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Full text of "The new advanced arithmetic"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at http : //books . google . com/| Digitized by VjOOQiC '^w>. ^\- fX^-^ X U- 'jy\ ^-'.i. v; ■H-vv Digitized by VjOOQIC Et)e StannarD Snieis at ^at))ematiciS ^ ^-'^ -y^'^ THE. NEW '^ ^i^ ADVANCED ARITHMETIC JOHN W. COOK raxswxtiT or illihoib btatx kobxai. CKiraMirr MISS N. CROPSEY ASSISTANT SUPSBINTSKDENT OF CITT SCHOOLS, INDIANAPOLIS, INDIANA } V V SILVER, BURDETT AND COMPANY NEW YORK BOSTON CHICAGO c Digitized by VjOOQICx u\. f J ^ ' <^-' >tr^O^ ■4^i^ "Wt-W^^^ GoPTmiOHT, 1892. 1896, 1903, 1904, 1906, Br SIL^^», BURDETT A^'D COMPANY. sducatios iiBa# Digitized by VjOOQIC PREFACE TO THE REVISED EDITION. A NUMBER of changes have been thought desirable in the plan of "The New Advanced Arithmetic"; all of them, however, are intended to recognize the latest and best phases of class-room- work in arithmetic. Some of these changes — more particularly those in the early part of the book — have been necessary the better to relate the Advanced book of the Series to the Elementary book, which has at the same time undergone revision. Much emphasis is given to Factoring, and its applica- tion to Cancellation, the Highest Common Factor, the Lowest Common Multiple, and Evolution. The equation is introduced in a simple form much earlier tlian in the previous edition. This is done in order Jx) simplify the treatment of other important topics which follow it. Percentage and its applications, as well as Mensuration, are treated more fully than before. A large number of new problems have been added, which will, it is thought, be found interesting and profitable. Cube Root, Compound Proportion, Equation of Pay- ments, and True Discount are among the subjects which have been omitted. Digitized by VjOOQIC iv PREFACE TO THE REVISED EDITION It is hoped that the revised book is fairly representa- tive of what is best and most progressive in present-day methods, and that teachers generally will find it well adapted to their needs. The authors and publishers desire to express their sin- cere appreciation of the assistance and cooperation given in the preparation of this edition by Dr. Robert J. Aley, Professor of Mathematics in Indiana University, and Oscar L. Kelso, Professor of Mathematics in the Indiana State Normal School. Digitized by VjOOQIC PREFACE TO FIRST EDITION. It has seemed to the authors of the Normal Goubsb m Number that there is room for another series of Arithmetics, notwithstanding the fact that there are many admirable books on the subject already in the field. The Elementary Arithmetic is the result of the expe- rience of a supervisor of primary schools in a leading Ameri- can city. Finding it quite impossible to secure satisfactory results by the use of such elementary arithmetics as were available, she began the experiment of supplying supple- mentary material. An effort was made to prepare problems that should be in the highest degree practical, that should develop the subject systematically, and that should appeal constantly to the child's ability to think. Believing that abundant practice is a prime necessity to the acquisition of skill, the number of problems was made unusually large. The accumulations of several years have been carefully re- examined, re-arranged, and supplemented, and are now pre- sented to the public for its candid consideration. Not the least valuable feature of this book is the careful gradation of the examples, securing thereby a natural and logical devel- opment of number work. No space is occupied with the pres- entation of theory, — that side of the subject being left to the succeeding book. The first thoughts are whaJt and hxniOj — these so presented that the processes shall be easily Digitized by VjOOQIC VI PREFACE TO FIRST EDITION. comprehended and mastered. Subsequently, the why may be intelligently considered and readily understood. The Advanced Arithmetic is the outgrowth of a somewhat similar experience in the class-room of a teachers' training- school. For many years an opportunity was afforded to study the effects upon large numbers of pupils of the current methods of instruction in arithmetic. The result of such observation was the conviction that the ratibnal side of the subject had been seriously neglected. An effort was made to supplement the ordinary text-book by a study of principles and by explana- tions of processes. The accumulations of fifteen years have been edited with all of the discrimination of which the authors are capable. Great care has been exercised in the presenta- tion of principles and in the formulation of processes, to the end that the learner shall have every facility for the use of his reasoning powers, and at no point be relieved from the proper exercise of his mental activity and acumen. It is hoped that the book may contribute somewhat to the move- ment, now so happily going on, that looks toward the dis- establishment of the method of pure authority, and the establishment of a method that makes its appeal to intelli- gence and reason. The authors desire to express their appreciation of the excellent suggestions offered by many friends; but especial thanks are due Professor David Felmley, of the Illinois State Normal School, for his discriminating criticisms and valuable assistance. THE AUTHORS. Digitized by VjOOQIC INTRODUCTION. Arithmetic amply justifies its place in the cumculum because of its utilitarian and culture values. No subject surpasses it in usefulness. It is so closely connected with our lives that the blotting out of all aritKmetical notions would stop at once all commercial and industrial activity. To be a member of the social world one must know something of arithmetic. On the culture side, arithmetic is fitted by its nature to do certain things. The three most important things that it can do well are : 1. To train in scientific reasoning. Arithmetic is a science ; that is, it is a body of organ- ized truths. Its conclusions are the results of logical reasonings. Every step is taken because of some pre- ceding step or fundamental assumption. The value of the training that comes from this sort of study can hardly be overestimated. 2. To train in concentration. The nature of arithmetical work is such that the whole mind must be given to the problem under consideration. If, in adding a column of figures, the student allows his mind to wander, he finds that he must pay the penalty for so doing. He must add the column again. No other subject in the common school course equals arithmetic in its power to train the attention to concentrated effort. 3. To train in accuracy, ▼il Digitized by VjOOQIC vm INTRODUCTION. Arithmetic is an exact science. The results of arith- metic are fixed and infallible. The multiplication table will never cease to be true. It is of great value to a growing mind to work at a subject in which the results are exact. The purpose of the teacher in teaching arithmetic should be to put the pupil in possession of arithmetic as a tool for use in everyday life, and, in addition, to give him all the training in scientific reasoning, concentration, and accuracy that the subject is capable of furnishing. The arithmetical operations employed in ordinary busi- ness affairs are simple, but they must be performed with absolute accuracy and with rapidity. They are based, pri- marily, upon the memory. Neither accuracy nor rapidity is possible without a thorough mastery of the primary work. This mastery is acquired ' through constant repetition of the old in connection with the acquisition of the new. The tables of the fundamental operations must be learned so thoroughly that* their application can be made without much conscious thought. To this end much of the work in the elementary arithmetic should be abstract and with rather large numbers. The mechanical side of arithmetic consists of the four fundamental operations of addition; subtraction, multipli- cation, and division. These need to be thoroughly mas- tered so that in the higher phases of work upon problems the whole strength of the mind may be given to the reasoning. In view of the nature of arithmetic and the purpose in teaching it, the following things should be observed by the teacher: 1. Arithmetic is a unity. AH the parts of arithmetic are so related that they Digitized by VjOOQIC INTRODUCTION. ix make a unity. This unity is a part of the larger unity, mathematics. At no stage of the development of the subject should the student learn anything which it will be necessary to unlearn at a later stage. If the student learns a definition of multiplication, it should be true not only throughout arithmetic, but also throughout the whole of mathematics. 2. There are certain truths wh^ch are organic. The organic truths of arithmetic are the threads of unity which run through it and about which it is organ- ized. The %cale relation, which is the fundamental thing in all number systems, is organic. It binds together simple numbers, compound numbers, and fractions. That "multiplying or dividing both dividend and divisor by the same number does not affect the quotient" is an organic truth which binds together division, fractions, and ratio. 3. Arithmetic is twofold^ — pure and applied. Pure arithmetic has to do with the properties of num- ber. It includes the fundamental operations, factors, multiples, divisors, powers, roots, and ratios, with all the principles and relations pertaining to them. Pure arith- metic could exist in perfection without any application. Applied arithmetic is pure arithmetic answering the ques- tions that come to man in his material experiences. The culture value of arithmetic is found in both its phases. The bread and butter value is found only in the applied side. The teacher must know that no applied arithmetic is possible except as it rests upon the sure foundation of the pure arithmetic. 4. There is no definite boundary/ between arithmetic and algebra. Algebra and arithmetic overlap. No one has yet been Digitized by VjOOQIC X INTRODUCTION. wise enough to mark the exact place where the one ends and the other begins. The simple form of the equation belongs as much to arithmetic as it does to algebra. There is no reason why the equation in arithmetic should be treated in a stilted, complicated manner. Its principles are as simple and easy as those of multiplication or divi- sion. It is to the student of arithmetic what improved machinery is to the farmer. The observation of the four facts above enumerated will have an important effect upon the method of teach- ing the subject. Among the many results, the following may be enumerated : 1. There will be teaching instead of hearing of lesions, 2. The teacher will see the end from the beginning, and will direct his instruction to that end. 3. The law of apperception will be followed, and each new subject will be related to the preceding ones. 4. In applied arithmetic the pupil will be led into the particular field of experience in which the problems occur, and made familiar with the activities in that field before attempting to apply the pure arithmetic. 5. The value of mental arithmetic will be appreciated, and the pupils given ample opportunity to grow through mental problems. 6. The value of direct, masterly methods of solution will be understood, and the pupils encouraged to use every possible short cut. 7. The pupils, when solving problems, will be urged to think first and to figure as little as possible. Digitized by VjOOQIC CONTENTS. CHAPTER L PAGB NUMEBATION AND NOTATION . 1-6 Arabic Notation .... 3 Roman Notation .... 5 Different Readings for a Number 6 CHAPTER IL The Fundamental Opeba- TION8 7-36 Addition and Subtraction . 7 Multiplication and Division . 11 The Law of Signs .... 16 Compound Numbers ... 19 Compound Addition and Subtraction 22 Compound Multiplication and Division 25 Extension of Addition and Subtraction 27 The Area of a Rectangle . 28 The Volume of a Rectangu- lar Solid 30 Miscellaneous Problems . 33 CHAPTER III. Factobs, Divisors, and Mul- tiples ...... 37-53 Tests of Divisibility ... 39 Factoring 40 Cancellation 41 The Highest Common Factor 43 The Lowest Common Mul- tiple 45 Miscellaneous Problems . 47 CHAPTER IV. Longitude and Time . . 54-61 Standard Time 59 CHAPTER V. Fractions 62-97 Reduction of Fractions . . 65 Addition of Fractions . . 69 Subtraction of Fractions . . 74 Multiplication of Fractions . 76 Division of Fractions ... 82 Complex Fractions ... 86 To Find the Part which One Number is of Another . . 87 To Find a Number when a Specified Part of it is given 89 Miscellaneous Problems . 89 CHAPTER VL Decimal Fractions . . 98-128 Reading Decimal Fractions . 99 Writing Decimal Fractions . 100 Reduction of Decimal Frac- tions 102 Addition of Decimal Frac- tions 105 Subtraction of Decimal Frac- tions . .• 106 Digitized by VjOOQIC Xll CONTENTS. PiiGS Multiplication of Decimal Fractions 106 Division of Decimal Frac- tions 109 The Metric System ... 112 Reduction of Metric Numbers 115 Specific Gravity . . . .117 Miscellaneous Problems . 119 CHAPTER Vn. * The Equation . . . 120-146 The Use of a; in Problems . 129 Equations containing Frac- tions 134 Verifying an Equation . . 135 Miscellaneous Problems . 138 CHAPTER Vin. Pbrcektaoe .... 147-162 Miscellaneous Problems . 160 CHAPTER IX. Applications op Percent- ' age 163-185 Profit and Loss 163 Commission 167 Commercial or Trade Dis- count 170 Taxes 174 Insurance 180 Property Insurance . . . 181 Life Insurance 182 Miscellaneous Problems . 183 CHAPTER X. Applications of Percent- age {continued) . 186-226 Interest 186 Notes ........ 195 FAOS Partial Payments .... 196 The Method of finding In- terest by Days .... 201 General Problems in Simple Interest 203 Banks and Banking . . . 205 Exchange 211 Stocks 214 Bonds 215 Miscellaneous Problems . 221 CHAPTER XI. Ratio and Proportion 226-241 The Relation of Numbers . 226 Ratio 228 Proportion 233 Miscellaneous Problems • 238 CHAPTER XII. Involution and Evolution 242-253 Powers 242 Roots 243 The Relation of the Squares of Consecutive Numbers . %ib The Relation of the Squares of Any Two Numbers . . 247 The Number of Figures in a Product 250 The Number of Figures in the Square Root of a Number 251 The Extraction of Square Root 252 CHAPTER Xin. Mensuration .... 254-302 Carpeting 257 Papering ....... 259 Digitized by VjOOQIC CONTENTS. xui PAOB Plastering 261 The Parallelogram .... 262 The Triangle ..... 264 The Right-angled Triangle . 266 The Trapezoid 269 United States Surveys . . 271 The Circle 272 The Area of a Circle ... 275 Solids 277 Wood Measure 277 Lnmber Measure . • • . 279 PAOB Masonry and Brickwork . . 281 The Inverse Problem in Men- suration 284 Prisms and Pyramids . . . 287 Cylinders, Cones, and Spheres 290 MiSCBLLANEOnS PROBLEMS . 294 General Reviews . . 802-313 Appendix 815-^27 o-'\ - ^' - ^ ' ' \-' •^ ' Digitized by VjOOQIC Digitized by VjOOQIC THE NEW ADVANCED ARITHMETIC. CHAPTER I. NUMERATION AND NOTATION. 1. Numeration is the naming or reading of numbers. 2. The group system is universally used in naming numbers. The numbers of a small group are given indi- vidual names, and then a systematic process of repetition follows. In naming numbers we use the group ten. The numbers from one to ten have distinct individual names. The names eleven and twelve seem to be individual, but in their early forms they were the combination of one and ten^ and two and ten respectively. The word thirteen is a combination of three and fen, the idea of three being in thir^ and of ten in teen. The same relation is seen in the remaining teenB, The word twenty is made from the com- bination two tena^ the idea of two being in twen^ and ten in ty. The same idea is in tliirty, forty, fifty, sixty, seventy, eighty, and ninety. Except for slight changes in form made to produce better sounding words, no new number name occurs after ten, until we reach ten tens, when the new name hundred appears. Repetition and combination then continue to ten hundreds, when the new name thousand occurs. In the number eight hundred seventy-four, we have a com- bination of the words eight, hundred, seventy, and four, 1 Digitized by VjOOQIC 2 NUMERATION AND NOTATION. After the word thousand is introduced, no new name is needed until the number one thousand thousand occurs, to which the name million is given. 'The next new number name is billion^ given to the number a thousand million. 3. It will be noticed that all this number naming centers about ten. The new names are for numbers ten times as large as those already named. A hundred is ten times a ten, a thousand is ten times a hundred, a million is ten times a hundred thousand, and a billion is ten times a hundred million. 4. For the reason that in our system everything centers about ten, it is called the decimal system (^decern — ^ten). 5. Ones are grouped into tens^ tens into hundreds^ hun- dreds into thousands^ and so on. Ones are units of the first order, tens are units of the second order, hundreds are units of the third order, and so on. 6. From the formation of our number names we see that ten units of one order make one of the next higher. Ten is, therefore, the scale (or radix) of our system. 7. Three orders form a period. The orders from left to right of any period are hundreds, tens, and units of that period. 8. Arrangement of orders and periods : Trillions. Billions. Millions. Thousands. Units. I ^ i I tgS tiSao '2s§ 'Sis I M'Co M'^p ^Ss ^uas M ^3*3.2 73-9© '09.2 "O^S '^ t^ m 5a. g Sfls ^ a B Sao ggS £S^ 30^ J2®3 3 ^ fi »S®5 Whh whoq khs Whh Who 86 6, 40 6, 38 2, 10 4, 579 Note, Learn the names of the periods in their order from left to righL Digitized by VjOOQIC ARABIC NOTATION. 8 9. Notation is the expression of numbers by means of characters. 10. There are three methods of notation in use: the word, the Arabic, and the Roman. 11. In the word method the number names are written out in full, e.ff. May eleventh, eighteen hundred sixty-three. ARABIC NOTATION. 12. The Arabic notation represents numbers by the use of the ten characters 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These character are called figures or digiU^ and are in' almost universal use. The Arabic notation possesses three marked character- istics : (1) Its number of characters corresponds to the scale of the system ; that is, there are ten characters, and the scale of the system is ten. This makes possible the same sort of combina- tion and repetition in the writing of numbers that occurs in the naming of numbers. (2) It uses the idea of place value ; that is, the same char- acter in different places in a number represents different values. In the number 6666 the first 6, beginning at the left, represents, because of its place, six thousand ; the second 6, six hundred ; the third 6, sixty ; and the fourth 6, six. (3) It has one character, (zero), used to fill vacant places. 13. Moving a character one place to the left multiplies its value by ten, while moving it one place to the right divides its value by ten. In the number 666 the middle 6 is ten times the right-hand 6 and one tenth of the left-hand 6. 14. To read a number, group the figures into periods, beginning at the right, and separating the periods by Digitized by VjOOQIC 4 NUMERATION AND NOTATION commas. Beginning at the left, read the number in each period as if it stood alone; then add the name of the period. 81,367,937 is read " eighty-one million, three hundred sixty- seven thousand, nine hundred thirty-seven." Note i. The name of the last period, units, is generally omitted. Note 2, In reading whole numbers the word and is unnecessary. EXERCISE I. Eead the following numbers : 1. 2345. 6. 250849. 11. 683471. 2. 4638. 7. 381307. 12. 829406. 3. 7912. a 408391. 13. 200619. 4. 3105. 9. 716004. 14. 100054. 5. 26853. 10. 500836. 15. 973070. EXERCISE 2. Before writing the following numbers, tell how each will appear when written. Thus, three thousand eight hundred seven is expressed by writing the following : three, comma, eight, cipher, seven. 1. Forty thousand six. 2. Ninety-seven thousand five hundred twelve. 3. Three hundred sixty-nine thousand twenty-four. 4. Four million eight thousand two. 5. Fifty-six million nineteen thousand thirty-three. 6. Eighty -one million five hundred thirteen thousand two hundred fifty-one. 7. Three hundred million ninety thousand four. 8. Five billion six million seven thousand eight. 9. Seventy-two billion six hundred thirty-five thousand two hundred fifty-one. Digitized by VjOOQIC ROMAN NOTATION. 6 • 10. One hundred three billion two million seventeen thousand one hundred four. To the Teacher. If needed by the pupils, the teacher should supply further exercises in reading and writing numbers. ROKAN NOTATION. 15. The Roman notation expresses numbers by means of the characters I, V, X, L, C, D, M, and . This notation is used very little. It is being rapidly re- placed in dates and chapter numbers by the Arabic notation, 1 = 1, V = 6, X = 10, L = 60, C = 100, D = 600, M = 1000. The use of these characters is determined by the following principles : PRINCIPLES. 16. Prin. 1. Repeating a letter repeats its value. II = 2, XX = 20. 17. Prin. 2. When a letter is placed after one of greater valtte^ the two express a number equal to the sum of their values. XV = 15, CI =101. 18. Prin. 3. When a letter is placed before one of greater value^ the two express a number equal to the difference of their values. (Limited to IV, IX, XL, XC, CD, and CM.) IX = 9, XC = 90. 19. Prin. 4. When a letter is placed between two others^ each of greater value^ its value is taken from the sum of their values. XIX = 19, XIV = 14. 20. Prin. 5. Placing a dash over a letter multiplies its value by a thousand. X = 10,000, M = 1,000,000. Digitized by VjOOQIC 6 NUMERATION AND NOTATION. EXERCISE 3. Express by the Eoman characters : 1. All numbers from one to one hundred. 2. 125. 5. 419. 8. 752. 11. 1776. 3. 263. 6. 599. 9. 1066. 12. 1799. 4. 379. 7. 643. 10. 1492. 13. 1904. To the Teacher. Use dictation exercises freely. DIFFERENT READINGS FOR A NUMBER. 21. For the reason that numeration and notation are both based upon the scale of ten^ any number of two or more places may be read in several ways. The number 2743 should be read "two thousand seven hundred forty-three.'^ This is the regular way of reading a number, and should always be used unless some other form is specifically demanded. The number 2743 may also be read in the following ways : Twenty-seven hundred forty-three. Two thousands seven hundreds four tens three units. Two thousands seventy-four tens three units. Two hundred seventy-four tens three units. These different ways of reading are the results of reductions or changes made by means of the scale relation. Two thousand may be thought as twenty hundred, and two thousand seven hundred may be thought as twenty-seven hundred. EXERCISE 4. Give three or more readings for each of the following : 1. 785; 963 1031; 406. 2. 2400; 2043 2004; 4301. 3. 5673; 6007 5060; 7190. 4. 70310; 65000 ; 30600; 40060. S. 70000; 70707 70007; 77070. Digitized by VjOOQiC CHAPTER II. THE FUNDAMENTAL OPERATIONS. 22. Addition, subtraction, multiplication, and division are called the Fundamental Operations. They are fre- quently referred to as the Four Rules. ADDITION AND SUBTRACTION. 23. Addition and subtraction are related as direct and inverse processes. Addition is the direct process of uniting numbers into a sum. Subtraction is the inverse process of finding a number when the sum of it and a given number is known. 24. The numbers to be added are called Addends, and the result of the addition is the Sum or Amount. 25. In subtraction the given sum is called the Minuend and the given addend is the Subtrahend. The addend to be found, or the result of the subtraction, is the Remainder or Difference. 26. Add 876 and 497. 875 497 875 and 497 are the addends, and 1372 is the sum. 1372 27. Subtract 497 from 1372. 1S72 1372, which is the sum of 497 and some other j^Q«. number, is the minuend. 497, which is the given -g=g one of the addends used in making 1372, is the sub- trahend. 875, which is the other addend used in making 1372, — or, which is the result of the subtraction, — is the remainder or difference. 7 Digitized by VjOOQIC 8 THE FUNDAMENTAL OPERATIONS. In subtracting 9 from 16 we may think " 9 from 16 leaves 7," or " 9 and 7 make 16/^ The latter form is preferable ; it is the one used in making change. PRINCIPLES. 28. Prin. 1. Only like numbers can be added or Bvb- traded. Number is either abstract or concrete. When number is unapplied, it is abstract 3, 8, 13, 125 axe abstract numbers. When number is applied, it is cona^ete, 3 bu., $8, 11 ft., 16 men, 160 A. are concrete numbers. All abstract numbers are like numbers. When concrete num- bers have the same name, they are like numbers. 5 sheep and 12 bu. cannot be added, nor can 5 cows be subtracted from $40. 29. Prin. 2. Addends may be used in any order in finding their sum. 3 + 5 + 8 = 3 + 8 + 5 = 8 + 3 + 5 = 5 + 3 + 8. 30. Prin. 3. Addends may be grouped in any order in finding their sum. 6 + 5 + 8 = 11 + 8 = 6 + 13 = 5 + 14. 31. Prin. 4. Adding the same number to^ or svhtrdct- ing the same number from^ both minuend and subtrahend does not change the remainder. 17 — 5 = 12. If 15 is added to both 17 and 5, they become 32 and 20 respectively. 32 - 20 = 12. 32. Prin. 5. Multiplying both minuend and subtrahend by the same number multiplies the remainder by that number. 11- 5= 6. Multiplying by 8, _§_§_§• 88 - 40 = 48. Digitized by V3OOQ1C ADDITION AND SUBTRACTION. 9 33. Prin. 6. The Bum of the Bubtrahend and the re- mainder equals the minuend. 361-164 = 197. 197 + 164 = 361. Note, This principle is used in proving the correctness of an ex- ample in subtraction. EXERCISE 5. (Mental.) 1. The addends are 5, 7, 8, 15. What is the sum ? 2. The addends are 18, 9, 11, 22. What is the sum ? By Principle 2 we may think them as 18, 22, 9, 11, and then by Prin- ciple 3 as 40, 20, thus grouping them so as to add with the least effort. 3. Add 7, 12, 13, 18, 21, 9. Mentally group as in problem 2. 4. Add 9, 17, 11, 13, 19, 21. 5. John has $35, and Henry $20. Each receives $60. How much more does John then have than Henry ? (Prin. 4.) 6. Mary had $24, and Lucy $30. They each trebled their money. How much more did Lucy then have than Mary ? 7. Does 19 from 42 leave 22? Why? 8. What is the sum of $8, 5 bu., 9 bu., $12, 4 bu., and $5? 9. Can you add 8 horses, 22 cattle, and 120 sheep? If these are owned by Mr. Horner, how many head of stock has he ? In what does the likeness now consist ? 10. The sum is 64. Four of the addends are 12, 13, 10, 15. What is the fifth ? U. In the subtraction 32 — 28, what effect does the adding of 15 to the minuend have upon the remainder ? 12. If 9 is added to the subtrahend in the problem 65 — 41, what is the effect upon the remainder ? 13. 8 + 6 + 24-18 + 20-30 = ? Digitized by VjOOQIC 10 THE FUNDAMENTAL OPERATIONS. 14. I bought oranges for 15 cents, lemons for 10 cents, eggs for 22 cents, and gave the grocer 60 cents. How much change should I receive? 15. Last year I saved $ 150. If this year I double both my income and my expenses, how much shall I save, at the same rate (Prin. 5) ? 16. 9 + 6 + ll + ?-16 = 30. 17. A has $ 30 more than B. B gives A $ 10. How much more does A then have than B ? 18. If each addend is increased by the same number, what is the effect on the sum ? 19. If all the addends are multiplied by the same number, what is the effect on the sum ? ao. 15 + 20-?-f25 = 42. EXERCISE 6. 1. How would you find the subtrahend when the minuend and the remainder are given ? Why ? 2. How would you find the minuend when the subtrahend and the remainder are given ? Why ? 3. 60241-42374 + 26082=? 4. 280621+460082-31892-42671 = ? 5. 529706-31793-64802 + 5729 = ? 6. A farmer had 861 A. of land, and gave 294 A. to his son. How many acres did he have left ? 7. C and D were 350 mi. apart. They traveled toward each other, C at the rate of 46 mi. a day, and D at the rate of 37 mi. a day. How far apart were they at the end of the second day ? Which had traveled farther ? How much ? 8. Four men together owned 786 cattle. E owned 227, and F, 339. G owned as many less than E as F did more than E. How many did the fourth man own ? Digitized by VjOOQIC MULTIPLICATION AND DIVISION. 11 9. A merchant paid $ 2500 for a quantity of silk. For other dry goods he paid $ 265 more than he did for the silk. For groceries he paid $ 683 less than what he had before expended. To one customer he sold goods amounting to f 5280 ; to another, $ 32S less than half as much ; to another, $ 2895 less than to the first, thus disposing of all of his stock. Did he gain or lose, and how much ? 10. A has f 650 ; B has $25 less than half as much ; C has 9 125 less than A and B together ; D has f 275 less than A, B, and C. How much have all of them together ? 11. From A to B is 492 mi. ; from B to C is half as many miles, plus 42. How would you find the difference in the two distances ? How many miles is it from A to C ? 12. Three persons bought a mill valued at $25,642. The first paid $ 6743.25 ; the second, twice as much ; and the third, the remainder. How much did the third pay ? 13. How many more days are there in the months of March, April, May, and June, counted together, than in the months of September and October ? 14. A lady bought ribbon for 36 cents, lace for $ 1.48, gloves for $ 1.75, and velvet for $ 1.27. She gave in payment a five- dollar bill, and received her change in nickels and cents. How many nickels did she receive ? 15. I bought several articles costing respectively 63 cents, 89 cents, 48 cents, f 1.38, $ 2.76, $ 4.75, and gave the merchant a $ 20 bill. What change should I receive ? MULTIPLICATION AND DIVISION. 34. Multiplication and division are related as direct and inverse processes. . 35. Numbers to be multiplied together are called factors. When there are two factors to be multiplied together, one of them is called the multiplicand, and the other the multi- Digitized by VjOOQIC 12 THE FUNDAMENTAL OPERATIONS, plier. The result obtained by multiplying two or more factors together is called the product. 36. To multiply one number by another is to do to the first (the multiplicand) what was done to unity to produce the second (the multiplier). To multiply 8 by 5 is to take 8 five times, for unity has been taken five times to produce 5. 37. If a product and one factor are given, the other factor is found by division. The number which corre- sponds to the product is called the dividend; the given factor is the divisor ; and the factor to be found, or the result of the division, is the quotient. 86x25 = 2150. 86 and 2^ are the factors, and 2150 is the product. 2150-5-86 = 25. 2150 is the dividend, 86 is the divisor, and 2b is the quotient. 38. To divide one number by another is to do to the first what must be done to the second to produce unity. To divide 12 by 4 is to do to 12 what must be done to 4 to produce unity. Unity is produced from 4 by separating it into four equal parts and taking one of them, or by taking one fourth of 4. Hence, to perform the division required, 12 must be separated into four equal parts and one of them taken, or one fourth of 12 must be found. PRINCIPLES. 39. Prin. 1. Factors may he used in any order in find- ing their product, 8x5x7 = 8x7x5 = 7x8x5 = 5x7x8. 40. Prin. 2. Factors may he grouped in any order in finding their product, 8x5x7 = 40x7 = 8x35 = 5x56. Digitized by VjOOQIC MULTIPLICATION AND DIVISION. 13 41. Prin. 3. Multiplying or dividing any factor of a product multiplies or divides the product by the same number. 7 X 8 = 56. Multiply 7 by 3 and we have 7 x 3 x 8 = 168 = 56x3. Note the effect on the product when the factor 8 is divided by 2. 42. Prin. 4. Multiplying the dividend or dividing the divisor multiplies the quotient. 48 -J- 16 = 3. Double the dividend and we then have 96 ^ 16 = 6 = 3x2. Note the effect on the quotient when the divisor is divided by 4. 43. Prin. 5. Dividing the dividend or multiplying the divisor divides the quotient. 64-5-8 = 8; 64-5-32 = 2. Here the divisor has been multi- plied by 4, and the quotient is then one fourth of its former value. The same effect is produced if the dividend is divided by 4. 44. Prin. 6. Multiplying or dividing both dividend and divisor by the same number has no effect on the quotient. Use 100 -5- 20 = 6 to illustrate this. EXERCISE 7. (Mental.) 1. A man has 400 sheep, which he puts in lots of 50 each. How many lots does he make ? 2. If he doubles the number of sheep in a lot, how many lots will there be ? What principle has been used ? 3. If he doubles his number of sheep and also the number in each lot, how many lots will there be ? What principle ? 4. If the lots are only half as large, then how many lots will there be ? What principle ? Digitized by VjOOQIC 14 THE FUNDAMENTAL OPERATIONS. 5. 7x6x9x5 = 7x9x6x5 = 63x30=? What prin- ciples have been used ? 6. 9x5x11x8 = 9x11x5x8 = 99x40=? 7. An 80 A. field was divided into 10 A. lots. How many lots did it make ? 8. A stagecoach went 6 mi. an hour. How many hours were required to go 30 mi. ? 9. A schoolroom contains 54 seats arranged in 6 equal rows. How many seats are there in each row ? 10. At 3 cents each, how many oranges can be bought for 30 cents ? U. If 5 bbl. of flour cost $20, what is the price per barrel ? 12. If a school of 42 pupils were divided into 6 equal classes, how many pupils would there be in each class ? 13. With divisor and quotient in each of problems 7 to 12, make a problem in multiplication ; with dividend and quotient make a problem in division. 14. How many oranges at 3 cents each should be given in exchange for 4 lb. of butter at 15 cents ? 15. How many bushels of oats at 30 cents should be given in exchange for 40 bu. of wheat at 75 cents ? EXERCISE 8. 1. A dealer bought 3 horses at $85 each; 5 at $96 each; 7 at $ 124.50 each. He shipped them to the city, the freight averaging $ 12.50 each. He sold the cheapest at $ 110.50 each, the second lot at $ 128 each, and the third lot at $ 164 each. If his personal expenses and the care of the horses amounted to $32, what did he gain by the transaction ? 2. A stone falls 16 ft. the first second, (16 + 32) ft. the second second, (16 + 32 -f- 32) ft. the third second, and so on. How many feet will it fall in 8 sec. ? Digitized by VjOOQIC MULTIPLICATION AND DIVISION. 16 3. Light travels 186,000 miles a second. It takes 498 sec. for a light wave to pass from the sun to the earth. What is the distance ? 4. A bushel of com in the ear weighs 70 lb. ; shelled, 66 lb. How many pounds of cobs are there in a crib containing 1800 bu. of ears ? 5. A car is loaded with 49 steel rails 32 ft. long, weighing 78 lb. to the yard. The weight of the car and its load is 62,592 lb. What is the weight of the car ? 6. If the above car be loaded with 512 bu. of shelled com (6Q lb. each), what will the car and its load weigh ? 7. What is the estimated number of words in a book con- tainilig 240 pages, each page averaging 350 words ? 8. A had 125 bu. of corn ; B had twice as much and 26 bu. more; C had as much as both A and E; D had as much as the difference between A's and B's quantities. The corn was sold at 40 cents a bushel. What did it bring ? 9. I bought 125 A. of land at $60 an acre. I spent $1250 to fence it. At what price an acre must I sell it to gain f 1000 ? 10. If 12 men can do a piece of work in 15 da., in how many days can 20 men do it ? U. Sixteen men undertook a piece of work that would take them 24 da. ; when it was half done 10 men left. In how many days could those remaining finish it ? 12.. A farmer bought 20 lb. of sugar at 6 cents a pounds 10 yd. of cloth at 75, cents a yard ; 2 lb. of tea at 50 cents a pound ; 15 lb. of coffee at 20 cents a pound. He sold the mer- chant an equal number of bushels of potatoes and apples, getting 40 cents a bushel for the potatoes, and 60 cents a bushel foi the apples, and received 70 cents in change. How many bushels of each did he sell ? 13. A merchant bought 48 yd. of cloth at 62 cents a yard, and 76 yd. at 75 cents a yard. He sold the former at 81 cents Digitized by VjOOQIC 16 THE FUNDAMENTAL OPERATIONS. a yard, and the latter at such a price as to gain $ 20.37 on the whole transaction. At what price per yard did he sell the cloth? 14. A number is increased by 185, the sum divided by 10, the quotient increased by 8, and the sum multiplied by 5 ; the result is 240. What is the number ? THE LAW OF SIGNS. -- 45. In an expression which contains the various signs of operation, +, — , x, and -3-, the operations of x and -3- are to be performed before the operations of + and — . 12-6x2 + 30-5-5 = 12-10 + 6 = 8. 46. Any succession of x and -f- designates operations which are to be performed from left to right in thie order of the signs. 12 X 6-^8 -^-3x5 = 72-^8-^3x5 = 9-^3x 5 = 3x5 = 15. 47. Any succession of + and — designates operations which are to be performed from left to right in the order of the signs. 18-8 + 9-7 = 10 + 9-7 = 19-7 = 12. 48. By means of one of the principles of addition and subtraction, we may bring all the +'s to the first part of the problem and all the — 's to the last part. 18-8 + 9-7 = 18 + 9-8-7. 49. The brackets [], parentheses (), braces { }, and vinculum are called symbols of aggregation. Inclosed expressions are called bracketed expressions. Operations within the bracketed expressions must be performed first. 12-[3 + 2] + 18-^(3 + 6) = 12-5 + 18^9 = 12-6 + 2 = 9. Digitized by VjOOQIC THE LAW OF SIGNS. 17 EXERCISE 9- \ 1. 17-^x2 + 36-f.3x5-42^6=P/ "/ -^ X ^ -l'34-;3 ^h 2. 63-^7x>-28x9-^8,+ 18)f4-^9===?B^ 3. 51^l)'xl&)-30xV-^U+18H-a]<'l23-16H-8 = ? ((^ 4. (8-2T3)X(6 + 7-9) = ? 5. 4x[6-{11-(5h-3)}+2] = ? 6. 19 + 3x3-64^8 + 6x4 = ? 7. [30^5x2 + 9x2]-^[10 + 60-^12] = ? a [60 -^ 6 + 4 X 3] -f- [16 - 4] X [90 x 2 ^ 3 + 40] = ? 9. (86,429 - 4786) - (7612 - 482) = ? 10. 86,429 - 4786 - 7612 - 482 = ? 11. The divisor is 879, the quotient 46, and the remainder 23. What is the dividend ? 12. The remainder is 279, and the subtrahend 673. What is the minuend ? 13. The product is 4212, and the multiplier 78. What is the multiplicand ? 14. The minuend is 964, and the remainder 278. What is the subtrahend ? 15. The product is 196, and the multiplicand 39. What is the multij)lier ? In the following problems a = ^, 6 = 4, c=^Qy and d = 10. ah n>eans a x 6 and is 6 x 4 = 20. 3 c means 3 x c and is 3 x 6 = 18. ta + ^)c=(6 + 4)6 = 9x6 = 64. 16. ab + c=^? 19. (a + c)(6 + d)=? 17. abc^d^? 20. (a6 + c(f)-^10=? 18. {a'\-c)d='^ 21. od -^ (6 + c) = ? 22. 3a6-j-(c + d + 4) = ? 23. a5 + 6c + cd— (4a + 66) = ? 24. (ac + 2)^16 + (M + 8)-^c = ? 25. abed -^3x4x5 = ? Digitized by VjOOQIC 18 THE FUNDAMENTAL OPERATIONS. 26. 3a + d — 6c = ? 27. (6 + c)a-^d-|-3c-46 = ? 28. c(a + cO-*-(^ + <^) + <^-3c = ? 29. (a6 + c(i)-t-(6 + c-|-d) — 6 + ff6c = ? 30. (a6c-12d + cd)H-(&cd-3cd) = ? EXERCISE 10. (Mental.) 1. 4 times 6 are how many times 8 ? 2. 6 times 8 are how many times 12 ? 3. 4 times 14 are how many times 8 ? 4. 6 times 12 are how many times 16 ? 5. 8 times 9 are how many times 4 ? 6. 9 times 12 are how many times 6 times 3 ? 7. 12 times 8 are how many times 4 times 6 ? a 5 times 8 plus 4 times 8 are how many times 4 times 2? 9. 3 times 16 plus 4 times 6 are how many times 2 times 9? 10. 4 times 16 less 3 times 4 are how many times 13 ? 11. 6 times 13 plus 2 times 6 are how many times 25 ? 12. 4 times 17 plus 3 times 9 are how many times 19 ? 13. 4 times 19 plus 2 times 7 less 3 times 17 are how many times 13 ? 14. 6 times 16 are how many times 10 ? 15. 4 times 16 plus 6 are how many times 7 ? 16. If 2 oranges cost 10 cents, what will 7 oranges cost? 17. If 9 yd. of muslin cost 108 cents, what will 7 yd. cost ? la If 11 books cost $4.40, what will 6 books cost ? 19. If a man travels 72 mi. in 4 hr., how many miles will he travel in 5 hr., going at the same rate ? 20. If 12 A. of oats yield 480 bu., how many bushels will 17 A. yield at the same average ? 21. If $24 buys 12 yd. of cloth, how many yards will $36 buy at the same price ? Digitized by VjOOQIC COMPOUND NUMBERS. 19 22. If 11 cd. of wood cost $44, how many cords can be bought for $72 at the same price? for $84? for $92? for $68? 23. If 7 men can do a piece of work in 10 da., in how many days can 2 men, working at the same rate, do the work ? 5 men ? 14 men ? 24. If 8 men can do a piece of work in 12 da., how many men would be required to do the same work in 6 da. ? in 8 da. ? in 16 da. ? in 24 da. ? 25. Sold 3 doz. eggs at 12 cents and 2 lb. of butter at 24 cents. What is the change out of a dollar ? 26. Sold 5 articles at 10 cents, 2 articles at 12 cents, and 2 at 10 cents. Find the change out of a dollar. COBIPOUND iniMBERS. 50. A simple number is one in which the scale is uni- form. In our system this scale is 10. 51. A compound number is one in which the scale is not uniform. 62. 3 6w. 3 pk. 7 g«. 1 p^. is a compound number. The sim- ple number 3371 may be written 3 thousaiids 3 hundreds 7 tens 1 unit In the simple number 10 units make a ten, 10 tens a hundred, and 10 hundreds a thousand. In the above compound number 2 pt. make a quart, 8 qt. a peck, and 4 pk. a bushel. The scales, read from left to right, are 10-10-10 and 4-8-2 respectively. 53. 3371 may be changed to units thus : 3 thousands = 10 hundreds x 3 = 30 hundreds. 30 hundreds + 3 hundreds = 33 hundreds. 33 hundreds = 10 tens x 33 = 330 tens. 330 tens 4- 7 tens = 337 tens. 337 tens = 10 units x 337 = 3370 units. 3370 units + 1 unit = 3371 units. Digitized by VjOOQIC 20 THE FUNDAMENTAL OPERATIONS, 3 bu. 3 pk. 7 qt. 1 pt. may be changed to pints thus : 3bu. = 4pk. x3 = 12pk. . 12pk.-|-3pk. = 15pk. 16pk.=8qt. xl5 = 120qt. 120qt.-|-7qt. = 127qt. 127 qt. = 2 pt. X 127 = 254 pt. 254 pt. + 1 pt. = 255 pt. The above operations are similai* and are based on precisely the same principles. In changing the simple number the multi- plicand is always 10, because that is the uniform scale. In changing the compound number the multiplicands are 4, 8, and 2 in order, because these numbers are the variable scale. EXERCISE II. (Mental.) Note. The compound number tables are given in the Appendix. Those not already known should be learned. 1. In 3 ft. 5 in. what is the scale ? 2. Give the scale in 3 gal. 3 qt. 1 pt. 1 gi. 3. Give the scale in 8 rd. 2 yd. 2 ft. 11 in. ^. Give the scale in 25 cu. yd. 15 cu. ft. 76 cu. in. 5. How many pecks in 5 bu. 3 pk. ? 6. How many feet in 8 yd. 2 ft. ? 7. How many quarts in 10 gal. 3 qt. ? 8. How many inches in 5 ft. 11 in. ? 9. Eeduce 5 hr. 32 min. to minutes. 10. Reduce 5 da. 8 hr. to hours. U. Eeduce 2 lb. 8 oz. to ounces. 12. Reduce 6° 36' to minutes. 13. Reduce 7 pk. 3 qt. to quarts. 14. How many gills are there in 3 gal. 3 qt. ? 15. How many yards are there in 4 rd. 3 yd. ? Digitized by VjOOQIC COMPOUND NUMBERS. 21 EXERCISE 12. 1. Reduce 18 gaJ. 3 qt. 1 pt. to pints. 2. How many hours axe there in January ? 3. How many seconds in 21** 3r 22" ? 4. How many square feet in 4 sq. rd. 20 sq. yd. 8 sq. ft. ? 5. How many feet in 40 rd. ? 6. How many cubic feet in 73. cd. 6 cd. ft ? 7. How many ounces in 3 T. ? 8. Eeduce 17 da. 17 hr. 17 min. to minutes. 54. 389 units may be changed to higher denominations, thus : 389 units = ^ tens = 38 tens 9 units. 38 tens = f| hundreds = 3 hundreds 8 tens. 389 units = 3 hundreds 8 tens 9 units. 389 in. may be changed to higher denominations, thus : 389 in. = ^ ft. = 32 ft. 5 in. 32 ft. = ^ yd. = 10 yd. 2 ft. 389 in. = 10 yd. 2 ft. 6 in. These two processes are similar. They differ only in the divisors used. In making the reductions in the simple num- ber the divisor is always 10, because that is the scale. In making the reductions of the compound number the divisors are 12 and 3, because in this particular number the scale is 12,3. EXERCISE 13. 1. Reduce 341 pt. Dry Measure to higher denominations. • 2. Reduce 341 pt. Liquid Measure to higher denominations. 3. Reduce 1289 oz. to higher denominations. 4. How many weeks, days, and hours are there in 1365 hr. ? 5. Reduce 12,567" to degrees, minutes, and seconds. 6. Reduce 1867 rd. to miles and rods. Digitized by VjOOQIC 22 THE FUNDAMENTAL OPERATIONS. COBIPOUND ADDITION AND SUBTRACTION. 55. Add: (1) (2) bu. pk. qt. 337 3 3 7 236 2 3 6 420 4 2 734 7 3 4 1727 19 1 1 Explanation. The first (right hand) column in (1) when added makes 17 units, which equals fj tens, or 1 ten and 7 units. We write the 7 units and add the 1 ten to the second column. The second column makes 12 tens, which equals \^ hundreds, or 1 hundred 2 tens. We write the 2 tens and add the 1 hun- dred to the third column. The third column makes 17 hun- dreds, which we write. In (2) the first column makes 17 qt., which equals ^ pL, or 2 pk. 1 qt. We write the 1 qt. and add the 2 pk. to the second column. The second column makes 13 pk., which equals ^ bu., or 3 bu. 1 pk. We write the 1 pk. and add the 3 bu. to the third column. The third column makes 19 bu., which we write. 56. These problems show that the addition of compound numbers is very similar to that of simple numbers. In addi- tion of simple numbers we carry one for every 10, because 10 is always the scale. In addition of compound numbers the carrying is likewise determined by the scale. In the first column of (2) we carried one for every 8, and in the second column one for every 4, because these numbers make the scale for this problem. 57. From 7 yd. 2 ft. 1 in. take 3 yd. 2 ft. 9 in. yd. ft. in. 7 2 1 3 2 9 3 2 4 Digitized by VjOOQIC EXERCISE 14. T. lb. oz. rd. yd. ft. 4 860 5 2. Add: 20 4 2 6 980 11 17 3 1 7 1030 14 19 5 2 8 670 10 27 4 1 COMPOUND ADDITION AND SUBTRACTION 23 Explanation. The scale connecting inches and feet is 12, hence we say, 9 from 13 (= 12 -f 1) leaves 4. The scale con- necting feet and yards is 3, hence we say, 2 from 4 (=3 + 1) leaves 2. We have already used 1 out of the 2 in making the subtraction in the first column. In the third column we have, 3 from 6 leaves 3. This is seen to be very similar to subtraction of simple numbers. The only difference is that the variable scale must be kept in mind and properly used. 1. Add: bn. pk. qt. pt. da. hr. min. sec. 3. Subtract: 19 3 1 4. Subtract: 27 19 15 30 13 2 5 1 10 23 45 50 5. A man was born July 19, 1836. How old was he Jan. 1, 1904? Since January is the first month and July the seventh, the problem is to take 1836 yr. 7 mo. 19 da. from 1904 yr. 1 mo. 1 da. 6. A note dated May 4, 1899, was paid June 19, 1902. How long did it run ? 7. By using the date of your birth and to-day, find your exact age. a New York is in 74® 3" west longitude, and Boston IV 3' 30" west longitude. Boston is how far east of New York ? 9. Chicago is in 87® 36' west longitude. How far is it west of New York ? of Boston ? 10. Albany is 298 mi. east of Buffalo, and Chicago is 589 mi. west of Buffaio. What is the distance from Albany to Chicago ? Digitized by VjOOQIC 24 THE FUNDAMENTAL OPERATIONS. U. Berlin is 13** 23' 43" E., and New Orleans 90° 3' 28' W. What is the difference of their longitudes ? 12. Boston is in 42** 21' 24" N. latitude. The latitude of New York is 40° 42' 43" N. Boston is how much farther north than New York ? 13. London is in latitude 61° 30' 48" N. It is how much farther north than Boston ? 14. New Orleans is 29° 67' N., and Eio Janeiro 22° 64' S. What is their difference in latitude ? 15. rind the time from July 6, 1896, to Sept 10, 1903. 16. Find the time from March 12, 1896, to Oct. 18, 1902. 17. Find the time from June 16, 1894, to April 6, 1904. la Find the time from Aug. 21, 1897, to May 16, 1904. 19. Find the time from Sept. 12, 1898, to Dec. 25, 1904. 20. Find the time from Oct. 28, 1893, to June 19, 1902. 21. From 2 sec. 612 A. 73 sq. rd. take 1 sec. 638 A. 96 sq. rd. 2^ sq. yd. /22. Find the difference between 16 cu. yd. 18 cu. ft. 1276 cu. in. and 7 cu. yd. 23 cu. ft. 1628 cu. in. 23. Bought 600 lb. of sugar. Sold 124 lb. 6 oz., 73 lb. 13 oz., 48 lb. 9 oz., l73 lb. 14 oz. How much was left ? 24. Bought 624 cd. of wood. Sold 76 cd. 7 cd. ft., 116 cd. 14 cu. ft., 124 cd. 6 cd. ft. 12 cu. ft., 283 cd. 4 cd. ft. 10 cu. ft.. How much was left ? 25. What is the difference between 8 lb. Apothecaries' weight and 6 lb. 7 oz. 4 dr. 2 sc. 16 gr.? 26. A is in longitude 124° 42' 36" E., and B is 67° 49' 24" E. What is their difference in longitude ? 27. From a cask containing 38 gal. the following amounts were drawn : 4 gal. 3 qt. 1 pt., 7 gal. 2 qt., 12 gal. 1 qt. 1 pt., 8 gal. 3 qt. 1 pt. How much was left in the cask ? 28. Find the difference in time between Jan. 21, 1896, and July 28, 1848; between Aug. 12, 1876, and May 10, 1890. Digitized by VjOOQIC COMPOUND MULTIPLICATION AND DIVISION, 25 29. From 12 lb. 9 pz. 7 pwt. 11 gr. take 7 lb. 10 oz. 15 pwt 18 gr. r 30. At what date will jou be 26 yr. 4 mo. 18 da. old ? 31. At what date will\the United States be 150 yr. 9 mo. 26 da. old? COBIPOUND MULTIPLICATION AND DIVISION. 58. Multiply 3 gal. 3 qt. 1 pt. by 9. gal. qt. pt. 3 3 1 9 34 3 1 Explanation. 1 pt. x 9 = 9 pt. 9 pt. = 4 qt. 1 pt. Write the 1 pt. in the result and add the 4 qt. to the next partial product. 3 qt. X 9 = 27 qt. 27 qt. -f 4 qt. = 31 qt. 31 qt. = 7 gal. 3 qt. Write the 3 qt. in the result and add the 7 gal. to the next partial product. 3 gal. x 9 = 27 gal. 27 gal. + 7 gal. = 34 gal. Hence the result of the multiplication is 34 gal. 3 qt. 1 pt. It should be noticed that this multiplication proceeds exactly as simple multiplication. The only difference is one of scale. Instead of the uniform scale 10 of simple numbers, we have the variable scale of whatever compound number we happen to be considering. 59. Divide 34 gal. 3 qt. 1 pt. by 3. gal. qt. pt. 3 )34 3 1 11 2 1 Explanation. 34 gal. -f- 3 = 11 gal. and a remainder of 1 gal. Write the 11 gal. in the result and unite the 1 gal. with the 3 qt. 1 gal. = 4 qt. 4 qt. + 3 qt. = 7 qt. 7 qt. -5- 3 = 2 qt. and a remainder of 1 qt. Write the 2 qt. in the result and unite the 1 qt. with the 1 pt. 1 qt. = 2 pt. 2 pt. + 1 pt. = 3 pt Digitized by VjOOQIC 26 THE FUNDAMENTAL OPERATIONS. 3 pt. -!- 3 = 1 pt. Hence the result of the division is 11 gal. 2 qt. 1 pt. Here again it is seen that the only difference between com- pound and simple division is in the scale. EXERCISE 15. 1. Multiply 3 gal. 2 qt. 1 pt. by 5 ; by 8 ; by 9 ; by 12. 2. Multiply 7 yd. 2 ft. 8 in. by 7; by 10; by 11. 3. Multiply 46 bu. 3 pk. by 15; by 24; by 38 ; by 49. 4. 1 bu. 5 qt. 1 pt. of oats is divided equally among 15 horses. How much does each receive ? 5. 15 bu. 3 pk. 4 qt. of oats were divided into 4 equal piles. What amount was there in each pile ? 6. Divide 24 gal. 3 qt. 1 pt. 2 gi. of vinegar into 7 equal parts. What is the amount in each part ? 7. If 36 men work an average of 12 da. 7 hr. 30 min. each, what is the total amount of time they all work ? 8. A ship sails from New York, longitude 74° 0' 3" W., and makes an average daily easting of 9® 24' 36''. What is her longitude at the end of 7 da. ? 9. Twenty-seven cans hold 10 gal. 3 qt. 1 pt. each. How much do they all contain ? 10. How much wheat is there in 25 loads, each containing 36 bu. 3 pk. ? 11. Sold 28 loads of oats, each containing 74 bu. 3 pk., at 22 cents a bushel. What was the amount received ? 12. Multiply 15 hr. 24 min. 38 sec. by 42. 13. Multiply 16° 17' 22" by 76. 14. Multiply 128 lb. 7 oz. by 56. 15. Bought 160 A. of land at $65 an acre, and 80 A. at $75 an acre. Sold 120 A. at $80 an acre and the remainder at $66.50 an acre. What was the gain or loss ? Digitized by VjOOQIC EXTENSION OF ADDITION AND SUBTRACTION. 27 EXTENSION OF ADDITION AND SUBTRACTION. 60. The live stock on three farms is as follows : On A's farm 29 horses, 43 cattle, 211 sheep ; on B's, 18 horses, 67 cattle, 510 sheep ; 6n C's, 22 horses, 61 cattle, 425 sheep. How many horses, cattle, and sheep are there on the farms ? horses cattle sheep 29 43 211 18 67 510 22 51 425 69 161 1146 61. It will be noticed that three columns are needed, corre- sponding to the three kinds of live stock named. If the letters h., c, and s. are used instead of the words horses, cattle, and sheep, then 29 h. + 43 c. + 211 s. represents the stock on A's farm, 18 h. + 67 c. -f 510 s. that on B's farm, and 22 h. + 51 c. + 425 s. that on C's farm. The problem then might be written as follows: 29 h. 4- 43 c. -h 211s. 18h.+ 67 c. + 510 s. 22h.-|- 51c. + 425 s. 69h. + 161c. + 1146 8. EXERCISE 16. 1. Add 33 h. + 61 c. + 67 s., 51 h. + 73 c. +98 s., 27 h. +31 c. + 121 s., and 16 h. + 70 c. + 500 s. 2. Add 11 h. + 19 c. + 217 s., 13 h. + 431 s., 35 h. + 17 c, and 27 c. + 243 s. 3. From 77 h. + 96 c. + 500 s. take 39 h. + 43 c. + 273 s. 4. From 91 h. + 370 c. + 900 s. take 73 h. + 821 s. In the following problems the same principles apply, but the letters will not be considered as abbreviations. Digitized by VjOOQIC 28 THE FUNDAMENTAL OPERATIONS. 5. Add 7aj-|-9y + 32, 9aj + 42; + llw, 17aj + 10y + 43 + 3 tu, and 9 a? -h 4 w. Arrange for additiou thus : 17x + 10y + 42f+ 3ti7 9a; + 4w 6. Add6a + 46 + llc,15a + 13c + 6d, 1164-17c + 13d, andl6aH-106H-6d. 7. From 13a + 196 + 17c take 9a + 36 + 15c. a From 11 a? -1-13 2^-1- 10 intake 8 a; + 92;. 9. Add lla; + 7y + 102, 9aj + 52^ + 32;, 7a;+132;, lly+92, and 21 z. 10. From 13a-|-96 + 17c take 9a + llc. U. Simplify, that is, unite all the terms that can be united : 82/ + 9ajH-72;H-13 2 + lla?H-62^ + 62^H-7a; + 5a?-|-122;. 12. Simplify 18a + 10c + 5a + 136 + 12c + 9aH-6 6. 13. Add 9 »y -h 11 a6, 11 xy-\-6ah, and 13a^ + 21a6. 14. 8a?H-llaj-9a;H-3aj — 10a?= ? 15. 13a-6aH-27a-30a + 3a=? 16. 152^ + 202^-27y + 9y-82/ = ? 17. Simplify 8a;H-7y-6a; — 3y-|-9aj — 2y. 18. Simplify 18 a + 17 6 + 7 a-10 6-20a. 19. Simplify 21a; + 7y + 32; — 4y — 9a; + 102. 20. Simplify 11 a^ + 13 a6 — 5 a^ + 6 a6 — 12 a6. THE AREA OF A RECTANGLE. 62. A figure like the top or the side of a chalk box is called a Rectangle. 63. The figure ABOB is a rectangle. AB or BO is its length, and AB or BOis its width. Digitized by VjOOQIC THE AREA OF A RECTANGLE, 29 It will be noticed that this rectangle is made up of 8 rows of 13 squares each, or of 13 rows ^ of 8 squares each, and therefore contains 8 x 13 or 104 squares. If each square is 1 in. long, the rectangle is said to be 8 in. by 13 in., and it contains 104 square inches. If each square is 1 ft. long, the a \ rectangle contains 104 square feet. ^ Rectanqlb. 64. The number of square units in a rectangle is called its area. From the above it is evident that the area of a rectangle is equal to the product of its length and width. The length and width are called the two dimensions of the rectangle. 65. The area of a rectangle is the product of its two dimensions. The two dimensions must be expressed in the same units, and the area is square units of the same kind. EXERCISE 17. (Mental.) 1. A floor is 10 X 15 ft. What is its area? Note. 10 X 15 ft. is read " 10 by 16 ft. ," and means that the dimensions are 10 ft. and 15 ft. respectively, 2. What is the area of a floor 30 x 100 ft. ? 3. The floor of a room 20 ft. long contains 300 sq. ft. How wide is the room ? 4. How many square feet are there in a flower bed 12 Xl6 ft.? 5. A porch contains 240 sq. ft. and is 8 ft. wide. How long is it? 6. A city lot 60 ft. front contains 6000 sq. ft. How deep (long) is the lot? 7. A room is 15 x 20 ft. What is the area of the floor? Digitized by VjOOQIC 30 THE FUNDAMENTAL OPERATIONS. 8. What is the combined length of the walls of the above room? 9. If the above room is 10 ft. high, how many square feet are there in its walls? 10. A flower bed containing 10 sq. yd. is 15 ft. long. How wide is it? 11. How far is it around a city lot 40 x 120 ft.? 12. What is the area of the above lot? THE VOLUME OF A RECTANGtHAR SOLID. 66. A solid in the shape of a chalk box, shoe box, or brick is a Rectangular Solid. Each of its sides is a rec- tangle. It has length, width, and height. These are called its three dimensions. yyyyyyyyy > ^-^-^>">">^>^ ^ zzz: O 'yyy y y y y z: A IlKCTANGULAB SOLID. 67. The solid ^5 aZ> is a rectangular solid. AB is the length, BQ the width, and CZ> the height. The length of this solid is 9, the width 5, and the height 4. It is seen that if the solid were cut along the lines marked, it would be divided into a number of cubes. The solid is made up of 4 layers of 45 cubes each, for in a single layer there are 9 rows of 5 cubes each, or 45 cubes. The tota,l num- ber of small cubes in the solid is 45 x 4, or 180 cubes. 68. The number of equal cubes in a solid is called its Volume or Solid Contents. In the above solid, if each cube is 1 in. long, then the solid contains 180 cubic inches. 69. The volume of a rectangular solid is the product of its three dimensions. The dimensions must be expressed in like units, and the volume is cubic units of the same kind. Digitized by VjOOQIC THE VOLUME OF A RECTANGULAR SOUIK 31 EXERCISE la (MentaL) 1. An ovdmaiy brick is 2 x 4 x 8 in. What is its Tolaine? 2. What is the Tolnme of a box 4 x 6 x 10 in. ? 3. Howhighmii8ta4xlOin.boxbetoo(mtatnS20cQ«in.? 4. A box is 6 in. square. How deep must it be to contain 360 cu. in. ? 5. What aie the scdid ocmtents of a piece of timber 6 x 10 xSOin.? EXERCISE 19. 1. How many acres are there in a field 60 x 48 rd.? 2. Find the volume of a box 51 x 33 x 80 in. a A field containing 35 acres (acre=160 sq. rd.) is 112 rd. long. How wide is it ? 4. A cellar is 18 x 20 ft It contains 120 cu. yd. How deep is it ? 5. How many cubic yards of air are there in a schoolroom 36 X 42 X 16 ft ? 6. In a certain window, each pane is 14 in. wide and con- tains 392 sq. in. How long is each pane ? 7. What is the width of the desk-top upon which I write ? Its area is 999 sq. in., and its length 37 in. a The floor of a room is 26 ft. wide and contains 806 sq. ft. How long is it ? How many square yards of linoleum must I buy to cover it ? 9. A 40- A. field is 1320 ft square. How wide a strip along one side contains an acre (43,560 sq. ft.)? How many corn rows, 3 ft. 8 in. apart, can be planted in this strip the long way? 10. A man bought a piece of land 40 rd. long and 24 rd. wide. He bought a second piece containing as many square rods but measuring 30 rd. wide. What was its length ? He paid f 72.50 an acre (160 sq. rd.) for the two pieces. What did they cost ? Digitized by VjOOQIC 32 THE FUNDAMENTAL OPERATIONS. 11. A town lot having a depth of 161 ft. and containing 19,642 sq. ft. sold for $ 1952. What was the price per front foot? Note. The depth of a lot is the distance it extends back from the street. A foot front is a strip 1 ft. wide fronting on the street and extend- ing back the full depth of the lot. 12. A drawer is 21 x 10 x 4 in. What is its capacity ? 13. A drawer contains 1680 cu. in. Its bottom is 21 x 10 in. How many cubic inches are needed to cover the bottom 1 in. deep ? How many such layers will fill the drawer ? What is the depth of the drawer ? ^-^4. 231 cu. in. equal 1 gal. A wagon tank is 10 ft. long and 33 in. wide, inside measure. 360 gal. fill it to what depth ? 15. What is the capacity of my schoolroom, 30 x 24 x 15 ft. ? 16. A cubic foot of air at 70** Fahrenheit weighs 525 gr. How many grains of air are there in the above schoolroom ? 17. 7000 gr. equal 1 lb. How many pounds of air are there in your schoolroom ? 18. The coined gold of the world is estimated at 10,800 cu. ft. To what height will it reach, if piled uniformly so as to cover your schoolroom floor ? J.9. What is the volume of a pine sill 8 x 12 in. x 24 ft. ? Note. In describing timbers the dimensions are given in this order : thickness, width, length. 20. Dry pine weighs 29 lb. to the cubic foot. What is the weight of the sill ? 21. How many men are needed to carry it, each man being able to lift 232 lb. ? 22. Joliet limestone weighs 160 lb. to the cubic foot. What is the weight of a stone step 6 in. x 3 ft. x 12 ft. ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS, 38 MISOELLANEOUS PROBLEMS. EXERCISE 20. (Mental.) 1. Mr. Jones paid $68 for one horse and $ 100 for another; he sold them both for $ 187. What was his gain ? 2. A boy shot an arrow 80 ft. up the road, and another 90 ft. down the road. His brother brought them to him. How far did he walk ? 3. A merchant sold silk at $ 2.50 a yard. He lowers the price 90 cents a yard, and sells at 20 cents above cost. What does the silk cost him a yard ? 4. 