238 345; 416 2.t 5 S10J 1 2 6* 2 311] 1173} 2 15 14 158 1 13 11 213 9 2 010 516 2i 319 5; 11 2 1J L 9 7 8 2 13 61 219 O.J 3 7 3j 562 4 Oil 319 2^ 49 3. : 18 7 90 PRACTICE. 5. d. 20 8 5 11 14 81 21610 118 5 21 01U 218 2 16 10 5 1 27 1 8. d 9 3^ 9 4^ 311; 9" 1 6i 15 6; 5. d. 14 8 13 7 3 710 92 318 If 14 BIO! 314 4j 7 6 5 4 1 5 s. 16 12 9 18 9 d 4 8 6 3 6 Oi 20. 4 ml. 7 fur. 12 po. 21.84ml. 6 fur. 10 po. 22. 4 yds. 2 ft. 6 in. 23. 25 da. 10 ho. 30 min. 24. 7 wk. 5 da. 16 ho. 25. 22 ho. 48 miu. 36 sec. 99. Ex. 9. Find the value of 527 cwt. 3 qrs. 7 ft) at 5, 11s. 8d. per cwt. To avoid the long process of multiplying by 527, we may in this example first find the value of 527 cwt. at the given price ; and take then 3 qrs. 7 lb at the given price ; and add the results. Thus *. A 10 1 8 l 527 5 price of 527 cwt. at 5 per cwt. 10s. ls.8d. 2 qr. at 5, 1 1, 8 p.cwt. " i r - i 7 ib 2635 263 43 2 1 10 18 15 7 6 4 10 H 2 qr. Iqr. 7ft L 2946 19 This method really combines two questions in Practice, and may be used with advantage whenever the aliquot parts for the shillings and pence in the price are few. EXERCISE VIII. 1. 239 cwt. 2 qrs. 16 tb 2. 412 tons, 14 cwt. 1 qr. 3. 193 tb 4 oz. 8 dwt. 4. 4541b7oz.ll dwt. 5. 325 ac. 3 ro. 29 po. 6. 1426 ac. 3 ro. 3 po. 7. 724 qrs. 6 bu. 2 pk. 8. 95 bu. 1 pk. 1 gall. 9. 713 galls. 2 qt. 1 pt. 10. 315 qts. 1 pt. 3 gi. 11. 347 yds. 2 qr. 3 na. 12. 250 yds. 1 qr. 2 na. 13. 536 yds. 2 ft. 8 in. 14. 115yds. 1ft. 6 in. 15. 425 mi. 3 fu. 8 po. 16. 374 mi. 7 fu. 16 po. 17. 723 da. 6 ho. 14 mi. 18. 117 wk. 5 da. 15 ho. 7 010; 317 L 209 11 7 21410 5 211 784 314 411 36 2 2918 5; 309^ 21 7 PRACTICE. 91 100 EXERCISE IX. MISCELLANEOUS. 1. What does a contractor pay weekly for wages who employs 546 labourers at 15/9 each ? 2. Find the price of 8 pieces cloth, each 36 yards, @ 1, 3s. 6d per yard. 3. Bought soap at 4|d. per K> : what is that per cwt. ? 4. Bought 56 hampers apples @ 16/2 each : how much cost the whole ? 5. What cost 236^ gross bottles @ 1/3^ per dozen ? 6. A farmer rents 129 acres @ 3, 17s. ~6^d. : what is his total rent ? 7. What is my nett income, if my taxes are 1/10^ on my gross in- come of 320 ? 8. What is the freight of 7 trucks, each 6 tons, 3 cwt. 2 qrs. @ 1, 6s. 7d. per ton ? 9. What is the expense of making 147 miles 3 fur. 24 po. railway @ 2345, 10s. 6d. per mile ? 10. What did I pay for 3 cwt. 2 qrs. butter @ 15/4 per stone ? 11. What cost 937 yards ribbon @ 3|d. per yard? 12. A farmer sold 39 oxen @ 14, 10s. 6d. each, and 256 lambs @ 1, 2s. 6d. each : how much money did he get ? 13. In a railway train there were 79 first-class passengers @ 2, 17s. 6d. each ; 193 second-class @ 1, 14s. 8d. ; and 256 @ 19/8 each : what were the receipts from the train ? 14. What will I make by selling 26 stones starch @ 11 |d. per Ib ? 15. A bankrupt paid 9/10 a pound on a debt of 2456, 17s. 6d. : what were his assets ? 16. Find the price of 123 yards 3 qr. 3 na. cloth @ 3/8 per yard ? 17. If a dollar is 4/2^, how many pounds are in 798 dollars ? 18. The cost of a vessel was 758296 francs : what is that sum in sterling money, if the franc is 9|d. ? 19. If a sovereign weighs 5 dwt. 3 gr., what is the weight of 25000 sovereigns ? 20. Bought 27| yds. flannel @ 1/5 ; 18 doz. pairs of stockings @ 1/10 per pair ; 156 yds. linen @ 2/6| ; 596 yds. calico @ 7|d. per yd. : what was the amount of the bill ? 21. What cost 172 doz. and 80 bo. sherry at 42/ per doz. ? 22. What cost 3 qr. 17 Ib sugar @ 3, 18s. 6d. per cwt. ? 23. What did I pay in all for 25^ R> beef @ 9^d. ; 16 R> 8 oz. cheese @ lO^d. ; 23^ loaves bread @ 6d. ; and 2 R> 11 oz. tea @ 4/9. 24. Find the duty on a puncheon (84 galls.) whisky @ 9/6^ per gall. 25. Find the nett weight of 217 cwt. 2 qr. 25 Ib, allowing 13 Ib per cwt. as tare (deduction for weight of package). 26. Find tne value of a silver tea-service weighing 325 oz. 6 dwt. @ 5/6 per oz. 27. If I spend 13/8^ daily, what do I save out of an annual income of 350 guineas ? 28. A company of seven miners find a nugget of gold weighing 23 R> 8 oz. : what does each make by it, if gold sells @ 3, 16s. lO^d. per oz. ? 29. If a labourer's wages are 17/6 a week, what does he earn in 6 weeks 4 days ? 30. An apprentice, whose wages are 60 a year, dies after a service of 7 weeks 3 days : how much falls to be sent to his friends ? 92 RULE OF THREE. RULE OF THREE. 101 1. By Multiplication and Division. Ex. 1. If 4 ft> tea cost 16/, what cost 16 It) ? Here the price of a certain quantity is given, and we wish to know the price of so many times that quantity. 16 ft) is 4 times 4 ft), therefore the price of 16 lb will be 4 times the price of 4 Ib ; that is, 4X 16/, which is 64/, or ,3, 4s. Questions of this sort, in which the quantity whose price is sought is so many times the quantity whose price is given, are solved by multiplication. In all such questions there are three numbers given, two being of the same kind, and the third of a different kind ; hence the name applied to the solution is the " Rule of Three." A fourth quantity is in all cases sought, which is of the same kind with the third given. The Rule of Three is chiefly useful for the finding of prices ; but it will be seen from the examples that it is applied also to questions in which money is not involved. EXERCISE I. ; 1. If 3 yds. cost 17/, what cost 18 yds ? 2. If 16 tb sugar cost 6/, what cost 48 tb ? 3. If 2 quarts cost 1/6, what cost 2 galls. ? 4. If a labourer earns 25/in 13 days, what will he earn in 13 weeks ? 5. A coach goes 19^ miles in 3 hours, what distance will it go in 15 hours ? 6. If 45 men can build a wall in 18 days, in what time will 9 men do it ? 7. If 3 books cost 3/9, how many may be bought for 18/9? 8. If 6 acres produce 2t> bu., what will 43 acres of 4 same land pro- duce? 9. If 44 acres rent for 17, what will be the rent of 34 acres ? 10. How many yards of cloth at 3/6 are worth 27 yds. at 14/ ? 11. If ^ tb tea cost 2/3, what cost 8 tb? 12. If | tb cost 7/1 A, what will be got for 3, 16s. ? 10 4 Ex. 2. If 12 yards cost .2, 8/, what is the price of 4 yards \ Here the quantity whose price is sought is an even part o that whose price is given. Since 4 yards is the third part of 12 yards, the price of 4 yards will be the third part of that of 12 yards. Now the d of .2, 8/ is 16/, which is the answer. Such examples of the Rule of Three are wrought by Division. EXERCISE II. 1. If the cost of printing 128 pages be 17, 4/8, what cost the print Ing of 32 pages ? RULE OF THREE. 93 2. If 32 cwt. cost 36, what cost 4 cwt. ? 3. If 1 lb tea cost 4/6, what is that per oz. ? 4. If 3 dozen oranges are bought for 2/, what could I buy 4 oranges for? 5. If 7 men reap a field in 4 days and 2 hours, how many days will 28 take ? 6. If a railway train goes 200 miles in 8 hours, in what time will it go a journey of 40 miles ? 7. I read a book of 365 pages in 15 days ; how many pages do I get over every 3 days ? 8. A ship with a crew of 160 men was provisioned for 84 days ; how long would these last a crew of 960 ? 9. If 21 yards cost 6, Os. 9d., what will 5 yards cost ? 10. A firm expends on wages 61, 12/ a week for 75 men ; what does another expend, employing only 25 men at same rate ? , 11. If 63 qrs. of wheat weigh 18 tons, find the weight of 7 qrs. 12. A farmer rents 275 acres for 400 : if he adds 55 acres at same rate, what was his rent then ? 103 Ex. 3. If 8 ft cost 28/, what cost 11 ft) ? Here the quantity whose price is sought neither contains, nor is contained in, the quantity whose price is given, an even number of times. We therefore find the price of 1 ft, as an intermediate step, the number 1 being contained in both the quantities. Thus, since 8 lb cost 28/, 1 ft cost J of 28/ ; and since 1 ft cost J of 28/, 11 ft cost ^ of 28/ ; that is, 1, 18s. 6d. Such examples of the Eule of Three are wrought by Multi- plication and Division combined. EXERCISE III. 1. If 7 sheep cost 8, 15/, what should be paid for 11 ? 2. If 5 books cost 6/8, what cost 9 ? 3. For 7 doz. wine I paid 9, 19s. 6d. ; what would I have got 3 doz. for? 4. If I walk 17 miles in 5 hours, in what time shall I walk 267^ miles ? 5. If an apprentice earns 3/9 in 5 days, how long must he work foi 3, 7s. 6d. ? 6. My income is 120, and I pay taxes 3, 4s. 7d. : what should an income of 420 pay ? 7. If 13 men mow 7 ac. 2 ro. 25 po. in a given time, what will 19 men do in same time ? 8. If 29 sheep are worth 3 oxen, how many sheep are worth 21 oxen ? 9. If 11 silver spoons weigh 9| oz., what is the weight of 2^ dozen ? 10. If 12 lb butter cost 5/6, what cost 3 st. 3 lb ? 11. When wheat is at 37/6 per qr., what should I get 5 bushels for? 12. If the carriage of a parcel by railway for 17 miles be 10d., what should it be for 32 miles ? and how far should it go fo* 2/11 ? 104 105 94 RULE UF THREE. Ex. 4. If 8 books cost 18/, what cost 25 ? and what cost 23 f This is a case like the last, and could be wrought by multi- plying by 8, and dividing by 25 and 23 respectively. But since 25 contains 8 three times and 1 over, the price of 25 may be found by taking three times the price of 8, and adding to it the price of 1. Thus : price of 24 is 3Xl8/= 2 14 price of 1 is of 18/= 2 3 therefore price of 25 is 2163 Again : to find the price of 23, since 23 is 241, say, price of 24 is 3X 18/ = 2 14 and price of 1 is J of 18/ = 2 3 therefore price of 23 = 2 11 9 EXERCISE IV. 1. If 8 oz. of tea cost 2/8, what cost 9 oz. ? 2. If 3 yds. of cloth cost 17/3, what cost 7 yds. ? 3. For 8 chairs I paid 5, 4s. : what cost 15 ? 4. 9 cwt. sugar cost 16, 16s. : what cost 26 cwt. ? 5. 16 reams of paper go to 300 copies of a book : how many copies may be got out of 36 reams ? 6. If I pay 8, 5s. for 45 yds., what is the price of 50 yds. ? 7. If a coach go 42 miles in 7 hours, how many miles will it go in 10A hours ? 8. If 35 gall, cost 20, 3s. 6d., what cost 10 gall. ? 9. If 4 lb coffee cost 6/8, what should I get 27 R> for ? 10. If 2 tons 3 cwt. cost 4, 10s., what cost 16 tons 2 cwt. 2 qr. ? 11. If 4 ac. 2 ro. are rented for 15, what is the rent of 23 ac. 2 ro. 20 po. ? 12. If a box, 2 ft. sq. and 3 ft. deep, contain 100 oranges, how many should be in one 2 ft. long, 3 broad, 7 deep ? 2. By Proportion. Ex. 1. What number contains 9 as often as 12 contains 6 ? Since 12 contains 6 twice, and 18 contains 9 twice, 18 is the number sought. The number of times that one number contains another is called the ratio of the two numbers ; thus the ratio of 12 to 6 is 2, and of 18 to 9 is 2. In the question, the ratio of 12 to 6 is equal to the ratio of 18 to 9 ; and the four numbers are on that account said to be proportional, and to form a simple proportion. That 12 contains 6 as often as 18 contains 9 is usually ex- pressed more shortly thwa * PROPORTION. 95 As 12 is to 6, so is 18 to 9. Or 12 is to 6, as 18 to 9. By symbols : 12 : 6 : : 18 : 9. The four numbers of a proportion are called its terms. EXERCISE V. State the term wanting to the following proportions, and read each proportion as completed : 1. 4: 2. 6:1 2 : } 8: . 5. 36 : . 6. 21 : 12 : : 2 7: : 7:( ). 9. 18: 9 : ( ). 10. 7:3 6::1 5 27: 8: 3. 24: 4 49 3 :3 1 '6 6: 3 . 7. 8: 8 4 16 : :1 20 : : 2 : ( ). 11. 14 : 4 5 : ( ). 12. 108 : 1 2 : : ?' ' 2: P 13. 3: 14. 2 : 15. 6 : 16.7: 25.4: 26. 3 : 27. 6 : 28. 8: 37. ( ) 38. ( ) 12:: 8:: i8 : 53: I : 6 8 . c 11 | 7 '} : i : . 9 : 1 16. 17. 20. 18. 1 8. 19. 27. 20. 1 18. 29. ' 48. 30. ! 25. 31. 11 35. 32. 1< 18. 41. 44. 42. 9 :36 : 3:52: 9:3: 2: 4: | ( : 16. 21. 36 : 32. 22. 56 ( : 8. 23. 84 : ( : 7. 24. 42 : : 13 : 39. 33. 21 : : 8 : 32. 34. 12 : : 18 : 3. 35. 54 : : 63 : 9. 36. 35 : ):: 8:40. 45. ( ) J:: 3:36. 46. ( ) 12 7 12 6 : 4 i 3 8 10 : 2 :: $ ): 0: 4: 3: 8: JO: 7 2. 1. 7. 12. 10. 7. 12. 4. 5 1 39. ( ) 40. ( ) 42 7: 9: 21. 43. 63. 44. \ . i 5 : : 18 : 6. 47. ( ) *::54: 6. 48. ( ) : 8 ::1J : : i $2: )6 : 12 12 Ex. 2. What sum of money contains 7, as often as 12 oz. contains 3 oz. 1 Since 12 oz. contains 3 oz. four times, and .28 contains 7 four times, 28 is the sum sought ; thus : 12 oz. : 3 oz. : : 28 : 7. Quantities may be proportional as well as numbers. But the two terms in each ratio must be of the same kind, as only such can contain each other. EXERCISE VI. 10. 11. 12. 17. 18. 19. 20. 9 lb : 3 ft> : : 12 : ( ) 18 cwt. : 6 cwt. : : 60 : ( ) 16 yds. : 4 yds. : : 4 : ( ) 15 ac. : 3 ac. : : 16s. : ( ) 2o yds. : 5 yds. : : ( ) : 6s. 9 ac. : 36 ac. : : ( ) : 12s. ml. : 22 ml : : (' ) : lOd. >. ::( ):. 15 ho. : 60 ho. :36. 7 ml : ( ) : : 8 : 24. 8 wk. : ( ) : : 6/8 : 2. 8 in. : ( ) : : 32/ : 4/. 45 galls. : ( ) : : 63 7 5. 6 da. : 1 da. : : 3 gs. : ( ) 6. 10 ml. : 2 ml. 4 f. : : lOd. : ( ) 7. 7 gall. : 3 gall. : : 10, 10 : ( ) 8. 8 cub. ft. : 1 c. ft. : : 9 : ( ) 13. 24 in. : 6 in. : : ( ) : 9d. 14. 5 qrs. : 30 qrs. : : ( ) : 72. 15. 27 lb : 3 lb : : ( ) : 4/6. 16. 10 dwt. : 25 dwt. : : ( ) : 20/. 21. 4 years : ( ) : : 600 : 150. 22. 32 qrs. : ( ) : : 5 : 1, 5. 23. 96 yds. : ( )::!,!: 14/. 24. 72 tons : ( ) : : 132 : 12. 96 RULE OF THREE. : 16 ml. : : 2$ : lOd. 29. ( : 23 qrs. : : 3 A : 14<L 30. 27. ( ) : 3 ac. : : 6/8 : 3. 31. 28. ( ) : 2 days : : 15/ : 3/9. 32. : 1 qt. : : 27 : 9. : 2 pk. : : 8, 10s. : 2, 2/6. : 1 cwt. : : 100 : 5. 8 oz. : : 7d. : 2d. JLU f If 12 oz. contains 3 oz. as often as 28 contains 7, then 3 oz. is contained in 12 oz. as often as 7 in 28, also, 28 contains 7 as often as 12 oz. contains 3 oz., and 7 is contained in 28 as often as 3 oz. in 12 oz. Every proportion may thus be expressed in four ways : either of the two ratios may be put first, which gives two ways ; and either of the two terms of each ratio may be put first, which gives two ways for each ratio ; making four ways in all. Thus : (1.) 12 oz. : 3 oz. : : 28 : 7. (3.) 28 : 7 : : 12 oz. : 3 oz. (2.) 3 oz. : 12 oz. : : 7 : 28. (4.) 7 : 28 : : 3 oz. : 12 oz. EXERCISE VII. Read each of the proportions in Ex. v. and vi. in its four different orders. 108 In any proportion, such as 12 : 6 : : 8 : 4, the 1st and 4th terms are called the extremes, the 2d and 3d are called the means. . The product of the means is 6X8=48. Now, as the first extreme is as many times greater than the first mean, as the second extreme is less than the second mean, the product of the extremes must be the same as that of the means ; which it is, for 12X4=48. This is a second way, therefore, in which a proportion may be tested ; the product of the numbers whicn form its extremes is equal to the product of the numbers which form its means. EXERCISE VIII. Test each of the proportions in Exercises v. and vi, proceeding thus : 4 : 2 : : 8 : 4. " Four times 4 are 16, product of the extremes ; two times 8 are 16, product of the means." 109 Since the lstX4th = 2dx3d, it follows that when three terms of a proportion are given, we can find the remaining term. Thus : , , 2dX3d , Istx4th , 1st X 4th , , 2dx3d lst== --' 2d= -~' 3d= --' 4th --" PKOPOKTION. 97 Ex. 3. What is the first term in the proportion : ( ) : 15 : : 12 : 20 ? 15X12_ lst= 20 l.( ): 2. ( ): 7. 45 : | 8. 55 : i 13. 14: 14. 15 : 19.2:1 20. 4 : i 21. 12 : 22. 21 : EXERCISE IX. ) : 12 : : 15 : 9. 5. ( ) : 60 ) : 9 : : 30 : 22.}. 6. ( ) : 64 :: 792: 990. 9. 25 : ( ) : : 175 :49. 11. 143 : : : 495 : 594. 10. 572 :( ) : : 360 : 95. 12. 252 : : : ( ) : 400. 15. 60 : 45 : ; : ( ) : 15. 16. 177 : 59 23. 37 : 150 : ; 24. 10 : 19 : : <?n Qfi K9 : : : 8 : 20. ' : : 14 : 63. :985: :100: : 84 : 210. ; 162 : 288. ) ::11:9. i:21. 17. 234 : 252 : : ( ):28, :37i 18. 19:32::( ):428, : ( ) 27. 18 : 29 : ( ) 28. 48 : 41 i. 96 :52 ::18 :( ) 29.113:84 26. 20 : 24 ; 30. 912 : ] 234 : ( 576 : ( :3289:( i::57:( JLU The first and second terms must not only be of the same kind, but, when compound, must be reduced to the same name ; the third term, when compound, must also be reduced to its lowest name. The fourth term, when found, is of the same name as that to which the third has been reduced ; and must be brought back, when necessary, to its own highest name. 1. 