LIBRARY OF THE UNIVERSITY OF CALIFORNIA GIF^T 01^ Class APPLETOJfS' MATHEMATICAL SERIES. MENTAL ARITHMETIC. BY G. P. QUACKENBOS, LL. D., AUTHOR OF '*A2f ENGLISH GRAMMAR'," "FIRST LESSONS IN COMPOSITION;" "ADVANCED COURSE OF COMPOSITION AND RHETORIC;" u A NATURAL PHILOSO- PHY;" "ILLUSTRATED SCHOOL HISTORY OF THE UNITED STATES;" " PRIMARY HISTORY OF THE UNITED STATES," ETC. NEW YORK: D. APPLETON AND COMPANY, 549 & 551 BROADWAY. 1878. [ADVERTISEMENT.] APPLETONS' MATHEMATICAL SERIES. BY G. P. QUACKEHBOS, LL. D., A Primary Arithmetic. Beautifully illustrated; carries the beginner through the first four Rules and the simple Tables, combining mental exercises with examples for the slate. 16mo. 108 pages. 22 cents. An Elementary Arithmetic. Reviews the subjects of the Primary in a style adapted to somewhat maturer minds. Also embraces Fractions, Federal Money, Reduction, and the Com- pound Rules. 12mo. 144 pages. 40 cents. A Practical Arithmetic. Prepared with direct reference to the wants of Common Schools, giving special prominence to the branches of Mercantile Arithmetic. 12mo. 336 pages. 80 cents A Mental Arithmetic. Designed to impart readiness in mental calculations, and extending them to the various opera- tions needed in business life. Introduces short methods, and new and beautiful processes. 16mo. 168 pages. 35 cents. A Higher Arithmetic. I2mo. 420 pages. $1.10. Entered, according to Act of Congress, in the year 1868, by D. APPLETON & CO., In the Clerk's Office of the District Court of the United States for the Southern District of New York. PREFACE. THE study of Mental Arithmetic has two principal objects in view, to discipline the mind and impart quickness and accuracy in mental calculations. To secure these objects in the highest degree and in the pleasantest way to both teacher and learner is the aim of this little volume. It is intended for pupils who have gone through a Primary Arithmetic, and know how to read and write numbers up to thousands inclusive ; and may be used either by itself to succeed the Primary or Elementary, or as an auxiliary to the Elementary or Practical, on alternate days. Among the more important features of the present work are the following: 1. The gradual and inductive mode of unfolding the subject, with the aid of rigid but clear analyses expressed as briefly as possible. 2. The introduction of necessary definitions, too often excluded from Mental Arithmetics. 3. The great variety and prac- tical bearing of the Examples ; the careful avoidance of obscurity in wording them, and the exclusion of all questions involving im- possibilities or absurdities. 4. The presentation of the Metric Sys- tem, hitherto confined mainly to text-books on written Arithmetic. 5. The teaching of short methods and processes actually used in the counting-room. 6. The extension of mental calculations to more of the operations of every-day business life than has hitherto been at- tempted; such as equation of payments, stockjobbing, U. S. securi- ties, taxes, duties, &c. The value of this last feature, it is believed, can hardly fail to be appreciated in this practical age. The interest and profit with which classes will use this work will depend entirely on the thoroughness with which the successive steps are taken. Review again and again if necessary, and let nothing pass till it is mastered. Short lessons should be given, to be pre- pared beforehand. The books should be closed at recitation, the 153.