17x8x2x5 = ? 5. (17 + 8)-*-(27-22) + 8x5^-4 = ? 6. 29 + 19+21 + 6 = ? 7. 8aj + 9aj-llaj + 27aj=? 8. 10 + 3x4-6x5-^ 10-15 = ? 9. Find the perimeter of a room 8 yd. long and 20 ft. wide. Note, The perimeter of anything is the distance around it. 10. K the above room is 10 ft. high, find the area of the two end walls. 11. Find the area of the two side walls in the above room. 12. A, traveling 8 mi. an hour, and B, 7 mi. an hour, met in 10 hr. How far apart were they at first ? 13. 144 is the product of 9 and what other number ? 14. What number divided by 9 gives a quotient of 16 with a remainder of 6 ? 15. The sum of two numbers is 27 ; the smaller is 13. What is 4 times the larger ? 16. The difference between two numbers is 9. If the larger is 13, what is 12 times the smaller ? 17. 26x5x3x4 = ? Digitized by VjOOQIC 34 THE FUNDAMENTAL OPERATIONS. 18. What is 4 times the difference between 19 and the sum of 7 and 3? 19. Albert has 6 times 2 marbles less than 52, and Edward has 7 times 2 marbles more than 48. How many has each ? 20. John has 8 times 2 marbles more than both Albert and Edward. How many has John ? 21. 36^9x54-5-15x4-1-20 = ? 22. I have 4 times as many marbles as the sum of 5 and 3 is contained times in 64. How many have I ? 23. If you have 5 times as many marbles as I have, how- many have you ? 24. If a man gains 10 mi in 5 hr., how long will it take him to gain 30 miles ? 25. I have a number in mind which, divided by 5, gives 2 times 15. What is the number ? 26. If 72 is divided by some number, the result is 4. What is the number ? 27. 13a;-5a;-|-24aj-17aj = ? 28. 21y-|-9y-14y-8y=? 29. (3x5 + 4x15)h-(15-2x5) = ? 30. 18^24-(27-3)^3-|-13 = ? 31. How many feet are there in 9 yd. 2 ft. ? 32. How many ounces in 5 lb. 15 oz. ? 33. How many pints in 3 gal. 2 qt. ? 34. Eeduce 9 bu. 3 pk. to pecks. 35. How many minutes in 5 hr. 38 min. ? 36. If a horse travels 6 mi. an hour, how far will he travel in 3 da. of 10 hr. each ? 37. An automobile averages 15 mi. an hour. How far will it go in 6 da. of 12 hr. each ? 3a An automobile makes 800 mi. in 5 da. of 10 hr. each. What is the average rate per hour ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 35 39. A train runs from Chicago to Louisville, a distance of 320 mi., in 10 hr. What is the rate per hour ? 40. If this distance could be made in 8 hr., how much per hour would the train increase its rate ? 41. If it takes a train 6 hr. to run from Indianapolis to St. Louis at the rate in Problem 40, what is the distance between the cities ? 42. Find the cost of 5 bags of coffee of 72 lb. each, at 20 cents a pound. 43. A farmer bought 36 bags of clover seed, each containing 2 bu. What did it cost at $ 4 a bushel ? 44. 200-8x15 + 12x6 = ? 45. 13 + 17x4-29 + 11x3=? 46. A farmer sold 6 cows at $35 each, and 4 horses at $125 each. What did they bring him ? 47. I rode in a railway train 5 hr. at the rate of 32 mi. an hour, and then in a carriage 4 hr. at the rate of 8 mi. an hour. How far did I ride ? 4a When oranges cost 3 cents each, a boy gave 60 cents for 10 oranges and 15 apples. What did the apples cost apiece ? 49. A woman exchanged 10 qt. of berries at 7 cents a quart, and 40 cents in money, for 22 lb. of sugar. What did the sugar cost a pound ? 50. What is the cost of 14 hats and 10 caps, if hats are $3 each, and caps are worth half as much as hats ? 51. A man has 72 bu. of potatoes, and spends f 40 in buying more potatoes, at 50 cents per bushel. How many bushels does he then have ? 52. In each of 5 schoolrooms there is an average of 48 pupils ; there are 131 boys in the 5 rooms. How many girls are there ? 53. 17a + 33a-16a + 26a = ? 54. 736-236 + 196 + 76-466 = ? 55. (26-4 + 8) x5 + 12x9 + 4-300-i-4=»? Digitized by VjOOQIC 36 THE FUNDAMENTAL OPERATIONS. 56. (31-19 + 8) x42-7x5-360^9 = ? 57. (15x74-10x8-60-45 + 40)^(2x5) = ? 58. What will the answer to Problem 57 be, if the parenthesis about 2 X 5 is removed ? 59. What will the answer to Problem 57 be, if both paren- theses are removed ? 60. A man buys 12 head of cattle ; 4 of them cost $30 each, 5 of them f 32 each, and the remainder $ 40 each. How much do they all cost ? 61. A man receives $ 2 a day, and his boy half as much. What will both earn in 5 wk. ? 62. At the rates of the above problem, what will 5 men and 10 boys earn in 8 da. ? 63. The front wheel of a carriage is 8 ft. in circumference. How many revolutions will it make in going 1 mi. ? >^'64. If the rear wheel of the above carriage is 10 ft. in cir- cumference, how many more revolutions will the front wheel make than the rear wheel in going a mile ? 65. A man sold coal for $ 450, and gained 50 cents a ton ; the coal cost him $ 390. How many tons did he sell ? 66. In the above problem, what was the cost of the coal a ton? 67. Reduce 61 pk. to bushels and pecks. 6a Reduce 279 min. to hours and minutes. 69. How many days are there in March, April, and May ? 70. How many days are there from July 10 to August 20 ? 71. Reduce 86 liquid pints to higher denominations. 72. Find the sum of 3 score and 6 doz. 73. One half a gross plus 4 doz. minus 5 score is how much less than 49 ? 74. 37aj + 18aj-45aj4-32a;-19aj = ? 75. 46w;-21w7 + 15w;-(38w74-16m7 — 15t(;) = ? 76. 36M?-tl00w;-(63w-|-17w;) + 12w;]=»? Digitized by VjOOQIC CHAPTER III. FACTORS, DIVISORS, AND MULTIPLES. DEFINITIONS. 70. Numbers are integral^ fractional^ or mixed. 71. An Integral Number, or Whole Number, is a number of whole units. 72. Integral numbers are classified as composite^ prime^ even^ and odd. 73. A Factor of a number is one of the two or more integral numbers which, being multiplied together, will produce that number. It is consequently a Divisor of that number. 74. A Common Factor of two or more numbers is a factor which occurs in each of them. 3 is a common factor of 6, 12, and 15. 75. A Composite Number is a number that has integral factors besides itself and 1. 4, 10, 35, are composite numbers, because 4=2 x 2, 10=2 x 5, and 35 = 5 X 7. 76. A Prime Number is a number that has no integral factors except itself and 1. 1, 2, 3, 5 are prime numbers. 77. All numbers have 1 for a factor. When 1 is the only common factor of two numbers, they are said to be 37 Digitized by VjOOQIC 38 FACTORS, DIVISORS, AND MULTIPLES, prime to each other. Numbers which are prime to each other are not themselves necessarily prime numbers. 9 and 49 are prime to each other, but each is itself a com- posite number. 78. An Even Number is a number that contains 2 as a factor. 4, 6, 8, 10, etc., are even numbers. 79. An Odd Number is a number that does not contain 2 as a factor. 3, 5, 7, 9, etc., are odd numbers. Note, Readiness in factoring depends upon knowing certain properties of numbers, hence they should be carefuUy noted. PRINCIPLES. 80. Prin. 1. A factor of a number is a factor of any numher of times that number. This is obvious, since every time the number is repeated the factor is repeated. 5 is a factor of 10. It must then be a factor of any number of lO's. 81. Prin. 2. A common factor of two numbers is a factor of their sum. Since each number is some number of times the common factor, their sum must be some number of times the common factor. 3 is a factor of 6 and of 9. It must then be a factor of 16. For 6 is two 3*s and 9 is three 3's, hence the sum of 6 and 9 is five 3's. 82. Prin. 3. A common divisor of two numbers is a divisor of their difference. Since each is some number of times the common divisor, if they differ it is because one of them contains the common Digitized by VjOOQIC TESTS OF DIVISIBILITY. divisor more times than the other; hence their difference divisible by the common divisor. If 5 is a common factor of two numbers, as 15 and 25, each must be composed of 5's. If one exceeds the other, it must be because it has more 5's ; hence the difference must be 5's. TESTS OF DIVISIBILITY. 83. Any nuvnher is divisible by 2 if its right-hand figure is 2, 4, 6, 8, or 0. 84. Any number is divisible by 3 if the sum of its digits is divisible by 3. 263,457 is divisible by 3 because the sum of 2, 6, 3, 4, 5, and 7 is divisible by 3. 85. A number is divisible by 4 if the number expressed by the two right-hand digits is divisible by 4. 39,484 is divisible by 4 because 84 is divisible by 4. 86. A number is divisible by 5 if the right-hand digit is or 6. 87. Any number is divisible by 6 if divisible by 2 and 3. 27,684 is divisible by 6 because it is even, and, therefore, divisible by 2, and the sum of its digits is divisible by 3. • 88. A number is divisible by 8 if the number expressed by the three right-hand digits is divisible by 8. 6,252,144 is divisible by 8 because 144 is divisible by 8. 89. A number is divisible by 9 if the sum of its digits is divisible by 9. (See Appendix, § 24.) 8,645,373 is divisible by 9 because 8 + 6 + 4 + 5 + 3 + 7 + 3 = 36, a multiple of 9. 90. A number is divisible by 10 if the right-hand figure is 0. Digitized by VjOOQIC 40 FACTORS, DIVISORS, AND MULTIPLES. FACTORING. 91. Learn to apply readily tlie tests of divisibility. Try the successive prime numbers, beginning usually with 2. 92. If the right-hand figure is 0, the factors 2 and 5 are readily recognized. In such numbers as 25,000, each implies 2 and 5, hence the factors may be read off at once : 2, 5, 2, 5, 2, 5, 5, 5. 93. Numbers ending in 25, 50, or 75 may be factored by inspection, by remembering that each hundred contaiu/S 4 25^s. Illuatration. 1575 = 1500-1-75. 1500 = 60x25. 75 = 3 x 25. 60 X 25-1-3 X 25 = 63 X 25 = 3 X 3 X 7 X 5 x"5. EXERCISE 21. Write the prime factors of numbers to 100, in the following form : 4 = 2x2. 6 = 2x3. 8 = 2x2x2. This expression may be read, "4 equals 2 times 2," or "the prime factors of 4 are 2 and 2." Learn the prime numbers to 100 so that they can be repeated easily in ten seconds. Give the prime factors of the following : I 1. 2. 3. 4. 102 201 301 400 105 203 304 403 108 217 310 407 120 221 319 427 125 247 323 437 150 250 343 451 152 259 361 469 164 287 371 473 176 289 380 481 186 , 299 391 497 Digitized by VjOOQIC CANCELLATION. 41 8. 6. 7. a 600 660 826 10,660 626 676 833 16,824 629 700 851 24,860 639 703 869 66,626 683 731 899 73,000 596 749 900 100,000 600- 767 917 121,212 611 799 940 265,850 629 800 950 640,000 637 804 976 1,000,000 CANCELLATION. 94. One of the most important applications of factoring is cancellation. It should be used freely in all mathe- matical work to which it will apply. 95. The principles upon which cancellation depends are those of multiplication and division. The two im- portant ones may be restated thus : (1) Dividivtg any one of a series of factors hy any num- her divides their product hy that numiber. (2) Dividing dividend and divisor hy the same number does not change the quotient. EXERCISE 22. 1. Divides x9x 6 by 72. Solution. 8x9x6 _92<j^ rj^j^ ^^^^ f^^j^ dividing both divi- dend and divisor by 8. 9 V 6 2~-ii = 6. This results from dividing both dividend and divisor by 9. Hence, 8 x 9 x 6 -^ 72 = 6. The solution should appear as follows : gx^xJ^A 9 Digitized by VjOOQIC 42 FACTORS, DIVISORS, AND MULTIPLES. 2. Divide 24 X 45 X 60 by 18 X 20. 12 X 36 X 51 „ • 24x18x34"" 4. Divide 27 x 35 x 52 by 18 x 7 x 13. 5. Divide 92 x 87 X 57 x 69 by 23 x 23 x 19 X 29. 6. Divide 140 x 169 x 510 by 39 x 68. 7. How many baskets of eggs, each containiag 12 doz., at 15 cents a dozen, will pay for 8 bolts of cloth, each containing 24 yd., at 30 cents a yard? ' a If 63 books cost $ 126, what will 125 books cost ? Solution. ^^^^^^^^^ = f 2 x 126 = ^ 250. 63 Since the question asks for the cost of certain articles, we begin with #126, writing it above a short horizontal line. If 63 books cost $126, each book will cost one sixty-third of $ 126, which is expressed by writing 63 below the line as a divisor. 125 books will cost 125 times this number of dollars, which is expressed by writing 125 above the line as a factor of the dividend. Cancelling the common factors and completing the work, the result is $250. 9. If 15 men can do a piece of work in 7 da., in how many days can 21 men do the same work ? 10. If 24 men dig a ditch in 18 da., how many would be required to dig the same ditch in 27 da. ? 11. If 11 T. of hay can be made from 5 A., at the same rate, how many tons can be made from 65 A. ? 12. If 12 A. of land produce 720 bu. of com, how many acres would be needed to raise 1800 bu. at the same rate ? 13. If 26 horses eat a certain quantity of grain in 39 da., how many days would it last 338 horses ? 14. If a certain quantity of grain lasts 46 horses 34 da., how many horses would eat the same amount in 391 da. ? 15. Sixty-four men can do a piece of work in 57 da., work- ing 9 hr. a day. In how many days can 38 men do the same work, working 8 hr. a day ? Digitized by VjOOQIC THE HIGHEST COMMON FACTOR, 48 16. If 42 men do a piece of work in 18 da., working 10 hr. a day, how many men can do the same work in 90 da., working 7 hr. a day ? 17. If 91 men can do a certain amount of work in 54 da., working 9 hr. a day, how many hours a day must 162 men work to perform the same labor in 39 da. ? la (18x35x63x71)^(81x49x142) = ? 19. (48x72xl05xl9)-i-(144x35x57) = ? 20. (31x65x91x40)-^(62xl3x7x25) = ? THE HIGHEST COMMON FACTOR. 96. Factoring may be applied to the finding of the Highest Common Factor (H. C. F.) of two or more numbers. 97. The Highest Common Factor of two or more num- bers is the highest (i.e. greatest) factor common to all of them. 7 is the H. C. F. of 14, 28, and 49. 98. The Highest Common Factor of two or m^re numbers is found ly resolving the numbers into their prims factors and talcing the product of all factors common to all of the numbers. EXERCISE 23. 1. Find the H. C. F. of 360, 540, and 840. Solution. 860 = 2x2x2x3x3x6. 640 = 2x2x3x3x3x5. 840 = 2x2x2x3x5x7. The prime factors common to the three numbers are 2, 2, 3, and 6 Hence, the H. C. F. = 2 x 2 x 3 x 6 = 60. 2. Find the H. C. F. of 2205 and 1875. 3. Find the H. C. F. of 11,550 and 15,400. 4. Find the H. C. F. of 8190 and 7800. Digitized by VjOOQIC 44 FACTORS, DIVISORS, AND MULTIPLES. 5. Find the H. C. F. of 190, 1330, and 6270. 6. Find the H. C. F. of 81 and 75. Solution. It is at once seen that the factor 6 is found in 75 and not in 81. Therefore 5 cannot be a factor of the H. C. F. Divide 75 by 5, and the problem then is to find the H. C. F. of 81 and 15. For the same reason we may divide 5 out of 15, thus reducing the problem to the deter- mination of the H. C. F. of 81 and 3. The H. C. F. of 81 and 3 is at once seen to be 3. 7. Find the H. C. F. of 3600 and 4500. Solution. By inspection 100 is seen to be a common factor of 3600 and 4500. 100 is, therefore, a factor of the H. C. F. After removing this factor, we need to find the H. C. F. of 36 and 45. The H. C. F. of 3i3 and 45, by the use of the principle developed in Problem 6, is at once seen to be 9. Hence, the H. C. F. of 3600 and 4500 is 100 x 9, or 900. In the following problems, use, whenever possible, the methods of Problems 6 and 7. As far as i)ossible solve them mentally. Find the H. C. F. of 8. 18 and 45; 27 and 63; 32 and 54. 9. 220 and 120 ; 540 and 630 ; 810 and 270. 10. 90, 60, and 120 ; 80, 140, and 260 ; 3200 and 4800. 11. 155 and 248 ; 165 and 231 ; 120 and 168; 187 and 221. 12. 720 and 840; 190 and 950; 910 and 650; 3400, 6800, and 8500. 13. In fencing a lot 36 x 192 ft. with planks of equal length, what is the greatest length of plank that can be used ? 14. If, in the above problem, the fence is 5 boards high, what will the lumber for it cost at 5 cents a board ? 15. If, in Problem 13, the distance between the posts is one half the length of the boards, how many posts will be required, and what will they cost at 15 cents apiece ? 16. The sides of a triangular lot are 70, 112, and 126 ft. respectively. In fencing it with planks of equal length, what is the length of the longest possible plank that can be used ? Digitized by VjOOQIC THE LOWEST COMMON MULTIPLE. 45 17. Two distances of 165 mi. and 450 mi. are portioned off into equal daily journeys. If these portions are the largest possible, what is the length of each daily journey ? 18. Three men travel, in equal daily journeys, distances of 192 mi., 720 mi., and 264 mi. respectively. If all three travel at the same speed, what is the greatest possible daily journey of each? 19. Under the conditions of the above problem, how many days will each man be upon his journey ? THE LOWEST COMMON MULTIPLE. 99. Finding the Lowest Common Multiple (L. C. M .) of two or more numbers is another application of factoring. 100. One number is a Multiple of another when it has the other for a factor. 24 is a multiple of 2, of 3, of 4, of 6, of 8, and of 12, because each of these is a factor of 24. 101. A Common Multiple of two or more numbers is a multiple of each of them. 36 is a common multiple of 18, 12, 9, and 6. 102. The Lowest Common Multiple (L. C. M.) of two or more numbers is the lowest (least) multiple of each of them. 42 is the L. C. M. of 21, 14, and 7. 103. "The Lowest Common Multiple of two numbers which are prime to each other is their product. 104. The Lowest Common Multiple of two or more numbers must contain all their prime factors. 105. To find the L. C. M. of two or more numbers, resolve each number into its prime factors and find the product of all the different factors, using each factor the highest number of times it occurs in any of the nmnbers. Digitized by VjOOQIC 46 FACTORS, DIVISORS, AND MULTIPLES. EXERCISE 24. 1. Find the L. C. M, of 18, 63, and 72. Solution. 18 = a x 8^ x ^. 63 =^ X a X 7. 72 = 2x2x2x3x3. The different prime factors are 2, 8, and 7. 2 occurs three times, 3 two times, and 7 one time. Hence, theL. CM. is2x2x2x3x3x7 = 504. 2. Find the L. C. M. of 7, 12, 21, 24, and 36. Solution. Since 7 and 12 are factors of 21 and 24 respectively, they may be omitted. The problem is thus shortened, as now we need only consider 21, 24, and 36. 21 = 3 X 7. 24 = 8 X 2 X 2 X 2. 36 = 3x3x2x2. The L. C. M. is 3 X 8 X 2 X 2 X 2 X 7 = 504. Whenever possible, such rejections as noted in Uus problem should be made. Solve mentally as many as possible of the next five problems: Find the L. C. M. of: 3. 3, 4, and 6; 5, 10, and 7; 3, 9, and 12. 4. 8, 10, and 12 ; 7, 13, and 21 ; 9, 15, and 20. 5. 3, 4, 5, and 6 ; 5, 6, 8, and 10; 7, 14, and 25. 6. 20, 40, and 60 ; 30, 45, and 60 ; 48, 60, and 72. 7. 18, 20, 36, and 40 ; 16, 32, 64, and 90 ; 21, 42, 63, and 70. 8. 642 and 963 ; 815 and 1141 ; 707, 1414, and 2121. 9. 234, 738, 1025, and 1845; 1386, 3220, and 2240. 10. What is the least number which, when divided by 3, 4, and 5, will each time leave a remainder of 1 ? (What is their L.C.M. ?) 11. What is the least number which, when divided by 5, 6, 7, and 9) will each time leave a remainder of 2 ? of 3 ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS, 47 12. Three men starting from the same point travel 18, 24, and 27 mi. per day respectively. How far from the starting point is the first station at which all three will stop ? 13. Along a road are telephone poles 440 ft. apart, trolley poles 120 ft. apart, and lamp posts 240 ft. apart. At the be- ginning they are three abreast. How far from the starting point does this first occur again ? 14. Find the L. C. M. of 10, 12, 18, 30, 36, and 45. Solution. By the principle of Problem 2 above, strike out 10, 12, and 18. There are then left for consideration 30, 36, and 45. 45 must be in the L. C. M. All factors of 30 not already in 45 must also be in the L. C. M. 15 is the largest factor of 30 which is in 45. Hence 2 is the factor of 30 not in 45 ; it must therefore be a factor in the Ij .C. M. with 45. The largest factor of 36 which is in 45 is 9. The other factor is 4. As we have already used one 2, we reject one of the 2*s in 4 and use the other one. Hence the L. C. M. is 45 x 2 x 2 = 180. Use the above method in finding the L. C. M. of the following : 15. 8, 16, 30, 48, 60, 75. 23. 27, 38, 45, 54, 60, 76, 90. 16. 7,15,28,42,75,90. 24. 28,30,40,56,60. 17. 21,45,63,72,84. 25. 51,68,78,88,91. la 28,44,56,70,88. 26. 23,46,69,92. 19. 39,52,64,78,91. 27. 15,30,45,50,60,90. 20. 42,58,84,91,98. 28. 3,7,13,25,49,56. 21. 36,62,65,72,84. 29. 5,11,13,29. 22. 4, 5, 6, 8, 12, 16, 2b, 30. 30. 6, 8, 12, 16, 24, 36, 144. MISCELLANEOUS PROBLEMS. EXERCI3E 25. 1. The minuend is 86,431 ; the remainder is 27,563. Find the snbtrahend. 2. Find the prime factors of 5544. 3. The sum of four addends is 11,876 ; three of them aare 2641, 3479, and 3964. Find the fourth addend. Digitized by VjOOQIC 48 FACTORS, DIVISORS, AND MULTIPLES. 4. What is the quotient of 21 x 27 x 35 x 11 divided b^ 49 X 81 X 55 ? 5. A farmer has two farms of 160 A. 86 sq. rd. and 320 A 16 sq. rd. respectively. He reserves 174 A. 154 sq. rd. and divides the remainder equally among his 4 children. How much does each child receive ? 6. How much wood is there in a pile 32 ft. long, 12 ft. wide, and 6 ft. high ? 7. How many dozen bottles, each holding 2 qt. 1 pt., can be filled from 105 gal. of vinegar ? a How many sheets of paper will be required to make a 16mo book of 864 pages ? Note, See table in Appendix, § 21. 9. The dividend is 97,831 ; the quotient is 361. What is the divisor ? 10. A field containing 11 A. is 55 rd. long. How wide is it ? U. What is the cost of excavating a cellar 24 x 18 X 8 ft., at f .80 a cubic yard ? 12. Eighty hogsheads of molasses containing 63 gal. each, worth 55 cents per gallon, were exchanged for barreled pork (200 lb. to the barrel) at 11 cents per pound. How many barrels of pork were received ? 13. Find the value of (17 x 3 - 6 x 5) ^ (420 -3 X 139). 14. A man bought 40 A. of land at $ 22.50 per acre, and 65 A. at $ 30 per acre. He sold 75 A. at $ 36 per acre and the remainder at f 20 per acre. What did he receive for the land ? How much did he gain ? 15. A man's income is f 2250 per year. He spends f 250 for rent, $ 650 for provisions, f 250 for clothing, $ 150 for books and recreation, f 75 for taxes, $ 225 for life insurance, and $ 300 for incidentals. In how many years can he save $4200? 16. Find the H. C. F. of 65, 91, 143, 351, and 585. Digitized by VaOOQlC MISCELLANEOUS PROBLEMS. 49 17. If a = 12, ft = 20, c = 8, and d = 5, find the value of (ad ~ 5 c) 6 -f- (6c - oc?) + odd -*■ 100. 18. With the same values as above, find the value of 3ad-f-6+56c-h(3a-12). 19. Simplify ^x-^-'dy •\'bz + 4:y + llz +17 x + ^y + lU. 20. Find the L. C. M. of 8, 9, 11, 13, 26, 27, and 65. 21. Find the least number which, when it is divided by 13, 14, and 15, gives in each case a remainder of 11. 22. Six head of cattle worth $ 35 each, and 9 horses worth $110 each, were given for 15 A. of land. What was the land worth per acre ? 23. A man bought an equal number of sheep and cows for $ 4095. The sheep cost f 3.75 each and the cows $ 31.25 each. How many of each did he buy ? 24. Of what number is 865 both divisor and quotient ? 25. Of what number is 1259 both subtrahend and remainder ? 26. 256 taken forty-five times and the product subtracted from a certain number leaves a remainder of 139. Find the number. 27. Two cogwheels, containing 105 and 165 cogs respectively, are working together. After how many revolutions of the larger wheel will the wheels be in the same position as at starting ? 28. Find all the divisors of 420. 29. A room is 30 x 24 x 15 ft. Find how many square yards there are in the walls and ceiling. 30. How many chalk boxes 4x4x8 in. will fill the above room? 31. A farmer has a farm of 80 A. which is cultivated as follows : 25 A. in corn, 20 A. in wheat, 10 A. in meadow, 10 A. in oats, 2 A. in garden and truck patch, and 13 A. in pasture and woodland. His wheat yields 15 bu. per acre, the corn 47 bu. per acre, and the oats 38 bu. per acre. How much grain does he raise ? Digitized by VjOOQIC 50 FACTORS, DIVISORS, AND MULTIPLES. 32. Wheat is worth 77 cents per bushel, corn 41 cents per bushel, and oats 32 cents per bushel. What is the value of his grain? 33. His meadow makes 1^ T. of hay per acre. Hay is worth 9 8 per ton. What is the value of the hay ? 34. This farmer keeps 3 good cows, which give an average yield of 24 qt. of milk per day. What is the value of this milk per year at 2^ cents a quart ? 35. He also keeps 60 hens, from which he gets an average of 20 eggs per day. What are these worth for the year, if sold ' at 12 cents per dozen ? 36. What is the value of the board for the farmer, his wife, and two children for a year, rating it at f 2.50 per week for each one ? 37. The farmer spends for clothing for himself and family f 78.50, for medical service $ 13.75, for newspapers and books $5.50, for taxes $43.31. What is the total outlay for these items ? 38. Find the difference between the total value of the products named above and the total outlay. 39. What is the natural gas bill for 1 yr. at 25 cents per M, if an average of 5^ M is used each week ? 40. What is the cost of 3 lines of tile across a field 60 rd. wide, at a total cost of f 1.15 per rod ? EXERCISE 26. (Mental.) 1. 5x7x8=? (Arrange thus: 6x8x7=40x7=280.) 2. 6x9x5 = ? 3. Ilx8x 5 = ? 4. 11 X 17 = ? In multiplying by 11, the product may be written or read at once, by placing between the units and tens of the multipli- cand their sum. Thus, 17 X 11 = 1 (7 + 1) 7, or 187. 27 x 11 = 2 (2 + 7) 7 = 297, etc. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 51 5. 23xll=«? 6. 71x11 = ? 7. 85x11 = ? Solution. 86x11 = 8(8 + 6)6 = 086. a 79x11 = ? 9. 89X11 = ? 10. 96x11 = ? 11. 7x12x8x5 = ? 12. 5^x12 + 7x5 = ? 13. If 8 yd. muslin cost 56 cents, what is the cost of 12 yd. T 14. What is the cost of 8 oranges at 3 cents each and 13 lemons at 2 cents each? ^ 15. What is the H. C. F of 12, 15, and 36 ? 16. 5x13x8 = ? 17. 792-5-24=? Solution. 24 = 4x6. 792 -i- 24 = 792 -4- (4 x 6). 792 + 4=198; 198 + 6 = 33. Hence, 792 -^ 24 = 33. 18. 252^28 = ? 28 = 4 X 7. 19. 364^28 = ? 20. 936^36 = ? 21. 84x25 = ? Solution. 26 = ^. Hence, 84 x 25 = 84 x J^J^ = ^^ = 2100. To multiply by 25, annex two ciphers and divide by 4. 22. 126x25 = ? 23. 841x25 = ? 24. 967x25 = ? 25. 1231 X 25 = ? 26. What are the prime factors of 42 ? of 65 ? of 78 ? 27. How many pecks are there in 13 bu. 3 pk. ? 2a8aj + llaj-|-6aj+? = 35a?. Digitized by VjOOQIC 52 FACTORS, DIVISORS, AND MULTIPLES, 29. If 25 bu. of wheat bring $20, what will 75 bu. bring? (75 is three times 25.) 30. If 16 men earn $27 per day, how muoh will 48 men earn per day ? 31. If it takes 20 men 32 da. to do a piece of work, how long will it take 10 men ? 40 men ? 32. If A can do a piece of work in 2 da., and B can do it also in 2 da., how long will it take the two together to do it ? 33. Wheat is 77 cents per bushel and oats 23 cents per bushel. If $50 are paid for an equal number of bushels of each grain, how many bushels are bought ? 34. What number increased by 12 and the sum doubled makes 38? 35. What number diminished by 13 and the remainder mul- tiplied by 5 makes 45 ? 36. The sum of three numbers is 61 ; two of them are 20 and 21. What is the third ? 37. Of what number is 9 both divisor and quotient ? 38. Of what number is 13 one of the five equal addends ? 39. Of what number is 4 one of the three equal factors ? 40. Of what number is 15 both subtrahend and remainder ? 41. A man earns $12 per week. He spends $5 per week for room and board, 80 cents per week for car fare, $ 1 a week for building and loan dues, and $ 2.20 a week for clothes and incidentals. How much does he save per week ? 42. How much will he save in a year (52 weeks) ? 43. If, in Problem 41, the man's wages are increased $1 per week, what will be the effect on his yearly savings ? 44. What is the value of 480 eggs at 15 cents per dozen ? 45. What is the value of 8 turkeys, averaging 12 lb. each, at 10 cents per pound ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 53 46. A load of wheat brought $ 21.30 at 71 cents per bushel. How many bushels were there in the load ? 47. K a carpenter receives $423 for 9 months' work, how much should he receive for 3 months' work ? 48. How much greater is 192 -*■ 8 than 253 -f- 11 ? 49. What is the cost of 4 barrels of sugar at 5 cents per pound, each barrel weighing 215 lb. ? 50. What number doubled, and increased by 18, is then one fourth of 120? Digitized by VjOOQIC CHAPTER IV. LONGITUDE AND TIME. 106. In what direction does the earth rotate upon its axis ? How long a time is required for one rotation ? 107. Any particular point on the earth's surface describes a circle during a complete rotation. How many degrees are there in a circle ? Through how many degrees does any point on the earth's surface pass in 24 hr. ? In 1 hr. ? (360° -5- 24.) Inlmin.? (15° -j- 60 = i° = 15'.) In 1 sec? 108. How long does it take the earth to turn through 15°? 30°? 90°? 120°? 180°? 270°? 109. Which has noon first, Boston or New York ? Boston has noon first because it is east of New York, and, as the earth turns from west to east, Boston will be brought under the direct ra^s of the sun sooner than New York. Which has noon first, Buffalo or Chicago ? Indianapolis or Pittsburg ? Cincinnati or St. Louis ? Denver or Washington ? Salt Lake City or San Francisco ? 110. A watch showing correct time at Indianapolis is taken to Washington, D.C. Will it be too fast or too slow? Why? 111. A man traveled from Evansville, Ind., until his watch was 1 hr. fast. In what direction and how far did he travel? A man traveled from Fort Wayne, Ind., until his watch was 1 hr. 20 min. slow. In what direction and how far did he travel ? 112. On arriving at Denver from Terre Haute, Ind., will a man turn his watch forward or backward ? Why ? 64 Digitized by VjOOQIC LONGITUDE AND TIME. 55 113. When it is 9 o'clock a.m. here, what is the time 45° east of here ? 60** west of here ? 114. The Meridian of a place is the circumference of a great circle passing through it and the north and south poles of the earth. When the sun is directly above a meridian, all places upon that meridian within the sun's rays have noon. Before it is noon again at a place on the meridian, the earth must make about one revolution on its axis. The time occupied in making this revolution is divided into 24 hours ; during this time a given point on the earth's surface has made a complete circle (360**). 115. The Longitude of a place is its distance in circular measure either east or west of a fixed meridian. The longitude of a place cannot exceed 180°. 116. The fixed meridian from which longitude is reck- oned is the Prime Meridian. The prime meridian gener- ally used is that passing through the Royal Observatory at Greenwich, England. 117. Table of Longitudes. Paris 2^ London O*' New York 74° Boston .71° Chicago . . . . . .87° New Orleans 90° San Francisco 122° Berlin 13° St. Petersburg 30° Calcutta 88° Pittsburg 80° St. Louis 90° Cincinnati . . . . . . 84° Eome 20' 22' E. 5' 38' W. 0' 3' 'W, 3' 30' 'W. 36' 0' w. 3' 28' 'W. 26' 16' w. ' 23' 43' E. : 16' 0' E. 19' 2' ^• 2' 0' W. ) 12' 11' w. 26' 0' W. 'i 27' 14' E. 1 Digitized by VoOOQlC 56 LONGITUDE AND TIME. 118. The difference of longitude between two places which are both east longitude or both west is found by compound subtraction. The difference in longitude be- tween two places of which one is east longitude and the other west is found by compound addition. Since two places cannot be more than 180° apart, if the sum is greater than 180°, it must be subtracted from 360° to give the actual difference in longitude. EXERCISE 27. Use the above table, and find the difference of longitude : 1. Of Paris and New York. Solution. Long, of Paris, 2° 20' 22" E. Long, of New Y ork, 74° 0' 3'^ W. Difference, 76° 20' 26" 2. Of New Orleans and San Francisco. 3. Of Berlin and Calcutta. 4. Of San Francisco and Calcutta. 5. Of London and St. Louis. Beckoning Difference in Time. 119. When it is noon at any place, what time is it 15** east of that place? 30^ E.? 45° E.? 90° E.? 180° E.? 15° west? 30° W.? 45° W.? 90° W.? 120° W.? 180° W.? 15' E.? 15' W. ? 30° 30' E. ? 45° 45' W. ? 60° 30' 15" E. ? 90° 45' 30" W. ? 120. Prove that the following statements are true : A difference of 15° in the longitude of two places makes a difference of one hour in their time. A difference of 15' in the longitude of two J)laces makes a difference of one minute in their time. A difference of 15" in the longitude of two places makes a difference of one second in their time. Digitized by VjOOQIC LONGITUDE AND TIME. 57 EXERCISE 28. 1. What is the difference in time of two places whose difference of longitude is 36** 42' 30''? Solution. Since a difference of 15° in the longitnde of two places makes a difference of 1 hr. in their time, a difference of 36° of longitude makes a difference of 2 hr. in their time, with a remainder of 6°. 6° = 360'. 360' H- 42' = 402'. Since a difference of 16' in the longitude of two places makes a difference of 1 min. in their time, a difference of 402' of longitude makes a difference of 26 min. of time, with a remainder of 12' of longitude. 12' = 720". 720" + 30" = 760."- Since a difference of 16" of longitude makes a difference of 1 sec. of time, a difference of 750" of longitude makes a difference of 50 sec. of time. Therefore, a difference of 36° 42' 30" of longitude makes a difference of 2 hr. 26 min. 50 sec. in time. The solution consists in dividing the difference of longitude by 15 and calling the result hours, minutes, and seconds. 15 )36° 42' 30" 2 hr. 26 min. 50 sec. Find the difference of time, when the difference of longitude is: 2. 94** 17' 45". 4. 6** 56' 46". 6. 48' 45". 3. 112° 48' 15". 5. 64° 0'50". 7. 150° 12' 42". Beckoning Difference in Longitude. 121. If the time at A is an hour later than at B, what is the difference of their longitudes ? if 2 hr. later ? if 5 hr. later ? if 1 min. later? 5 min. ? 10 min. ? 1 sec. ? 5 sec. ? 20 sec. ? Is A east or west of B ? How do you know ? Change the word later to earlier, and ask the same questions. 122. Prove the truth of the following statements : A difference of an hour in the times of two places shows a difference of 15° in their longitudes. A difference of a minute in the times of two places shows a difference of 15' in their longitudes. A difference of a second in the times of two places shows % difference of 15" in their longitudes. Digitized by VjOOQIC 58 LONGITUDE AND TIME. EXERCISE 29. 1. The time in one town is 2 hr. 35 min. 22 sec. earlier than in another. Which is farther west ? How many degrees, etc. ? Solution. Since the time is earlier in the first town than in the sec- ond, the sun will not reach the meridian of the first until it has passed the meridian of .the other ; the first town is, consequently, farther west. Since a difference of 1 sec. in the times of two places shows a difference of 16" in their longitude, a difference of 22 sec. in their times shows a difference of 330" of longitude, which equals 6' 30". Since a difference of 1 min. in the times of two places shows a difference of 15' in their longitude, a difference of 35 min. in their times shows a difference of 525' in their longitude. 525' + 5' = 530' = 8° 50'. Since a difference of 1 hr. in the times of two places shows a difference of 15° in their longitude, a difference of 2 hr. in their times shows a difference of 30^ in their longi- tude. 30° + 8° = 38°. Their difference in longitude is 38° 50' 30". Why begin with the lowest denomination ? The solution consists in multiplying the difference of time by 16 and calling the result degrees, minutes, and seconds. 2 hr. 35 m. 22 sec. 15_ 38° 50' 30" In the following problems A and B represent places the difference of whose times is given. Find the difference of their longitudes, and tell which is farther east. 2. 4 hr. 25 min. 15 sec. A's time is later. 3. 6 hr. 40 min. 18 sec. A's time is earlier. 4. 1 hr. 59 min. 59 sec. A's time is later. 5. 12 hr. B's time is earlier. 6. 11 hr. 24 sec. A's time is earlier. Make a rule for each of the two general processes. 7. When it is noon at Paris, what is the time at St. Peters- burg? at San Francisco? a When it is 6 a.m. at London, what is the time at New York ? at Cincinnati ? at Eome ? Digitized by VjOOQIC STANDARD TIME, 69 9. When it is 35 min. past 3 p.m. at Berlin it is 34 min. 41^ sec. past 8 p.m. at a second city. Find the name of the city by consulting the table of longitudes. 10. When it is 4 p.m. at Chicago it is 40 min. 35 sec. past 1 P.M. at a second city. Find its name in the table. 11. A ship's chronometer indicates that the time at Green- wich is 25 min. past 3 p.m. By observations the captain ascer- tains that it is noon where the ship is. What is the longitude of the ship ? To the Teacher. Many problems may be formed from the table of longitudes. STANDARD TIBfE. 123. The railroads of the United States put stand- ard time in operation Nov. 18, 1883. The scheme was first proposed by Professor Abbe in 1878, and President Barnard of Columbia College worked the plan out in detail in 1882. By this plan the United States is divided into four north and south strips, each theoretically 15° in width. The time of the whole strip is determined by the time of a particular meridian of the strip. 124. The four standards are the times upon the 75th, 90th, 105th, and 120th meridians. These meridians are 15°, or 1 hr., apart. The difference in time between any two places in two adjacent strips is always just 1 hr. Each meridian theoretically governs a strip 7 J° wide upon each side of it ; but, as a matter of fact, the railroads have found it necessary to change time at some division point or railroad center. This results in irregular boundaries for the strips. The map on page 60 shows the strips as they actually exist. The Central time strip is very wide in the southern part of the United States. Digitized by VjOOQIC G I F T C Digitized by VjOOQIC STANDARD TIME. 61 EXERCISE 30. • By use of the map solve the following problems mentally : 1. When it is noon in the Central strip, what is the time in the Eastern ? Pacific ? Mountain ? 2. Will a traveler need to change his watch in going from Cleveland to Galveston ? 3. On arriving in Indianapolis, I find my watch 1 hr. slow. From what strip did I come ? 4. A train arrives at Ogden at 5 p.m. It leaves at 6.15 p.m., after having stayed but 15 min. Explain. What is the difference in standard time : 5. Between Los Angeles and St. Louis ? 6. Between Savannah and Ft. Worth ? 7. Between Boston and Denver ? a Between Tacoma and Jacksonville ? 9. Between Duluth and Mobile ? 10. Between Ogden and Buffalo ? 11. How often must a man reset the hands of his watch in traveling from Boston to Denver ? 12. When it is 8 p.m. standard time at Denver, what is the standard time at Ft. Wayne, Ind. ? 13. When it is 11 a.m. here, what is the time at Philadel- phia ? At Portland, Oregon ? 14. Is our real (sun) time faster or slower than standard time? 15. When it is 1 p.m. at Baltimore, what is the time at South Bend, Ind. ? At Pueblo, Col. ? 16. When it is midnight at San Francisco, what is the time at Little Rock ? Nashville ? Richmond, Ind. ? Richmond, Va. ? St Augustine ? Digitized by VjOOQIC CHAPTER V, FRACTIONS. DEFINITIONS. 125. The integral unit 1 is the primary unit of all number. 126. Other units are derived from the primary unit by multiplication or division. Thus, from the unit 1 lb. the unit 2000 lb., or 1 T., is derived by multiplying the unit 1 lb. by 2000. Also, from the unit 1 ft. we may derive the unit 3 ft., or 1 yd., by multiplying the unit 1 ft. by 3. Again, from this unit 1 yd. may be derived the unit 1760 yd., or 1 mi., by multiplying 1 yd. by 1760. Taking the unit 1 ft. again, we may derive the unit 1 in., by dividing 1 ft. (12 in.) by 12. 127. Such units as 3 ft., 2000 lb., 1 in., etc., are called derived integral units. The abstract numbers 3, 8, 9, 12, 25, etc., are derived integral units when considered as derived from 1, 2, 3, 4, or 6. 128. An Integral Number is a number that is composed of integral units. The number 72 contains the unit 1 seventy-two times, the unit 3 twenty-four times, the unit 8 nine times, the unit 9 eight times, the imit 12 six times, and so on. 129. A Fractional Unit is one of the equal parts of the primary unit 1. •^, -^^ and ^ are fractional units. 62 Digitized by VjOOQIC FRACTIONS, 68 130. All fractional units are therefore derived units, being derived from the unit 1. 131. The denominator of a fraction always designates the scale, that is, it shows the number of fractional units into which the primary unit 1 is divided. 132. In the fraction \ the scale is 3, for 3 thirds make 1. In I the scale is 4. From the standpoint of scale | is like 3 pk., the scale in each case being 4, because 4 fourths make 1, and 4 pk. make a bushel. 133. A Fractional Number is a number that is composed of fractional units. f , ^, f|, and -^ are fractional numbers. The fractional number -^ may be regarded as containing the fractional unit ^ eight times, or ^^ four times, or ^ twice, or ^ once. -^ may be regarded as 8 times -^^y or as ^ of 8. The latter view is the one most frequently used in practice. This view originates from an attempted division, in which the numerator is the dividend, the denominator the divisor, and the fraction the indicated quotient. Whichever view is taken, the relation to division holds. 134. A Proper Fraction is one whose numerator is less than its denominator. Its value is, therefore, less than one. f , ^, and \^ are proper fractions. 135. An Improper Fraction is one whose numerator is equal to or greater than its denominator. Its value is equal to or greater than one. f , ^, f are improper fractions. 136. A Simple Fraction is one whose numerator and denominator are both simple numbers. f, I, and ^ are simple fractions. Digitized by VjOOQIC 64 FRACTIONS. 137. A Mixed Number is one composed of an integral number and a fractional number. 7f , 7f , and 1^ are mixed nmnbers. Notice that there are two kinds of units in a mixed number. The number 3J is made up of the units 1 and \, This is like a compound number of two denominations in which the scale is 5. 138. A fraction is in its lowest terms when its numera- tor and denominator are prime to each other. h A> h A ^® fractions in their lowest terms. PRINCIPLES. 139. Prin. 1. Multiplying the numerator of a fraction by an integer multiplies the fraction by the integer. Since a fraction represents a division, this is merely the first part of Prin. 4, § 42. — <-^ = — Since the number of fractional units in the frac- 7 7 tional number is multiplied by the integer, while their size is unchanged, the fraction is multiplied by the integer. 140. Prin. 2. Dividing the numerator of a fraction by an integer divides the fraction by the integer. This is the first part of Prin. 5, § 43. "^ = -• Since the number of fractional units in the f rac- tional number is divided by the integer, while their size is unchanged, the fraction is divided by the integer. 141. Prin. 3. Multiplying the denominator of a fraction by an integer divides the fraction by the integer. Show that this is the second part of Prin. 5, § 43. 2 2 = — The number of fractional units is not changed, 3x3 9 but the unit in | is one third as large as the unit in \ Digitized by VjOOQIC REDUCTION OF FRACTIONS. 65 142. Prin. 4. Dividing the denominator of a fraction by an integer mvltiplie% the fraction by the integer. This is like what principle of division ? Why ? 7 7 = -. The number of fractional units is unchanged, 8-^-2 4 but the unit in | is twice as large as that in |. 143. Prin. 6. Multiplying both terms of a fraction by the 9ame number does not change its valite. Show the relation of this to a principle in division. 1 1x4 4 - = - 7 = :r« There are four times as many fractional 2 2x4 8 ^ units in I as in I, but each is only \ as large. 144. Prin. 6. Dividing both terms of a fraction by the same number does not change its valv£. What principle of division is this ? - = "^ = - • There are only \ as many fractional units in ^ as in 1^ but each is 4 times as large. REDUCTIOir OF FRACTIOITS. EXERCISE 31. (Mental.) 1. In $ 5 there are how many quarters ? Solution. In $ 1 there are 4 quarters, and in ^ 6 there are five times 4 quarters, which are 20 quarters. 2. In $7 there are how many tenths of$l? in$8? in $ 9 ? in $ 16 ? in f 86 ? 3. How many quarter yards are there in 5 yd. ? in 12 yd. ? in 15 yd.? in 25 yd.? in 63 yd. ? Digitized by VjOOQIC 66 FRACTIONS, 4. Change 7 to fifths ; 8 to thirds ; 12 to eighths ; 15 to sixteenths. 5. Tell how to reduce any integer to an equivalent fraction having any denominator. EXERCISE 32. 1. In 7f there are how many fourths ? Solution. In 1 there are 4 fourths, and in. 7 there are 7 times 4 fourths, which are 28 fourths. 28 fourths and 8 fourths are 31 fourths. Hence, 7f = 31 fourtiis, or ^^. This problem is like the reduction of 7 bu. 3 pk. to pecks, the scale in each case being 4. Change the following to improper fractions : 2. 2|. 7. 12^. 12. 85|^. 17. 18|. 22. m\. 3. 3f. a 15f. 13. n-^. 18. 41|. 23. 66|. 4. 3|. 9. 20||. 14. 34^^. 19. 26|. 24. 81^. 5. 63^. 10. 25|. 15. 512^^. 20. 62f. 25. 83f. 6. 8|. 11. 64f. 16. 6J. 21. 33|. 26. 47fJ. EXERCISE 33. Change : 1. 8tollths. 7. 9tol5ths. 13. 18f to 7ths. 2. 6| to 5ths. 8. 10 to 18ths. 14. 34^^ to 17ths. 3. 12tol0ths. 9. llfto6ths. 15. 46^^ to 25ths. 4. 12|to3ds. 10. 15to21sts. 16. 50^ to 30ths. 5. 7to8ths. 11. 14^tol0ths. 17. 66|^ to 12ths. 6. 7|to8ths. 12. IT^^toieths. 18. 72^tol8ths. EXERCISE 34. (Mental.) 1. Change 24 quarters of a dollar to dollars. Solution. In $1 there are 4 quarters; in 24 quarters of a dollar there are as many dollars as there are fours in 24. There are 6 fours in 24 ; hence, in 24 quarters of a dollar there are $6. This problem is like the reduction of 24 pk. to bushels. Digitized by VjOOQIC REDUCTION OF FRACTIONS. 67 Similarly, reduce the following fractions to integral numbers : 2. ^, j^; v. ¥• 7. ^, ^, II, V. 3. V, ^, V, ¥• 8- M, ¥, M. H- ♦. M, ft, M. «• 9- H^, W, H^, ¥ii^- «• «> H. H, H- 10. ifA, ^., i^, i^. 6. H> M, ¥, H. 11- H> H^. Wi W- EXERCISE 35. 1. Eeduce ^ to a mixed number. Solution. In one there are |, and in ^ there are as many ones as there are fours in 27. There are 6| fours in 27 ; hence, there are 6| ones in V. Tliis problem is exactly like the reduction of 27 qt. to gallons and quarts. Change the following fractions to integral or mixed numbers : 2. ¥> ¥, ¥• 10. i^, w. ^' 3. ^, -V, ff. 11. w. w, w. 4. fi, ff , ¥• 12. ei, ffi, tM. 5. H, ft, ^' 13. M, m %¥• 6. fi, tt, **. 1*. M> M, ¥^^- 7. W> W, ¥^. 15. W, W, W- 8- W, W, ¥f . 16. m^ HF- ». W> W> ¥f . 17. 7| ft. are how many thirds of a foot ? ^ of a foot equal how many feet ? la 12| yd. equal how many fifths of a yard ? ^ of a yard equal how many yards ? 19. $17^ equal how many dimes ? $-W equal how many dollars ? 20. 15f lb. are how many eighths of a pound? ^ of a pound are how many pounds ? To the Teacher, Give many dictation exercises until facility in solution is acquired* Digitized by VjOOQIC 68 FRACTIONS. -h EXERCISE 36. 1. Eeduce |4 to its lowest terms. Solution. — = ^ "^ = -. Both terms of the given fraction may be* divided by 6 by Prin. 6, § 144. The terms of the resulting fraction are prime to each other ; hence, { is the fraction in its lowest terms. 2. Reduce |f to its lowest terms. Solution. By factoring both numerator and denominator, the fraction takes the form : —^ — - — - — ^ — . By cancelling the factors common 2x2x2x2x6 to both terms, we have { as its form in the lowest terms. In practice, the cancellation should be performed without the complete factoring of the terms. Thus, o 5 The first result was obtained by dividing both terms by 8, and then the terms of this result were each divided by 2. Reduce the following to their lowest terms, by cancelling all factors common to both numerator and denominator : ♦• M. M, M, T%- 8- n, T^r. m, m- ^' iti tii M> fx' ^* 1T7> sW? ¥60* ^* T4> ■9T> W> ¥8* ^^' ¥2T> ¥7T* Express each of the following in the simplest form : 11. -Hof $1, $M. $AV ^W 12. -H^ of a yard, ^ yd., \^ yd., \^ yd. 13. ^ of a bushel, if bu., ^^^ bu., |f bu. 14. :^ of a mile, ^ mi., ^f mi., ^ mi. Digitized by VjOOQIC ADDITION OF FRACTIONS, 69 EXERCISE 37. 1. Reduce ^3^ to eOths. Solution. L=,J^^ 35 12 12 X 5 60 Both terms of the given fraction may be multiplied by 6, by Prin. 5, § 143. How was the multiplier 6 determined ? In reducing ^ to 72ds, what is the multiplier ? Reduce : 2. ^to46ths. 3. |to56ths. 4. |to36ths. 5. ^toSOths. 6.' ^tolOOths. 7. ^to55th8. a |to60ths. 9. 3^to96ths. 10. ftolSths. 11. |^to48th8. 12. 3^to96th8. 22. |4to343ds. 13. 32ytol20ths. 23. ^to378th8. 14. ^ to 105tbs. 24. H to 860tb8. ' 15. ^ to 144tbs. 25. 3^ to lOOOths. 16. fjtol28ths. 26. ^tol350ths. 17. ^tol08ths. 27. 11^ to 2816ths. 18. i^to333ds. 28. flf to 2118ths. 19. ||to360tbs. 29. ^ to 1563ds. 20. f^to630ths. 30. f^to2524ths. 21. |ito924ths. 31. ^to2600th8. ADDITION OF FRACTIONS. 145. Find the sum of 3 pk. and 4 pk. ; of 3 peaches and 4 peaches ; of 3 fifths and 4 fifths ; of f and |. 146. Find the sum of 5 chairs and 3 chairs ; of 5 chairs and 3 apples ; of ^f and f ; of ^ and |. Can 5 chairs and 3 apples be added ? Why ? 147. At first thought it might seem that j- and f cannot be added. But ^ and J differ from 5 chairs and 3 apples in that Digitized by VjOOQIC 70 FRACTIONS. they can be brought to a common scale, and therefore can be added. The lowest scale to which ^ and | can both be brought ^^^*- 6^5x4^20 7 7x4 28' 3^3x7^21 4 4x7 28" Hence, f + f = If + fi = H- 148. Fractions with different denominators must be reduced to a common scale, or denominator, before they can be added. 149. The most convenient scale to use is the L. C. M. of the given denominators. EXERCISE 38. 1. i + i=? 5. | + f = ? 9. 1 + ^ + 1=? 2- i + | = ? 6. 1 + 1 = ? 10. 1 + 1 + 1 = ? 3. i + ^ = ? 7. f + i=? 11. i + i + i = ? 4. i + | = ? a i + f=? 12. | + | + f=? EXERCISE 39. 1- i + l + l = ? 12. 1 + 1 + ^ + 1^ = ? 2. i + f + f = ? 13. f + ^ + i + tt = ? 3- f + i + f = ? "• i + f + f+l + f=? *• 1 + 1 + 1 = ? 15.f + i + f + A=? 5. i + f+J+i2ff = ? >^^^ + ^ + ^ + ^ + ^ = ? 6- f + * + l + i = ? ". l + f + T»ir + H + A = ? 7- i + f + i + f = ? 18. | + * + H + A = ? a l + f + i + A = ? 19- f+TAr + A + A = ? 9- i + f + * + i = ? 20. 1 + 1 + 1 + ^ + 1^ = ? 10- f + i + f + f + 4 = ? 21. ^ + 1 + ^ + ^ + 1=? 11. | + i + * + A = ? 22. 1 + 4 + 1 + 1 + ^ = ? Digitized by VjOOQIC ADDITION OF FRACTIONS. 71 24. i + i + A+A + A = ? 28. |^ + ^ + ^+i| + J^ = ? 25. t + f + | + i35, = ? 29. H.f + | + | + ,2^ + ^ = ? 26. | + | + A = ? 30. t + A + M + M + H = ^ Short Method. 150. J + ^ =5 ? Since the denominators are prime to each other, their L. CM. is their product. 12 is, therefore, the denominator of the fractions when reduced to a common scale. Since the numerator is 1 in each case, the denominator of the second becomes the numerator of the first reduced fraction, and the denominator of the first becomes the numerator of the second. Hence, j + j = ^ + 3^ = ^. EXERCISE 40. (Mental.) 1. i + i=? 4. J + i = ? 7. tV + A = ? 2. i + i = ? 5. ^ + A = ? a J + i = ? 3. i + f=? 6. iV + l^=? «• i + ^ = ? Is this method applicable to fractions whose denominators are not prime to each other ? In such cases is the result in its lowest terms ? Modify the preceding rule for such cases as ^ + ^. Solution. ^ + J = 2 x (^ + J). Give the results rapidly for the following problems : 10- i + i = n9; 13- f + « = ? 16. | + A = ? 11. 1 + ^ = ? 14. ^jr^ = ? 17. A + A = ? 12. ! + ♦=? 15. ^ + ^=? la ^^-A = ? To the Teacher. Dictate many similar problems. Digitized by VjOOQIC 72 FRACTIONS. EXERCISE 41. (Mental.) To the Teacher, Have the pupils practice adding small fractions mentally until facility is acquired. Solution. J + f = lJ. l + t = f i + A = ii + « = lH- ♦• * + l + i + H-A = ? EXERCISE 42. In adding mixed numbers do not reduce them to improper fractions. 1. Add7|,9f,3H. Solution. 7* + 9| + 3^1 = 7J| + 9fJ + 8H = Wf» = SIH- ' With this as a model, solve : 2. 5i + 12| + 4^ + 8H- 3. 12Jf + 26H + 33H-86f ' 4. 61J + 124H + 96H + 216J. 5. 77^ + 66,3^ + 1181 + 46^ 6. 126H + 99f + 231f + 184f. 7. 286^ + 324J + 789^ + 612|. 8. 177A + 268ii + 317^ + 4391. 9. 48,^ + 64^ + 92^ + 354ft. la 291fr + 37-^ + 63^. Digitized by VjOOQIC ADDITION OF FRACTIONS. 73 11. Four rolls of carpet contain respectively 98| yd., 97| yd., 112| yd., 96^ yd. What is the total amount ? 12. How many yards of border would be required in papering this room ? Floor plan, (Dimensions given are in feet.) 1% I'll l*/4 «<i kV, 5»/a t% 9'/f fs% 10 13. In a box weighing 12| lb., a grocer packed for shipment 15 J lb. of ham, ff of a pound of tea, 3| lb. of coffee, 6^ lb. of sugar. What was the total weight of the package ? 14. The United States coins weigh as follows : cent 48 gr., 5-cent piece 73 J- gr., dime 38^ gr., quarter-dollar 96^ gr., half-dollar 192^^ gr., dollar 412^ gr., quarter-eagle 64^ gr., half- eagle 129 gr., eagle 258 gr., double eagle 516 gr. What is the weight of the entire series ? A man has in his purse 2 silver dollars, 3 half-dollars, 15. 6 dimes, and 7 5-cent pieces. (Addition.) What is the weight of the whole ? Digitized by VjOOQIC 74 FRACTIONS. 16. How many acres are there in 4 tracts of land, the first containing 88| A. ; the second, 112| A. ; the third, 146J A. ; and the fourth, 39^^^ A. ? 17. Find the sum of 124J lb., 316| lb., 518J lb., 209^ lb., and 77 J lb. la Find the amount of coal in 5 car loads weighing as follows : 28| T., 29| T., 30^ T., 27^^ T., and 31f T. 19. A mail carrier traveled 12y^ mi. on Monday, 11| mi. on Tuesday, 13 J mi. on Wednesday, lOf mi. on Thursday, 9^^ mi. on Friday, and 14}^ mi. on Saturday. Find the whole dis- tance traveled in the 6 da. SUBTRACTION OF FRACTIONS. 151. From 8 cakes take 5 cakes; from 8 fourths take 5 fourths ; from f take J. From ^ take |. This cannot be done unless some changes are made. Why ? By Prin. 5, § 143, these fractions may be made alike ; that is, they may be reduced to the same scale. Solution. *-} = H-H = «. EXERCISE 43. Solve the following in the same way : 1. i-|=? 11. H-f=? 21. A-i=? 2. |-i = ? 12. A-f = ? 22. M-A = ? 3. |-i = ? 13. ^-| = ? 23. H-H = ? 4. f-f = ? 14. A-i = ? 24. ii-H = ? • 5. f-i = ? 15. |-^ = ? 25. H-A=? 6. f-| = ? 16. t2^-A = ? 26. H-3ftr = ? 7. i-| = ? 17. H-A = ? 27. H-A = ^ 8. f-i = ? 18. H-*=? 28. H-H=? 9. A-l = ? 19. M-i^ = ? 29. H-'H = ^ 10. A-4 = ? 20. i|-|=? 30. M-H = ? Digitized by VjOOQIC SUBTRACTION OF FRACTIONS. 75 ShMi Method. 152. ^ — I = ? By reasoning like that of § 150, the result is seen to be the difference of the denominators divided by the product of the denominators. Hence, EXERCISE 44. (Mental.) Give results rapidly : 1. i-i = ? 14. T>5-A = ? 27. |-^ = ? 2. i-i = ? 15. tV-T»5 = ? » f-i=? 3. i-t = ? 16. it-^=? 29. |-A = ? 4. i-i=? 17. A-A = ? 30. |-A = ? 5. i-i = ? la |-| = ? 31. ^-A = ? 6. i-i = ? 19. |-| = ? 32. f-t=? a i-| = ? 21. |-^=? 34. t-A = ? 9. |-T>5 = ? 22. |-| = ? 35. |-^ = ? 10. |-A = ? 23. |-T«r=.? 36. f-T«r = ? 11. i-i = ? 24. ^-^ = ? 37. 4-A = ? 12. i-^ = ? 25. J-| = ? 3a i-A = ? ^ A-tV = ? 26. f-i=? 39. H-tt = ? EXERCISE 45. 2. 18|-9| = ? SowTiON. 18§-9| = 18A-9« = 17}»-»H = SH- OT, the following : 18| = 18A=17}» 94= »« = _««. Digitized by VjOOQIC 76 FRACTIONS. Solve the following in the same way : ^ 3. 124^-98f = ? a 92ff-46^ = ? 4. 317^:^-2681 = ? 9. 126ii-97|f = ? 5. 91^2^-481 = ? 10. 532i|-483H=? 6. 461^-178|i = ? U. 624^ -279^ = ? 7. 83A-59| = ? 12. 1217H-968H=? MULTIPLICATION OF FRACTIONS. 153. There are three general types of problems in the multiplication of fractions. I. The multiplicand may be a fraction and the multi- plier an integer. Thus, fx6 = Y; ix4 = |. At I of a dollar per bushel, what will 5 bu. of wheat cost ? $ix5 = f^ = $3f. At $1 per yard, what will 4 yd. cost ? $|x4 = $|. II. The multiplicand may be an integer and the multi- plier a fraction. Thus, 5x| = ^; 3xf = f = 2i. At $5 per yard, what will | of a yard cost? $5 x | = $-1^ = $3i. At the rate of 3 mi. per hour, how far will a man walk in f of an hour? 3 mi. x f = fmi. = 2\ mi. III. Both the multiplicand and the multiplier may be fractions. Thus, *x| = T^. At $1 per day, how much can a man earn in f of a day ? «|X| = $T^. 154. Each of the three types of multiplication noted above conforms to the definition of multiplication given in § 36. Digitized by VjOOQIC MULTIPLICATION OF FRACTIONS. 77 I. To multiply | by 5 is to do to | what was done to unity to produce 5. Unity was taken 6 times to produce 5, so | must be taken 5 times to produce the product. i + i + i + f -h I = J^. Five times | equals ^. II. To multiply 5 by | is to do to 5 what was done to unity to produce |. Unity was doubled and the result divided by 3 to produce |, so 5 must be doubled and the result divided by 2 6x2 10 3 to produce the required product. 5 x - = — — = -r - o o o III. To multiply f by f is to do to f what was done to unity to produce |. Unity was doubled and the result divided by 5 to produce ^, so ^ must be doubled and the result divided by 2 2 2x2 4 5 to produce the required product. - x - = . ., = t- • 3 5 3x5 15 EXERCISE 46. Note, The sign x should always be read ** multiplied by." It was invented in 1631. Solution. Multiplying the numerator of a fraction by an integer multiplies the fraction by that integer, Prin. 1, § 139. | x 7 = ^ = 5^. 2. |X4 = ? 9. 3Jj^x9 = ? 16. 8|f x21 = ? 3. f x5 = ? 10. 33^x11 = ? 17. 15|X32 = ? 4. |x6 = ? U. |xlO = ? 18. 26||xl3 = ? 5. Jx7 = ? 12. ^x6 = ? 19. 9^x14 = ? 6. |x9 = ? 13. T^x9 = ? 20. 35^x43 = ? 7. ix4 = ? 14. 5|x8 = ? 21. 52Jx49 = ? 8. fx8 = ? 15. 7fxl2 = ? 22. 69fx66=? Note, Observe that in these problems the number of fractional units is multiplied in each case. Digitized by VjOOQIC 78 FRACTIONS. EXERCISE 47. Solution. This is done by Prin. 4, § 142. Dividing Uie denominator of a fraction by an integer multiplies the fraction by that integer. In practice this result should be obtained by cancellation. The written work should appear thus: | x ^ =: - = 2}» 2 Solve : 2. ^x7. 14. \ixU. 26. lif X 72. 3. ^2^ X 8. 15. it X 19. 27. HJ X 64. 4. ii X 12. 16. II X 34 2a ^ X 48. 5. H X 13. 17. H X 10. 29. iif X 38. 6. il X 7. 18. If X 18. 30. ^ X 75. 7. If X 19. 19. ^ X 37. 31. ^^ X 75. a tt X 21. 20. VV X 25. 32. IH X 64. 9. If X 17. 21. fl X 19. 33. ^ X 49. 10. M X 26. 22. fl X 13. 34. Ill X 25. 11. II X 11. 23. If X 27. 35. ^ X 24. 12. AX8. 24. If X29. 36. T%X28. 13. :^X20. 25. ||X28. . 37. ^^ X 33. EXERCISE 48. In the following problems use Prin. 1, § 139, or Prin. 4, § 142, according to the nature of the problem. Solve: 1. I X 4. 7. 3| X 6. 13. tW^t X 125. 2. TAfX3. a 4^X5. 14. ^X81. 3. T^X2. 9. 1^X4. 15. ^X^49. 4. -5^X7. • 10. 3^X7. 16. Uff X3. 5. |x3. 11. 15|x3. 17. 21^x7. a 2|x4. 12. If X 5. la 60|x7. Digitized by VjOOQIC MULTIPLICATION OF FRACTIONS. 79 EXERCISE 49. It has been shown that the product of a multiplicand and the factors of a multiplier is the same as the product of the multiplicand and the multiplier itself. Thus : 25 x 6 = 26 x 3 x 2. 1. ^ X 12 = ? Solution. ifxl2=Jfx3x4=Vx4 (Prin. 4) = ^ (Prin. 1) =8|. In practice, cancel as far as possible. The written form of this problem should be : 9 Solve : 2. |xl2. 12. fix 66. 22. VftxlSO. 3. -^xa. 13. iiVx63. 23. |fix48. . 4. Hx28. 14. 1^x76. 24. Iff X 72. 5. ^ X 24. 15. il X 66. 25. ^ X 126. 6. i|x20. 16. iix28. 26. 1^x120. 7. HX36. 17. MX 66. 27. :^X218. a ^X24. . la |ix87. 28. 1^1x240. 9. i|x33. 19. HxlOO. 29. |f|x260. 10. Hx40. 20. tJ^x144. 30. fi|xl44. 11. if X 46. 21. T%X61. 31. tV;^x246. EXERCISE 50. 1. 8xf = -^. _SoLUTioN. Since the product of two numbers is the same whichever one is made the multiplicand, 8x| = |x8 = ^ (Prin. 1). But the same result would be obtained if the integer were multiplied by the numerator of the fraction, and the result divided by the denominator ; or divide by the denominator and multiply by the numerator ; that is, use cancellation as far as possible. Digitized by VjOOQIC 80 FRACTIONS. 2. 12xi = ? 10. 8xf = ? la 64x| = ? 3. 16xf = ? 11. 9xt = ? 19. 105x^ = ? 4. 21xf = ? 12. 12xf=? 20. 105x/y = ? 5. 28x|| = ^ ^3. 15xf = ? 21. 201x| = ? 6. 6bu. Xj = ? 14. 21xf = ? 22. 201 X| = ? 7. $24x| = ? 15. 24x| = ? 23. 1575xA = ?. 8. 10^xf = ? 16. 30x1 = ? 24. 609xf = ? 9. 40A. X^ = ? 17. 32X| = ? 25. 63x2| = ? EXERCISE 51. 