2. 3. 4. 6. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. EXERCISE X. 4 cwt. 2 qrs. 8 ft : 9 cwt. 2 qr. 24 lb : : 10, 3s. : (?) 3 ml. 3 fur. 18 po. : 1 fur. 21 po. : : 10/10.4 : ( ? ) 16 el. qrs. 2 nl. : 25 el. 1 qr. : : 1, 11s. 7Jd. ( ? ) 1 cub. ft. 36 cub. in. : 189 cub. in : : 2, 18s. lid. ( ? ) 1, 14s. 4d. : 38, 3s. lid. : : 4 sq. ft. (?) 7 qrs. 7 bu. 3 pk. : 4 bu. 1 pk. : : 20 guineas ( ? ) 22 ft tr. 2 oz. : 1 ft 7 oz. : : 2, 13s. 8d. ( ? ) 97 yds. 2 ft. 10 in. : 17 ft. 1 in. : : 22, 2s. 6Jd. ( ? \ 1 ton. 6 cwt. : 1 ton 16 cwt. : : 1, 9s. 3d. (?) 22 yds. 8 in. : 1 ft 5 in. : : 1, 13s. 4d. (?) 12 sq. ft. 72 sq. in. : 72 so. in. : : 1, Is. 4jd. ( ? ) 13 ac. 2 ro. 36 po. : 9 ac. 3 ro. 6 po. : : 5, Is. 8d. ( f * 1 ho. 48 min. : 2 da. 6 ho : : 8/4. ( ? ) 38 lb : 4 st. 10 :: 1, 3s. 9d. (?) 26 ft : 12 qr. 2 lb : : 4^d. ( ? ) 1, 3s. O^d. : lOJd. : : 34 wk. 5 da. (?) 20 galls. 2 qts. 1 pt. : 5 galls. : : 4, 14s. 6d. (?) 15 qu. paper 20 sh. : 19 sh. : : 3, Is. 8d. ( ? ) 39 qrs. 6 bu. 3 pk. : 2 qrs. 5 bu. 1 pk. : : 15. ( ? ) 2yrs. 20 da. : 25 da. : 54. (?) 3bu. 2pks. : 52.} bu. :: 12/9. (?) 1 ton 13 cwt. 3 qrs. : 2 tons 3 cwt. 3 qrs. : : 2, 16. ( ?) 9 oz. 9 dwt. : 1 ft 12 dwt. : : 7/10i. ( ? ) . 98 RULE OF THREE. Ill Ex. 4. If 6 ft) butter cost 5/3, what cost 3 st. 3 ft) ? This is a question in the Eule of Three, and may be wrought by multiplication and division, as already shown. It may be solved, however, by means of a statement of propor- tion, which in many cases facilitates the applying of the two processes. Thus 6 ft) are contained in 3 st. 3 ft) as often as the price of 6 ft) is contained in that of 3 st. 3ft> : hence lb st. Ib s. d. 6 : 3 3 : : 5 3 : 14 st. 6 ft). 14 12 4Mb 63 4th term= 45 * 63 =gJg=472^d. = l, 19s. 4J. u 2 Bule for statement. For the third term place that which is of the same kind as the answer. Find out from the question whether the answer is to be greater or less than the 3d term ; if it is to be greater, place the greater of the other two terms second ; if less, place the less second. Ex. 5. If 12 men build a wall in 8 days, how many will build it in 6 days ? Here the answer is to be so many men ; place 12 men in 3d place. Then, as it will take more men to build the wall in 6 days than in 8, the 8 must be placed second, and the 6 first, da. da. men. 112 Thus, 6 : 8 : : 12 : answer i-*P=16 men. EXERCISE XL-MISCELLANEOUS.! 1. If 4 pints of gooseberries cost 1/8, for how much may a party of 12 persons get a pint and a half each ? 2. If 16 copies of an arithmetic cost 5/, how much will it take to supply a class of 50 ? 3. If 4 galls, ale cost 28/, what cost 1 hhd ? 4. A farmer was offered 65 sheep for. 73, 2s. 6d., but he took only 20 : what did they cost him ? 5. Two dozen oranges cost 2/6, now many may I buy for 7d. ? 6. Six pairs shoes cost 34/6, what cost 6^ dozen pairs ? 7. What cost 48 yds. cloth, if 72 yds. cost 26, 5s. ? 8. What cost 15 casks sugar, at the rate of 5, 10s. for 4 ? 9. If | ft tea cost 1/1$, how much may be had for 27/? 10. If 8 oz sugar cost 2^d., what cost 7 ft.? 1 To be wrought by the direct application of the elementary rules, or by means of a statement of proportion, at the teacher's discretion. ntoroiiTiON. 90 11. If 11 men do a piece of work in 17 days, in what time will 25 men do it ? 12. A farmer has 36 lb bread for his 24 reapers, but he engages other 6 : how much does each get ? 13. If 2 lb 6 oz. meat cost 1/10, what is that per lb? 14. My railway fare is 6/3 for 30 miles, what will it be for 92 miles ? 15. If 35 oz. silver-plate cost 9, 3s. 9d., what cost 34 oz. ? 16. In a bundle of 26 sovereigns, 1 lose 9|d. on three, how much may I expect to lose in all ? 17. If I walk 9 ml. 6 fur. in 3 hours, in what time shall I walk 37 ml. 5 fur. at the same rate ? 18. If f stone of salt cost 3^d., what cost 1| cwt. ? 19. I have 1/6 to spend on apples, how many shall I get at 3 for 2d. ? 20. If a class writes 6 lines per lesson, in how many lessons will they fill a copy-book of 12 leaves, each having 9 lines ? 21. How many yds. linen can be had for 24, 18s. 8d., if 17 yds. cost 3, 2s. 4d. ? 22. If 22 ac. 2 ro. 30 po. let for 65 guineas, what is the rent of 173 ac. 3 ro. 30 po. of the same land ? 23. Eight men are engaged in ditching at 12/4 a day for the whole ; if 5 more are employed, what will be the daily expense ? 24. If a ton of coals cost 19/6, what is that per ^ cwt. ? 25. What is the weekly rent of a room at 19/10 for 17 days ? 26. A field of 38 ac. was ploughed in 16 days : in what time would a farm of 175 ac. 20 po. be ploughed ? 27. If 100 gain 4, 10s. in a year, what is the gain for the same time on 22, 10s. ? 28. A tradesman's wages are 168 per year : how much is that for 25 days ? 29. I can go a journey of 63 miles in 18 hours, which the coach goes in 7 hours : how many times faster can I go by coach than by walking ? 30. How long will a field, which has pasture for 6 score sheep for 15 days, graze 4 score and 5 ? 31. Two slates have the same area. The one is 8 inches by 14 in., the other is 10 in. broad : what is its length ? 32. A parcel is carried 15 miles for 2/6 ; if I pay 19/3, how far should it be carried ? 33. A parcel is carried 15 miles for 3/6 : for what should it be carried 172 1 miles ? 34. The railway van delivers a parcel weighing 1 ton 4 cwt. 3 qr. for 5/6 : what is the weight of a parcel for which the charge is 17AOJ! 85. If 4 tons 14 cwt. cost 9, 5s. 6d., what cost 14 tons 16 cwt. ? 36. I bought 28 lb sugar at 5d. per lb : wishing to change it for sugar at 5<]d., what quantity should I get? 37. If 4 guineas weighed 21 dwt. 12 gr., what is the value of 10 oz. 14dwt. 6 gr. of gold? 38. The current of a river runs at the rate of 1\ miles in 3 hours : in what time will a boat drift to the sea 245 miles off? 39. If the current reaches the sea, 150 miles down, in 45 hours, in what time will it pass a village 24 miles down ? 40. What cost 1000 pencils at 5 for 3d. ? 113 100 RULE OF THREE. EXERCISE XII. MISCELLANEOUS- -continued. 1. A floor is covered with carpet, 25 yards of 5 qr. wide : liow many yds., 3 qr. wide, would be required ? 2. If the sixpenny loaf weighs 4 Ib when wheat is 32/ per quarter, what should it weigh when wheat is 27/ ? 3. If the 4 Ib loaf sells for 7.^d. when wheat is at SO/, what should it sell at when wheat is 34/ ? 4. How many yds. cloth at 10/6 per yd. should I receive in exchange for 4 pieces of 7 yds. each at 8/6 per yd. ? 5. What is the price of a pipe of wine at 41/3 for three gallons? 6. A bankrupt pays 250 on a debt of 350 : if my claim is 15, how much shall I get ? 7. Bought linen at 7 yds. for 1, 2s. UJd. : how much may be bought for 4, Os. 11 id. ? 8. A pipe fills a cistern of 280 gall, in 31 hours : in what time will it fill a 20 gall, cask ? 9. How many pounds sugar at 4|d. should I receive in exchange for 2 casks butter, each of 25.^ Ib, at 10d. per K> ? 10. What cost 24 bales cotton, each 90A Ib, bought at the rate of I/a for 7 ft 1 11. If a tree 20 feet high cast a shadow 30 feet long, how much longer will be the shadow of one 50 feet high ? 12. If 60 men can build a house in 90^ days, how much longer will it take 15 men to build it ? 13. What weight Avoirdupois is equal to 9 Ib 6 oz. Troy, if an oz. Troy is to an oz. Avoirdupois as 192 to 175 ? 14. A. and B. contribute to the capital of a business 1955 and 1720 respectively. The profits for the first year are 910 : B. receives a salary of 175 for management ; how should the rest be divided ? 15. A goods train runs 10 miles an hour for 7 ho. 30 min. : in what time will the express train make the distance at 36 miles per hour? 16. A tradesman has a yearly wage of 75 guineas : if he leave his situation on 18th August, how much has he to receive ? 17. If the diameter of a circle is to its circumference as 113 to 355, what length of plank will measure across a tank 45 feet round ? 18. Two plots of ground are equal in area : if the one is 28 feet long by 22 broad, and the other 77 feet long, what is its breadth ? 19. What cost 3 Ib raisins at 6, 7/6 per barrel of 100 Ib ? 20. How many yards cloth at 15/6 per yard should be exchanged for 230 yards at 17/6 per yard ? 21. If a bankrupt owes 279, 10/, and his estate realizes 200, how much will he pay per pound ? 22. A man takes 7 steps for a lad's 13 : in a distance of 8 miles, if the lad takes 19,500 steps, how many more does he take than the man? 23. A pound Troy, silver, is coined into 66 shillings : what is the value of 1 Ib Avoirdupois ? 24. Bought iron at 37, 10/ per ton : how must it be sold to gain the price of 1 ton upon 15 ? 25. If 10 sterling are equal to 256 francs, how many francs should be got for 36 guineas ? 26. At the rate of 16/8 sterling for 1 Canadian currency, what is the sterling value of 1000 currency } COMPOUND RULE OF THREE. 101 27. Find the rent of 106 ac. 2 ro. 30 po. of land at the rate of 25 for 23 ac. 1 ro. 27 po. 28. A ditch is measured with a line supposed to be 22 feet long, and found to be 530 yards long ; but the line is discovered to be 23 feet 6 inches long : what is the true length of the ditch ? 29. Of 75 men engaged for 24 days to do a piece of work, only 65 make their appearance : how much longer time will be required ? 30. 29, 17s. 6d. was collected as rate on a property at 9^d. per pound : find the rental. 31. How many yards of carpet, f yard wide, will be required for a hall, now covered with 87^ yards l| yard wide ? 32. If 6 men or 10 women can do a piece of work in 12 days, in what time will 3 men and 7 women do it together ? 33. Twenty-five printers can do a job in 12 days ; but three are sick : how long does that delay it ? 34. What length of board, ! , 9| inches wide, will make a square foot ? 35. How many gallons water must I add to a puncheon of whisky which cost 39, 4/, to sell it at 9/ per gallon, and neither gain nor lose? COMPOUND RULE OF THREE. Ex. If 16 cwt. are carried 45 miles for .3, 10s., how far ought 36 cwt. be carried for ,15, 15s. ? Here the distance required depends on two things (1.) The weight to be carried, and (2.) the price to be paid. The question therefore resolves itself into two others : (1.) If 16 cwt. are carried 45 miles for a certain price, how far ought 36 cwt. to be carried 1 which gives the statement 36 cwt. : 16 cwt. : : 45 ml. ; and (2.) If .3, 10s. pay the carriage for 45 miles, for wbat dis- tance should .15, 15s. pay? which gives the statement ,3, 10s. : ,15, 15 : : 45 ml. These statements are accordingly combined, thus : 36 cwt. : 16 cwt. : : 45 ml. 3, 10 : 15, 15. The third term in the combined statement being 45, the first term of the whole is the product of the two several first terms ; the second of the whole the product of tbe two several second terms ; and the fourth term = 45 Xl6x315 = go ^ 36X70 A statement of proportion, consisting of two or more state- ments in simple proportion, is said to be in compound pro- portion ; and the questions solved by it are sometimes said because involving more than one question in Rule of Three to be in Compound Rule of Three. 102 COMPOUND RULE OF THREE. Rule. Let the third term be of same kind as the answer. Arrange each pair of similar terms as in simple Rule of Three. Multiply all the first terms together for the compound first term, and all the second terms together for the compound second term ; then work as in simple Rule of Three. H5 EXERCISE I. 1. If 24 tons are carried 68 miles for 8, 10s. 6d., for what money will 14 tons be carried 120 miles ? 2. If I travel 144 miles in 4 days of 9 hours, in what time will I travel 560 miles, walking 8 hours a day ? 3. And liow far will I travel in 15 days of 10 hours daily ? 4. If 27 persons consume 252 lb bread in 6 days, how many pounds will 146 persons consume in 5 days ? 5. And how many persons will consume 1 ton weight in 14 days? 6. If 20 men can mow a field of 81 ac. in 9 days, how many men can mow 36 ac. in 10 days ? 7. If 16 men can build a wall 60 ft. long, in 45 days of 8 hours each, in what time will 35 men build a wall 145 ft. long, working 10 hours a day ? 8. A household of 18 persons spends 35, 15s. in 6 weeks : at same rate what should maintain a household of 14 persons for a quarter of a year? 9. If 5 labourers earn 12, 18s. 9d. in 16 days, how much will 12 labourers earn in a week of 6 days ? 10. If 18 yds. cloth, yard wide, cost 25, 7s. 6d., what cost 11 yds. 5 quarters wide ? 11. If 250 gain 28 in two years, in what time will 675 gain 150 guineas ? 12. If the fourpenny loaf weighs 72 oz. when wheat cost 77/ per quarter, what should the threepenny loaf weigh when wheat is at 81/ per quarter ? 13. And what is the price of wheat when the sixpenny loaf weighs 4 lb? 14. If it cost 3, 12s. 6d. to paper a room 21 ft. square and 14 high, what is the cost to paper one 15 it. square and 9 ft. high ? 15. I borrowed 50 for 6 months when money was at 5 per cent. : how much should I lend for 10 months in return at 4 per cent. ? 16. It costs 16/6 to supply 14 men with bread for 9 days when wheat is at 12/6 per bushel : what is the price of wheat to furnish 25 men for 7 days at same cost ? 17. To fill a tank of water, 30 ft. long, 8 ft. deep, and 24 broad, cost 3, 2s. 6d. : what cost it to fill another 6 ft. greater every way? 18. A wheel, 15 feet round, makes 52 revolutions a minute, and goes 75 miles in 5 hours : what distance will be traversed in 8 hours by another, which is 17| feet round, and revolves 45 times a minute? 19. If a boatful of herrings, containing 5000, be sold for 17, 13s. 6d., what should a boatful of 8560 bring, three of the latter being equal to in value to five of the former ? 20. A block of stone, 9 ft. x 6 ft. x 4J ft. weighs 5^ cwt, what is the weight of a stone -|d longer each way f 21. If the wages of 18 workmen amount to 78, 14s. for 22 days, MEASURES AND -MULTIPLES. 103 *hat will the wages of 25 apprentices coine to in 45 days, if 7 work- men receive as much as 13 apprentices ? 22. If 100 gain 4 in one year (1.) what sum will gain 36 in four years? (2.) what will 500 gain in six years? (3.) in what time will 400 gain 2 ? 23. A besieged town of 2500 men had provisions for 60 days at the rate of 30 oz. daily for each man : (1. ) how long will the stock last if they are reinforced by 500 men, and receive 24 oz. daily? (2.) what rate of distribution will make it last for 40 days ? 24. A family of 5 persons spend 18, 15s. in 3 weeks : (1.) what will it cost to maintain, at the same rate, a household of 18 persons for a year? (2). for what time will 225 maintain a family of 6 persons? (3.) how many persons will 1000 guineas maintain for 5 yrs. 20 wks. ? . 25. If 45 men cast a ditch, 48 feet long 9 broad 6 deep, in 24 days of 9 hours each (1.) What length of drtch will 75 men cast in 15 days 10 hours each, 6 feet broad and 8 feet deep ? (2.) How many men would cast a ditch, 6 feet broad 20 long and 8 deep, in 15 days of 8 hours each ? (3.) In how many days of 8 hours each, would 15 men cast a ditch 12 feet long by 8 broad and 4 deep ? (4.) How many hours daily, for 10 days, would 16 men take to cast a ditch 20 feet long by 8 broad and 4 deep ? LI 6 MEASURES AND MULTIPLES. Prime Numbers. A number which cannot be broken up into factors is called a prime number ; but 1 is not counted as a factor. A number which can be so broken up is called a composite number. Thus 3, 5, 7 are prime numbers ; and 4, 6, 8 are composite numbers. To find whether a number is prime or composite, we divide it by any of the simple numbers which we think it will contain ; bearing in mind the rules already given for this purpose in sect. 53. EXERCISE I. "Resolve the following numbers into prime factors : 1. 27, 46, 64. 4. 156, 240, 236. 7. 1456, 1728, 2445. 