& 4 CONTENTS. question read but once, and the scholars should have no intimation as to which of their number will be called on to solve it. The analy- ses given in the Models should be followed (unless better ones can be devised), with distinct articulation and in correct language. Let the answer always be distinctly stated, when it is reached, in con- nection with the denomination, as in the Model. Whenever any particular form of analysis has become perfectly familiar, it will be well to omit it in the case of some of the questions, and require im- mediate answers, as well to encourage quickness of thought as to economize time. A few questions from previou^kssons, to be an- swered thus promptly, will be found useful at each recitation. The Author can only hope that this work may meet with as cordial a reception as has been so kindly extended to the other Numbers of the Series. NEW YORK, May 22, 1868. . , CONTENTS. PAGl CHAPTER FIRST, ADDITION, 5 CHAPTER SECOND, SUBTRACTION, 13 CHAPTER THIRD, MULTIPLICATION, .... 20 CHAPTER FOURTH, DIVISION, 28 CHAPTER FIFTH, FRACTIONS, 38 CHAPTER SIXTH, FEDERAL MONEY, .... 72 CHAPTER SEVENTH, REDUCTION, 75 CHAPTER EIGHTH, THE METRIC SYSTEM, 89 CHAPTER NINTH, THE COMPOUND RULES, . 94 CHAPTER TENTH, MISCELLANEOUS EXAMPLES, 100 CHAPTER ELEVENTH, PERCENTAGE, 117 CHAPTER TWELFTH, INTEREST . .... 127 CHAPTER THIRTEENTH, DISCOUNT, ..... 140 CHAPTER FOURTEENTH, STOCKS, U. S. SECURITIES, 143 CHAPTER FIFTEENTH, MISCELLANEOUS EXAMPLES, . 150 MENTAL AEITHMETIO. CHAPTER FIRST. ADDITION. [!T is supposed that the pupil has learned the Addition, Sub- traction, Multiplication, and Division Tables. Let them be re- viewed, as presented under Chapters 1, 2, 3, and 4, until he can say them promptly and perfectly, backward as well as forward.] ADDITION TABLE. 1 and 2 and S and 4 and 5 and 1 are 2 1 are 3 1 are 4 1 are 5 1 are 6 2 are 3 2 are 4 2 are 5 2 are 6 2 are 7 3 are 4 3 are 5 3 are 6 3 are 7 3 are 8 4 are 5 4 are 6 4 are 7 4 are 8 4 are 9 5 are 6 5 are 7 5 are 8 5 are 9 5 are 10 6' are 7 6 are 8 6 are 9 6 are 10 6 are 11 7 are 8. 7 are 9 7 are 10 7 are 11 7 are 12 8 are 9 8 are 10 8 are 11 8 are 12 8 are 13 9 are 10 9 are 11 9 are 12 9 are 13 9 are 14 10 are 11 10 are 12 10 are 13 10 are 14 10 are 15 6 and 7 and 8 and 9 and 10 and 1 are 7 1 are 8 1 are 9 1 are 10 1 are 11 2 are 8 2 are 9 2 are 10 2 are 11 2 are 12 3 are 9 3 are 10 3 are 11 3 are 12 3 are 13 4 are 10 4 are 11 4 are 12 4 are 13 4 are 14 5 are 11 5 are 12 5 are 13 5 are 14 6 are 15 6 are 12 6 are 13 6 are 14 6 are 15 6 are 16 7 are 13 7 are 14 7 are 15 7 are 16 7 are 17 8 are 14 8 are 15 8 are 16 8 are 17 8 are 18 9 are 15 9 are 16 9 are 17 9 are 18 9 are 19 10 are 16 10 are 17 10 are 18 10 are 19 10 are 20 6 ADDITION. SECTION 1. Addition is the process of uniting two or more numbers in one, called their Sum. Two and one are three / we have added 2 and 1, and 3 is their sum. Addition is denoted by an erect cross -f , called Plus, placed between the numbers to be added. 2 + 1 is read two plm one, and means that 2 and 1 are to be added. Two short horizontal lines = denote equality. 2 -f 1=3, is read two plus one equals three, and means that the sum of 2 and 1 is 3. 