1^ fxf = A- SomTioN. If the multiplier were 3 instead of }, the product would be { by Prin 1. But } is ^ of 3. Hence, to get the true product, divide f by 4, which is done by Prin. 3, or by multiplying the denominator of | by 4. Notice, however, that the same result would have been obtained by multi- plying the numerators together for a new numerator, and the denomina- tors together for a new denominator. Thus : 3 3_3x3_ 9 6 4 6x4 20 This method may be farther shortened by the use of cancellation. Solve: 2. fXf 5. ttxf ^- HxH. ^x2=l. 6. «Xf 12. T«yVXiJ. f ? « 7. MxH. 13- WxM- o 8. HXf 14. |XtV- 3. |Xf. 9. Hxf 15. IT^Xf 4. Axf 10. Mx^. 16. 24|xf 1. 8|xl5 = 129. EXERCISE 52. Solution. | x 16 = 9. 8x16 = 120. 120 + 9 Form. 8J 16 9 120 129 = 129. Digitized by VjOOQIC* MULTIPLICATION OF FRACTIONS. 81 A 2. 8fxl6 = 137|. Solution. | x 16 = 9 j. 8 x 16 = 128. 128 + 9§ = 137J. FoRii. ^ 16^ 128 137f Multiply : 3. 12ibyl0. 6. 584^ by 50. 9. 15| by 12^. 4. 16fby24; 7. 624^ by 86. 10. 86| by 27|. 5. 125^3fby48. a 11| by 8^. 11. 1248^^ by 492J; Note. Reduce the mixed numbers to improper fractions. 12. What is the cost of -^ of an acre of land at $ 88 an acre ? 13. What is the cost of ^^ of a quire of paper at 23 cents a quire ? 14. What is the cost of 4| cd. of wood at $4.75 a cord ? >^^. What is the cost of 2^ yd. of broadcloth at $ 2| a yard ? 16. What is the cost of 34^ T. of coal at $2.57 a ton ? 17. What is I of I? |of^of|^? ^i^lB, Whatisf of ^A^of IJof 50? 19. What is ^ of 2| of || of 5^ of 12? 20. Find the cost of 3^ yd. of cloth at $3.75 ; of ^ yd. at $2.40; of 15| yd. at $3.20. 21. Bought 4^ cd. of wood at $ 3.50 ; 5f cd. at $4.40 ; ^ cd. at $4.56. Sold the wood at an average price of $5^ a cord. What was the gain ? 22. A farmer gathered 42^ loads of corn, averaging 33| bu., from his 25- A. field. What was the total yield ? 23. Multiply 17f by 14^; 26| by 29f ; 132^5^ by 68|; 694| by 87f 24. A steamer ran at an average rate of 386| mi. for 5| suc- cessive days. What distance was covered ? 25. Find the cost of 469| bu. of wheat at 60f cents. Digitized by VjOOQIC 82 FRACTIONS. 26. Find the cost of 36J lb. of coffee at 28| cents. 27. A father worked 6 da. at 9 2} per day, his son 5 da. at 9 1}, his daughter 4 da. at 9 1. What were their total earnings for the week ? 2a Whatistheareaof adoor3ift.x7ift.? DIVISION OF FRACTIONS. 155. There are three general types of problems in the division of fractions. I. The dividend may be a fraction, and the divisor an integer. Thus, t-^2 = f If 2 bu. of corn cost 9 1, find the cost of 1 bu. ^p|-f-2=$|. II. The dividend may be an integer, and the divisor a fraction. Thus, 3 -5- f = 4. If 1 bu. of wheat cost $|, find how many bushels may be purchased for $ 3. $3 ^ $f = 4. III. Both the dividend and the divisor may be frac- tions. Thus, A-*-* = |. If 1 bu. of potatoes cost $|, find how many bushels may be purchased for $^. $^-.$| = | = 2J. 156. Each of the above types conforms to the definition of division given in § 38. I. To divide f by 2 is to do to f what must be done to 2 to produce imity. 2 must be multiplied by \ to produce unity, so ^ must be multiplied by \ to produce the required quotient. 6 5^ 6 Digitized by VjOOQIC DIVISION OF FRACTIONS. 83 II. To divide 3 by f is to do to 3 what must be done to f to produce unity, f must be separated into 3 equal parts, and 4 of these parts taken. Since f must be multiplied by | to pro- duce unity, 3 must be multiplied by ^ to produce the required quotient. o a III. To divide ^ by | is to do to ^ what must be done to \ to produce unity. Since \ must be multiplied by ^ to pro- duce unity, -^ must be multiplied by f to produce the required quotient. ±^^^^^h^^ 10^6 ip 2 4* 2 EXERCISE 53. 1. |-*-4=f Solution. This is done by Prin. 2. Dividing the numerator of a fraction by an integer divides the fraction by that integer. a. |H-3 = ^. Solution. This is done by Prin. 3. Multiplying the denominator of a fraction by an integer divides the fraction by that integer. Note. Problems 3-20 are all solved by Prin. 2. 3. ^--6=? 9. f|^16=? 15. f^-f.l5=? • 4. 11^3=? 10. 1^-^17=? 16. fi-13=? 5. 1^^7=9 u. |^^19=? 17. f|-!-19=? 6. 11-^12=? 12. ||-f.l7=? 18. J^-^24=? 7. ||h-9=? 13. ^-j.29=? 19. 4f-^7=? 8. 1^ + 13=? 14. ^-^33=? 20. 53?i+19=? Note, Problems 21-36 are all solved by Prin. 3. 21. 3^-f.4=? 26. |-f-8=? 31. 5|^8=? 22. f-i-7 = ? 27. |-^ll = ? 32. 7|-f-12=? 23. ||-^9=? 28. 3:^-5-12=? 33. 8f-^9=? 24. 2^-J.10=? 29. -^^1^=^? 34. T^-5-13=? 25. 41 + 13=? 30. 3^-s-7=? 35. J^-i-6= ? Digitized by VjOOQIC 84 FRACTIONS. EXERCISE 54. Note carefully whether Prin. 2 or Prin. 3 is used in solving the following problems : 1. H + 6 = ? 6. 7^ + 3=? 11. 334-t-4=? 2. f + 6=? 7. 21|-^13=? 12. 709^ -f- 16=? 3. H-^3 = ? a H^u=? 13. a^-8-6=? 4. 2J^5=? 9. H-s-15=? 14. 201^ -H 3= ? 5. 2| + 4=? 10. 33J^3=? 15. 107|-t-13=? 1- A-^12=A. EXERCISE 55. Solution. ^5 -i- 12 = = A + (4x3) = A + 3 (Prin. 2) = A (Prin. 3). Note. In each of the following problems both Prin. 2 and Prin. 3 are used. a. i|-i-12 = ? 7. If-!- 14 = ? 12. ||-f.38=? 3. ^^-20=? 8. }^-i-26=? 13. ff-!-86=? 4. ^1 + 26=? 9. 1^ + 34=? 14. 7i-!-33=? 5. ^^-14=? 10. 11-^39=? 15. 28i-j-94=? 6..H + 18=? U. H-i-36=? EXERCISE 56. 16. 202^-5-42=? 1. 4-1-1 = 6. Solution. 4 = J/. Now, 12 thirds -h 2 thirds = 6, or ^ -1- f = 6. Solution. Using the same process as above, the problem becomes In Problems (1) and (2) above, the result in each instance could have been obtained by inverting the divisor and multiplying. Thus, 4^| = 4x 1=12^ = 6; and ^ -J. I = ^ X I = t|. Digitized by VjOOQIC DIVISION. 85 Solve Problems 3 to 14 by making the dividend and divisor alike and finding the quotient of the results ; and Problems 15 to 40 by inverting the divisor and using cancellation. 3. 7-!-|=? 7. |-Hi=? u. f-Hf =? 4. 12-5-1 = ? a f-^i = ? 12. *+A=? 5. 10-!-^ = ? 9. |-*-i = ? 13. f-^«=? 6. 8+| = ? 10. 1-^1=? 14. A+ + =? Divide : X5. 18 by ^. 17. 32 by il- 19. 63by^. 16. 24 by If. ia 46 by if 20. 36 by ff . 21. How many boxes, each holding J of a quart, can be filled from 8 qt. of berries ? 22. How many yards of cloth, at f of a dollar a yard, can be bought for $ 12 ? 23. If a man can dig a ditch -^^ of a rod in length in an hour, in how many hours can he dig 91 rd. ? 24. At 2| cents each, how many apples can be bought for 52 cents ? Reduce the mixed number to an improper fraction. 2 j = J^. 25. At f 2^ a yard, how many yards of cloth can be bought for $63? 26. At ^p 3^ a day, how many days must a man work to earn $144? 27. At $24| an acre, how many acres can be bought for »724? 28. If one horse costs $124J"J, how many horses can be bought for 9 3990 ? 29. i -5-1=? 33. 2^ H.l| = ? 37. 2i^4i=^? 30. ^-j-|=? 34. ^ ^2| = ? 38. |-f-|^=:? 31. t-^^ = ? 35.1^^-31 = ? 39. A-^H=? 32. H-«-f=? 36. .f-5-|=? 40. H-*-A = ? Digitized by VjOOQIC 86 FRACTIONS. 41. At $ ^ a pound, how many pounds of cofEee can be bought for$3|? 42. If a man travels 4| mi. an hour, in how many hours will he travel 23 J mi. ? 43. At $ 2^ a yard, how many yards of cloth can be bought for$6|? 44. If each bag holds If bu., how many bags will be needed to hold 40J bu. of oats ? 45. At $ IJ a yard, how much cloth will $ | buy ? 46. If an acre of land will yield 23^ bu. of wheat, how many acres are necessary to yield 18744 bu. ? 47. (|^|)H.ax|) = ? 49. I Of lof^S;,-!.! Of f Of T^ = ? 50. (|Xi|)-^(i|-^i) = ? 51. Divide H ^y 9; If by 25; i^ by 10. 52. Divide 6 by |; 15 by |. COMPLEX FRACTIONS. 157. The numerator of a fraction is a dividend, and the denominator a divisor. 158. Since the dividend and divisor may be fractions, the numerator and the denominator may, one or both, be fractions. Thus, * i 6 V^' T 159. Such fractions as have a fraction in one or both terms are called Complex Fractions. Digitized by VjOOQIC COMPLEX FRACTIONS. 87 160. Such forms of expressions are used for convenience merely, since each could be expressed by means of the division sign (-J-). 161. Complex fractions should be read as follows : " The complex fraction whose numerator is | and denominator |." T' T' 7' I' ii' 162. The longest straight line used separates the numer- ator from the denominator. This line may be regarded as a sign of division. The expression above it is the divi- dend, and that below it the divisor. EXERCISE 57. Solve the following as problems in division. Reduce : 1. i» 5. X. 9. S_lti4. 2. 2*. 6. ^-±1. 10. M^i. 3. 1. 7. ?Jii. 11 lJ<ii. 12 fx8 6i-!-f 4 1^L±. 8 t±i. 12 l2i^. TO FIND THE PART WHICH OITE NUMBER IS OF ANOTHER. EXERCISE 58. 1. 3 is what part of 7 ? Solution. 1 is ^ of 7, hence 3 is f of 7. This is equivalent to using the flist number as the numerator of a fraction and the second number as the denominator. 2. 5 is what part of 12? 6, of 17 ? 11, of 22? 6, of 20? 8, of 24? 9, of 36? 17,ofl2? 19, of 7? Digitized by CjOOQ iC 88 FRACTIONS. 3. I is what part of 5 ? Solution. 5 = J^. {is the same part of f^ that 2 is of 15. 2 is 3^ of 15 ; hence, } is ^ of 5. This is equivalent to } -r- 5. 4. f is what part of 8 ? ^ is what part of 12 ? |^ is what part of 10 ? i is what part of 9? 3^^. is what part of 8 ? ^ is what part of 6 ? 5. 1^ is what part of 4 ? ^ is what part of 14 ? ^ is what part of 126? 6. 2 J is what part of 6 ? 3f is what part of 4 ? 6| is what part of 10 ? Change the mixed numhers to improper fractions. 7. J is what part of f ? Solution. | = J^. t = M* I^ is the same part of H that 24 is of 25. 24 is Jf of 25 ; hence, | is i| of {. This is the same as f -r- f What was done in the above problem? Make a rule for such cases. 8. I is what part of -f ? 9. |is what part of I? 10. -^ is what part of ^ ? 11. With each of the following pairs of numbers, find the part which the first is of the second, and give the results rapidly : (l)i.f (9)ff a7)7A,8f (2)i,i. (10) if ■ (18)8J,10f (3)|,i. (ll)4|,5i. (19) 12,!^, 15J. (4)i,|. (12) hH- (20)19i,14|. (5) h 6. (13) h 9. (21) 21i, 25|. (6) I, 8. (14) 9, i. (22) H. A- (7) 8, 6i. (15) 4, f (23) H. M- (8) 6J, 8 (16) 5|, 4|. (24) 22^, 46^. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 89 TO FIND A NUMBER WHEN A SPECIFIED PART OF IT IS GIVEN. EXERCISE 59. 1. 15 is f of what number ? Solution. Since 16 is J of the required number, \ of that number is | of 16. } of 16 is 6. I of the required number is 8 fives, which are 40. Hence, 16 is f of 40. This is equivalent to =^* Rnd the number of which : 2. 48i8f. 9. 324 is ^. 16. 41| is ^Af. 3. 36is^. 10. 441 is It. 17. 46§ is ^7^. 4. 72 is If U. fisf la i^is^^. 5. 75 is If. 12. lisf 19. 91|is^. 6. 84isT^. 13. 3} is 3^. 20. 23fisH- 7. 90 is H- 14- ^ is M- 21. 34^ is li. a 125is||. 15. ISfis^^. 22. 23|^is^7. 23. A has $3| and B $7f A's and B's money together is ^ of C's. How much has C ? 24. A house cost | as much as the lot; both cost $896. Find the cost of each. 25. A certain farm contains 340 A., which is || of a sec- tion of land. How many acres are there in a section ? 26. A merchant sold goods to the amount of $316.80 on Monday ; this was ^ of his sales on Tuesday, which were ^ of his sales on Wednesday. What were the aggregate sales for the three days ? MISCELLANEOUS PROBLEMS. EXERCISE 60. (Mental.) (Use no written work in the solution of these problems.) 1. Find the L. C. M. of 4, 5, 6; of 8, 12, 20; of 5, 15, 30, 40 ; of 4, 6, 8, 10, 12, 15 ; of 12, 15, 18, 20, 30. Digitized by VjOOQIC 90 FRACTIONS. 2. Find the L. C. M. of 12, 18, 24; of 15, 30, 40, 45 j of 18, 24, 30, 36 ; of 60, 90, 105, 120; of 80, 120, 160, 240. 3. Change to whole or mixed numbers 4^, ^-, ^y f^* W l_60 4. Change to improper fractions 9|, 10^^, 12^, 15f, 18|, 5. Add I and i; ^ and ^; I and f ; f and f ; |, f, and |; f , \, ^, and ^. 7. Multiply I by 4, fj by 24; ^ by 45; ,81 by 44; ^ by 8. a Find I of 16; -j^ of 36 ; ^^ of 10 ; ^^ of 13; ^ of 7. 9. Findfof f; |ofi|; ^of t^V; i^of A- XO. Multiply f by ^; 2i by 3^; 6^ by 2^; 4 by ^; ^^ byf 11. Divide 4 by ^; 7 by |; 10 by ^; 12 by t^-; 6 by ^; 7 by V^r; 8 by f ; 6 by H; 15 by tJj; 14 by I; 18 by f ; 20 by |;10by^;9byH; 8byH;7byH. 12. Divide H by 5; H by 6; tSj by 16; ^ by 18; ^ by 36; Hby48; Hby61; Mby87; Mby39; Mby72. 13. Divide 1 by i; 1 by J; 1 by ^J^; 1 by |; 1 by ,5^; 1 by H;lby||;lbyT!^. If 1 be divided by any fraction, what will the quotient be ? 14. Divide|byi; |by|; | by ^;^hjU; ^hJ^. 15. f + t=? f-t = ? |X|=? i^i = ? fiswhat part of I ? I is what part of i ? 16. 7 is what part of 13 ? 8, of 19 ? 11, of 44 ? 18, of 27 ? f is what part of 6 ? of 9 ? of 15 ? | is what part of 4 ? of 7 ? of 10? of 12? of 15? 17. fis what part of 4? of^Aj? off? of||? ^3^ is what part of ^? of 3^? ofj^i? of 2^? of3|? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. X93 la 14f is f of what number ? 19. f of 15 is f of what number ? 20. f of I of 15 is I of ^ of 9 times what number ? 21. f of 21 is f of what number ? 22. \^ of 1^ is :^ of 5 times what number ? 23. John lost ^ of his marbles, and has 15 left. How many had he at first ? 24. Mary's money is f of Laura's, and both have 90 cents. How much has each ? 25. A raised 150 bu. of potatoes, which was \ less than what B raised. How many did B raise ? 26. James has $ 84, which is \ less than B's money. How much has B ? 27. A walked 120 mi., which is \ more than B walked, and ■^ less than C walked. How far did B and C walk ? 28. 63 is j^ more than what number ? It is | less than what number ? 29. Arrange the fractions |, -J-J, -^y ^, in the order of magnitude. 30. f of water is oxygen. What is the weight of the oxygen in a gallon of water weighing 8^ lb. ? 31. \ of air is oxygen. How much oxygen is there in a cubic foot of air weighing 525 gr. ? 32. If 15 gold pens cost ^26, what is the cost of 3 gold pens? 33. If 21 sheep are worth $56, what are 3 sheep worth ? Is it necessary to find the cost of one sheep ? 34. If 24 men consume a barrel of flour (196 lb.) in 2 wk., how many pounds do 3 men consume in the same time ? 35. If 32 horses in 3 da. consume 60 bu. of oats, how many bushels do 16 horses consume in 1 da. ? Digitized by VjOOQIC > 90 ^ FRACTIONS. ^^ V bu. of oats are worth 9f-bu. of corn, 4 bu. of oats jow much corn ? bbl. of flour cost $ 25|, what do 16 bbl. cost ? it noon on a certain day a l5-f t. jMjle casts a 12-ft. shadow, what length of shadow is cast by a pole 45 ft. long ? y-\39. A can dig a ditch in 6 da. What part of it can he dig in nda.? f B can do the same work in 4 da. What part can both dig / in 1 da. working together ? ^ ^ 40. A and B can trim ^ of a hedge in 1 da. In what time can they trim ^ of the hedge ? f ^of the hedge ? ^ /^ 41. A house and lot cost $ 5200. The lot cost -^ as much as the house. What did each cost ? 42. -J^ is f of what number? 43. What is the cost of 6^ yd. of cloth at f 2^ a yard ? of 4f yd. at $ 2| a yard ? of 7^ yd. at f If a yard ? 44. How many pounds of material can be bought for $5^ at f f a pound? for $ 4| at $ ^^^ a pound? for f 8f at $ ^^^ a ' pound ? for $ 10^ at f | a pound? 45. If A had $ 4 and lost $ |, what part of his money did he lose? if he had f 5 and lost f | ? if he had $ 6^ and lost $4? if he had $12f and lost $7? if he had $ 10| and lost f 6i? 46. What is the cost of ^ of a yard and ^ of a yard at f f ayard? of | + iat$|? of f + ^atf^^? of i + |at$|? 47. A grocer bought equal lots of eggs at 9 cents per dozen and 10 cents per dozen. He sold them at 12 cents per, dozen, clearing 60 cents. How many dozen did he buy? / '^ .■ 48. Sarah's age is f of Mary's and | of Ruth's. The sum of their ages is 46 yr. How old is each ? Find an expression for each in sixths of Sarah^s age. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 9^ EXERCISE 61. 1. The product of three numbers is 124|. Two of the numbers are 7^ and 8 J. What is the third ? 2. What number divided by | of ^2^ equals 132f ? 3. What number diminished by 253^^ leaves 84| J ? 4. What is the L. C. M. of 60, 125, 180, 225, 250 ? 5. Bought 8^ yd. of cloth at $3^ a yard, 12| yd. at $2f, and 18 J^ yd. at $4^. How many bushels of corn at 26f cents a bushel will pay the bill ? e. i5x^x|xl068 = ? 7. What is the cost of 28 lb. of butter at 37^ cents a pound, 84 bu. of corn at 43| cents a bushel, 135 bu. of oats at 25 cents a bushel, 160 bu. of rye at 62^ cents a bushel? . 8. A man made ^. of his journey the first day, -^ of it the second day, f of it the third day, and had 17f miles left to go, What was the length of his journey? ,^ 9. A man left ^ of his estate to his eldest son, y^ of it to "^ his second son, and the remainder to his third son. The share of the second was f 520 less than that of the third. What was the value of the estate? What was each son's share? io. Bought a farm of '324 J A. of land for $15,646. If the house and a house lot of 12 A. were counted at $3150, what price per acre was paid for Jhe rest ? U. I of a number exceeds \ of it by 112. What is the number? 12. I of f of a number is 291 less than | of | of it. What is the number ? 13. Owning ^^ of a farm, I sold \ of my share. If what I had left was worth $219, what would the whole farm be worth at the same rate ? 14. I bought a house and lot for $5784, paying ^ as much for the lot as for the house. How much did I pay for each ? Digitized by VjOOQIC 94 FRACTIONS. 15. f of A's farm equals ^ of B's. Together they have 551 A. How many has each ? ^ 16. How many bushels of grain can be bought for $1260| at 31 J cents a bushel ? at 56 J ? at 66| ? at 87^ ? 17. If b^\ A. of land cost $ 1200, what is the price per acre? la How long will 225 lb. of meat last a erew of 5 men, at the rate of 1^ lb. per day for each man ? 19. A man's crop of oats weighed 66,677 lb. He sold it for 31 J cents a bushel. Counting 32 lb. for a bushel, what was the value of his crop ? 20. A crop of wheat weighing 90,144 lb. brought ^1252. Counting 60 lb. to a bushel, what was the price per bushel for wheat sold ? 21. ^ + i\ + A-^of,^ = ? / 22. Bought 8^ yd. of broadcloth at $4 J a yard, 5 J yd. of cassimere at f 2J a yard, 5J yd. of silk at $ 2^ a yard, and paid, for all with corn ^ 41| cents a bushel. How many bushels were required ? 23. Find the sum, difference, and product of 4^ and 6^. 24. Find the quotient arising from dividing the sum of 8^ and 5f by their difference. 25. Divide I of 6| by I of ^ of 12|. 27. A cistern having a capacity of 88 J bbl. contained 63^- bbl. After 15^^ bbl. were pumped otfi, the cistern was what part full ? 28. A had a journey of 43^7^ mi. to make. After traveling 36^ mi., what part of the journey remained ? 29. 23 is what part of 48^? of 61| ? of 58f ? 30. ^ is what part of 87 ? of 169 ? of 46^^^ ? of 83^ ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 96 31. 7f is what part of 15f ? of 28| ? of 72J. ? 32. Divide 76 into two such parts that | of the first shall equal f of the second. 33. Divide 417 into three such parts that the first shall be ■^ of the second, and the third shall be ^ of the second. 34. How is the value of a proper fraction affected by adding the same number to both terms? Why ? 35. How is the value of an improper fraction affected by adding the same number to both terms ? Why ? 36. How is the value of an improper fraction affected by subtracting the same number from both terms ? Why ? Of a proper fraction ? Why ? 37. What is the circumference of a wheel that makes 24| revolutions in 400 ft. ? 3a If another wheel has a tire 2 ft. shorter, it makes how many more revolutions in the same distance ? 39. Divide f 6600 among 4 persons so that the second shall receive twice as much as the first, the third three times as much as the second, and the fourth four times as much as the third. 40. (3i + 2|)x(4i-l|)-h|of^ = ? 41. 2\ is what part of 6,^? of 8f ? of 9J ? of 10| ? 42. f is 3^ of what? 7| is ff of what? r| is ^^ of what? 43. Divide I of I by f of IJ. 44. Divide If of 1\ by ff of ff of ff . 45. Add 11 \, 26f , 69f , 142H, 317^. 46. Add 24|, 63f, 96H, 38|, 1243^^. , 47. Solve: 64^2^-421-1; 69|| + 26||; 3153^^- 278f 4a Solve : 84f x 63| ; 215 f^ x 49| ; ^ X ^. Digitized by VjOOQIC 96 FRACTIONS. 49. Solve: 12f ^1; 64^-^5; 65^-h6; 126}f ^7. 50. A. J. Brown bought of Worley & Treadway, July 10, 1903, the following items : 8 lb. coffee @ 30 cents, 40 lb. sugar @ 6J cents, 20 bars of soap @ 4 cents, 3 brooms @ 30 cents, 1 16-lb. ham @ 14 cents, ^ doz. oranges @ 30 cents. Make a bill and receipt it. Solution. Bloomington, lu/fA^ 10, |C|03. To Worley & Treadway, Dr. So 8 Wv. cx>||*>t/ l+O Wv. (MAXJXX/V llo Wv. Kxvmy @ 30 ^ " M- •' 30 " m- '■ 30 $2 2 2 M-0 50 80 ^0 2M- 15 $8 ^^ 3^uXa^ 10, '03. ID'oaXoo^ ^ S'u^cudUtMxo^. 51. Make a bill of the following items : A. R. Brown bought of G. G. Johnson 8 gal. oil at 12^ cents, 14 brooms at 20 cents, 1 pkg. gold dust 25 cents, 3 cakes soap at 8^ cents, 5 gal. gaso- line at 10 cents, 16 lb. sugar at 6\ cents, 300 lemons at 1^ cents, 2 pineapples at 35 cents. 52. Make a bill of the following: P. A. Coen & Son sold to Augustine & Co., Feb. 20, 1 qt. mucilage 50 cents, 1 jar paste 20 cents ; March 2, 1 qt. ink 50 cents ; March 13, 10 rm. cap at f L35; April 6, 30 lb. paper at 7 cents; April 9, 2 qt Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 97 ink at 50 cents; April 13, 24 quinine bottles at 4j^ cents; April 17, 90 penwipers at If cents, 1 floor brush $3; May 6, 2 gal. benzine at 20 cents ; May 12, 31 lb. paper at 7 cents, 5 rm. cap at $1.35; May 14, 5 gal. turpentine at 50 cents; May 16, 5 rm. cap at f 1.35. 53. How many bushels of corn can be bought for $1936.40 at 23^ cents a bushel ? 54. Sold 1836 bu. of wheat at 61| cents and invested the proceeds in corn at 24J cents. How many bushels did I buy ? Digitized by VjOOQiC CHAPTER VI. DECIMAL FRACTIONS. DEFINITIONS. 163. A Decimal Fractional Unit is one of the decimal divisions of the primary unit 1. 1^' T^> ^^^ 10^0 ^^^ decimal fractional miits. 164. A Decimal Fractional Number, or a Decimal Frac- tion, is a number composed of decimal fractional units. ^f A> tW> H^y jAfSy an<i ^jnftr are decimal fractions. 165. The denominator of a decimal fraction is 1 with some number of ciphers annexed. If the unit of a fractional number is not a decimal division of the unit 1, the fraction is called a Common Fraction. The denominators of common fractions are not I's with ciphers annexed. f , f , and 14 ^^^ common fractions. 166. Decimal fractions are usually expressed by the use of the decimal point. ^ = .3, ,2^ = .7, ,1^ = .14, and fM = 3.16. The first place to the right of the decimal point is tenths; the second place, hundredths; the third place, tkoiLsandthSf etc. Note. The decimal point was first used some time in the seventeenth century. 98 Digitized by VjOOQIC READING DECIMAL FRACTIONS. 99 167. A pure decimal is composed of decimal fractional units only. .12, .231, .03 are pure decimal fractions. 168. A mixed decimal is composed of integral units and decimal fractional units. 12.7 is composed of 12 integral units and 7 decimal fractional imits. READING DECIMAL FRACTIONS. 169. .73 is read " 73 hundredths " ; 2.5 is read " 2 and 6 tenths" ; 3.14 is read "3 and 14 hundredths" ; 29.6 is read "29 and 6 tenths." 170. Use and only in reading mixed decimals, and then at the decimal point. 26.3 is read " 26 and 3 tenths " ; 2.15 is read " 2 and 15 hun- dredths " ; 5.004 is read " 5 and 4 thousandths." If the and is used carelessly, "two hundred and fifteen thousandths" may mean 200.015 or .215. There will be no confusion if the and is used properly. 171. The following names of the orders to the right of the decimal point should be committed to memory : First, tenths' order. Second, hundredths' order. Third, thousandths' order. Fourth, ten-thousandths' order. Fifth, hundred-thousandths' order. Sixth, millionths' order. Seventh, ten-millionths' order. Eighth, hundred-millionths' order. Ninth, billionths' order. Digitized by VjOOQIC 100 . DECIMAL FRACTIONS, EXERCISE 62. Bead the following numbers : 1. .5, .05, .004, .0006, .15, .014, .124, .1006. 2. .00008, .000009, .0000009, .0106, .1013, .075. 3. .00000002, .000000003, .00100002. 4. .000000007, .0000000001, .00300407. 5. .26, .264, .3864, .029, 56.27, 500.05. 6. .0294, .00874, .087463, 327.505. 7. .0056849, .00046928. 172. Like whole numbers, decimals may be read in several ways. 28.75 may be read as follows : 28 and 75 hundredths, 2875 • hundredths, 287.5 tenths, or 2.875 tens. The first form should always be used tmless some other is specifically required. Give two or three readings for Problems 5, 6, and 7 in Exercise 62. WRITING DECIMAL FRACTIONS. 173. To gain facility in writing decimal fractions we must be thoroughly familiar (a) with the names of the orders, and (b) with the number of each order, counting from the decimal point toward the right. 174. What is the number of each of the following orders: hundredths' ? millionths' ? tenths' ? ten-thousandths' ? ten- millionths' ? hundred-thousandths' ? hundred-njillionths' ? billionths'? .. ^^t, ^j ic>/^r'' „^ Name eaoll^pf the following orders: '4th, 8th, 1st, 2d, 7th, 5th,3d,''9th,'6th?''" Digitized by VjOOQIC WRITING DECIMAL FRACTIONS. 101 EXERCISE 63. 1. Write in figures 469 ten-milliouths. Solution. In order that a number shall express ten-millionths, its right-hand figure must stand in the seventh order to the right of the deci- mal point In 469 there are three figures ; hence, four ciphers must precede the numerator. To express this number, consequently, write: decimal point, four ciphers, 469. Thus, .0000469. Tell how each of the following numbers is expressed before you make any figures. 2. Write in figures 17 hundred-thousandths. This number is expressed by writing decimal point, three ciphers, one, and seven. 3. Write in figures 6 tenths; 16 hundredths; 43 thou- sandths ; 28 ten-thousandths ; 467 thousandths. Write in figures : 4. 29 hundredths ; 921 thousandths ; 7 ten-thousandths ; 8 hundred-thousandths ; 4 millionths ; 6 billionths ; 38 ten-thou- sandths; 4562 hundred-thousandths. 5. 419 millionths; 306 hundred-thousandths; 96 ten- thousandths ; 8158 hundred-millionths ; 59001 billionths. 6. 23006 and 40007 millionths ; 29000 and 29 thousandths ; 29029 thousandths ; 307 million and 307 millionths ; 307000307 millionths. 7. 379 tenths; 5824 hundredths; 69708 thousandths; 524896 hundred-thousandths. a Three million seventeen thousand eight hundred- billionths. 9. Seven hundred ten-thousandths; seven hundred ten thousandths; nine thousand two hundred-millionths; nine thousand two hundred millionths. 10. Eight and eight thousandths. U. Twenty-three and sixty-one ten-thousandths. Digitized by VjOOQIC 102 DECIMAL FRACTIONS. REDUCTION OF DECIMAL FRACTIONS. 175. The scale in decimal fractions is the same as in whole numbers ; that is, 10 units of one order make one of the next higher. Decimals may be thought of as an extension of the decimal system below unity. 176. Eeduce.8 to tenths j 24 to thousandths; .5 to hun- dredths ; .75 to ten-thousandths ; .0624 to millionths. Reduce .2600 to hundredths; .050800 to ten-thousandths; 18.000 to hundredths, to tenths; 75.063 to millionths, to thousandths. Reduce 4.6 to hundred-thousandths; reduce the resulting fraction to hundredths ; 87.2 to billionths ; the resulting frac- tion to millionths, to tenths. 177. Annexing ciphers to the right of a decimal does not change its value. Why ? What principle of fractions explains this? Reduction op Decimal Fractions to Common Fractions, AND TO their LoWEST TeRMS. EXERCISE 64. 1. Reduce .16 to a common fraction in its lowest terms. Solution. . 16 = il = J:^^:± = i- . 100 100 -h 4 26 2. .375 = f. Solution. .376 = JIJ = ^^6 -. 125 ^8, 1000 1000 -T- 126 8 Reduce to common fractions in their lowest terms : 3. .75. 6. .075. 9. .0012. 12. 3.08. 4. .0125. 7. .515. 10. .085. 13. 3.075. 5. .364. 8. .205. 11. .04725. 14. 29.205 15. .8J, .08^, .008^. 17. .43f, .00431, .58^. 16. 1.6$, .16J, .016f. la .56i, .068|, .081^. Digitized by VjOOQIC REDUCTION OF DECIMAL FRACTIONS. 103 19. 16i. 21. .37^. 23. .87J. 25. 4.44^. ao. .18}. 22. .62i. 24. .22|. 26. 6.33^. Reduction of Common Fractions to Decimal FBACTiONa EXERCISE 65. 1. Eeduce f to a decimal fraction. FOKM. 8)3 000 Solution. | = | of 3. But 3 = 3.000. ^^ Hence, \ of 3 = \ of 3.000 = .376. 2. Beduce j^ ^ ^ decimal fraction. 19mTL Solution, xb = ri^ of 4. But 4 = 4.000. izo24^ Hence, liy of 4 = ji^ of 4.000 = .032. .032 Anothbr Solution. • -— = . ^ -^ " 125 126 X 8 What principle of fractions is this sec Eeduce to decimal fractions : 3. i- 4. f 5. ^^ 6. f 7. 1- a f 9. ¥• 10. A- u. A- 12. A- 13. A. 14. A- IS. A- 16. liV- 17. tt- 18. H- 1000"' * md solution based uponf 19. ^ 27. tIt. ao- A- 2a T%. 21. f*. 29. H- 22. 1^. 3o: v^. 23. IJ. 31- i^- 24. ^. 32. ttf. 25. H- 33- lirr- 26. ^V 3*. T^. J\ro£e. The foregoing fractions all reduce to pure decimals because each reduced numerator is divisible by its denominator. A study of the denom- inators shows that the only factors of the denominators not found in the corresponding numerators are 2 and 5. But these are the factors that are introduced Into the numerator with each reduction, that is, with the annexation of each cipher. Hence, it is possible to carry the reduction far enough to make each numerator divisible by its denominator. To the Teacher. Dictate many problems until this principle become? plain. Digitized by VjOOQIC 104 DECIMAL FRACTIONS. EXERCISE 66. 1. Reduce | to a decimal fraction. Solution. } = } of 2 = | of 2.0 = .6] = .66} = .666}. Form. 8 )2.0 ; or 8 )2.00 ; or 8 )2.000 . .6} Ml .666| 2. Eeduce ^ to a decimal fraction, Solution. J = J of 5 = J of 6.0 = .8J ; or J of 5.00 = .88J ; or J of 5.000 = .888^ ; and so on. Form. 6 )5.0 ; or 6 )5.00 ; or 6 )5.000 ; and so on. .8} .88^ .888^ Such results as are found in each of the above problems, (1) and (2), are called complex decimal fractions. A Complex Decimal Fraction is one composed of decimal and common fractional units. .8|, .26}, .715), .0083} are complex decimal fractions. In such cases the division may be continued to any designated order. Eeduce the following, extending the decimals to ten- thousandths : 3. fj. M ' - att. - 13. f ls. # *• A. -A-K '^'' A*, f 19. A. 5. A. 10. H- • ^^^15. f 20. tV. 6- H. u- M. 16. f 21. «. •>■ A-! 12. f 17. il- 22. A. Digitized by VjOOQiC / ADDITION OF DECIMAL FRACTIONS. 105 ADDITION OF DECIBCAL FRACTIONS- EXERCISE 67. Decimals are added like simple numbers. In arranging for addition the points must form a column. Add: 1. 2. a 4. .626 4.073 126.0009 4006.092 .0984 26.0084 482.1872 9.069701 .4907 69.00462 600.50983 683.086409 .00864 83.01879 17.008469 34.189367 .09769 60.00043 6.098726 1086.049078 5. Add .00862, 4.04378, 73.096, 168.00097, 49.287005, 83.460037. 6. Add 77.02081, 94.09069, 88.00799, 686.060098, 897.0609, .084858, .087857, .3060686, .76978. 7. Add .04069, .008972, .0934, .0083462, .027309, .5302681, .05003701. 8. Add 7 tenths, 56 hundredths, 93 thousandths, 329 hun- dred-thousandths, 8052 millionths, 42067 ten-thousandths, 43 hundredths, 98 ten-millionths. 9. Add .3^, .007|, .17^^^, and .0862|. SOLDTION. ,^ = .3126 .0071 = .0074 .17A =.1718A .08621 = .08621 .5779JI 10. Add .3J, .0291, .056^, .24^, .183f, .86^. Digitized by VjOOQIC 106 DECIMAL FRACTIONS. SUBTRACTION OF DECIMAL FRACTIONS. EXERCISE 68. Subtract as in simple numbers. In arranging for subtrac- tion the points must form a column. 1. .0861 - .0295 = ? 9. 400. -.00004=? 2. .7043 - .4806 = ? 10. .00583 - .0000583 = ? 3. .0461 - .00356 = ? U. .7t-.008J = ? Annex ciphers to the minuend Solution. until it has as many decimal places 71 _ .7331 as the subtrahend. .0084 = .0084 4. 4.02603 - .9078 = ? ~~^72&A 5. 26.1059-19.74308=? 12. .43| - .0047^ = ? 6. 461.083024-86.59260834=? 13. .0084| - .00023^ = ? 7. .023 - .000465 = ? 14. 42.08^ - 34 0574^ = ? a 92.-.06479 = ? 15. .8035| - .045^ = ? MULTIPLICATION OF DECIMAL FRACTIONS. 178. There are three general types in the multiplication of decimal fractions. I. The multiplicand may be a decimal fraction, and the multiplier an integer. Thus, .14 x 4 = .66. II. The multiplicand may be an integer and the multi- plier a decimal fraction. Thus, 14 x .04 = .56. III. Both the multiplicand and the multiplier may be decimal fractions. Thus, .14 x .4 = .056. 179. The solution of each of the three types of decimal multiplication is a result of the application of the general definition of multiplication given in § 36. Digitized by VjOOQIC MULTIPLICATION OF DECIMAL FRACTIONS. 107 I. To multiply .14 by 4 is to do to .14 what must be done to unity to produce 4. Unity must be taken 4 times to produce 4, so .14 must be taken 4 times to produce the required product Four times .14 equals .56. II. To multiply 14 by .04 is to do to 14 what must be done to unity to produce .04. .04 is produced from unity by divid- ing by 100 and multiplying by 4, or by multiplying by 4 and dividing by 100, so the required product is obtained by multi- plying 14 by 4 and dividing by 100. 14 x .04 = ^r^ = .56. III. To multiply .14 by .4 is to do to .14 what must be done to unity to produce .4. Unity must be multiplied by 4 and divided by 10 to produce .4, so .14 must be multiplied by 4 and divided by 10 to produce the required product. .»x.4 = ii^=f=.«5e. PRINCIPLES. 180. Prin. 1. Any number may he multiplied by 1 toith any number of 0'« annexed, by moving the decimal point to the right as many places as there are 0'« in the multiplier. Thus, 2.34 X 10 = 23.4 ; 2.34 x 100 = 234. 181. Prin. 2. Any number may be divided by 1 with any number of 0'« annexed, by moving the decimal point to the left as many places as there are 0'« in the divisor. Thus; 342 -J. 10 = 34.2; 34.2 -^ 100 = .342. What principles of fractions show that the above principles are true ? 182. When the multiplier is an integer, the product is like the multiplicand. .14x4 = .56. Since the multiplicand is himdredths, the prodact is hundredths. By the use of Prin. 1 and Prin. 2, the multiplier may always be made integral. Digitized by VjOOQIC 108 DECIMAL FRACTIONS. EXERCISE 69. 1. 14x.04 = .56. Solution. Move the decimal point of the multiplier to the right two places to make the multiplier integral. To counteract the effect of this on the product move the decimal point of the multiplicand to the left two places. Now the problem is .14 x 4 = .66, as is shown in Problem 1. 2. .14 X. 4 = .056. Solution. As in Problem 2 above, make the multiplier integral by moving the decimal point to the right one place, and in the multiplicand move it one place to the left. The problem now becomes .014 x 4 = .056. In each of the above problems there are as many decimal places to the right of the decimal point in the product as there are in both the multiplicand and the multiplier together. Multiply : 3. .75 by .42. 6. 28.056 by .057. 4. .7093 by .49. 7. .800694 by .17. . 5. 6.028 by .072. a .006294 by .00863. 9. .0976 by 24^. Change ^^ to a decimal fraction. 10. 25864 by .03972. 11. 50638 thousandths by 9026 hundredths. 12. 5063087 ten-millionths by 6204 thousandths. 13. 48 tenths by 48 ten-thousandths. 14. 9 hundredths by 7 thousandths.' 15. 16 ten-thousandths by 6 thousandths. 16. 8 thousands by 9 thousandths. 17. 17 hundreds by 4 hundred-thousandths. 18. 23 tenths by 4. 19. 18 ten-thousandths by 4 hundredths. 20. What is the product of tenths and thousandths? of thousandths and hundredths ? of hundreds and thousandths ? of thousands and hundredths ? Digitized by VjOOQIC r DIVISION OF DECIMAL FRACTIONS. 109 21. .125 is what part of 1 ? of 1 ten ? 22. .375 is what part of 1 Z .0375 is what part of .1 ? 23. .0025 is what part of .01 ? 24. .0625 is what part of .1 ? 25. .006 is what part of .01 ? 26. .00075 is what part of .001 ? 27. .00037 is what part of .001 ? 2a Read .00125 as a part of .01. 29. What is the cost of 864 bu. of oats at f 0.41 per bushel ? 30. What is the cost of 17 horses at f 112.375 each ? 31. What is the cost of 384 A. of land at f 67.65 each ? 32. What is the cost of 18.56 yd. of cloth at f 2.5625 a yard ? 33. What is the cost of 29 books at $0.0625 each ? 34. What is the cost of 465.375 bu. of wheat at f 0.9175 per bushel ? 35. At 6 cents a pound, how much will 350 lb. of old iron bring ? 36. In one rod there are 5.5 yd. How many yards are there in 7.25 rd. ? 37. Find to the nearest cent the value of 243.6 bu. wheat at 76.5 cents per bushel. 38. A man hauled 15 loads of coal, averaging 42.33^ hun- dredweight per load. How much coal did he haul ? DIVISIOK OF DECIBCAL FRACTIONS. 183. There are Wieri general types in the division of decimal fractions. I. The dividend may be a decimal fraction and the divisor an integer. Thus, .216 -i- 6 = .086. II. The dividend may be an iateger and the divisor a decimal fraction. Thus, 216. -i- .6 = 860. Digitized by VjOOQIC 110 DECIMAL FRACTIONS. vi III. Both the dividend and the divisor may be decimal factions. Thus, .216 -!- .06 = 3.6. (1) .216 -^ 6 = .036. Since the dividend is thousandths, the quotient is thousandths. 216 apples divided by 6 gives 36 apples for a quotient. (2) 216. -^ .6 = 360. ; for if the dividend and divisor be multi- plied by 10, the quotient will not be changed ; and the problem becomes 2160 -^ 6 = 360. (3) .216 -5- .06 = 3.6 ; for if the dividend and divisor both be multiplied by 100, the problem becomes 21.6 -s- 6 = 3.6, as ex- plained in (1) above. When the dividend and divisor are both decimals, the quo- tient contains as mauy decimal places to the right of the deci- mal point as the decimal places of the dividend exceed those of the divisor. Thus, 1.296 ^ .6 = 2.16. 184. The solution of each of the types of decimal divi- sion is a result of the application of the general definition of division given in § 88. I. To divide .216 by 6 is to do to .216 what must be done to 6 to produce unity. ^ of 6 produces unity, so the required ^ .036 quotient is found by taking | of .216. .216-5-6 = i of .^;[^ = .036. ^ II. To divide 216 by .6 is to do to 216 what must be done to .6 to produce unity. .6 must be multiplied by ^^ to pro- duce unity, so 216 must be multiplied by ^ to produce the 36 ^r. required quotient. 216 -j- .6 = ^;^ x ^ = 360. P III. To divide .216 by .06 is to do to .216 what must be done to .06 to produce unity. .06 must be multiplied by J^ to pro- duce unity, so .216 must be multiplied by -^ to produce the .036 ^^ required quotient. .%l^ X ^ = 3.6. P Digitized by VjOOQIC + DIVISION OF DECIMAL FRACTIONS. Ill 6. .042864-^.24 = ? 7. .008399 -f. 37 = ? 8. ,010867 -^ 46 = ? 9. 4.20638-5-55 = ? 10. 436.0095-5-77 = ? EXERCISE 70. 1. .7658-5-7 = ? 2. .04536-5-6 = ? 3. .157032^9 = ? 4. 71.40636 -f. 12 = ? 5. 146.0736-5-16 = ? 11. .875 -5- .5 = ? Solution. The dividend is the product of the divisor and the quotient, and the product contains as many decimal places as both the multipliei and the multiplicand. Hence, the number of decimal places in the quotient is equal to the number of decimal places in the dividend minus the number in the divisor. As there are three decimal places in the dividend and one in the divisor, there are two in the quotient. 12. 87.5 -5- .005 = ? Solution. Annex ciphers to the dividend until the num- ber of decimal places equals the number of those in the divisor. 13; .6241 -5- .79 = ? 14. 1.0276 -5- .028 = ? 15. 44.814-^.97 = ? 1§. .39071 -^ .0089 = ? Form. .6 ).876 1.76 FOBM. .00 6)87.600 17600. 17, 20. .02336081-^.00583 = ? 21. 2099.274-^.3607 = ? 22. 26624.32 -f. .4379 = ? 23. 5481 -f. .063=? 24. 48760^.0092 = ? 25. 766300 -5- .00079 = ? 26. 133574^.000329 = ? 091512 -5- .0124 = ? 18. 7.5522-5-2.46 = ? 19. .153032^.00376 = ? 27. 24980020 -f. .0406 = ? 28. .7 -h 43 = ? Carry the quotient to 5 decimal places. 29. 31. 4.6 -J- 58 = ? Carry to 6 decimal places. 63. -5- 97 = ? Carry to 4 decimal places. .1 -5- 329 = ? Carry to 6 decimal places. 32. 4. -f- 586 = ? €arry to 8 decimal places. Digitized by VjOOQIC 112 DECIMAL FRACTIONS. 33. If 30.75 yd. of calico are valued at 9 153.76, find the value of 1 yd. 34. If 1.25 of a bushel of wheat cost $ 1.00, what is the value per bushel ? 35. If 6.75 lb. of sugar cost 40.5 cents, what is the price per pound ? > THE METRIC SYSTEM. 185. The Metric System of weights and measures is named from the meter ^ which is the primary unit^ and from which all other units of the system are derived, 186. This system, on account of its extreme simplicity, has been adopted by nearly all civilized nations. It was adopted first by France in 1799. The United States and Great Britain are notable exceptions; however, its use has been legalized in these two countries. 187. The meter is approximately one ten-millionth part of the distance on the earth's surface from the equator to the pole. 188. The multiples and submultiples (fractions) of the meter, and the units derived from it, are indicated by prefixes. The Greek prefixes deka^ hecto^ hilo^ and myria^ are used to indicate the multiples of the units ; and the Latin pre- fixes, ded^ centi, and milli^ are used to indicate the frac- tional parts of the units. Milli means thousandth, centi means hundredth, deci means tenth, deka means ten, hecto means hundred, kilo means thousand, and myria means ten thousand. 189. Hence, this is a system in which the scale is uni- form and is 101 It is, therefore, a decimal system. Digitized by VjOOQIC THE METRIC SYSTEM. 113 Linear Units. 190. In the following table it is evident that the scale is 10. Note the abbreviations. TABLE. 10 millimeters (mm.) = 1 centimeter (cm.) 10 centimeters = 1 decimeter (dm.) 10 decimeters = 1 meter (m.) 10 meters = 1 deli^ameter (Dm.) 10 dekameters = 1 hectometer (Hm.) 10 hectometers = 1 kilometer (Em.) 10 kilometers ss 1 myriameter (Mm.) SuBFACE Units. 191. The surface units are squares, each of whose sides is a linear unit. It follows that 100 of each order make one of the next higher. The scale is 100. TABLE. 100 sq. mm. = 1 sq. cm. 100 sq. cm. = 1 sq. dm. 100 sq. dm. = 1 sq. m. 100 sq. m. =1 sq. Dm. 100 sq. Dm. = 1 sq. Hm. 100 sq. Hm. = 1 sq. Km. Note 1. The square meter is used in the measurement of small surfaces, as the square yard is used in the ordinary system. Note 2. The square dekameter is called an are when used in land measure, and the square hektometer is called a hectare when so used. Digitized by VjOOQIC 114 DECIMAL FRACTIONS. Units of Volume. 192. The volume units are cubes, each of whose edges is a linear unit. In the following table, 1000 of each denomi- nation make one of the next higher. The scale is 1000. TABLE. 1000 CO. mm. = 1 cu. cm. 1000 cu. cm. = 1 cu. dm. 1000 cu. dm. = 1 cu. m. = 36.316 cu. ft Note 1. The cubic meter is the unit most commonly used. Note 2, When the cubic meter is used in measuring wood, it is called a stere. The stere is a little more than a quarter of a cord. Note S. One tenth of a stere is a decistere, 10 uteres make a dekastere (Dst.). Units op Capacity. 193. The primary unit of capacity is the liter. The liter is equal to a cubic decimeter. The scale is 10. TABLE. 1 milliliter (ml.) = 1 cu. cm. 10 ml. = 1 centiliter (cL) 10 cL = 1 deciliter (dl.) 10 dl. =1 liter (1.) = 1 cu. dm. 10 1. =1 dekaliter (Dl.) 10 Dl. = 1 hectoliter (HI.) 10 HI. =1 kiloliter (KL) = 1 cu. m. 10 KL =1 myrialiter Units op Weight. 194. The unit of weight is the gram^ which is the weight of one cubic centimeter of pure water at the temperature of greatest density. The scale is 10. Digitized by VjOOQIC REDUCTION OF METRIC NUMBERS. 115 TABLE. 10 milligrams (mg.) = 1 centigram (eg.) 10 eg. = 1 decigram (dg.) 10 dg. = 1 gram (g.) 10 g. =1 dekagram (Dg.) 10 Dg. = 1 hectogram (Hg.) 10 Hg. = 1 kilogram (Kg.) = wt. 1 cu. dm. of water 10 Kg. = 1 myriagram (Mg.) ^^^^ 10 Mg. = 1 quintal (Q.) 10 Q. =1 tonneau (T.)= wt. 1 cu. m. of water 195. TABLE OF EQUIVALENTS. 1 decimeter = 3.93 in. 1 meter = 39.37 in. = 3.28 ft. 1 kilometer = } mi. (nearly) lare = ^ of an acre 1 stere = 1 cu. m. = 35.32 cu. ft. 1 liter = .908 dry quart, or 1.067 liquid quart 1 gram = 16.432 gr. 1 kilogram (Kilo)= 2.204 lb. Avoirdupois (2\ lb.) REDUCTION OF METRIC NUIIBERS. 196. A metric number may be reduced from one de- nomination to another by moving the decimal point to the right or to the left according to Prin. 1 and Prin. 2, §§ 180, 181. EXERCISE 71. 1. Eeduce 6453 m. to kilometers. Solution. Since 1000 m. = 1 Km., 6453 must be divided by 1000 to get the number of Km.; therefore, move the decimal point to the left three places. Hence, 6453 m. = 6.453 Km. a. Reduce 29.463 Dl. to deciliters. • From the table, 1 Dl. = 100 dl. Hence, 29.463 Dl. = 2946.3 dl. Digitized by VjOOQIC 116 DECIMAL FRACTIONS. 3. Eeduce 26,709,853 cu. dm. to cubic centimeters; to cubic meters ; to decisteres ; to dekasteres. 4. Reduce 47,073 1. to centiliters ; to dekaliters ; to kilo- liters ; to deciliters. 5. Reduce 279,436 Dg. to grams ; to milligrams ; to myria- grams ; to quintals. 6. Write 4 Km.^ 5 Hm., 2 m.^ 8 cm. as meters. 7. Write 73 HI., 5 1., 17 cL as liters ; as kiloliters. a Reduce 59.74 Kg. to myriagrams ; to grams ; to milli- grams. 9. Reduce 75,432 mg. to hectograms; kilograms; deka grams; decigrams. 10. Reduce 2.15 cu. m. to cubic millimeters ; to steres. U. In 314 1. there are how many cubic decimeters? hecto- liters ? cubic meters ? cubic dekameters ? 12. In 75,413 cu. cm. there are how many liters ? hectoliters ? cubic meters ? 13. What is the weight, in grams, of 34.25 cu. cm. of water ? in hectograms ? in kilograms ? in centigrams ? 14. What is the weight in kilograms of 24 HI. of water ? 15. In 347.2946 HI. of water, how many cubic meters are there? What is its weight in tons ? in grams? in milligrams ? 16. Find the sum of 6 Dm., 5713 dm., 97 m., 75 Km., 4934 mm., and .34 Mm. ' 17. Multiply 59.135 g. by 24; by .13; by 3.04. 18. Divide 3645 HI. by 9 cl. ; by 90; by 4.5. 19. What is the weight of the water in a tank, if it takes 54 min. to empty it at the rate of 12 dl. per minute ? 20. if the tank in Problem 19 were filled with wine at $9.00 per dekaliter, what would the contents be worth ? Digitized by VjOOQIC SPECIFIC GRAVITY. 117 SPECIFIC GRAVITY. 197. Weigh a piece of lead very accurately and then place it in a jar which is brimful of water. Weigh accurately the water which runs over, and divide the weight of the lead by the weight of the water which ran over. The quotient is the specific gravity (sp. gr.) of lead. It is the number by which the weight of a bulk of water must be multiplied to produce the weight of an equal bulk of lead. 198. The Specific Gravity (sp. gr.) of any substance is the relation that exists between the weight of a given bulk of it and the weight of an eqiuxl bulk of water, for water is taken as the standard of reference. Note. One cubic foot of water weighs 1000 oz., or 62} lb. Avoirdupois. 199. TABLE OF SPECIFIC GRAVITY. The specific gravity of gold is 18.6 The specific gravity of lead is 11.4 The specific gravity of silver is 10.5 The specific gravity of copper is 8.8 The specific gravity of iron is 7.2 The specific gravity of slate is 2.5 The specific gravity of milk is 1.03 The specific gravity of ice is .92 The specific gravity of oil is .90 The specific gravity of cork is .25 Note, It the specific gravity of cork is .25, this means that a cubic foot of cork weighs one fourth as much as a cubic foot of water. Therefore, cork thrown upon water does not sink. EXERCISE 72. 1. What is the weight of 1 cu. ft. of lead ? Solution. Since lead has a specific gravity of 11.4, the weight of J cu. ft. of it equals 1000 oz. x 11.4 = 11,400 oz. = 712.5 lb. Avoirdupois. Digitized by VjOOQIC 118 DECIMAL FRACTIONS. 2. What is the weight of 1 cu. ft. of cork ? 3. What is the weight of 6 cu. ft. of iron ? 4. What is the weight of 3 cu. ft of ice ? 5. What is the weight of 5 gal. of milk ? 6. A piece of slate weighs 200 lb. How many cubic feet does it contain ? Solution. A bulk of water weighing 200 lb., or 3200 oz., would con- tain 3.2 cu. ft. The specific gravity of slate is 2.5. Hence, the piece of slate contains 3.2 cu. ft. -»- 2.5 = 1.28 cu. ft. 7. A piece of cork weighs 5 lb. How many cubic inches does it contain ? a How many ounces Avoirdupois are there in 5 qt. of oil ? 9. If 20 cu. ft of ebony weighs 1500 lb., find its specific gravity. 10. A piece of copper is 6 in. long, 4 in. wide, and 2 in. thick. How much does it weigh ? 11. A piece of silver equal in bulk to 3 qt. of water weighs how many pounds ? 12. If ijiarble (specific gravity 2.83) is worth $26 per cubic inch, find the value of a block weighing 625 Kg. 13. Find the weight of 1 cu. dm. of copper. Solution. 1 cu. cm. of water weighs 1 g. The specific gravity of copper is 8.8. Hence, 1 cu. cm. of copper weighs 8.8 g. 1 cu. dm. =a 1000 cu. cm. Hence, 1 cu. dm. weighs 8.8 g. x 1000 = 8800 g. 14. Find the weight of 3 cu. cm. of gold. 15. Find the weight of 11 cu. cm. of silver. 16. Find the weight in kilograms of 8 cu. dm. of iron. 17. Find the weight of 7 1. of oil. la Find the weight of 450 cu. cm. of slate. 19. Find the volume of 22.8 Kg. of lead. Solution. The specific gravity of lead is 11.4 ; then 22.8 Kg. of lead have the same bulk as 2 Kg. of water. 1 cu. dm. of water weighs 1 Kg., and 2 cu. dm. weigh 2 Kg. Hence, the volume of 22.8 Kg. of lead is 2 cu. dm. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 119 20. What is the volume of 460 g. of ice ? 21. What is the volume of 1500 Kg. of slate ? 22. What is the volume of 500 g. of cork ? 23. What is the weight of silver equaling the bulk of 420 ^%, of gold ? 24. What is the weight of iron equaling the bulk of 3700 eg. of cork ? 25. A cubic foot of water weighs 62^ lb. What is the weight of 12 cu. ft. of ice ? y 26. What is the weight of a cubic foot of Joliet limestone which is 2| times as heavy as water ? of dry pine, ^ as heavy ? ^ 27. A gallon of water weighs approximately S\ lb. What is the weight of a gallon of mercury 13 f times as heavy ? of a gallon of milk ? y^ 2a Standard silver is -^ pure. How many grains of silver are there in the standard dollar of 412 J gr.? ^ 29. An iceman delivered a block of ice 1 ft. long, f of a foot wide, and f of a foot thick, and charged for 50 lb. What was the shortage in weight, a cubic foot of water weighing 62J lb., ice weighing ^^ as much ? 30. A steel beam is 16 ft. long, 2f in. thick, and 14 in. wide. What is its weight, its specific gravity being 7.84 ? 31. What is the weight of a block of limestone 6^ ft. X 1\ ft X 4| ft, its specific gravity being 2.62 ? MISCELLANEOUS PEOBLEMS. EXERCISE 73. 1. Change .00875 to a common fraction. 2. Change ^ to a decimal fraction. 3. How many square feet are there in a piece of ground 86.48 ft long and 39.6 ft. wide ? 4. If a man travels 29.6 mi. a day, in how many days will he travel 1016.088 mi. ? Digitized by VjOOQIC 120 DECIMAL FRACTIONS. 5. Change ^ to a decimal fraction. 6. What is the cost of 473.5 bu. of corn, at .47 of a dollar a bushel ? 7. At .47 of a dollar a bushel, how many bushels of corn can be bought for $ 222.545 ? Qa Add 34f thousandths, 568J' hundredths, 634^ tenths, ^ of a hundi^edth, and ^ of a ten-thousandth. 9. From ^ of a tenth take ^ of a thousandth. ^ 10. Change .QOJ^ to a common fraction. pO I ^-^ 11. Change .OJ lo a common fraction. ^ ^ 12. Change -j^ to a decimal fraction of three orders. ' " 13. If 97 books cost f 317.675, what will each cost ? 14. If 38.5 bales of cloth cost $3048.43, what is the price of each bale? 15. If a pair of shoes costs $2,625, how many pairs can be bought for $49,875? 16. At 83 cents a yard, how many yards of cloth can be purchased for $61,005? 17. At $.04$ per pound, how many pounds of sugar can DC purchased for $1.89? la Define a decimal fraction. How does it differ from a common fraction ? 19. Find the L. C. M. of 18, 24, 36, 40, 180. 20. Define a pure decimal ; a mixed decimal. 21. 23fx4f = ? 83Jx25^=:? 3^X24.08 = ? 22. 693}^7 = ? 826f-^9=:? 938^^^-15 = ? 23. Tell how to change a decimal fraction to a common fraction. Give two ways of changing a common fraction to a decimal fraction. 24. ^1>LM^? :2§i2<i = ? Digitized by VjOOQIC % MISCELLANEOUS PROBLEMS. 121 25. What common fractions can be changed to pure deci- mals? 'Explain in two ways why this is so. 26. 2\ is what part of 15? of 20? of 31? of 6|? of 8}? of 18|? of 50? of 100? ' 27. The liquid gallon contains 231 cu. in., and the bushel contains 2150.42 cu. in.; find to the nearest thousandth the number of liquid gallons in a bushel. 28. A cubic foot of fresh water weighs 62.5 lb., and cast iron is 7.21 times as heavy as water, bulk for bulk. How many cubic feet of cast iron weigh 3000 lb.? Simplify : 29. ?i«lii;and?i:=ii. 2iof6i' 2f^| 32. | + lJof2f^i + A' 33. 3J-Ao£2J-1.125. 2i-i^ Qfl f • ^ot3i + tf ^35^1720- {f of (28-7^)1] -i-[40i + (A-|.f)]. ' 36. An adult inspires 30 cu. in. of air at an ordinary inspi- V ration. If they breathe 18 times per minute, how long will it take 50 adults to breathe once all the air in your schoolroom ? 37. If each expiration vitiates a cubic foot of air, how rapidly should the air be changed in the above room ? 38. Allowing 1\ cu. ft. to the bushel, what is the capacity of a bin 6 ft. X 8 ft. x 10 ft. ? 39. A bushel of com in the ear occupies 2\ cu. ft. To what height must a rail crib 9 ft. square be filled to hold 300 bu. ? 40. How many bushels of corn can be bought for $1936.40 at 23^ cents a bushel ? Digitized by VjOOQIC 122 DECIMAL FRACTIONS. 41. Sold 1836 bu. of wheat at 61 1 cents, and invested the proceeds in corn at 24| cents. How many bushels did I buy ? 42. Built a fence around a square lot 400 ft. on a side The posts were placed 8 ft. apart. The boards were 16 ft. long and each contained 8 ft. of lumber. If the fence was 5 boards high, what was the cost of the material (not counting the nails), if the posts cost 18 cents each and the lumber cost $18.50 for 1000 ft. ? 43. What is the cost of the lumber for flooring and ceiling a room 24 ft. and 3 in. wide and 84 ^t. long, if the lumber costs 3^ cents a square foot ? 44. A field is 40 rd. (16J ft.) wide and 80 rd. long. What will it cost to plow the field at $2.25 an acre (160 sq. rd.) ? EXERCISE 74. (Mental.) 1. John divided an orange, giving Henry ^ and Sarah f . How many fifths did he give away and how many did he have left? 2. How much are ^, f , and ^ ? / 4. a + i)x2 + |-3xf = ? (^ ' 5. .5 + .3X 5-1.6 = ? 6. A man gave to some children f 3.00, which was f of his money. How much had he left ? 7 At f J per yard, how many yards of cloth can be pur- chased for f 3| ? 8. .7 of what number is 21 ? 9. .15 of 80 is what? 10. 16 is how many hundredths of 200 ? 11. How many times 6 in | of 40 ? 12. Howmany times8in .12of 800? 13. How many times f of 12 in f of 72 ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 123 14. How many times .5 of 24 in .6 of 60 ? 15. How many ties will be used in a mile of railroad, if there is 1 tie for every 2 ft.? 16. If a horse takes 1800 steps in a mile, and makes each step every half second, what is his rate per hour ? 17. If cord wood is piled 6 ft. high, how many cords are there in a pile 20 ft. long ? 18. If cord wood is piled 8 ft. high, how long must a pile be to contain 20 cd.? 19. 1^ of f of ^ of a number is 11. Find the number. . 20. I of 1^ of ^ of a number is 10. What is the number? 21. (.8 + 1.2) ^(1.4 -.6) = ? 22. (^-|)x7 + JofT% = ? 23. Find the least number of boys that could be formed into groups of 3 or 4 or 5 boys. 24. |a: + |iB + ia;-fa; = ? 25. |a + fa-fa + |a = ? 26. 1.3 6 + 5.7 6 - 4.3 6 + 2.7 6 = ? 27. .25 y + .15 y - .08 y - .16 y + .04 y = ? 28. At 3 yd. for 2 cents, how many yards can be bought for 9 cents ? 29. At 3 lb. for $ 1.00, how many pounds of coffee can be boughtfor f 2.50? 30. f of I of a number is 750. What is the number? 31. f of a number plus ^ of | of 24 is 24. What is the number? 32. How much com will 13^ A. produce, if 4J A. produce 180 bu.? 33. When 5 qt. of milk cost ais much as 6 loaves of bread, what is the cost of milk per quart, if bread is worth 4 cents per loaf? Digitized by VjOOQIC 124 DECIMAL FRACTIONS. 34. A boy earns $ .80 a day, and his father 1| times as much. How long will it take both to earn $ 42.00 ? 3». aofM + *off)-hf + ^of(t + i)=? 36. (62 X .5 + 20 X .8 + 3) X 12^.3 = ? 37. {(1.7 - 1.3) X (2.8 + 2.2) + 316] -^ .07 = ? 38. (36 X 2 + 4 X 7 ^ 12.5) x 5 + 1 of 12 - | of 32 = ? 39. ^ of II of 28 is ^ of what number ? 40. At 48 cents a dozen, how much will 10 oranges cost ? 41. Find the L. C. M. of 5, 6, 7, 10, and 12. 42. Find the G. C. D. of 540, 630, and 720. jot ^ ofH _o • lof^s^of^-- 44. What two consecutive integers multiplied together make 110? 132? 420? 600? 45. Of what number is 2.65 one of the four equal addends ? 46. Of what number is 8.575 both remainder and subtrahend ? 47. Of what number is 2.5 both quotient and divisor ? 48. Of what number is 9.637 both multiplicand and product? 49. Name all the divisors of 48; 60; 72; 75; 84 EXERCISE 75. x}^ 1. Multiply 8 hr. 23 min. 46 sec. by 6. -^ / 2. Divide 50 hr. 22 min. 36 sec. by 6. 3. (24 - 6) H- 3 = ? 24 - 6 -^ 3 =c ? ^ 4. (3 4-4) x2 = ? 3 + 4x2 = ? '/^ '" /^ -A 5. 8x2-(6-2)H-2x44-2xl3^6 = ?r .cyJ^^-'' .> \j 6. A man's crop of oats brought him $ 493.90, the selling price being 22 cents a bushel. The field in which he raised them contained 45 A. What was the average yield per acre ? V 7. Bought wheat at 63 cents a bushel, paying for it $ 2844.12. How many bushels were bought ? j^ Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 125 ^ a At 21 cents a bushel, how many bushels will $ 188.74 buy? A 9. The President of the United States receives $60,000 a year. How much is that for each day of 1904 ? 10. An Illinois farmer sold his farm, consisting of 320 A., at $ 82.50 an acre, and invested the proceeds in western land at $ 30 an acre. How many acres did he buy ? U. A man sold a piece of property for $ 3825 ; another for $4682; a third for $5620. After paying a debt of $6327 he invested the remainder in land at $ 65 an acre. How many acres did he get ? 12. Make a receipted bill of the following items, using the name of a classmate as purchaser with your own as seller : 24^ lb. sugar at 6^ cents. 18 lb. coffee at 27J cents. 15 bu. potatoes at 36|^ cents. 15^ lb. butter at 25 cents. 13. If $ 80 is paid for the labor of 12 men for 6 da., at the same rate what should be paid for the labor of 18 men for 24 da. ? ^ 14. A and B start from Chicago and Louisville, traveling 16 and 17 mi. per hour respectively. How soon will they meet? How far from Chicago will they meet, the distance between the cities being 320 mi. ? <^ 15. A fast train starts 2\ hr. after a slow one. Their rates are 50 and 30 mL respectively. When will the fast train over- take the slow one ? 16. I bought 60 articles at a total cost of $8765. I sold them at a profit of $ 22.50 each, and invested the proceeds in land at $ 75.00 per acre. How many acres did I buy ? 17. Find the prime factors of 1369. -Q^ 5040 X 999 X 198 « 18 X 37 X 27 "* Digitized by VjOOQIC 126 DECIMAL FRACTIONS. 19. I owe a certain sum to A and 3 times as much to B. The two debts are f 172.80. I am to pay these debts in a year by equal monthly payments. How much must I pay to each man a month ? 20. Find the least number to be added to 1137 to make the result divisible by 23. 21. 1^ and I of a certain sum of money added together make f 266. What is the sum of money ? 22. A man left one child ^ of his property, another child | ^ of his property, and the remaining $300 to a third child. How much did he leave, and what did each child get ? 24. Using your own name as buyer and some classmate as seller, make out a bill for the following goods and receipt it : 19 yd. calico at 6J cents. 36 yd. sheeting at 12^ cents. 28 yd. flannel at 37^ cents. 27 yd. ribbon at 24 cents. 25. G. G. Edwards bought from Hines & Co. the following items: ^ j^ ^^^ ^^ n^ ^ents. 42 lb. mutton at 14^ cents. 12 lb. lamb at 16f cents. 30 lb. pork at 12^ cents. Make a bill and receipt it. 26. Henry Ives bought from Ayers & Co. the following bill : 15 doz. shirts at $9.00 per dozen. 18 doz. undershirts at $4.50 per dozen. 50 pairs shoes at $ 2.25 per pair. 30 suits of clothes at $8.15 a suit Make out the bill and receipt it. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 127 27. Eeduce | of a foot to the fraction of a rod. Solution. 1 ft. is J of a yard, f of a foot is f of J of a yard, which is 5^ of a yard. Since there are J^l yd. in a rod, J of a yard is ^ of a rod, and 1 yd.^is ^ of a rod. ^ of a yard is ^ of |^ of a rod, which is j|j of a rod. Short form : f x J x |«r = x|^. 28. 6 ft. are what part of a rod ? of a mile ? 29. ^ of an inch is what part of a yard ? of 2 yd. ? of 2Jyd.? 30. 2 ft. 3 in. are what part of a yard ? of a rod ? 31. 1 ft. 9 in. are what part of a yard ? of a rod ? 32. 4 yd. 2 ft. are what part of a rod? of a mile? 33. f of a foot is what part of a rod ? 34. 2J ft. are what part of a rod ? 3| ft. ? 4^ ft. ? 35. 3^ of a rod is what part of a mile ? 2^ rd. ? 3| rd. ? 