2. 95, 56, 121. 5. 198, 432, 560. 8. 2384, 5408, 7280. 8. 87, 115, 140. 6. 375, 820, 972. 9. 6372, 4116, 4716. 10. Write down the prime numbers in order from 1 to 50. 11. Do. from 50 to 100. (12.) Do. from 100 to 150. 104 MEASURES AND MULTIPLES. 117 Greatest Common Measure. One number is called a measure of another when it is con- tained in it evenly ; thus 6 is a measure of 12. One number is called a common measure of two or more numbers, when it is contained in them evenly ; thus 3 is a measure of 18 and 24. One number is called the greatest common measure of two or more numbers, when it is the greatest number which is con- tained in them evenly ; thus 6 is the greatest common measure of 18 and 24. The greatest common measure can often be found by inspec- tion ; when it cannot, we proceed as follows : Ex. Find the greatest common measure of 237 and 395. Divide the greater (395) by the less 237)395(1 (237) ; then the first divisor (237) by the 237 first remainder (158) ; then the second 158)237(1 divisor (158) by the second remainder 15g (79) ; which is a final divisor. 79)1 W2 Now 79 is the greatest common measure 158 of 79 and 158. But whatever measures any two numbers measures their sum ; therefore 79 is the greatest common measure of 158 and 237. On the same prin- ciple, the greatest common measure of 158 and 237 is the greatest common measure of 237 and 395, the given numbers. Therefore 79, the last divisor, is the greatest common measure required. Rule. Divide the greater of the two numbers by the less, the first divisor by the first remainder, the second divisor by the second remainder, and so on. The last divisor is the greatest common measure required. If there be three or more numbers given, find the greatest common measure for the first two, then for that greatest com- mon measure and the third number, and so on. EXERCISE TI. Find the greatest common measure of the following numbers : 1. 122 and 427. 5. 2387 and 2563. 9. 5705 and 6559. 2. 148 and 185. 6. 2002 and 2509. 10. 18996 arid 29932. 3. 285 and 465. 7. 2145 and 3471. 11. 5415 and 30105. 4. 576 and 744. 8. 6465 and 7335. 12. 34789 and 85937. 13. 36, 84, 720. 17. 1241, 1428, 1853. 14. 45, 63, 108. 18. 925, 1475, 5680. 15. 256, 372, 522. 19. 1092, 1716, 2940. 16. 244, 472, 636, 20. 3081, 5451, 6255, 8703. MEASURES AND MULTIPLES. 105 Least Common Multiple. One number is a multiple of another when it contains it evenly, and a common multiple of two or more numbers when it contains them all evenly. One number is the least common multiple of two or more numbers, when it is the least number that contains them evenly. If the numbers are prime to each other, the least common multiple is got by finding their product ; thus, the least common multiple of 11 and 13 is 11 X 13 = 143. But, if they have some common factor, the least common multiple is got by throwing out the greatest common factor and multiplying the remaining factors together ; thus, the least common multiple of 24 and 36 is, by throwing out the factor 12 from one of the numbers, 24X3 or 2X36 = 72. Ex. Find the least common multiple of 15, 24, 30, 42, 72. 15, 24, 30, 45, 72 15, 45, 36 ~~ Arrange the numbers in a line. 2 Strike out 15 and 24, since any 3 multiple of 30 and of 72 will be i, 1 i r j.1 O a multiple also of them. Divide by the first prime factor ' of more than one of the numbers, that is, by 2; writing the quo- 2X3X3X5x4 = 360 Lc.m. tients, and any of the numbers not divisible by 2, in a line below ; thus, 15, 45, 36. As 2 is not a common measure of these, divide by the next prime, which is 3 ; giving 5, 15, 12. Divide again by 3, which is a common measure ; giving 5, 5, 4. Divide by the next prime which is a common measure, that is 5 ; giving 1, 1, 4. These numbers have no common measure. The divisors and the remaining factors are the only necessary factors of the least common multiple required. Rule. To find the least common multiple of two num- bers : Divide one of them by the greatest common measure, and multiply the other by the quotient. To find the least common multiple of more than two num- bers : Throw out any number which is a measure of another ; divide the rest by 2 as often as it is a common measure, bringing down the undivided numbers into the line of quotients ; then" by 3 and the other primes in order, till the division is exhausted ; multiply the divisors and the remaining quotients for the least common multiple. 106 VULGAR FRACTIONS. EXERCISE III. Find the least common multiple of the following numbers : 1. 16, 20. 6. 63, 108. 11. 14, 21, 28. 16. 288, 360, 1728. 2. 32, 48. 7. 98, 156. 12. 18, 32, 56. 17. 720, 336, 1736. 3. 56, 64. 8. 391, 659. 13. 32, 44, 52. 18. 6, 9, 15, 18, 20. 4. 120, 144. 9. 703, 1036. 14. 17, 29, 53. 19. 1, 2, 3, etc., to 9. 5. 72, 132. 10. 1548, 2537. 15. 90, 100, 125. 20. 5, 7, 9, 12, 15. 21. 25, 60, 72, 35. 23. 3, 7, 8, 9, 11, 49, 55. 22. 14, 54, 63, 81. 24. 12, 16, 24, 36, 48, 72, 144. 119 VULGAR FRACTIONS. Notation of Fractions. Any part of a whole number is called a fraction. Thus, if the line A D be divided into three equal parts, any one part, as A B, is one-third of the whole, denoted J, and any two parts, as A c, are two-thirds of the whole, denoted . The lower of the two numbers is the name, or, as it is called, the denominator of the fraction, and shows the size of the parts into which the whole is divided ; the upper is the number, or, as it is called, the nume- rator of the fraction, and shows the number of these parts which the fraction contains. Fractions denoted by a numerator and denomi- nator, like J or , are called vulgar fractions, to dis- tinguish them from a certain kind of fractions, which as will be noticed further on may be denoted in a different way. Any part of a number larger than one is also a fraction. Let the whole line AC be composed of AB = 1 and BC = 1, and therefore be = 2. Divide A c into three equal parts, A D being one : divide A B into three equal parts, A D will be found to be two. Thus of 1 is equal to J of 2 ; and both are denoted f . EXERCISE I. Read the following fractions (1.) as fractions of unity ; and (2.) as fractions of their own numerators. i. f, *, T 9 *> &, &, H. 3. y, ft, v, to m- 2. fc H, ifc to i*> T*I- 4 - to Ik tt> tt> *& --B VULGAR FRACTIONS. 107 A whole is equal to the sum of all its parts ; thus the line A D above is equal to its three thirds together, A B-J-B c-f-c D. Fractions, whose numerator and denominator are the same number, such as f , f, or , etc., denote one unit, broken up into 4, 5, or 6 parts respectively ; and therefore they are all equal in value to one another. >Q Improper Fractions and Mixed Numbers. If we have two whole numbers, and divide each into three equal parts, we have six parts each the third part of one. If we take four of these, we have a fraction made up of one whole number and one-third more, and denoted f. A fraction greater than one, and whose numerator is consequently greater than its denominator, is called an improper fraction. An improper fraction may always be resolved into a whole number, or into a whole number and a fraction, which is called a mixed number ; and so a whole number, or a mixed number, may always be resolved into an improper fraction. Ex. 1. How many whole numbers in 2 ^ ? and in %f 1 Since seven sevenths are one whole number, there will be as many whole numbers in *p as there are sevens in 28, that is, 4 ; which is the answer. And as many whole numbers in %f as there are sevens in 29, that is 4}- ; which is the answer. Rule. To change an improper fraction to a whole, or to a mixed number, divide the numerator by the denominator, EXERCISE II. Change to whole or mixed numbers : i. v, , v, v, if- 4. v, ft, is, w, m- 2. * 2 , W, W, W- 5- w, W, W, ** 3. V, It, ffc II & 6. v 9 i 3 , <W, ft, Hi- Ex. 2. How many sevenths in 5 ? and in 5 J ? Since there are seven sevenths in 1, there will be 5 times as many in 5, that is 3 ^. To 35 sevenths add the 6 sevenths already given in 5|, and the total number will be ^. Rule. To change a whole number, or a mixed number, to an improper fraction, multiply the whole number by the given 108 VULGAR FRACTIONS. denominator, and (in a mixed number) add the given numera- tor. Place the denominator below. EXERCISE III. 1. How many ninths and elevenths in 3, 6, 8, 9, 13, 16? 2. Write with 8 and 12 as denominators : 4, 6, 7, 10, 12, 15 ? Change to improper factions : 3. 24, 8ft, 9J, 120$. 5. 17ff, 29|, 45&, T2&. 4. 8ft, 73^, 125tf, 200ft. 6. 342, 74^, 2ftf . ; 121 Equal Fractions of Different Denominators. Let the line A c be divided into three equal parts, A B being $. Let each of these third-parts be sub- divided into two, making six parts in all : A B will contain . Thus f ^ ; and a fraction is not altered in value, if its terms be multiplied or divided by the same number. Rule. To change a fraction to higher terms, multiply both by the same number : to change it to lower terms, divide both by a common factor. Note. In all operations with fractions, they should, as a rule, be reduced to their lowest terms. The common factor to be taken as divisor may often be found by inspection ; if not, find the greatest common measure, and divide both by it. EXERCISE IV. 1. How many ^ths in $, \, I, \, T V, T V, & ? 2. How many ^ths in fe f, f , , , |, &, A, &, ft, ft, f|, ft ? 3. How many T ^ths in \, f, f, f, f, ft, ^, ft, ft, ft, f|? 4. Reduce to their lowest terms &, f , f-f, |f, ^^ ft|. 5. Also, T ^, ftf , Ml, ftft, Hft, mi, MM, iV^ 6. Also, H|, flfl, |Mi, m I IM, -B Common Denominators. ^x. 1. Change the series of fractions J, |, J, ^ to another series having the same denominator. The least common multiple of the denominators is the new denominator required. It is obvious that 12 contains them all ; so that all the frac- tions have to be brought to ^t VULGAR FRACTIONS. 109 Multiplying the terms of the first by 6, of the second by 4, of the third by 3, we have, as the series of fractious required, T2> T2> V2"> T2* Ex. 2. Change f , J, to fractions of the same denominator. The denominators being prime to each other, the common denominator is f X 4 X 5 = | 3X4X5 = 60. k x3x5 = 45 Then, multiplying the terms of each ^ x 3 x 4 = 4 8 fraction, so as to bring it to ^th, we have Jtule. To change fractions to a common denominator : find the least common multiple of the given denominators for the new denominator, and multiply each numerator by the number of times the new denominator contains the old. EXERCISE V. 1. !, *, A; *, t, A; i, i A, H; *, &, If 2. *,f,i; f,*,f; *,**; *>*,*; f,*,A- 3. A, H, tt ; > ^ A, A ; A, H, H ; i, I, H, if. 4 7 8 11 13. 5 JT 13. 56 29 _4\ 194 8 S4 * 6* l~8> IT> 1TO; 60 > 8J ~tt) "24 > TDIT^ TJlfOJ TGU^ BTTd^ 8TTO' I Addition and Subtraction. Ex. Find the sum and the difference of J and f . The fractions must first be brought to the same name : |-J Thus, as 21 sh.+16 sh. are 37 sh., so f J + Jf = JJ, or 1 And, as 21 sh.-16 sh. are 5 sh., so ^Tbte. The denominator is not changed, since adding or sub- tracting quantities does not change their name. Rule. To add or subtract fractions, change them to a common denominator, add or subtract their numerators, and place the common denominator below. EXERCISE VI. l. i + 1 + A* f + A + &> f + i + 8, 1 + f + I + H- 2- A + A + A* T S * + A + H> A + A + M, M + *f + H- 3. 2| + 44 + 6|, 7f + 9A + l^ii, If + A + 1 *. I - f . A - A, H - A, H - H T 9 ^ - A- 5. 8* - 21, 9ft - 7|, 42^ - 264, 158f i 79f- : Multiplication of Fractions. Ex. 1. Multiply A- ^7 3. Since 4 X 3 = 12, ^ 5 - X 3 = i^ = If ^ |. This is also the result for 3 X r-. 110 YULG All F 11 ACTION S. Ex. 2. Multiply ^ by f, T% X 3 = ^ ; but, as the multiplier is only the fifth part of 3, this product is 5 times too great, and the product required *'B'* Rule. To multiply a fraction by a whole number, multiply the numerator, or divide the denominator, by the whole number. To multiply a fraction by a fraction, multiply the numerators together, and also the denominators together, cancelling when possible. If one of the terms be a mixed number, change it to an improper fraction. Note. ^ of f is another way of denoting the fraction got by multiplying T ^ by f ; and is called a compound fraction. EXERCISE VII. 1. |f x 4, 7, 8, 14, 18, 6, 24, 48, 150. 2. f x 12, 16, 4, 42, 7, 26, 36, 21, 84, 256. 3. 288 x f, If, &> H, *, A, ft, Hi tt> ffl. 5. Ifxf, 2{fxlfc, 7fx6, lO&xlOf, 6. 5xof, 4fx|of^, f of2x|, *** In connexion with this and the following section, the pupil may revisf sections 52 and 53. Division. Ex. 1. Divide by 2. Since 4-=-2 is 2, -f-s-2 is f. If the numerator does not con- tain the divisor, multiply the denominator instead ; thus -J-v-2, being equal to f , is also equal to $. Ex. 2. Divide by J. A_i_2=_:L. But, as we are to divide only by the third part of 2, we have divided by a number 3 times too great, and therefore the quotient is 3 times too small. The quotient re- 4x3 quired will therefore be g^, that is, f 'or If. Rule. To divide a fraction by a whole number, divide the numerator, or multiply the denominator, by the number ; to divide a fraction by a fraction, invert the divisor, and multiply the fractions. EXERCISE VIII. 1. T 4&-M2, 4, 16, 8, 48, 96, 192, 14, 36, 40. 2. tf-H. *Ki, flH-H> tm > 2J-M& 5H-tf. 3. * of I-M of |, -I of 5H-& of 3$, i|^f of 9f . 4- 34!-^, A of f|-l of fc T T of SS-5-t, H of VULGAR FRACTIONS. Ill .26 Reduction. Fractions of quantities often require to be reduced to a higher or a lower name. Ex. 1. What part of a shilling is ^-th of a pound ? Pounds are reduced to shillings by multiplying by 20, therefore ^ f = |J sh. or f sh. Ex. 2. What fraction of a pound is f of a shilling ? Shillings are reduced to pounds by dividing by 20 ; so Ex. 3. What is the value of Pounds are reduced to shillings by multi- 4 plying by and then by 12. Multiplying 20 by 20, we get 5 for the shillings ; multiply- 15j80^ 5 ing the remainder of shillings by 12, we get 75 4 for the pence. ^~ Ex. 4. Express 2/3J as a fraction of a 12 pound. ^ )6C)(4 Farthings are reduced to pounds by divid- 60 ing by 4+12+20, or 960. The number of J ^Jf farthings in 2/3J- being 109, the required fraction therefore is Rule. To reduce a fraction of a quantity from one name to another : If to a lower, multiply by the number of times the lower name is contained in the higher ; if to a higher, divide by that same number. Note. Fractions of different names must be reduced to the same name, before they can be added or subtracted. EXERCISE IX. 1. What part of a shilling & of a penny is T \, &, ^, -^ S) ^3. 2. Reduce to oz. and to dwt. lb tr., fib, lb, -*f lb, ^ lb. 3. Express as fractions of a foot and of an inch, T % yd., yd., -fa yd, 4. Reduce to quarts and gills, gall., ^ gall., ^ pk., ^ pk. 5. What part of a is f sh., 3 T % sh., f cr., f cr., ^ sixp., ^ hlfcr. 6. Reduce to quarters bu., ^ bu., -|| bu., pk., f pk. 7. Express as fractions of a yard f qr., | nl., f qr., & nl. 8. Reduce to cwts. V oz., %$ lb, 3 ^ 6 lb, 2 | qrs., 72 ^ lb. 9. Reduce to acres 2| ro., f ro., 25| po., T \ po. 25f sq. yds., & yds 10. Find the value of |f, , ^ cr., ^ flo., f half-gum., ^ guin. 11. Also of ^ yr., ^ day, ^ wk., ^ ac., T % ml., ^ ^ tr. 1 1 2 VULGAR FRACTIONS. 12. Also of ^ fur., f| cwt., f gall., f- sq. yd., cub. foot. 13. Express 7/10$ as fraction of , also as fraction of 17/6. 14. What part of an acre is 2 ro. 27 po., also 15 po. 20 yd. 6 sq. ft. f 15. Reduce 16 wk. to years, and 5 bu. 3 pk. to qrs. 16. Reduce 7 oz. 16 dwt. 20 gr. to ft), and 15 cwt. 1 qr. 25 tb to tons. 127 EXERCISE X.-MISCELLANEOUS. 1. Find the difference between 3 guineas and 4. 2. Multiply the sum of and by their difference. 3. From bu. were given away ^ pk. and gall. : what remained? 4. How often will the price of 4 ells Eng. exceed that of 4 yds ? 5. If 3 of a loaf is divided equally among 12 children, what share of the whole loaf does each get ? 6. What is the difference between of ff and of & ? 7. What number has 16 for its ths ? 8. What number added to i+| + & will make 2 ? 9. What number is contained in three times ? 10. I read of a book in an hour : when shall I finish it? 11. If a train goes 74 miles in 2 hours, what is that per minute? 12. What part of 5 guineas is 3 of 5| half-crowns ? 13. A -cwt. tea in J-lb packages is further subdivided into 6 equal parts each package : what part of the whole does a family get which receives 7 shares ? 14. How many 3.} lb loaves are required to give 100 poor people each Ulb of loaf? " 15. What part is 13/6f of 3, 7s. 10M., and of 3 guineas ? 16. If my property is only ^ of my debts, what is that per pound ? 17. From 340 yards cloth take away |, and then of remainder : how much is left? 18. What number is reduced to 64 when $ of it are taken away? 19. Divide the sum of 2$ and 3.4 by their difference. 20. What number multiplied % 3 will give 2 ; and what number divided by it will give ? 21. Bought 45 shares at 105^, and sold them at 106 : find the gain. 22. What part is a square of If inches a side,of one of 3| inches ? 23. In a bag of 1000 sovereigns, each is light by fa dwt., find their total value. (See 88, qu. 11.) 24. If I hold f of a house, whose value is 2760, 10s., and sell | of my share, what value remains to me ? 25. What number will multiply of llf so as to give 1 ? 26. Find the price of 35f- stones sugar @ 12/4 per 2^ stone ? 27. If f tt> cost 25/6, what cost 3| cwt. ? 28. What is the breadth of an acre of land 47^ yards long ? 29. In a school of 100 pupils, of whom f are boys, 7 boys and 4 girls are absent : what part of each is present ? 30. If beer is distributed at the rate of 4^ gallons to 9 persons, what will a family of four persons get ? 31. What part of 14 days 10 hours is of 2^ days ? 32. How much of a mile remains if 150|y fathoms be cut off? 33. What part of 1 is of 6/10, -| of f crown, and f florin ? 34. Add together inches + f foot + yard. VULGAR FRACTIONS. 113 35. What number has 9^ for ^ of its eighth part ? 36. If I gain ^ in a day, what part of a crown do I gain per hour ? 37. How many cub. ft. in a box 4^ ft. long, 3 broad, and 7^ deep ? 38. After walking 15^ miles I had^still of my journey before me : what was its entire length ? 39. If a stopcock empty a cistern in 6 hours and another in 9 hours, in what time will both together do it ? 40. And in what time will both do it, if No. 2 begins to run after No. 1 has run for 2 hours ? 41. Divide the quotient of ^fa and j-jjW by their product. 42. If ^ of a property is worth 7g > what is ^ worth ? 43. A shepherd said that if he had as many more sheep, and half as many more, and quarter as many more, his flock would number 132 : what was its actual number? 44. If 2 persons buy 1 Ib tea, and one pays 2/11 for f of it, how many oz. does the other get, and what does he pay ?" 45. If a labourer can mow a field in 7f$ days, how much of it can he mow in 1 day ? 46. What number is that of which the third part of its quarter is 47. How many steps of 2f feet each are in a quarter of a mile ? 48. And how many more steps, if each is only 2 feet ? 49. If from 1, I give away 5s. 9d., then 2/3, and then -fa of the 1, what part of it remains to me ? 50. What number is that to f of which if 9 be added, there will result 19 ? 51. What remains of 1000 Ib troy, after subtracting , , , and | of it ? 52. If 56 labourers get each -| florin per hour, how much do they all get together in 6 days 8 hours, working 10 hours daily ? 53. What part of 4 da. 5 ho. 20 min. is of 3 da. 16 ho. 10 min. ? 54. What fraction is 1 ton of 3 cwt. 1 qr. 16 Ib and of 3 ton 17 cwt. 1 qr. ? 55. What cost 27| yards at llf d. per yard ? 56. A. can collect a given sum in 6 da., B. in 8, C. in 9, and D. in 10 : in what time will they do it together ? 57. A field 47$ yards x 27f yards is equal to another of 29 yards long : what is its breadth ? 58. If a farm of 276 acres is rented at 478, 10/, what is that per acre ? 59. If I can walk 20 miles in 5 hours, and my friend can do it in 6 hours : starting from opposite ends at the same time, how far are we from each other after 1 hour? and in what time from starting should we meet ? 60. Divide ^ acres among a family of 7 persons, giving to the four oldest jt of a share each more than to the three youngest. 61. What is the difference between the fourth proportionals to |, 62. From a certain field its third part was cut off, but 8 acres were added, making it now 172 ac. 2 ro. : what was its original size ? 63. A tradesman bought 13| Ib tea for 2| guineas, and retailed it at 2|d. per oz. : what did he gain or lose ? %* Work the questions in Practice, Exercises vii. and viii. H 1&I9 114 DECIMAL FRACTIONS. DECIMAL FRACTIONS. Notation. In any number, as 348, the first place is units, the second tens, and the third hundreds ; each place being ten times the value of the place to its right. If the notation were extended to numbers having places of lower value than units, the place to the right of units would be tenths, the next hundredths, the next thousandths, and so on. Such a notation is actually in use. Let there be figures after the 348, marked off from it, for distinction's sake, by a point, thus, 348-888 ; the 8 immediately after the point denotes A, the next 8 denotes yg^, and the next 8 denotes 10 9 00 . The figures after the point are therefore really fractions, whose numerator alone is written, and whose denominator, 10 or powers of 1 0, is understood. The value of each place is known by its position from the point, that is, from the unit's place ; so that each place must always be represented, if not by a figure, then by a cipher. Thus, if T% is denoted by '8, y^ (which is j%+tfer) is denoted by -08, and j^ (which is J^+^+j^) is denoted by '008. But ciphers occurring after figures are of no use ; thus, 8 or ^ is the same as "80, or I%+TOTT- These fractions are called decimal fractions, from the deno- minators being either 10 or powers of 10. Rule. The number of ciphers in the denominator of a decimal fraction is equal to the number of figures after the point. EXERCISE I. 1. Write with denominators : 7, '07, 3'06, '009, -209, '5763. 2. 36-2008, -7064, -0009, '0101, -006001, -090005. 3. -061, -250, -300145, '007201, -000051, -0000001. 4. Write without denominators : ^, ffo, T ^, f $, *$, -ft?. ** itTo* Tiro 1 1000* looflo* Tooo* loooo 1 oo o 5 o> loooooo* " IffOOO* TITOS TuTTO 1000000* 100000* 1000000* Equivalent Vulgar and Decimal Fractions. Ex. 1. Express f as a decimal fraction. If we add ciphers to both terms till the denominator may be cancelled, we have t=f$$==TT5ifo = "375 ; which is just the quotient we should get, by adoling ciphers to the numerator, and dividing by the denominator. DECIMAL FRACTIONS. 115 Rule. To change a vulgar fraction to a decimal, add ciphers to the numerator, and divide by the denominator. EXERCISE II. Change to decimal fractions. 1- f> t, I, A, A, & H- o 9 o 69 15 93 1 83 * TWV> fSt 3Tff> lff> S&OOJ ^~617> 64' Ifo. 2. Change "78 to a vulgar fraction. 78 = /& = f&. Rule. To change a decimal to a vulgar fraction, write the decimal denominator below, and reduce to lowest terms. EXERCISE III. Change to vulgar fractions : 1. 0.25, -875, -68, -36, -780, -375, -008, '02, -068. 2. -072, -100, -144, -00628, '0560, -0081, 2-00125, 6-00408. .30 Interminate Decimals. Ex. 1. Change i to a decimal fraction. Adding ciphers to the numerator by rule, and dividing by deno- minator, J = "222222, etc., the 2 repeating itself for ever. Such a decimal is called a repeating or recurring decimal, and is denoted by a point over the repeating figure ; thus : f == '2. A recurring decimal, when changed to a vulgar fraction, will therefore have 9, instead of 10, for its denominator. EXERCISE IV. 1. Change to decimal fractions, I, tf, J, I, V V. 2. Change to vulgar fractions, '3, '1, '4, 1'6, 3'3, 4'S. Ex. 2. Reduce / T to a decimal fraction. By the rule T T T = 636363, etc., the 63 repeating itself. Such a form of decimal fraction, where more than one figure repeats itself, is called a circulating decimal, and is denoted by a point over the first and last figures of the part which repeats itself ; thus : A = '63. Rule. A circulating decimal is changed to a vulgar fraction by writing for denominator as many nines as there are repeating figures ; thus: GSssSfssA 116 DECIMAL FRACTIONS. EXERCISE V. 1. Change to decimal fractions, T 9 T , ?, A, !?, *V, 2. To vulgar fractions, -64, '024, -72S, '0198, 2 57, .Re. 3. Reduce if to a decimal fraction. By the rule, i|='43181S18, etc., the 18 repeating. Such a deci- mal, where part only repeats itself, is called a mixed decimal, and is denoted by a point over the first and last figures of the part repeat- ing ; thus iJ=-43i8. To reduce '43i& back to a vulgar fraction : -43i&=-43+'OOl$. Now, -43=i*oV, and -OOite^o of '18 or of {&, that is, = 9 J8ij. Therefore, 43x99+18 43(1 00-1)+ 13 4300-43 + 18 4318 -43^ - - Rule. Subtract the finite part of the^ decimal from^the whole decimal given, and below the difference write as many 9's as there are figures repeating, with as many ciphers as there are figures in the finite part. EXERCISE VI. 1. Change to decimal fractions : i. u, m, n, if, Ttir, ill.. 2. Change to vulgar fractions : 2. '272, '025, -0045, "0286$, '3666, '3666. \* For practical purposes, interminate decimals are little used, as all necessary accuracy may be secured by carrying out the fraction a few places. We shall, therefore, exhibit the rules for operation with finite decimals alone, Addition and Subtraction. 9. To add or subtract decimal fractions, (1.) write the numbers so that places of the same name 293*406 shall be under each other, and proceed as in whole 29'06 numbers. 7'093 Note. Any decimal may be extended by the !59?i addition of ciphers to the right ; but if it be a re- 430'0674 peating or circulating decimal, the repeating part is used for that purpose. <2.) lOl Ol 24-078 37-542 EXERCISE VII. 1. 72-093+391-7+805'006+r094+48'0008+730-0514. 2. 63'904-flO'09-j-240-099+381-0001+l-0904+51'280i. 8. 79-6+82-7214+301-73+293-+26'64+31-125+-0004. DECIMAL FRACTIONS. 1 1 7 4. 83*7024+36-620l44*0001+7*3+29-2i+25-23+7'0108. 5. 256'704-f2*0093+47-6002+39'0804+2*09+3*014. 6. 7-30S2+31*0041+7'0001+38-009+25'4+42*72+-i. 7. 94-748-08, 74-002 39'C09, 41-128-601. 8. 37-000419-071, '00098 '000041, 2'7041 '0047. 9. 24-6 7-3, 125-1 76-009, '00821 -000047. L 3 2 Multiplication. Ex. Multiply (1.) -75 by -5, and (2.) '075 X '5. 75 X -5= T Vb X T s n =vVD 5 =*375. 075 X -5=1^ X ^ s = I g^ =-0375. Rule. To multiply decimals, multiply as for whole numbers; and point off in the product as many decimal places as there are in both factors together, prefixing ciphers if necessary to make up the number. Note. A decimal is multiplied by 10, 100, or 1000, by carrying the point to the right one, two, or three places respectively. EXERCISE VIII. 1. 730.x -84, -093, '006. 7. 17827 X '00006, '0905, 3'0075. 2. 4-709 X *38, 1-72, -0024. 8. 73'04 x 27'02, 56*009, '4056. 3. 36-001X76, -076, 7'006. 9. 684*6 X2'56, '784, '003. 4. 84-008X1000, 3003, '093. 10. 2*847x10000, 100, *001, '64. 5. 258X-075, 3'005, 24'01. 11. 'OOOSX'Oo, *7, '009, *732. 6. 1824X182'4, '0002, '195. 12. 'C00091X-004, -71, 7000, 1*4. Division. Ex. 1. Divide 19'305 by -65. It will not alter the quotient if both divisor and dividend be multi- plied by the same number. Multiply both by 1000 ; then 19 -305 _ 19305 ~~ 7 65 ~650~~i proceeding with the division as in whole numbers, and pointing the quotient when the fractional part of it occurs. Ex. 2. Divide -000042 by -007. Multiply both terms by 1,000,000 to remove the fractions ; then "We may check the correctness of the position of the point in the Quotient, by observing that the dividend should contain as many ecimal places as the divisor and quotient together. Rule. To divide decimals multiply both divisor and dividend by the larger of the two denominators, and divide as in whole numbers. 118 DECIMAL FRACTIONS. EXERCISE IX. 1. 256-7236, -174, -006 7. 1-^*76, '009, 2-56, -1. 2. 3-4894-1-84, 12'62, -0007. 8. 2'708-h*33, 5*07, 40'602. 3. '000639, '09, '009, '0009. 9. 853-096-^-037, lOOO'l, -298 4. '0006390, 900, 90'09. 10. 7'9-7-39*68, '85, '0027. 5. 5000 1, -05, -0025. 11. 305*081-5-3456, '29, '528. 6. *85642-*74, *96, '0056. 12. 3476-h*0008, '094, 3476'07. Reduction. Decimal fractions of quantities often require to be reduced to a higher or to a lower name. Ex. 1. What part of I/ is '025 ? Pounds are reduced to shillings by multiplying by 20 ; therefore -025=-025 X 20 sh.s='5 sh. Ex. 2. What part of a pound is *37o sh.? Shillings are reduced to pounds by dividing by 20 ; therefore Ex. 3. What is the value of -0875 ? *0875 90 Pounds are reduced to shillings and pence by multi- plying by 20 and by 12. Multiplying by 20 we get I/ 1'7500 and a remainder ; multiplying the remainder by 12, 1% we get 9d. Answer 1/9. 9*0000 Ex. 4. Express 2/3 J as decimal of a pound. Farthings are reduced to pounds by dividing by 960 or (4 X 12 X 20). The number of farthings in 2/3$ being 109, the sum is $g$, which, reduced to a decimal, is '113573 nearly. Rulo. To reduce a decimal fraction of a quantity from one name to another : If to a lower, multiply by the number of times the lower is contained in the higher ; if to a higher, divide by that number. Note. Decimal fractions of different names must be reduced to the same name before they can be added or subtracted. EXERCISE X. 1. What part of a shil. & of a penny is '75, -296, '0085, '54. 2. Reduce to oz. and dwt. '396 ft tr., *094 ft, '11875 ft, *0792 ft. 3. Express as parts of a foot and inch -294 yd., '0576 yd., -0075 yd. 4. As parts of a quart & of a gill, '0375 gall., '0063 pk,, -1859 gall. 5. What part of a pound is '275 sh., '945 cr., 6'275 flo., 9'736 hfl-cr. 6. What part of a quarter is '98 bu., -095 bu., 8' 625 pk., '986 pk. 7. What part of a yard is -825 qr., 2-76 nl., -0856 qr., *125 nl. 8. What part of a cwt. is 6'75 oz., 15'375 R>, -8930 qr., -0824 ft. 9. What part of an acre is 1'36 ro. '86 ro., 1S'32 po., 12*96 sq. yd. DECIMAL FRACTIONS . 119 10. Find the value of -784, 2 "0086, '98 cr., '656 hf.-sov., '8 guin. 11. Also of '0872 year, -3768 wk., '175 ho., 4-085 da., '756 min. 12. Also of -279 fur., *936 cwt, '785 galls., '0025 tons, '6248 cu. ft. 13. What part of a pound, and of 13/9, is 7/6, 8/9, 5/10 J, 17/6, 12/9*. ? 14. ,, of an acre, &of5Jac., is 3 ro. 7po., 14 po. 15yd., 5yd. 6 s.ft. ? 15. oflyr.,&ofl yr. 175 da.,is 10da.6ho., 27 wk. 5da.,5h. 10m.? 16. of Icwt.,&oflc.2qr.6ft>,isl7ft>6oz.,3qr.l5ft>, 10c.lqr.4ib? L35 EXERCISE XL MISCELLANEOUS. 1. Add '375+*675 guin. +'792 cr.+'125 fl. 2. Divide f of 8*236 by -138 of A. 3. My age is 1/075 of my brother's ; if I am 30, what is he ? 4. What number is that of which '45 is 25 ? 5. If | yards cost 1-235, what cost 3'7896 yards? 6. What is the area of a grass plot which is '296 yds. less than a pole ? 7. What decimal of 2} yds. is 1 ft. 7 in. ? 8. What part of a gallon is '08935 quarter? 9. Find the price of 28*6 st. butter at 16/9* for 1'75 st. 10. Find the area of a field '0876 miles by '0056 miles. 11. Find the weight of four packages, of which one is "276 ton, another -025 cwt,, a third 75'8 ft), and the fourth 1'96 qr. 12. From 3*285 of 16/5, take 1'3 of 17/6. 13. Reduce 3 da. 6 ho. 30 min. to decimals of a week and of a year. 14. How many imperial acres in a farm which measures 295*65 ac. Scotch, if the ac. Scotch be 1-261183 of the acre imp.? 15. Bought 3-75 cwts. for -0125 per R> : find the whole price. 16. Find the number which, taken twelve times, is '1728. 17. Divide the average of 3'079, 4-276, 5*60548 by '006. 