36. Eeduce to the fraction of a rod : .12 of a foot ; .015 of a foot ; .08 of a yard ; .7 of an inch. 37. Eeduce to the fraction of a mile : .26 of a foot ; .375 of a yard ; .875 of a rod. 3a Eeduce ^ of a foot to the fraction of 2 mi. ; of 3| mi. "^as. Change 3 yd. 2 ft. 3 in. to the decimal of a rod. Solution. 3 in. are J of a foot ^ =: .25. 2.26 ft. = J as many yards = .76 yd. 3.75 yd. = ^ as many rods = .68+ rd. 40. Change 4 yd. 2 ft. 6 in. to the decimal of a rod. (Eeject all terms below fourth place.) CL Change 3 rd. 5 yd. 1 ft. 8 in. to the decimal of a mile. 42. Change to the decimal of a mile : 4 rd. ; 6 rd. ; 20 rd. ; 10 rd. 3 yd.; 50 rd. 5 yd. 2 ft.; 80 rd. 2 yd. 2 ft. 9 in. 43. From 3 mi. 10 rd. 3 yd. take 1 mi. 27 rd. 5 yd. 2 ft. 11 in. 44. Eeduce 87,653 sq. in. to higher denominations. 45. Eeduce -^ sq. mi. to lower denominations. Digitized by VjOOQIC 128 DECIMAL FRACTIONS. 46. Eeduce ^ cu. yd. to cubic feet and cubic inches. 47. Divide 8 mi. 186 rd. 4 yd. by 7. 48. Divide 71 A. 82 sq. rd. into 8 equal parts. In the next ten problems a = .5, 6 = 1.6, c = 3, dwm2. 49. ac-^6d4-(o + &)c^(l=t? 50. {acd -J- a6) X (ac 4- 6d — c) = ? -- ax^.cd c ,ad o '^ -6+7-^ + 26 = ^ 52. (a + b + c + d)d-t-(a+c-b) + 2ac^f 53. (ac + M)fac-^^-/'a6c+^ = ? 54. f3ab + ^ + 5abcd = ? 55. [16a6-(6cd-8o)]-<-/^2 + ^=5? 56. a6cd + [4ac-(M-3a)]+2^=? cd a 6 Digitized by VjOOQIC CHAPTER VII. THE EQUATION. THE USE OF QCi IK PROBLEMS. If a? + 3 = 8, what does x stand for ? The question might be read, " What number added to 3 makes 8 ? " The answer is evidently 5. 201. In the following problems such expressions as Zx^hxy etc., mean 3 times Xy 5 times x, etc. x means 1 time x, or 1 x, EXERCISE 76. Find what number x stands for if: 1. aj+7=10. 5. 7x=21. 9. 3aj+6aj=32. 2. 9+«=16. 6. 3aj+ 5=29. 10. 9aj-2a;=35. 3. aj-3=12. 7. 7 +2aj=17. U. 7aj+3aj-5a;=30. 4. 12-a;=2. a 17-2aj=13. 12. 5a;-2aj-Ml =17. 202. John and James together have 42 marbles. James has 5 times as many as John. Kow many has each ? Solution. John has a certain number, James has 6 times a certain number, together they have 6 times a certain number. We know that together they have 42 marbles, so that 6 times a certain number is 42. A certain number is, therefore, J of 42, or 7. 5 times a certain number is 5 X 7, or 35. So John has 7 marbles, and James 35 marbles. 203. Instead of representing John's marbles by a certain numh'.r we might have represented them by a?, or by any other convenient letter or symbol. 129 Digitized by VjOOQIC 180 THE EQUATION. Solution. Let x = John's marbles. 6 a; = James's marbles. 6 x = all the marbles. 42 =r all the marbles. 6aj = 42. a; = J of 42 = 7, John's marbles. 5 a; = 5 X 7 = 35, James's marbles. DEFINITIONS. 204. An Equation is a statement in mathematical sym- bols that two expressions stand for the same number. 6 a? = 42 in the problem above is an equation. 6 a? and 42 each stand for the total number of marbles. 205. The Members of an equation are the two equivalent expressions, as 6 a: and 42 in the equation above. 206. Equations are used to find the value of unknown numbers represented by a;, y, 2, etc. 207. To %olve an equation is to find the value of the unknown number involved. 208. In a school of 45 pupils there are 7 more girls than boys. How many are there of each ? Solution. Let x = number of boys. Then, since, etc., x + 7 = number of girls ; and « + flc + 7 = number of pupils in school But 46 = number of pupils in school Therefore a: + a + 7 = 46. 2x + 7 = 46. 2a; = 38. X = 19, number of boys. X + 7 = 26, number of girls. 209. The various steps in the above solution were made in accordance with the following : Digitized by VjOOQIC N THE USE OF X IN PROBLEMS. 131 AXIOMS. 1. Things equal to tTie same thing are equal to each other. 2. If equals he added to equals^ the sums are equal, 3. If equals be subtracted from equals^ the remainders are equal. 4. If equals be multiplied by equals^ the products are equal. 5. If equals be divided by equals^ the quotients are equal. The equation a5 + aj + 7 = 45, m§ 208, is in accordance with Axiom 1. The equation 2 a? = 38 comes from the equation preceding it, by subtracting 7 from both members of the equation, and is in accordance with Axiom 3. The equation aj = 19 comes from the preceding equation, by dividing each side by 2, and is in accordance with Axiom 5. The equation a; + 7 = 26 comes from the preceding one by adding 7 to both members, and is in accordance with Axiom 2. EXERCISE 77. 1. Six times John's age exceeds four times his age by 22 yr. How old is he ? Let X = the number of years in John's age. If we say, let x = John's age, we treat x as a mere quantity of time, not as a number of time units. 2. f 21,000 is divided among three children so that the first receives twice as much as the second, and the second twice as much as the third. What is the share of each ? Let X = number of dollars in the share of the third. 3. Thomas, Eichard, and Henry have 72 marbles. Thomas has twice as many as Richard. Henry has twice as many as both the others. How many has each ? 4. How old am I, if three times my age four years ago ex- ceeds twice my present age by 27 yr. ? Digitized by VjOOQIC 182 THE EQUATION. 5. Equal weights of sugar and flour were bought for 63 cents. The sugar cost 5 cents per pound, the flour 2 cents. How many pounds of each were bought ? Let X = number of pounds of each. 6. The perimeter of a rectangular field 80 rd. long is 280 rd. What is its width ? 7. The perimeter of a rectangular field, twice as long as it is wide, is 180 rd. What is its length ? a It takes 70 ft. of border to inclose a square room. What are its dimensions ? 9. A room 27 ft. wide and x ft. long requires 99 sq. yd. of matting. What is the value of aj ? 10. A Sunday-school collection in dimes, nickels, and cents amoimted to 200 cents. There were three times as many nickels as dimes, and five times as many cents as nickels. How many were there of each ? U. Grace is 5 yr. older than May. May is 2 yr. older than Ethel. The sum of their ages is 42 yr. What is the age of each ? 12. A father is four times as old as his son. Five years ago he was seven times as old. What is the father's age ? Solution. Let x = number of years in son's age. Then (why ?) 4 a: = number of years in father's age. Then (why ?) x — 6 = number of years in son's age 6 yr. ago. Then (why ?) 7 (at — 5) = number of years in father's age 6 yr. ago. Then (why ?) 4 x — 5 = number of years in father's age 6 yr. ago. Hence (1) 7 (x - 5) = 4 x - 6. (2) 7 X - 36 = 4 X - 6. (3) 7 X = 4 X - 6 + 36. (4) 7 X - 4 X = 35 - 5. (6) 3 X = 30. (6) X = 10. (7) 4 X = 40, number of years in father's age. What was added to each member of (2) ? What was subtracted from each member of (3) ? Digitized by VjOOQIC THE USE OF X IN PROBLEMS. 133 13. A man of 35 is seven times as old as his son. In how many years will he be twice as old ? Solution. Let x = number of years hence, when the father's age will equal twice the son's age. Then 5 + x = number of years in sou's age at required time ; and 2 (5 + a^) = number of years in father's age at required time. But 36 + X = number of years in father's age at required time. Hence (1) 2 (6 + «) = 35 + x. (2) 10 + 2a5 = 35 + x. (3) 2a;-aj = 36-10. Note, In subtracting 10 and x from each member of Eq. (2) , we cause each of these terms to pass to the other member of the equation with change of sign. This transfer of a term to the opposite member with change of sign is called transposition. 14. A had 8 dollars more than B. After paying B 12 dollars, A has only \ as many as B. How much had each at first ? 15. John had 40 marbles more than Fred. After giving Fred 50, John has only \ as many as Fred then has. How many marbles had each boy at first ? 16. A debt of $ 102 is paid with an equal number of ten- dollar, five-dollar, and two-dollar bills. How many bills were paid in all ? 17. In paying 27 cents for an article, I tendered some dimes and received an equal number of cents as change. How many dimes did T tender ? la Harry and Walter, 62 mi. apart, ride toward each other. Harry, starting at 9 a.m., rides 2 mi. per hour faster than Walter, who started at 8 a.m. They meet at noon. What is the rate of each? 19. Take some number, double it; add 20, divide by 2, take away the first number ; you have 10 left. Why is this ? If you had added 30, instead of 20, how many would you have left ? 20. Take some number, multiply by 6, add 30, divide by 3, subtract 4, divide by 2, take away the first number; you have 3 left. Explain. Digitized by VjOOQIC 134 THE EQUATION. EQUATIONS CONTAINING FRACTIONS. EXERCISE 78. 1. \ of what number = 4 ? 2. \ of what number = 3 ? 3. ^ of what number = 7 ? 4. ^ of what number = 10 ? 5. ^ of a number -f J of the same number = 21. What is the number ? 6. I of a number — ^ of the same number = 10. What is the number ? 7. f of a number = 7. What is twice the number ? 8. f of a number = 11. What is four times the number ? 9. ^ = 5. Find the value of x, 3 By what number must we multiply each member to obtain the value of a? ? 10. ^ = 36. Find the value of a?. 5 By what number must we multiply each member to obtain the value of 4aj? What axiom is involved ? 11. ? 4- 1^ = 17. Find the value of a. 3 5 Solution. To make the first fraction integral, we must multiply it by 8, or some multiple of 3 ; to make the second fraction integral, we must multiply it by 5, or some multiple of 6 ; to preserve the equality of the members, we must multiply both by the same multiplier. 15 is the least common multiple of 3 and 5. Multiplying the first fraction by 15 (first by 3 to suppress the denominator, and then by 6), we have 5x. Multiply- ing the second fraction by 15 (first by 6, and then by 3), we have 12 x. Multiplying the second member by 15, we have 255. Our equation now stands : bx + \2x = 266. 17 x = 255. x = 15. Note, This process of transforming a fractional equation into an integral equation is called clearing of fractions. Digitized by VjOOQIC VERIFYING AN EQUATION, 135 Solve, finding the value of x: 12. :^ + 'l? = 31. 19. 2j?_| + | = 22. 5 4 3 2 5 13 ^_H0 = ^_1 20. ^a? + ia:-iaj = 25. '32' X Note. \x means J- 14 ^_?^ = 1 '43* 21. |aj + iic-^« = 82. 15. 5-6 = --2. 22. 2fa? = 105. 16. 11-1-5 = ^+8. ^ 24. 3ia;-2|ic = 45. 7 7 ^^- 2 + i-^i = ^^- 25. 5ia.-3ia. = 44. 18. 5 + 5_| = 22. 26. aj + 2ic + ^ = 36. 4 2 5 5 VERIFYING AN EQUATION. 210. If the members of an equation are alike in form, or if they are reducible to the same form, the equation is called an Identical Equation, or an Identity. . 9 = 9, 5 + 2 — 4 = 3, and 5a; — 7 = 3a? — 7 + 2a?are identities. 211. A solution is verified by substituting for x in the given equation the value of a:, as found in the solution, and performing all indicated operations in each member. If the equation reduces to an identity, the solution is correct. lUustratuM. 3^2^ 7^^18^. 5 11 10 66 + 44aj + 70aj+ 60 = 198aj. 114ic + 126 = 198aj. 126= 84 aj. Digitized by VjOOQIC 136 THE EQUATION. VerifiGation. 6 ' 11 10 3 + 3 ,10i + 6_ 27 6 ' 11 - "10" 6 16i_ 6"^ 11 " 27 = 10' 6 + 3- 6 + 2 27 = Io- 27 ■ 10= 27 "10' 212. If, upon substituting for x its supposed value, the equation becomes an identity, the value of x is said to Botiafy the equation. EXERCISE 79. Solve and verify : 1. Divide 90 into two such parts that one shall be 3J- times the other. 2. Divide 100 into two such parts that one shall be 2\ times the other. 3. A horse was sold for $80, at a gain of \ of the cost. What was the cost ? 4. A is 12 yr. older than B. \ of A's age = | of B's. "What is the age of each ? 5. If to John's age there be added its half, its third, and its fourth, the sum is 26 yr. What is his age ? 6. If to Mary's age there be added its half, its third, and its fifth, the sum is 2^ times her age. What is her age? What is the matter with the foregoing problem ? 7. If to A's age there be added its double, its half, and its third, the sum lacks 7 yr. of four times his age. What is his age? Digitized by VjOOQIC VERIFYING AN EQUATION 1S7 8. Two barrels contained respectively 42 and 50 gal. of oil. After the same amount was drawn from each, the first contained ^ as much as the second. How much was drawn from each ? 9. A campaign pole 84 ft. high broke at such a point that the top was \ of the stump. What was the height of the stump? Let X = the height of the stump. 10. A campaign pole 100 ft. high broke at such a point that the top was 6 ft. longer than the stump. What was the length of the stump? 11. The sum of two numbers is «, the difference d. What are the numbers ? Solution. Let x = the smaller number. Then « + d = the greater number, and x-^x-^ d — 8, 2 x + d = ». 2x = 8-d. X = ^~ , smaller number. 2 x + d = 111^4. 2^ = L±J, greater number. 2 2 2 In solving the preceding problem we have solved every problem in which the sum and difference of two numbers are given to find the numbers ; for s and d may be any numbers. By substituting the values of s and d in any particular prob- lem of this type, we avoid a formal solution. 12. The sum of two numbers is 42, their difference 12. What are the numbers? Solution. Greater number = ^i^ = l^jMg ^ 54 _ 27. 2 2 2 SmaUer number = i:=-^ = ^^ = - = 16. Digitized by VjOOQIC 138 THE EQUATION. 13. Find, at sight, the numbers whose sum and difference are given: Sam. DiflSsrenoe. Sum. Difference. (1) 30 20 (7) 88 12 (2) 30 6 ( 8) 33 3 (3) 22 8 ( 9) 62 12 (4) 66 6 (10) 26 6 (5) 44 6 (11) 13 9 (6) 23 7 (12) 61 19 MISCELLANEOUS PROBLEMS. EXERCISE 80. 1. Find the sum of $83.2, $632.7, $504.9, $473.3, $712.5, $490.04. 2. Find the sum of $6041.072, $4003.926, $9621.863, $7028.414, $8631.372, $36027.496, $48971.022. 3. What is the cost of : 7 articles at $ 8.464 each ? 36 articles at $15,842 each? 329 articles at $76,575 each? 974 articles at $83,125 each? 87 articles at $479,375 each ? Use your name as buyer and that of a classmate as seller, and make out a bill for all these articles and receipt it. 4. If 23 bbl. of flour cost $155.25, what is the price per barrel ? 5. If 725 A. of land cost $49571.875, what is the price per acre ? 6. What is the difference between $7000 and $2874.664 ? 7. A man received $6126.82 for his farm, $2579.12 for his stock, and $1966.47 for his grain. He bought a house for Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 139 $3582.96; furniture for $1391.65; a horse for $164.25; a carriage for $164.28; and harness for $36.80. How much money did he have left ? 8. A man purchased a library for $763.65^, paying an average price of $2.34^ per volume. How many volumes did he buy ? 9. Find the entire cost of the following articles : 1 desk, $28.50; 1 bookcase, $68.30; 1 half dozen chairs, $18.25; 1 rocker, $12.70; 1 bedstead, $29.50; 1 bureau, $29.58; 1 washstand, $ 11.76 ; 1 stove, $ 37.49 ; 1 table, $ 24.76 ; 1 lounge, $19.46. 10. At 47^^ cents each, how many bushels of com can be purchased for $343.98 ? 11. A paid $ 491.75 for a pair of horses, and $278.90 for a carriage. How much more did he pay for the horses than for the carriage ? 12. $6215.824 is how much more than $1987.948 ? 13. A street-car company bought 864 mules, paying $79,200 for them. What was the average price ? 14. At $17.00 each, how many calves can be bought for $5695? 15. A man bought six farms. For the first he paid $6012.07; for the second, $4631.26; for the third, $3712.84; for the fourth, $8067.53; for the fifth, $7824.86; for the sixth, $ 6098.94. What did he pay for all ? 16. A merchant sold a man the following articles: sugar, $1.40; coffee, 97 cents; tea, 83 cents; salt, 48 cents; flour, $1.85; apples, $2.38; potatoes, 86 cents; molasses, 85 cents. He received in payment a twenty-dollar bill. What amount of money should he return ? 17. What is the entire cost of the following articles: one horse, $ 116.87 ; one buggy, $ 129.40 ; a set of harness, $ 28.90 ; a whip, $2.55; one wagon, $65.75; one blanket, $3.78; one sleigh, $36.47? Digitized by VjOOQIC 140 THE EQUATION, 18. Change 873 yd. to a compound number. 19. Change 2 ft. 3 in. to a fraction of a mile. 20. Add: 4 bu. 3 pk. 5 qt. 1 pt.; 6 bu. 2 pk. 7 qt.; 12 bu. 1 pk. 6 qt. 1 pt. ; and 23 bu. 3 qt. 21. A railway train, running at the average rate of 34 mi. 68 rd. 4 yd. 2 ft. per hour, went from A to B in 9 hr. What is the distance between the two places ? 22. Multiply 217 rd. 4 yd. 2 ft. 7 in. by 23. 23. The area of a floor is 25 sq. yd. 6 sq. ft. 83 sq. in. What is the entire area of 12 such floors ? 24. How much land is there in 9 fields, if each contains 63 A. 47 sq. rd. 26 sq. yd. ? 25. A lady sold 400 eggs at 20 cents a dozen, and took her pay in coffee at 16 cents a pound. How many pounds of coffee did she get ? 26. Divide 583 bu. 3 pk. 7 qt. of corn into 16 equal parts. 27. How many 40 gal. barrels of water will a cubical cistern contain that is 10 ft. deep ? 2a Multiply 45 A. 24 sq. rd. 18 sq. yd. by 38. 29. 37 equal quantities of land contain 37 sec. 201 A. 88 sq. rd. 23 sq. yd. 2 sq. ft. 72 sq. in. What does each contain ? 30. A railway train, moving at a uniform rate, ran 307 mi. 299 rd. 3|- yd. in 9 hr. What was the rate per hour ? 31. How many revolutions will a carriage wheel, whose cir- cumference is 11 ft. 4 in., make in going a distance of 1 mi. 125 rd. 4 yd. 10 in. ? 32. Divide 41 rd. 4 yd. 10 in. by 4 yd. 2 ft. 8 in. 33. If 80 cu. yd. 4 cu. ft. 848 cu. in. of earth were removed in 28 equal loads, how much did each load contain ? 34. How many piles of wood, each containing 2 cd. 75 cu* ft., can be made from 93 cd. 12 cu. ft. ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 141 35. 3.46 mi. = what, in lower denominations ? 36. Change 4 in. to the decimal of a mile. 37. 3 yd. 1 ft. 8 in. are what part of 225 rd. 4 yd. ? What is the sum of the following numbers ? ml. rd. yd. ft. in. 1 182 4 1 7 2 309 5 2 9 6 169 3 4 8 274 2 2 11 15 1 2 6 39. Divide 66 sq. yd. 6 sq. ft. by 2 sq. yd. 7 sq. ft. 40. How many lots, each containing 4 A. 36 sq. rd., can be formed from 50 A- 112 sq. rd. ? 41. From a 40 gal. barrel of vinegar a merchant sold to one man 4 gal. 3 qt. 1 pt. 1 gi. ; to a second 5 gal. 2 qt. 3 gi. ; and to a third 13 gal. 1 qt. 1 pt. 2 gi. What amount was left in the barrel? 42. What quantity of oats will 15 bins contain, if the capacity of each is 186 bu. 3 pk. 7 qt. ? 43. Change 2 yd. 1 ft. 11 in. to -a fraction of a rod. 44. f of a mile = what, in lower denominations ? 45. 1^ of a rod = what, in lower denominations ? 46. How much wood is there in 24 piles of wood, each con- taining 9 cd. 86 cu. ft. ? 47. Multiply 18 cu. ft. 724 cu. in. by 46. 4a Reduce 2 mi. 180 rd. 3 yd. 2 ft. 10 in. to inches. 49. How many feet are there in 321 rd. 4 yd. 1 ft. ? ^ Reduce 87,889 in. to integers of higher denominations. 51. Reduce y^ of a square mile to integers of lower denomi- nations. 52. Reduce .028 of an acre to lower denominations. Digitized by VjOOQIC 142 THE EQUATION. 53. 108 sq. iu. is what part of a square rod ? 54. 2 sq. yd. 5 sq. ft 64 sq. in. is what part of 16 sq. yd. 2 sq. ft. 76 sq. in. ? Express the result as a decimal fraction. 55. Reduce 4813 ft to integers of higher denominations. 56. Change 429 yd. to a compound number. 57. A loaded truck weighs 6 T. 4 cwt. 20 lb., and the truck / itself weighs 1 T. 15 cwt. How many 4 lb. packages would load the truck? 58. How many square yards of tin will be required to line the sides and bottom of a bin 6 ft. x 15 ft. and 8 ft. deep ? 59. If 4 bbl. of grapes produce 150 1. of wine, how many hectoliters will 72 bbl. produce ? 60. If a family consumes 3 1. of milk a day, at 4 cents a liter, what is the milk bill from Feb. I'to July 4, 1904, inclusive ? 61. A man walking 8 hr. a day travels 1260 Km. in 35 da. How many meters does he walk in an hour ? 62. The wheel of a locomotive is 4.5 m. in circumference. How many turns should it make in a second in order that the velocity of the locomotive may be 64.8 Km. an hour ? 63. How many revolutions will a carriage wheel whose cir- cumference is 14 ft. 8 in., make in going 2 mi. 84 rd. ? 64. Change 6 gr. Av. to the decimal of a pound. 65. What are 7 loads of hay, each weighing 2460 lb., worth at f 8.25 a ton ? 66. The roof of a hall is to be divided for decorative pur- poses into square panels. What is the largest size of panel that may be used, if the roof is 210 ft. x 154 ft. ? 67. The product of a number multiplied by 6 is how many times the product of the same number multiplied by •^? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 143 69. A man sets his watch according to Chicago local time. After having traveled for some time he finds that it is 1 hr. 24 min. 30 sec. faster than the local time where he is. What is his longitude ? -^. Which is the greater, fff or 3.14159 ? 71. A man buys three times as many sheep as cattle, and gets a total of 2^0 animals. How many of each has he ? (Use a?.) 72. A square cistern, whose bottom is 8 ft. on a side, is 12 ft. deep. How many gallons of water are there in it when it is I full? ^ 73. Ilaj + 18==5a?-hl26. -^Find^. y 74. Sheep cost one seventh as much as cows. If an equal number of sheep and cows are purchased for f 960, and the sheep cost $ 4.00 each, how many of each are there ? 75. The sum of two numbers is 720 ; one of the numbers is three times the other ; find them. (Use a?.) 76. When eggs are sold at the rate of 2 for 3^ cents, what is the cost of 4J gross ? 77. If 4 persons in a tour of 3 mo. spend f 1600, what would be the cost, at the san\e rate, to a party of 11 persons traveling 15 mo. ? ^ 78. A court 6 yd. 2 ft. 7 in. long, and 5 yd. 2 ft. 5 in. broad, is to be paved with square tiles. Find the largest possible size tilg^that could be used, and how many tiles would be required. 79. In exchange for 568 articles at $ 1.25 each, I gave f 460 and 750 articles. What was the cost of each of the latter articles ? 80. Is 1109 a prime number? How many trials must be made to determine the answer to this question? 81. The minuend is twelve times | X .8 ; the remainder is \ of .11 -^ ^. What is the subtrahend ? 82. Three times a certain number plus five times the same number, diminished by 217, is 223. Find the number. (Use a?.) Digitized by VjOOQIC 144 THE EQUATION. 83. If a wheel makes 2| turns in 1 min. 17 sec.^ how many times will it revolve in 7 hr. ? 84. I have to travel 210 mi. What fraction of my journey have I still before me after traveling 8| hr., at 16 mi. an hour ? 85. Make a bill of the following items, and receipt it : 32 lb. of sugar, at 6 J cents. 48 yd. of calico, at 8^ cents. 28 bu. of potatoes, at 37^ cents. 18 bu. of apples, at 87^ cents. 32 yd. of cloth, at 75 cents. 3 doz. plates, at 50 cents. 86. What is the cost of 3| reams of letter paper at 12J cents a quire ? 87. What is the time from March 11, 1899, to. Jan. 19,- 1904? ^es. Change 88,537 sq, ft. to square yards, etc. ^^, How many days are there from Jan. 25, 1904, to Dec. 11 of the same year ? «-^^-crXA/v-<xX 90. A steamboat going downstream is propelled 12 mi. an hour by steapi, and 320 ft. a minut.e by the current. In what time can she go 175 mi. ? In what time will she go the same distance upstream ? 91. A can do a piece of work in 10 hr., and B in 15 hr. In what time could they do it, working together ? Sol DTI ON. A will do ^ of the work in 1 hr. B will do ^ of the work in 1 hr. •^■¥^ = \. They will both do \ of the work in 1 hr. Hence, they will do the work in 6 hr. 92. A can plow a field in 12 hr., and B in 18 hr. In what time could they both plow the field ? 93. A can do a piece of work in 6 hr., B in 8 hr., and C in 9 hr. In what time could all three do it, working together ? Digitized by VjOOQIC /T^^ -^ -^ ^ j VC-^u^ ^ /JVC-^U^ MISCELLANEOUS PROBLEMS, *^ (/ 145 ^ 94. Two men working together can do a piece oi work in ^ 5 da. One of them working alone can do it in 15 da.^JL^j'^^^ In how many days can the other one working alone do the^J iT" work ? o /^^ ^ 95. A cistern has a supply pipe which will fill it in 10 hr., ^^ and an outlet pipe which will empty it in 12 hr. If both i pipes are le£.t open, how long will the cistern be in filling ? (^ vy96. In a pacing race two horses started together and went armiie. The time of the faster was 2 min. 6 sec, and of the other 2 min. 6^ sec. How much was the winning horse ahead at the finish ? 97. Sound travels at the rate of 1120 ft. a second under ordinary conditions. If the report of a gun is heard 3J sec. after the flash of the discharge is seen, what part of a mile is the observer from the gun? "C^^"*^i2^ 98. What is the longitude of Quebec, if the time there is 5 min. and 42 sec. past 1 p.m., when it is noon at Chicago? 99. Water flows into a tank through three pipes. The first would fill it in 3J hr., the second in 4J hr., and the third in 5J hr. In what time will the three pipes fill it ? 100. A 52-gal. oil barrel was f full ; 13 gal. were drawn out. What fraction of its capacity did the barrel then contain? • Change this fraction to a decimal. vl 101. At the time of her marriage, 8 yr. ago, Mrs. S. was 10 yr. younger than her husband. Her age is now f of his. What was her a^e at the time of her marriage ? (Use x,) 102. In a school of 575 pupils, the number of boys is ^ of the number of girls. How many are there of each ? (Use x.) 104. An iron beam 16 ft. long, 2\ in. wide, and 8 in. thick weighs 900 lb. This piece of iron is how many times as heavy as an equal volume of water? Digitized by VjOOQIC 146 THE EQUATION. 105. What part of | of f is f of J^? Change the result to a decimal fraction. ^ 106. Shingles are sold in bundles, each containing the equivalent of 250 shingles 4 in. wide. If shingles are laid 4J in. to the weather, how many bunches must be bought for a roof 20 ft. X 30 ft. ? What is the cost of laying them at 80 cents per square? Note, A '•^ square ** contams 100 sq. ft. 107. A rug 16 X 12 ft. is placed in the middle of a floor 19 X 15 ft. What is the width and area of the uncovered strip? 106. The length of one degree of longitude at the 40th par- allel is 53.063 mi. How far apart on this parallel do two men live whose noons are just 1 min. apart ? 109. If 8 men can do a piece of work in 12^ da. of 8 hr. each, in how many 10 hr. days can 5 men do the same work ? 110. If one bushel of wheat will make 40 lb. of flour, how many barrels of flour can be made from the contents of a bin full of wheat, the dimensions of the bin being 10 ft. x 6 ft. x 4 ft.? HI. Find the value of /^l - — 4- ?i V ^• d 112. (4.4 - .00027) X 2.1 x .005 -t- .000005 = ? 113. When it is 4 hr. 20 min. p.m., 65** 2b^ west longitude, what is the time 17** 20' east longitude? 114. The Capitol at Washington is 751 ft. long and 384 ft. wide. How many acres does it cover? 115. A can hoe a row of corn in a certain field in 30 min., B in 20 min., and C in 35 min. What is the least number of rows that each can hoe in order that all may finish at the same time, supposing they start together ? 116. A owns ^ of a ship's cargo, valued at f 493,000; B owns ^ of the remainder; C owns -^ as much as A and B; and D owns the remainder. How much does each own? Digitized by VjOOQIC CHAPTER VIII. PBRCBNTAGB. 213. i = :^=.60. l = 22t=.33i. 2 100 3 100 ^ 1 26 o. 2 66| ^^j i = i00 = -2^- 3 = 100 = -^^*- 3_ 75 _ .. 5 83i «oi l-m-'o- f-i55-^«»- 5 100 8 100 ^ 214. In § 213 each common fraction has been changed to special forms of decimal fractions called hundredths* 215. The phrase per cent means hundredths. Hence, each of the common fractions may be changed to per cent. ^ = .50 = 50 per cent ; f = .75 = 75 per cent ; f = .28^ = 28^ per cent. ' 216. In the business world, the phrase per cent is usually represented by the symbol %, which is read "per cent." 25 per cent — 25%] ^2^ per cent = ( 217. \:==,2b = 25 per cent = 25% ; or, in short, \ = 25%. I = .834 = 83^ per cent = 83^%; or, in short, f = 83^%. . 147 Digitized by VjOOQIC 148 PERCENTAGE. ^ERCISE 81. Express as hund^dfls %nd then as per cent : 1- h *A^i h\^.h h ^. h 7?u> A. M- 2* A> A> A> A> A> A? A> A> A> "Atj M* ^' TT> A> A> 1T> TO TO TO Too* TTT* V^ ^^ ^ V ;- -^ EXERCISE 82. 1. 8 is what per cent of 15 ? Solution. Two things are to be done in solving this problem. (1) We are to find what part 8 is of 16. (2) The resulting fraction is to be changed to hundredths. The first will require a review of the method of finding the part that one number is of another. (See Exercise 58.) The second will require a review of the methods of changing a common fraction to a decimal. (See Exercise 65. ) 8 is A of 15. A = -63J = 53J%. Find what per cent the first number is of the second in each of the following pairs : 2. 6: 20. J :> 9. 18:64. 'V - 15. 6 35. 3. 6:8. / -5 10. 42:63. 16. 6 4. 4. 8:10,? C' 11. 7 : 56. 17. 14 12. 5. 11:20.,-^. / 12. 1j:24. 18. 24 6. 6. 18:25. fP^ 7. 21:30.* 13. 21:66. 14. 3 : 36. 19. i 20. 1 8. 6 : 30. 21. i h Solution. } = }. \ = ofi. = J. Jisfoft. » = 60}%. Hence, i is 66}% 22. |:i. 24. i:f. 26. i-.i. 23. J:f. 25. k--\- 27. .4:. 26. Solution. .4 = .40. .40 is J§ of .25. « = = «» = 160%. 28. .018 :.2. 30. 007^ : .03. 29. .024 :.l. 31. 2i:3i. Digitized by VjOOQIC PERCENTAGE. ^ 149 32. A boy having 20 marbles lost 3 of them. What per cent of his marbles did he have left ? ( ' . * ' 33. In a scho^ of 42 pupils, 7 were in one class, 14 in a second, 6 in ^^hi^, 12 in i fourth, and the remainder in a fifth. Give the per cent of the schobl in each class. 34. A man owning f of a mill sold ^ of it. What per cent of his share did he sell ? What per cent of the mill did he still own ? 35. A received a salary of f 125 a month. His board cost him $ 20, his clothing $ 5, his other expenses f 30. Each of these items is what per cent of his income?- He saves what per cent of his income ? EXERCISE 83. 1. $ 17 is what per cent of f 24 ? Solution. .$ 17 is i} of $ 24. J} = .TOf = 70f %. Therefore, $ 17 is 7010/^ of $24. 2. 2 is what per cent of 450 ? Solution. 2 is ^jj or ^J^ of 450. ^J^ = .00 J = ^%, Therefore, 2 is ^% of 460. Find what per cent the first number is of the second in each of the following problems : 3. 23 : 69. -: ■ 9. 84:64. IS. 99 : 451. 4. 36:81. -.'■■ 10. 91 : 28. 16. 328 : 1076. 5. 54:88. 11. 96 : 124. 17. 256 : 72. 6. 69:52. 12. 29 : 37. 18. 500 : 128. 7. 66:92. 13. 125:625. 19. 836 :*1000. 8. 72:80. 14. 140 : 720. 20. f:T^- Solution, f ■*- 1^ : mn of A- = JxV=ti = 1.60? = --im% Therefore, } is a. |:H. 22. f:H. 23. i:H. 24. .02:. 25. 25. 26. .15 : .126. 3.5 : 7.16. Digitized by VjOOQIC 150 PERCENTAGE. 27. 2J:5f 29. .0128:. 456. 31. t:^- aa 7H:24H. 30. 7T^:36f. 32. ^:f Solution. ^-j.J = ^xJ = A = -^SJ = 53J%. Therefore, ^ ^ 63J% of J. 33. 3J:28. 35. ^^ : .6625. 37. $24: $84. 34. 24:70f 36. 3.75 ; 7^. 3a $63: $40. 39. 125 lb. : 370 lb. 48. $ 10.24 :$ 1280. 40. 84 A.: 640 A. 49. 375 men : 12,000 men. 41. 130 sheep : 1200 sheep. 50. | : |. 42. 12f days : 19 days. 51. 3^:7^. 43. 1000 rd. : 25 rd. 52. 3 qt. 1 pt. : 5 gal. 2 qt. 44. 6 ft. : 324 ft. 53. 40 sq. rd. : 8 A. 45. 136:624. 54. 3 yd. : 8 rd. 46. 38:112. 55. 2^ 30': 10°. 47. 2 yd. 2 ft. 3 in. : 12 rd. 56. $ 624 : $ 12. 57. A man bought a farm for $ 6250. He paid cash $ 1250. What per cent of the purchase price remained unpaid ? 58. A man had 24 cd. 6 cd. ft. of wood. He sold 4 cd. 4 cd. ft. What per cent of his wood was left ? 59. A boy purchased 10 copies of a newspaper for 20 cents and sold them for 30 cents. What rate per cent did he gain ? Solution. 30 cents — 20 cents = 10 cents. Therefore, the gain was 10 cents. 10 cents = } of 20 cents. But, | = 50%. Hence, he gained 60% of the cost. 60. Three pounds of candy is what per cent of 12 lb. ? Solution. 3 lb. = J of 12 lb. J = 26 %. Hence, 3 lb. candy is 26 % of 12 lb. 61. John's top spins for 50 sec, and James's spins for 40 sec. The difference is what per cent of 50 sec. ? Of 40 sec. ? 62. A man walks on an average 4 mi. per hour, and an auto- mobile runs at an average of 20 mi. per hour. The speed of a man in walking is what per cent of the speed of an automobile ? Digitized by VjOOQIC PERCENTAGE. 151 63. The population of a city at a given time was 40,000 inhabitants. A year afterwards it was 48,000. What was the rate per cent of gain in the year ? 64. An apple orchard produced 60 bu. of apples one year, and the next it produced but 48 bu. What was the rate per cent of decrease the second year ? 65. A man invested $ 12,000 in pork and sold it for f 13,500. What rate per cent did he gain ? 66. If 75 pupils study grammar and 80 study arithmetic, what per cent more pupils study arithmetic than grammar ? EXERCISE 84. ' 1. Express 8% as a common fraction in its lowest terms. Solution. 8 % = .08 = jf^ = ^. Therefore, 8 % = ^j. 2. Express 350% as a mixed number in its simplest form. Solution. 350 % = 3. 60 = 3i. Therefore, 350 % = 3i. 3. Express ^^i% as a common fraction in its lowest terms. Solution. 66i%= .66J = ^ = |. Therefore, 66J% = f 100 3 Express the following as common fractions, and reduce to lowest terms : 4. 36%. 12. 41|%. 20. 1|%. 28. 2J%. 5. 25%. 13. m\%. 21. 3^%. 29. .25%. 6. 50%. 14. 43^%. 22. f%. 30. .7%. 7. 75%. 15. 1000%. 23. ^%. 31. .08^%. 8. 38i%. 16. 465%. 24. ff%. 32. 2,26%. 9. 62^%. 17. 116|%. 25. 1%. 33. .0^%. 10. 83^%. la 1%. 26. ^%. 34. .uut7(^. U. 225%. 19. ^%. 27. 1%. 35. .00J%. Digitized by VjOOQIC 162 PERCENTAGE. EXERCISE 85- ,' -T^ 1. Find 6% of 650. Solution. 6% = .06. 650 x .06 = 39. Often the rate per cent can be readily expressed as a common fraction. 2. Find 5% of 720. Solution. 6% reduces to a convenient common fraction. 5% = ^. ^ of 720 = 36. In the following choose either of the above methods. Find: 3. 26% of 589. 21. .2% of 163. 4. 35% of 1230. 22. .25% of 7826. 5. 43^% of 1584 23. \% of 7826. 6. 52J% of 6825. 24. .125% of 5624. 7. 86% of 42,563. 25. |% of 5624. a 117% of 324^. 26. ^% of 3162. 9. 125% of 861f. 27. ^% of 4563. 10. 250% of 936.8. 28. .06|% of 58,635. U. 1000% of 78.32. 29. ^% of 58,635. 12. 17% of .4. 30. .06i% of 32,064. 13. 23% of .625. 31. ^% of 32,064. 14. 31% of 3^. 32. .0625% of 24,638. 15. \% of 125. 33. ^(fo of 24,638. • 16. \% of 324. 34. f% of 896.24. > 17. 1% of 762. . 35. .625% of 896.24. 18. ^% of 1284. 36. 3^% of 756. 19. |^% of 825. 37. 41f % of 756. 20. \^% of 867. 38. 39|% of 7824. Digitized by VjOOQIC PERCENTAGE. ^/ 163 / ,i^ EXERCISE 86. t ^ ' 1. Find 7% of f. Solution. 7% = 3^. } x rJrr = A^- Therefore, 7% of f = ;^. Find: 2. 16% off. 9. 72%of2J. 15. 3^% of f 3. 24% of 3^. Note, 2J = J. le. 4^% of ff. 4. 30% of 3^^. 10. 95% of 3f Note. ^^xif. 5. 42% of ^. U. 124% of if. 17.. 8i% of H- 6. 56%of^. 12. 500% of 15^. 18. 13^% of f|. 7. 63% of |. 13. 1% of |. 19. 18f % of j^. 8. 76% of f|. 14. f% of H. 20. 16f% of If. EXERCISE 87. (Mental.) ^ 1. What is 4% of 60? 7% of 80? 5% of 90? 12% of 400? 11% of 900? r|^ -^ 2. What is 6% of i25? of 12? of 120? of 1200? of 15? of 150? of 1500? 3. What is 1% of .24 ? of 36 ? of 480 ? of 4800 ? of 72 ? of 7200? 4. What is f% of 1200? of 120? of 12? of |? of f ? off? 5. What is 1% of 75? f% of 640? 3^% of 3300? f% ofi? f%off? |%ofH? 6. What is 10% of 2500 lb. ? 16% of $4000? 7% of 71 mi. ? f % of 120 A. ? f % of 2500 bu. ?, I' ^^^ , ' EXERCISE 88. ,, Solve mentally as many of this list as j)ossible. 1. Find 37^% of 96. Solution. 37i% = f^ = ?^i^^^i?i = ?. ^'^ 100 100-5-12J 8 f of 96 = 36. Therefore, 37}% of 96 = 36. Digitized by VjOOQIC 154 PERCENTAGE. 2. Fiii(il2i% of 72; of 144; of 60; of 240; of |; of A. 3. Find 62^% of 2400; of 320; of .048; of 3^; of^^; of 84,000. 4. Find 40% of 250; TBfs of |; 87^% of A; 66|% of .081; 25% of 16; 6^% of .32; 8^% of 132; 50% of |; 60% off 5. Find 37^% of 96; of 120; of 144; off; of .64. 6. Find 33^% of 27 ; of 81 ; of 122 ; of 650 ; of f ; of .018. 7. Find 16f% of 84; of 120; of 135; of 225; of |; of ^', of .0144. 8. Find 6|% of 45; of 80; of 140; of 328; of J; of ||; of .18. 9. Find 18f % of 160, of 324; 31^% of 256, of 320; 43|% of 180, of If. 10. Find 56i% of .0288; 68|% of ^. 11. Find 20% of 165; of f ; of .72. 12. Find 40% of 821; 80% of .096. ^ ^-tlT • EXERCISE 89. ^^^ 1. Find 7% of 325. Solution. 325 ^„, ._ , ^. ^. . ^, ^ - ,^. ,. /v« 7% = .07, and the operation is that of multiphca- ' - tion of decimal fractions, Aiu.70 Find: 2. 9 % of 426. 5. 20 % of 630 bu. 3. 13 % of 612. 6. 23 % of 1824 mi. 4. 17 % of $725. 7. 33^% of 756 A. a 125 % of 67.2 rd. 9. 37^% of 3^; of .0688; of 432. 10. I % of 7563 ; | % of 1200. ^» U. 37i% of £24 16s. 8d. 12. 33^ % of 15 lb. 9 oz. 18 pwt 13. 25% of lOrd. 2 ft. 4 in. Digitized by VjOOQIC PERCENTAGE. 7^ . 165 14. 72% of 76 cwt. 76 lb. 15. 75% of 440 sheep. 16. A cistern with a capacity of 84 bbl. is 41f % full. How many barrels does it contain ? 17. How much is 200% of a quantity ? 400%? 1000%? 250%? 75%? 37i%? 83^%? mi%'^ 41|%? 18. Find 27% of $864.50. 19. rxnd6f% of $965.80. 20. What is I % of $ 1286.43? 21. What is 18f % of $1680.48? 22. What is i% of $972.84? 23. What is 2^% of 7824 bu.? f@> A merchant bought a stock of goods for $8324.60. The charge for transportation was 1J% of the cost. What was the entire cost ? Lis. A owed B a certain sum of money. After paying him 20 % of the debt, 25 % of the remainder, 50 % of what then remained, and 83 J % of the third remainder, what part of the debt was still unpaid ? 26. What is the interest on $ 468.15 for one year at 7 % ? Note, Interest is the amount paid for the use of money, and is com- puted at a given per cent of the amount loaned, called the principal^ for one year. 27. What is the interest on $1236.50 for two years at 6^% per year? 2a What is the interest on $2580 for 4 yr. at 6^% per year? v^. A farmer owns a section of land. 25 % of it is meadow, 33| % of the remainder is corn land, Zl^ % of the remainder is pasture, 80 % of the remainder is wheat land, and the rest is oat land. (Make a diagram 8 in. on a side, and show the several tracts.) Digitized by VjOOQIC 156 PERCENTAGE. 30. A piece of cloth containing 36 yd. was found to have lost 3^% of its length by shrinkage after sponging. How much did it lose in length ? 31. A man's income is $ 1500. He pays 46| % of it for his household expenses, 20 % of it for general expenses, and 13^ % of it for personal expenses. What are his expenses for the year ? How much does he save ? 32. In a school of 650 pupils, 52 % were girls. How many boys were there ? 33. A schoolhouse is 96 ft. long. Its width i§ 83J % of its length. How wide is it ? i- EXERCrSE 90. 1. 5 is ^ of what number ? Solution. Let x = the number. Then, Ja; = 6, and x = 10. Hence, 6 is i of 10. But, J = 60%. Then this problem might be read : 5 is 60% of what number ? If a; = the number, then 60 % of a = 6. But, 60 % = .60. Hence, .60 a; = 6, and a; = 6 -s- .50 = 10. 2. 14 is 7% of what number ? Solution. Let x = the number. Then, 7 % of a; = 14, or .07 x = 14. Then, a; = 14 -^ .07 = 200. Hence, 14 is 7 % of 200. 3. 16 is 6|% of what number? Solution. Let x = the number. Then, 6J% of a; = 16. But, 6|% = .06f . Hence, .06| x = 16, and a; = 16 -^ .06| = 1600 -i- 6| = 240. Hence, 16 is 6j% of 240. These three solutions show that in such problems the required number is found by division. The divisor is the rate per cent expressed decimally, and the dividend is the given number. Solve the next ten problems mentally. 4. 48 is 10% of what? 12% of what? 25% of what? 33^% of what ? 50% of what ? 5. 60 is 1% of what ? |% of what ? f % of what ? ^% of what ? Digitized by VjOOQIC PERCENTAGE. 167 6. I is 6% of what? f% of what? 25% of what? 100% of what ? 62 J % of what ? 7. $ 21 is 25% less than what ? 8. $ 18 is 91% less than what ? ^-^^^^ 9. $ 150 is 50% more than what ? -^ ^ 10. 60 A. is 30% of what? 11. 75 mi. is 25% of what? 12. ^ of f of a yard is 20% of what ? 13. f of f of f of a bushel is 16f % of what ? 14. f 6.24 is 18% of what ? Solution. 18 % of « = $ 6.24 ; or, . 18 a; = $ 6.24. Hence, x = 1 6.24 ^ .18 = ^ 34.66J. 15. 750 rd. is 125% of what ? 16. 18 gal. 3 qt. 1 pt. is 6f % of what? 17. $ 62.50 is 15f % of what ? fia 4 sq. rd. 16 sq. yd. is 5|% of what ? r' ^ 19. 4 cu. ft. 428 cu. in. is 73f % of what? 20. 67^ 30' is 18|% of what ? 21. i^isf% of what? Solution. Let x = the number. Then, } % of x = ^\ or ^ of « = A. Hence, x = tV x * J^ = ^^= 24 A. Hence, A is J % of 24^. 22. 980 A. is 51% less than what ? 23. $ 6820 is 241% more than what ? 24. Seventy-two men deserted from a regiment, leaving 92^% of the whole number. How many men were there in the regiment before the desertion ? 25. The Koman pace of 5 ft. is 8J% of how many feet? 26. A dozen eggs is 3^% of how many eggs? 27. The distance from Indianapolis to Kichmond is 68 mi. ; this distance is 40% of how many miles ? 400% of how many miles ? Digitized by VjOOQIC 158 PERCENTAGE. 28. $ 141.57 is 18% of what number ? 29. 84 rd. is 125% of what number? 30. ^ is 37^% of what number ? v/ai. 5 lb. 3 oz. 6 pwt. is 33^% of what? 32. 35 bbl. is 41|% of the capacity of a cistern. How many barrels will it hold ? 33. $ 315.09 is 18f % of what number of dollars ? 34. The width of a pane of glass in my window is 14 in., which is 48^% of the length. How long is the pane ? 35. 75 m. is 200% of how many meters ? 36. Find the number of which 263 is |% ; of which 79 is 37. 826 is T^% of what number ? 38. 964 is 500% of what number ? 1000% of what numbei;? 133i%? Xo9. Sold 445.5 lb. of sugar, which was 44% of what I had left. How much had I at first ? 40. Sold a farm for $ 12,860. f of this amount is 50% of the cost of the farm. The gain is what per cent of the cost ? 41. What number increased by 20% of itself equals 126 ? 42. What number diminished by 20% of itself equals 126? Ql3y A railway train, running at an average rate of 35 mi. an hour, for 2f hr., passes over 35% of the conductor's run. What is the length of his division ? 44. What number increased by 25% of itself equals f ? Solution. 26 % = J. Therefore, f a — f . Hence, 45. What number diminished by 75% of itself equals ^^ ? 46. If 340 is added to a number, the result is 117% of the number. What is the number ? 47. If 520 is subtracted from a number, the result is 86% of the number. What is the number ? Digitized by VjOOQIC / PERCENTAGE. 159 4a A town is found to have gained 824 in population in 5 yr. This is an increase of 8%. What was the population of the town 5"yr. ago? 49. A traveler having gone 384 mi. has completed 84% of his journey. How much farther has he to go ? 50. A farmer having plowed 36 J A. finds that he has finished 56% of his field. How large is it ? 51. A farmer contracted to deliver to a dealer 1800 bu. of com. Upon delivering his grain, he found that he had over- estimated the amount of corn in his crib by 4%. How many bushels did it actually contain ? 52. A merchant being obliged to vacate his room sold his stock at a discount of 11% of the cost and realized $23,568.46. What did the goods cost him ? 53. In a certain school there are 168 boys, who form 42% of the whole school. How many girls are there in the school ? Solution. Let a; = the number of pupils in whole school. 42 % = .42. rheref ore, .42 a; = 168, and a; = — = 400. .42 The number of girls = 400 - 168 = 232. 54. The distance from the earth to the moon is 240,000 mi. This is 200 % of what distance ? 55 $3825 is 11% more than what? 15% less than what? 56. What number increased by 18% of itself equals 3379.52? 57. What fraction diminished by 30 % of itself equals ^ ? 58. f -J- f equals 24 % of what ? 6 1. 7 59. g^ y^ equals 14^ % of what number ? ^ 60. In a certain city there are 15% more Germans than Swedes. The former number 5589, and the latter form 9 % of the entire population. What is the population of the city ? Digitized by VjOOQIC 160 PERCENTAGE. 61. The weight of a man is 180 lb., which is 10|^% of the weight of a horse. Find the weight of the horse. t 62. A traveler, having gone 24 mi. 124 rd. 4 yd., has com- pleted 37^% of his journey. What distance has he yet to travel ? 63. A has a tract of land containing 82 A. 120 sq. rd. This is 5% more than B's land. How much has B? y 64. A wholesale merchant's sales were 6f % less in 1903 than in 1902, when they aggregated $824,960.50. What were his sales in 1903? ^ 65. H. M. Senseney lost by fire 3720 tons of coal. This was 62 % of his stock. What was his stock ? V 66. H. M. Senseney carried 25 % more stock at the time of the fire than Parker Bros. What amount had Parker Bros.? • 67. The uninjured portion of H. M. Senseney's stock was 18 % less than the amount required for a year's supply to a manufacturing establishment. How much did it consume annually? ^ J ^ , ^, ^ ^^W MISCELLANEOUS PROBLEMS. ^ i EXERCISE 91. ^" 1. Find23i% of $628.50. 2. What is the interest on $ 1296 for one year, at 5^ % per annum ? 3. If I paid $ 71.28 for the use of $ 1296 for one year, what is the rate of interest ? Note, Rate is the number of hundredths of the principal paid for its use for one year. 4. 17^ is what part of 49 ? Change the result to per cent. 5. I is what part of ^ ? What per cent ? 6. A man earns $ 15 a week. He pays each week $4.50 for board, 70 cents for car fare, an average of f 1.25 for clothing, and $3.20 for all other expenses. How much can he save in a year ? This is equal to the interest on what sum at 5 % ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS, 161 7. Multiply one thousand one ten-thousandths by four thousand three millionths. a Divide one thousand one hundred one millionths by one ten-ihousandth. ^^^ Add the following numbers : 34,883,469, 55,273,289, 52,678,979, 46,864,278, 54,489,858, 47,791,697, 34,963,248, 46^15,798, 68,866,337. (jj). Arrange these problems as given below, and place the differences at the right, thus : 75,063 - 38,156 = 36,907. * 84,152 - 68,237 = 91,005 - 42,807 = 63,254-27,809 = 83,274-58,695 = 91,352-63,806 = 74,083 - 35,108 = Find the sum of the minuends ; do the same with the sub- trahends; with the remainders. To the sum of the subtra- hends add the sum of the remainders. If no error has been made, what should the last sum ^ual ? 11. A man deposited $8650 in a*bank on May 1. During the month he drew the following checks against his deposit : $650.70, f 329.85, $48, $64.50, $1,540.90, $1937.26^ $76.80, $2170.40. What was his balance on June 1 ? /I2) The following is a copy of A's bank account for June : B^nce to his credit June 1, $584.60; June 2, deposited $275.25 ; June 4, drew a check for $ 146.85 ; June 6, deposited $64.50; same day drew a check for $186.15; June 10, de- posited $ 225 ; June 15, drew a check for $324.10, and on June 18 for $462.90; June 21, deposited $240; June 25, drew a check for $72.12. What was his balance July 1 ? ^ (74 -16) -(62 -36)=? (74 -16) -62-36= ? ^. 562-F79-(324-M48)=? 562 + 79-324 + 148=? Digitized by VjOOQIC ^ 162 PERCENTAGE. )Bought of A 974 bu. of oats at 19^ cents; 1328 bu. of cfewl^at 28^ cents; 1726 bu. of wheat at 58 cents. Sold him 30 A. of land at $39 an acre. Which was indebted to the other ? How much ? 16. Two railway trains start at the same time from the opposite ends of a division 212 mi. long. One runs at an average rate of 29 mi. an hour, and the other at 24 mi. How far apart will they be at the end of 3 hr. ? 4 hr. ? 5 hr. ? 6hr.? 17. A fkock train has 29 cars. Each car contains 19 cattle, whose average weight is 1450 lb. They sell for $5.25 a hun- dred. What do they bring ? la Change ^ to a 5-place decimal. 19. Add 5 mi. 180 rd. 4 yd. 2 ft. ; 16 mi. 79 rd. 3 yd. 1 ft. ; 26 mi. 136 rd. 2 yd. 2 ft. 8 in. ; 29 mi. 278 rd. 5 ft. 10 in. ; 46 mi. 316 rd. 1 yd. 1 ft. 10 in. . 20. Multiply 2 lb. 10 oz. 16 pwt. 15 gr. by 36. 21. Reduce 10 sq. yd. to a fraction of an acre. 22. If 18 gal. of water is mixed with 22^ gal. of grape juice, the water is what per cent of the mixture ? 1/ 23. 1,049,760 is the product of three factors, two of which are 216 and 15. What is the third? 24. What is the difference between J% of $1800 and 33 J% of the same ? V 25. The value of a house is 87|% of the value of the lot. Together they cost $9645. What is the value of each ? 26. What is the weight of a dozen silver spoons, each weigh- ing 3 oz. 5 pwt. 7 gr. ? » 27. Simplify ^^i^ 28. Write the answer to the following question: What common fractions -can be changed to pure decimals ? Digitized by VjOOQIC T CHAPTER IX. APPLICATIONS OF PBRCBNTAGB. Profit and Loss ; Commission ; Commercial or Trade Discount; Taxes; Insurance. 218. The methods of calculation taught in percentage are applicable to the solutions of many practical problems. PROFIT AND LOSS. 219. The Cost of an article is the expenditure involved in its purchase or production, and is usually expressed in money. 220. The Selling Price of an article is the amount of money which the buyer pays the seller for it. 221. The Profit on an article is the excess of the selling price over the cost. 222. The Loss on an article is the excess of the cost over the selling price. 223. There are three general types of problems pertain- ing to profit^ and three to Iobb, 224. The types pertaining to profit are : I. Given the cost of an article and the rate of profit, to find the profit. II. Given the cost of an article and the profit, to find the rate of profit. III. Given the rate of profit and the profit, to find the cost of an article. Note. The first one is direct^ and the others are inverse, 163 Digitized by VjOOQIC 164 APPLICATIONS OF PERCENTAGE. 225. There are three general types of problems pertain- ing to loss ; state them. 226. TYPE PROBLEMS. I. If wheat costs 55 cents per bushel, and is sold at a gain or profit of 20% on the cost, find the gain and the selling price. Solution. 20% of 65 cents = 11 cents. 65 cents + 11 cents = 66 cents. Hence, the gain is 11 cents, and the selling price 66 cents. II. If corn costs 45 cents per bushel and is sold at 55 cents per bushel, find the rate per cent of gain. Solution. 55 cents — 45 cents = 10 cents. 10 cents is J^, or j, of 45 cents, j = .22i = 22 j %. , Hence, the rate per cent of gain is 22| %. III. If the rate of profit is 1^% on the cost, and the profit is $2.80, find the cost. Solution. Let x = the cost. Then, .14 a; = ^ 2.80 ; whence, a; = $ 2.80 + . 14 = 1 20. Therefore, cost = $ 20. EXERCISE 92. (Mental.) 1. Cost, $12; rate of gain, 25%: find gain and selling price. 2. Cost, $25; gain, $8: find rate per cent of gain and selling price. 3. Eate of loss, 40%; loss, $20: find cost and selling price. 4. Eate of gain, 12^% ; selling price, $45: find cost and gain. 5. Cost, $ 1.44 ; loss, 18 cents : find rate per cent of loss and selling price. 6. Eate of gain, 16|%; selling price, $63: find cost and gain. ;„ vrt;^ ^^^ 7. Cost, $96; rate of loss, 6J% : find loss and selling price. Digitized by VjOOQIC PROFIT AND LOSS. 165 y/ a Cost, $63; gain, $7: find rate per cent of gain and selling price. ^/ ^ . i, 9. Cost, $540; selling price, $ 600 : find gain and rate per cent of gain. , SU -^ % 10. Cost, $225; rate of loss, llj^% : fiind loss and selling U. Gain, f91; rate of gain, 7%: find cost and selling ^^ 12. Cost, %2S^\ rate of gain, 8% : find gain and selling price. i ^ 3^- 13. Cost, $36; selling price, $42: find ^in and rate" per ^ cent of gain. w ^ J 14. Selling price, $52; cost, $48: find gain and rate per 3 cent of gain. v^^ . A. 15. Selling price, $560; fatevof l(Jse, 20% : find^st and loss. / -^^^ ^16. Rate of gain, 30%; selling price, $390: fina cost and ^m. ( t 17. Gain, $81^ rate of gain, 11J^% : find cost and selling price. '^^-r- t . •^•xV 18. Loss, $21; cost, $63: find rate per cent^or iSsss and 197 Cost, $1800; rate of gain, ^^\% : find gain and selling price. ^t^ ^L 20. Selling price, $3000; cost, $1800:* find gain and rate per cent of gain. > /^ r >> ^ ^1 21. Loss, $250; selling price, $750: find" cost and rate^ per cent of loss. 22. Gain, 46 cents ; rate of gain, 23% : find cost and selling price. 23. Cost, 84 cents; rate of gain 7^% : find gain and selling price. 24. Gain, 6 cents; rate of gain, 1\% • find cost and selling price Digitized by VjOOQIC 166 APPLICATIONS OF PERCENTAGE. EXERCISE 93. 1. A man invested $2680, and made a profit of 23^%. How many dollars did he gain ? 2. A farmer raised in a single year 6400 bu. of oats, 6250 bu. of corn, dnd 42 T. of hay. He sold the oats at 37^ cents a bushel, the corn at 40 cents, and the hay at f 6 a ton. ^ The landlord received | of the value of the crop, which was 7% of the value of the farm. What was the farm worth ? 3. A house, costing $12,250, was destroyed by fire. The insurance company paid the owner $10,500. He lost what per cent of his investment ? 4. A merchant marked his goods at an advance of 25% above the cost. The market declining, he reduced the selling price 10%, and made a profit of $1280 on his sales. What was the cost of the goods sold ? 5. If ^^ of an arti^e is sold for what the whole cost, what is the rate per cent of gain ? ^ ^ J ' Solution. H *" A = A» which is the gain. Now, ^ is what per cent of ^2 ? ^j ^ A = f = .71f = 71f %. Hence, the rate of gain is 71f %. 6. A man bought a quantity of apples at 75 cents a bushel. If there was a waste of 12%, at what price must he sell the remainder to make 25% in the transaction ? 7. A merchant sold 450 yd. of cloth at a profit of 24%, realizing a profit of $2^3^ Find the cost and selling price per yard. 8. Find the prices at whioh ihe following articles may be sold so as to bring a profit of 30% ; tea, costing 50 cents ; coffee, costing 30 cents; shoes, costing $2; cloth, costing $1.50; eggs, costing 15 cents ; butter, costing 20 cents. 9. If a span of horses, costing $465, be sold for $ 620, what is the rate per cent of gain ? Digitized by VjOOQIC COMMISSION. 167 10. If the purchaser sold the same horses for $520.00, what was the rate per cent of loss ? U. Selling price, $9850; cost, f 10,000: find rate percent of loss. 12. Fifty acres of oats yielded an average of 45 bu. last year. This year the aggregate yield from the same field is 2500 bu. What is the per cent of gain ? ) 13. A merchant bought 350 bu. of potatoes at 45 cents. He lost 15% of them. For how much a bushel must he sell the remainder to gain 25% of the cost of the whole ? ) 14. Bought a bankrupt stock for f 3280. Sold it for f 4100, and deducted 5% for cash payment. What was the gain per cent ? 15. In our school there was a total enrollment last year of 560. This year it amounted to 610. What is the per cent of gain? 16. Cost, $28.60; rate of gain, 23%: find selling price without finding the gain. 17. Cost, $824.60; gain, $206.15: find per cent of gain and selling price. 18. Gain, $2400; rate per cent of gain, 16: find selling price without finding cost. 19. Selling price, $860; rate of profit, 25%: find the profit without finding the cost. 20. Selling price, $428.50; rate of loss, 15% : find cost. 21. A farmer raised 6824 bu. of grain. By fertilization he increased the yield the next year by 8|%. What was the increased yield ? COMMISSION. 227. Commission is a sum paid to a person or firm for the transaction of business for another. This transaction may be purchasing, or it may be selling. Digitized by VjOOQIC 168 APPLICATIONS OF PERCENTAGE. 228. The person for whom the business is transacted is called the Principal. 229. The person transacting the business for another is called an Agent, Commission Merchant, or Broker. 230. Agents collect debts and do other kinds of trans- actions for their principals. 231. Commission is one of the applications of percentage, because the agent's compensation is usually some per cent of the amount of the purchase or of the sale. 232. There are three general types of problems in com- mission : I. Given the value or amount of the transaction and the rate per cent of the commission, to find the commission. II. Given the amount of the transaction and the com- mission, to find the rate per cent of commission. III. Given the rate per cent of commission and the commission, to find the amount of business transacted. Note, A vast amount of the business of the country is done by com- mission, and ahnost all of it is illustratiye of the first general type. EXERCISE 94. 1. An agent sold a house and lot for f 3400, and was paid a commission of 2% of the selling price. Find the commission. Solution. 2% of $3400 = $68. Hence, the commission was $68. 2. An agent collected a debt of f 742, charging 3% com- mission. Find the commission. Solution. 3 % of $ 742 = $ 22.26. Hence, the commission was $ 22.26. 3. An insurance agent insured a man's life for f 3000, and received 2^% commission. What was his commission? Digitized by VjOOQIC COMMISSION. 169 4. An agent bought for his principal 500 bbl. of apples at f 2.50 per barrel, and was paid his commission at the rate of 2^%. What was his commission ? 5. An agent sold for his principal a farm of 80 A., at $ 75 per acre, on 2% commission. What was his commission ? What sum of money did the agent send to his principal as the proceeds of the sale ? 6. An agent purchased for his principal 200 mules, at an average price of f 110 each, on a commission of 1^%. If the agent paid out $40 for labor, how much money must the principal send him to pay for the transaction? 7. If an agent received a commission of $ 11.40 for trans- acting business to the extent of $ 456, find the rate per cent of his commission. Solution. $11.40-^ $456 = .02}. Hence, the rate of commission is 2i%. 8. An agent transacted business to the extent of f 845. His principal paid him $36.90, f 20 of which was to pay for labor employed by the agent. What was the rate per cent of his commission ? 9. An agent's commission for selling a house and lot was f 240. The proceeds of the sale were $7760. What was the rate per cent of the commission ? 10. An agent's commission on a sale was $324.60; the rate of commission was 3 %. What was the amount of the sale ? U. A sent $ 1250 to an agent with which to buy a town lot and pay his commission at 2^ %. What was the cost of the lot? What was the commission? 12. An agent received a consignment of corn amounting to 2864 bu. He sold it at 39 cents a bushel. He paid freight, $75.92, and charged 2 J % commission. What was his commis- sion, and what amount did he remit to the principal ? Digitized by VjOOQIC 170 APPLICATIONS OF PERCENTAGE. 13. A principal sent to his agent 324 bbl. of flour, with directions to invest the proceeds of the sale in wheat, after deducting his commission oi 2\% for selling, and 2\ % for buying. The flour sold for $6.25 per barrel. How many bushels of wheat at 83^ cents could the agent buy ? Solution. $ 6.25 x 324 = $ 2026. 21 % of $ 2026 = $ 50.626. $2025- $ 60.625 = % 1974.375. If sc = the number of dollars expended for wheat ; then, 1.02J a; =$1974.375. Whence, a; = ^ 1974.375 _ ^ 1930,93 to the nearest cent. $1930.93 -r-$. 83^ = 2317.11 to the nearest hundredth. Hence, the agent bought 2317.11 bu. 14. What is an agent's commission for collecting f 1264.50 at If % ? What amount should be sent to the principal? 15. An agent is to pay a debt of f 836 for a merchant. What amount should the merchant remit, if the agent's com- mission is J % ? 16. An agent's commission on an investment at 1|% ^^.s f 52. What amount should be sent him to cover investment and commission ? COMMERCIAL OR TRADE DISCOUNT. 233. Manufacturers and wholesale dealers usually issue schedules of the prices of their goods. These schedules are called Price Lists. 234. These price lists do not usually give the price at which shopkeepers purchase the goods, but give prices from which discounts are made. 235. The deduction from the list price of goods is called Commercial or Trade Discount. It is usually computed in per cent. 236. As commercial discounts are given for different reasons, such as the amount purchased or the time of pay- ment, more than one discourit is sometimes allowed. Digitized by VjOOQIC COMMERCIAL OR TRADE DISCOUNT. 171 237. When more than one discount is allowed, the first is computed upon the list price, the second upon the re- mainder, and so on. 238. The amount of a bill after all discounts are de- ducted is called the Net Amount. EXERCISE 95. 1. On a bill of goods amounting to $ 60 a discoimt of 5% was allowed for cash payment. What was paid for the goods ? Solution. 5% of $60 = $3. $60 -$3 = $67. 2. Find what was paid for a comet, the list price of which is 9 42, bought subject to a discount of 15% for cash. 3. If the catalogue price of a bicycle is $ 75, find the cost of one purchased at a dealer's discount of 30% for cash. 4. I can buy a piano for $ 375, and have 90 da. in which to pay for it. But the merchant will discount at 12% for cash. By the latter offer, what will the piano cost me ? 