18. If I walk 3*789 miles an hour, how far will my friend walk in 5 hours, if he goes 1*075 miles for my one ? 19. The French metre is 39*37079 inches: how many in 147*895 yards ? 20. What decimal is 2 galls. 3 qts. 1 pt. of 14'576 pks. ? 21. How much carpet 1*25 yds. wide will cover a floor 22'3x 19*45 ft., and having 2 oriel window spaces, each 4J feet by 2|? 22. If I pay 9jd. per as income-tax, what part is that of my whole income ? and what part of my income should I save by a re- duction of 2 Jd. per ? 23. A regiment of 560 men has on its sick-list '295 of the whole : how many men are fit for service ? 24. By what fraction of itself does '00125 ac. fall short of 7'86 sq. yards ? 25. What decimal fraction multiplied by f of 71 gives J of I of \ ? 26- In a town of 240756 inhabitants, it was found that *0475 of the whole could not read, and only '575 of those able to read could write : how many were there of each ? 27. From 3'265 of 17/6 take 1*3 of 2778. 1 2 DECIMAL FRACTIONS. 23. St. Peter's Cathedral is 437 ft. high : what fraction is that oi St. Paul's, which is 340, and of the cathedral of Strasbourg, which is 574? 29. What is the average length of the first four months of the year ? and by what fraction of a day does it differ from the average length of the second four ? 30. If I bought 2 c wt. 2 qrs. 16 ft> sugar at 3*0296 per cwt., and sold it at '035 per Ib, what did I gain or lose on the whole, and on each pound ? 31. How often is 2*375 contained in *6 guineas? 32. What fraction is 1 Ib troy of 1 Ib avoir., and vice versa ? 33. How much is | of -00295 of 8, 2s. 6d.? 34. From Hamburgh to Bremen is22J German miles, or 109 J English miles : what fraction is a German mile of an English one ? 35. If the cost price of a book is 2/3i, and the selling price 3/4, what fraction of the former represents the profit? 36. In a school of 100 boys, 80 girls, and 58 infants, there waa absent 12 boys, 9 girls, and 8 infants : what fraction of each was absent, and what fraction of the whole school? Also, how many per cent, of the school were absent ? 37. Find the value of 4r- x 2 of '9*5 + ^T^ x rb- *07o 84 4*07 38. The French litre is '220097 gallons: express the bushel in litres. 39. At 3 '875 per acre, what is the rent of a farm which is equal to one field 193*85 yards square ? 40. If the diameter of a circle is to the circumference as 1 to 3*1416, what is the difference in length between the rims of two wheels whose diameters are 5 and 5f feet respectively ? 41. And how much oftener will the former revolve in a journey of 3*65 miles than the latter? 42. What ratio does the fourth proportional of 3, 4'75, 5'095 bear to that of -6, '063, and -0005 ? 43. Divide twice the sum of 1*0006 and 1-0606 by 5 times their difference. 44. A. walks 2*5 feet each step, B. 2*785: when B. has gone a mile, what part of a mile has A. still to go? 45. What fraction is "6 of 2| ells of 1;07 of 3 yards ? 46. In multiplying any number by *36, what is the difference in the product (expressed as a vulgar fraction of the multiplicand) if we multiply by 2 and by 4 decimal places respectively ? 47. What cost 8 rounds of beef weighing in all 2*64 cwt., at 1/4 J for 2-25 Ib? 48. Divide 576*58 guineas among 3 men and 4 women, giving each man 1*75 of a woman's share. 49. To give 7 persons 1/8J each out of half a guinea, what frac- tion of a crown do I want ? 50. A cubic foot of water weighs 62 Ib 7 oz, 4 dr. : what is the INTEREST. 121 weight of water in a cistern 6'2 feet long, 4*5 broad, and 3*75 deep ? and how many quarts may be filled of it, if a quart weighs 2 '25 Jfo avoir. ? 51. Gold coined has ^th alloy: the weight of a sovereign being 02139 lb, what would be the value of a purse of 670 sovereigns were the gold pure ? 36 INTEREST. Simple Interest. If I borrow ,50 from the bank, I have to pay so much for the use of its money ; if I pay .50 into the bank, I receive so much from it for the use of my money. The sum which produces this profit is called the Principal ; what is paid for the use of the principal is called the Interest ; and the principal, with the interest added to it, is called the A mount. Interest is calculated at so much every year for every 100, or per cent. (%), as it is called. Thus 5 per cent. =5 for 100, or .1 for 20. 4 =4 ,100; 1 25. 3 =3 100, 1 33 J. 2J = 2^ 100, 1 40. 2 =2 100, 1 50. Ex. 1. Find the interest on 275 for a year at 4 per cent. This is a question in the Rule of Three ; thus, if 100 gain 4, what will 275 gain ? 100 : 275 : : 4 : Answer=ll. Ex. 2. Find the interest of the same sum at the same rate for three years. This will be three times the interest for one year ; or, stating it in double Rule of Three 100 : 275 : : 4 / A , QQ I , g > Answer =33. If the time given be less than a year, express it as a fraction of a year ; thus, had it been five months, the second statement would have been 1 : ^ ; if it had been 239 days, it would have been 1 : f^f . It is here supposed that the principal remains the same during the five years ; not accumulating by the addition of the interest each year. Interest given on this supposition is called Simple Interest. 122 INTEREST. Questions in Interest can always be solved by Rule of Three, or by the following rule derived therefrom : Rule. To find the simple interest of a,ny sum, multiply the principal by the rate per cent., the number of years or part of a year, and divide by 100. EXERCISE I. 1. What is 4 per cent, of 150, 250, 375, 20, 1000, 25/. 3 15s ? 2. 5 per cent, of 275, 60, 400, 3, 6, 10s., 36 ? 3. ,,2 per cent, of 10, 300, 2, 10s., 75, 875, 2000 ? 4. Find the interest of 256, 10s. for 1 year at 3 per cent, per ann 5. Of 4562, 17s. 6d. for 1 year at 4.J per cent, per ann. 6. Of 675, 19s. 4d. for 3 years at 5 per cent, per ann. 7. Of 89, 14s. 8Ad. for 8 years at 2 per cent, per ann. 8. Of 560, 15s. for 9 months at 3 per cent, per ann. 9. Of 849, 13s. 6d. for 15 weeks at 4 per cent, per ann. 10. Of 2000, from March 15 to November 18, at 5 per cent. 11. Of 1625, 9s. 8Jd. for 189 days at 5 per cent. 12. Find the amount of 125, 10s. for 8 years at 4 per cent 13. Of 97, 16s. 2d. for 8 months at 2i per cent. 14. Of 87, 15s. lOd. for 12 weeks at 5 per cent. 15. Of 216, 9s. 3d., from January 16, to May 30, at 4 per cent. J.O I Ex. 1. What principal, invested for 10 years at 4 per cent., will bring in a sum of .240 ? In the given time, and at the given rate, 100 of principal will bring interest ,40, so that the question is equivalent to this : If 100 give 40, what will give 240, same time and rate ? And the statement will be 40 : 240 :: 100 : Answer=600. Ex. 2. What sum will amount to 1250 in 8 years at 4 per cent, per annum ? In the given time, and at the given rate, 100 will amount to 132, so that the question is equivalent to this : If 100 amount to 132, what will amount to 1250, same time and rate ? And the statement will be 132 : 1250 :: 100 : Answer=946, 19s. 4jff. Ex. 3. At what rate per cent, will 900 amount to 1116 in 6 years ? In the given time 900 gains 216, and therefore in one year 36, so that the question is equivalent to this : If 900 gains 36 in one year, what will 100 gain ? And the state- ment will be 900 : 100 : : 36 : Answer=4 per cent INTEREST. 123 Ex. 4. In what time will 1500 amount to 1980 at 4 per cent. 1 At the given rate 1500 will gain 60 interest in one year, so that the question is equivalent to this : If 1500 gain 60 in one year, in what time will it gain 480, the sum required to make up the amount ? And the statement is 60 : 480 :: 1 year. Answer =8 years. 1 EXERCISE II. 1. What sum must be lent at 4 per cent, on April 1, to bring in for interest 3, 17s. 4d. on May 28 ? 2. What principal lent at 4^ per cent, for 7 mo. will yield interest 17, Is. 9d. ? 3. What capital sum at 5 per cent, will bring a yearly income of 250? 4. What sum must be lent at 6 per cent, for 1 year 2 months to amount to 56, 3s. 6d. ? 5. What principal will amount to 1109, 11s. 3d. in 3 years at 2, 15s. per cent. ? 6. What sum will amount to 3376, 8s. 11 Jd. in 9 years 7 months at 4, 11s. 8d. per cent, per annum? 7. If 350 gain 60 simple interest in 4 years, what has been the rate per cent, gain ? 8. A capital of 780 brings a return of 126, 15s. in 5 years, simple interest, find the rate per cent. gain. 9. In what time will 25 become 27, 3s. 9d. at 5 per cent, simple interest ? 10. How long must 7200 be lent at 4 per cent, simple interest to amount to 9760 ? 11. If I lodge 180 in bank, how long must I let it lie at 3 per cent. to gain 24 ? 12. What rate of interest must I receive on a sum of money which I wish to double itself in 12 years ? 1 If I denote interest, P principal, r, rate, and t t time, the rule for find- ing Simple Interest may be expressed thus : r Pxrxt 100 From that formula, other formula may be devised for working each of the four cases now given, thus : / X 100 1. Where Principal is sought, its interest being given, P= 2. Where Principal is sought, its amount being given, P= 3. Where rate is sought, 100+rx* JxlOQ Pxt 4. Where time is sought, t= p 124 INTEREST. 138 Compound Interest. Ex. If, however, I lodged 275 in the bank for 3 years at 4 per cent., I should get more than the simple interest as cal- culated, sect. 136. For at the end of the first year the interest would be added to the principal, and the amount would be the principal for the second year. At the end of the second year the interest on the amount would again be added, and the new amount would be the principal for the third year, and so on. Interest calculated in this way is called compound interest. To find it, we have to work three questions in simple interest. (1.) Int. of 275 for 1st year at 4 p. c. = 11. (2.) Int. of 286 for 2d year at 4 p. c. = 1 1, 8s. 9|d. (3). Int. of 297, 8s. 9|d. for 3d year at 4 p. c.=ll, 17s. llrf^d. And the amount I should then draw from bank would be 309, Gs. 9y|^d. The simple interest of the sum was 33, showing a difference of 1, 6s. 9yf Tfd. And if the interest were paid half-yearly, as it is sometimes, I would receive still more ; for at the end of the first half-year, the interest would be added to the principal, which w r ould make the interest for the first year greater than before, and consequently the amount for the second year, and so on. Rule. To find compound interest, find the amount for the first year, and make that the principal for the next ; then find the amount for the second, and make that the principal for the third ; and so on for the number of years. EXERCISE III. 1. Find the compound interest of 65, 14s. for 3 years at 4 per cent. 2. Of 378, 10s. 7d. for 2 years at 5 per cent. 3. Of 100 for 1 year at 4 per cent, payable half-yearly. 4. How much more would it be if interest be paid quarterly. 5. Find the amount of 750, 10s. for 1 year 9 months at 2^ per cent, compound interest. 6. Find the amount of 1250 for 2 years 100 days at 4 per cent, compound interest. 139 The computation of compound interest may be effected more simply thus : The interest of 1 for 1 year at 4 per cent, being '04, the amount will be 1*04. Then as 1 is to its amount for a year, so is any other prin- cipal to its amount for a year ; hence DISCOUNT. 125 1 : 1'04 : : 1'04 : 1'04 2 amount for 2 years. 1 : ^1'04 2 : : i'04 : r04 3 3 years. etc. etc. Then as 1 : 275 : : l'Q^ : ^l-04 ? x275, the amount of .275 for 4 years at 4% = 309, 6s. 9^(1. Interest for the time= 309, 6, 9 T f ir -275=34, 6, 9y T . Rule 1. To find amount at compound interest, raise the amount of l for one year to the power denoted by the number of years, and multiply by the principal ; to find the interest, subtract the principal from the amount. From that follows immediately Rule 2. To find the principal that will yield a given amount at compound interest, divide the amount by the amount of 1 for the number of years. Thus - i-04 a T7 *^ = -^75, principal giving the amount. Note. If the interest is paid half-yearly or quarterly, calcu- late as above, with half-yearly or quarterly terms respectively. EXERCISE IV. 1-6. Perform by this method the questions in Ex. iii. 7. What sum will amount to 1019, 10s. 2d. in 6 yr. at 5^ per cent. ? 8. What sum will amount to 1351, 7s. 10 ^ in 3 yrs. at 4 per cent, interest, payable half-yearly ? 9. I invest 5000 for 3 years at 5 per cent. : how much more should I receive if the interest be paid half-yearly than if paid annually? .40 DISCOUNT. Merchants' accounts are not generally paid ready money, but either after a certain time (six or twelve months), or by bill, in which the buyer promises to pay at a date agreed upon. The merchant may take this bill to a banker, who will if the buyer's credit be good give ready money for it, charging a certain percentage for paying the money before it is due. This charge is the banker's discount, and he is said to discount the bill The ready money received for the bill is called its present value. Ex. What is the present value of 400, payable 9 months hence, interest being 4 per cent. ? 126 DISCOUNT. 100, at 4 per cent., would amount in 9 months to ,103 ; in other words, ,100 is the present value of 103, payable 9 months hence, interest 4 per ceut. Therefore, as 103 is to 4'K), so is the present value of 103 to that of 400 : 103 : 400 : : 100 : Answer = 3887 nearly. Rule. As the amount of 100 for the time is to the sum, so is 100 to the present value required. To find the discount, subtract the present value from the sum : 400 -388, 7s. =11, 13s. ; or, making a statement in Proportion, we have the following : 103 : 400 : : 3 : Answer = 11, 13s. Rule. As the amount of 100 for the time is to the sum, BO is the discount of the former to that of the latter. In practice, however, discount is never reckoned in this way, but exactly as interest ; thus the discount on 400, pay- able 9 months hence, at 4 per cent., would be the interest on that sum for the given time. The common discount is therefore somewhat greater than the true discount. In discounting bills, bankers calculate interest for the num- ber of days the bill has still to run ; but three days are allowed, called days of grace y after a bill is nominally due, before it be- comes legally due. EXERCISE. 1. Find the common and the true discount on 450, due 8 months hence, at 4 per cent. 2. On a bill of 536, 10s., drawn and discounted March 14, payable Sept. 10, at 3 per cent. 3. On a bill of 279, 15s. 3d., drawn March 1 at 9 months, discounted April 20, at 3 per cent. 4. On a bill of 1000, dated May 15, for 8 months, and discounted August 24, at 4 per cent. 5. Bought sugar at 2, 18s. per cwt. and 6 months' credit : what should be allowed for present payment per cwt. at 3^ per cent. ? 6. I engage to pay 750, 14s. lOd. 10 months hence : find its present worth at 5 per cent. 7. What sum at 10 per cent, will amount to 325 in 2 years ? 8. A bill of 472, 16s., drawn May 8 at 3 months, is discounted June 19 at 6 per cent.# what is its present worth ? 9. Find the present worth of 875, 5s. 8d., drawn Feb. 25 at 7 months, discounted June 4, at 5 per cent. STOCKS. 127 L41 STOCKS. The sum required for constructing a railway or any other public work is called the capital. This is raised by means of a Joint-Stock Company, as it is called, in so many shares, it may be of 5, 10, or .50, according to the amount of the capital. The value of the share, as fixed at the outset of the scheme, is called par. If the scheme turn out a good one, many will want shares in it for the sake of the high interest yielded ; so that those who hold shares may sell them, if they choose, at a higher price than they paid for them. A ,50 share, e.g., may sell for 60 ; in which case the value is said to be 10 above par. On the other hand, if the scheme turn out a bad one, nobody will care to buy the shares ; and those who hold them, if they sell out, will have to do so at a loss. A share ,50 may have to be sold for 40, that is, 10 below par. Joint-Stock Companies are formed in the same way for other purposes, e.g., banks, mining operations, and the like. And the shares or stock of these companies are regularly sold in the market, at a higher or lower price according to the opinion entertained at the time of their value. There is a different kind of stock, called Government Stock, or the Funds. The Government has sometimes to borrow large sums of money for public purposes. As it cannot expect to get the loan from one person alone, it divides the amount into 100 shares ; and to each person who gives it 100, it binds itself to give a certain rate of interest, say 3 per cent. The rate of interest is not high, but that is compensated by the certainty of its payment. The person who lends 100 really buys a 100 share ; he cannot claim his money again, but he has got the right to 3 annual interest instead. He can sell his share, how- ever, in this 3 per Cent. Stock, or the 3 per Cents, as it is called. The selling price depends on circumstances, and is constantly varying. Thus, if public affairs are in such a state of peace and prosperity as to offer other safe means of investment and a higher rate of interest, the price will be lowered ; if otherwise, the price rises. Ex. 1. What should be paid for 1790 in the 3 per cents. at94|? i.e., if 100 bring 94|, what will 1790 bring? 100 : 1790 : : 94| : Answer =1689, 6s. 3d, 128 BROKERAGE. Ex. 2. How much stock at 94 may be purchased ftr .2573 brokerage Jth per cent ? i.e., if 94J buy a 100 share, what will 2570 buy ? 94J : 2570 : : 100 : Answer=2730, 8s. 2J Ex. 3. What interest will ;>9 got by buying 4 per cents. t 87^ ? i.e., if 87 bring interest 4, what will 100 bring ? 87i : 100 : : 4 : Answei = 4f5 per cent, or 4'584. EXERCISE. 1. What should be paid for 576. 10s. in the 3 per cents, at 941 and at 89$ ? 2. Find the buying price of 295 stock in the 3 per cents, at 77g and at 84, brokerage J per cent. ? 3. What cost 475 in 4 per cents, at 86$, and at 91 ? 4. How much stock at 98$, and at 96, may be bought for 1280, brokerage per cent. ? 5. What amount of stock at 83$ and at 89$ may be purchased for 2000, brokerage per cent. ? 6. If I invest 600 in stock at 107$ and at 111$ (brokerage & per cent.), what amount will I hold ? 7. What interest will be got by buying India stock 9A per cent, at 179.1, and at 168$ ? 8f Do., by buying 3 per cents, at 934, and at 96$ ? 9. Do., stock giving 6 per cent, at 124$, and at 136? 10. Bought 3 per cents, for 1000 at 94$ and sold at 96$, what did I gain ? 11. What annual income is got by investing 2500 in 3 per cents. 90$, and at 101 J ? 12. How much must be invested in the 3$ per cents, at 96$ and at 99 to yield an annual income of 350, brokerage per cent. ? 142 BROKERAGE. The selling of stock, both Government stock and that of private companies, is a regular business carried on by Brokers or Stockbrokers ; their market is called the Exchange or Change ; and their charge for transacting business Brokerage. This is generally Jth per cent., that is 2s. 6d. for every 100 value bought or sold. Similarly, buying and selling of goods of all kinds is very largely carried on upon commission. Thus a merchant may import provisions from a firm or company of merchants abroad, and sell them on commission ; or a manufacturer may have INSURANCE. 129 agents through the country selling his goods on commission. The rate of commission is much higher than that of brohrage ; varying from 2j per cent, to as high as 10 per cent. EXERCISE. 1. Find the commission on 1495, 10s. at 3] per cent. 2. On 295, 17s. 6d. at 1 per cent. 3. On 793, 18s. 9d. at per cent. 4. At $ per cent, what is the brokerage on insuring goods to the value of 1560, 13s. 4d. ? 5. Sold by auction goods which fetched 375, 10s. 6d. ; what sum would be paid to the owner by the auctioneer after deducting his com- mission of 5 per cent. ? 6. What is the cost of collecting debts to the amount of 794, 16s. allowing 25/ per cent, to the collector ? INSURANCE. Insurance is the name applied to the percentage paid on property to secure it against damage. Thus there is Fire In- surance, to secure it against risk from fire ; Hail and Storm Insurance, to secure the farmer's crops against damage from storms ; Insurance against sea-risk, to secure vessels and their cargoes against shipwreck ; and Life Insurance, to secure a man's family or friends against risk of poverty at his death. The percentage is paid annually in all these cases, except sea-risk, and is called the Premium of Insurance. On this a Government duty is charged of so much per cent. ; all sums intermediate between hundreds being charged as the higher number of hundreds. The deed of agreement between the person who insures and the Insurance Company is called the Policy of Insurance. The premium for fire insurance is Jth per cent., or 2/6 for .100 ; but higher rates are charged on property exposed to more than average risk. The premium of life insurance increases with the age of the person insured, its amount being fixed by calculations based on the average duration of life. EXERCISE. 1. What must be paid for insurance of 1495 at 2f per cent. ? 2. For insurance of 790 at 4 per cent. ? 3. For insurance of 2485, 17s. 6d. at 2/6 per cent. ? 4. What is the expense of insuring a ship worth 3000 at 3 guineas per cent., duty 3/6 per cent. ? 130 PROFIT AND LOSS. 5. A person aged 32 insures his life for 800, what is the expense at 3, 13s. 6d. per cent. ? 6. What sum will insure a cargo valued at 1790 guineas at 7 guineas per cent. ; duty, 3/ per cent. ; commission per cent. ? Ex. What sum must be insured to cover 470 at 2^ guineas pel cent. ; duty, 2/6 per cent. ; commission, A per cent. ? The expenses of insurance are 2, 12s. 6d. +2/6 + 10/ per 100; BO that every 100 insured produces 100- 3, 5s. ; that is 96, 15s. Then 96, 15s. : 470 : : 100 : 485, 10s. 7 T 6 w- 7. What must be insured to cover 2570 at 4 guineas per cent. ; duty, 3/6 per cent. ; brokerage, per cent. ? 8. To cover 4875, 10s. at 5^> guineas per cent. ; duty, \ per cent. ; commission, | per cent. ; extra charges 1 per cent. ? 9. What sum insured will cover total loss of house property worth 2765, 10s. at 3$ guineas per cent. ; duty, 3/6 per cent. ; commission, ^ per cent. ? 144 PROFIT AND LOSS. Ex. 1. Bought tea at 3/6 per lb, and sold it at 4/ per ft) : what is the gain per cent. ? As the cost price is to ,100, so is the gain upon the cost price to the gain upon ,100 ; i.e., 3/6 : 100 : : 6d. : to the gain per cent. = 14, 5s. 8fL Ex. 2. Bought tea at 3/6 per ft), what must I sell it at to gain 10 per cent. ? As 100 is to the cost price, so is the selling price of 100 to the selling price required ; i.e., 100 : 3/6 : : 110 : selling price, or 3/10 J. EXERCISE. 1. Bought butter at 3, 15s. per cwt., which I sold at 4, 4s. 6d. : what was the gain per cent. ? 2. Bought cloth at 4/9 per yard, and sold it at 5/2 : find the gain per cent. 3. Sold sugar at 4^d. per lb, for which I paid 1, 17s. 4d. per cwt. : what was the gain per cent. ? 4. Bought coffee at 6, 15s. 6d. per cwt., what should I have sold it for to gain 1\ per cent. ? 5. Bought linen at 3/6 per yard, and lost 4 per cent, on it : find the selling price. 6. How must wine, which cost 21/6 per gall., be sold to gain 4.^ per cent. ? 7. Bought 74 yards carpet at 2/10 per yard, and sold the whole for 27, Us. : find the gain or loss per cent. 8. If 10 yards are found to be damaged, at what per cent, profit must the remainder be sold so as not to lose upon the whole ? SQUARE HOOT. 1 3 1 9. Sold oranges at 1/1 per dozen, with a gain of 3 per cent. : what did they cost ? 10. Find the cost price of a book on which a bookseller gained 9 per cent, by selling it at 25/6. 11. Bought paper at 1, 10s. 6d. per ream, which I sold at 1/9 per quire : what was gained on 15 reams, and what was the gain per cent. ? 12. If 2^ per cent, be lost by selling bacon at 6d. per tb, what did it cost per cwt. ? AC SQUARE BOOT. The square of any number is that number multiplied by itself ; thus 25, being equal to 5 X 5, is the square of 5, sect. 38. The square root of any number is that number which, when multiplied by itself, has the given number for product ; thus 5 is the square root of 25. As the square of 5 denoted by 5 2 , so the square root of 25 is denoted by 25* or v'25. The square root of any number, if it be any of the first 12 numbers, is known from the multiplication table ; thus since 7 2 =49, ^49=7. Ex. Find the square root of 576 ? When the square root of a number exceeds 12, the method of finding it depends on the principle that " the square of any number is equal to the square of its two parts, together with twice their product ;" thus, 24 2 =20 2 -f 2X20X4+4 2 which is the same as 20 2 +(2x20+4)x4. To find the square root of 576, therefore, we separate it into two parts, the one containing the hundreds, 500, the other the units, 76. The nearest square number to 500 is 400, 2 5,76(24 which gives 20 for the first part of the root. 2 4 Subtract the 400 and 176 remains. By the 44) 176 above formula, if this be divided by twice the 176 first part of the root increased by the second part, the quotient will be the second part. But the second or units part of the root is so much smaller than the first, that twice the first part of the root may be taken as the approxi- mate divisor. Double the 2 tens, giving 4 tens or forty ; this is in 176, 4 times ; add this 4 to the 4 tens for complete divisor, making 44 ; 4 times 44 is 176. Complete root 24 ; as may be proved by multiplying it by itself. If the root consist of more than two places, divide the given number into periods of two figures, beginning with the units* place ; annex one period to each remainder for the successive 132 MENSURATION. dividends ; and for each trial divisor add the part of the root last found to the divisor preceding it. Thus the next trial divisor in the above example would have been 48. The square root of decimal fractions is found precisely as that of whole numbers ; only the periods of two figures are reckoned from the point. And if the square root is wanted of a number which has no even square root, it may be found ap- proximately by adding ciphers, and finding the root as of a whole number and a decimal fraction. The square root of a vulgar fraction is found by extracting the root of its numerator and of its denominator ; if these have no even root, it should first be reduced to a decimal fraction. EXERCISE. Find the square root of 1. 289. 9. 5184. 17. 6-25. 25. -06. 2. 361. 3. 484. 10. $f 11. 20736. 18. -0121. 19. -001156. 26. 6J. 27. 6|. 4. 676. 12. 55225. 20. 65-1249. 28. 2. 5. ttf. 13. 126736. 21. 2067844. 29. . 6. 1849. 14. 316969. 22. -00053361. 30. 6-5536. 7. 3136. 15- IWfti- 23. -030976. 31. 655-36. 8. 3969. 16. fftft. 24. 6. 32. 65-536. MENSURATION. Ex. 1. What is the area of an oblong plot 4 feet 9 in. long by 3 feet 6 in. broad ? The area being found by multiplying the length by the breadth (sect. 82), we multiply 4 feet 9 in. by 3 feet 6 in. If feet multiplied by feet give square feet, feet multiplied by inches will give twelfths of feet, and inches multiplied by inches will give one-hundred-and-forty-fourths of feet, or square inches. Multiplying first by 3 feet : 4 9 3 times 9 are 27 ; 3 and carry 2. _3 6 3 times 4 are 12, and 2 are 14. 14 3 Then multiplying by the 6 inches : 246 6 times 9 are 54 ; 6 and carry 4. - ^7-77 6 times 4 are 24, and 4 are 28 ; 4 and carry 2. ' Adding the partial products, we have 16 7' 6", i.e., 16 sq. ft., 7 twelfths of a square foot, called primes, and 6 one-hundred- and-forty-fourths of a sq. foot, or sq. inches, or seconds, as they are called, to distinguish them from the primes. Reducing the primes to sq. inches, the answer may also be expressed as 16 sq. feet 90 sq. in. MENSURATION. 133 The next lower names to primes and seconds are thirds and fourths, denoted by '" and . Because of the carrying by tivelves, this multiplication for areas is often called duodecimal. Rule. To find area, multiply the greater dimension by the highest name of the less, carrying by twelves ; then by the lower names in their order, setting the first result :n each pro- duct one place to the right : add the several products. To find solid content, multiply the product of the two greater dimensions by the least. EXERCISE I. V* Express the answers (1.) in sq. feet, primes, seconds, etc. ; (2.) in sq. feet and sq. inches. ft. in. ft. in. ft. in. li. ft. in. li. 1. 36 4x7 6 7. 745x276 2. 9 8x6 10 8. 936x187 3. 87x52 9. 18 62x740 4. 11 10 x 9 1 10. 25 10 9 x 6 3 5. 24 6 x 18 5 11. 16 80x789 6. 29 3 x 10 11 12. 14 61x985 13. How many sq. feet in a floor 26 feet 8 in. x 17 feet 6 in. ? 14. rina rhe area of a field 136 feet 8 in. x 78 feet 10 in. 15. There are two windows in ray room, each containing 12 panes, each pane 1 foot 7 in. x 10 in. Find the price of the whole at 1/10| per sq. foot. 16. What is the expense of carpeting a floor 21 feet 6 in. x 18 feet 9 in., at 3/7 per sq. yard ? 17. What will it cost to paper a room 11 feet 6 in. high and 86 feet 8 in. in circuit, at 8d. per sq. yard ? 18. A square plot of grass measures 18 feet 8 in. long, but in its centre is a square flower-bed 4 feet 6 in. long : what is the extent of the grass surface ? 19. What is the length of a paved court containing 186 sq. feet 10 sq. in., which is 9 feet 6 in. broad ? 20. The carpet of a room at 5/ per sq. yard cost 10 guineas. If the room was 17 feet 6 in. broad, find its length. 21. Find the content of a wooden cube 4 feet 8 in. of edge. Express this and the following answers in cubic feet and inches. 22. A block of granite measured 10 feet 8 in. long by 7 feet 4 in. broad, and 5 feet 6 in. thick : what was its content ? 23. How many cubic feet of air in a room 60 feet long by 25 feet 6 in. broad, and 14 feet 8 in. high ? 24. Find the depth of a cistern 4 feet 6 in. square, to <. ontain 84 cubic feet of water. 25. Find the expense of painting a chest (exclusive of the bottom) 6 feet 3 in. long by 4 feet 4 in. broad, and 4 feet 2 in. deep, at 1/2^ per sq. yard. 26. A school-room is 45 feet 8 in. long by 27 feet 6 in. broad : what 134 MENSURATION. must be its height to accommodate 220 pupils, allowing 80 cubical feet for each pupil ? 27. How many bricks 8i in. long, 3 in. broad, and 2 thick, would be required to fill a cubical space measuring 18 feet 8 in. long ? 28. What will it cost to remove a mass of earth 36 feet 4 in. long by 25 feet broad, and 14 feet deep, at 7d. per cubic yard ? 29. Find the weight of water in a tank 25 feet long by 16 feet 8 in. broad, and 12 feet 6 in. deep, if a cubic in. of water weighs 252*458 gr. 30. How many galls, water in a cistern 7 feet 6 in. long, 5 feet 4 in. broad, and 6 feet deep, one imp. gall, being equal to 277*274 cub. in. ? 14cO Ex. 2. What is the area of a rectangular field, whose length is 7 chains 40 links, and breadth 8 ch. 30 Ik. ? Reducing the dimensions to links, the area is 740X830 Ik. = 614200 sq. Ik. =6 ac. ro. 227 po. Note 1. If the field is not rectangular, but has its opposite sides parallel, multiply the length by the perpendicular breadth. Note 2. If only two of its opposite sides are parallel, multi- ply half the sum of these by the perpendicular breadth. Note 3. If the field is triangular, multiply the base of it by half the perpendicular height. Ex. 3. Close by a wall 8 ft. high flows a stream 6 ft. broad : what length of plank will just reach from the edge of the stream to the top of the wall ? In any right-angled triangle, the square of the * \ side opposite the right angle (A B) is equal to the sum of the squares of the other two sides (B o and AC). If then A B 2 =A c2+c B 2 , A B= VA cH-c B 2 ; i.e., the length of the plank =V8 2 +6 2 =VlOO= 10 ft. Note. In such a triangle, one of the shorter sides will be equal to the square root of the difference between the squares of the other two. Ex. 4. What is the circumference of a circle whose diameter is 3 feet? The circumference is the line containing the circle ; the diameter, any line drawn through the centre, and bounded by the circumference. Any line drawn from the centre to the circumference is called the radius, and is half the diameter. The circumference is 3'1416 times the diameter. If, there- MENSURATION. 