5. Coal retails at f 7.50 per ton, but it is subject to a dis- count of 10 % if sold by the carload. What is the cost of a carload of 22 T. ? How much is saved in this manner ? Note. The above problems illustrate Single or Direct Discount. In Problems 1 to 4 the articles were discounted for cash. The fifth was discounted because of the wholesale feature* There are other reasons for discounts. Goods may become shelf-worn, soiled from exposure to water, or out of fashion. Two or more of these causes may be present in one business transaction. EXERCISE 96. 1. If I buy a mandolin which is listed at $ 18 and receive a discount of 20% because I am a music teacher, and a further discoimt of 5% for cash, what does the mandolin cost me ? Solution. 20%of $18 = $3.60. $18 -$3.60 = $14.40. 5% of $14.40. = $0.72. $14.40 -$0.72 = $13.68. Hence, the mandolin cost me $13-68. Note. This is called a Double Discount and is usually called *^ 20 and 5 off.** It is not the same as 26 off. Digitized by VjOOQIC 172 APPLICATIONS OF PERCENTAGE, 2. An umbrella dealer bought a lot of slightly soiled um* brellas amounting to $1800, for cash, at a discount of 15% and 10%. What was the cost? 3. A piano catalogued to sell at $ 500 was sold at a triple discount of 20%, 15%, and 10% ; 20% for being the first sold in the community, 15% for cash, and 10% for a slight defect in the finish of the case. Find the cost. 4. Find the net cost of 15/>00 bags at $4 per M., less 25% and 15% off. 5. On May 1 a dealer bought a bill of goods amounting to $600, payable in 60 da., subject to a discount of 5% if paid at once; 3% if paid in 30 da.; or 2% if paid in 40 da. How much would settle this bill May 1 ? May 31 ? June 10 ? 6. Find the cost of 15 gross of essence of lemon at 50 cents per dozen, at 12% and 10% off. 7. Find the cost of 24 cases of cocoa at f 14 each, less 5% and 10%. 8. Find the cost of 3 reaping machines at f 190 each, less 33i% and 10%. 9. What single or direct discount is equivalent to 10% and 5% off? Solution. lOOo/, - IQo/, = 90o/^ b% of 90% = 4.\%. 90% - ^ % = 86 J %. 100 % - 86i % = U\ %. Hence, 10 % and 5 % off is the same as Hi % off. 10. To what single discount is the double discount of 25 and 10 equal ? U. Which is the better to the buyer, 25% and 5% off, or 20% and 10% off? 12. Which is the better: 40% and 10%, 30% and 20%, 25% and 25%, or 50% direct ? 13. Henry Metz of Trenton, N. J., bought of McClurg & Co., of Chicago, 40 boxes of paper at 16 cents ; 32 boxes at 12^ cents ; Digitized by VjOOQIC COMMERCIAL OR TRADE DISCOUNT. 173 28 boxes at 15 cents ; 30 boxes at 14 cents ; 2 quires blotting paper at 60 cents. Find the net cost, if the bill was discounted at 10% for wholesale, and 6% for cash. ^ 14. A dealer sent $ 19.53 in payment of a bill of hose, after it was discounted 7%. What was the list price of the hose ? Solution. Let x = the list price ; then, .93 x = $19.53, and x = $21. Hence, the list price was $21. 15. A carriage was sold for $195 after a discount of 22%. Find the list price. 16. After a double discount of 25% and 20%, a piano was * sold for f 360. Find the list price. Find the rate of discount on the following four problems : . 17. A suit of clothes marked down from f 10 to f 6.80. Solution. $ 10 - $ 6.80 = $ 3.20. $ 3.20 is ^^ of $ 10. But, ^flftflj = .32 = 32 %. The rate of discount is 32 %. ^^'^ 18. Axminster carpet marked from $ 2.25 per yard to $ 1.75. 19. Shoes marked from $5.50 to $3.50. 20. Table cloth from $1.10 to 88 cents per yard. 21. Bought a piano listed at $950. Discount 45%, and 5% for cash. What was the net amount ? 22. If the list price, in Problem 21, had been $ 880, and the discounts 48% and 6%, what would the net amount have been ? 23. Eented a piano, listed at $1080, at $6 a month, with the agreement that the rent should be applied to the purchase price if I decided to buy it. Kept it 8 mo., and bought it at a discount of 40, 8, and 4^. What did I pay in addition to the rent ? *^ 24. Find the net amount of a bill of $ 1524.60, the discounts being 16 and 4. 25. Find the net amount of a bill of $ 825, the discounts being 10, 4, and 2. 26. Which would you prefer, a single discount of 20%, or a discount of 16| and 4 ? Digitized by VjOOQ IC 174 APPLICATIONS OF PERCENTAGE. 27. Which is greater, a single discount of 25%, or a dis- count of 20 and GJ ? 28. Find the net amount of a bill of $ 650, discounted at lSf%. 29. The net amount of a bill discounted at 15 and 10 is $ 722.16. What is the gross amount of the bill ? 30. Find the net amount of a bill for $ 1260.50 discounted at 8J%. 31. What is the bill which, discounted at 5J%, yields a net amount of $648. 09 ? 32. The bill^is f 728.50; the net amount, f 684.79. What is the rate of di&count? 33. The amount of the bill is f 800; the net amount, $608 ; the first discount, 20%. What is the second discount ? Find the net amount of the following bills : 34. $325.56 at 18 and 3. 35. $464.92 at 21 and 6. 36. $ 681.20 at 15, 5, and 3. 37. $760.10at20, 8, and2. 3a $ 1241.10 at 12, 8, and 4. 39. 200 yd. of calico at 6 cents ; 225 yd. of muslin at 7 cents ; 50 yd. of broadcloth at $3.50; 80 yd. of silk at $1.40; 95 yd. of flannel at 39 cents ; 72 yd. of cashmere at 98 cents ; 140 yd. of gingham at 9 cents. The discounts were 16 and 5. 40. State the three general types of problems in Trade Discount. TAXES. 239. A city must provide for water, light, and a police force; must make streets and sewers; must educate its children and take care of its poor; must meet other ex- penditures for the common good of its citizens. These enormous expenditures require much money. Digitized by VjOOQIC TAXES. 175 240. A county must provide for roads; build bridges and a courthouse ; provide for education ; pay the sala- ries of its officers ; and meet many other kinds of expenses, all of which require much money. 241. The state must pay the expenses of its legislature, the salaries of the governor and other officers ; must sup- port its schools and the various asylums for the blind, the deaf and dumb, the insane, and other unfortunates. These things require much money. 242. The United States government must provide for pensions ; for the army and navy ; for improvements of rivers and harbors ; for the salaries of the President, con- gressmen, judges of the Federal courts, and ministers abroad ; for the Post-office Department, and for many other things. .The expenses of the United States government amount to more than $1,000,000 per day for each of the 365 da. in the year. 243. Money collected for the various purposes named above is called a tax. 244. A Tax is a sum of money levied by the authority of the law upon citizens and their property, for public purposes. 245. Taxes may be levied by the general government, by a state, county, city, township, or school district. 246. Many states levy a tax upon each voter without regard to the amount of property that he owns. Such a tax is called a Poll Tax. It is usually a small amount, rarely exceeding $2 a year. Digitized by VjOOQIC 176 APPLICATIONS OF PERCENTAGE. 247. Property may be either Personal or Real Estate. Personal Property consists of movables, such as money, securities, household goods, cattle, etc. Real Estate is that form of property which consists of lands and improvements put upon them. 248. A Property Tax is a tax assessed upon property. It may be general or special. Note. An example of a special tax is a paving tax, which is a tax for improving streets and sidewalks. Give other examples. 249. General taxes are usually assessed and collected as follows : The state legislature determines by its appropriation bills the amount to be expended for state purposes. A state officer, usually the Auditor of Public Accounts^ as- certains the amount of taxable property in the state from the reports of certain officers, called assessors, who are elected by the people in townships or other districts of territory. He thus discovers what each dollar's worth of property must pay, in order to make up the amount appropriated by the legislature. The same thing is done in each of the smaller political districts, — the county, town, etc. These rates of taxation are all reported to the proper officer, who calculates the amount of tax each individual and piece of real estate must pay, and puts it into a book. This book is given to a collector, who collects the tax and returns it to another officer, called a treasurer. 250. A problem in taxes is not always a type of per- centage problem, although it is customary to speak of the tax on each $ 100 worth of property. Digitized by VjOOQIC TAXES. 177 251. There are three general types of problems in taxes : I. Given the amount of taxable property and the rate of taxation, to find the tax. II. Given the amount of taxable property and the tax to be raised, to find the rate of taxation. III. Given the tax and the rate of taxation, to find the amount of taxable property. Note. The last type of problem is of slight practical use, for one never needs to find the amount of property he has by this method. EXERCISE 97. In the year 1902 Laporte County had property assessed at $ 24,471,090, and 5469 persons who were liable to pay poll tax. The rate of poll tax for the county was $ 1 for each poll. The rate of property tax for the county was 34J cents on each $ 100 worth of property. 1. How much poll tax did the county receive for 1902, not allowing for those who did not pay ? 2. How much property tax was received, if there were no delinquents ? 3. The state levy for 1902 was 29| cents on each $100 of valuation. How much did Laporte County pay toward the support of the state ? 4. Indiana has levied a tax of 2| cents on each $100 worth of property to support Purdue University, Indiana University, and the State Normal School. How much per year does La- porte County pay to support these institutions ? Laporte City, the county seat of Laporte County, has prop- erty valued at $3,916,365, and 1299 polls. 5. How much tax did Laporte City pay toward the support of the county for 1902 ? 6. How much toward the support of the state ? Digitized by VjOOQIC 178 APPLICATIONS OF PERCENTAGE. 7. How much toward the support of the three state educa- tional institutions for the year ? Michigan City, in Laporte County, has property valued at $3,289,115, and 1735 polls. 8. How much was Michigan City's portion of the county tax for 1902 ? 9. How much did the city pay toward the state tax for 1902? 10. How much toward the support of the three educational institutions for a year ? Hendricks County had property valued, in 1902, at $15,645,- 278, and 3616 polls, each assessed at $1. U. If the county council wished to raise f 71,412 by taxa- tion, what would be the tax on each $ 100 worth of property ? Solution. Since each poll was assessed at $ 1, 3616 polls would pay $3616 poll tax. $71,412 - $3616 = $67,796 = the tax to be raised on the property. $67,796 -^ $16,646,278 = .00433+. Hence, the tax on each $100 is $.433 to the nearest mill. The levy by the county council in Hendricks County for 1902 for county purposes was $.43^ per f 100. 12. What did a citizen of Danville, the county seat of Hendricks County, pay to support the county, if he owned 200 A. of land assessed at $ 20 per acre, and personal property valued at f 1500? (Do not overlook the poll tax.) 13. If a farmer in Washington township, Hendricks County, has property assessed at f 3450, what is his tax for a year to support the three educational institutions ? How much does he pay to support Indiana University alone, at ^i^^ of a mill on the dollar ? {-^ of a mill on the dollar is the actual rate.) The state of Indiana has (1903) property valued at approxi- mately $1,500,000,000. The Legislature of the state has/ for several years, levied a tax of f .29| on each f 100 worth of property and a poll tax of f 1 on each poll, for state expenses. Digitized by VjOOQIC TAXES. 179 14. On the above property basis, how much tax is due the state for 1903 ? 15. Of the above 29f cents, levied by the state, 11 cents is for the support of the public schools. How much (approxi- mately) is due the schools from this source ? 16. If a citizen of Pike County has a farm of 160 A. assessed at $25 per acre, and personal property assessed at $ 1400, find his total state tax. How much of this goes toward the support of the schools ? 17. A tax of f 15,377.42 is to be raised in a town, the taxable property of which is valued at f 1,672,386. If there are 528 polls, each taxed at f 2, find the tax on each f 100 worth of property for town purposes. 18. Using the rate of taxation found in the preceding prob- lem, find the tax paid by a citizen of the town who pays a poll tax and who owns property valued at f 7231. 19. Suppose the county in which the above town is located has property valued at $34,043,550 and has 10,589 polls, each taxed at $1 for county purposes. If the county wishes to raise f 149,690.52 by taxation, find the county tax on each 9 100 worth of property. 20. Suppose, a citizen of the above county has 240 A. of land assessed at $28 per acre, and personal property assessed at $ 1,543. Find his total tax for a year, if he pays poll tax. 21. New York City, in 1902, had property valued at $3,857,047,318 and the rate per cent of taxation for that year was 2.23744%. Find the amount of money raised by New York City for 1902 from property, not allowing for delinquencies. 22. If a man in New York City paid f 1686.90 as taxes for 1902, what was the valuation of his property ? Digitized by VjOOQIC 180 APPLICATIONS OF PERCENTAGE. mSURANCE. 252. Houses are liable to be destroyed by fire or tor- nadoes, ships may be wrecked, and persons die or may be injured. Because of these facts, companies are organized that, for a certain amount, agree to pay for property thus de- stroyed, to allow a person a stated amount if injured, or to pay to his heirs, or those dependent on him, a fixed sum in case of his death. Such companies are called Insurance Companies, 253. Insurance is security against financial loss on account of the destruction of property, or by the injury or death of a person. _ ^ ^ 254. Many kinds of insurance receive their names from the dangers against which they offer insurance. The com- mon forms of insurance are : (a) Fire and Lightning. (e) Accident. (6) Tornado. (/) Life, (c) Marine. (^) Live Stock, (rf) Plate Glass. (A) Property. 255. The company taking the risk is called the Insurer, or Underwriter. 256. The property upon which the risk is taken, or the person, is the Insured. 257. A Policy is a contract made between the insurer and the person securing the insurance. 258. The Premium is the amount paid to the insurer for assuming the risk. Digitized by VjOOQIC PROPERTY INSURANCE. 181 259. The underwriters assume the risk for the length of time specified in the policy. 260. In Property Insurance the premium is estimated at a certain per cent of the amount of the risk for a speci- fied time. 261. In Accident Insurance the premium is estimated at a certain amount for the protection afforded for a given time. 262. In Life Insurance the premium is estimated at a certain amount per year for each thousand dollars of insurance, and varies with the age of the insured. 263. There are three general types of problems in In- surance. Can you state them ? PROPERTY INSURANCE. EXERCISE 98. 1. What is the premium on the following policies : (a) $2500 for 3 yr., at i% a year? (6) $3650 for 5 yr., at \% a year? (c) $4680 for 2 yr., at |% a year? (d) $8250 for 1 yr., at 1\% a year? 2. Find the annual rate per cent of the premium on the following policies : Amount. Premium. Time. (a) $3600 $60 6 yr. (6) 34260 ,$68 4 yr. (c) $8400 $316 3yr. (d) $24600 $1230 2yr. Digitized by VjOOQIC 182 APPLICATIONS OF PERCENTAGE. 3. Find the amount insured in each of the following policies : Premium. Annual Rate. Time. (a) $68.24 1% lyr. (6) $760 li% 3yr. (c) $872.73 ^% Syr, Id) $463.40 Hfc 2yr. 4. A schoolhouse worth $ 32,400 is insured for | of its value at i% annually for 3 yr. Find the premium. 5. A ship worth $52,000 was insured for f of its value at 2^%. The cargo, worth $ 8640, was insured for | of its value at 3%. What was the whole premium ? 6. Paid$290.19forinsuringacargo worth $13,656. What was the rate per cent of the premium ? 7. My house being destroyed by fire, I received from the underwriters $ 2760, which was f of the amount of the policy. Insurance was f of the value of the house ; the premium was $66.24. Find the rate and the value of the house. 8. A factory worth $ 75,000 was insured for | of its value by an insurance company at 2J%. Not wishing to carry the entire risk, the company reinsured \ of the risk at 1^%, ^ of the risk at 2J%, and ^ of the remainder at 2%. What was the amount of the premium remaining for the first company ? LIFE INSURANCE. EXERCISE 99. 1. Find the annual premium of a policy of $ 8000 at $ 17.54 a thousand. 2. Find the annual premium on an endowment policy of $10,000 at $85.60 a thousand. Note, An endowment policy is one which runs for a certain term of years (instead of for life), and which at the end of the term, if in force, is entitled to a share in the profit of the company, in addition to its face amount. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. ^ 183 3. In 5 yr. I have paid premiums amounting to $472.50 on my life policy of $5000. What is the annual premium for $1000? 4. For 8 yr. I have made bimonthly payment3 of $7.86 ks premium on my poUcy in a mutual insurance company. What is the' aggregate of my p aymen ts?" The annual premium is $15.72 per $1000. What is the amount of my policy? 5. A person carried a policy for $ 6000 for 9 yr., paying annual premiums of $31.80 per $1000. At the end of that time he surrendered his policy for 23% of what he had paid. What did he receive ? , 6. A person insured his life for $7000 at an annual pre- mium of $36.40 per $1000. After making 7 payments he died. How much more did his heirs receive than he had paid? 7. A man carried a 20 yr. endowment policy for $2000 at an annual premium of $44.70 per $1000. His dividends aggregated 16|% of the premiums. How much more did he receive at the maturity of the policy than he had paid ? a If I pay an annual premium of $ 69.80 per $ 1000 on a 10 yr. endowment policy of $4500, what amount has been paid at the maturity of the policy ? MISCELLANEOUS PROBLEMS. EXERCISE 100. 1. Change yj^ to a decimal fraction. 2. When it is 45 min. past 8 a.m. at Buffalo, 78** 55' W., what is the time at Salt Lake City, 112^ & W. ? 4. Bought 36,824 bu. of oats at 18f cents and sold them at a gain of 12^%. What did they bring ? Digitized by VjOOQIC 184 APPLICATIONS OF PERCENTAGE. 5. Settled a bill of $436.50 at a discount of 16 and 5. What was the amount paid ? 6. Find the tax on a real estate assessment of $5640 and a personal property assessment of $3824, at 15 mills on the dollar. 7. What is the commission, at 2^%, on the sale of 320 A. of land at $ 72.50 an acre ? 8. Find the premium on a policy insuring a house worth $4860 for f of its value at 1^%. 9. Divide .00864 by 2.7. 10. Bought a stock of goods for $8645 and sold them for $9748. What per cent did I gain ? (Approximate.) 11. Crochet hooks were sold at 3 cents each, which was at a gain of 40%. Find the cost per gross. 12. Mohair shoe laces, listed at 28 cents per dozen, were bought at 25% discount and sold at 2f cents a pair. Find the rate per cent of gain. 13. The difference of time of two places is 3 hr. 40 min. 30 sec. If the place having the later time is in 15° 24' 12" east longitude, what is the longitude of the other ? 14. Find the cost, at $5.50 a* cord, of a pile of wood 8 ft. high, 4 ft. wide, and 36 ft. long. — 15. Sold a lot of goods for $650.40. After paying 2\% commission to the auctioneer, I find that I have made a net gain of 24%. What did they cost me ? 16. How far will a train go in 7 hr., if its average speed is 36 mi. 124 rd. an hour ? 17. The valuation of a town is $ 2,624,800. The tax levy is $32,810. Give the rate in mills. la A owns f of a mill, and sells f of his share to B, who then owns \ of the mill. What part of the mill did B own before his purchase ? B's share is now what part of A's ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 185 19. A bill, having been discounted 26 and 6, amounts to $ 855. What is its face ? Note. Employ cancellation when possible. 2 ^^ 20. A can do a piece of work in 7 da., B in 8 da., and C in 9 da. In what time can they do it, working together ? 21. If, in problem 20, A worked 3 da. and B 4 da., in what time could C finish the work ? 22. Sugar was selling at 20 lb. for a dollar. The price advanced 16|%. How many pounds less could then be bought for a dollar ? 23. Bought a house for $5680. After renting it for one year at $41 a month, and spending $121 for taxes and repairs, I sold it for $6000. What is the per cent of gain ? 24. What is the commission on the following sales ? 3640 bu. of oats at 18 cents. Commission 2%, 2500 bu. of wheat at 53^ cents. Commission 2\%. 3600 bu. of com at 27|^ cents. Commission 2^%. 25. I of 3^ is what per cent of | -i- 1 ? Digitized by VjOOQIC CHAPTER X. APPLICATIONS QP PBRCBNTAQB. Interest; Notes; Partial Pasrments; Banks and Bank- ingr; Exchange; Stocks; Bonds. INTEREST. 264. Many persons have occasion to use money when they do not have it at hand. If they are responsible, they can obtain the money from others by guaranteeing its return and by agreeing to pay a specified sum for such service. Such persons are called Borrowers. 265. The amount paid for such services is called in- terest. Interest is compensation for the use of money. The money borrowed is called the Principal. Note, Money paid for labor is usually called wages; that paid for the use of houses and lands is called rent. Wages and rent are usually estimated at a specified amount for a day, a month, or a year. Time, consequently, is an essential element in calculating such compensation. 266. The longer the time that one uses the money of another, the larger should be the compensation. The ordinary unit of time for estimating interest is one yeaf . 267. The Rate of Interest is a specified number of hun- dredths of the principal, paid as a compensation for its use for one year. Hence, interest is one of the applica- tions of percentage. 186 Digitized by VjOOQIC INTEREST. 187 268. When money is loaned at 6%, it is understood that the compensation is 6 hundredths of the principal for its use for one year, unless otherwise specified, 269. Interest is one of the most common of the applica- tions of percentage. Since the time for which money is borrowed may vary from a few days to many months or years, it is the most difficult of these applications. 270. Problems in interest involve four things : I. The sum of money upon which interest is paid. This is called the principal. II. The rate per cent of interest. III. The time for which the sum is loaned. IV. The interest itself. 271. Hence, there are four general types of problems in the subject of interest. The four things mentioned above are so related that if any three of them are given, the other may be found. The four general types are: I. Given the principal, the rate of interest, and the time, to find the interest. II. Given the principal, the rate, and the interest, to find the time. III. Given the principal, the time, and the interest, to find the rate. IV. Given the interest, the time, and the rate, to find the principal. Note, In business the first t3rpe of problem is the only one that is used to any considerable extent. It is rare that any one wishes to find the time, the rate, or the principal. 272. There are many methods of calculating interest. Per- sons whose business requires a considerable amount of such Digitized by VjOOQIC 188 APPLICATIONS OF PERCENTAGE. work frequently, though not always, supply themselves with * books containing "Interest Tables." With the assistance of these Tables, it is easy to solve any problem in interest, though not necessarily accurately. For the ordinary person one good method is all that is needed. Three methods will be here presented, but proficiency in one is recommended. 273. Method I. Illustrative Problem. What is the interest on $750 for 2 yr. 7 mo. 15 da. at 6% ? Solution. 2 yr. 7 mo. 15 da. = 2 yr. 7 J mo. = 2f yr. = -^ yr. •750 X xfij = interest for 1 yr. $7.50 3 Hence, i^H^ x -f- x ^ = #118.125 = interest for 2.yr. 7 mo. 15 da. 4 The accurate result m practice is #118.13. This method commends itself as simple in thought, and short, since cancellation can be used to good effect. The pupil should notice that the time must be expressed in years. For example : 2 yr. 8 mo. 21 da. = m yr. = w yr. Note. The year is counted as 12 mo. of 30 da. each. 274. Method II. Take the same problem. SoLunoir. »750. .06 #45.00 = interest for 1 yr. 2 190.00 = mterest f or 2 yr. 6 mo. = } yr. 22.50 = interest for 6 mo. 1 mo. = J of 6 mo. 3.75 = interest for 1 mo. 15 da. = i of 1 mo. 1.875 = interest for 15 da. % 118.13 = interest for 2 yr. 7 mo. 15 da. This method is a good one. It is sometimes known as the ** aliquot parts*' method. Digitized by VjOOQIC INTEREST. 189 276. Method III. Twelve per cent a year is the same as 1% a month. Hence, if the time is expressed in months and a fraction of a month, multiply the principal by the number thus found, expressed as hundredths, and the result will be the interest at 12%. Then take such part of the result at 12% as the given rate per cent is part of 12%. Take the same problem as in Method I. Solution. 2 yr. 7 mo. 16 da. = 81.6 mo. xJrr ^^ 31.5 = .315. ^760 X .316 = $236.26 = interest at 12%. Smce the rate is 6%, $236.25 -t- 2 = 1118.125 = the result at 6%. This is known as the *^ 12% method/* and is used in some banks. To the Teacher, Each of these three methods shoold be learned by the pupil. However, the pupil should be permitted, and even encouraged, to use the one method which appeals most to him, and he should practice no other. It is not necessary that there be uniformity of method among the pupils of any one class. EXERCISE 101. (Mental.) 1. What is the interest on $800 for 1 yr. at 6%? at 8%? at 9% ? at 49J, ? at 12% ? at 10% ? at 7% ? 2. What is the interest on $600 for 2 yr. at 3% ? at 5% ? \ at 7^% ? at 4^% ? at 6% ? at 8% ? )i| First find the interest for 1 jt, ^ a Substitute 3 yr. for 1 yr. in first problem, and solve. 4. What is the interest on $900 for 2 yr. 6 mo. (2^ yr.) aJ^6%? at4%? at8%? at3%? k What is the interest on $400 for 3 yr. 3 mo. (3 J yr.) at6%? a,t^%? at5%? at8%? at3%? 6. What is the interest on $ 300 for 3 mo. at 6% ? at 8% ? at 6% ? at 4% ? 7. What is the interest on $240 for 2 mo. 16 da. at 6%? at 10%? at7i%? a What is the interest on $150 for 15 da. at 6% ? at 8% ? at 6%? For 10 da. at 6%? at 6%? Digitized by VjOOQIC \i 190 APPLICATIONS OF PERCENTAGE. EXERCISE 102. Find the amount in each problem, giving the result to the nearest cent The annount is the interest plus the principal. Principal. Rate. Time. 1. 33600 i^% 6 yr. 10 mo. 16 da. Solution. Using Method I, 6 yr. 10 mo. 16 da. = 5f JJ yr. = 6} yr. Then, nm x ^ x ^ = ^^^l-^^ = interest. 2 The amount = $3600 + 1961.75 = ^4551.75. ^_^_^^ 3480 4% 3yr. 6 mo. 10 da. ^jC/^'JJ^^^ $650.40 6% 2yr. 7 mo. 15 da. ^^^'^'^^^^'^ 3864.36 6% 4yr. 8 mo. 6 da. y -'s. $1275.86 8% lyr. 9 mo. 5 da. 6. $1464.29 10% 2yr. 1 mo. 3 da. 7. $2580.47 7% 3 yr. 2 mo. 10 da. -^ $4580 6\% 4yr.' 6 mo. 2 da. -w^i^ ^^^. $5000 6i% 6yr. 3 mo. Ida. i -^ TlO. $6840.75 7J% 6 yr. 11 mo. 19 da. g^fl^^ $490.92 8% lyr. 3 mo. 6 da. TZ'^^J^^^^ $3794.08 9% 2yr. 4 mo. 7 da. ^13. $892.45 5% 3yr. 4 mo. 8 da. 14. $1234.16 6% 2yr. 3 mo. 12 da. Solution. Using Method III, 2 yr. S mo. 12 da. = 27.4 mo. Then, .274 is the multiplier. % 1234.16 X . 274 = $ 338. 16 = interest at 12 %. $338.16 -^ 2 = $169.08 = interest at 6%. 11234.16 + $169.08 = $1403.24 = amount. 15. $5871.48 7% 4yr. 1 mo. 20 da. 16. $89.26 3% 6yr. 8 mo. 25 da. 17. $13.55 5i% 5 yr. 10 mo. 16 da. Digitized by VjOOQIC INTEREST. la $1.05 mo 8yr. 2 mo. 24 da. 19. $10,874.80 4% 6 mo. 10 da. 20. $916.16 8% 9 mo. 18 da. 21. $1371.49 6% lyr. 11 mo. 22 da. 22. $1641.04 n Syr. 2 mo. 29 da. 23. $3200 10% 2yr. 7 mo. 1 da. 24. $93.94 • 12% 10 mo. 4 da. 25. $16,921.72 Hfc 3 mo. 11 da. 26. $879.40 4% 2yr. 4 mo. 10 da. 27. $783.12 6i% 3yr. 8 mo. 15 da. 2a $1864.60 7% lyr. 7 mo. 18 da. 29. $1460.20 7% 4 yr. 10 mo. 21 da. 30. $21.26 6% 4yr. 1 mo. 2 da. 31. $683.40 6% 6yr. 3 mo. 16 da. 32. $1660 mo 2yr. 11 mo. 22 da. 33. $830.80 mc 3yr. 5 mo. 25 da. 34. $98.07 H% Syr. 6 mo. 28 da. 35. $2664.90 H% 6yr. 7 mo. 7 da. Solution. Using Method II : 92664.90 .04J » 106. 1960 8.8496 #115.0456 = 6 interest for 1 yr. 191 1690.2736 = interest for 6 yr. 6 mo. = J yr. 57.5228 = interest for 6 mo. 1 mo. = J of 6 mo. 9.5871 = interest for 1 mo. 6 da. = J of 1 mo. 1.9174 = interest for 6 da. 1 da. = J of 6 da. .3196 = interest for 1 da. $759.6204 = interest for 6 yr. 7 mo. 7 da. 2654.90 $3414.62 = amount for 6 yr. 7 mo. 7 da. Digitized by VjOOQIC 192 APPLICATIONS OF PERCENTAGE. 36. $3371.10 37. $640 • 3a $756.50 39. $833.12 40. $794.80 41. $1026.26 42. $1230 43. $1426.80 44. $1862.60 45. $2045 46. $3870 . 47. $4312 'tt. $5680 49. $6875 ^. $5892 >^ 51. $66.40 52. $83.70 53. $168.10 54. $234.90 55. $361.12. > 56. $jt32.90\ "57. $663.08 58. $839.16 59. $748.90 60. $2126.40 09» ^% mo 4J% 5% 6% 5Wc 6% 6% 6J% H% 7% 7% 7% 7% 7% S% S% 9% 10% 10% 10% 12% 12% 5 yr. 5 mo. 11 da. 4 yr. 4 mo. 4 da. 5 yr. 3 mo. 8 da. 7 yr. 6 mo. 13 da. 8 yr. 8 mo. 7 da. 9 yr. 1 mo. 14 da. 4 yr. 2 mo. 16 da. 3 yr. 10 mo. 2 da. 7 yr. 5 mo. 19 da. 6 yr. 9 mo. 21 da. 2 yr. 11 mo. 20 da. 3 yr. 11 mo. 25 da. 5 yr. 10 mo. 22 da. 6 mo. 24 da. 3 mo. 22 da. 24 da. 18 da. ^ 2yr. 10 mo. 29 da. i ^^' 5 yr. 24 daJ c^ v 7 mo. 16 da/ \ y 11 mo. 22 da. ^ '^ 24 da./ •' 20da. V '-^ 26 da. je^^- 12 da. \^ Digitized by VjOOQIC INTEREST. 193 Time Determined by Dates. 276. In each of the above problems the time is given. In the follovring problems conditions for getting the time are given. From these conditions the time is obtained by com- pound subtraction. Thus, the time from Sept. 20, 1900, to July 12, 1904, is found as follows : Write for the minuend the year, number of month, number of day in the month of the later date ; and for the subtrahend, the corresponding numbers of the earlier date. Proceed as in subtraction of compound numbers, counting 30 da. for a month. Solution. 1904 7 12 1900 9 20 8 9 22 Hence, the difference in time is 3 yr. 9 mo. 22 da. After the time is obtained use any of the three methods suggested in §§ 273, 274, 276. EXERCISE 103. 1. What is the interest on $478.40, at 7%, from June 12, 1900, to Dec. 21, 1903? 2. What is the interest on 1^548.50, at 8%, from July 18, 1896, to May 5, 1900? 3. What is the interest on $93.80, at ^%, from Dec. 7, 1895, to Oct. 1,1902? 4. Find the amount of $2480, on interest at 6%, from Aug. 28, 1898, to Feb. 19, 1903. 5. Find the amount of $3728.30, on interest at 7%, from Feb. 28, 1898, to Jan. 14, 1902. Find the interest on : 6. $469.12, at 6%, from June 8, 1897, to Aug. 15, 1900. 7. $48.16, at 7%, from Dec. 15, 1900, to May 3, 1904. Digitized by VjOOQIC 194 APPLICATIONS OF PERCENTAGE. a $907.92, at 7%, from May 7, 1900, to Sept. 19, 1904. 9. $1359.06, at 7%, from Oct. 24, 1900, to AprU 6, 1904. 10. $750.20, at 5%, from March 12, 1902, to Nov. 28, 1905. 11. $609.47, at 5%, from Aug. 16, 1901, to June 1, 1905. 12. $1936.82, at 4^%, from Jan. 29, 1902, to Oct. 14, 1905. 13. $207.49, at 4^%, from Nov. 3, 1897, to July 28, 1903. 14. $2882.75, at 8%, from Feb. 15, 1900, to Oct. 11, 1903. 15. $69.20, at 8%, from April 1, 1898, to Jan. 22, 1903. 16. $7512.36, at 8%, from July 30, 1901, to Dec. 9, 1901. 17. $169.17, at 7^%, from Sept. 7, 1895, to March 18, 1902. la $273.48, at 7^%, from Dec. 19, 1896, to Nov. 30, 1900. -19. $428.10, at 6%, from Feb. 23, 1897, to Aug. 5, 1901. 20. $491.73, at 6%, from Nov. 16, 1901, to Nov. 28, 1901. ^. $636.80, at 6%, from Jan. 5, 1899, to Jan. 31, 1903. Find the amount of : 22. $19.12, at 5^%, from Aug. 12, 1900, to June 10, 1904. 23. $729.13, at ^%, from June 18, 1896, to Oct. 21, 1900. 24. $258.18, at 5^%, from Dec. 3, 1900, to May 1, 1902. , 25. $371.29, at 5^%, from March 25, 1899, to July 29, 1901. 26. $5§0, at 9%, from Jan. 16, 1897, to Sept. 10, 1901. ^ 27. $412.31, at 9%, from Oct. 2, 1898, to March 28, 1903. 2a $7539.06, at 3^%, from Feb. 19, 1899, to Nov. 6, 1900. 29. $117.59, at 3%, from April 29, 1898, to Dec. 19, 1903. 30. $396.16, at 6%, from July 5, 1899, to Feb. 2%, 1902. 31. $872.28, at 7%, from May 11, 1897, to Jan. 7, 1902. 3^. $250.10, at 7^%, from Nov. 17, 1896, to Aug. 24, 1902. 33. $1536.81, at ^%, from Sept. 23, 1899, to Jan. 1?, 1903 Digitized by VjOOQIC NOTES. 195 34. $2140.60, at 6%, from June 6, 1896, to April 23, 1900. 35. $16.31, at 8%, from Oct. 12, 1891, to March 5, 1901. 36. $589.76, at 7%, from Feb. 8, 1896, to Aug. 19, 1901. 37. $26,824, at 4%, from Dec. 18, 1898, to Feb. 21, 1902. 277. NOTES. $800. Boston, Mass., Jan. 8, 1902. Six months after date, for value received, I promise to pay to James B. Rogers, or order, Eight Hundred Dollars, with interest at seven per cent per annum. John T. Walkbb. The above promise is a Promissory Note. Walker is the Maker, and Rogers the Payee. The $800 is .the Principal. 278. From the foregoing, form a definition of a promis- sory note, and of maker and payee. 279. Promissory notes may be bought and sold like other forms of property. Some calculation is usually necessary to determine the value of a note. Why? 280. If James B. Rogers should sell the above note, he would write his name on the back. This is called indorsing the note in blank. It is an order to John T. Walker to pay it to the person who owns it at its maturity, July 8, 1902. 281. The indorsement might indicate some person upon whose order the note is to be paid. This is a special indorsement. 282. Under the laws of some states the above indorse- ments would make Rogers responsible for the payment of this note in case Walkeir should fail to do so. If he wishes Digitized by VjOOQIC 196 APPLICATIONS OF PERCENTAGE. to avoid such responsibility, he adds the words " without recourse " after his name. Note. If a note is payable to ** bearer," it may be exchanged without indorsement. If made payable to a particular person, and does not con- tain the words ** or order" qr ** or bearer," it is not negotiable, that is, it cannot be sold. 283. There are many forms of notes, but the differences are slight. If, instead of "six months after date," the words " on demand " occurred in the above note, it would be called a Demand Note. Write one. 284. If, instead of " I promise," it read " we or either of us promise," and two persons signed it, it would be a Joint- and-Several Note. PARTIAL PAYMENTS. 285. Sometimes the borrower instead of paying all the note at once makes a partial payment. He may make a number of such partial payments. 286. The United States Supreme Court has prescribed a rule for finding the value at any time of notes upon which partial payments have been made. It is called THE UNITED STATES RULE. I. The rule for casting interest^ when partial payments have been made^ is to apply the payment^ in the first place^ to the discharge of the interest then due, II. If the payment exceeds the interest^ the surplus goes toward discharging the principal, and the subsequent interest is to be computed on the balance of principal remaining dus. III. If the payment be less than the interest^ the surplus of interest must not be taken to augment the principal; but Digitized by VjOOQIC PARTIAL PAYMENTS. 197 interest continues on the former principal until the period when the payments^ taken together^ exceed the interest due^ and then the surplus is to be applied toward discharging the principal; and interest is to be computed on the balance as aforesaid. Note, The time periods for the United States Rule are all found by compound subtraction. EXERCISE 104. 1. A note of $ 850, bearing interest at 7 % , and dated March 1, 1902, has the following indorsements written on the back of it : Jan. 24, 1903, $86. Mch. 10, 1904, $175. What is due Dec. 29, 1904 ? Year. Month. Day. Solution. 1904 12 29 $860 1904 8 10 176 1903 1 24 86 1902 3 1 10 23 $86 1 16 176 9 19 850 The above is a convenient way to write the dates. Subtract the lowest from the one above it, the second lowest from the one above it, and so on. The principal =$850. The interest for 10 mo. 23 da. = 53.385 The amount for this time = $903,385 The first payment = 86. The second principal = $817,385 The interest on this for 1 yr. 1 mo. 16 da. = 64.527 The amount = $881,912 The second payment = 176. The third principal = $ 70(5.912 The interest on this for 9 mo. 19 da. = 39.724 The amount due to nearest cent, Dec. 29, 1904 = $746.64 Note. Payments less than the accumulated interest will rarely be made, since they do not diminish the interest-bearing portion of the debt. Digitized by VjOOQIC 198 APPLICATIONS OF PERCENTAGE. 2. A note whose principal is $500, dated March 1, 1902, and bearing interest at 6%, has the following indorsements : June 1, 1903, $ 65. Sept. 16, 1904, $124. What is due Jan. 1, 1906 ? 3. Principal, $1000. Date, Sept. 1, 1902. Rate, 8%. In- dorsements: ^- , ^o lAAO ^rrtt March 12, 1903, $ 75. June 18, 1904, $275. March 15, 1905, $360. What is due Sept. 1, 1905 ? 4. Principal, $ 1200. Date, July 1, 1902. Rate, ^%. Indorsements: ^arch 16, 1903, $160. June 12, 1904, $320. Aug. 5, 1905, $500. Rate, 1% What is due July 1, 1906 ? 5. Principal, $850. Date, Indorsements: _ , ^^ ^^^^ July 15, 1902 June 1, 1903, Dec. 12, 1904 May > > 10, 1901 $130. $46. $380. What is due May 10, 1905 ? Year. Month. Day. Solution. 1906 6 10 1904 12 12 1903 6 1 1902 7 16 1901 6 10 $860 380 46 180 1 1 2 10 6 4 6 16 11 28 9130 46 380 860 Digitized by VjOOQiC PARTIAL PAYMENTS, 199 The principal = $850.00 The interest for 1 yr. 2 mo. 6 da. = 70.24 The amount = $920.24 Payment = 130.00 The second principsd = $790.24 The interest for 10 mo. 16 da. is greater than the payment, $46. Hence, find the interest from July 15, 1902, to Dec. 12, 1904. This time is the sum of 10 mo. 16 da. and 1 yr. 6 mo. 11 da. The interest for 2 yr. 4 mo. 27 da. = 133.22 The amount = $923.46 The sum of the two payments = 426.00 The third principal = $497.46 The interest for 4 mo. 28 da. = 14.32 Amount due May 10, 1905 = $611.78 6. A note of $ 1800, bearing interest at 7%, and dated June 12, 1900, is indorsed as follows : March 21, 1901, $ 183.50. Oct. 12, 1902, $395.75. May 10, 1904, $583.45. What is due July 1, 1906 ? 287. THE MERCHANTS' RULE. I. Find the amount of the principal for the entire time. II. Find the amount of each payment from the time that it was made to the time of settlement. III. From the first amount subtract the sum of the amounts of the several payments.. Note i. This method allows interest on each payment*f or all of the time that it is in the creditor's possession. Note 2, It is frequently used among business men, but it has no legal standing and must be a matter of special contract. The amount due at settlement is less than by the United States Rule. Note 3. The time periods for the Merchants* Rule are found in this text by compound subtraction. Digitized by Google 200 APPLICATIONS OF PERCENTAGE. EXERCISE 105. 1. A note of $475, dated July 24, 1902, at 6% interest, had the following partial payments made: Nov. 4, 1902, $200; April 14, 1903, $ 150. What was due Dec. 30, 1903 ? Year. Month. Day. Solution. 1903 12 30 1903 14 1902 11 4 1902 24 1 6 $475 1 26 $200 8 16 $160 The principal = $476.00 The interest for 1 yr. 6 mo. 6da. = 34.04 The amount = $609.04 The first payment = $200. The interest for 1 yr. 1 mo. 26 da. = 11.66 The amount = $211.66 The second payment ^ $160.00 The interest for 8 mo. 16 da. = 6.33 The amount = $166.33 The sum of the amounts of payments = $366.88 The amount due Dec. 30, 1903 = $142.16 2. A note for $ 650, dated Jan. 10, 1903, and bearing interest at 5%, has the following indorsements: March 15, $125; July 12, $ 240 ; Oct. 5, $ 85. What was due Jan. 10, 1904 ? 3. A note for $ 785.40, dated May 1, 1902, and bearing in- terest at 6%, is indorsed as follows: Aug. 20, $180.20; Nov. 5, $250.80; Feb. 24, 1903, $236.50. What was due May 1, 1903 ? 4. Principal, $892.60. Date, July 25, 1902. Rate, 7%. Indorsements: Sept. 1, $325; Nov. 19, $175.50; Jan. 12, 1903, $ 90 ; May 10, $ 300. What was due July 1 ? 288. Originally the Merchants' Rule was concerned with problems in which the whole time to run was not to exceed one year^ and the time periods for which the payments Digitized by VjOOQIC METHOD OF FINDING INTEREST BY DAYS. 201 were on iqterest were much less than a year. But of late the use of this rule has been extended to include problems with the same conditions as the United States Rule. 289. The methods of calculating the time for both rules, as found above, obtain in all instances where a hank is not a party in the note. Indeed, many banks do not enforce the method of calculating interest by days even when par- ties to the note, except in connection with the discounting of notes for short periods. THE METHOD OF FINDING INTEREST BT DATS. 290. Interest is usually computed by banks, by days, when the time is short. In such cases find the actual number of days by the following Method. The actual number of days from March 10 to July 15 : In March, 21 da. more. In April, 30 da. In May, 31 da. In June, 30 da. In July, 15 da. 127 da. Hence, there are 127 da. from March 10 to July 16. EXERCISE 106. 1- Find the interest on 1^850, at 5%, from April 9 to July 12, same year. Solution. In April, 21 da. more. In May, 31 da. In June, 30 da. In July, 12 da. Hence, there are 94 da. from April 9 to July 12. 94 da. = ^A yr. = ^j^ 1^- The interest =*860xiJ^x^ = a 11.10. Digitized by VjOOQIC 202 APPLICATIONS OF PERCENTAGE. Find the interest: Principal. Time. Rate. 2. $640.80 March 15, 1902, to June 24, 1902 6% 3. « 1254.25 June 24, 1902, to Sept. 18, 1902 7% 4. * 796.13 Aug. 12, 1902, to Dec. 27,1902 5% 5. $2958 May 17, 1902, to Aug. 29, 1902 4^% 6. $4872.80 Sept. 28, 1902, to Jan. 5,1903 8% Solution. In Sept, 2 da. more In Oct., 31 da. In Nov., 30 da. In Dec, 31da; In Jan., 5 da. Hence, there are 99 da. from Sept. 28 to Jan. 6. 99 da. = ii^ yr. = JJ yr. Then, in short « 4872.60 X yjjy X Ji = • 107.20. Hence, the interest = % 107.20. Principal. Time. Rate. 7. $86.97 Jan. 5, 1903, to June 16, 1903 9% a $916.20 Oct. 12, 1903, to Feb.| 2,1904 10% 9. $8712.91 Dec. 27, 1903, to April 30, 1904 1\% Solution. In Dec, 4 da. more In Jan., 31 da. In Feb., 29 da. (Leap Year) In Mar., 31 da. In April, 30 da. 126 da. = the time period. The interest = « 8712.91 x tJJtt x JM = ♦ 226.90. Principal. Time. Bate. 10. $345.40 Feb. 5, 1903, to July 1,1904 5% U. $760.66 Nov. 14, 1903, to Feb. 24,1904 6% Digitized by VjOOQ IC GENERAL PROBLEMS IN SIMPLE INTEREST. 203 y GENERAL PROBLEMS IN SIMPLE INTEREST. 291. Simple interest is interest on the principal alone. 292. The four general types of problems in interest were mentioned in § 271. Thus far all the problems in interest have been of the first kind. The second, third, and fourth general types are not much used in business. Short lists of problems illustrating these types will be given here. ^ EXERCISE 107. . " ' GIVEN THE PBINCIPAL, INTEBEST, AND BATE PEB CENT, TO FIND THE TIME. 1. The interest on $ 324.60, at 8%, was $ 91.97. What was the time ? Solution. The interest on $324.60 for 1 mo., at 8%, is $2,164. To produce $91.97 at the same rate, the time must be as many times 1 mo. as $91.97 is times $2:164. $91.97 is 42^ times $2.164 ; hence, the time was 42} mo., which = 3 yr. 6 mo. 16 da. Principal. Interest. Bate. 2. $3794.08 $803.40 9%. Find time. a $892.46 $149.73 6%. Find time. 4. $1234.16 $169.08 6%. Find time. 5. $6871.48 $1701.10 7%. Find time. 6. $89.26 $18.04 3%. Find time. EXERCISE 108. t^ GIVEN THE PBINCIPAL, INTEBEST, AND TIME, TO FIND THE BATE PEB GENT. 1. The interest on $ 324.60 for 3 yr. 6 mo. 16 da. is $91.97/ What is the rate per cent ? Solution. The interest on $32^.60 for 8 yr. 6 mo. 15 da., at 1%, is $11.49f. To produce 191.97 in the same time, the rate must be as many times 1 % as $ 91.97 is times $ 11.49f . 1 91.97 is 8 times $ 11.49f ; hence, the required rate must be 8 %. /- Digitized by VjOOQIC 204 APPLICATIONS OF PERCENTAGE, Principal. Interest. Time 2. $650.40 $85.36 2 yr. 7 mo. 16 da. Find rate. 3. $864.36 $242.89 4 yr. 8 mo. 6da. Find rate. 4. $1275.86 $180.04 1 yr. 9 mo. 5 da. Find rate. 5. $ 1464.29 $306.28 2 yr. 1 mo. 3 da. Find rate. 6. $2580.47 $ 577.02 [ EXE 3 yr. 2 mo. ^/ Find rate. IRCISE 109. ^ / GIVEN THE INTEREST, RATE PER CENT , AND TIME, TO FIND THE PRINCIPAL. 1. What principal will produce $ 91.97 in 3 yr. 6 mo. 15 da., at8%? Solution. A principal of $1, with the above rate and time, will produce $ .28}. To produce $91.97, the principal must be as many times $1 as $91.97 is times $.28}; $91.97 is 324.6 times $.28); hence, the required principal is $ 324.60. Interest. Bate. Time. 2. $166.62 8% 3 yr. 9 mo. 17 da. Find principal. 3. $35.16 ^% 6 yr. 9 mo. 24 da. Find principal. 4. $89.68 6% 3 yr. 2 mo. 7 da. Find principal 5. $8.04 7% 2 yr. 4 mo. 18 da. Find principal. 6. $ 150.41 7% 2 yr. 4 mo. 12 da. > EXERCISE 110. Find principal. 1. At what rate per cent will any principal double itself in 10 yr.? Solution. At 1 %, the interest on any principal is yj^ of itself for 1 yr. Tlicn, in 10 yr. it will be ^, or ^^, of itself. Hence the rate must be 10 % tx> make the interest equal the principal, or to double itself. Or, in short, since any principal is 100 % of itself, divide the number 100 by the number of years, and call the result the rate per cent. 2. At what rate per cent will any principal double itself in 8 yr.? in 12 yr.? in 16§ yr.? in 20 yr.? Digitized by VjOOQIC BANKS AND BANKING. 205 3. At what rate per cent will any principal treble itself in 10 yr.? 8yr.? 12 yr.? 16f yr.? 22 yr.? Note. To treble itself, the interest in the given time must be twice the principal. 4. In what time will any principal double itself, at 3% ? Solution. Since 3 % of any principal is the interest for 1 yr., it will take as many years for the principal to double itself as 3 is contained times in 100. Hence, 33^ yr. is the time required. 5. In what time will a principal double itself at 4%? 4i%? 5%? 6%? 8%? 10%? 6. In what time will any principal treble itself at the rates mentioned in Problem 5, above ? 7. Bought a piece of land for $ 6825. Kept it 2 yr. 6 mo. 18 da., and sold it for $9841.65. What is the rate per cent of interest that the investment yielded ? Find the lacking numbers in the following problems : Principal. Bate. Time. Interest. Amonnt. a $834.60 7% ? $48.20 1 9. $560.40 ? 2 yr. 2 mo. $64,639 ? 10. ? 6% 7 mo. 15 da. ? $350.65 11. $2500 8% ? $656.67 ? 12. ? 9% 90 d. $72.56 ? 13. $150.60 ? 72 d. $2.26 BANKS AND BANKING. 293. A Bank is a financial institution which keeps money on deposit for individuals or firms ; which collects money; which loans a part of its capital or the deposits on hand ; which cashes checks and drafts, often charging a fee for doing so ; and which discounts notes. The national banks of the United States issue promissory notes in the form of " bank bills " which circulate as money. Digitized by VjOOQIC 206 APPLICATIONS OF PERCENTAGE. 294. A Check is a written order on a bank, made by a depositor, and payable whenever presented. It may be made payable to the bearer^ to the order of the bearer or payee (the person named in the check), or to self. 295. The following is a form of check used in drawing from a bank money which is on deposit : IBAVIB.INDL. M9lLCaE:N & COm Bajskers, «. .IXl&UyRS 296. An individual may obtain money from a bank by giving his note for it, or by selling to the bank a note made by another person. It is necessary that he give security for the payment of the amount of the note when due. 297. If a bank buys a note, it deducts interest on the face (or amount) of the note from the date of purchase to the date of maturity. 298. In some states, three days, called " days of grace," are allowed beyond the date fixed for payment in the note. About half the states of the Union allow the days of grace. In this text the three days are to be allowed^ unless otlle^ wise designated. A note is due on the day specified in it for payment; the date of maturity^ when three days of grace are allowed, is three days later. Digitized by VjOOQIC BANKS AND BANKING, 207 299. The charge made by a bank for advancing money jn notes is usually called bank discount. It is nothing but interest paid in advance. 300. Banks buy two kinds of notes : I. Non-interest-bearing notes. II. Interest-bearing notes. 301. To solve a problem concerned with a non-interest bearing note : (1) Find the date of maturity. (2) Find the term of discount in days. (3) Find the interest on the face of the note for the term of discount, calling it the bank discount. (4) Subtract the bank discount from the face of the note, and call the result the proceeds. 302. To solve a problem concerned with an interest- bearing note: Find the amount of the note at simple interest from date to maturity^ then ascertain the discount on the total amount (principal and interest) as above, and deduct. 303. Some notes are drawn up in days and some in months ; but they must be for short periods if they are to be sold at banks. If a note is drawn at 90 da., count 98 da. from date, to find the date of maturity. If it is drawn at 3 mo., count 3 mo. 3 da. from date, to find date of maturity. Thus, a npte dated July 15 at 90 da. is matured as follows : 16 da. more in July, 31 da. in August, 30 da. in September, and 16 da. in October make up the 93 da. Hence, the date of maturity is Oct. 16. If the above note be drawn at 3 mo. , count forward 3 mo. from July 15 to Oct. 15, and then add 3 da. Hence, the date of maturity is Oct. Id. Digitized by VjOOQIC 208 APPLICATIONS OF PERCENTAGE. EXERCISE HI. 1. A note of If 80, dated May 14, 1903, payable in 60 da., was discounted June 1 at bank at 7%. Solve completely. Solution. (1) The date of maturity is 03 da. after May 14, or July 16. (2) The number of days from June 1 to July 16 is 45 da., which is the term of discount. (3) The bank discount, or the interest on $80 for 46 da. at 7%, is 10.70. (4) $80 - $0.70 = $79.30, which is the proceeds. This all means that Mr. A gave Mr. B the above note on May 14; but on June 1, when the note was not due, Mr. B needed the money, and he sold the note to the bank. The bank could not collect this note until July 16. Hence, it charged Mr. B $0.70 for thus accommodating him, and gave Mr. B the remainder, $79.30. Now Mr. A must settle this note at the bank, and if he does not, Mr. B must do it, for he signed the note over to the bank when he sold it, thus becoming security for the pay- ment of it. 2. A note of $ 175, dated March 14, 1903, payable in 3 mo., was discounted May 1 at 7%. Solve completely. Solution. (1) The date of maturity is June 17. (2) The term of discount is 47 da. (3) The bank discount is $1.60. (4) The proceeds are $173.40. Solve the following, not adding days of grace in Problems 3 to 10: Face of Note. Time Discounted. Bate of DisQoont. 3. $826.00 60 da. 6% ^4. f 927.18 30 da. 6% 5. $264.83 46 da. 5% R f 169.47 90 da. 7% 7. $2968.61 4 mo. mc a $417.80 5 mo. Mc 9. $361.28 6 mo. 1% 10. $248.66 60 da. 7% 11. $1827.90 40 da. 6i% 12. $83.70 1 yr. 3 mo. 6% Digitized by VjOOQIC BANKS AND BANKING, 209 13. $850. BooNviLLE, Ind., July 21, 1903. Three months after date, for value received, I promise to pay to Katharine Sedgwick, or order, Eight Hundred Fifty- Dollars, at the City Bank. ,_. ^ ^ WiLLISTON CoOK. Discounted Aug. 15, 1903, at 7%. 14. $ 1200. * DBS Moines, Ia., Aug. 1, 1903. Four months after date I promise to pay to Eichard M. Johnson, or order. Twelve Hundred Dollars, value received. Discounted Oct. 15, 1903, at 6 % . Thomas R. Williams. 15. $128.50. Detroit, Mich., July 10, 1903. Sixty days after date, for value received, we promise to pay to Henry R. Sunderland, or order. One Hundred Twenty-eight ^^ * Arthur G. Hunting, Discounted Aug. 1, at 8%. Peter T. Small. 16. $480. Atlanta, Ga., Sept. 1, 1903. Three months after date, for value received, I promise to pay to Samuel S. Huston, or order, Four Hundred Eighty Dollars, with interest at 6% per annum. Discounted Sept. 18, at 8%. Jqhn L. Whitney. Note, Remember that the sum discounted is the amount to be paid at the maturity of the note ; hence, if the note bears interest, "first find the amount of principal and interest. 17. $1540. South Bend, Ind., Aug. 21, 1903. Two months after date, for value received, I promise to pay to Joseph H. Freeman, or order, Fifteen Hundred Forty Dol- lars, with interest at 6% per annum. Discounted Sept. 1, at 6^%. James S. Campbell. Digitized by VjOOQIC 210 APPLICATIONS OF PERCENTAGE. la $ 875. YiKCBNNBS, Ikd., July 10, 1903. Three months after date, for value received, I promise to pay to the Chemical National Bank Eight Hundred Seventy- five Dollars, with interest at 6% per annum. Discounted at 7%, July 10. Robert S. Dakikl. 19. $ 760. MiNNBAPOLiB, Minn., Aug. 3, 1903. Four months after date, for value received, I promise to pay to the National State Bank, or order. Seven Hundred Sixty Dollars, with interest at 6% per annum. Discounted at 7%, Aug. 3. James L. Atwood. 20. $1200. Jabpbr, Ind., May 11, 1903. Ninety (90) days after date, I promise to pay Gotlieb & Co., or order. Twelve Hundred Dollars, with interest at 5% per annum, value received. Discounted June 16, at 7%. Henry Machb. Note 2. In the District of Columbia, Delaware, Maryland, Missouri, and Pennsylvania, the day of discount^ as well as the day of maturity^ is counted in finding the term of discount. Note 2. There are four general types of problems in this subject The eighteen problems above illustrate the first and direct type. The student should give the three others. It happens rarely, if at all, that the rate per cent or the time is wanted. One might have a debt of 1374.85 to pay and have to borrow the money at bank to pay it, but one would hardly wish to know what the face of the note should be to have the pro- ceeds just 1374.85. 304. Mr. A, desiring to pay a debt of $ 460, went to a bank to obtain the money. Since he must receive $ 460 from the bank, it is evident that he must make his note for more than that amount. He wishes to borrow the money for 4 mo. The current rate of interest is 7%. If he should make his note for 9 1, the banker would give Digitized by VjOOQIC EXCHANGE. 211 him the difference between $1 and the interest on it for 4 mo. and 3 da., if days of grace are counted. The interest on $1 at 7% for 4 mo. and 3 da. is 23|^ mills. The proceeds of such a note would be 1 1 - $ .023|| = | .OTG^^^. In order that he shall receive $460, his note must be made for as many dollars as there are times $.976^^ in $460. 460^ .976tV= 471.27 ; hence, the note must be made for $ 471.27. It is not likely, however, that Mr. A would give his note for $471.27, but for $475, a sum sufficiently large that the proceeds may be at least $ 460. Proof, The interest on 1471.27 for 4 mo. and 3 da. at 7 % is $ 11.27+ ; hence, the proceeds of such a note would be |460. EXCHANGE. 305. If Mr. Gates of Greencastle should owe Mr. Job of Chicago $ 100, there are many ways in which the debt conld be paid. (1) Mr. Gates could go in person to Chicago and pay it This would cost him $ 8 or $ 10. (2) He could put the $100 in an envelope and send it to Mr. Job through the mails for 2 cents. This would be at great risk. (3) He could send it by registered letter through the mails. This would cost 8 cents for registration and 2 cents for postage. (4) He could send it by express at a cost of 26 cents. In any of these four ways the money would be actually sent to Mr. Job. (5) Mr. Gates could buy an express money order for 30 cents and send this to Mr. Job, and Mr. Job could go to the express office at Chicago and get his money. (6) Mr. Gates could buy a post-office money order and send it \jj mail. This would cost 30 cents. Digitized by VjOOQIC 212 APPLICATIONS OF PERCENTAGE. (7) He could purchase a hank draft on a Chicago or New York bank and send it by mail, and Mr. Job could get the draft cashed at a bank at Chicago. (8) He could purchase a telegraph order at a home telegraph office for $1.70 and telegraph the amount to the Chicago office, where Mr. Job could collect his money. (9) He could deposit the money in bank, write his personal check for the amount, and send this by mail to Mr. Job. Mr. Job could get the check cashed at the Chicago bank with which he does business. By any one of the latter five ways Mr. Cates would not actually send the money itself, but an order for it. 306. Exchange is a process by which payments are made between persons without the money being actually transmitted. 307. If the exchange is by bank draft, a sum called a Premium is usually paid for the draft. Drafts purchased in smaller cities on larger ones are usually at a premium. More purchases are made by small cities in large ones than are made by large cities in small ones. Hence, drafts for larger amounts must be sent to the large city to pay off debts than go to the small city for the same purpose. As a result, there is a balance of money which must be sent by the small city to the large one, to square the account. It costs the banks in small cities something to send the cash balances to the large cities ; hence they must charge a premium for drafts. 308. A draft may also be bought at par^ or its face value* It may even be obtained at a discount. Drafts purchased in large cities are often purchased at a dis- count, because the banks do not have to send the actual money to settle the balances ; besides, they want money and not more drafts. Digitized by VjOOQIC EXCHANGE. 218 309. The following is a form of bank draft : ^—' P^\$a (fimttftp^jttfamalplatik <!^^^!>g»«»g»ww^^»i«^ai«p^^ EXERCISE 112. 1. What is the cost of a draft on New York for $650, when exchange is at ^% premium ? Solution. \ % of % 650 = $ 1.30. ^ 660 + ^ 1.30 = % 661.30. There- fore, the cost is $661.30. What is the cost of a draft for : 2. % 185.40 on Chicago at ^1^% premium ? 3. % 467.32 on New Orleans at ^% premium ? 4. $ 1875.12 on Boston at ^% premium ? 5. $ 2694.38 on Pittsburg at ^% discount ? Solution. ^ % of % 2694.38 = % 2.69. 1 2694.38 - % 2.69 = % 2691.69 Therefore, the cost is $2691.69. 6. $ 10,850 on Philadelphia at |% premium ? 7. $ 791.16 on New York at \% discount ? 8. $ 2874.93 on St. Louis at ^% discount ? 9. % 167.91 on Cincinnati at ^% premium ? 10. $ 16,824.57 on San Francisco at J% premium ? Digitized by VjOOQiC 214 APPLICATIONS OF PERCENTAGE. STOCKS. 310. Let us suppose that a number of business men believe that a Street Fair would be a good thing for them financially, and would incidentally entertain the people of the city and the surrounding country. These men call a meeting of all who would be interested in such a venture. The meeting proves to be an enthusiastic one, and an or- ganization is formed, to be known as the "Street Fair Association." This association forms a Btock company and secures a charter from the state ; it is then known as an incorporated company. A board of directors is chosen, and they in turn choose a president, a vice president, a secre- tary, and a treasurer. The capital of the association is fixed at $10,000 and is divided into 2000 shares of i 5 each. The shares are made small with the hope of inducing many people to subscribe for them, so that in case of failure in the venture, the loss would not be great on any one subscriber. To any one who subscribes for one or more shares a stock certificate is given. This certificate sets forth the name of the associa- tion, its capital, the size and number of shares of the com- pany, the name of the subscriber, the number of shares subscribed for by him, and is signed by the proper official or officials. If enough shares are sold to encourage the directors to proceed, the payment of subscriptions is called for, so that the treasurer may have money with which to meet the expenses of the association. If the Street Fair is successful and the association makes some money above all expenses, this profit money, which is called a dividend^ may be divided properly and paid to the subscribers of stock. Thus a dividend is declared. Digitized by VjOOQIC BONDS. 215 311. But a street fair association may not be permanent, as is the case with some other kinds of stock companies. Apaong the more permanent stock companies are such organizations as gas companies, water works, electric light and heating companies ; almost all manufacturing com- panies, such as iron works, steel works, glass works, barrel works, breweries, distilleries, etc. ; many dry goods and furniture stores, grocery stores, banking companies, cream- eries, and railroad companies. 312. The Par Value of a share is the value named on the face of it. 313. The Market Value is the price at which a share can be bought or sold in the ordinary course of business. It may be more or less than par. 314. An Installment is a portion of the capital stock paid in by the stockholders. 315. An Assessment is a sum collected from the stock- holders to meet expenses of the company. 316. A Dividend is that part of the net earnings which is distributed among the stockholders. Note. By net earnings is meant aU earnings above expenses. 317. The dividend is usually expressed as a certain per cent of the capital stock. This rate per cent is used in calculating the dividend due any particular stockholder. Hence, this subject is an application of percentage. BONDS. 318. A stock company might need more money than the full amount of its capital. The company may borrow money, giving bonds as security. These bonds would be sold in the markets for money. Digitized by VjOOQIC 216 APPLICATIONS OF PERCENTAGE. 319. A state, county, township, or city may issue bonds as a means of securing money to meet the various expenses. 320. A county might wish to build a bridge or a court- house, which might be too great an expense for the people to bear in one year. The county could distribute tJie debt thus incurred over a period of several years by issuing bonds, which draw interest and are made payable to suit its convenience. 321. A Bond is a note issued by a corporation or gov- ernment ; it is secured, by a mortgage, or by the strength of the government issuing it. 322. A Broker is a person who buys and sells stocks and bonds for others, and charges a certain per cent of the par value of the stock or bond for his services. EXERCISE 113. 1. What is the cost of 24 shares of bank stock at 95 ? Note. 05 is called the quotation of this stock, and means $95 per share of $100. Solution. If 1 share costs $95, the cost of 24 shares = $95 x 24 = $2280. Note. Shares may be $5, $10, $20, $25, $50, $100, or $500 each, but in this book they are always $ 100 shares, unless it is otherwise stated. 2. What will 36 shares of Big Four railroad stock cost at 92? 3. What will 18 shares of Interurban railway stock cost at 75? 4. What will 300 shares of Coal Bluff mining stock cost at 115? 5. What should one receive for 56 shares of Fort Wayne stock at 185f ? Solution. Each $100 share would sell for $185.75. Hence, one should receive $185.75 x 56 = $10,402. Digitized by VjOOQIC BONDS. 217 6. If a man should sell 86 shares of stock in an automobile factory at 78J, what would he receive for them ? 7. What should a person receive for 14 shares of natural gas stock at 58 J ? 8. What will 164 shares of Belt Railroad stock cost at 164i? 9. What should one receive for 17 shares of Monon railway stock at 99i? 10. What should one receive for 28 shares of Baltimore stock at 102? 11. If I should pay $6600 for 60 shares of electric light stock, what would be the quotation ? Solution. |6600 -t- 60 = $ 110. Therefore, the quotation is 110. 12. What is the quotation, if 45 shares of Vandalia railroad stock sell for $3600? 13. What is the quotation of Indianapolis Belt railway stock, if 64 shares sell for $ 11,200 ? 14. What is the quotation of Muncie illuminating gas stock, if 46 shares sell for $ 2921 ? 15. How many shares of stamping works stock at 86^ can I buy with $3460 ? Solution. One share costs $86}. $3460 -h $ 86} = 40. Hence, I can buy 40 shares. 16. How many shares of E. & T. H. stock can I buy with $2853 at 79 J? 17. How many shares of French Lick Hotel stock can I buy with $4036.50 at 87|? 18. How many shares of Southern railway stock can I buy with $2104.50 at 91^? In the problems thus far no account has been taken of broker- age. But it is almost necessary for any one who wishes to buy or sell stocks to apply to a broker in order to know the stocks Digitized by VjOOQIC 218 APPLICATIONS OF PERCENTAGE, on the market and their prices. If a purchase or a sale is made through a broker, he must be paid for his time and efforts. 19. What will 18 shares of Evansville Gas and Electric Light stock cost at 95|, brokerage \ ? Solution. Since the quotation and brokerage are each per cents of the par value, their sum is the total cost. 96} + i = 95}. Hence, the entire cost of 1 share is 995}. Hence, the cost of 18 shares is $95} x 18 = $1719. 20. If I sell 24 shares of Princeton Water Works stock at 82|y through a broker, and pay him \y what is the net amount I receive ? Solution. 82} — } = 82}. Hence, I receive $82} net for each share. $82} X 24 = $ 1980. Hence, the net amount of the sale is $ 1980. Note. When buying stock through a broker, add the rate of brokerage to the quotation, and when selling^ subtract the rate. 21. What is the total cost of 34 shares of Bedford Electric Light stock at 52|, brokerage \ ? 22. What must I pay for 14 Vincennes sewer bonds at 88^, brokerage \ ? 23. What is the net amount received for 72 Fort Wayne courthouse bonds sold at 102f , brokerage | ? 24. If I should sell, through a broker, 88 shares of Vigo National Bank stock at 132, brokerage ^, what amount would I receive ? 25. What is the brokerage in the 19th problem ? Solution. } means } % of the par value of the stock. The par value of 18 shares is $ 1800. i % of $ 1800 = $ 2.25. Hence, the brokerage is $2.25. 26. Wliat is the brokerage in Problem 20 ? 27. What is the brokerage in Problem 22 ? 28. What is the brokerage in Problem 24 ? 29. If I pay $1719 for bonds at 95f, brokerage ^, how many do I buy? Solution. 96| + J = 96^. $1719 -*- $95^ = 18. Therefore, I buy 18 bonds. Digitized by VjOOQIC BONDS. 219 30. If I received $ 1980 for bonds at 82|, brokerage ^, how many did I sell ? 31. If I received $2104.50 for Southern railway stocks at 91f , brokerage ^, how many did I sell ? 32. 6% stock bought at 75 yields what rate per cent on the investment ? Solution. Since one share yields $6 income, and costs |76, the rate on the investment = $6 ~ $ 76 = .08 = 8%. 33. 5% stock bought at 125 yields what rate per cent on the investment ? Solution. One share yields $5 income annually, and costs $126. Then, the rate per cent on the investment = $ 6 -f- $ 126 = .04 = 4%. 34. What rate per cent on an investment does 8% stock yield, if purchased at 160 ? at par ? 35. What is the quotation of 6% stock, if it pays 5% on an investment ? Solution. 6% stock draws $6 per share. Now, |6 is 6% of what sum ? 1 6 -i- .06 = 1 120. Hence, the quotation is 120. 36. What is the quotation of 6% stock, if it pays 8% on the investment ? 37. If stock is quoted at 90 and pays 5% on the investment, what kind of stock is it ? Solution. One share costs $ 00, and draws 6 % of $ 90, or $ 4. 60. But, $ 4.60 is ^% of % 100, the par value. Hence, it is 4^% stock. 38. What rate per cent on an investment does 4^% stock pay, if purchased at 90 ? 39. If 5% stock pays 6% on the investment, what is the quotation ? 40. If stock is purchased at 120 and pays 10% on an invest- ment, what kind of stock is it ? 41. At what rate must 4 % bonds be bought to yield annu- ally 5 % of the investment ? 4^ % bonds, to yield 6 % ? 7 % bonds, to yield 5 % ? 8 % bonds, to yield 5| % ? Digitized by VjOOQIC 220 APPLICATIONS OF PERCENTAGE, 42. Which is the better investment, 5 % bonds at 95, or 6 % bonds at 102^, if the brokerage on the first is 1^ %, and on the second 1| % ? 43. Tell which is the better investment in each of the fol- lowing cases : (a) 4^ % bonds at 92, brokerage 1\%\ or 5^ % bonds at 106, brokerage If. {h) 8 % bonds at 124, brokerage 2\%\ or 6^ % bonds at 110, brokerage 1|^%. (c) 3 % bonds at 62, brokerage | % ; or 10 % bonds at 185, brokerage If %. 44. A has a farm of 240 A., which yields him an annual rental of $ B\ an acre. A real estate agent sells it for $ 75 an acre, charging him 3% commission. Reserving $43.33, A invests the net proceeds in insurance stock at 83, brokerage ^%. His annual income is increased $203. What is the rate per cent of the semiannual dividends ? 45. A sold 42 100-dollar school bonds bearing 7 % interest, Sit 5 % premium. The brokerage was $39. He sent the net proceeds to a Denver broker to invest in silver-mine stock at 45% premium, brokerage 2%. This stock yielded a semi- annual dividend of 12%. How much was his annual income increased? What rate per cent did the first broker charge him ? What was the unused balance? 46. An electric light company, whose capital stock was $50,000, earned above all expenses $3212.50 during the first half of the year. If it passed $212.50 to the repair fund, what semiannual dividend can it declare? What will a per- son receive who owns $ 2150 worth of the stock ? 47. A manufacturing company, whose capital stock was $ 250,000, bought new machinery worth $ 4000, and levied an assessment on its stockholders to meet the expenses. What was the rate per cent of the assessment ? What did A pay, who owned $7800 worth of stock? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 221 48. A milling company, whose capital stock was $30,000, declared a semiannual dividend of 4^ %, and passed $864.50 to a reserve fund. What were tte net earnings ? 49. How many shares of Illinois Central stock can be bought for $ 7087 at 7 % discount, brokerage \%? 50. What amount must be invested in United States bonds, bearing 4% interest, at 18^ premium, to secure an annual income of $ 600, brokerage being |^ % ? What per cent of the investment is the income ? 51. How many bonds at 83 J, brokerage \ %, can be pur- chased for $ 5761.50 ? 52. How much must I invest in bank stock, paying a semi- annual dividend of 4 %, and selling at 160, brokerage \ %, to yield an annual income of $320? 53. Bought 75 shares of railroad stock at 28 and sold it at 30 J, brokerage ^ % in each case. What was the profit ? 54. Sold at a discount of 1\% shares which I had pur- chased at a premium of | %, losing $600, brokerage being \(fo on the sale and |^ % on the purchase. How many shares were there ? MISCELLANEOUS PROBLEMS. EXERCISE 114. 1. A note for $428.50, dated Aug. 15, 1903, is to run 90 da., without interest. It was discounted Oct. 1, at 8%. What were the proceeds ? 2. A does J^ of a piece of work in 6 da., B f of the re- mainder in 5 da. and C finishes the work in 10 da. How long would it have taken the three working together ? 3. A merchant bought a bill of goods for $1875.60, on 90 days' time. Finding that he could get a discount of 5 % of the whole amount by paying cash, he borrowed the requisite Digitized by VjOOQIC 222 APPLICATIONS OF PERCENTAGE. amount from a bank. For what must he draw a note for 90 da., current rate being 8 %, to pay the bill? How much did he gain by the process ? (Bank Discount.) 4. A merchant bought a bill of goods for $1674.20. He was allowed 5 % off fpr cash. What did he pay ? 5. A stock train has 29 cars, each containing 19 cattle, whose average weight is 1460 lb. They sell for $5.25 a hundred. What do they bring ? 6. A man bought a half-section of land at $81.50 an acre. He gave one note for $12,650, a second for $5830, paid $3600 in cash, and gave a third note for the remainder. What was the face of the third note ? 7. The above notes were all dated July 1, 1900. The first was due in 2 yr. and bore 6% interest. Falling heir to $15,000, the man discounted this note on Jan. 15, 1901, at 7 % bank discount. What did he pay for it ? a The second of the above notes was due in 3 yr., and bore interest at 6%. On Sept. 25, having sold his wheat crop, he discounted the second note by bank discount at 8%. What did he pay for it ? 9. The third of the above notes was due in 4 yr., and bore interest at 5%. He paid no interest until the maturity of the note. What amount was then due ? 10. He borrowed the $3600, in Problem 6, at a bank by giving his note for 4 mo. at 7%. What was the face of his note which yielded him that amount ? 11. Find the amount of the following bill: 24 Arithmetics @ 95 cents. 16 Readers® $1.20. 20 Geographies @ $ 1.40. 18 Grammars @ 75 cents. 42 Spellers @ 22 cents. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS, 223 The dealer made a discount of 33^ % from the regular price, and a further discount of 5 % of the balance for cash. 12. Bought a bill of $ 748.80, the seller making a discount of 33^ % and 5 %. What was the amount to be paid ? 13. What fraction, being multiplied by | of |, gives J of -j^ as a product ? 14. Bought a piano for $560 on March 5, 1903. The agree- ment is that I shall pay 6% interest on the purchase price, with the privilege of making payments upon which interest shall be allowed at the same rate. I made the following pay- ments: May 25, $175; July 18, $160; Nov. 12, $125. What was due March 5, 1904 ? 15. A man bought a business lot 60 ft. x 160 ft., at $4^ a square foot. He erected a six-story building, costing $ 50,000, on the lot. In the first story there are two storerooms, which rent for $1500 each. There are 20 rooms on each of the other floors. Those on the second floor average $20 a month. There is a reduction of $ 3 a month as the stories ascend. If insurance, taxes, and care of the building amount to $ 3500 a year, what interest will the owner receive on his investment ? 16. If 1^ % is paid for collecting the rents in the building, what is the amount of the collector's commission per month ? 17. What time will be required for $460 to earn $ 64.80, at 7%? la What principal earns $56.40 interest in 2 yr. 5 mo. 15 da., at 6% ? 19. What is the difference between London local time and New York local time ? If a cablegram is sent from London at 8 A.M., and one hour is employed in getting it to its desti- nation in New York, at what time will it reach there ? 20. Bought 5% school bonds at 102. What rate (fo of in- terest do they pay on the investment ? Digitized by VjOOQIC 224 APPLICATIONS OF PERCENTAGE. 21. Bought 40 shares of National Bank stock at 160. Thej pay a semiannual dividend of $160. What is the rate of dividend ? ' What rate do they pay on their cost ? 22. I send to a commission merchant $1000 to be invested in wheat at 57f cents a bushel, after deducting his commission at 2%. How many bushels will my remittance buy ? 23. A man whose watch shows Chicago local time finds that it is 36 min. 30 sec. faster than the local time of the place in which he is. What is his longitude ? (See Table, § 117.) 24. In a school of 750 pupils, the number of boys is 87^% of the number of girls. How many boys are there in the school ? 25. An auctioneer sold the following articles at a farmer's sale. What is his commission at 2% ? 5 horses, averaging $65. 8 cows, averaging $36.50. 2 wagons, one for $18.75, the other for $31.60. 20 tons hay at $9.75. 3 plows at $5.31. 1 harrow at $4.90. 26 hogs at $8.30. 26. At the above sale a discount of 5% was made for cash. What amount would discharge the obligation of the man who bought the horses and the cows ? 27. It was a condition of the above sale that purchasers might pay in one year without interest. The man buying the wagons and the hay discounted his note at 8% in bank, at the end of 9 mo. 18 da. What did he pay ? 28. The man buying the hogs borrowed the money from a bank, at 7%, for 90 da., with grace, and paid cash for them. What was the face of his note ? 29. Eeduce: |ij| x ^ -^ | of | of ,^. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS, 225 30. In what time will a principal of $864.12 amount to $1040, at 7% interest? 31. The time from 9 o'clock to 15 min. 20 sec. past 10 is what part of a day ? 32. What is the area of a city lot 50 ft. x 150 ft. ? What is it worth at $6000 an acre ? 33. An auctioneer's com missions for a year, at 2|^ % , amounted to $3124.80. He was employed 265 da. What was the daily average of his sales ? 34. A manufacturer received from his foreign agent 50 bales of wool, 250 lb. each, invoiced at 36 cents per pound ; and 36 bales, 300 lb. each, invoiced at 31 cents a pound. What was the duty, at 25 % ? Digitized by VjOOQIC CHAPTER XI. RATIO AND PROPORTION. THE RELATION OF NUMBERS. EXERCISE 115. (Mental.) 1. 1 is what part of 3 ? of 5 ? of 11 ? 2. 2 is what part of 6 ? of 10 ? of 16? 3. 3 is what part of 4 ? Anb. 8 is } of 4. 4. 3 is what part of 8 ? of 9 ? of 12?. 5. 4 is what part of 6 ? of 13? of 20? 6. 10 is what part of 10 ? of 5 ? Anb. 10 is 1 time 10. 10 is 2 times 6. 7. 12 is what part of 20 ? of 30 ? of 8 ? 8. 16 is what part of 20 ? of 30 ? of 45 ? of 10? 9. If 5 pencils cost 15 cents, what will 20 pencils cost ? Solution. Since 20 is 4 times 5, 20 pencils will cost 4 times as much as 5 pencils, or 16 cents x 4 = 60 cents. 10. If 8 cd. of wood cost $ 25, what will 16 cd. cost ? 24 cd. ? 32 cd. ? U. If 11 sheep cost $31, what will 33 cost ? 12. If 24 bu. of wheat cost $18.50, what will 12 bu. cost? 13. If 36 lb. of sugar cost $2, what will 9 lb. cost ? 12 lb. ? 18 lb. ? 72 lb. ? Digitized by VjOOQIC THE RELATION OF NUMBERS. 227 14. If 7 apples cost 5 cents, how many can be bought for 25 cents ? Solution. Since 26 cents is 6 times 5 cents, 5 times as many apples can be bonght for 25 cents as for 5 cents. Hence, 7 apples x 6, or 36 apples can be bought for 26 cents. 15. If 3 horses cost $ 200, how many horses can be bought for $1000? for $1200? for $1800? 16. 3i is what part of 10? of 20? of 30? 17. 4^ is what part of 9? of 13J? of 18? la i is what part of 2 ? of 6 ? of 11 ? 19. f is what part of 5 ? Solution. }-i-5 = fxJ = ^. Hence, } is ^ of 6. 20. ^ is what part of 2? of 6? of 7? 21. 3 qt. is what part of a gallon ? of 3 gal. ? of 6 gal. ? 22. 6 in. is what part of a foot ? of 3 ft. ? of 5 ft. ? 23. $6| is what part of $20 ? of $40 ? of $50 ? 24. If 6| lb. of meat cost 90 cents, what will 20 lb. cost ? 401b.? 25. If 8 doz. eggs cost $1.10, how many dozen can be bought for $3.30? for $4.40? 26. If 3^ yd. of cloth cost $8, what is the cost of 14 yd.? of 21 yd. ? of 35 yd. ? 27. I is what part of f ? Solution. |^f=|xf=| 3 6^66 Hence, } is | of f . 28. i is what part of f ? of ^? of ^? 29. If I yd. of ribbon cost 60 cents, what is the cost of f yd. ? Solution. f-5-} = |xi = f. { yd. is J times } yd. Hence, the cost is 60 cents x } s 76 cents. Digitized by VjOOQIC 228 RATIO AND PROPORTION. 30. If 6 articles cost $33, what will 15 articles cost ? 31. If a train runs 133 mi. in 4 hr., how long will it be in running 399 mi. ? 632 mi. ? 32. If 15 men earn $ 42 a day, how much can 30 men earn in a day ? 45 men ? 75 men ? 33. If 12 men earn $49 a day, how many men will earn $24.50 a day? $98 a day? 34. If it costs $13 for hauling 27 loads of dirt, how many loads can be hauled for $4^ ? for $26? for $39 ? 35. How many books can be purchased for $ 5.60, if 2 books cost 70 cents ? 36. If 3 lb. of candy cost 35 cents, how many pounds can be bought for $2.80? 37. If I yd. of silk cost $3, how many yards can be bought for $18? for $27? 38. How many oranges, at 4 cents each, must be given in exchange for 12 pineapples at 16 cents each? at 12 cents each ? at 20 cents each ? 39. If I can walk 2 mi. in 36 min., how far can I w4lk in 3 hr. ? in 6 hr. ? 40. If 18 men require 42 da. to do a piece of work, how many men can do it in 21 da. ? RATIO. 323. The part one number is of another is called the Ratio of the first to the second. 3 is i of 6, or the ratio of 3 to 6 is J. 324. What is the ratio of 3 to 4 ? 5 to 10 ? 12 to 8 ? 8 to 4? 6 to 12? 8 to 24? 13 to 26? Digitized by VjOOQIC RATIO. 229 325. The ratio of 7 to 14 is written 7 : 14 or ^. The sign : has the meaning of -*-. NoTB. The fractional writing of ratio is rapidly replacing the other form. 326. Since a number can only be a part of a like num- ber, ratio can exist only between like numbers. There is no ratio between $5 and 4 bu. 327. Ratio tells the numerical relation of two numbers, and is, therefore, always abstract. 328. The numbers whose ratio is to be found, are the Terms of the ratio. The first term is the Antecedent, and the second is the Consequent. In the ratio 9 : 12 or ^, 9 is the antecedent and 12 is the consequent. 329. Since a ratio is a fraction, the antecedent being the numerator and the consequent the denominator, all principles of fractions become principles of ratios, if for the words fraction^ numerator^ and denominator^ we substitute the words ratio^ antecedent^ and consequent respectively. 330. The principle. If both numerator and denominator of a fraction are multiplied or divided by the same num- ber, the value of the fraction is not altered — becomes. If both antecedent and consequent of a ratio are multiplied or divided by the same number, the valvs of the ratio is not altered. The ratio ^ = the ratio f , or the ratio J|. Digitized by VjOOQIC 230 RATIO AND PROPORTION. 331. The principle: Multiplying the antecedent multi-. plies the ratio, comes from what principle of fractions? Change all the principles of fraction% into corresponding principles of ratios. 332. A ratio is in its simplest form when the terms are prime to each other. The ratio ^ in its simplest form becomes |. It is reduced to this form by dividing both terms by 6. EXERCISE 116. (Mental.) Beduce the following ratios to their simplest forms : 1. 6:12; 28:24; 9:54. 3. 18:27; 24:42; 76:57. 2. 7:35; 66:44; 16:56. 4. 49:56; 84:63; 60:72. 5. |:6; M; |:A- 6. What is the antecedent, if the ratio is f and the conse- quent is 12 ? 7. What is the consequent, if the ratio is ^ and the ante- cedent is 12 ? 8. One train runs a certain distance in 40 min., and another runs the same distance in 1 hr. What is the ratio of their speeds ? EXERCISE 117. 1. A man's salary in two successive years is $875 and $1200. What is the ratio of the first to the second ? 2. The ratio of the school enumeration in a city in 1902 to that of 1903 is ||. The enumeration m 1902 was 6237. What was the enumeration in 1903 ? 3. If -^ = 1, what is a?? 7246 9' 4. IflM§ = ^,whatisa:? X 13 Digitized by VjOOQIC RATIO. 231 THERMOMETER SCALES. 333. There are two kinds of thermometers in general use, the Fahrenheit and the Centigrade. The freezing point on the Fahrenheit is 32®, and the boiling point is 212®. The freezing point on the Centi- grade is 0®, and the boiling point 100®. The number of degrees from freezing to boiling on the Fahrenheit (F.) is, therefore, 180, and that on the Centi- grade (C.) 100. 334. From the above statements it is at once seen that the ratio of any number of degrees above freezing on the Fahrenheit thermometer to the corresponding number of degrees above freezing on the Centigrade thermometer is 180 : 100, or 9 : 6. Or, conversely, the ratio of the Centi- grade scale to the Fahrenheit is 6 : 9. EXERCISE 118. 1. What is the temperature by a Centigrade thermometer, when the Fahrenheit indicates a temperature of 86° ? Solution. 86° — 32° = 54°, the number of degrees above freezing. Let X = the degrees Centigrade. 54^9 9 X = 6 X 54. 9 Hence, the Centigrade reading is 30°. 2. When the temperature is 36° C, what is the Fahrenheit reading ? Solution. Let x = the degrees Fahrenheit above freezing. -^ = ?. 36 6* aj = MiL2 = 7x9 = 68. 5 63° + 32° = 96°. Hence, the Fahrenheit reading is 96°. Digitized by VjOOQIC 232 RATIO AND PROPORTION. When the temperature on either thermometer is below 0, the fact is indicated by the use of the sign — . Thus, "5**F. below 0," is written "- 5** F." 3. What is the Centigrade reading, when the Fahrenheit reading is - IS** ? Solution. 13° + 32° = 46°, the number of degrees below freezing. Let X = the degrees Centigrade. 46^9 X 6* 9 x = 46 X 6. 9 Hence, the Centigrade reading is — 26°. 4. What is the Fahrenheit reading, when the Centigrade reading is — 15° ? Solution. Let x = the degrees Fahrenheit below freezing. 16 6' aj = li2L2 = 3x9 = 27. 6 82*'-27° = 6°. Hence, the Fahrenheit reading is 6^ 5. Change 72° F. to Centigrade. 6. Change 25° C. to Fahrenheit. 7. When it is - 18° F., what is it Centigrade ? a When it is — 30° C, what is it Fahrenheit ? 9. When it is 180° F., what is it Centigrade ? 10. Change blood heat, 98° F., to Centigrade. 11. - 20° C. is what Fahrenheit ? 12. When it is 61° F., what is it Centigrade ? 13. When it is 61° C, what is it Fahrenheit ? 14. Change 83° C. to Fahrenheit. 15. Change 38° F. to Centigrade. Digitized by VjOOQIC PROPORTION. 233 PROPORTION. 335. An equality of two ratios forms a Proportion. |. = ^ is an equality of the ratios f and ^, and is a propor- tion. It is read, "3 is to 6 as 5 is to 10," or "the ratio of 3 to 6 equals the ratio of 5 to 10." 336. The proportion may be written thus : 3 : 6 = 5 : 10, or 3 : 6 : : 5 : 10. Note. The : : between the ratios has the meaning of =. The equality sign (=) is now generally used instead of the double colon ( : : ). 337. In the proportion | = ^, 3, 6, 5, and 10 are the Terms, and are called the firsts second^ thirds and fourth terms respectively. 838. The first and fourth terms are the Extremes, and the second and third are the Means. 339. 7 = ^ is a general proportion. It states that the ratio of a to 6 equals the ratio of c to d. PRINCIPLES. 340. Prin. 1. In a proportion.^ if the first term is equal to., or greater or less than the second., the third term is cor- respondingly equal to., or greater or less than the fourth. In the proportion ^ =i - , if a is greater than b, then - is greater d b than 1, and since - = ?, ^ is greater than 1 ; which can only d b d be so when c is greater than d. If a is equal to b, then ^ = 1 ; and since^ = ^, ^ = 1 ; which shows that c = d. ^ d b d c If a is less than b, then - is less than 1; and since - = -, b d b - is less than 1 ; which can only be so if c is less than d. d Digitized by VjOOQIC 234 RATIO AND PROPORTION. 6 X If — = ---, is a? equal to, or greater or less than 20 ? Why ? 9 36 If - = — , is a equal to, or greater or less than 36 ? Why ? 7 X M TT = ^> what is the value of a? ? Why ? 11 30 ^ 341. The product of the first and fourth terms of a proportion is equal to the product of the second and third terms. If ? = -, the equality will not be destroyed if we multiply both sides of the equation by 6 x (f . This gives : -xhx d ^- Xb xd, which, by the cancellation of common factors, d becomes axd^cxb. 342. This principle may be stated thus : Prin. 2. In any proportion the product of the extremes is equal to the product of the means. 343. By means of this principle, we can find any term of a proportion if the other three terms are given. 1. Find the first term of a proportion, if the second, third, and fourth terms are 30, 42, 35. Solution. Call the omitted term z, X 42 Then, the proportion is ^ = —• By Prin. 2, X = 80x42. 6 6 X -.30X^?« 3^ Digitized by VjOOQIC PROPORTION. 235 2. Find the third term in a proportion in which the first, second, and fourth terms are 11, 45, and 90. Solution. Call the omitted term x. 11 X Then, the proportion is ■— = ^• 46 90 ByPrin. 2, 46a; = 11x90. 2 ll_xR2^22 Note. Because in problems in proportion there are always three things given, the subject was formerly called the rule of three, • EXERCISE 119. Find the missing term in each of the following proportions : 1. 6:10 = 15:a?. 8. 55:77 = a?:42. 2. 12: 39 = 0?: 91. 9. 8:15::a;:24. 3. i:aj = |:f. 10. 15 : 36: ; 72 : a?. 4. a?: 125 = 72: 108. U. a;: 48 : : 60 : 75. 5. 26 A.: 36 A. = 13 T.: a?. 12. 30:aj::|:2f 6. $625: $825 = ®: $33. 13. f:|::64:a?. 7. 31b.:a? = f yd.:33^yd. 14. .36 : a? : : .125 : 4. 15. $8: $2^:: $144: a;. 16. 42bu.:36bu. ::a;:3pk. 17. 184 mi. : a; : : 75 mi. : 526 mi 18. a;:320 A. ::f A. :| A. 19. 3200 lb. : 200 lb. : : 96 lb. : x. 2D. 468 rd. : 920 rd. ::aj: 2760 rd. 21. i:{i::S:x. 22. 2J: 7::. 0084: a?. 23. 63 gal. : 90 gal. : : a; : $ 120. 24. 75 lb. : 40 lb. : : 60 da. : x. Digitized by VjOOQiC 236 RATIO AND PROPORTION. 344. Proportion may be used in the solution of a great variety of problems, in fact, in all problems in which three numbers are given with which to find a fourth, provided two of the numbers have the same ratio to each other that the third has to the missing number. EXERCISE 120. 1. K 12 yd. of cloth cost $ 36, what will 40 yd. cost, at the same rate ? Solution. Denote the required cost by x; write it and the other quantities in two lines which correspond, thus: 12 yd., $36. 40yd.,* fix. Write ^ as the first ratio, and then determine by the conditions of the problem, whether x must be greater or less than 36. Since 40 yd. will cost more than 12 yd., x must be greater than 36. Of) Now, by Prin. 1, we know that our second ratio must be — , and our proportion is 12 36 * io^T 12« = 36x40. 12 2. If 60 bu. of corn cost $20, what will 600 bu. cost, at the same rate ? 3. If a man travels 48 mi. in 12 hr., how many miles can he travel in 60 hr., at the same rate ? 4. If 17 horses cost $ 1360, what will 39 horses cost, at the same rate ? 5. If the cost of 1360 sq. yd. of plastering is $340, what will 3824 yd. cost, at the same price ? 6. If a steeple 124 ft. high casts a shadow 93 ft. long, how long a shadow will a steeple 216 ft. high cast, at the same time and place ? Digitized by VjOOQIC PROPORTION. 237 7. If a steeple 216 ft. high casts a shadow 162 ft. long, how long a shadow will be cast by a steei^le 124 ft. high, at the same time and place ? 8. Suppose the shadow cast by a 216 ft. steeple to be 162 ft. long, what is the length of a steeple that cast's a 93 ft. shadow, the other conditions being equal ? 9. If a 124 ft. steeple casts a 93 ft. shadow, what is the height of a steeple that casts a 162 ft. shadow, under the same conditions ? 10. If 25 men can do a piece of work in 12 da., in how many days can 10 men do the same work ? Solution. 12 da., 25 men. xda., 10 men. 12 As we are required to find a number of days, write — as the first ratio. Since it took 25 men 12 da., it will take 10 men longer, so that x is greater than 12. Then, by Prin. 1, 1? = 15. X 25 10x= 12x25. ^ = 122125 = 80. 11. If 10 men can do a piece of work in 30 da., how many men can do the same work in 12 da. ? 12. If a principal of f 325 earns $ 76.50 in a given time, what will a principal of $ 648 earn in the same time ? 13. A room is 24 ft. wide and 28 ft. long. How long must a room 18 ft. wide be to contain the same area ? 14. If 50 A. of land produce 2800 bu. of corn, how many bushels will 76 A. yield, at the same rate ? 15. If a man can do a piece of work in 36 da., working 10 hr. a day, in how many days could he do it working 12 hr. a day ? 16. A railway train runs 429 mi. in 8 hr. and 15 min. Hqw far would it run in 10 hr. and 20 min., at the same rate ? Digitized by VjOOQIC 238 RATIO AND PROPORTION. 17. If 3 lb. 5 oz. of butter cost $1.06, how much would 8 lb. 12 oz. cost, at the same price ? 18. A garrison of 320 men is supplied with provisions for 60 da. How long would the remainder of the stock of supplies last the rest of the garrison, if 40 men were withdrawn at the end of 20 da. ? 19. Two cog wheels are geared together. The larger has 42 cogs, and the smaller 16. How many revolutions does the smaller make while the larger makes 24? 20. If 120 bu. of oats are necessary to seed 40 A. of land, how many bushels will be required to seed 195 A. ? 21. If 2| bbl. of pork cost $23,625, how much will 8^3^ bbl. cost? 22. If S^ A. of land yield 172 bu. of wheat, how much will 18.75 A. yield? 23. If the interest on $375.50 for a certain time is $52.57, what is the interest on $ 680 for the same time ? MISCELLANEOUS PROBLEMS. EXERCISE 121. 1. At 4 P.M. in Boston, is it earlier by sun time or later at Washington, D.C. ? 2. At 9 A.M. in Indianapolis, is it earlier or later by sun time in St. Louis ? 3. Express in its simplest form the ratio of 3J to 2J ; of 2ito4i; of 4^to7f. 4. If the antecedent of a ratio is 9 and the ratio is f , find the consequent. 5. If the consequent of a ratio is 13^ and the ratio is f, find the antecedent. 6. What is the ratio of 121 to 165 ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS 239 7. When it is 5 a.m. at Philadelphia, it lacks 2 hr. of noon at London. What is the difference in longitude ? a Solve for a;: ^ = ^. ^ 15 Solution. Multiply both terms of the first ratio by 8, and both of the second ratio by 2. Then, y^ = |^; whence 9 a; = 21, and x = 2 J. 9 Find a:- f a? _ 12Kg. 10. K 5 cu. ft. are equivalent to 4 bu., how many cubic feet are there in a bin which holds OS bu. ? (Solve by relation of number.) U. If 50 bu. of apples cost $37.50, find the cost of 12^ bu. 12. What is the ratio of 3 m. to 20 dm. ? 13. Why is the cost of 36 bu. of apples equal to f of the cost of 24 bu. of the same kind ? 14. Why is the cost of 75 bu. of potatoes equal to ^ of the cost of 93f bu. ? 15. Is f of the value of a flock of sheep equal to f of | of the value of the flock ? Why ? 16. If a traveler goes from Cincinnati to St. Louis, carry- ing Cincinnati sun time, should he set his watch forward or backward to have St. Louis sun time ? 17. John has three times as many marbles as Henry, and James has 2^ times as many as John; they all have 92 marbles. How many has each ? Solution. Let x = the number of Henry^s marbles. Then 3 x = the number of John's marbles, 7} X = the number of James's marbles, • and 11} a; = the number of marbles all have. But 02 = the number of marbles all have. Then 11} x = 92, whence a; = 8, 3a; = 24, 7}x = 60. Hence, Heniy has 8, John has 24, and James 60 marblei. Digitized by VjOOQIC 240 RATIO AND PROPORTION. \B, Make a true prop^Mrtion from the quantities $ 12, 75 HI., 300 HI., and «48. \9. Name two numbers whose ratio is equal to the ratio of 24 to 10; of 2^ to 3^. 20. If 24 cows consume 196 bu. of grain in a given time, how many bushels will 42 cows consume in the same time ? 21. If 13 sheep cost $52^^ find the cost of 91 sheep. 22. The sum of two numbers is 996, and the larger is twice the smaller. Find the numbers. (Use x.) 23. If the sum of two numbers is 650 and the larger is 100 more than the smaller, find the numbers. 24. If the weight of a common brick is 4J lb., how many bricks should a hauler load to have as nearly as possible 2 T. ? 25. What is the value of 1 T. 4 cwt. of soap, if 4 oz. cakes cost 45 cents per dozen ? 26. The extremes of a proportion are 6 and 37^. One of the means is 12 ; find the other. 27. If the means of a proportion are ^ and f , and one of the extremes is ^, find the other. 28. Find the sum of 24 consecutive numbers, beginning with 125. 29. 24 is added to 48 ; their sum is the same as the product of what two factors less than 10 ? 30. Find the difference in time between Canton, China (113'' 14' east), and Greenwich, Eng. 31. Find the ratio of 4 gal. to 2 CU. ft. (A gal. contains SSl ca. in.) 32. Which is the greater ratio, t^ or |? 33. Which is the greater, the ratio of $2.50 to $3.75, or that of 8 ft. to 12 ft. ? 34. A merchant failed and paid 60 cents on the dollar. How much would a creditor receive whose bill was $ 1750 ? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 241 35. Arrange the following ratios in the order of their mag- nitude : II, If, and }|. 36. Divide $372 among 3 boys and 4 girls, giving to each boy I as much as to each girl. 37. Bought 250 gal. of wine at $2.40 per gallon; paid for freight $15.40, and $74.60 for duties. If ^ of the wine was lost by leakage, at what price must the remainder be sold to gain $80 on whole transaction ? 38. Solve for a?: i^4.?^ = 31. 5 4 39. ^4-10 = 5^-1. Find x. 3 2 5v 7v 9v 95 ^. , «. 3£±i_32^ + 2^=a;. Find «. 5 8 3 42. Divide 90 into two such parts that one part shall be 3| times the other. 43. A horse was sold for $80, at a gain of \ of the cost. What was the cost ? 44. Find the number of bushels in 2 T. of salt (sp. gr. 2.15). 45. Find the number of cubic inches in 3 lb. aluminium (sp. gr. 2.64). 46. What is the specific gravity of a liquid weighing 8 lb. per gallon ? 47. What is the longitude of a place whose time is 36 min. 45 sec. after 2 p.m. when it is noon at Greenwich ? 48. A father's age is now 2^ yr. more than that of his son, and the sum of their ages is 46 yr. Find the age of each. 49. Express 60° C. in the Fahrenheit scale. 50. Express 66'' F. in the Centigrade scale. Digitized by VjOOQIC CHAPTER XII. INVOLUTION AND EVOLUTION. POWERS. 345. A Power of a number is the product arising from using the number one or more times as a factor. 3 is the first power of 3. 3 X 3 = 9, the second power of 3. 3 X 3 X 3 = 27, the third power of 3. 2 X 2 X 2 X 2 s= 16, the fourth power of 2. 346. The process of finding a power or of raising a number to a power is called Involution. EXERCISE 122. (Mental.) 1. What is the second power of5?6?8?9?12? 2. What is the third power of 2? 4? 5? 10? 3. What is thefourth power of 1 ? 3? 5? 10? The second power of a number is called its Square. 9 = 3 X 3 is the square of 3. 4. Find the square of 7; 10; 11; 13; 16. The third power of a number is called its Cube. 8 = 2x2x2isthe cube of 2. 5. Find the cube of 3; 6; 7; 8; 20; 30; 40; 50. 347. Powers are indicated by means of Exponents. In the expressions 6^, 5^, and 5*, the numbers 2, 3, and 4 242 Digitized by VjOOQIC ROOTS. 243 are exponents, and indicate that 5 is to be used two, three, and four times respectively as a factor. 5^ is read " 5 squared " or " 5 to the second power.'' 5* is read " 5 cubed " or " 5 to the third power." 5* is read " 5 to the fourth power." EXERCISE 123. 1. rind83^ 1212; 1752. 1352. 1962^ 2. Find 258; 318. 74. ig4. 5*^ 3. Find36^ 632; 9^2. 952. 2161 A number ending in 5 may be squared by multiplying the number in tens' place by one more than itself, and annexing 25 to this product. 662 = (6 X 7)26 = 4225. 862 = (8 x 9)26 = 7225. 4. Find mentally 252 ; 35^; 45^; 55^; W. The rule above for numbers ending in 5 may readily be extended to numbers of three places. 1252 = (12 X 13)25 = 15,625. 5. Findl352; 165^; 205^; 365^; 805^; 895*. ROOTS. 348. A Root of a number is one of the equal factors whose product is the number. 349. The Square Root of a number is one of the tuoo equal factors whose product is the number. The square root of a number is frequently called its second root, 350. The Cube Root of a number is one of the three equal factors whose product is the number. Cube root is the same as third root. Digitized by VjOOQIC 244 INVOLUTION AND EVOLUTION. 49 = 7 X 7. 7 is the square root of 49, because it is one of the two sevens whose product is 49. 64 == 8 X 8. Why is 8 the square root of 64 ? 64 = 4 X 4 X 4. Why is 4 the cube root of 64 ? 64 = 2x2x2x2x2x2. What root of 64 is 2 ? Why ? 351. The process of finding the root of a number is called Evolution. It is the inverse of involution. 852. V4 = 2. V25 = 5. Vl44 = 12. The sign placed before these numbers is called the Radical Sign. It indi- cates that a root of the number is to be found. 353. ^ = 2. V64 = 4. </8l = 3. </32 = 2. The ex- pression placed above the sign is called the Index. 354. The index is read as its corresponding ordinal number. "v^S is read, " the third root of 8," or " the cube root of 8." The radical sign, when used alone before a number, indicates that its square root is to be extracted. Read </l6; </32; V2|; -v/^; •v/32; ^128. Note. The radical sign was originally in the form y/. The extended form V has the force of a sign of aggregation. EXERCISE 124. 1. Find V81. Solution. 81 = 3x3x3x3 = 9x9. 9ia one of the two equal factors, and is the square root of 81. 2. FindVl764. Solution. 1764 = 2x2x3x3x7x7. There are three pairs of equal factors. One factor of each pair most be a factor of the square root. Hence, the square root is 2 x 3 x 7 = 42. Verification, 42 x 42 = 1764. 3. Fmd V2025. Solution* 2025 k6x6x3x3x8x3. The square root is 6 x 3 x 8 :46. Digitized by VjOOQIC THE SQUARES OF CONSECUTIVE NUMBERS. 245 4. Find V1225; V576; Vi96; V4900. 5. Find V9216; Vlp^S; Vll,025; Vi4;JOO. 6. Find V12,100; V237ri6; V6p36; V94,864. 7. FindV270i; V3364; V4096; V4356. In extracting the square root by this factoring method, any unpaired factors may be left under the radical sign and their root found by methods to be given later. 8. Find a/162. Solution. 162 = 2 x 9 x 9. The factor 2 is unpaired. The square root of 162 is 9\/2. This is read ^^9 times the square root of 2.*' 9. FindV27. Solution. 27 = 3 x 3 x 3. V27 = 3\/3. 10. Find V288; V432; V405; Vi250. 11. Find V726; V980; Vi728; Vi452. 12. Find V3l25; V972; V2048; VI620. The following are correct to three decimals. They shmUd be memorized. ^ __ ^ ^^^ V3 = 1.732 V5 = 2.236 13. V8=V2 x2x2 = 2V2 = 2x 1.414 = 2.828. 14. FindV72; Vl08; V98; V75; Vl25. 15. FindV450; V200; V1200; V405; V648. THE RELATION OF THE SQUARES OF CONSECUTIVE NUMBERS. 355. The square of a number is very intimately related to the square of the following number. 3« = 9. 4* = 16. Digitized by VjOOQIC 246 INVOLUTION AND EVOLUTION. .S56. The square of 3 may be represented by spaces arranged in the form of a square. Fig. 1. Fia. 2. FiQ. 3. 357. The square of 4 may be similarly represented. Instead, however, of making a new square we may use the old one, and by putting 3 spaces at the bottom, 3 at the side, and 1 in the corner, we have the square of 4. The second figure shows the arrangement. 358. The square of 5 may be m;ade from that of 4 by placing 4 spaces at the bottom, 4 at the side, and 1 in the corner. The third figure shows the arrangement. 359. Evidently this process could be continued indefi- nitely. Let us notice carefully what has been done. 4« = 3« 4- 3 4- 3 H- 1 = 3« 4- (2 X 3) + 1 = 16. 6« = 4« + 4 -h 4 + 1 = 4«H- (2 X 4) + 1 = 25. 6« = 5« + 5 + 5 + l = 5«+ (2x5) + 1 = 36. 360. From the above we see that if we have the square of an integer, we can get the square of the next higher integer by adding one more than twice the integer to its square. ^ 6* = 36. 7* = 36 4- (2 X 6) 4- 1 = 49. 8» = 494(2x7)H-l=64. Digitized by VjOOQIC THE SQUARES OF ANY TWO NUMBERS. 247 EXERCISE 125. 1. 8' = 64. Make in succession the squares of 9, 10, 11, 12, and 13. (13^ = 169.) 2. 20« = 400. 21« = 400 4- (2 X 20) 4- 1 = 441. Make the squares of the numbers from 22 to 30 inclusive. (You should get 302 = 900.) 3. 45^ = 2025. Form the squares of the numbers from 46 to 51 inclusive. 4. 132* = 17,424. Form the squares of the numbers from 133 to 140 inclusive. (140* = 19,600.) 5. 245* = 60,025. Form the square of the numbers from 246 to 250 inclusive. THE RELATION OF THE SQUARES OF ANT TWO NUMBERS. 361. We have seen how we can pass from the square of an integer to the square of the next following integer. Let us now see how we can pass from the square of one integer to the square of any other larger integer. 362. 4^ = 16. 7^ = 49. These squares may be repre- sented as follows : A' 7' Fio. 1. Fig. 2. If we study these two squares, we shall see a relation between them. The square of 7 is made from the square Digitized by VjOOQIC 248 INVOLUTION AND EVOLUTION. of 4 by placing at the bottom 3 horizontal rows of 4 spaces each, at the side 8 vertical rows of 4 spaces each, and in the corner a square of 9 spaces. Hence, 72 = 42 H-(3x4) + (3x4)H- 32 = 42 + 2x(3x4) + 32 = 49. 363. It should be noticed that the number of rows of spaces at the side and the bottom, and also in the square at the corner, is the difference between the given integer and the one whose square is sought. In the problem above we start with the integer 4 and we seek the square of 7. Hence, 3 is the number of rows of spaces at bottom and side. 5' 9' Fia. 1. Fig. 2. 364. From these figures we see that 92=52 H-(4x5)+(4x5) + 4» = 62 + 2 X (4x5) +42 = 81. EXERCISE 126. 1. Make a diagram and verify that 9« = 6«+(3x6) + (3x6)+3» = 6« + 2x(3x6)+3«. Digitized by VjOOQIC THE SQUARES OF ANY TWO NUMBERS. 249 2. Make a diagram and verify that 12« = 8«+ (4 X 8) 4- (4 X 8) +4? = 8»H-2x(4x8) + 4«. SOLUTIOK. 12 =8 + 4. 12«=(8 + 4)2 = 8« + 2x (4x8)+4«. 3. 25« = 625. rind28«. Solution. 28 — 26 = 3. 28« = 252 + 2 X (3 X 25) + S« = 625 + 150 + 9 = 784. 4. Pind26l 26 = 204-6. Solution. 26« = 202+2x6x20 + 6« = 400 + 240 + 36 = 676. 26 = 2 tens + 6 units. 26« = C2tens)2 + 2 x 2tensx6units+(6units)«. 365. The square of any number made up of tens and units is the square of the tens + twice the product of the tens and the units -h the square of the units. 5. Find27«. (27 = 20 + 7.) 6. Find 36* (36 = 30 + 6.) 7. rind84«. (84 = 80 + 4.) a Square 34; 46; 67; 73. 9. Square 86 ; 92 ; 97 ; 106. (106 = 100 + 6.) 10. Square 108; 114; 123; 206. Verify the following by doing the things indicated : U. 3»+ 4»= 6*; 6' + 12*=13* 12. 8* + 16«=17*; 7' + 24* = 26». 13. 60' + ll' = 61»; 48* + 56' = 7ff'. 14. 39« + 80» = 89»; 66» + 72» = 97». Digitized by CjOOQ IC 250 INVOLUTION AND EVOLUTION. THE NUBiBER OF FIGURES IN A PRODUCT. 366. How many figures are there in 356 x 87 ? Solution. 87 is more than 10 and less than 100. Hence, 356 X 87 is more than 356 x 10 and less than 356 x 1 00. 356 x 10 = 3560. This product contains 4 figures, but it is less than 356 x 87. Hence, the required product must contain at least 4 figures. 356 x 100 = 35,600. This product contains 5 figures, but it is greater than 356 x 87. Therefore, the required prod- uct cannot contain more than 5 figures. The conclusion is that the product of 365 and 87 must contain either 4 or 5 figures. 367. How many figures are there in 8743 x 387? 387 is greater than 100 and less than 1000. 8743 X 100= 874,300, which contains 6 figures. 8743 X 1000 = 8,743,000, which contains 7 figures. Now, as the real product is between these two, it must contain either 6 or 7 figures. 368. In the first problem the sum of the number of digits in the two factors is 5, and we find that the number of digits in the product is either 5 or 4. In the second problem the sum of the number of digits in the two factors is 7, and we find that the number of digits in the product is either 7 or 6. 369. The number of digits in the prodiict of any two numbers is either the sum of the number of digits in the two numbers or one less than that sum. EXERCISE 127. (Mental.) 1. How many figures are there in 29*? 292 = 29x29. In 29 there are 2 figures. Hence, in 29^ there are either (2 + 2 =) 4 figures or (4 — 1 =) 3 figures. Digitized by VjOOQIC NUMBER OF FIGURES IN THE SQUARE ROOT. 261 2. How many figures are there in 876 x 961 ? 8793 x 341 ? 3. How many digits in 3652? 54329 18,736x121? 4. How many digits in 5641^ ? 78,431^ ? 5. How many digits in 763,421* ? 3,698,743 x 6942 ? THE NUMBER OF FIGURES IN THE SQUARE ROOT OF A NUMBER. 370. Finding the number of figures in the square root of a number is the inverse of the preceding process. If there are 6 digits in a square, there are 3 digits in its square root. This is true, because the square of a number of 3 digits contains either 6 or 6 digits. 371. The number of digits in the square root of any square is usually determined by separating the given square into groups* of 2 digits each, beginning at units. The left-hand group may contain either 1 or 2 digits. 1 67 96 16 shows the grouping of 1679616, and indicates that its square root contains 4 digits. 372. The grouping must always begin at units. In the case of mixed decimals, it proceeds both ways from units. 9 15. 06 25 shows the grouping of 915.0625. 373. The grouping is conveniently shown by the use of an accent mark. Instead of 1 67 96 16, we may use 1'67'96'16. EXERCISE 128. Group the following numbers and determine how many digits there are in each square root: * 1. 529; 1296; 6561; 56644. 2. 390625; 5764801; 987656329. 3. 234.09; 145.2025; 1.008016. 4. 550183936; 5256250000; 191810713444. Digitized by VjOOQIC 252 INVOLUTION AND EVOLUTION. THE EXTRACTION OF SQUARE ROOT. 874. The extraction of the square root is the inverse of the squaring of a number, as shown in § 365. EXERCISE 129. 1. FindV4489. Solution. Since 4489 is a four-place number, its square root must be a two-place number. Since the square of tens is hundreds, the ten of the square root of 4489 must be found in 44 hundreds. The largest square in 44 hundreds is 36 hundreds. Its square root is 6 tens. Subtracting 3600 from 4489 leaves as a remainder 889, of which we have still to find the root. If the original number (4489) is a square, 889 is twice the product of the tens and units of the root plus the square of the units (§ 365). The tens' term of the root, we have seen, is 6. Twice the tens is 12 tens, or 120. The uhit figure of the square root, therefore, must be one which, by multiplying 120 and also itself, will yield 889. By inspection we find that 7 is probably the number. Adding 7 to 120 (making It 127), and multiplying by 7, we obtain 889. The square root of 4489, therefore, is 67. Long Form. 4489160 + 7 60« = 3600 Short Form. 4489 1_67 36 2 X 60 = 120 7 889 889 127 1 889 889 127 127 X 7 = 2. Find V288,369. Solution. This problem shows the extension of the preceding process to larger numbers. By examining the number, we find that its root is a three-place number. 637 1067 I 7469 7469 Digitized by VjOOQIC THE EXTRACTION OF SQUARE ROOT. 263 3. Find V841; V7921. 4. Find V6889; V355,216. The square root of a fraction is the fraction formed from the square root of its terms. V25 6 \49 ■ '49 V49 7 If the fraction is not in its lowest terms, it should be reduced. If the terms are not squares, reduce the fraction to a decimal before extracting its root. \162 V81 9 - -^ ^.> ^' ^-^ N^> ^0 The square root of a decimal is found like that of a whole number, proper attention being paid to the pointing. The grouping must begin at the decimal point, and the right-hand group must contain two digits. When necessary, a is annexed to fill out this group. 6. Find Vl93. Solution. If we desire to find tbe result to three decimal places, we annex three ciphers and group as foUows : .lO'SO'OO' 1.439 16 83 I 380 249 869 I 8100 7821 7. Find V2g to two decimal places. ('28.00'00'.) a Find to two decimal places : V5; V^; Vl9. 9. Find to two decimal places : VJ| ; V| (| = .6666) ; V| 10. Find V^SMi VdlS.Offife; Vl.OdflWie. Digitized by VjOOQIC CHAPTER XIII. MENSURATION. DEFINITIONS. 375. Two lines extending from the same point form an Angle. 376. The size of the angle depends on the difference in direction of the two lines, or the amount of opening between them. The size of the angle does not depend on the length of the lines. 377. The lines BA and BO extending from the point B form an angle which is read "the angle ABO^ In reading an angle, the middle letter must always be the one at the point from which the lines forming the angle extend. Of the two letters at the ends of the lines, either may be read first. The above may be read "the angle -4JBC/' or "the angle cba:' Read the following angles : To the Teacher, A pair of scissors or a pair of dividers may be used to advantage in illustrating angles. Digitized by VjOOQIC MENSURATION. 256 378. If from any point P in a line AB the line P(7 is drawn, two angles are formed. These are the j_ angle APC and the angle CPB. Bead the following pairs of angles: -4f R- 379. In such pairs of angles as the above, if the two angles of any pair are equal to each other, each of them is a Right Angle. In the last figure, the angle R8A equals the angle AST, and each of them is, therefore, a right angle. 380. The sides of a right angle are said to be perpen- dicular to each other. 381. What kind of angle is formed at the comer of a book cover ? At the corner of a carpenter's square ? At the corner of a sheet of paper ? Point out six right angles in the school- room. Fold a piece of paper so as to make two creases perpen- dicular to each other. How many right angles are there on the lid of the chalk box ? Count all the right angles on the chalk box. A Plane is a surface against which a straight edge will fit in all directions, as the desk top or the blackboard. 383. A plane surface bounded by four straight lines is a Quadrilateral. The figure ABDG is a quadrilateral. How many angles has it? Read its angles. e.g. "the angle ABDJ' etc. ^ ^^ Digitized by VjOOQIC 256 MENS URA TION. 384. When every angle of a quadrilateral is a right angle, the figure is a Rectangle. The figure ABCD is a rectangle hecause the angle at each corner is a right angle. Draw a rectangle. Draw a rectangle twice as long as it is wide. Point out six or more rectangles in the schoolroom. ^ Rbctanolm. 385. When all the bounding lines of a rectangle are of the same length, the figure is a Square. 386. The length of a rectangle is frequently called its Base, and the width its Altitude. 387. We already know from § 65 that the area of a rectangle i$ the product of its base and altitude. EXERCISE 130. (Mental.) 1. How far is it around a rectangle 20 ft. x 15 ft.? 2. How far is it around a rectangular field 35 rd. x 20 rd. ? 3. A room is 16 ft. x 18 ft. How far is it around the room? 4. A town lot is 40 ft. x 60 ft. How far is it aroimd it ? 5. How many posts will be required to fence this lot, if the posts are 10 ft. apart ? 6. How many feet of wire will make a five-wire fence around this lot ? 7. How many posts are required for a fence across one end, if they are 5 ft. apart ? a How many posts are needed for a side and an end, if the posts are 5 ft. apart ? 9. What is the area of a floor 15 ft. x 30 ft? Digitized by VjOOQIC CARPETING. 257 10. How many square yards are there in a floor 30 ft. x 24 ft.? (30 ft. X 24 ft. = 10 yd. X 8 yd.) 11. How many square yards are there in a floor 36 ft. x 48 ft.? 12. How many strips of carpet 1 yd. wide are required for a floor 30 ft. x 42 ft., the strips to be laid lengthwise ? 13. How many yards of carpet are required in Problem 12? 14. How many acres are there in a field 40 rd. x 40 rd. ? 15. How many yards of carpet 1 yd. wide will cover a hall 9 ft. x42ft.? CARPETING. 388. The usual widths of carpets are 27 in. and 36 in. Mattings are usually 1 yd. in width. Oilcloth for floors is generally sold by the square yard. 389. "A. yard of carpet" means orie yard of length without any regard to the width. 390. In finding how many yards of carpet are required for a room, it is necessary to decide which way the strips of carpet are to be laid, and then to determine the number of strips required. A fractional part of a strip must be reckoned as a whole strip, because a dealer will not split a strip without charging for all of it. EXERCISE 131. How many yards of 27 in. carpet are required to carpet a rOom 12 ft. x 16 ft., the strips to run lengthwise ? Solution. We find the number of strips by dividing the width of the room by the width of the carpet. 27 in. = 2J ft. 12 -^ 2J = 5J. Hence, 6 strips are needed. Each strip is 16 ft., or 5| yd., long. 6 strips = 51 yd. X 6 == 32 yd. Digitized by VjOOQIC 258 MENSURATION. i H« (^ Find how many yards will be required for the above •om, if the strips run crosswise. 3. A room 18 ft. x 28 ft. is carpeted with 27 in. carpet, ow many yards are needed if the strips run the long way ? 4. How many yards are needed for the above room, if the strips run the short way ? 5. A room 15 ft. x 18 ft. is carpeted with 27 in. carpet, and is surroimded with an 18 in. border. How many yards of border and how many yards of carpet will be needed ? Solution. A room carpeted in this manner will appear like tlie figure. Tlie border is cut as shown, so that the length of border re- quired is the total distance around the room. In this problem the dis- tance around the room is 15 ft. x 2 + 18 ft. X 2 = 66 ft., or 22 yd. Hence, 22 yd. of border is required. (Borders are of various widths, but are generally J yd., f yd., or } yd. wide.) The carpet must be calculated for the space inside the border. Double the width of the border must be taken from both the length and the width of the room to give the dimensions of the space inside the border. In this problem the dimensions inside the border are 12 ft. x 15 ft. If the strips run the long way, (12-i-2J = 5J) 6 strips are required. Each strip is 16 ft., or 5 yd., long. 5 yd. x 6 = 30 yd., the amount of carpet needed. If the strips run the short way, (15 -r- 2J = 6J) 7 strips are needed. Each strip is 12 ft, or 4 yd., long. 4 yd. x 7 = 28 yd., the amount of carpet needed. 6. What is the cost of carpeting a room that is 16 ft. by 19 ft., the carpet being a yard wide, and costing $1.12^ a yard, if there is a loss of 1\ yd. in matching the pattern, the carpet running the longer dimension of the room ? Note. How many breadths are needed for this room ? How much is to be turned under at one side ? Make a plan of the room, using a scale of an inch for a foot, and mark the breadths. Digitized by VjOOQIC PAPERING. 259 7. How many yards of carpet, each strip being f of a yard wide, the loss in matching being 6 in. in each strip except the first, are needed for a room 22 ft. long and 18 ft. wide, the strips running the long way? How many, if the strips run crosswise? Make a plan of the room for each case. Which plan is more economical ? Why ? ^ (8?) What would be the cost of covering your schoolroom with cocda matting at 30 cents per square yard ? /cocda n 9. What would be the cost of carpeting it with ingrain car- pet 36 in. wide, if the " design " in the carpet is 26 in. long ? Note. How many times is the design repeated in one strip ? How much must be cut ofE or turned under at the end ? at the side ? 10. A room is 18 ft. x 24 ft. The border is | yd. wide and the carpet 27 in. wide, and each costs $ 1.50 per yard. The strips are laid the long way, and there is no loss in matching. What is the total cost of carpeting the room ? PAPERING. 391. Wall paper is of various widths, but the usual width is 18 in. It is put up in single rolls of 8 yd. and double rolls of 16 yd. Border paper is of various widths, and sells by the yard. 392. Paper hangers generally calculate the amount of paper required for the room without openings, and then deduct one half roll for each door and window. No de- duction is made for the border. EXERCISE 132. 1. How much paper is required for a room 18 ft. x 15 ft. x 12 ft., there being 1 door and 3 windows ? Digitized by VjOOQIC 260 MENSURATION. SoLUTiOH. The length of the 4 walls is 18 ft X 2 + 16 ft. X 2 = 66 It The area of the 4 walls is 66 X 12 = 792 sq. ft., or 88 sq. yd. The area of the ceiling is 18 X 16 = 270 sq. ft, or 30 sq. yd. 88 sq. yd. + 30 sq. yd. = 118 sq. yd. A roll 8 yd. long and j^ yd. wide contains 4 sq. yd. 118 -H 4 = 29^. As only whole rolls can be bought, the nmnber re- quired is 30. Deducting 2 rolls for the door and 3 windows, we have 28 rolls as the number required. The distance around the room is QQ ft., or 22 yd., which is the amount of border required. 2. Find the cost of papering a room 18 ft x 20 ft x 12 ft. with paper costing 15 cents a roll and the border 8 cents a yard. Make deductions for 1 door and 2 windows. 3. A room is 24 ft. x 18 ft. x 12 ft The border for the ceil- ing costs 15 cents a yard, and the paper for the ceiling 22 cents a roll. The border for the walls costs 18 cents a yard, and the paper for the walls 25 cents a roll. The room has 3 windows and 2 doors. Find the total cost of paper. 4. Find the cost of papering the 4 walls and ceiling of your schoolroom at 5 cents a roll, and 3 cents per yard for border for walls. 5. Measure a room at home and calculate the amount of paper needed for the walls and ceiling ; also find the length of border. 6. A room is 16 ft by. 20 ft., with walls 12 ft. high. There are 3 windows 2| ft. by 6 ft., and 2 doors 2 ft. 8 in. by 8 ft. 4 in. If wall paper costs 22 cents a roll, and border 6 cents a yard, what is the cost for papering walls and oeiling ? Digitized by VjOOQIC PLASTERING. 261 PLASTERINO. Plastering is done by the square yard. In making estimates, it is the usual custom to deduct from the total area of walls and ceiling one half the total area of all openings, and then express the result by the nearest square yard. EXERCISE 133. 1. What is the cost, at 22 cents a square yard, of plastering the walls and ceiling of a room 36 ft. x 48 ft. x 12 ft., no de- ductions being made for openings ? 2. " Develop " the several plastered surfaces in Problem 1 on the scale of 6 ft. to 1 in. on the blackboard, or 6 ft. to \ in. on your tablet. Development. End, 36'. Side, 48'. End. 36'. Side, 48', ir Ceiling, 36' X 48'. Make a similar diagram for each problem in this set. 3. What is the cost, at 2^ cents a square yard, of plastering a cottage containing 6 rooms, 2 of which are 14 ft. by 15 ft., 2 are 10 ft. by 12 ft., and 2 are 13 ft. by 15 ft., the ceilings being 10 ft. high, and no allowance being made for openings ? 