135 fore, -the diameter is 3j feet, the circumference is 3^X3*1416 = 10-995 feet. Ex. 5. Kequired the area of the same circle 1 The area of any circle =3*1416 times square of the radius. The area required is therefore l 2 X31416 = 9*62115 sq. ft. Note. If the area were given, the radius would be found by dividing the area by 3*1416, and taking the square root of the quotient. Ex. 6. The trunk of a tree which was cut down measured 4 feet across one of its ends, and 12 feet in length : what was its content ? A body of this shape is called a cylinder. Content of a cylinder=area of base X length. Therefore area required is 2 2 X 31416X12 = 1507968 ft. Ex. 7. A glass globe is 14 inches in diameter : what is the area of its surface ? Area of surface=circumferenceXdiameter=3'1416Xi> 2 . Required surface is therefore 14 2 X3*1416=4 sq. ft. 387 in. Note. If the area of surface were given, the diameter would be found by dividing the area by 3*1416, and extracting the square root of the quotient. Ex. 8. What is the content of the same globe ? The content of a globe ='5236 times the cube of the dia- meter. The content required is therefore 14 3 X '5236 = 14367584 cubic inches. L49 EXERCISE II. 1. What is the area of a rectangular field 22 ch. 45 Ik. by 17 ch. 29 Ik. ? 2. A room is 12 ft. long by 9 broad : what length of line will stretch between its two opposite corners ? 3. A circular plot measures 27 '65 feet across its widest part : what length of netting would enclose it ? 4. Find the area of a rhomb-shaped field, 28 '6 poles long and 25 poles in perpendicular breadth. 5. The spoke of a cart-wheel measures 3^ feet from centre to riia : how many revolutions will the wheel make in a mile ? 6. A line 75 ft. long, attached to the top of a flag-staff 60 ft. high, was fastened in the ground : at what distance was it from the bottom of the staff? 7. Find the breadth of a rectangular field of 16 ac. 2 ro. 21 po- whose length is 26 ch. 61 Ik. 136 MENSURATION. 8. A circular field mecisures 729 sq. yds. in area : what is the length of the side of a square field of the same area ? 9. Find the area of a circular plot, diameter 20 yds. 10. Find the area of a quadrilateral plot, having 2 parallel sides of 15 and 21 feet respectively, and 12 feet in perpendicular breadth. 11. An oblong field, 18 ch. 75 Ik. long by 25 ch. 30 Ik. broad, is under potatoes and turnips, and the potatoes extend over '375 of its length : find the area under both. 12. What is the radius of a circular pond 7854 sq. ft. in surface ? 13. If a square field is laid off on a base of 12 ch. 40 Ik., what is its area ? and if a circle is traced within it on a diameter of the same length, what area is lost in the corners ? 14. A diamond-shaped grass plot was found to contain 272 sq. ft. ; if its length was 17 ft., what length of iron fence would divide it into two parts broadways ? 15. How many cubic feet of air in a balloon 2 ft. 7 in. in diameter? 16. What is the length of a moor containing 2 sq. miles in area, and 17 ch. 44 Ik. broad ? 17. How long a tether will give a cow an acre of pasture ? 18. Two fields, one oblong, the other square, are measured and found to contain the same area. The length of the one is 32 po., and its breadth 18 po. : what is the length of a side in the latter? 19. A pillar, 14 ft. high, was 2 feet across its end : find its solid content. 20. Find the surface of a circular table-cover, the table measuring 5 ft. 3 in. across by its centre, and the cover hanging 4 inches over the edge. 21. A four-sided field contained 3 ac. 1 ro. 38 po. in area, and its two parallel sides were respectively 25 and 37 poles : find its breadth. 22. A carriage-wheel revolves 900 times in a distance of 2 '5 miles, find the length of a spoke. 23. What cost the gilding of a ball for the vane of a church spire, 3 ft. 6 in. in diameter, at 4^d. per sq. inch ? 24. How many feet of plank would cover a well whose mouth mea- sured 4*75 feet across its centre, leaving a round hole one foot across for air ? 25. A cheese measured 2 ft. 8 in. across one of its ends ; if it was 9 in. in depth, what was its content ? and its weight, allowing 10.^ cub. in. per lb ? 26. What superficial content of paper would be required to cover a pair of 21-inch globes ? 27. An oblong garden, 3 ch. 25 Ik. broad by 3 ch. 60 Ik. long, is surrounded by a walk 10 ft. 6 in. broad : find the expense of paving the walk at 2/7 per sq. yd. 28. A point at the end of one of the sails of a wind-mill is distant from the centre 27 ft. 9 in. : through what distance will it travel in an hour, at the rate of 2| revolutions a minute ? 29. A spherical metal boiler was 10 ft. 3 in. in radius : how many galls, water will it contain, if one gall. =277 '274 cub. in. ? 30. And if the watt r it contains just fills a circular pond 2 ft. 6 in. in depth, what is the diameter of the pond ? 150 MISCELLANEOUS EXERCISE. 137 MISCELLANEOUS EXERCISE. I. 1. Express in words (1) the sum of one hundred and seven millions five hundred and eighty-four thousand and twenty ; one hundred and ten thousand five hundred and two ; thirty-seven thousand and five : (2) the difference between nine hundred and sixteen thousand and nine, and fifty-six millions and three. 2. A man spends 155, 5s. 7d. per year : how much will he lay by in 37 years out of 200 per annum ? 3. Find the value of (1.) 7 tons 14 cwt. 2 qr. 25 Ib hay, at 3, 10s. 6d. per ton ; and (2.) 2 tons 7 cwt. 1 qr. 15 Ib, at 1, 3s. 4^d. per cwt. 4. If 24 oxen require 6 acres turnips to supply them for 10 weeks, how many acres would supply 6 score sheep for 15 weeks, if 3 oxen eat as much as 10 sheep ? 5. Divide 35 by -0175, and -0175 by 35. 6. A bill of 760 is due 7 months hence : find its present value at 5 per cent, per annum. 7. Find the interest on 189, 16s. 6d. for 341 days at 34 %. 8. If I gain 16 per cent, by selling 98 yards of cloth for 23, 13s. 8d., what was the buying price per yard ? 9. Find the tare on 84 hhd. sugar at 30 K> per hhd. 10. Find the square root of 10-624 to 3 dec. places. 11. How much carpet will cover a room 12 ft. 6 in. long by 14 ft. 9 in. broad ; and what will be the cost at 5/6 per square yard ? O 'Jo 12. Find the sum, difference, product, and quotient of -4- and 1 . 13. How many yds. cloth would be needed for the clothing of 10000 soldiers, if each coat took If yds., a pair of trousers 1 yds., and a waistcoat f yds. ? 14. If the thirteenth part of 5 yds. 2 qr. 3 na. cloth be divided by \, what results ? 15. How many parcels of sugar of 2 K>, 1 ft>, lb, | Ib, can be made out of a cask containing 8 cwt. 2 qr., the number of each being the same? 16. Exchanged 40 yds. muslin, worth 2/6 per yd., for 30 yds. linen : what was the linen valued at per yd. ? 17. If 1 K> weight standard gold were worth 46, 14s. 6d., how much should one sovereign weigh ? 18. What is the rent of 1200 ac. 3 ro. at 1, 8s. 6d. per acre ? 19. If 2 cwt. 3 qr. 21 Ib sugar cost 12, 3s. 4d., what is the value of 17 cwt. 2 qr. 14 Ib ? 20. Reduce $ of 16/4| to the decimal of 1, 9s. 10|d. 21. Find the amount of 5433, 13s. ll^d. for 5 years 5 months at 2J per cent. 22. A circular tank, eight feet in depth, contains 10000 galls. : find its diameter. i n i OK K o r>i 23. 24. What sum will purchase 820 stock in the five per cents, at 108? 25. Divide 2850 between A, B, and 0, giving ^ of B.'s share to A, and to c 300 more than to A and B together. 138 MISCELLANEOUS EXERCISE. II. 1. Multiply f of -175 by -285714, and divide the result by -00425 2. If 7| yds. cost 7, 18s. 4d., what cost 49A yds. ? 3. Make out the bill for 40 chests of cloves at 2/1 each ; 35 bags coffee at 2, Os. 6d. per bag ; 71 bags saltpetre at 1, 5s. 6d., per bag : and 5 casks sugar at 2, 6s. 6d. per cask. 4. How many yards of stuff 3 qrs. wide will line a cloak 5 A yds. in length, and 1 J yds. wide ? 5. What rate of income-tax will yield 38. Is. lid. on an income of 570, 16s. 6Jd.f 6. Find the value in inches and fractions of an inch of '0003551 13G of a mile. 7. In a school of 360 children, 10 pay 6d. a week, 80 pay 4d., 104 pay 3d., 75 pay 2d., 91 pay ld. : what is the average sum paid by each child per week ? 8. How much per cent is 7/6 of 4 guineas ? 9. Find the square root of 7 to 4 decimal places, and multiply the result by^j^.. 10. In what time will the interest of 325 at 31 % per annum pay a debt of 67, 12s. ? 11. What length must be taken from a rectangular field 66 yds. broad to cut off from it two acres ? 12. How long would it take to count a million of sovereigns at the rate of 80 a minute, for 12 hours each day ? 13. I exchange 4375 yards for pieces of 3 qr. 2 nl. : how many should I receive ? 14. Find the value of (1.) 3068 articles at 1. 15s. 7fd. ; (2.) 217 at 5, 19s. 6Ad. 15. If 120 bushels oats serve 14 horses for 56 days, how many days will 90 bushels serve 6 horses ? 16. Find the sum of the greatest and the least of the fractions ^-, H> M> 5 the sum of the other two ; and the difference of these sums. 17. A bill of 894 is drawn February 16, 1860, at 7 months' date, what will be the immediate discount at 5 per cent. ? and what the discount on 1st June same year ? 18. \Vhat sum will amount to 162, 8s. in 5 years at 4 per cent, simple interest ? 19. Find the sum of -09 of 1, 3s. 2d., and -51 of 19 : and what part of 5 that sum is. 20. If a loaf weighing 48 oz. cost 8p. when wheat is at 60/ per qr., what should be the price of wheat when a 6d. loaf weighs 38 oz. 8 dr. ? 21. Find the interest on 215, 12s. for 3 years 73 days at 4 % per annum. 22. A invests 1000 in the 3 per cents, at 84 ; B the same sum in the 4 per cents, at 110 : find their respective incomes. 23. A circular plate of gold, 3 inches in diameter and ^-inch thick, is extended by hammering so as to cover 5 square yards : find its present thickness. 24. Bought oranges at 20 for a shilling, and sold them at Id. a piece : what was gained per cent. ? 25. Find to 4 decimal places the square of the sum of the square roots of -25, -025, 3'6, and 14 -4. MISCELLANEOUS EXE11CISE. 139 III. 1. Find the value of (2346784 x 53-4-583) x (107298 4-18 x 79). 2. A person's quarterly income is 135, 10s. and his daily expendi- ture 2, 5s. : how much will be his debt for the two years and a half ending June 30th ? 3. Find the least number that will contain 225, 255, ^89, 1023, and 4095, without remainder. 4. If 3 men or 4 women can do a piece of work in 56 days, in what time will one man and one woman (working together) do it ? 5. What must be added to '356 of 2, 17s. 6d. to make up J| of 8, 9s. 7^d. ? 6. If you have 1000 money in the three per cents, at 83, and ex- change it into the security of shares at 233 each, on which a dividend is paid annually of 7, 13s. 4d. : what difference will it make on your income ? 7. Required the amount of 400 in 3 years 35 days at 3f /o P er annum. 8. Five tubes have an internal diameter of 1 inch, 1-2 inches, 1'4 inches, 1/6 inches, 1*8 inches respectively: how high will a pint of water stand in each, a pint containing 35 cub. in. ? 9. When will an investment of 1000 gs. at 6 per cent, double itself ? 10. Find the discount (true and common) of 132 payable at the end of 3 months at 3 % 11. What is the price of the 3| per cents., when 3930 invested in them produces 130 per annum ? 12. The pavement of a street is 15 ft. broad ; and from a point in the street 9 ft. beyond the pavement, a ladder 40 ft. long just reaches to the top of a house : what is the height of the house ? 13. If 25, 11s. 3|d. pay the carriage of 15 tons 16 cwt. 14 K> for 240 miles : what weight should be carried 180 miles for the same sum ? 14. What will 19, 13s. 9d. a day amount to in a solar year of 365 days 5 ho. 48 min. ? 15. What number is that from which if you deduct - , and to the remainder add the quotient of J 7 by 5J, the sum will be Hf + 10^ r ? 16. What is the value of the recurring decimal 3*4545 ? 17. A piece of cloth when measured with a yard measure two-thirds of an inch too short, appears to be 10 yds. long : what is its true length? 18. How many pieces of gold leaf, 4 inches square, must be bought to cover one face of a diamond-shaped kite, 2 feet broad and 4 feet long (diagonally) ? 19. A quadrilateral field has two parallel sides ; one is 67 chains, the other 5*8 chains, and the perpendicular distance between them 7'4 chains : find the acreage of the field. 20. To what other pairs of numbers is the mean proportional be- tween 6 and 24 also a mean proportional ? 21. The diameter of a circular enclosure is 370 yards : what will 9 wall going round it cost at 9/6 per yard ? 22. A field of grass is rented by two persons for 27. The one keeps in it 15 oxen for 10 days, the other 21 oxen for 17 days : find the rent to be paid by each. 23. What fraction of if of 5 tons 17 cwt. 6 K> is 1 ton 2 cwt. 6 lb? 1 4 MISCELLANEOUS EXERCISE. 24. Three merchants make a stock of 700, and their profits are respectively 231, Os. 5fd., 64, 3s. 3f^d., and 39, 8s. 7Ad. : how much did each contribute ? 25. A and B exchange goods, A gives B 15 cwt. of hops, the retail price of which is 58/ per cwt., but which he reckons at 3, 3s. per cwt. ; B gives A 12 barrels of beer, retail value 1/2 a gallon, but the value of which he raises in proportion to the increased value of the hops : how much must be paid in money ? IV. 1. Find the square root of 20|. 2. The diameter of a well is 375 ft. and its depth 22'5 ft. : what did it cost in sinking at 3/7 per cub. yd. ? 3. Bought at 193, 12s. and sold for 216, 13s. 4d. : find the gain %. 4. Find the interest on 1199, 19s. 6d. from April 1, 1858, to Jan. 9, 1859, at 3J per cent. 5. If I add f of a pound to ^ of a guinea, into how many shares, each 4 of a shilling, can the sum be divided ? 6. If 1 oz. tea cost 375d., what cost 17'28 lb ? 7. Find the following bill : 20 doz. copybooks at 15/ per dozen ; 1000 quills at 5/6 per hundred ; 125 inkstands at 7jd. each ; 24 doz. lead pencils at 2/9 per doz. ; 64 boxes steel pens at 1/8 per box. 8. If 40 men require 20 worth of bread in 10 days, when wheat is at 63/ per qr., how long would 90 worth serve 54 men, when wheat is at 56/ per qr. ? 9. How many pounds of tea at 5/6 per lb must be exchanged for 293 yds. silk at 3/4 per yd. ? 10. The annual deaths in a town being 1 in 45, in the country 1 in 50 ; in how many years will the number of deaths out of 18675 persons in the town, and 79250 persons in the country, amount to 10000 ? 11. What is the value of a quarter of oats if 17g qr. cost 33^ ? 12. Reduce iVi%& t its lowest terms, and divide it by ^ of 4f . 13. A person who began business 5| years ago has increased his capital at the rate of 15 per cent, per annum simple interest, and it now amounts to 5960 : what had he at first ? 14. The length of a room is 20 ft. 6 in., its breadth 15 ft. 9 in., and its height 10 ft. 6 in. ; what will it cost for plastering, the ceiling at 8d. a yd., and the rendering (on the walls) 3d. a yd. ? Allow for a door 6 ft. 9 in. by 4 ft. 2 in., and a fireplace 5 ft. 6 in. by 5 ft, 3 in. 15. What is the side of a square field of 48 ac. 10 po. 22 yds. 49 ft. ? 16. Find the nett weight of 64 hhds. sugar, each 5 cwt. 2 qr. 10 lb ; tare 5 lb per cent. 17. A person's weekly expenditure is 15, 5s. : what must be his daily income, so that at the end of eleven years he may have saved 425, 18s. 8d., supposing that the first is leap year? 18. Find the value of (1.) 