4. What will it cost to paper the walls and ceilings of this cottage with paper at 20 cents a roll, deducting 400 sq. ft. for baseboard, allowing for 14 windows and for 7 doors, the paperer's charge being 20 cents a roll, and the border costing 6 cents a yard, no deductions being made for border ? Digitized by VjOOQIC 262 MENSURATION. TH£ PARALLELOGRAM. 394. Two lines which are everywhere the same distance apart are Parallel. The rails of a railroad track are parallel. 896. The Distance between a point and a line is the length of the perpendicular from the point to the line. A R- 8 The line AS is perpendicular to ET. the distance from A to ET, The length of AS is 396. The distance between two parallel lines is the length of the line connecting them ^ which is perpen- dicular to both of them. AB is the distance between MN and PQ, and CD is the dis- tance between LE and ST, 397. A Parallelogram is a quadrilateral whose opposite sides are parallel. The opposite sides are also equal in length. PQ, the distance between the parallel sides AD and BC, is the altitude of the parallelogram when BC is the base. ES, the distance between the parallel sides AB and DC, is the altitude, when DC is used as the base. Q A Parallelogram. Digitized by VjOOQIC THE PARALLELOGRAM. 263 398. If the parallelogram ABCD is made of cardboard and is cut along the altitude PQ^ and the pieces placed as in Figure 2, a rectangle is formed. The length of B Q Fig. 1. this rectangle is BO and its width is PQ. Its area is BO y. PQ. Evidently the rectangle and the parallelogram have the same area. 399. Hence, the area of a parallelogram is the product of its base and altitvde. Verify the above by actually making a cardboard parallelo- gram and cutting it as indicated. EXERCISE 134. 1. Find the area of a parallelogram whose base is 40 ft., and whose altitude is 27 ft. 2. Find the area of a field in the shape of a parallelogram, base 86 rd. and altitude 80 rd. ^ F_ 3. ABOD is a paral- lelogram. AB = 16 ft. ^Z)=45ft. Pe=12ft. ^z i Find the area. ^ 4. How many feet of wire will make a five-wire fence about the above parallelogram ? Digitized by VjOOQIC 264 MENSURATION. THE TRIANGLE. 400. A Triangle is a plane surface inclosed by three straight lines. 401. Triangles are of various shapes. When one side is perpendicular to another side, the triangle is a Right- angled Triangle. The third triangle above is such a triangle. 402. The corners of a triangle are called its Vertices. Any one corner is a Vertex. Note, The word vertex is also applied to a comer of a quadrilateral or any figure bounded by straight lines. 403. Any side of a triangle may be called its lase. 404. The altitude of a triangle is the distance from a vertex to the opposite side. 406. Every triangle has three altitudes. BC is base, and AD is altitude. AG is base, and BD is altitude. Digitized by VjOOQIC THE TRIANGLE. 265 AB is base, and CD is altitude. In this figure ABG is the triangle, BC is the base, and AD is the altitude. The altitude is outside the triangle. Notice that AD is the perpendicular from A to the opposite side BO, and that BG had to be extended, or produced. 406. Cut out of cardboard two equal triangles and place a pair of equal sides together, thus forming a parallelogram. If the triangles are like M and iV, then they will make the parallelogram PBRS. Evidently either triangle is one half the parallelogram. 407. The area of the parallelogram is BR x PK, The area of the triangle PBR is ^J^l^LJ^. BR is the base of the triangle, and P^is its altitude. 408. Hence, the area of a triangle is one half the product of its base and altitude. Digitized by VjOOQIC 266 MENSURATION. EXERCISE 135. 1. Cut out five paper triangles, measure accurately theii bases and altitudes, and determine their areas. 2. Determine the area of any triangular spaces in the school yard. Draw each on some convenient scale. Find the area of the following triangles : Base. Altitude. Base. Altitude. 3. 4 ft. 2ift. 8. 20 ft. IT^ft. 4. 2 yd. l\it. 9. 18f ft. 6ift. 5. Sfft. 4|ft. 10. 9fft. 14|ft. 6. 4^ ft. 2fft. 11. 6 ft. 14f ft. 7. Sfft. Hyd. 12. 4|yd. Sfyd. THE RIGHT-ANGLED TRIANGLE. 409. The side opposite the right angle in a right-angled triangle is called the Hypotenuse. 410. The other two sides of a right triangle are called its Legs. The legs are frequently given individual names, and are called Base and Perpendicular. These names are interchangeable. Either leg may be called the base, and then the other leg is the perpendicular. The triangle ABC is a right triangle. AB is the hypotenuse, and AC and BC are the legs. BC is the base, and AC is the perpendicular. 411. The sides of a right triangle are related in the following remarkable manner : The sum of the squares upon the legs is equal to the square upon the hypotenuse* This is known as the Pythagorean proposition. Digitized by VjOOQIC THE RIGHT-ANGLED TRIANGLE. 267 Vebifications of the Pythagorean Proposition. 412. I. Draw a right triangle, ABG, on a sheet of pasteboard. Draw the squares on the three sides and subdivide as shown in this figure, by ex- tending the sides of the largest square through the smaller squares, and drawing a line at right angles to the longer extension. With a sharp knife cut out the five pieces and place them in the posi- tions 1', 2', 3', etc. We thus see that the square of the hypote- nuse is equal to the sum of the squares of the legs. II. Draw two equal squares upon cardboard. Mark off equal distances from the corners as shown in the figures. Cut along the lines in each square and thus remove the triangles. It A E GO r^' n » \ 'i 1 L 2 M -Q will be found that the four triangles cut from the first square are equal to the four cut from the second square. Since the two squares were equal, the remainders left after removing the four triangles from each square must be equal. Digitized by VjOOQIC 268 MENSURATION. Now. take any one of the triangles and it will be found that the side of the larger square is equal to the length of the hypotenuse, and that the sides of the other two squares are equal to the base and perpendicular, respectively. This shows that the square upon the hypotenuse is equal to the sum of the squares upon the two legs. EXERCISE 136. 1. Find the length of the hypotenuse. Solution. By the Pythagorean proposition, we have : A* = 52 + 12« = 25 + 144 = 16d. h = Vl69 = 13. 2. Find the base. LUTiON. By the Pythagorean proposition, w ^r^^ So] 6 e have : 58+62 = 132. 26+62 = 169. 62 = 169 - 25 = 144. 6 = Vl44 = 12. Find the missing part in each of the following right tri- igle Base. Altitude. Hypotenuse. 3. 8 in. ? 10 in. 4. 20 ft. 15 ft. ? 5. 224 yd. ? 260 yd. 6. ? 272 mi. 363 mL 7. 192 rd. 144 rd. ? Digitized by VjOOQIC THE TRAPEZOID. 269 Make diagrams of each of the following problems and solve : 8. The top of a ladder that is 30 ft. long rests against a telegraph pole 24 ft. from the ground. How far is the foot of the ladder from the foot of the pole ? 9. A and B start from the same point at the same time. A travels north, and B east, the former traveling at the rate of 4 mi. an hour, and the latter 3 mi. How many feet apart are they in 16 min. ? 10. A rope is attached to the top of a 96 ft. pole. It touches the ground 28 ft. from the foot of the pole. What is its length ? 11. What is the diagonal of a rectangle whose dimensions are 6 yd. and 8 yd. ? Note, The diagonal of a rectangle is the straight line joining opposite vertices. 12. A 13 ft. ladder rests with its top against a window sill .12 ft. from the ground. How many feet from the wall is the foot of the ladder ? 13. Against the top of a pole 15 ft. high are braced in opposite directions a 17 ft. ladder and a 25 ft. ladder. How far apart are the feet of the ladders ? 14. A field is 80 rd. x 60 rd. How much farther is it from one corner to the opposite along the fence of the field than it is along the diagonal? j^ 15. ABOD is a parallelogram. J5C=40rd. ^B=13rd. BE=5 vd. ^l Find the area of the parallelogram. THE TRAPEZOID. 413. A Trapezoid is a quadrilateral only two of whose sides are parallel. Ag^ ^— .jp* The figure ABCD is a trapezoid. /• '^^^--^.^^^ \ i [D is parallel to BC. AE is the ^Z-i lllllii^'c AD is parallel to BG, altitude or width of the trapezoid. ^ ^ Teafb2oh>. Digitized by VjOOQIC 270 MENSURATION. 414. By joining A and (7, we divide the trapezoid into two triangles. If 5(7 is the base of the triangle ABC^ then AE is its altitude. If AD is the base of the triangle ACD^ then OF is its altitude. The two altitudes AE and OF are equal, because each of them is the distance between the parallel lines AD and BO. 415. The area of the triangle ABCi^ ^^ ^ ^^ > The area of the triangle ACB is ^ - But since GF^ AD X AE AE^ we may write the area of the second triangle : The area of the trapezoid is the sum of the areas of the two triangles, and is ^O xAE _^ AD xAE ^ ^j^ ^^ be written (^^+^^^)x^^. 416. Hence, the area of a trapezoid is one half the product of the sum of the parallel sides and the altitude, EXERCISE 137. 1. If, in the figure of § 413, AD = 20 rd., i5(7 = 48 rd., and AE = 32 rd., what is the area ? 2. If AD =^65 rd., BO =75 rd., and AE = 4:S rd., what is the area ? 3. The parallel sides of a trapezoid are 120 rd. and 216 rd., and the width is 80 rd. What is the area ? 4. AD = e5tt, BO =95 ft. AB = 50it, BE = SOit Find the area. Digitized by VjOOQIC UNITED STATES SURVEYS. 271 UIHTED STATES SURVEYS. 417. Most of the lands of the United States west of the original thirteen states, except the tract between the Ohio and the Tennessee rivers, are surveyed in accordance with the following system : In each great survey district there is run a Principal Meridian and an east and west line called a base line. On each side of the Principal Me- ridian, at distances of 6 mi., are north and south lines called range lines^ which divide the land into strips 6 mi. wide called ranges. 6 5 4 3 2 1 7 8 9 10 11 12 18 17 16 15 14 13 19 20 21 22 23 24 30 29 28 27 26 25 31 32 33 34 35 36 Fig. I. 418. By east and west lines parallel to the Base Line the ranges are divided into townships 6 mi. square. A township is designated by giving its number and direction from the Base Line, the number and position of its range, and the name or number of the Principal Meridian. The writer is in township 9 North, Range 1, West of the 2d Principal Meridian. 419. Townships are subdivided into 36 sections num- bered as in Figure I. Sec. 28. 420. The sections are divided into halves and quarters; the quarters into halves and quarters, and so on, as in Figure II. 421. Tracts are described thus : (1) W. J Section 28, T. 24 N. 2 E., 2d P.M. Fig. IL 1 3 1 4 2 Digitized by VjOOQIC 272 MENSURATION. (2) S. J S.E. \, Section 28, etc. (3) N.W. \ N.E. J, Section 28, etc. (4) W. J N.E. \ S.E. J, Section 28, etc. 422. How many acres are there in each of these tracts? What fraction of a section is each ? EXERCISE 138. Draw a 6 in. square representing a section. Mark in it each of the following tracts, and tell how many acres each contains : 1. The N.E. i of the N.E. J. 2. The S. \ of the N.W. \. 3. The S. i of the N.W. \ of the S.W. \. 4. The E. \ of the N. ^ of the S.E. \ of the S.E. \. 5. The N. \ of the S.E. J of the S.E. \ of the N.W. \. 6. The N.W. \ of the S.W. \ of the N.E. \ of the N.E. \, 7. The N.W. \ of the N.W. \ of the N.W. \. a The N.W. \ of the N.W. \ of the N.E. \. 9. The N. \ of the S.W. \ of the N.W. J of the S.E. \, 10. The S. \ of the S. ^ of the S.W. \ of the N.W. J. THE CIRCLE. 423. A Circle is a plane figure bounded by a line all points of which are equally , distant from a point within called the Center. 424. Mention ten circles, such as the top of a cup, the bottom of a lamp chimney, etc. 425. The bounding line is called the Circttmference. Digitized by VjOOQIC THE CIRCLE. 273 426. Any straight line from center to circumference is called a Radius. A straight line passing through the center and terminating at both ends in the circumference is called the Diameter. 427. A portion of the circle set off between two radii is called a Sector. The part of the circumference bounding a sector is called an Arc. 428. In the figure (§ 423) the curved line is the circum- ference, the point the center, 0(7 a radius, AB a diameter, the part AOC a sector, and the curved line ^(7 an arc. 429. In general, we represent the circumference by (7, the diameter by 2>, and the radius by R. (I)=2R.^ 430. Measure, as accurately as you can, the circumfer- ence and diameter of each of five circles. In each case divide the circumference by the diameter. Each result will be 3 and a fraction. The fractions will not be exactly the same, because of inaccuracies in measuring. 431. The quotient of the circumference divided by the diameter ((7-5- i)) is always the same, and is represented by the Greek letter ir (Pi). 432. The value of ir to four decimal places is 3.1416. For all ordinary measurements ^\ is a sufficiently accurate value of IT. Unless otherwise directed, you will use 3^ for TT. 433. Since -^ = tt, it follows that C = ttD. Hence, the circumference of a circle is the product of it and the diameter, 434. Since D = 2 ff, we have C^^irR. Hence, the circumference of a circle is two times the product of IT and the radius. Digitized by VjOOQIC 274 MENSURATION. 435. Since (7=: irD, then - = D. TT Hence, the diameter of a circle is the quotient of the circum- ference divided by tt. EXERCISE 139. (Mental.) 1. How would you find the diameter of a tree? of a round column ? 2. The diameter of a stovepipe is 7 in. What is its circum- ference ? 3. The circumference of a wheel is 44 in. What is its diameter ? 4. How can you draw a circle ? 5. A tree is 66 in. in circumference. What is its diameter ? 6. The diametey of a grindstone is 35 in. What is its circumference ? 7. The radius of a bicycle wheel is 15 in. What is its cir- cumference ? 8. How many times is the circumference of a 7 in. wheel contained in that of a 14 in. wheel ? Note, A 7 in. wheel is one whose diameter is 7 in. 9. How many times is the circumference of a 10 ft. circle contained in that of a 15 ft. circle ? 10. How many times is the circumference of a 5 in. wheel contained in that of a 30 in. wheel? EXERCISE 140. 1. How long is the tire on a 53 in. wheel ? 2. The tire on a wheel is 196 in. in length. What is the radius of the wheel ? 3. How many revolutions will a 56 in. carriage wheel make in going 1 mi. ? Digitized by VjOOQIC THE AREA OF A CIRCLE. 275 4. A bicycle is said to be geared to 70 in. when each revolu- tion of its pedals propels it a distance equal to the circumference of a 70 in. wheel. How many xevolutions of the pedals will move the wheel 1 mi. ? 5. What is the circumference of the largest circle that can be laid out in a field 36 rd. square ? THE AREA OF A CIRCLE. 436. Cut from a round potato or turnip a thin circular slice. Cut it in two and divide each semicircle into 8 equal wedges. Be careful not to cut the rind. Straighten the rind of each semicircle and fit the wedges together. 437. The circle is now approximately a parallelogram. Its base is half the circumference, 2TrB , or ttjB. Its alti- tude is B ; hence, its area is irR x -B = TriP, or 2 7rR xB. Digitized by VjOOQIC 276 MENSURATION. The form Trip is the one in general use. Since ^irR = (7, the last form becomes — - — . 438. Hence, the area of a circle is the square of the radvus multiplied by tt. 439. It is to be noticed that the figure is not an exact parallelogram. If each semicircle had been divided into 16 equal wedges, the figure would have been more nearly a parallelogram. If each semicircle had been divided into 1600 equal wedges, the parallelogram would be perfect beyond the detection of the eye. In every case the area is 7riJ2. 440. The sector AOB is the same part of the circle that the arc AB is of the circumference. The area of the circle is one half the product of circumference and radius; hence, the area j[ ^ of a sector is one half the product of its arc and radius. EXERCISE 141. 1. If R in the diagram of § 437 is 7 in., what is the length of AB ? What is the area of the circle ? 2. What is the area of one face of a silver dollar, the diame- ter being 1^ in. ? 3. The face of a watch is If in. in diameter. What is its area? Solution. ttB^ = S}x (i)^ Hence, the area is 2Jf sq. in. 4. How many square feet are there in a circular flower bed whose radius is 7 ft. ? Digitized by VjOOQIC SOLIDS. 277 5. A circular pond has a diameter of 250 yd. How many square yards are there in its surface ? 6. What is the difference in area between a field 40 rd. square and one 40 rd. in diameter ? 7. What is the area of a sector in a 28 in. circle, the length of its arc being 17| in. ? SOLIDS. I 441. A Solid is any inclosed portion of space. 442. A Cube is a solid inclosed by six equal squares. 443. The Edges of the cube are the lines in which the squares meet. The squares are the Faces. How many faces has a cube ? How many edges ? How many corners ? 444. A Rectangular Parallelepiped is a solid bounded by six rectangles. A shoe box is a good example of such a solid. A cube is a rectangular parallelopiped. 445. The three dimensions of a rectangular parallelo- piped are lengthy breadth^ and thickness. Instead of thick- ness, this third dimension is often called the altitude. We already know (§ 69) that the volume of a rectangular parallelopiped is the product of its three dimensions. WOOD MEASURE. 446. Cord wood is stacked in piles which are in the form of rec- tangular parallelopiped s. Digitized by VjOOQIC 278 MENS URA TION. EXERCISE 142. 1. How many cords of cord wood are there in 3 piles of wood, eacll being 4 ft. high, the first being 36 ft. long, the second 42 ft. long, and the third 73 ft. long ? Find its cost at $4.75 per cord. Note, Where cord wood is piled 4 ft. high, there is 1 cd. for every 8 ft of length. 2. Each of the following piles is 4 ft. high. Find how many cords each contains. (a) 83 ft. long. (d) 47^ ft. long. (6) 69 ft. long. \e) 61 ft. 5 in. long, (c) 35^ ft. long. (/) 93 ft. 10 in. long. Find the cost of each pile, at f 5.25 a cord. 3. How much cord wood is there in each of the following piles ? What is the cost at 1^4.75 a cord ? (a) 2Q ft. long, 6 ft. high. Solution. The number of cubic feet in the pile is 26 x 6 x 4. Since 26 X 6 X 4 there are 128 cu. ft. in a cord, the- number of cords is loa ' 13 3 Form. jZ^ x ^ x jt 13x3 39 _ .. X^^ " 8 " 8 ** 8 Shice each cord costs $4.76, the total cost is $4.76 x 4} = $23.16{. If cost alone is wanted, the form is 13 3 »4.75x;Zg X g X jt ^ $4.75 x 13 x 3 _ $186.26 _^g3 ^^ X^ 8 8 * * ^* 8 (5) 54 ft. long, 7 ft. high, (c) 61 ft. long, 7 ft. high. Digitized by VjOOQIC LUMBER MEASURE. 279 (d) 42 ft. 8 in. long, 6 ft. 4 in. high. Note. Call inches twelfths of a foot, and change mixed numbers to improper fractions, thus : 42 ft. 8 in. = 42} ft. = ^^it. 6 ft. 4 in.=6J ft=J^ft. ^^^i^ ' ^q ^ L^'^^ ' (c) 86 ft. 3 in. long, 7 ft. 6 in. high. If) 124 ft. 5 in. long, 8 ft. 6 in. high. ig) 97 ft. 6 in. long, 5 ft. 9 in. high. Qi) 224: ft. long, 12 ft. high. LUMBER MEASURB. 447. The unit of lumber measure is the board foot^ 1 ft. long, 1 ft. wide, 1 in. thick. 448. Lumber that is less than one inch thick is counted as if an inch thick. If lumber is more than an inch thick, the excess is taken into account. 449. How many cubic inches are there in a board foot? How many board feet in a cubic foot ? 460. What is the width of an inch board that contains as many board feet as it is feet long ? of a 2 in. plank ? of a 3 in. stud? 451. What is the end area of each of the above pieces ? 452. What is the end area of a 6 x 8 in. sill ? How many board feet in each foot of its length? What divisor have you employed ? 453. How many board feet in a beam 10 in. x 12 in., 24 ft. long? 454. Show the truth of the following : Number of board feet = thickness x width x lengthy 12 455. Thickness and width are expressed in inches and length is expressed in feet. Lumber bills are regularly made out in these units. Digitized by VjOOQIC 280 MENS URA TION. EXERCISE 143. 1. What is the cost of 16 sills 6 in. x 8 in. x 18 ft. @ $ 18 per thousand feet ? FORM. 8 $18 X j:0 X g X 3 X 18 ^ ^18 X 8 X 18 ^ ^ OQ... ^125 Solution. Since a number of dollars is required, I write J|18, the cost of 1000 board feet. I express the cost of one board foot by writing 1000 as a divisor. I express the cost of a stick 1 ft. long, 1 in. wide, and 1 in. thick by writing 12 as a divisor. I multiply by 18 because the sill is 18 ft. long ; by 8 because it is 8 in. wide ; by 6 because it is 6 in. thick ; by 16 because there are 16 sills. 6 X 8 X IS The number of board feet in each sill is expressed by . I write 16 as a multiplier because there are 16 sills, 1000 as a divisor to get number of thousand feet, then mutliply by 18 because the number of dollars paid must be 18 times the number of thousand feet. 2. What is the cost of the following bill of lumber at $ 21 a thousand (M)? Ssills, 8x10, 16 ft. long; 48 studs, 2 X 4, 18 ft. long; 22 joists, 2 X 10, 16 ft. long; 50 rafters, 2 x 4, 14 ft. long; 24 joists, 2 X 8, 16 ft. long. 3. Find the cost of the following bill of lumber, at $ 16.50 per M : 32 common boards, 8 in. wide, 14 ft. long; 65 fence boards, 6 in. wide, 16 ft. long ; 16 comer posts, 4 x 4, 18 ft. long; 7 sills, 6x8, 14 ft. long; 46 rafters, 2 x 6, 16 ft. long ; 36 joists, 2 X 8, 18 ft. long. Digitized by VjOOQIC MASONRY AND BRICKWORK, 281 / 4. Find the cost of the following bill of lumber : 86 pieces maple flooring 1 x 3 x 16 @ f 40 per M ; 86 pieces ash flooring 1 x 3^ X 16 @ $ 36 per M ; 72 pieces clear pine boards 1 x 10 X 16 @ f 32 per M ; 24 rafters 2 x 6 x 16 @ $ 18 per M ; 36 joists 2 X 10 X 22 @ f 21 per M. r 5. What is the cost of 46 planks, 2 in. thick, 10 in. wide, and 18 ft. long, at $ 22 per M ? 6. What is the cost of 65 2^ in. oak planks, 12 in. wide and 16 ft. long, at $ 36 per M ? 7. How many posts set 8 ft. apart are required for wire fencing around a field 800 ft. square and for two cross fences dividing the field into four equal squares ? Show that the cost of these posts at 8 J cents each is $ 49.75. Vs. Find the cost of the following bill of lumber at $21.50 per M : 5siUs, 8x10, 18 ft. long; 36 joists, 2 X 10, 16 ft. long; 42 studs, 2 X 4, 22 ft. long; ^ •, 70 boards, averaging 9 in. wide and 14 ft. long. ^ \ MASONRY AND BRICKWORK 456. Stone walls are generally measured by the perch^ which contains 24| cu. ft. When stone is laid in mortar, a perch of the wall con- tains 22 cu. ft. of stone and 2| cu. ft. of mortar. 457. The usual rule in measuring stone walls is to measure the distance around the walls on the outside, thus counting the corners twice, and to make no allow- ance for openings. ; ^. v ' digitized by CjOOQIC 282 MENS URA TION. 458. Brick work is estimated by the thousand brick. In brick work, allowance is made for openings, and the corners are counted but once. An ordinary brick is 8 in. X 4 in. x 2 in. 459. When laid in mortar : 7 bricks build 1 sq. ft. of wall, 1 brick, or 4 in., thick. 14 bricks build 1 sq. ft. of wall, 2 bricks, or 9 in., thick. 21 bricks build 1 sq. ft. of wall, 3 bricks, or 13 in., thick. 28 bricks build 1 sq. ft. of wall, 4 bricks, or 18 in., thick. EXERCISE 144. 1. How many thousand bricks are used in a foundation wall 30 ft. X 24 ft., the wall being 4 ft. high and 2 bricks thick? Solution. 2 (30 ft. +24 ft) =108 ft., distance. around the waU. 108 x 4 = 432, the number of square feet in exposed surface. The comers have been measured twice in the above result. A waU 2 bricks thick is 9 in., or } ft., thick. One surface of a corner is 4 x f = 3 sq. ft. The surface of the 4 corners is 3 x 4 = 12 sq. ft. 432 - 12 = 420 sq. ft. 14 bricks are required for each square foot of surface. 14 X 420 = 6880, the number of bricks required. 5880 = 5.88 thousand. 2. How many bricks are used in a wall 80 ft. long, 4 ft high, and 3 bricks thick ? 3. How many perch are there in the wall of a cellar 20 ft. x 18 ft. X 9 ft., the wall being 1 ft. thick ? 4. A rectangular cistern, 15 ft. x 12 ft. x 10 ft., is walled with one thickness of brick. How many bricks are required ? 5. How many perch of masonry are there in the foundation of a building 80 ft. x 20 ft., the foundation being 8 ft. high and the wall 2 ft. thick ? Digitized by VjOOQIC MASONRY AND BRICKWORK. 283 EXERCISE 145. A comer lot is 66 ft. x 132 ft. 1. How many loads of earth will be required to raise its level 18 in. ? Note. A load is 1 cu. yd. 2. How many square feet of sod will cover it ? 3. How many posts are needed for the fence, the posts being 6 ft. apart? 4. How many feet are there in the 2x4 stringers 12 ft. long, required for the fence ? 5. The fence is a tight board fence, 5 ft. high. How many feet of 10 ft. inch lumber are required ? 6. A 6 ft. walk is built in front and on one side of the lot. How many square feet of surface are there in it ? 7. The filling cost 25 cents a load, the sod 8 cents a square yard, the posts 12^ cents each, the stringers $2.25 per hun- dred feet, the boards for the fence $ 30 per thousand feet, and the walk 60 cents a running foot, the labor on the fence $ 37.50. What was the total cost of improving the lot ? 8. The brick street on the front and side of the above lot cost $ 5.80 a lineal foot. What was the cost of the improve- ment, this lot being assessed for one half the cost ? Note. In city improvements, the city pays for the intersection of streets, and of alleys with streets. 9. The foundation of the house on fse/t. the above lot is like this : The outside walls are 2 bricks thick, the cross walls 1 brick thick. The walls are 30 in. in height. How many thou- sand bricks are required for the foun- dation? The measurements given are outside. ^*-^- 10. If the above foundation is made of stone and the walls are 1 ft. thick, how many perch of masonry are required ? ti ^ Digitized by VjOOQIC 284 MENSURATION. THE INVERSE PROBLEM IN MENSURATION. 460. Thus far we have considered the direct problem ; that is, we have had the dimensions given to find the surface or volume. In the inverse problem we are to find some lacking dimension, having given the siu'face or vol- ume and all dimensions but one. EXERCISE 146. 1. The area of a rectangular field is 9 A.; its length is 60 rd. What is its width ? Solution. 9 A. = 160 sq. rd. x 9. Let X = tUe width in rods. 60 X = area in rods. 60«=160 X 9. 60 The width of the field is 24 rd. 2. The area of a circle is 616 sq. ft. What is its radius ? Solution. Let x = its radius. irx^ — its area. xx2 = 616. x=\/l96 = 14. Hence, the radius is 14 ft 3. The volume of a rectangular box is 600. cu ft. It is 8 ft wide and 6 ft. deep. How long is it ? Solution. Let x = its length. 8 X 6 X = its volume. 8 X 6x = 600. Hence, the box is 12| ft. long. *=8^=^'*- Digitized by VjOOQIC THE INVERSE PROBLEM. 285 4. A pile of wood is 4 ft. wide, 4 ft. high, and 27 ft. long How many cords are there in 15 such piles ? What is it worth at $ 4.50 a cord ? 5. A bin is 8 ft. wide, 12 ft. long, and 7 ft. high. How many bushels of shelled corn will it hold, estimating a bushel as I cu. ft. ? 6. A bushel measures exactly 2150.4 cu. in.; using this figure, find the error in the result of Problem 5. T. Bought a city lot containing -| of an acre, at $ 2.2b a square yard. What did it cost ? 8. If a field containing 71 A. 82 sq. rd. is divided into 8 equal parts, what will each part contain ? 9. Bought 36 2 X 4 16-ft. studs, 48 2 x 8 18-ft. joists, 4 silb 8x8, 16 ft. long, at $ 18.50 per thousand ; 1250 ft. flooring at $32; 1460 ft. sheathing at $20, and 1480 ft. siding at $ 33.50. If the bill were divided into 4 equal pay- ments, what would each amount to ? 10. How many paving stones, 4 ft. 4 in. long and 3 ft. wide, will be needed to make a 3 ft. walk, 186 ft. 8 in. long ? U. How many bricks, of ordinary size, will be required to pave a court 16 ft. wide and 80 ft. 8 in. long ? 12. If a man travels at an average rate of 4 mi. 25 rd. 5 yd. an hour, how many hours will be required to travel 175 mi. ? 13. A cellar is 18 ft. 6 in. by 24 ft. 4 in. and 5 ft. deep. How many loads, \ of a cubic yard each, will the excavated earth make ? 14. A man, having 44 A. 96 sq. rd. of land, sold 5 A. 92 sq rd. What part of the land does he still own ? 15. A cubical tank, 10 ft. square at the base, has a capacity of 8000 gal. What is its height ? 16. In excavating a cellar 18 ft. x 22 ft., 286 loads of earth were removed. How deep was the excavation ? Digitized by VjOOQIC 286 MENS URA TION. 17. A triangular field, whose altitude is 60 rd., contains 3 A. What is the length of the base ? 18. The entire surface of a cube is 3456 sq. in. What are the dimensions of the cube ? 19. A bin 10 ft. X 20 ft. holds 960 bu. On the basis of | cu. ft. to the bushel, how deep is the bin ? 20. A man has two fields, one a rectangle 40 rd. x 60 rd., the other a square of the same area. Which field will require the more fencing, and how much more ? 21. What is the diameter of a circle containing 22 A. ? 22. A rectangular field twice as long as it is wide contains 20 A. What are its dimensions ? 23. A race track 30 ft. wide is built around a circle whose radius is 500 ft. What is the length of the center of the track ? 24. What is the area of the track in the above problem ? 25. In a room 30 ft. X 40 ft. x 12 ft., what is the distance from an upper corner to the opposite lower corner ? 26. A granary is 4 ft. X 12 ft. x 6 ft. How many bushels of wheat will it hold ? (1 bu. = | cu. ft.) 27. How deep must a 5 X 6 ft. box be to hold 100 bu. of wheat ? 28. A corn crib is 30 ft., long and 4 ft. wide. How many bushels of corn does it contain, when it is filled to a uniform depth of 8 ft. ? (1 bu. corn in the ear = 2\ cu. ft.) 29. How wide must a wagon bed 14 ft. long and 30 in. deep be, to hold 40 bu. of corn in the ear ? 30. A rail com pen is 9 f t. x 9 ft. X 10 ft. How many bushels of corn will it hold ? 31. Measure your schoolroom and find how many bushels of wheat it would hold if filled to a depth of 4 ft. 32. In digging a cellar 36 ft. long and 9 ft. deep, 360 loads of earth were removed How wide was the cellar ? Digitized by VjOOQIC PRISMS AND PYRAMIDS. 287 PRISMS AND PTRAmDS. 461. A solid whose ends are equal and parallel, and whose sides are rectangles, is a Right Prism. 462. The two ends of such a solid are called its bases. The rectangular sides of a prism are called its faces. 463. The convex surface of a prism is the sum of the areas of its faces. 464. The volume of a prism is the product of the area of its base and its height, 465. A Right Pyramid is a solid with one base, and with its faces equal triangles which meet at a point called the vertex. 466. The slant height of a right pyramid is the distance from the vertex to the middle of the base of one of its faces. The height of a pyramid' is the perpendicular dis- tance from the vertex to the base. 467. The convex surface of a pyramid is the sum of the areas of its triangular faces. 468. 27ie volume of a pyramid is one third the product of its base and height. 469. Make out of cardboard a right prism 5 in. square and 6 in. high. The following is the plan : 6 in. 6 in. Sin. Sin. 6 in. Digitized by VjOOQIC 288 MENSURATION. 470. Also make a pyramid 5 in. square and 6 in. high. The faces of this pyramid will be equal triangles having a base of 5 in. and an altitude of 6J in. The following is the plan: 5 m, 471. Use the pyramid as a measure, and see how many times it is contained in the prism. Use any convenient material, such as bran or oats. This shows that the volume of a pyror mid is one third the volume of a prism of the same dimensions. EXERCISE 147. 1. The base of a right prism is a right triangle whose legs are 3 ft. and 4 ft. The height of the prism is 10 ft. Find its convex surface, its total surface, and its volume. Solution. The hypotenuse of the triangular base is V3'^ + 4« = \/26 = 6. The convex surface is made up of three rectangles, each 10 ft. long and with bases of 3 ft., 4 ft., and 5 ft. respec- tively. If these rectangles are placed side by side, they will make one rectangle 10 ft. by 12 ft. The figure shows what is known as the ** developed convex surface." It is seen that the length of the rectangle, which is the developed surface, is the distance around the prism ; that is, its perimeter. The area of the convex surface is the prod- uct of perimeter and altitude, or 1£0 sq. ft. The base of the prism is a right triangle whose legs are 3 ft. and 4 ft, Digitized by VjOOQIC PRISMS AND PYRAMIDS. 289 S X 4 respectively. The area of the base is ^ sq. ft. = 6 sq. ft The two bases have a combined area of 12 sq. ft. The total surface of the prism is 12 sq. ft. + 120 sq. ft. = 13£ sq.ft. The volume of the prism is the product of the base and altitude, which in this case is 60. The volume is 60 cu. ft. 2. The base of a prism is a triangle whose sides are 13, 14, and 15 ft The altitude is 30 ft. What is the convex surface ? Develop it. 3. In the above problem, the area of the base is 84 sq. ft. Find the entire surface and the volume. 4. What is the convex surface of a square pyramid, each side of the base being 12 ft., and the height 8 ft. ? Find also the total surface and the volume. Solution. In the figure which represents this pyramid, CD is 12 ft., and AO is 8 ft. BO is half of 12 ft., or 6 ft. AOB is a right triangle. AB = V^ 0^ + -B 02 = V82 + 6« = 10. The slant height of the pyramid is AB, which is 10 ft. Each of the four face triangles has a base of 12 ft and an altitude of 10 ft. The area of each is, therefore, 60 sq. ft. The area of the four is ^40 sq.ft. J the convex surface. The area of the base is 144 sq. ft. The total surface is 240 sq. ft. + 144 sq. ft. = 384 sq.ft. The volume is ^^ ^ ^ cu. ft. = 384 cu.ft. 5. Find the cost of painting an eight-sided church spire at 20 cents a square yard, the sides of the base being 6 ft., and the slant height 75 ft. 6. A building, 24 ft. square, has a pyramidal roof 5 ft. high. How many square feet of tin will cover it ? 7. Each side of the base of a square pyramid is 8 in., and its altitude is 18 ft. Find its volume. 8. The altitude of a pyramid is 15 ft., and its base is a rectangle 4 ft. x 5 ft. Find its volume. Digitized by VjOOQIC 290 MENSURATION. 9. Find the entire surface and volume of a parallelopiped 8 ft. X 5 ft. X 3 ft. 10. What is the cost of a stick of timber 21 in. square and 36 ft. long, at 30 cents a cubic foot ? CTLIin)ERS, CONES, AND SPHERES. 472. A Cylinder is a solid whose ends are equal parallel circles, and who6e lateral surface is uniformly curved. 473. When the ends are perpendicular to the curved surface, the cylinder is a Right Cylinder. (Only right cylinders will be considered here.) 474. The lateral or convex sur- face of a cylinder is equivalent to a rectangle whose length is the circumference of the cylin- der, and whose height is the altitude of the cylinder. We may imagine the surface of a cylinder unrolled and thus see that it is a rectangle, or we may take a rectangular sheet of paper and roll it into a cylinder. 475. To find the convex surface of a cylinder^ multiply the circumference of the base hy the altitude, 476. To find the volume of a cylinder^ multiply the area of the base by the altitude, 477. A Cone is a solid whose base is a circle, and whose convex (curved) surface tapers uniformly to a point called the vertex. Digitized by VjOOQIC CYLINDERS, CONES, AND SPHERES. 291 478. The 9lant height of a cone is the distance from the vertex to any point in the circumference of the base. 479. The altitude of a cone is the perpendicular distance from the vertex to the base. 480. The convex surface of a cone, if developed, is a sector of a circle. The arc of the sector is the circum- Dbveloped Sitrfacb. ference of the base of the cone, and the radius of the sec- tor is the slant height of the cone. 481. The convex surface of a cone is one half the product of the circumference of the base and the slant height, 482. The volume of a cone is one third the product of the area of the base and the altitude. 483. A Sphere is a solid every point of whose surface is equally distant from a point within it called its center. A baseball is a good example of a sphere. 484. The surface of a sphere is 4 tim^s the product of the square of its radius and ir. It is ordinarily written 4 7riJ2. 485. The volume of a sphere is | times the product of the cvhe of its radius and ir. It is generally written |> irK?^ Digitized by VjOOQIC 292 MENSURATION. EXERCISE 148. 1. A cylinder 6 in. high has a radius of 2^ in. What is its convex surface ? its volume ? Solution. The convex surface is a rectangle whose width is 6 in. and whose length is the circumference of the cylinder. Circumference = 2J x 2 x t in. = 6 x x in. Area of rectangle or convex surface = 5 x x x 6 sq. in. = 30 x sq. in. Area of base of cylinder = (2^)2 x t sq. in. = ^ x t sq. in. Volume of cylinder = ^ x x x 6 cu. in. ^ -^ x cu. in. Make this cylinder of cardboard. Make the rectangle 6 in. x 15^ in., allowing enough for lapping and pasting. Make a circle 2^ in. in radius for the base. Pattern. "V ^7 V — V 2. A cone 6 in. high has a base 2^ in. in radius. What is its convex surface ? its volume ? Solution. Before we can get its convex surface, we must know its slant height. The slant height is the hypotenuse of a right triangle whose legs are 6 in. aiid 2\ in. respectively. Slant height = V^TW in. = V^ in. = ^ = ^i in. The circumference of the base of the cone is 2 J x 2 x x in. = 6 t in. Convex surface = a sq. in. = ^ x sq. in. Area of base = (2 J) 2 «• sq. in. = ^4 ^ sq. in. Volume as ^^^ n cu. in. « ^ x cu. in. Digitized by VjOOQIC CYLINDERS, CONES, AND SPHERES. 293 Make this cone of cardboard. Pattbkn. 0^5 isif/i^xn. Fill the cylinder with bran or oats, and see how many times you can fill the cone from it What have you verified ? 3. A cylindrical cask 4 ft. in diameter is 6 ft. deep. How many gallons will it hold ? 4. How much tin will be required to line the above cask ? 5. Find the surface of a sphere 8 in. in diameter. 6. How many square inches of paper will be required to wrap 20 globes, each 4 in. in diameter, provided no allowance is made for waste ? 7. What is the volume of the largest sphere that can be cut from a cube, each edge of which is 20 in. ? a A flag pole is 80 ft. long, is 12 in. in diameter at the base, and tapers uniformly to a point. What is its volume ? 9. What will it cost to cover the pole in Problem 8 with canvas, at 12 cents a square yard ? 10. A cylindrical cistern 12 ft. in diameter holds 250 bbL of water. How deep is it ? 11. An oil vat is 50 ft. in diameter and 22 ft. high. How many barrels of oil will it hold ? 12. If iron weighs 450 lb. to the cubic foot, what is the weight of an iron ball 3 ft. in diameter ? Digitized by VjOOQIC 294 MENSURATION. MISCELLANEOUS PROBLEMS. EXERCISE 149. 1. What is the length of the sweat band in a 6^ in. hat ? 2. How high must a 3 in. tin cup be made to hold one pint ? 3. One gallon = 231 cu. in. Measure pails, tin cups, coffee barrels, and other cylinders and determine their capacity in cubic inches. 4. What is the volume of a new lead pencil ^ in. x 7 in. ? 5. Around a circular pond 500 ft. in diameter is a gravel walk 30 ft. wide. What is the area of the walk ? What is the area of the entire circle, including walk and pond ? the area of the pond only? 6. How many square inches of tin are needed to make a quart cup 4 in. in diameter, making no allowance for seams ? 7. How many barrels of 31 J gal. each will a cylindrical water tank hold, if it is 14 ft. in diameter and 12 ft. 10 in. deep, inside measure ? 8. Mercury is shipped from the mines in cylindrical steel bottles holding 100 lb. each. If these bottles are 4 in. in diameter, what must be their depth ? (Mercury is 13.6 times as heavy as water.) 9. What is the weight of a dry pine log 12 ft. long and 30 in. in diameter ? (Specific gravity of dry pine = .48.) 10. How many cubic inches are there in a cylindrical tile 12 in. long, outside diameter 8 in., inside diameter 6 in. ? 11. How many square inches are there in the entire surface of a cylindrical block 6 in. in diameter and 6 in. high ? How many cubic inches in its volume? Which is the larger, surface or volume, in a 4 in. x 4 in. cylinder ? in an 8 in. x 8 in. cylinder ? 12. The diameter of a cylindrical tank is lOJ ft., and its length is 30^ ft. How many gallons will it hold? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 295 13. Shingles are sold in bundles, each containing the equiva- lent of 250 shingles 4 in. wide. If shingles are laid 4^ in. to the weather, how many bunches must be bought for a roof 20 ft. X 30 ft. ? What is the cost of laying them at 80 cents per square ? (A " square " contains 100 sq. ft.) 14. A rug 16 ft. X 12 ft. is placed in the middle of a floor 19 ft. X 15 ft. What is the width and area of the uncovered strip? 15. What is the area of a 4 in. circle ? of an 8 in. circle ? Divide the latter area by the former. The first is what part of the second ? 16. What is the area of a 5 in. circle? of a 10 in. circle? The second is how many times the first ? 17. If the radius {R) of one circle is twice the radius (r) of a smaller circle, irB^ is how many times irr^ ? 18. How large a water-pipe is needed to carry four times as much water as a 3 in. pipe can carry ? 19. What is the total cost of the following piles of cord wood, at $ 4.75 a cord ? 1 pile, 8 ft. high, 22 ft. long. 2 piles, each 6^ ft. high, 31 ft. long. 1 pile, 9J ft. high and 32J ft. long. 20. A square cistern, whose bottom is 8 ft. on one side, is 12 ft. deep. How many gallons of water are there in it when it is I full ? 21. Change ||^ of a cubic yard to cubic feet and cubic inches. 22. What is the cost of the Brussels carpet, at $ 1.63 a yard, to cover the floor of a room that is 22 ft. long and 19 ft. wide, the strips to run the long way ? 23. How many bushels of oats will a rectangular bin con- tain that is 6 ft. long, 4 ft. wide, and 5 ft. 8 in. high ? Digitized by VjOOQIC 296 MENSURATION. 24. A farmer bought the following tracts of land; all lying in the same section: The N. \ of the S.E. \ of the S.W. \. TJie S. ^ of the N.E. \ of the S.W. \. The N. i of the S.W. ^ of the S.E. \. Draw a diagram of the section and show his purchase. What did it cost at $ 62.50 an acre ? 25. Find the cost of the posts and fencing necessary to build a four-board fence around the land just described : posts being worth 22 cents each, and placed 8 ft. apart, and the fencing being 1 in. thick, 6 in. wide, 16 ft. long, and costing $18.25 per M? 26. How many 40 gal. barrels of water will a cubical cistern contain that is 10 ft. deep ? 27. What is the volume of a prism whose base is 7^ sq. ft. and height 10 ft. ? of a pyramid of the same dimensions ? 28. What is the volume of a prism whose altitude is 6 ft and whose base is a square each of whose sides is 2 ft. ? of a pyramid of the same dimejasions ? 29. What are the volume and convex surface of a pyramid whose altitude is 6\ ft., and whose base is a square with a diagonal of 8 ft. ? 30. Find the volume of a pyramid whose base is an equi- lateral triangle, each side measuring 6 ft., and whose altitude is 24 ft. 31. Compare the volume of this pyramid with that of a pyramid of the same height, but whose base is 6 ft. square. 32. A gas receiver in a city is a cylinder 84 ft. in diameter ; on an average during the year, gas to the depth of 4 ft is consumed each night. How many cubic feet of gas are con- sumed each night ? 33. Find the length of a hand rail for a straight stairway having 20 steps, each 6^ in. high and 9 in. wide. Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 297 EXERCISE 150. 1. Multiply 5.62 by .00016 to the nearest 1000th. 2. Divide the product of 15, 18, 24, 25, and 49 by the product of 12, 21, 30, 14, and 10. 3. Find the H. C. F. of 749, 791, and 4662 by the factoring method. 4. At $1.90 a rod, find the cost of fencing a piece of ground 63.5 rd. long and bb,^ rd. wide. 5. By how much is the fraction ^ increased or diminished when 7 is added to each term ? 6. Explain why the 10th of April and the 10th of July of any year fall on the same day of the week. 7. Write the number of next year in the Roman system. 8. If the cost of sending a telegram of 10 words is 25 cents, and 2 cents for each additional word, find the cost of sending a telegram of 32 words. 9. The sum of two numbers is 1014, and the larger is 78 more than the smaller. Find the two numbers. (Use x,) 10. If I purchased an equal number of cows and sheep for <$450, each cow costing $45 and each sheep $5, find how many of each I bought. 11. Simplify : {(49 + 16) x54-20{x{95^(20-15)x4{. 12. Simplify : [49 + 16 X 5 + 20] X [95 -s- 20 + 15 X 4]. 13. Simplify : .05 -?- .005 . .07 -?- .007 .12j: -?- 6^ .004 --:04"*" 40 -5- .04 50 -^33^' 14. Two men are 1^ mi. apart, and set out to meet each other. The one walks 5 mi. while the other walks 4 mi. How far must each walk before they meet ? Digitized by VjOOQIC 298 MENSURATION. 15. A tax collector's commission was $ 360. If his rate of commission was 2%, what amount did he collect? 16. One fourth of the money received by a fruit seller was profit. What was the rate per cent of his profit ? 17. A room is 24 ft. long, and its width is to its length as 3 to 4. What is the area of the floor ? 18. The area of a floor is 720 sq. ft., and its length is to its width as 5 is to 4. Find the dimensions in feet. 19. A rectangular field contains 45 A. Its width is to its length as 8 to 9. Find its dimensions in rods. • 20. Find the square root of -^^, and express the result in the form of a decimal fraction, correct to the nearest 1000th. 21. If I sell f of a barrel of apples for what it cost, what is the rate per cent of profit ? 22. The sum of two numbers is 400, and \ of one of them is \ of the other. Find the numbers. 23. What is the reciprocal of \ carried to the nearest thou- sandth ? Note. J is the reciprocal of }. 24. How many kilometers are there in 60 mi. ? How many kilograms in 150 lb. Avoirdupois ? How many liters in 80 liq. gal. ? (See § 195.) 25. Mr. K endowed a professorship with a salary of $ 1800 per year. What sum must he invest at 5% per annum to pro- vide for this salary ? 26. A has a farm which is 200 rd. wide. If he sells 15 A. off one end, how many rods shorter is his farm than before ? 27. A note of $360, dated July 27 at 90 da., was discounted at bank on Sept. 1 at 7%. What were the proceeds ? 28. On a note of $850 at 6%, dated April 25, 1903, are the following payments : July 1, 1903, $150; Nov. 30, 1903, $240. What was due Oct. 1, 1904 ? (Use the U. S. Rule.) Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 299 29. Solvefor «: 13a; 4- 26 — 34 = 5a? + 56. ; 30. Solvefora?:g.^:^^g^"^Q = 4-g. 3 6 2 31. A coal bed 8 ft. thick covers an area equal to 400 A. Allowing 25 bu. to the ton, how many 30-T. carloads are in the bed? 32. If I sell stock at 75 that cost 80, what rate per cent do I lose ? 33. Simplify: Jl_^ + ^^^_j. 34. The list price of a piano is $460. What will it cost if the regular discount of 15% is allowed, and 10% off for cash ? 35. Find the simple interest on $ 750 from March 26, 1901, to Nov. 10, 1904, at 5%. 36. Write a promissory note so that it shall be negotiable ; so that it shall not be negotiable. 37. A banking company purchases 3 800,000 city bonds at a premium of 2^%. They sell ^ of them at 5% premium, \ of them at 4^% premium, and the remainder at 4% premium. If^ it costs i% to handle them, find the profit to the bank. 3a The list price of a piano is $540. A dealer purchases it at a discount of 40% from list. At what price must he sell it to make 25% on cost to him ? 39. $720 is my annual income from 4^% stock. What is it worth at $120 per share ? 40. Eive gallons of pure milk should weigh how many pounds and ounces ? 41. What is the length of an edge of a cube whose surface is 18 sq. ft. and 54 sq. in. ? 42. I sell ^ of my farm. After deducting agent's commission, $ 240 at 5%, I gain 20% on the part sold. Find cost of farm. 43. Show that 144 lb. Avoirdupois are equivalent to 175 Ib^ Troy. What is the ratio of a pound Avoirdupois to a pound Troy? Digitized by VjOOQIC 800 MENSURATION. 44. What is the ratio of an ounce Avoirdupois to an ounce Troy? 45. An empty bottle weighs 4200 dg., and when filled with . water it weighs 1.288 Kg. How many liters does the bottle hold? 46. Simplify: V^xf^^^- 192 30 47. One iron bar is 10% longer than another. What per cent shorter is the second than the first? 48. A rectangular field is 200 m. long, 18 dm. wide. How many hectoliters of water would fall on it in a rainfall of 3 cm. ? 49. A can do a piece of work in 8f hr., A and B together can do it in 4^ hr., and A and G can do it together in 4 hr. How many hours will it take B and G to do the work? 50. How much will it cost to plaster the walls and ceiling of a room 27 ft. long, 15 ft. wide, and 12 ft. high, at 25 cents a square yard, allowing 432 sq. ft. for doors and windows? ^ 51. Find the circumference and area of a circle whose diam- eter is 2 ft. 4 in. 52. I have a bin 36 ft. long, 10 ft. wide, and 9 ft. high, which is filled with shelled corn. A commission merchant sells it for me at 24^ cents a bushel, charging 1^ % commis- sion, and remits to me the balance. What is the amount of his remittance? 53. I own 30 shares of Building and Loan stock upon which I pay 50 cents a share monthly. After paying for 4 yr., I draw out my investment and receive $936. What rate of interest do I receive ? Note. Find average time. 54. A vessel sailed from a port directly on a line of latitude a certain distance, then sailed due north a certain other dis- tance, when the captain found his chronometer 40 min. slow. In what direction had he first sailed and how many degrees? Digitized by VjOOQIC MISCELLANEOUS PROBLEMS. 301 65. 4J^ in. is what decimal of a rod? 56. What is the value of 8 sills, each 10 in. x 10 in., and 16 ft. long, at $ 19.60 a thousand feet? 57. A man bought a horse and a carriage for $280. f of the cost of the carriage was f of the cost of the horse. What was the cost of each? 5a My farm is the northeast quarter of a section. How many acres does it contain ? I paid $80 an acre for it. What did it cost ? It has diminished in value 8 %. What is its present value? 59. For the use of the above farm my tenant pays me \ of the oats raised, ^ of the corn, and $4.75 an acre for meadow and pasture. Last year the S.E. \ was sowed in oats, the W. ^ was planted in corn, and the K.E. \ was meadow and pasture. The oats yielded an average of 51 bu., and sold for 18 cents. The corn yielded an average of 54 bu., and sold for 25 cents. The taxes and repairs were $ 200. What per cent of its present value did it pay? 60. How many barrels (31^ gal.) will a cylindrical cistern hold, its diameter being 8^ ft. and its depth 10 ft.? 61. A delivery pipe 3 in. in diameter has what percentage of the capacity of a pipe whose diameter is 3| in. ? 62. A man whose watch shows Chicago time finds that it is 27 min. 36 sec. slower than local time. What is his longitude? 63. A schoolroom is 15 ft. x 60 ft. x 72 ft. The ventilator is 2 ft. X 2 ft. What must be the velocity of the air in feet per second, to change the air in 8 min. ? 64. The above schoolroom is lighted from the long sides. If the window space is 10% of the floor space, how many 4 ft. X 9 ft. windows are there on each side ? Digitized by VjOOQIC 802 GENERAL REVIEWS. 65. Put the following in the form of a bill, supplying names: 36 collars at 16f cents; 6 shirts at $1.50; 12 pairs cuffs at 25 cents; 15 handkerchiefs at 30 cents; 8 pairs hose at 45 cents ; 8 ties at 35 cents ; 3 suits underwear at $ 2.50 ; 1 hat at $2.50; 2 pairs gloves at $ 1.25. 66. What is the cost of an article which is sold for $225, after a deduction of 10% from the marked price, it having been marked so as to gain 33 J % ? GENERAL. REVIEWS. EXERCISE 151. I. 1. Define Multiplication; Denominator; Decimal Fraction; Interest; Sphere. 2. What common fractions can be changed to pure deci- mals? Why? 3. 824»=? ^ a- 1 vc 1.18 3.64 «.. Simplify _x—. 5. Derive a rule for dividing by a fraction. 6. Find the interest on $824.60, at 7%, from June 12, 1901, to Sept.. 5, 1904. ,7. How many gallons of water will a cylindrical cistern hold whose diameter is 1\ ft., and depth 9^ ft. ? 8. Change f of a square mile to integers of lower denom- inations. 9. What is the cost of the lumber and posts to fence the N.W. I of the S.W. \ of the K.E. J of a section with a four- board fence, posts 8 ft. apart and costing 23 cents each, fep^cing at $ 18;50 per M ? 10. What is the difference between seven hundred and two thousandths and seven hundred two thousandths ? Digitized by VjOOQIC GENERAL REVIEWS. JJQ3 II. 11. Divide 3744 into three parts in the proportion of h h ft- 12. Define Ratio; Proportion; Commission; Notation; Division. v' 13. How many square feet of sheet iron J of an inch thibk can be made from a cylindrical shaft 20 ft. long and 4 in. in diameter ? 14. If 52 men can dig a canal 355 ft. long, 60 ft. wide, and 8 ft. deep, in 15 da., what will be the length of a canal 45 ft. wide and 10 ft. deep, which 45 men can dig in 25 da. ? 15. At 90 cents per yard, how much will it cost to carpet a room 20 ft.x27 ft. with carpet 2\ ft. wide, allowing. 1 ft. waste on each cut for matching? . . 16. How many gallons of liquid will a hollow sphere hold whose inner diameter is 22 in. ? 17. A 45 ft. ladder placed between two poles reaches one of them 24 ft. from the ground, and the other 28 ft. How far apart are they ? 18. What is the difference of time between two places whose difference of longitude is 46° 18' 46" ? 19. If ^Zy of an acre of land cost f 33|, what will 36| A. cost ? 20. A,, B, C, and D together own a tract of land 2 mi, square. A owns f as much gs B; B | as much as C; C | as much as D. How many acres has each ? (Let x = D's part.) m. 21. Define Antecedent; Consequent. -i- 7 18\. 3 22/ . , 2a Find the h. C. M. of 174, 485, 14,065. 22 M-^25 ■ 63 Digitized by VjOOQIC 804 GENERAL REVIEWS. M. Give the rule for " pointing " the quotient in division of decimals, and give the explanation of it 25. Change 18 mi. 124 rd. 4 yd. to feet. 26. A public square is surrounded by a walk 2 rd. wide. The area of the walk is an acre. What is the area of the square ? Make a figure. 27. For what amount shall a 90 da. note be made in order that the proceeds shall be $358.60, interest at 7% ? (Bank Discount.) 28. State the three general types of problems in commercial discount. 29. -^ is what per cent of -^ ? 30. Find the cost of the following at $ 17.50 a thousand : 24 studs 2 X 4, 18 ft. long. 32 joists 2 X 10, 16 ft. long. 8 siUs 8 X 10, 14 ft. long. 1520 ft. common fencing. IV. 31.. Define Subtraction ; Minuend; Subtrahend; Bemainder. 32. Divide 83,600 by 37 J. Multiply 664 by 83 J. 33. Give tests of divisibility by 3, 4, 8. 34. If a schoolroom is 15 ft. high, how many square feet of floor must it have to furnish 60 person 300 cu. ft. of air ? If the length is to the breadth as 4 to 3, what is each dimension ? 35. At what advance must goods be marked so that the mer- chant may discount the marked price 20 % and 5 ^, and still make 14 % ? 36. Forty men agree to do a piece of work in 50 da., but after working 9 hr. a day for 30 da. only half the work is completed. How many additional men must be employed to finish the work on time by putting in 10 hr. a day ? Digitized by VjOOQIC GENERAL REVIEWS. / 306 37. What is the capacity in barrels (31|^ gal.) of a cylindrical cistern whose diameter is 10 ft and depth 12 ft. ? (ir = ^. Employ cancellation.) 3a V860473.02986 = ? 39. Find the value of the following lumber at 9 21 a thousand : 4 6x8 sills, 16 ft. long. 26 2 X 8 joist, 18 ft. long. 30 2 X 4 studs, 22 ft. long. 18 2 X 6 rafters, 20 ft. long. 40. Find the premium on the following policies of insurance: $ 2100, at IJ %. 9 28000, at 2| %. 9 3150, at f %. 41. When do you conclude that a number is prime? Why ? 42. The proceeds of a note for $265.50, discounted on June 12, 1891, at 7%, were $263. When was the note due ? 43. The proceeds of a 90 da. note for $480 were $470.08. What was the rate of discount ? (With grace.) 44. A merchant bought goods for $729. What must they be marked in order that he may give a trade discount of 10%, lose 10% on bad debts, and still gain 10% ? 45. A circular piece of land 16 ft. in diameter is to be divided into 3 equal parts, the inner part being a circle, and the second and third parts being circular strips. What is the diameter of the inner circle ? What is the width of each of the circular strips ? What is the area of the whole circle ? What is the diameter of the inner circle ? 46. Give two ways of changing a common fraction to a decimal. Change yf^ to a decimal, and explain each step. Digitized by VjOOQIC 806 V GENERAL REVIEWS, 47. A can do a piece of work in 2f da. ; B, in 3| da. ; and C, in 4^ da. How long would it take them to complete the job, working together ? If $ 6 is paid for the whole work, what is the share of each? 48. When it is noon in Boston, what time is it at San Francisco ? 49. At what rate must 4% bonds be purchased to yield b\% on the investment ? 50. Write the tables of linear, square, and cubic measures. VI. 51. A man traveled at the rate of 3 mi. 165 rd. an hour. How far did he go in 36 hr. ? 52. How many revolutions will a wheel, whose diameter is 4J ft., make in rolling 3 mi. ? 53. A man walks a certain distance at the rate of 4^ mi. an hour, and rides back at the rate of 7^ mi. an hour. If it takes him 8 hr. to go both ways, what is the distance ? 54. Find the cost of 25 pieces of scantling 5 x 3^ in., 16 ft. long, at $ 10.25 per M. 55. Solve the following by cancellation : If 15 men in 12 da. of 10 hr. each can dig a ditch 180 rd. long, 6 ft. wide, and 4 ft. deep, how many hours a day must 10 men work, to dig a ditch 200 rd. long, 8 ft. wide, and 2 ft. deep in 10 da. ? 56. Add : 5 A. 120 sq. rd. 21 sq. yd. 6 sq. ft. 12 A. 96 sq. rd. 18 sq. yd. 7 sq. ft. 22 A. 83 sq. rd. 25 sq. yd. 4 sq. ft. 17 A. 74 sq. rd. 28 sq. yd. 8 sq. ft 57. Find the interest on $ 1580, at 7J %, from Dec. 18, 1902, to May 2, 1904. 58. What is the value of a pile of wood 360 ft. long, 12 ft wide, and 6 ft. high, at $3.20 a cord ? Digitized by VjOOQIC GENERAL REVIEWS. 307 59. Sent to a commission merchant $894 to be invested after deducting his commission at 3%. What is the amount of his commission ? 60. What must be the depth of a cylindrical vessel whose diameter is 6 in., in order that it may hold a gallon ? vn. 61. Put the following items into the form of a receipted bill : R. D. Smith bought of Cole Bros., Newark, N.J., on June 1, 1904, 16 yd. silk, at $ 1.85. June 13, b% yd. cotton cloth, at 9 cents. June 15, 8 yd. broadcloth, at $ 2.25. June 20, 24 yd. carpet, at 96 cents. July 1, 31 yd. matting, at 40 cents. July 12, 5 sets curtains, at $ 3.85. 62. A man bought a house and lot for $ 5088. f of the cost of the house was ^ of the cost of the lot. What was the cost of each ? (Let x = cost of the lot.) 63. At $ 4.75 a cord, what is the total cost of the following piles of cord wood ? 18 ft. long, 6 ft. high. 23 ft. long, ^ ft. high. 17 ft. long, 7 ft. high. 64. Change 2 rd. 4 yd. 2 ft. to the decimal of a mile. 65. What must be the rate of taxation in a town to yield a net return of $ 16,660, if the real estate is assessed at $ 531,000, the personal property at $200,182.80, 7% of the tax being uncollectible, and the collector's commission being 2% ? 66. In what time will f 469.50 yield $ 36.80 at 7% ? 67. What is the volume of a sphere whose diameter is 1\ in.? 68. If a block of stone 18 in. long, 4 in. wide, and 2 in. thick, weighs 12 lb. 15 oz., what is the weight of a block of the same material 2^ ft. long, 2 ft. wide, and 9 in. thick ? Digitized by VjOOQIC 808 GENERAL REVIEWS. 69. If it costs $120 to build a wall 40 ft. long, 14 ft. high, 1 ft. 6 in. thick, what will it cost, at the same rate, to build a wall 180 ft. long, 21 ft. high, and 1 ft. 3 in. thick ? 70. A lake whose area is 45 A. is covered with ice an inch thick. Find the weight of the ice in tons, if a cubic foot weighs 920 oz. Avoirdupois. vin. 71. 2 is what part of 5J ? ^ is what part of fj ? .015J is what part of .6f ? 72. Add^,fof5i,f-^|,^x3i. 73. Give the rule for " pointing " the product in Multiplica- tion of Decimals, and explain it. 74. Change 61,368 sec. to integers of higher denominations, and give the reasons for two reductions. 75. ^J^ of an inch is what part of a rod ? 24 76. What number multiplied by jj will give 2 for a product ? 77. Change i^ of a mile to integers of lower denominations. 78. What will it cost, at 24 cents a square yard, to plaster a hall 46^ ft. wide, 82 ft. long, and 24 ft. high, no allowance being made for openings ? 79. Bought 42 shares of stock at 105^, received a 4% dividend, and sold the stock at 103|. The gain was what per cent of the investment ? 80. A railroad train moves a mile in 66 sec. What is its rate per hour ? IX. 81. What is the area of a triangular piece of land whose base is 124 rd. and whose altitude is 236 rd.? Digitized by VjOOQIC GENERAL REVIEWS. 309 82. Find the number of acres in the following tracts of land: a. N. ^ of S.W. ^ of a section. 6. S. i of N.W. J of S.E. f c. S.W. J of S.E. J of KW. 1. Make a figure showing each tract. 83. A is 26 rd. 4 yd. north of C. B is 13 rd. ^ yd. east of C. What is the distance from A to B ? 84. From ^ take the sum of | x |i and | x ^, and divide the result by 2^. 85. How many 40 gal. barrels will a cylindrical cistern hold, of which the diameter is 9^ ft. and the depth 10 ft. ? 86. Which is the better investment, 5% bonds at 98, or 4 % bonds at 95 ? Give the rate per cent of interest on each investment. 87. What is the per cent of gain if ^ of an article is sold for 1^ of its cost? 88. Find the cost of 8 $100-bonds, bearing A\(fo interest, which yield an annual income of 6J% of the investment. 89. Bought 15 railroad bonds at 1\ ^o discount, brokerage If %. For what must a 90 da. note be drawn, interest at 8 %, to obtain the amount of the purchase at a bank? (With grace.) 90. Find the square root of 7 to within .001. X. 91. Define a ratio. Define each term of a ratio. What is the difference between a ratio and a fraction? Define a pro- portion, and each term. By what principle is any term of a proportion found when three are given ? 92. A can dp a piece of work in 6 da.; B, in 7; and C, in 8. In what time can they do it working together? Digitized by VjOOQIC 310 GENERAL REVIEWS. 93. What is the capacity of a hollow sphere whose oatside diameter is 15 in., and whose walls are ^ of an inch thick? 94. A man bought four articles for $896.45. For the first he gave $21 more than for the second ; for the second, $62.50 more than for the third; for the third, $81.75 more than for the fourth. What did each cost? (Let aj = the cost of the fourth article.) 95. Sold 160 acres of land at $ 87.50, commission 2\ % - di- rected the agent to invest the proceeds in 5% bonds at 98, reserving his commission at 2%, and returning the surplus of less than $ 100. How many bonds did he purchase, and how much did he return ? 96. 'V|"=? 97. The interest on two sums of money for 4 yr. and 8 mo. at 6 % was $256. | of the first sum equaled the second. What was the interest on each sum ? (Use a?.) 98. A steamer can sail 10 mi. an hour with the current, and 5 mi. an hour against it. What is the rapidity of the current ? How long a trip up stream and down can it make in 6 hr.? 99. Find the volume of a cone whose altitude is 15 in., and radius of base 3^ in. 100. Sent to a commission merchant $894 to be invested after deducting his commission at 3%. What is the amount of his commission ? XI. 101. Give the laws of the Roman l^otation. 102. \ of A's money equals f of B's. If A has $ 65 more than B, how much has each ? (Let x = B's money.) 103. How many rolls of paper are needed to cover the walls and ceiling of a room 16 ft. by 18 ft., and 11 ft. high, deductions being made for 3 windows 3 ft. 2 in. by 7 ft. 4 in., 2 doors 3 ft. by 8 ft. 2 in., and a 10 in. baseboard ? Digitized by VjOOQIC GENERAL REVIEWS. 311 104. Find the value of a pile of cord wood 7 ft high and 36 ft. long, at $ 5.25 a cord. 105. A room is 15 ft. by 18 ft., and 10 ft. high. What is the length of a line extending from an upper comer diagonally through the room to an opposite lower corner ? 106. What is the diameter of a circle containing 20 A. of land ? What is the area of a strip 18 ft. wide, lying next to the circumference and reaching around the field on the outside ? 107. i±i+i2iix--5^=? I-I 1^ + f H 106. What is the weight of a stone roller 8 ft. long and 8 ft. in circumference (ir = ^), the specific gravity of the stone being 2.3? 109. What is the weight of a grindstone 4 in. thick and 30 in. in diameter, the hole being 2 in. in diameter, the specific gravity of the stone being 2.143 ? 110. How many bushels of corn in the ear will a crib hold that is 46 ft. long, 8 ft. wide, and 10 ft. high, counting the bushel at f of true capacity ? XII. HI. What is the face of a 60 da. note, the proceeds of which are $ 2654.38 when discounted at a bank at 7% ? (With grace.) 112. A and B can do a piece of work in 24 da. A can do | as much as B. In how many days can each do it alone ? 1 2 Let X = the time it takes B. - = part B does in 1 da. — = part A does in Ida. ^ ^* 113. A rectangular field contains 12^ A. Its width is | of its length. What is the distance around it ? (Let x = length.) 114. After paying | of a debt and | of the remainder, I owe $ 430.37^ less than at first. What was the debt at first ? Digitized by VjOOQIC 312 GENERAL REVIEWS. 115. Find the cost of the Brussels carpet for a room 20 ft X 24 ft., the strips running the long way, 8 in. being lost on each strip in matching, and costing $ 1.12 a yard. 116. A woodhouse is 16 ft. x 16 ft. If filled with wood to a height of 7 ft., what is the wood worth at $ 5.50 a cord ? 117. By selling an article for $ 565, 1 gain 13%. What did it cost me ? 118. How many gallons of water will a hollow sphere hold whose interior diameter is 3^ ft. ? 119. A piece of land in the form of a trapezoid is 120 rd. between its parallel sides, one of which is 45 rd. long, and the other 60 rd. long. What is the land worth at f 62.50 an acre ? 120. A merchant sold a customer 7 pieces of cloth, each con- taining 50 yd. He made a reduction of 20% from the retail price, and a further reduction of 5% for cash. The retail price was 40% above cost. He received $532. What was the retail price per yard ? XIII. 121. A tank can be filled by keeping one pipe open 4 hr., or by keeping a second pipe open for 5 hr. The tank has a pipe by means of which it can be emptied in 2^ hr. In what time will the tank be filled if the three pipes are left open ? 122. A farmer has a 40 A. field in the form of a square. He has it planted in corn, the rows being 3 ft. 6 in. apart. The first row is 3 ft. from the line. How far does he walk in plowing it once, taking a row at a time ? 123. Five men in a factory accomplish as much as 8 boys. What part of a man's work does a boy do ? Change this result to per cent. What per cent of a boy's work does a man do ? 124. The diameter of a cylindrical tank is 10^ ft., and its length is 30^ ft. How many gallons will it hold ? Digitized by VjOOQIC GENERAL REVIEWS. 813 125. I spent 26% of my money, 33^% of the remainder, and 8^% of the remainder. I then had $550. How much did I have at first ? 126. A man was paying rent at the rate of $ 15 a month. He borrowed an amount from a Building Association to build a house. He made a monthly payment of $18 for 6 yr., when his house was paid for. How much more than his rent did the house cost him, counting interest on his money at 6%? 127. A house IS 30 ft. by 40 ft. The cistern connected with its roof is cylindrical in shape, 9 ft. deep, and has an average diameter of 10 ft. At the end of a rain the cistern was found to be half full. How many inches of rain had fallen ? 128. Find the cost of lumber and posts for fencing the E. ^ of the S.E. J of a section with a four-board fence, posts costing 18^ cents, and lumber $ 19.25 per thousand. 129. How many bushels of oats will a bin contain that ii 10 ft. high, 8 ft. wide, and 32 ft. long ? 130. How many shares at 83, brokerage ^%, can be pur. chased for $4571.88? Digitized by VjOOQIC Digitized by Google APPENDIX. HULTIPUCATION TABLE. 2 3 4 6 6 7 8 9 10 11 12 4 6 8 10 12 14 16 18 20 22 24 6 9 12 16 18 21 24 27 30 33 36 8 12 16 20 24 28 32 36 40 44 48 10 15 20 25 30 35 40 45 50 55 60 12 18 24 30 86 42 48 54 60 66 72 14 21 28 35 42 49 66 63 70 77 84 16 24 32 40 48 56 64 72 80 88 96 18 27 36 46 54 63 72 81 90 99 108 10 20 30 40 60 60 70 80 90 100 110 120 11 22 33 44 66 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 The factors are in the top row and the left column. The product is at the intersection of row and column. LINEAR MEASURE. 12 inches (in.) =1 foot (ft.). 3 feet = 1 yard (yd.). 5^ yards = 1 rod (rd.). 320 rods = 1 mile. 1760 yards = 1 mile. 5280 feet = 1 mile. 315 Digitized by VjOOQIC 816 APPENDIX. a SQUARE BfEASURE. 144 square inches = 1 square foot. 9 square feet = 1 square yard. 30 J square yards = 1 square rod. 160 square rods = 1 acre (A.). 640 acres =1 square mile. 36 square miles = 1 township. Note, In land surveys a square mile is called a section. 4. CUBIC M£ASUR£. 1728 cubic inches = 1 cubic foot. 27 cubic feet = 1 cubic yard. 16 cubic feet = 1 cord foot. 8 cord feet ) ^^^ , . „ }• = 1 cord (wood). 128 cubic feet ) ^ ^ 100 cubic feet = 1 cord (stone). 5. SURVEYORS' LONG MEASURE. 7^ inches = 1 link. 100 links = 1 chain. 80 chains = 1 mile. 6. SURVEYORS' SQUARE MEASURE. 1 square chain = 16 square rods. 10 square chains = 1 acre. 7. LIQUID MEASURE. The units of Liquid Measure are the gallon, the quart, the pint, and the gill. The primary unit is the gallon^ which con- tains 231 cu. in. The other units are divisions of the gallon. Digitized by VjOOQIC APPENDIX. 317 4 gills (gi.) = 1 pint (pt.). 2 pints = 1 quart (qt.). 4 quarts = 1 gallon (gal.). Note. The pint is divided into 16 equal parts, each of which is called an ounce. This measure is used by apothecaries. The teacher should show a 1 oz., a 2 oz., and a 4 oz. bottle. In computing the capacity of tanks and cisterns, the barrel of 31^ gal. is the unit. 8. DRY MEASURE. The units of Dry Measure are the bushel, the peck, the quart, and the pint. The primary unit is the btLshd, which contains 2150^ cu. in. The other units are divisions of the bushel. 2 pints = 1 quart. 8 quarts = 1 peck (pk.). 4 pecks = 1 bushel (bu.). Note. The standard bushel is 18} in. in diameter and 8 in. deep. It contains 2150{ cu. in., or nearly IJ cu. ft. 9. AVOIRDUPOIS WEIGHT. 16 ounces (oz.) = 1 pound (lb.). 25 pounds = 1 quarter (qr.). 4 quarters = 1 hundredweight (cwt.). 100 pounds = 1 hundredweight. 20 hundredweight = 1 ton (T.). Note 1. Avoirdupois weight is used in weighing ordinary articles of merchandise. Note 2, The long ton (2240 lb.) is used in the United States custom houses and in some states in weighing coal and iron. 28 lb. = 1 qr., long ton. Note 3. 62J lb. Avoirdupois = 1000 oz. = the weight of a cubic foot of distilled water. Note 4. 196 pounds of flour = 1 barrel. 200 pounds of pork or beef = 1 barrel Digitized by VjOOQIC 318 APPENDIX. 10. TROY WEIGHT. 24 grain (gr.) = 1 pennyweight (pwt.). 20 pennyweight = 1 ounce. 12 ounces =.1 pound. Note, Troy weight is used in measuring gold, silver, precious stones, jewels, etc. The Troy pound is little used. Gold and silver bullion are sold by the ounce; gold ornaments by the pennyweight; jewels by the carat (3.2 grains). The word carat is also used in the sense of twenty-fourths in stating the purity of gold. Gk)ld 14 carats fine is Jf gold, JJ alloy. 11. APOTHECARIES' WEIGHT. 20 grains = 1 scruple (3). 3 scruples = 1 dram (3). 8 drams = 1 ounce ( S ). 12 ounces = 1 pound (lb). Note. Apothecaries^ weight is used in mixing medicines. The metric system is rapidly displacing Apothecaries* weight in pharmacy. 12. COMPARISON OF WEIGHTS. The Troy pound is divided into 6760 equal parts callecl grains. 7000 grains equal a pound Avoirdupois, and 5760 grains the pound Apothecaries'. 1 pound Troy = fj^^ = |4t of 1 lb. Avoirdupois. 1 ounce Troy = -^^ = \^ of 1 oz. Avoirdupois. The ounce and pound Apothecaries' equal the ounce and pound Troy, respectively. Digitized by VjOOQIC APPENDIX. 819 13. APOTHECARIES' FLUID BfEASURE. 60 minims (m.) = 1 fluid dram (fl. 3). 8 fluid drams = 1 fluid ounce (fl. |). 16 fluid ounces = 1 pint (0.). 8 pints = 1 gallon (cong.). 14. EN6USH MONET. 4 farthings (qr.) = 1 penny (d.). 12 pence = 1 shilling («.). 20 shillings = 1 pound (£,). £ = libra = pound, d. = denaritts, Latin for " penny." qr. = quadrans = fourth. Note. The Troy pound of silver was originally coined into 240 silver pennies of 24 gr. (1 pwt.) each. A cross was stamped so deep that the penny was readily broken into fourths (farthings). The present value of the pound sterling is $4.8666. The gold coin of this value is called the sovereign. The shilling is coined of silver ; the penny and half-penny are of copper. The guinea {£l8.) and crown (5«.) are no longer coined. English gold coins are 22 carats fine. 15. FRENCH MONET. The franc, worth 19.3 cents in United States money, is the unit. The scale is decimal. 10 millimes (m.) = 1 centime (c). 10 centimes = 1 decime. •10 decimes = 1 franc (f .). 16. GERMAN MONET. The unit is the mark, or reichmark, worth 23.8 cents in United States money. 100 pfennige (pf.) = 1 mark (RM.). Digitized by VjOOQIC 320 APPENDIX. 17. TIME BfEASURE. 60 seconds (sec.) = 1 minute (min.). 60 minutes = 1 hour (hr.). 24 hours = 1 day (da.). 7 days = 1 week (wk.). 365 days = 1 common year (yr.). 366 days = 1 leap year. 100 years = 1 century. Note 1, If the year is not the last in the century, and if its number is divisible by 4, it is a leap year. If it is the closing year of a century, it is not a leap year unless its number is divisible by 400. Note 2. The months containing 30 da. are April, June, September, and November. The months containing 31 da. are January, March, May, July, August, October, and December. February contains 28 da. in a common year, and 29 da. in a leap year. 18. CIRCULAR MEASURE. 60 seconds (") = 1 minute C). 60 minutes = 1 degree (**). 360 degrees = 1 circumference. 19. COUNTING. 12 ones == 1 dozen (doz.). • 12 dozen = 1 gross. 12 gross = 1 great gross. 20 ones = 1 score. 20. PAPER MEASURE. 24 sheets of paper = 1 quire. 20 quires = 1 ream. 2 reams = 1 bundle. 5 bundles = 1 bale. Digitized by VjOOQIC APPENDIX. 321 21. A sheet folded in 2 leaves is called a folio. 4 leaves is called a quarto, or 4to. 8 leaves is called an octavo, or 8vo. 12 leaves is called a duodecimo, or 12mo. 16 leaves is called a 16mo. 32 leaves is called a 32mo. 22. IIARIKERS' MEASURE. 6 feet = 1 fathom. 120 fathoms = 1 cable length. 80 cable lengths = 1 mile. 23. THE INTERNATIONAL DATE LINE. Travelers crossing the Pacific Ocean westward set their time forward a day on crossing the 180th meridian. Islands in the equatorial portion of the Pacific, having been colonized by Europeans coming from the east with the trade winds, have the same reckoning as the American continent, although some of them are west of the 180th meridian. Australia, New Zealand, and the neighboring islands originally colonized by the Dutch have the time of Asia, one day in advance of those mentioned above. On many charts is shown the International Date Line separating these lands. It passes through Behring Strait, thence southwest, east of Japan, but west of the Philippines, thence east, southeast, and south to the east of JTew Zealand. Prior to 1867 this line passed east of Alaska. 24. DIVISIBILITY BY 9. A number is dicisihle by 9 if the sum of its digits is divisible by 9. To understand this test examine the nature of the decimal system. Digitized by VjOOQIC 822 APPENDIX. 10= 9 + 1. 20 = (2x 9)4-2. 30= 27 + 3. 100 = 99 + 1. 200 = (2 X 99) + 2. 300 = 297 + 3. 1000 = 999 + 1. 2000 = (2 x 999) + 2. 3000 = 2997 + 3. From this partial table it is evident that 1 of any order exceeds some multiple of 9 by 1; 2 of any order exceeds some multiple of 9 by 2 ; hence, we may say : A digit in any place expresses a number that exceeds some multiple of 9 by as many ones as the digit expresses when standing alone. Since a given number is the sum of the numbers expressed by its several digits (349 = 300 + 40 + 9), it follows that any number exceeds some multiple of 9 by the sum of the ones expressed by its separate digits. If this sum is a multiple of 9, the given number is the sum of multiples of 9 and is, therefore, divisible by 9. (Prin. 2, § 81.) 25. DIVISIBILITY BY 11. A number is divisible by 11 if the sum of the ones expressed by the digits in the odd orders equals the sum of the ones expressed by the digits in the even orders, or if the difference of these sums is a multiple of 11. The following statements will make this test clear : (1) One ten is one less than a multiple of 11 ; two tens are two less ; and any number of tens are as many less as there are tens. A similar statement may be made for thousands, hundred- thousands, ten-millions, etc. But tens, thousands, hundred- thousands, ten-millions, etc., occupy orders whose numbers counting from the right, are even. Hence, a digit standing in an order whose number is even, expresses a number which is as many less than a multiple of 11 as the number of ones expressed by the digit. Digitized by VjOOQIC APPENDIX, 323 (2) In a similar manner it may be shown that a digit stand- ing in an order whose number is odd expresses a number which is as many more than a multiple of 11 as there are ones expressed by the digit. (3) If these remainders are equal, they balance each other, and the number is a multiple of 11. If one set of remainders exceeds the other by a multiple of 11, it must follow that the whole number is a multiple of 11. 26. FINDING THE G. C. D. BY DIVISION. Illustrative Example. Find the G. C. D. of 91 and 325. FOBM. 91)325(3 273 * 52)91(1 52 39)52(1 39 13)39(3 39 Explanation. The G. C. D. of these numbers cannot be greater than 91. If 91 will divide 325, it is the G. C. D. of 91 and 325. The quotient is 3, and the remainder 52 ; hence, 91 is not their G. C. D. Since a divisor of a number is a divisor of any of its multi- ples, the G. CD. of these numbers must be a divisor of 273. Since a divisor of two numbers is a divisor of their difference, the G. C. D. must divide 52 ; hence, it cannot be greater than 52. Since 52 is a divisor of itself, if it will divide 91, it will divide 273, by Prin. 1, § 80, and 325, by Prin. 2, § 81. The quotient is 1, and the remainder 39 ; hence, 52 is not the G. C. D. sought. Since the G. C. D. of 91 and 325 must divide 52 and 91, it most divide 39, by Prin. 3, § 82 ; hence, it cannot exceed 39. Digitized by VjOOQIC 824 APPENDIX. If 39 will divide 62, it will divide 91, by Prin. 2; 273, by Prin. 1; and 325, by Prin. 2. The quotient is 1, and the remainder 13 ; hence, 39 is not the G. C. D. sought. Since the G. C. D. must divide 39 and 62, it must divide 13, by Prin. 3. Since 13 will divide itself and 39, it will divide 62, by Prin. 2 ; 91, by Prin. 2; 273, by Prin. 1 ; and 326, by Prin. 2 ; hence, 13 is the G. C. D. of 91 and 326. Rule. To find the G. C. D. of two or more numbers : Select two of the numbers and divide the greater by the less, and the less by the remainder, if there is one. Continue the process until there is no remainder. The last divisoi' wiU be the O. C. D. sought. Compare this divisor with a third number, proceeding in the same m>anner, and thus continue until aU of the numbers are dis- posed of, Note. Observe that this method discovers numbers that are smaller than the given numbers, and yet that have the same G. C. D. EXERCISE. Find the G. C. D. of the following: 1. 340 and 678. 2. 333 and 703. 3. 633, 697, and 779. 4. 1266, 1870, and 8613. 5. 7944, 12,247, an,d 13,902. 27. COBIPOUND INTEREST. 1. A man borrowed f 350 at 7% interest, agreeing that if the interest was not paid at the end of the first year it should be added to the principal to make a new principal for the second year, and that the interest should be added thus each year until the debt was paid. If he paid nothing until the end of 3 yr. and 3 mo., how much was then due ? Digitized by VjOOQIC APPENDIX. Solution. Ist principal $360 1st year's interest 24.60 2d principal 9374.60 2(1 year's interest 26.216 3d principal ♦ 400.716 8d year's interest 28.049 4th principal $428,764 Interest for 8 mo. 7.602 Amount $436,266 1st principal 360 Interest 186.26 825 2. Compoiind Interest is interest upon a principal that is in- creased at regular periods by its accumulated interest. 3. Interest may be compounded at the end of any period agreed upon, instead of annually, as above. Savings banks usually allow compound interest upon deposits, crediting the interest semiannually. Rule. To calculate compound interest : 1. At the end of each period increase the principal for that period by the interest accumulated during the period, 2. From the final amount subtract the first principal. EXERCISE. Find the compound amount and interest: 1. Of $628, for 3 yr., at 6%. 2. Of f 1200, for 4 yr., at 8%. 3. Of f 1680.60, for 2 yr., at 10%, compounding quarterly. r~-A. Of f 2660, for 3 yr. 8 mo. 26 da., at 6%, compounding semiannually. Digitized by VjOOQIC 326 APPENDIX. TABLE. Showing the amount 0/ $1 at compound interest from 1 year to 10 years^ at 3, 4, 4J, 6, 6, and 7 %. Years. 8%. 4%. 4i%. 5%. 6%. 7%. 1.030000 1.040000 1.045000 1.050000 1.060000 1.070000 1.060900 1.081600 1.092025 1.102500 1.123600 1.144900 1.092727 1.124864 1.141166 1.167625 1.191016 1.225043 4 1.125509 1.169859 1.192519 1.215506 1.262477 1.310796 1.159274 1.216653 1.246182 1.276282 1.338226 1.402552 1.194052 1.265319 1.302260 1.340096 1.418519 1.600730 1.229874 1.315932 1.360862 1.407100 1.503630 1.605781 1.266770 1.368569 1.422101 1.477455 1.593848 1.718186 1.304773 1.423812 1.486095 1.551328 1.689479 1.838459 10 1.343916 1.480244 1.552969 1.628895 1.790848 1.967151 Illustrative Example, Find the compound interest of $ 100 for 8 yr. 4 mo. 12 da., at 5%. Solution. Amount of J 1 for 8 yr. at 5 %, Amount of $ 100, Interest of $ 147.74 for 4 mo. 12 da., Amount, Compound interest, $1.4774 100 $147.74 2.70 $160.44 100.00 960.44 EXERCISE. With the aid of the table, find the amount and compound interest : PrincipaL Time. Rate. 1. $760 4yr. 6 mo. ^% 2. $ 5000 9 yr. 7 mo. 10 da. 6% 3. $1275.46 8 yr. 2 mo. 18 da. 7% Digitized by VjOOQIC APPENDIX. 327 28. ANNUAL INTEREST. 1. Annual Interest differs from compound interest in one particular. Interest as it accrues does not draw compound interest, but simple interest. 2. If a note provides that interest is payable annually, it means that unpaid interest at the end of any period shall draw simple interest until paid. A problem will make the difference clear. Illustration, A note of $400 is due in 4 yr. 6 mo. The interest at 7% is payable annually. If nothing is paid until the note is due, what will the interest amount to ? The interest on $ 400 for 4 yr. 6 mo. =$126. The $ 28 due at the end of the first year draws interest for 3^^ yr.; the $28 due the second year, for 2^ yr. ; the third, for 1^ yr. ; and the fourth, for ^ yr. There will then be due, in addition to the $ 126, the interest on $ 28 for 3^- yr. + 2\ yr. -h 1* y r. -h ^ yr. = the interest for 8 yr. = $ 15.68. $ 126 + $ 15.68 = $ 141. 68. Make a rule from the above analysis. EXERCISE. Find the interest : Principal. Time. Bate. 1. $350. 3 yr. 6 mo. 6 da. 6% 2. $400 4 yr. 7 mo. 9 da. 6% 3. $480 2 yr. 8 mo. 12 da. 7% 4. $610.40 4 yr. 10 mo. 16 da. 7% 5. $560 6 yr. 6 mo. 18 da. 7% 6. $85.50 3 yr. 11 mo. 19 da. 7% 7. $128.20 4 yr. 2 mo. 21 da. 8% 8. $649 3 yr. 4 mo. 24 da. 8% 9. $763.60 6 yr. 7 mo. 20 da. 5% 10. $840 6 yr. 1 mo. 1 da. 5% Digitized by VjOOQIC Digitized by VjOOQIC ANSWERS TO PROBLEMS IM THE NEW ADVANCED ARITHMETIC. Exeroise 6. a 43,949. a 105. 13. 61. 4. 666,140. 9. f 133 gain. 14. 2 nickels and 5. 438,840. 10. $3275. 4 cents. 6. 567 A. U. 780. 15. $9.11. 7. 184 mi. ; C 18 mi. 1 , 12. $5412.25. Exercise 8. 1. $293.50. 6. 50,496 lb. U. 32 da. 2. 1024 ft. 7. 84,000 words. 12. 12 bu. 3. 92,628,000 mi. 8. $380. la 90^. 4. 25,200 lb. 9. $78. 14. 215. 5. 21,824 lb. 10. 9 da. Exercise 9. 1. 54. a 200. la 26. 24. 10. 2. 37. 9. 74,613. 17. 12. 25. 8000. 3. 100. 10. 73,649. la 110. 26. 1. 4. 12. u. 40,457. 19. 154. 27. 7. 5. 20. 12. 952. 20. 8. 28. 11. 6. 40. 13. 54. 21. 5. 29. 120. 7. 2. 14. 686. 22. 3. 30. 1. 15. 5. 23. 64 Exercise 12. 1. 151 pt 4. 1277 sq. ft. 7. 96,000 oz. 2. 744 hr. 5. 660 ft. a 25,517 min. a 77,482". a 9424 cu. ft. COFTSIGHT, 1908, BT QlLTKR, BlTHPlCTT AJ»P OOMFAlfT. Digitized by VjOOQIC 2 ANSWERS. Exercise 13. 1. 5 bu. 1 pk. 2 qt. 1 pt. 4. 8 wk. 21 hr. 2. 42 gal. 2 qt. 1 pt. 5. 3^ 29' 27". 3. 80 lb. 9 oz. Av. 6. 5 mi. 267 rd. Exercise 14. 1. 26 T. 1542 lb. 8 oz. 16. 6 yr. 7 mo. 6 da. 2. 86 rd. 1^ yd. or 86 rd. 17. 9 yr. 9 mo. 20 da. 1 yd. 1 ft. 6 in. 18. 6 yr. 8 mo. 25 da. 3. 6 bu. 3 qt. 1 pt. 19. 6 yr. 3 mo. 13 da. 4. 16da.l9hr.29min 40 sec. 20. 8 yr. 7 mo. 21 da. 5. 67 yr. 5 mo. 12 da. . 21. 613A.137sq.rd.4isq,yd. 6. 3 yr. 1 mo. 15 da. 22. 7 cu. yd. 21 cu. ft. 1476 cu. in 8. 2** 56* 33". . 23. 179 lb. 6 oz. 9. 13^ 34' 57". 16** 31' 30". 24. 23 cd. 5 cd. ft. 12 cu. ft. 25. 2 lb. 4 oz. 3 dr. 4 gr. 10. 887 mi. 26. 56° 53' 12". 11. 103** 27' 11". 27. 4 gal. 1 qt. 1 pt. 12. 1** 38' 41". 2n 1 46 yr. 5 mo. 23 da. 13. 9** 9' 24". aO> ■ 13 yr. 8 mo. 28 da. 14. 52** 51'. ' 29. 4 lb. 10 oz. 11 pwt. 17 gr. 15. 8 yr. 2 mo. 4 da. 31. April 30, 1927. Exercise 15. 18 gal. qt. 1 pt. 4. 2 qt. 1 pt. 1. 29 gal. 5. 3bu. 3pk. 7qt. 32 gal. 2 qt. 1 pt. 6. 3 gal. 2 qt. 2 gi. 43 gal. 2 qt. pt. 7. 443 da. 6 hr. 8. go J, ^^n ^ 2. 55 yd. ft. 8 in. 78 yd. 2 ft. 8 in. 56 yd. 2 ft. 4 in. 9. 10. 11. 293 gal. 2 qt. 1 pt 918 bu. 3 pk. $460.46. 701bu. Ipk. 12. 647 hr. 14 min. 36 sec. 3. 1122 bu. 13. 1237° 59' 52". 1776 bu. 2 pk. 14. 7192 lb. 6 oz. 2290 bu. 3 pk. 15. $1180 gain. Digitized by VjOOQIC ANSWERS. Exercise 16. 1. 127^ + 235c + 786s. 11. S2x + 19y + 32z. 2. 59^ + 63c + 891s. 12. 32a + 196 + 22c. 3. 38^ + 53cH-227s. ' 13. 39a^-|-37a6. 4. 18^ + 370c + 79«. 14. 3a?. 5. 42x-\-19y-\'llz + lSw. 15. 7a. 6. 37a-f-256 + 41c + 23d 16. 9y. 7. 4a-|-166 + 2c. 17. 12x + 2y, 8. 3a;+133^ + l«. 18. 5a + 76. 9. 27 a; -h23y + 62 «. 19. 12aj + 32^-hl3». 10. 4a + £ l6 + 6c. 20. 6xi/ + 7ab. Exercise 19. 1. 18 A. a 31 ft. 89| sq. yd. 14. 21 in. 2. 134,640 cu. in. 9. 33 ft. 10 rows. 15. 400 cu. yd. 3. 60 id. 10. 32 rd. long, f 870. 16. 6,670,000 gf. 4. 9 ft. 11. $16. 19. 16 cu. ft. 5. 896 cu • yd. 12. 840cu. in. 20. 4641b. 6. 28 in. 13. 210 cu. in., 8 21. 2 men. 7. 27 in. layers, 8 in. 22. 2880 lb. Ezerciae 21. 1. 2. 3. 4. 5. 2,3,17. 3, 67. 7,43. 2,2,2,2,6,6.2,2,6,5,6. 3, 6, 7. 7,29. 2, 2, 2, 2, 19. 13, 31. 3, 6, 6, 7. 2,2,3,3,3. 7,31. 2, 6, 31. 11, 37. 23, 23. 2,2,2,3,6. 13, 17. 11,29. 7,61. 7,7,11. 6, 5, 6. 13, 19. 17, 19. 19, 23. 11, 63. 2, 3, 5, 6. 2, 6, 6, 6. 7,7,7. 11,41. 6,7,17. 2, 2, 2, 19. 7,37. 19,19. 7,67. 2,2,2,3,6,5. 2, 2, 41. 7,41. 7,53. 11,43. 13,47. 6, 5, 7. 17, 17. 2,2,5,19. 13,37. 17,37. 2, 3, 31. 13, 23. 17, 23. 7, 71. 7, 7, 13. c Digitized by VjOOQIC 4 ANSWERS. 6. 7. 8. 2, 5, 5, 13. 3, 6, 6, 11. 2, 2, 2, 2, 2, 2, 3, 6, IL 3, 3, 3, 5, 5. 7, 7, 17. 2, 2, 2, 2, 23, 43. 2, 2, 5, 6, 7. 23, 37. 2, 2, 6, 11, 113. 19, 37. 11, 79. 6, 6, 6, 5,89. 17,43. 29,31. 2, 2, 2, 5, 5, 5, 73. 7, 107. 2, 2, 3, 3, 6, 6. 2, 2, 2, 2, 2, 5, 5, 5, 5, 5. 13, 69. 7, 131. 2, 2, 3, 3, 7, 13, 37. 17, 47. 2, 2, 6, 47. 2, 6, 6, 7, 17, 48. 2,2,2,2,2,6,6. 2, 6, 6, 19. 2,2,2,2,2,2,3^/2,2,2,5,6,5,6. 2, 2, 3, 67. 3, 6, 6, 13. 2, 2, 2, 2, 2, 2, 5, 6, 5, 5, 6, 6. Exercise 22. 2. 180. 9. 6 da. 15. 108 da. 3. If 10. 16 men. 16. 12 men. 4. 30. 11. 143 T . 17. 7hr. 5. 108. 12. 30 A. 18. 5. 6. 4650. 13. 3 da. 19. 24. 7. 32 baskets. 14. 4 horses. Exercise 23. 20. 52. 2. 16. 4. 390. 8. 9,9,2. 3. 3860. 5. 190. 9. 20,90,270. 10. 30,20,1600 . 15. 76 posts. $11.40. 11. 31,33,24,17. 16. 14 ft. 12. 120,190,130,1700. 17. 15 mi. 13. 12 ft. 18. 24 mi. 14. $9.60. 19. 8 da., 30 da., 11 da. Exercise 24. 8. 1926,5705,4242. 9. 239,850; 5,100,480. 10. 61. 16. 6300. 21. 32,760. 26. 2J6. 11. 632; 633. 17. 2520. 22. 1200. 27. 900. 12. 216 mi. 18. 3080. 23r-l0^0. 28. ^ 382,200. 13. 2640 ft. 19. 17,472. 24. 840. 29. 20,735. 15. 1200. 20. 221,676. 25. 408,408. 3a/ 144. Digitized by VjOOQIC ANSWERS. Exercise 25. X 68,868. 11. $102.40. 2. 2,2,2,3,3,7,11. 12. 126 bbL 3. 1792. 4. 1. 13. 7. 5. 76 A. 67 sq. rd. 14. $3300, $460 gain. 6. 18 cords. 15. 12 yr. 7. 14doz. 16. 13. a 27 sheets. 17. 16. 9. 271. la m- 10. 32 rd. 19. 26a! + 30y + 30«. 20. 154,440. 22. $80. 2«. 748,226. 26. 11,669. 21. 2741. 23. 117. 25. 2618. 27. 7. [2,3,4,5,6,7,10,12. 31. .1866 bu. 36. $620. 2a 14, 15, 20, 21, 28, 30. 32. $834.36. 37. $141.06. . 35, 42, 70, 84, 106, 140, 210. 33. $120. 38. $686.29. 29. 2 >60 sq. yd. 34. $219. 39. $71.60. 30. ] L45,800. 35. $73. 4a $207. Exercise 27. 2. 32*»22'47". a 74*^55' 19". 4. 149''14'43". 5. 90** 6' 33". Exercise 28. 2. 6 hr. 17 min. 11 sec. 5. 4 hr. 16 min. 3^ seo. 3. 7 hr. 31 min. 13 sec. 4. 27 min. 47^ sec. 6. 3 min. 15 sec. 7. 10 hr. 50| sec. Exercise 29. 2. 66^8' 45"; A East. 3. 100** 4' 30" ; B East. 8. 4. 29^ 59' 45"; A East. 5. 180** ; A East. 9. 6. 165^6'; BEast. 10. _ f 1 hr. 51 min. 42^ sec. P.M. 11. [ 3 hr. 40 min. 53^ sec. A.M. 1 hr. 4 min. 22^ sec. a.m. 22 min. 38^ sec. a.m. 6 hr. 50 min. 11^ sec. A.M. Calcutta. San Francisco. 51** 15' W. Digitized by VjOOQIC 2. V. 3. -^. *• ¥• 6. ^. 1. ff. 2. ^. 7. W- a 1^. 10. 1^. U. A^. 5. <^. a V- Ezerciae 32. 12. H**- 14. W- 15. iffl. la J^. 17. ^. 18. 4*. 19. S^. 20. -4^. 21. Aft. 22. -Z^JL, 23. i|A, 24. 4^. 28. 4|i, 26. 4ftL, Exercise 33. 7. i^. 10. ^. 13. i^. la ifji. a ij^. 11. Vtf- 1*- W- 17. 2^. 9. ^. 12. w- "• HF- 18- HF 2. 4f, 6i, Hi. a 16, 9f , 4. 4. 6A, 6ff, 9f a 6i, 6tI^, 6. a 6i, 7, 6^. 17. ^ ft., 9 j ft. la 1^ yd., 13f yd. Exercise 35. 7. 8t3^> 12,^, 13^. 12. 5, 3|f|, 5. a 17M, 23, 22,%: 13. 2iff,2ff^,2H*. 9. 7^, 6f|, 6H. 14. 4, 4, 7. 10. 6f^, 6H, 7. 15. 9, 5ff. 8- 11. 6^, 6, 9. la 122H, 109^. 19. 173 dimes; $18^. 20. -4^ lb., 12f lb. Exercise 36. 3. h H. H. i- «• h h A. f 9- 4, H. li- 4. i, i, H. I- 7. I, i, I, «. 10. H. M- »• I. h h f 8. f, f, I, f 11. «J,«|,$i,ff 12. Jyd., iyd., ^ yd., ^yd. la ^ bu., f bu., f bu., ^ bu. 14. ^ mi., f mi., ^ mi., | mi Exercise 37. a ii- a M- "• !%• 20. m- 26. VWif. a H- 9. ^. 15. Hf 21. m- 27. MH- 4. a- 10. tf. la VW- 22. m- 28. H«. a ^^ u- H 17. ,>^. 2a irW 29. mi- a ^■ 12. M- la Hi- 24. m- »• MM- 7. H- la T^iV. la m- 25. iWr- »• sW- Digitized by VjOOQIC Exercise 38. 1. 1. 4. lA- 7. u. 10. 1^. 2- li^- 5. ih 8. H- XL f 3- M- & lA. 9. 2. Exereise 39. 12. 2,1^. 1. IH. 9. liM- 17. 2^ . 24. 2m- 2. 2i. 10. 2^4- 18. 2HI- 25. 3|. 3. IH- 11. 2^' 19- iM*- 26. 2^. *• 2jh- 12. 3^. 20. 4^. 27. 2^. 5. IfH- 13. 2W*- 21. 2Hf 2a 3H. «• Siiyff. 14. SH- 22. 2||e 29. 3^. 7. 1^. IS. 3HM- 23, 3fH. ; 30. 3fi*; 8- m- 16. 2WM- Exercise 42. /\ 2. 31^. a 1203^. 14. 18293^ gr. 3. 159m- 9. 560t|. 15. 2i4iH gr. 4. 499ff. 10. 392t!4V- 16. S86^ A. 5. 307f|. 11. 404f|yd. 17. 1247^ lb. 6. 641^. 12. 7'6{iit. 18. 148i j T. 7. 2012^. 13. 39^ lb. Exercise 43. 19. 73J mi. 1- tV 9. T^- 17. ^. 24. i. 2. «. 10. A- la ^. 25. Mf. 3- if- 11 i*- 19. A- 26. m- 4. ii- 12. H- 20. f f . 27. i4. 5- M- 13. lAr- 21. tV- 28 ^. 6- H- 14. M- 22. ^. 29. WV- 7. ^• 15. H- 23. A. 30. hhs- 8- M- 16. tW- Digitized by VjOOQIC J ANSWERS. Exercise 45. a 26A. 6. 282|f. 9. 28H- U. 344,V^ 4. 48H. 7. 23^. lO 49^. 12. 248AW. 5. 42H. a 45|^. Exercise 46. 2- 2t. a 6f . 13. 4^. la 360^. 3. 3f. 9. 6^. 14. 43*. 19. 131f. 4. 4^. lO 6^. IS. 91*. 20. 1644^. 5. 6i. U. 8|. 16. ISZT^r. 21. 2590*. 6. 6f 12. B^. 17. 608|. 22. 3913*. 7. 3^. Exercise 47. a. 2f 10. %. 17. If 24. 6*. 31. 4*. 3. 3^. 11. 2f 18. 3*. 25. 3|. 32. 26f. 4. 5J. 12. f 19. 4*. 26. 66*. 33. If 5. % 13. H- 20. 2|. 27. 31f. 34. 23^. 6. 2f 14. 5f 21. 5*. 28. If. 35. 4^. 7. 9i. 15. 41- 22. 2f. 29. 17. 36. 3^. a 5|. 16. 29|. 23. 3*. 30. 5*. 37. 26A^ 9. 12|. Exercise 48. 1. 2|. S. 4i- 9. 2^. 13. 70*. 16. 2m- 2. H. 6. 9*. lO |. 14. 4*. 17. 147|. 3. f 7. 19*. 11. 47*. 15. 4*. la 426*. 4. 1^. 8. 20f. 12. 4*^. Exercise 49. 2. 7i. a 18f 14. 72*. 20. 97*. 26. 66,^. 3. 5f 9. 26|. 15. 24*. 21. 31*. 27. 46*. 4. 22. 10. 38i. 16. 20*. 22. 113f. 28. 137,^ 5. 22^. 11. 32|. 17. 66*. 23. 39^. 29. 113*. 6. l^. 12. 53|i. 18.. 46^. 24. 56f 30. 95 J|. 7. 33^. 13. 16^. 19. 21^. 25. 7^. 31. 9o|. Digitized by VjOOQIC ANSWERS. f Exercise 50. 2. 10. 7. $16. 12. 8f 17. 21f 22. 80f. 3. 12. B. 6^. 13. 12. 18. 66. 23. 126. 4. 18. 9. 12 A. 14. 9. 19. 26. 24. 1015. 5. 26. 10. 6|. 15. 9f . 20. 23^. is. 147. 6. 4bu. U. 6. 16. 25. 21. 134. Exercise 51. 3. |. 6. f. 9. f . 12. AV M- 15. ♦• f- 7. |. 10. f|. 13. If. 16. '22}. «• H- 8- H- 11. If 14. Exercise 52. ^' 3. 128. 14^. $22.16|. 22. 1422f bu. 4. 376. 15. $66.86. [249f; 5. 6028. 16. $88.57i|. 23. 784|i; 6. 29,238. 17. TIT' ii' 9083|i; 7. 53,682^. 18. H^T- 1 60,783^. 8. 96J. 19. ^m- 24. 2190|mi. 9. mu- $12.60; 25. $286.16^. 10. 2369f. 20. $11.10; 26. $10.23J^. U. 615,395f| ^ $60.^0. 27. $28.46. 12. $51^. 21. $ 12.63i. 2a 25fsq. ft 13. lOH^. Exercise 53. 3. A. 9. A. 15. A-. 21. ^. 26. A. 31. H- 4. ^. 10. 7^. 16. A- 22. A. 27. ^. 32. H. 5. ^ 11. A- 17. A- 23. ^^. 2a rk- 33. M- 6. ^. 12. A. 18. f 24. ^. 29. dhr 34. 'tIt 7. ^. 13. Tty. 19. f 25. ii. 30. i^. 35. ItV a A. 14. iSo- 20. A- Digitized by CjOOQ iC 10 iUVrSFTfifiS. ExerciM 54. 1. A. 2- A- 3- «• • 6. 2f 7. If. la 9. A. 12. Exercise 55. Hi. 47A la s^. 14. 67iV a. A- 3. ^. 7. V^. e. ^. XL 9. ^. 12. 10. ^. 13, Exercise 56. A- 14. f 15. T^^. 16. 4«. 3. 9f 4. 14f 5. llf la 47,3^. 19. 66. 20. 38. 6. 20. 9. 2|. 12. 7. 2J. 10. 2^. la a 3|. u. l^f. 14. 26. 42 da. 34. If 27. 295V A. 35. ^. 2a 32 horses. 36. 1^. If 6A. 15. 20. 16. 26i. 17. 34f. 42. d^hr. 43. 3iyd. 44. 22 bags. 21. 12 boxes 22. 14|yd. 23. 156 hr. >. 29. A. 30. 2^. 31. f 37. |. 3a H- 39. f ««• SOtMt A. 47. 1. 24. 20 apples. 32. 2^. 25. 28 yd. 33. 1|. 50. |:f 51, 40. If 41. 11^ lb. A; A; Tk- 52- 4a 2A. 4a 1t%. 7i; 24. Exercise 57. 1. H. 2. |. 3. A 4. if. 6. 400. 7. 1. a 11. 9- If. Exercise 58. 10. ||. u. f *• A» -hf ^> Tf^» A» tV- a 1T> tt> A* f Digitized by VjOOQIC ANSWERS. 11 11. (1) (2) (3) (4) (5) (6) (7)||. (8)«. (10) f*. (11) i¥^. (12) ^»r- (13) ^. (14) ^. (15) M- (16) ff . (17) H- (18) fH. (19) *f* (20) H. (21) VW. (22) M • (23) |. (24),Wr. 2. 64. 3. 64. 4. 100. . 5. 93. 22. 81. 23. «16^. 6.. 192. 7. 95. 8. 265. a 1170. 24. Exercise 59. 10. 1680. 11. f 12. f . 13. 18. $336 lot; $ 560 house 14. 15. 18. 2H. 15. 100. 19. 200. 16. 100. 20. 56^. 17. 100. 21. 104^^. 25. 640 A. 26. $1284.80. 1. 2^. 3. 2. 58^^. 4. $3080; $880 eldest; $ 840 second ; I $ 1360 youngest, r $ 4097 house ; • \ $ 1687 lot. r A, 152 A. I B, 399 A. llf^. 26. 151^11^ bu. 27. 10^; 14. 15. 21. 22. 23. 24. 25. mi irfr- 28. 29. Exercise 61. 338^. 5. 543^3^ bu. 7. $181.00. 4500. a 19529f 8. 208f mi 10. $40. 11. 6272. 12. 1440. 13. $1505|. 17. $21 J. 18. 24 da. 19. $651.14^. 20. 83^) 75T5 1 8 6 ft 5 iftOS' ft207* f II I * T80 J 16. 4034f bu. 2241^ bu. 1891i bu. 1 1440^ bu, i^Hf. \m or Hi; ai- 20 7 30. >^ft* 16,^ ft. Digitized by VjOOQIC 12 ANSWERS. aa 3,^ or im; «• tt- 6380i; 3^. 4,1 Mi *♦• 2A- 4a 13,684Hi f»200; VX. A; 45. 674,1^. Ia- 39. $400; ^. 46. 3485V,,. [27|i; $1200; n; f2iVW; 49. 12H; $4800. 42. ] 16|; 47. 1. 96M; lOf*- 4a 69ifi. [36^. 118^. SL i P 10.26. S2. $4187. 53. 8240 bu. 54. 4668^ bu. Exercise 63. 3. .6, .15, .043, .0028, .467. 4. .29, .921, .0007, .00008, .000004, .000000006, .0038, .04562. 5. .000419, .00306, .0096, .00008168, .000059001. [307.000307. 6. 23006.040007, 29000.029, 29.029, 307000000.000307, 7. 37.9, 58.24, 69.708, 5.24896. a .00003017008. 10. 8.008. 9. .0700, .710, .00009002, .009200. 11. 23.0061. Exercise 64. 3. *• u. m^. 19. *. 4. lAr- M. 3^. 20. ^ 5. iftV M. 3A. 21. 1- 6. A. 14. 29^ . 22. |. 7. m- !*• h T^> ihs' 23. f 8. ■fgV 16. Ihh-h- 24. f 9. 2 5 0* 1^- A» T8W» A* 25. H' 10. Tinr* IS- ^,^,^' Exercise 65. 26. 6*. 3. .25. 11. .3125. 19. .04. 27. .024 4. .126. 12. .0626. 20. .18. 2a .232. 5. .76. 13. .15. 2i; .74. 29. .425. 6. .376. 14. .36. 22. .02. 30. .08. 7. .626. 15. .05. 23. .84375. 31. .04. 8. .875. 16. .28. 24. .09376. 32. .3576. 9. 1.875. 17. .44. 25. .359375. 33. .004. 10. .1876. la .76. 26. .015626. 34. .0066. Digitized by VjOOQIC 3. .0833. 4. .4166. 5. .5833. 6. .9166. 7. .3888. ANSWERS. Exercise 66. 18 a 9. .3214. 10. .4857. 11. .8854. 12. .1428. 13. .4285. 14. .5714. 15. .7142. 16. .8571. 17. .5909. la .5555. 19. .3076. 20. .2941. 21. .9473. 22. .0434. 1. 1.32043. 2. 222.10524. 3. 1231.805114. Exercise 67. 4. 5818.476545. 5. 377.896412. 6. 1843.4890516. 7. .75902231. a 6.0010518. 10. 1.721Ji. 1. .0566. 2. .2238. 3. .04254. 4. 3.11823. 5. 6.36282. Exercise 68. a 374.49041566. 7. .022535. a 91.93521. 9. 399.99996. 10. .0057717. 12. .42695. 13. .00817|i. 14. 8.024175. 15. .7582||. Exercise 69. 3. .315. 15. .000009. 2a \. 4. .347557. la 72. 29. 354.24. 5. .434016. 17. .068. 30. $1,910,376. a 1.599192. la 9.2. 31. f 25,977.60. 7. .13611798. 19. .000072. 32. $47.56. a .00005431722. 21. h^' 33. $1.81f 9. 2.3729. 22. if. 34. $426.9815625 10. 1027.31808. 23. i. 35. $21. 11. 4570.58588. 24. h 36. 39.875 yd. 12, 3.1411391748. 25. f 37. $186,354. 13. .02304. 2a f. 3a 63,5001b. 14. .00063. 27. ^nf Digitized by VjOOQIC u ANSWERS. 1. .1094. 2. .00756. 3. .017448. 4. 5.95063. 5. 9.1296. 6. .001786. 7. .000227. a .000236 + 9. .076479 + 10. 5.6624 + 13. .79. 5. 9. Exercise 70. 14. 36.7. 15. 46.2. 16. 43.9. 17. 7.38. la 3.07. 19.. 40.7. 20. 4.007. 21. 5820. 22. 60,800. 23. 87,000. 24. 5,300,000. 25. 970,000,000. 26. 406,000,000. 27. 615,271,428^. 2a .01627. 29. .079310. 30. .6494. 31. .000303. 32. .00682593. 33. $5. 34. 80^. 35. 6^.. U. 12. 13. Exercise 71 3. 25,709,853,000 cu. cm. 10. 25,709.853 cu. m. 257,098.53 dst. 2570.9853 Dat. 4. 4,707,300 cl. 4707.3 Dl. 47.073 KL 470,730 dl. 2,794,360 g. 2,794,360,000 mg. 279.436 Mg. 27.9436 Q. 4502.08 m. 7305.17 1., 7.30517 Kl. 14. 5.974 Mg. 15. 59,740 9. 59,740,000 mg. .75432 Hg. .075432 Kg. 16. 7.5432 Dg. 17. 754.32 dg. la 4,050,000;40.5H1.;810H1. 19. 2,150,000,000 cu. mm. 2.15 stares. 314 cu. dm. 3.14 HI. .314 cu. m. .000314 cu. Dm. 75.413 1. .75413 HI. .075413 cu. m. 34.25 g. .3425 Hg. .03425 Kg. 3426 eg. 2400 Kg. ' 34.72945 cu. m. 34.72945 T. 34,729,450 g. 34,729,450,000 mg. 79,123.234 M. 1419.240 g., 7.68756 g., 179.7704 g. . 64.8 Kg. 20. $68.32. Digitized by VjOOQIC ANSWERS. 15 Exercise 72. 2. 250oz. 12. $336,440.99. 23. 23.84 cu. cm. 3. 22501b. 14. 55.5 g. 24. 1.0656 Kg. 4. 172.51b. 15. 115.5 g. 25. 6901b. 5. 688^^11 oz. 16. 57.6 Kg. 26. 162.5 lb., 30 lb. 7. 552.96 cu. in. 17. 6.3 Kg. 27. 113^ lb., 8.5841b 8. 150fjoz. 18. 1.125 Kg. 28. 371.25 gr. 9. 1.2. 20. 500 cu. cm. 29. 14.0625 lb. 10. 15^j lb. 21. 600 cu. dm. 30. 1829^ lb. 11. 65.7951b. 22. 2cu.dm. 31. 6208.85+ lb. Exercise 73. 1. ^. 15. 19 pair. 30. 1. 2. .28125. 16. 73.5 yd. 31. If. 3. 3424.608 sq. ft. 17. 42 lb. 32. 0. 4. 34.327 + da. 19. 360. 33. 2. 5. .0125. 21. 102|; 2140; 34. 1. 6."$ 222.545. 77.056. 35. 16ff|. 7. 473.5 bu. 22. 99j^; 91||; 38. 384 bu. 8. 69.154. . 62f|f. 39. 832V ft. 9. .01%. 24. 21|; tL- *<>• 8240 bu. 10.^. 26. 4,'i,i^;f,?,A, 41. 45683i^bu. U. A- ^,i^. 42. fllO. 12. .638.^ 27. 9.309 gal. 43. $142.59. 13. $3,275. 28. 6.657 cu. ft. 44. $45. 14. $79.18. 29. 1,1^. Exercise 75. 1. 50 hr. 22 min. 36 sec. 2. 8 hr. 23 min. 46 sec. 3. 6,22. 8. 898.76 bu. 13. $576. 4. 14,11. 9. $136.61. 14. 10 hr.; 150 mi 5. 13f ^ 10. 880 A. 15. 3flir. 6. 49fbu. 11. 120 A. 16. 134|| A. 7. 4514.48 bu. 12. $15.80f 17. 37 and 37, Digitized by VjOOQIC 16 ANSWERS. la 19. 20. 21. 22. 23. 24. 49. 50. 55,440. !.60 to A ; ; 10.80 to B. 13. «210. r«3.€ Uio. 25. $18.59. 26. $573. 29. Yhj, y}-^, -j-Jy. 30. I, A. f$2250; $460; 31. ^, -^. I $1500; $300. 32.705. $22.67. 42. 43. 44. 45. 46. 47. 4a 3.5. 6. ^- ih 2 640* 33. ^. 34. ^, |-J, ff. 35. 36. 37. 3a 40. 41. 1 02 Af TTff> TIT; T7y> 1 10 0> TT15> 1080' 2112IT' 14060' 2 6fi0* 2 40 0> OftoOO* .8787. .0125. .0125, .01875, .0625, .0329, .1595, .2517. 1 mi. 302 rd. 2 yd. 1 ft. Tm. 2 sq. rd. 7 sq. yd. f sqfft. 29 sq. in. 290 A. 145 sq. rd. 13 sq. yd. 6 sq. ft. 108 sq. in. 19 cu. ft. 493f cu. in. 1 mi. 72 rd. 2 yd. 5| in. 8 A. 150 sq. rd. 7 sq. yd. 5 sq. ft. 9 sq. in. 51. 9f 53. .36. 55. 7. 57. 2. 52. 10. 54. .27f 56. 1.5625. 5a 2^. 3. 7. 15. 10. Exercise 76 3. 8. 6. 2. 9. 4. 10. 5. U. 12. 6. 2. Exercise 77. 15. 16. 11 yr. $12,000, 1st; $6000, 2d; $3000, 3d. 8, Richard ; 16, Thomas ; 48, Henry. 60, John's marbles ; 20, Fred's. 18 in all. 39 yr. 91b. 60 rd. 60 rd. IT^ft 33 ft. 17. la 10. 5 dimes, 15 nick- els, 75 cents. ' 11 yr., Ethel; 11. 13 yr., Mary; 18 yr., Grace. 14. $20, A's; $12, B's. 3. [8 mi., Walter; \ 10 mi., Harry. Digitized by VjOOQIC ANSWERS. 17 Exercise 78. 12. 20. 15. 16. la 40. 21. 106. 24 60. 13. 6. 16. 21. 19. 60. 22. 40. 25. 24. 14. 12. " 17. 36. 20. 60. 23. 15. 26. 10. Exercise 79. 1. 20,70. 4. 32yr., A's , 6. Any age. 9. 48 ft. 2. 28^, 71f . 20 yr., B's. 7. 42 yr. 10. 47 ft 3. »64. 5. 12 yr. a 26 gal. Exercise 80. 1. $2896.64. a 326. 14. 335. 2. »120,325,li55. 9. » 280.30. 15. $36,347.50. 3. $148,492.11. 10. 728 bu. 16. $10.38. 4. $6.75. U. $212.85. 17. $383.72. 5. $68,375. 12. $4227.876. la 158 rd. 4 yd. 6. $4125.336. 13. $91f 19. j^. 7. $5332.47. 20. 47bu.0pk. 6qt. 21. 307 mi. 299 rd. 3 yd. 1 ft. 31. 649. 6 in. 32. 47. 22. 15 mi. 211 rd. 1 yd. 2 ft. 33. 2cu.yd.23cu.ft.524cu.in. 5 in. 34. 36. 23. 10 sq. rd. 6 sq. yd. 2 sq. ft. 35. 3 mi. 147 rd. 1 yd. 3.6 in. 60 sq. in. 36. .0000631. 24. 479 A. 110 sq. rd. 22 J sq. yd. 37. ^^. 25. 41 J lb. 38. 33 mi. 297 rd. 1 yd. 2 ft. 7 in. 26. 36 bu. 1 pk. 7\i qt. 39. 24. 27. 187^^ bbl. 40. 12. 2a 1715 A. 134 sq. rd. 18 sq. 41. 16 gal. 2 gi. yd. 4 sq. ft. 72 sq. in. 42, 2804 bu. 2 pk. 1 qt. 29. 1 sec. 5 A. 71 sq. rd. 17 sq. ^>. i¥ir- yd. 7 sq. ft. 26|| sq. in. 44. 137 rd. 2 ft. 4f in. ao. 34 mi. 68 rd. 4| yd. 45. 4 yd. 2 ft. 8 in. Digitized by VjOOQIC 19 ANSWERS. 46. 232 cd. 16 cu. ft. 50. 1 mi. 123 rd. 4 yd. 2 ft. 7 in. 47. 31 cu. yd. 10 cu. ft. 472 cu. 51. 116 A. 58 sq. rd. 6 sq. yd. in. 4 sq. ft. 72 sq. in. 4a 162,502 in. 52. 4 sq. rd. 14 sq. yd. 4 sq. 49. 5309J ft. ft. 97.92 sq. in. 53. Y^. 64. .00085714. 75. 180 ; 540. 54. .16. 65. $71.03 J. 76. $10.92. 55. 291 rd. 3 yd. 66. 14 ft. square. 77. $22,000. 2 ft. 6 in. 67. 13. 7a 19 inT" square; 56. 78rd.0yd. 6a 1. 143. 57. 2230. 69. 108M2'30"W. 79. 33|f^. 5a 47|sq. yd. 70. The same. 80. Yes; 8. 59. 27 HI. 71. 60 cattle, 180 81. 7.174. 60. $18.60. sheep. 82. 55. 61. 4500 m. 72. 5026J^gal. 83. 900. 62. 4. 73. 18. 84. J. 63. 814^ 74. JOr^ 85. $57.76. 86. $8.33^. 89. 321 da. 87. 4 yr. 10 mo. 8 da. 90. 11^ hr. ; 20ff hr. 88. 2 A. 5 sq. rd. 6 sq. yd. 92. 7|hr. ^1 sq. ft. 108 sq. in. 93.' 2^ hr. 94. 7ida. 98. 7r9'30"W. iDa. 400 girls, 175 95. 60 hr. 99. li^hr. boys. 96. 10^ it. 100. -^,.35. 103. ly^. 97. ^. 101. 22 yr. 104. 7.20. 105. j%',.S. HI. a. 112. 9239.433. 106. (19 J) 20 bunches; $4.80. ua 9 hr. 61 min. p.m. 107. 1^ ft. ; 93 sq. ft. 114. ^^ A. 108. 13.26575 mi. 115. A, 14 rows; B, 21; C, 12. 109. 16 da. 116. A, $87,000; B, $210,000; no. 32|fbbl.(lbu.=fcu.ft.). C, $89,100; D, $106,900. Digitized by VjOOQIC ANSWERS. Exercise 81. 1. . 2. 3. .60 60% m 8i% 18A 18A% .80 80% .411 41|% 45A 46A% .12^ 12i% .68^ 68i% 38A 38A% .37i 37i% .21f 21f% 94^ 94^% .87i 87i% .35^ 35^% •385?r 38A% .161 16§% ■m 134% 20^ 20|% .14^ 14|% .461 46§% 794 794% .42^ 42f% >.27J 27J% \.38f 38f% \31i 31i% 034 .34% ^7+ 67|% OliS^ liiW% .71f 71f% .15 15% .^} 93|% .36 35% \ .66 65% \ Exercise 82. . 2. 26%. 10. 66f%. 17. 116|%. 25. 834%. 3. 76%. u. 124%. IS- 400%. 26. 374%. 4. 80%. 13. 624%. 19. 50%. 28. 9%. 5. 55%. 13. 374%. 20- 225%. 29. 24%. e. 72%. 14. 84%. 22. 88|%. 30. 25%. 7. 70%. 15. 14f%. 23. 106|%: 31. 75%. a 16§%. 16. 125%. 24. 40%. 32. 85%. 9. 33i%. A 33. 1st, 16|%; 2d, 334%; 3d, 14^%; 4th, 28^%; 5th, 74% 84. 60%; 33i%. 35. 16%; 4%; 24% ; 56%. Exorcise 83. 3. 33i%. 9. 1314%. 15. 21ff%. 22. 90H%. 4. 44|%. 10. 325%. 16. 3041^%. 23. 134%. 5. 6lT*r%. 11. 774f%. 17. 355f%. 24. 8%. 6. 132,!^% 12. 784^%. 18. 390|%. 25. 120%. 7. 71H%. 13. 20%. 19. 83f%. 26. ^m% a 90%. 14. 19^%. 21. 6541%. 27. 414f%. 19 Digitized by VjOOQIC 20 ANSWERS. » 31tti%. 3a 167i%. ♦7. 4i%. 56. 6200%. 29. 2mo- 39. 3m%- 4a i%. 57. 80%. 30. 20H*%. 40. 13i%. 49. 3i%. sa 81^%. 31. 187^%. 41. 10|%. 50. 76^%. 61. 20%, 26%. 32. 63^%. 42. 66^%. 51. 43|4%. 62. 20%. 33. 11H%. 43. 4000%. 52. 15H%. 63. 20%. 34. 34tV%. 44. 1M%. 53. 3i%. 64. 20%. 35. 65|%. 45. 21H%- 54. 6T«r%. 65. 12i%. 36. 60%. 46. 33U%. 55. 25%. 66. n%- 37. 284%. . Exercise 84. ♦. A. u. 2i. la TtTF* 24. YTV' 30. -jT^. 5. f U. A- 19. ^. 25. j^. 31. nW. 6. f 13 A- 20. lAr- 26. -g^. 32. :^. 7. |. 14 a- 21. A- 27. ^. a ^. 15 10. 22. Tir- 2a :^. 34. 4oita. 9. f 16, m- 23. TTFS' 29. f^. 35. gttoo- 10. f 17. li- Exercise 85. 3. 153.14. 12. .068. 21. .326. 30. 20.04. 4. 430.6. 13. .14376. 22. 19.565. 31. 20.04. 5. 689.04. 14. 1.03^. 23. 19.665. 32. 15.39876. 6. 3571.76. 15. .25. 24. 7.03. 33. 16.39875. 7. 36604.18. 16. .405. 25. 7.03. 34. 5.6016. 8. 379.665. 17. 4.572. 26. 14.756. 35. 6.6016. 9. 1077.1876. 18. 6.36. 27, 2.535. 36. 3.16. 10. 2342. 19. 5.96. 2a 39.09. 37. 316. U. 783.2. 20. 7.S8^. 29. 39.09. 3a 3067.008. Exercise 86. 2. .06. 7. .56. 12. 78.6. 17. .08i. 3. .112. a .72. 13. Th- la .13, 4. .21^. 9. 1.68. 14. .006^. 19. .16i. 5. .3192. 10. 3.04. w. A- 20. A- 6. .26f. 11. .60. 16. .04. Digitized by VjOOQIC ANSWERS. 21 Exercise 2. 38.34. 12. 5 lb. 3 oz. 6 pwt. 23. 195.6 bu. 3. 79.56. 13. 2 rd. 8 ft. 10 in. 24. $8428.66. 4. » 123.25. 14. 54 cwt. 54 lb. 25. 5%. 5. 126 bu. 15. 330 sheep. 26. '$32.77. 6. 419.62 mi. ' 16. 35 bbl. 27. $160.75. 7. 252 A. la $233.42. 28. $645. a 84 rd. 19. $61.81. 30. 1.2 yd. 9. :^; .0258; 162. 20. $8.58. 31. $1200; $300. 10. 50.42; 9. 21. $315.09. 32. 312 boys. 11. £9 68.Sd. 22. $2.43. Exercise 90. 33. 80 ft. 15. 600 rd. 32. 84 bbl. 51. 1730H bu. 16. 283 gal. 1 pt. 33. $1680.48. 52, $26,481.42. 17. $400. 34. 29 in. 54. 120,000 mi. la 78 sq. rd. 35. 37im. 55. J $3445.95; $4500. 23^ sq. yd. 36. 105,200^11,850. 19. 5 cu. ft. 37. 129,800. 56. $2864. 1360 cu. in. 38. 192.8; 96.4; 57. 1; Hi if 20. 360^ 723. 58. H- 22. 2000 A. 39. 1458 lb. 59. 200. 23. $2000. 40. 25%. 60. 54,000. 24. 1008 men. 41. 105. 61. 1650 lb. 25. 60 ft. 42. 157.5. 62. 40 mi. 207 rd. 26. 360 eggs. 43. 280 mi. 4 yd. 2 ft. 6 in 27. r 170 mi. 1 17 mi. 45. 46. If 2000. 63. 78 A. 129-J-l sq. rd. 28. $786.50. 47. 371 4|. 64. $772,163.03. 29. 67^ rd. 48. 10,300. 65. 6000 T. 30. A. 49. 73| mi. 66. 4800 T. 31. 15 lb. 9 oz. 18pwt. 50. 64HA. 67. 2780|f T. Digitized by VjOOQIC 22 ANSWERS. Exercise 91. 1. $147.70. S. 175%. 8. 11.01. 2. f 71.28. 6. $278.20; 9. 442626963. 3. ^% $6564. U. $1831.66. 4. s^%. 7. .0004007003. 12.' $197.23. 13. 32 ; Impossible, subtrar 20. 104 lb. 5 oz. 18 pwt 12 gr. hend too large b; 40. 21. ^A. 14. 169; 466. 22. U^fc. 15. I owe A $399.49 23. 324. 16. 53 mi. ; mi. ; 53 mi. ; 24. $ 594. 106 mi. 25. $5144, lot; $4601, house. 17. $41,944.88. 26. 3 lb. 3 oz. 3 pwt 12 gr. 18. .47058. 27. f 19. 126 mi. 31 rd. 3 yd .lft.4in. Exercise 93. 1. $623.10. a 65 ^,39 ^,$2.60, 14. 18|%. 2. $29,440. $1.95, 19^^, 15. 8H%- 3. 14^%. 26^. 16. $35.18. 4. $10,240. 9. 33J%. 17. 25%; $1030.76. 6. $1.06H- 10. 16^%. la $17,400. 7. $2.26 cost, 11. l^fo- 19. $172. $2.79 selling 12. 11^%. 20. $504.12. price. 13. 66^^. 21. 7421.1 bu. Exercise 94. 3. $76. 10. $10,820. 4. $31.26. U. $1219.51; $30.49. 5. $120; $6880. 12. $25.13; $1015.91. 6. $22,370. 14. $17.39; $1247.11. 8. 2%. 15. $838.09. 9. 3%. 16. $3262. Digitized by VjOOQIC ANSWERS. 23 Exercise 95. 2., f 36.70. 4. $330. 3. « 62.60. 5. « 148.60; $16.60. ' Exercise 96. 2. $1377. 6. $71.28. 10. 32^%. 15. $250. 3. $306. 7. $287.28:11. The first 16. $600. 4. $3&26, a $342.00. 12. 60% is the better $676; $682; $688. 13. $16.92. . la 22|%. 5. 19. 36^%. 20. 20%. 25. $698.64. 3b. $1167.14. 35. $U5.25. 21. $496.38. 26. The same. 31.. $684. 36. $633.67. 22. $430.14. 27. The same. 32. 6%.' 37. $648.24. 23. $621.33. 2a $628.13. 33. 5%. 38. $964.60. 24. $1229.44. 29. $944. 34. $268.96. Exercise 97. 39. $347.10. 1. $6469. a $13,027.63. 16. $17.02; $6.94. 2. $'84,017.41. 9. $9767.71. 17. 85.63^ on $100. 3. $72,697.57. Itt $904.61. la $63.92. 4. $6729.66. 12. $24.83. 19. 40.86^ on $100. 5. $14,745.19. 13. $.96; $.36. 20. $34.76 (county tax> 6. $11,618.66. 14 $4,460,000. 21. $ 86,299,1 19.6L 7. $1077. 15. $1,660,000. 22. $76,396. Exercise 1. (a) $37.60. 2. (a) i%. 3. (a) $8530. (6) $46.63. (b) Ifc (6) $20,000. (c) $62.40. (c) H%. (c) $16;616.20. (d) $103.13. (d) 24%. (d) $9930. 4 $324. 6. 2J%. a $418.75. 5. $1147.80. 7. 1.6%; $6620. Digitized by VjOOQIC 24 ANSWERS. Ezereiae 9S 1. 1. fl40^. 4. $377.20; 6. $5216.40. 2. f856. $3000. 7. $510, 3. f 18.90. S. $39496. & $3141. Ezerdfle 100. 1. .008. 9. .0032. la W;f 2. 32 min. 16 sec. 10. 12f%. 19. 1200. past 6 A,M. 11. $3.08f 20 2ii|da. 3. 31,^. 12. 67^%. 21. Ada. 4. » 7767.66. la 39" 43' 18" W. 22. 241b. 5. $348,327. 14. $49.60. 23. 12^%. 6. f 141.96. 15. $511.40. 24. $ 67.95. 7. f580. 16. 254 mi. 228 rd. 2S. 19H%. & f 48.60. 17. 12^ mills. Exercise 102. 2. $646.13. 18. $1.60. - 32. $1908.40 . 47. $5386.26 3. $736.77. 19. $11,068.13. 33. $1048.02 . 4a $8023.63. 4. $1107.26. 20. $974.78. 34. $113.27. 49. $7147.71. S. $1456.90. 21. $1507.11. 36. $4840.15 . 5O.->$^6020.31. 6. $1770.57. 22. $2120.63. 37. $737.32. 51. $66:71. 7. $3167.49. 23. $4027.66. 38. $896.10. 52. $83.99. a $6714.96. 24. $ 103.46. 39. $1099.96 . 53. $207.29. 9. $6663.38. 25. $17,087.88. 40. $1088.21 . 54. $330.11. 10. $10,416.47 . 26. $962.46. 41. $1494.33 . 55. $381.62. U. $640.67. 27. $942.84. 42. $1488.98 . 56. $476.23. 12. $4697.48. 28. $2077.79. 43. $1728.06 . 57. $666.83. 13. $1042.18. 29. $1960.20. 44. $2697.36 . 5a $843.82. 15. $7672.68. 30. $26.46. 45. $2880.38 , 59. $766.39. 16. $107.30. 31. $768.73. 46. $4688.91 . 60. $2134.90. 17. $17.93. Digitized by VjOOQIC ANSWERS. ^S Ezeroiae 103. X. f 118.06. 14. $843.04. 26. $822.73. 2. $166.62. 15. $26.62. 27. $578.88. 3. $35.17. 16. $215.35. 28. $7991.29. 4. $3146.88. 17. $82.86. 29. $137.48. 5. $4740.33. la $80.96. 30. $458.95. 6. $89.68. 19. $114.30. 31. $1256.55. 7. $11.41. 20. $0.98. 32. $368.32. 8. $277.52. 21. $155.59. 33. $1868.12. 9. $328.21. 22. $23.02. 34. $2639. io. $139.20. 23. $897.96. 35. $28.57. 11. $116.65. 24. $277.61. 36. $818.08. 12. $323.21. 25. $417.71. 37. $30,230.66. 13. $63.56. Exftrcise 104. 2. $414.97. 3. $49498. 4. $413.47. 6. $1251.33. Exercise 105. 2. $220.33. 3. $147.48. Exercue 106. 4. $27.94. 2. $10.79. 4. $15.15. 7. $3.52. 10. $24.27. a $20.97. 5. $38.45. a $28.76. 11. $12.76. Exercise 107. 2. 2 yr. 4 mo. 7 da. 4. 2 yr. 3 mo. 12 da. 6. 6 yr. 8 mo. 25 da. 3. 3 yr. 4 mo. 8 da. 5. 4 yr. 1 mo. 20 da.- 2. 5%. 3. 6%. Exercise 108. 4. 8%. 5. 10%. 6. 7< Exercise 109. 2. $548.49. 3. $93.78. 4. $469.10. 5. $48.19. 6. $907.91. Digitized by VjOOQIC 26 ANSWERS. ExeroiBe 110. 2. 12J%; 8i%; 6%; 5%. 3. 20%; 25%; 16f%; 12%; 9^^%. 5. 26 yr.; 22|yr.; 20 yr.; 16|yr.; 12iyr.; 10 yr. S. 50 yr. ; 44f yr. ; 40 yr. ; 33| yr. ; 25 yr. ; 20 yr. 7. 17|%. a 9 mo. 27 da.; $882.80. 9. 4^%; $615.04 10. $337.98; $12.67. 12. $3224.44; $3296.99. 11. 2 yr. 9 mo. 12 da. ; $3056.67. 13. 7^ % ; $152.86. 3. $8.25; $816.75. 4. $3.86; $923.32. 5. $1.66; $263.17. S. $2.97; $166.50. 7. $54.42; $2914.09. ExeroiBe 111. a 9. 10. u. 12. $13.93; $403.87. $12.64; $348.64. $ 2.42; $246.23. $14.19; $1813.71. $6.32; $77.38. la Oct. 24, 70 da., $11.57; $838.43. 14. Dec. 4, 50 da., $10; $1190. 15. Sept. 11, 41 da., $1.17 ; $127.33. 16. Dec. 4, 77 da., $8.34; $479.10. 17. Oct. 24, 53 da., $14.89; $1541.28. 18. Oct. 13, 95 da., $16.41 ; $872.15. 19. Dec. 6, 125 da., $18.85; $756.73. 20. Aug. 12, 57 da., $13.47; $1202.03. 2. $185.59. 3. $468.25. 4. $1879.81. Exercise 112. a $10,863.56. 7. $789.18. a $2870.14. 9. $168.25. 10. $16,858.22. Exercise 113. 2. 3. 4. a $3312. $1350. $34,500. $6751. 7. a 9. 10. $815.50. $26978. $1687.25. $2856. 12. 80. la 175. 14. 63^. la 36. 17. 46. la 23. 21. $1802. 22. $1240.75 Digitized by VjOOQIC ANSWERS. 27 23. $7362. 27. $1.76. 24. $11,605. 2a $11. 26. $3. 30. 24 41. 80,75,140,142^. 42. Second. 43. 1st, 1st, 2nd. 44. 3^ %• 45. $402; H%; «108. 46. 6%; $129. 47. lf%; $124.80. 31. 23. 3a 34. 5%,8%. . 39. 36. 75. 40. 4a $2214.50. 49. 76. 50. $17,793.75; 3 51. 69. 52. $6410. 53. $150. 54. 300. 6%. 83|. 12%. .372% Exercise 114. 1. $424.12. 14. $117.03. 24. 2. 5f|da. 15. 17.49%. 25. 3. $1819.42; 16. $24.75. 26. $56.18. 17. 2yr. 4da. 27. 4. $1590.49. 18. $382.37. 28. 5. $41,944.88. 19. 4 hr. 55 min. 29. 6. $4000. 37|sec.;4hr. 30. 7. $12,716.34. 4min.22|sec. a $5351.17. A.M. 31. 9. $4800. 20. 4^%. 32. 10. $3686.15. 21. 8%; 5%. U. $58.74. 22. 1697.6 bu. 33. 12. $474.24 23. 96'' 42' 30" W. 34. 13. H- Exercise 117. 1. if 2. 7371. 3. 1610J. 360 boys. $21.98. $586.15. $241.26. $208.79. 420f 2 yr. 10 mo. 27 da. .0623. 7600 sq. ft.; $1033.06. $471.67. $1962. 4. 3761| 5. 22f C. 6. 77° F. 7. 27i''C. Exercise 118. a -22''F. 9. 821" C. 10. 361° C. 11. U. la -4''F. 16i°C. 141f F. 14. IS. 181f F. 3i°C. Digitized by VjOOQIC 28 ANSWERS. Exeroise 119. 1. 25. 7. 24f lb. 13. 60. 19. 61b. 2. 28. a 30. 14. 11.52 , 20. 1404 rd. 3. M- 9. 12f 15. $45. 21. 13|. 4. 83i. 10. 172f 16. 3ipk. 22. .0252. 5. 18 T. u. 38f. 17. 1288 mi. 23. $84. 6. f25. 12. 84. la 288 A. 24. 32 da. Exercise 120. 2. f240. a 124 ft 14. 4256 bu. 19. 63. 3. 240 mi. 9. 216 ft. 15. 30 da b. 20. 585 bu. 4. $3120. u. 25inen. 16. 537| mi. 21. $77.25. 5. $956. 12. f 152.53. 17. $2.80. 22. 900 bu. 6. 162 ft. 13. 37^ ft. la 45^da. 23. $95.20. 7. 93 ft Exeroise 121. • 1. Earlier. 21. $366. 3a $62,girls; 2. Earlier. 22. 332; 664. $4H,boy8. 3. i,i^.m' 23. 275; 375. 37. $3.50. 4. 16. 24. 909. 38. 20. 5. 31f 25. $360. 39. 6. 6. H- 26. 18|. . 40. vw. 7. 76". 27. 2f 41. 12. 9. «3|. 2a 3276. 42. 20; 70. 10. 72i cu. ft. 29. 8 and 9. 43. $64. 11. f9.37f 30. 7 hr. 32 min. 44. • 23.81 bu. 12. H- 56 sec. 45. 31.42 J cu. in. 16. Backward. 31. ^• 46. .957. 18. $12:$48 = = 32. f 47. 39** 11' 15" E. 75 HI: 300 HI. 33. The same. 4a 10 yr. ; 36 yr. 19. 1:4; 4:5. 34. $1060. 49. 140** F. 20. 343 bu. 35. M,H,«- 50. 18f C. Digitized by VjOOQIC ANSWERS. 29 Exercise 123. 1. 6889; 14,641; 30,625; 34,225; 38,416. 2. 15,625; 29,791; 2401; 104,976; 3125. 3. 1296; 3969; 8281; 9216; 46,656. 5. 18,225; 27,225; 42,025; 133,225; 648,025; 801,025. ExerciBe 124. 4. 35; 24; 14; 70. 6. 110; 154; 256; 308. 5. 96; 125; 105; 120. 7^ 52; 58; 64; 66. 10. 12 V2; 12 V3; 9V5; 25 V2^ U. 11 V6; 14 V5; 24 V3; 22 V3. 12. -25 V5; 18 V3; 32 V2; 18 V5. 14. 8.484; 10.392; 9.898; 8.660; 11.180. 15. 21.21; 14.14; 34.64; 20.124; 25.452. Exercise 129. 3. 29; 89. 8. 2.44; .83; 2.21. 4. 83; 596. 9. .84; .81; .44. 5. H; A; fi; «; A- lo. .U; 30.25; i.oo4. 7. 5.29. Exercise 131. 2. 32 yd. 4. 78 yd. 7. 59f yd. 61| yd. 3. 74Jyd.. 6. f 44.44. 10. $112.88. Exercise 132. 2. $7.13. 3. $18.38. 6. $7.77. Exercise 133. 1. $91.52. 3. $120.64 4. $43.2a Exercise 134. 1. 120 sq. yd. 2. 43 A. 3. 60 sq. yd. • 4. 610 ft. Digitized by VjOOQIC 30 ANSWERS. a 6 sq. ft 4. 3f sq.ft. 5. Sff sq.ft. 6. 5^sq.ft 3. 6 in. 4. 25 ft. 5. 132 yd. 1. 6^ A. ExeroiBe 135. 7. f sq. yd. a 175 sq.ft. 9. 60|f sq. ft. Exercise 136. 6. 225 mi. 9. limi. 7. 240 rd. 10. 100 ft. a 18 ft. 11. 10 yd. 15. 480 sq. rd. or 3 A. Exercise 137. a 21 A. a 84 A. 10. 70f sq. ft 11. 43f sq. ft 12. 8f sq. yd. 12. 5 ft la 28 ft 14. 40 rd. 4. 3200 sq. ft Exercise 138. 1. 40A. a 20A. 5. 5A. 7. lOA. a 5A. 2. 80 A. 4. 10 A. a 2^ A. a 10 A. la 10 A. 1. 166^ in. Exercise 140. 2. 31^^ in. a 360. 4. 288. 1. 22 in. ; 154 sq. in. 2. Iff sq.in. Exercise 141. 4. 154 sq. ft. a 5. 49,107| sq. yd. 7. 5. 113|rd. 342^ sq. rd. 124J sq. in. Exercise 142. 1. 18fcd.;$ 89.66. 2. a. lOf cd.; $54.47. h, 8f cd.; $45.28. c. 4^^cd.; f 23.30. d. 5|| cd.; $31.01. e. 7||cd.; $40.30. /. llffcd.; $61.58. h. ll|f cd. ; $56.11. c. 134cd.; $63.38. d. 8|cd.; $40.11. €. 203^cd.; $96.02. /. 33^^ cd.; $156.98. g. 17^ cd.; $83.22. h. 84 cd.; $399. Digitized by VjOOQIC ANSWERS. 81 Exercise 143. 2. $62.89. 4. $93.56. 6. $93.60. a $62.59. 3. $52.72. 5. $30.36. 7. 597 posts. Exercise 144. 2. 6720. a 27^ perches. 4. 3687. 5. 129||. Exercise 145. 1. 484. 5. 1980 ft. a 5157. 2. 8712 sq. ft a 1224 sq.ft. 10. 17|^ perches. a 66. 7. $437.87. 4. 528 ft. a $574.20. Exercise 146. 4. $227.81. la 95iff (96). 22. 40x80rd. 5. 537| bu. 14. Hfi- 23. 3237|ft. 6. 2|bu. 15. lOffft. 24. 97,114^ sq. ft. 7. $7260. 16. 19|ft. 25. 51.42 ft. a 8 A. 150 sq. rd. 17. 16 rd. 26. 230|bu. • 7 sq. yd. 5 sq. la 24 in. 27. 4^ ft. ft. 9 sq. in. 19. 6 ft. 28. 426|bu. 9. $38.38. 20. The first re- 29. 30f in. 10. 43. quires 4.04 rd. 30. 360 bu. XI. 5808. more. 32. 30 ft 12. 42.88 hr. 21. 66.9 rd. Exercise 147. 2. 1260 sq. ft. a 624 sq. ft. a 158sq. ft; 3. 1428 sq.ft.; 7. 2fcu. ft. 120 cu. ft 2520 cu. ft. 8. 100 cu. ft. 10. $33.08. 5. $40. Digitized by VjOOQIC 82 ANSWERS. a 564.24 gal. 4. 100^ sq.ft. 5. 20l| sq. in. 6. 1006^. Bzereiae 148. 7. 4190^ cu. in. a 20^cu.ft. 9. 1^1.68. Exercise 149. 10. 111.66 in. 11. 10,262.4 bbL 12. 6364^ lb. 1. 21ffin. 5. 49,971f sq. ft. 2. 4^ in. 6. 70^ sq. in. 4. II cu. in. 7. 4691 bbl. 11. 169^^ sq. in. ; 169^^ cu. in. a 9. 16^ in. 1767^ lb. 264 cu. in. Surface; volume. 12. 19,764 gal. 13. 20 bundles; 1^4.80. 14. 1| ft. ; 93 s^. ft. 15. 4Tr; 16wy\. 16. ^tt; 25 IT] 4. 17. 4. la 6 in. 19. » 130.57. 20. 5026|^gal. 21. 24 cu. ft 1296 cu. in. 10. 22. » 107.58. 23. 108|bu. 24. »3750. 25. $506.88. 26. 1870|^gal. 27. 75cu. ft.; 25 cu. ft. 2a 24 cu. ft. ; 8 cu. ft. 29. 56 cu. ft. ; 67.53 sq. ft 30. 124.704 cu. ft 32. 22,176 cu. ft. 33. 222.03 in. 1. .0009. 2. 7f 3. 7. 4 $452.20. 5. ^, diminished. 6. 7x13. a 69^. 9. 468 ; 546. 10. 9. 11. 26,220. 12. 8647|. la 133|. Exercise 150. 14. I mi.; | mi. 15. $18,000. 16. 17. la 19. 20. 21. 22. 23. 331%. 432 sq. ft 24 ft. X 30 ft. 80 rd. X 90 rd. .3636. 33|%. 160; 240. .556. 24. 25. 26. 27. 28. 29. 30. 96 Km. ; 68.06 Kg. ; . 75.7 1. $36,000. 12 rd. $356.01. $511.39. 8. 4. ai. 148,684f T. Digitized by VjOOQIC ANSWERS. 83 32. 6^%. 46. 33. IJ. 47. 34. $351.90. 4a 35. $136.46. 49. 37. $13,000. 50. 38. $405. 51. 39. $19;300. '.0. 43 lb. 4.6 oz. 52. 41. 21 in. 53. 42. $5700. 54. 43. HJ- 55. 44. m- 56. 45. .868 L 57. 11%. 108 HI. $27.25. 88 in.; 616 sq. in. $625.51. 15%. 10*^ E. .02272. $20.80. $105; $175. 160 A.; 5a $12,800; $11,776. 59. 4.623%. 60. 134.8 bbL 61. 62. 63. 64. 65. 66. 70- 80^ 41' W. 33 ft. 9 in. 6 on each side. $41.40. $187.50. Exercise 151. a a 9. 10. 11. 569,476,224. 4. OfJ. 6. $186. 426 A. 106 sq. rd. 20 sq. yd. 1 sq. ft. $271.26 (use 16 ft. board 6 in. wide). 13. 14. 15. 699.3. 1260; 1134; 1360. 83^1 sq. ft. 646^ ft. $75.60. 16. 24^ gal. 17. 73.28 ft. la 3hr.5min.l5^ sec. $2621.30. 400 A., A. 480 A., B. 720 A., C. 960 A., D. 19. 20. 22. 23. 84,390. 25. 97,098 ft. 26. 324 sq. rd. 27. $365.20. 29. Si^%. 30. $59.64. 32. 2229^; 47,000. 34. 1200 sq.ft.; 30 X 40 ft 35. 50%. 36. 14 men. 37. 223.9 bbl. 38. 927.617. 39. $35.28. 40. $26.25; $600; $19.69. ,47. 7. 2934.91 gaL 72 sq. in. 42. July 30, 1891. $990. 9.237 ft. ; 1.909 ft.; 1.478 ft. .024. 47. liWffda.; $2.72; $1.72; $1.56. 4a 8 hr. 34 min. 29 sec. A.M. 49. 72^^. 51. 126 mi. 180 rd. 52. 1120. 45. 46. Digitized by VjOOQIC 34 ANSWERS. 53. 22imi. 81. 91.45 A. 108. 5854y\ lb. 54. »5.98. 82. 80 A.; 20 A.; 109. 218.268 lb. 55. IS^hr. 10 A. 110. 1962| bu. 56. 58 A. 56 sq. rd. 83. 165.02 yd. 111. $2687.30. 4 sq. yd. 36 84. m- 112. 60 da. ; 40 da sq. in. 85. 132.61 bbl. iia 180 rd. 57. $162.61. 86. 5A%;4^%. 114. $469.50. 5a $648. 87. im%' 115. $82.88. 59. $26.04. 8a $576. 116. $77.00. GO. 8Jin. 89. $1529.74. 117. $500. 61. $107.33. 90. 2.6457. 118. 168 gal. 62. $1728; $3360. 92. 2Mda. 119. $2460.94. 63. $52.47. 93. 6.91 gal. 120. $2 per yd. 64. .0089. 94. $291.55; 121. 20 hr. 65. 2*%. $270.55; 122. 94^ mi. 66. 1 yr. 1 mo. 13 da. $208.05; 123. 62^%; 160% 67. 220.98 cu. in. $126.30. 124. 19,764 gal. 68. 682 lb. 3 oz. 95. 136; $55.44. 125. $1200. 69. !i|;676. 96. .816. 126. $264.88. 70. 4696.31 T. 97. $153.60; 127. m^' 71. 1; Hi -023. $102.40. 128. $549.12. 72. ^m- 98. 2^ mi. ; 20 mi. Posts 6 ft. 74. 17 hr. 2 min. 99. 192.5 cu. in. apart, and 6 48 sec. 100. 26.04. in. 12 ft. 75. 188«- 102. $117; $52. lumber. 76. 3A. 103. 24. 129. 2048 bu. 77. 124 rd. 2 yd. 1 104. $41.34. 130. 55 shares. ft. 4 in. 105. 25.47 ft. 78. $266.16. 106. 63.816 rd. 5 79. 1.54%. 1.388 A. 80. 55j^ niL 107. tm- Digitized by VjOOQIC ^ ^^ 2yicZ^t^c>{ // oi ?n^. >> (fi (/^o s, , 1 Digitized by VjOOQIC Digitized by VjOOQIC i'B 35860 Digitized by VjOOQ IC Digitized by VjOOQiC