15 reams 9 qu. 6 sh. paper at 1, 6s. 9d. per ream ; (2.) 6 tons 7 cwt. 2. qr. at 3, 10s. 7d. per cwt. 19. A ship, with a crew of 32 men, has provisions for 45 days, at a daily allowance of 2 lb per man. It picks up another crew of 16 men : what allowance will make the provisions last 40 days ? 20. Divide the sum of (11^- -35)-:-(-05-^) and T ^+6'007 by the difference between f (1-35- 72) and 5'0004. 21. A merchant has teas worth 4/6 and 3/6 per tt> respectively, which MISCE LL ANEOUS EXERCISE. 1 4 1 I he mixes in the proportion of 2 Ib of the former to 1 Ib of the latter. He sells the mixture at 4/4 per Ib : what does he gain or lose per cent. ? 22. If I invest 1200 in the 3 per Cents, at 72, what is my income ? and how much per cent, do I get for my money ? 23. Find the cost of covering with gravel, at 7d. per sq. yd., a path 3 ft. wide, round the outside of a bed whose diameter is 9 It. 24. If I am liable for a bill of 380 due 3 months hence, and I pro- pose to pay at once, partly in cash, and partly with a bill for 152 due 4 months hence, what sum must I pay down, interest 4 % ? 25. A, B, and c join in an enterprise to which they each contribute in the proportion of 3, 3, 3 respectively. A pays down 220, 10s., B 205, 4s., and c 213, 5s. : what must each pay to the others, or receive from them, to make the proportion of capital accurate ? V. 1. The population of New York in 1830 was 203007 ; in 1840, 312710 ; and in 1845, 371102 : find the rate per cent, of increase each interval, and on the whole period. 2. The interest on a certain sum lent for 85 days at 4, 6s. 8d. per cent, per annum is 3, 13s. 8d. : what is the principal ? 3. What is the present worth of 120, payable thus : 50 in 3 mo., 50 in 5 mo., and the rest in 8 mo., discount at 5 per cent, per ann. ? 4. Divide the square root of -00093636 by 2^. 5. A reservoir is 56 ft. 8 in. long by 17 ft. 6 in. broad : how many cub. feet of water must be drawn off to sink the surface 2 ft. 6 in. ? 6. A number of men proceed on an expedition, with provisions for nine days, at the rate of 1 Ib 2 oz. for each man per day. The quan- tity furnished was 5062 Ib 8 oz. : required the number of men. 7. If A can do as much work in 5 hours as B in 6 hours, or as c in 9 hours, how long will it take c to complete a piece of work, one half of which has been done by A working 12 hours and B working 24 hours ? 8. (1.) Multiply 2^ by 16, and divide the lesult by f of 2f- ; and (2.) Reduce p x -JZ to its simplest form. 9. (1.) What number of shillings and pence is equivalent to '6 of 3, 5s. 8d. ? and (2.) what decimal of 99, 3s. 4d. is ff | of 233, 9s. 8d. ? 10. If 204 men build a wall of 306 feet long, 8 feet high, and 3 feet thick, in 42 days of 6 hours each, in how many days of 8 hours each will 188 men build a wall 6 feet high and 18 inches thick, round a rectangular enclosure whose length is 319 feet and breadth 97 feet ? 11. Make out an account for : 1 piece flannel 28^ yds., at 3/4 per yd. ; 35 yds. calico, at 5.^(1. per yd. ; 3^ doz. pairs stockings, at 18/6 per doz. ; 7 pairs gloves, at 3/3 per pair ; 12 yds. Irish linen, at 5/6 per yd. ; 4 pairs muslin curtains, at 12/8 per pair. 12. If 29,040 copies of a paper be printed, each of 3 sheets, and each sheet 3.^ feet by 2 feet, how many acres will the edition cover ? 13. A tradesman marks his goods with two prices ; one for ready money, and the other for one year's credit, allowing discount at 5 per cent. If the credit price be 12/3, what ought to be the cash price ? 14. A person loses 10 per cent, by selling cloth at 15/ a yard : how should it have been sold to gain 20 per cent. ? 15. If I lay out 1270 in the 3 per Cents at 92.} ; and, after allow- ing the simple interest to accumulate for two years, I sell out at 93, 142 MISCELLANEOUS EXERCISE. aud invest the sum in debentures at 104 paying 4 per cent. : neglect- ing all fractions of a penny, what is my income ? 16. A triangular field contains 10 ac. 2 ro., and measures 8 cli. 75 Ik. along its base : what is its perpendicular height ? 17. A bankrupt had 214, 17s. 6d. of good debts, and the following bad debts : 340, 8s. 4d., 60, 13s. 6d., and 19, 4s. 6d., for which he receives respectively 8/, 4/, and 16/ per pound : his liabilities amount to 1200 : how much can he pay in the pound ? 18. Find the difference between | [ ^ and (ft-^)+W + Jft). 19. A person bequeaths an annuity of 100 a year : what sum must he invest in 3 stock at 97 to do so ? 20. A semi-circular area whose base is 15 yards is covered with carpet 2 feet wide, what will it cost, at 3/6 per yard, allowing 11 -6 sq. yards for waste ? 21. If a family of 9 persons spends 300 in 8 months, how much money will serve 17 persons for 11 months at the same rate of expen- diture ? 22. A cubic inch of water weighs 252 '458 grains, and the weight of an imp. gallon is 10 Ib av. Find (to 3 dec. places) the number of cub. in. in an imp. gall., there being 7000 grains in the Ib av. 23. What sum will amount to 194, 16s. 10. iu2| yrs., at 4% per an.? 24. Goods are purchased at 28. 10s. 6d. per cwt. ; trade profits are 15 per cent, on invested capital ; the income-tax due on these at 9d. per pound amounts to 24, 3s. 6d. : how many cwt. were purchased ? 25. If a lump of iron 16 cwt. 1 qr. 5 Ib 5 oz. be rolled into a cylindrical bar 12 ft. long, find the diameter of the bar (to three places of decimals). A cubic foot of iron weighs 7788 oz. VI. 1. Divide 1175 into 4 shares, which shall have the proportions of \, f , f , . 2. Find the compound interest of 2500 in 4 yrs. at 4%. 3. Two ships sail, one due north at the rate of 12 miles an hour, and the other due east at the rate of 5 miles an hour : how far apart will they be in 6 hours ? and when will they be 69 miles apart ? 4. A boy buys a suit of clothes for which he will pay in a year at so much per week. There are 2 yds. cloth at 11/4 per yd., the trimming costs 7/> and the making 15/3 : what has he to pay per week? 5. By selling goods at 3, 14s. 6d. a cwt., which cost me 50/ per cwt., I gained 2 guineas : what quantity did I buy ? 6. For the rent of a farm of 27 ac. 3 ro. 27 po. at 7, 10s. 8d. per acre, 139| yds. velvet at 1, 19s. 4d. a yard were taken : what money was returned? 7. A ship's company take a prize of 1001, 19s. 2d. which is divided accord- ing to their pay and time on board. The officers and midshipmen have been on board 6 months, the sailors 3 months ; the officers receive 40/, the mid- shipmen 30/, the sailors 22/ a month. There are 4 officers, 12 midshipmen, and 110 sailors : what will be each man's share? 8. In the year of the Great Exhibition of 1851, London was supplied with butter by 215000 cows, whose produce was 17210 tons 15 cwt. 3 qr. 23 ib 8 oz. for the year : what did a single cow produce? 9. The carpeting of a room 32 ft. 2 in. long at 5/9 per sq. yd. came to 20, find the breadth of the room. 10. The price of bread is 8d. a loaf, and of butter 1/1 \ a pound : how many loaves of bread are equivalent to 5J cwt. butter? 11. The population of Dundee in 1821 was 30575; in 1821-31 it increased per cent. ; in 1831-41. 38S2 per cent. : what was it in 1831 aud 18411 MISCELLANEOUS EXERCISE. 143 12. (a.) A'S income, after deducting income-tax at 1/2 per pound, is 1000 : what was it before deduction? (b.) B'S income, after paying income-tax on half his income at 9d. per pound, and on the other half at 10d., was 576, 5s.: what was it before ? (c.) The income-tax is raised from 7d. to 1/4 per pound ; if c's clear income after paying the tax is 500 before it is raised, what is it after ? 13. Divide -14 by 7, 140 by '07, and "014 by 7000 ; and give the sum of the quotients as a vulgar fraction. 14. The length of a hollow iron roller is 3 feet, the exterior diameter 2 feet, and the thickness of the metal f of an inch : find its content, and how often it will turn from end to end of a gravel walk 65 yds. long. 15. (a.) If 4 per cent, is lost by selling linen at 2/9 per yard, how must it be sold to gain 10 per cent. ? (6.) By selling cheese at 3, 6s. 6d. per cwt., 12 per cent, was gained : what was the prime cost ? 16. The interest on 754, 6s. 8d. for 8 mo. 10 da. is 23, 5s., what is the rate per cent, per ann. ? (Reckon 28 days, one month.) 17. A marble slab, 6 ft. 3 in. long, by 2 ft 8 in. broad, and 4 in. thick, weighed 8 cwt. 1 qr. 20 Ib, and cost 4, Os. 6|d. : how much was the cost per cub. foot, and what is the weight of a cub. foot of marble? 18. When the 3J per cents, are at par, the 3 per cents, at 92, and a stock which pays 3| per cent, is at 104, which is the best, and which the worst ? 19. A barters sugar with B for flour worth 2/3 per stone, but uses a false weight of 13 Ib to the stone : what value should B set on his flour that the exchange may be fair? 20. A bankrupt owes 900 to three creditors, and his whole property amounts to 675 ; if the claims of two of the creditors are 125 and 375 respectively, what will the remaining creditor receive for his dividend? 21. If 3 men, working 11 hours a day, can reap 20 acres in 11 days, how many men working 12 a day will reap a rectangular field, 360 yds. long and 320 wide, in 4 days ? 22. A watch, which is 10 min. fast on Tuesday at noon, loses 2 rain. 11 sec. per day : what time will it show at 5 a.m. the following Saturday? 23. What sum will amount to 194, 16s. l^d. in 2| years at 4 % comp. int. ? 24. The present worth of a sum due 11 mo. hence, when discounted at 4 % per ann., is 2212 : what is its present worth, discounted at 5% per ann. ? 25. Find the product of (f of f of H) *y (f of $f of f) : add the result to the difference between '014 and -^ ; and express the result decimally. VII. 1. (a.) Whether is heavier, a pound of gold or a pound of feathers, also an ounce of gold or an ounce of feathers, the one being troy and the other avoir- dupois weight? In both cases express the one as a decimal of the other. (&.) What must be insured at 4 per cent, on goods worth 2450, so that in case of loss the worth of the goods and of the premium may be recovered ? 2. If 1 Ib Tr. is coined into 46 sovereigns, and 1 Ib av. into 48 halfpence, what is the difference in weight between a sovereign and a halfpenny ? 3. One man can do a piece of work in 5 days, a second in 7, and a third in S^, in how many days can they perform it when working together ? 4. (a.) What is the difference in gain per cent, between selling goods at 2d. which cost ld., and selling goods at 2d. which cost 2d. ? (5.) If a person, selling cloth at 15/6 per yard, gain 26, 10s. per cent, on outlay, what does he lose per cent, when he sells the same at ll/ per yard ? 5. The assets of a bankrupt estate are just sufficient to pay a dividend of 16/7 per pound, but the expense of realizing amounts to 1100, which reduces the dividend to 14/9 per pound : what was the debt? 6. A reservoir is supplied by one pipe and emptied by another. The sup- ply-pipe would fill it in 5 hours, and the escape pipe would empty it in 5 hours ; required the time of filling it when both are opened together. 7. Of every 24 oz. of gold, only 18 oz. in jewellery and 22 oz. in sovereigns are really gold. A sov. weighs 123 gr. When a goldsmith offers chains for their weight in soy., what does he charge per oz. for workmanship? 8. Siippose a railway train, proceeding at the rate of of a mile in a minute 144 MISCELLANEOUS EXERCISE. to be audible at a distance of 2J miles, how long exactly will its noise pre- cede it sound travelling at the rate of 1130 feet per second? 9. The rise of interest from 3J to 4 per cent, increase:? a person's nett in- come (after deducting income-tax of 7d. per pound) by 485, 8s. 4d. : what is the principal sum from which his income is derived ? 10. A person sells out 1725, 3 per cents., at 92J, and buys 3J per cents, at 95J, brokerage fcth per cent : what is the alteration on his income ? 11. Standard silver contains 37 parts of silver and 3 of alloy. Now, 5/6 just weighs an oz. Tr. : what weight of pure silver is in 100 ? 1 2. The weight of water being 1000 oz. av. per cub. ft. , what weight of water will an inch (area) pipe discharge in a day, flowing at the rate of 2 ml. pr. ho. ? 13. Find the number of cubic inches in a cube of which the edge is 2 ft. 5 in. long : find also the length of the diagonal of the cube. 14. Add f of ^ to & of 3-$ ; multiply the sum by the difference between and ^f a nd divide the product by 15 times the difference of -^fg- and -5 J^. 15. A tunnel ^ of a mile long is excavated at the rate of ^ of a yard per day : in how many years will it be completed ? 1C. (a.) If cloth be bought at 15/3 per ell of 5 qrs., and sold at 15/6 per yd., what is gained on an outlay of 47, 12s. 6d? (&.) If sugar be bought at 2, Is. 9d. per cwt, and sold at 6Jd. per ft, what part of every 100 is gain ? 17. The following bill was paid by a number of persons contributing 1, 16s. 7d. each: how many were there? 23 articles at 9, 2s. lid. per score ; 54 at 11/8 per score ; 37 at 12/6 per dozen ; 15 at 4Jd. each ; and 11 at 7Jd. each. IS. A manufacturer employs 60 men and 45 boys, who work respectively 10 and 8 hours per day during 5 days of the week, and half the time on the remaining day ; if each man receives 6d. per hour, and each boy 2d. per hour, what is the amount of wages paid in the year of 52 weeks ? 19. A wall 700 yds. long was to be built in 29 days. At the end of 11 days, IS men had built 220 yds. : how many additional men had to be engaged to work at the same rate, that the wall might be completed in the given time ? 20. A square, whose side is 500 feet, has a circular garden within it 400 feet in diam. : what will it cost to pave the part outside the garden at 2/ per sq. yd. ? 21. I buy a set of watches at 50/ each. I sell them at a profit of T ^th prime cost, but for ready money deduct 5 per cent ; for every 8 sold for ready money, 6 are sold without discount : find the gain per cent, on the money invested ? 22. (a.) It is half-past 3 ; at what hour will the hands of the watch first meet ? (6.) At what time between 1 and 2 are the hands together, in opposite directions, and at right angles respectively ? 23. Find the cost of covering a roof with lead at IS/ per cwt on the follow- ing data : the length of the roof is 43 feet ; the breadth 32 feet ; the gutter- ing is 57 feet long and 2 feet wide; the former requires lead at 9 '831, the latter at 7 373 ft> to the square foot. 24. A cistern 12 feet long, 2 ft. 4 in. wide, and 9 in. deep, contains pulp for making paper. If half its volume is lost in drying, how many sheets 8 in. by 6 in. will be obtained, if 300 sheets in thickness go to the inch ? 25. A man has an income of 400 a year, and the income-tax is 9d. a pound. A duty of ld. per Ib is taken off sugar :_what must be the yearly consumption of sugar in his family that he may -jus* save Iris income-tax ? /r^C/ 0# '"; ffUBJVEn ft oar ., , , ^rF&frj&g' EDINBURGH : T. CONSTABLE, PBINTER TO THE QUEEN, AND TO TUB UNIVEHS1TT. UNIVERSITY OF CALIFORNIA LIBRARY, BERKELEY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW Books not returned on time are subject to a fine of 50c per volume after the third day overdue, increasing to $1.00 per volume after the sixth day. Books not in demand may be renewed if application is made before expiration of loan period. JAN 26 1928 50w-8,'26 YA 02405 THOS. LAURIE, EDUCATIONAL PUBLISHER, GEOGRAPHY. V AXWELL'S FIRST LESSONS IN GEOGRAPHY, with Questions. By the Author of " Henry's first History of England," fcc. Pp. 96, 6d. ' AXWELL'S GEOGRAPHY OF THE BRITISH EMPIRE. Pp. 64, 4d. iAXWELL'S GENERAL GEOGRAPHY. Pp. 168, cloth, Is. SCRIPTURE LESSONS. ENRY'S FIRST SCRIPTURE LESSONS: Two Parts, 8d. each ; cloth, Is. 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