1. What is the sum of 2 and 3 ? 2 and 9 ? 2. What is the sum of 2 and 4 ? 2 and 7 ? 3. What is the sum of 3 and 1 ? 3 and 4 ? 4. What is the sum of 3 and 8 ? 3 and 6 ? 5. What is the sum of 4 and 5 ? 4 and 2 ? 6. What is the sum of 4 and 9 ? 4 and 10 ? 7. How many are 5 and 8 ? 5 and 1 ? 5 and 5 ? 8. How many are 5 and 9 ? 5 and 3 ? 5 and 6 ? 9. How many are 6 and 2 ? 6 and 9 ? 6 and 5 ? 10. How many are 6 and 1 ? 6 and 6 ? 6 and 8 ? 11. Add 7 and 7. 7 and 2. 7 and 4. 7 and 9. 12. Add 7 and 8. 7 and 1. 7 and 5. 7 and 10. 13. Add 8 and 8. 8 and 6. 8 and 3. 8 and 4. 14. Add 8 and 1. 8 and 9. 8 and 2. 8 and 7. 15. 9 + 4= how many? 9 + 9? 9 + 1? 9 + 0? 16. 9 + 7= how many? 9 + 2? 9 + 6? 9 + 3? 17. What is the value of 10 + 1? 10 + 10? 10 + 9? 10 + 5? 5 + 10? 10 + 8? 10 + 4? 1+9? 1+5? 18. How many are 2 and 5? 5 and 2? How muchis6 + 4? 4 + 6? 3 + 7? 7 + 3? ADDITION. 7 19. When two numbers are to be added, does it make any difference which is taken first ? 20. How many are 4 and 8 ? How many are 1 and 3 and 8 ? How many are 2 and 2 and 8 ? 21. How many are 3 and 9 ? How many are 1 and 2 and 9 ? How many are 2 and 1 and 9 ? 22. How many are 8 and 10? How much is 3 + 5 + 10? How much is 2 + 2 + 4 + 10? 23. How many are 6 and 7 ? How much is 3 + 3 + 7 ? How much is 1+4 + 1 + 7? 24. How many are 9 and 8? How much is 5 + 4 + 8? How much is 3 + 5 + 1+8? 25. How many are 8 and 5 ? How many are 8 and 3 + 2 ? How many are 8 and 4 + 1 ? 26. How many are 10 and 7? 2 + 8 and 4 + 3? 10andl+6? 1+6 + 10? 27. Mary and her seven sisters just filled a bench ; how many did the bench hold ? 28. You have four fingers on each hand ; how many fingers have you on both hands ? 29. A newsboy bought five daily and seven week- ly papers ; how many papers had he in all ? 30. If Ruth has seven pins in one pin-cushion, and six in another, how many pins has she in both ? 31. Guy pulled six ears of corn on Monday, and ten on Tuesday ; how many did he pull both days ? 32. Ten birds were sitting on a steeple, when six more alighted ; how many were there then ? 33. Henry had two cents, and his father gave him ten more. Louis had ten cents, and he found two more in the street. Which then had the most ? ADDITIC 34. Put four marbles in a bag already containing ten, and how many will you have in the bag ? 35. A boy caught six perch, three pickerel, and ten shiners ; how many fish did he catch in all ? 36. A baker gave nine loaves to one poor family, and five to another ; how many did he give both ? 37. How many trees stand beside my lane, if there are ten on one side and three on the other ? 38. A mother, having three sons aged two, five, and eight years, gave each as many dollars as he was years old ; how many dollars did she give all three ? 39. How many books had Paul, if his father gave him 3, his mother 6, his brother 1, and his sister 7 ? 40. A farmer had four ducks, six geese, and as many chickens as he had ducks and geese put to- gether ; how many fowls had he in all ? SECTION 2. 1. How many are 3 and 2 ? 13 and 2 ? 33 and 2 ? 63 and 2 ? 93 and 2 ? 103 and 2 ? 2. What does 2 + 3 equal? 22 + 3? 43 + 2? 53 + 2? 83+2? 183 + 2? 193 + 2? 203 + 2? 3. What is the sum of 4 and 5 ? 5 and 4? 55 and 4 ? 54 and 5 ? 74 and 5 ? 75 and 4 ? 4. How much is 7 + 2? 107 + 2? 207 + 2? 507 + 2? 807 + 2? 1007 + 2? 4007 + 2? 5007 + 2? 5. Add 2 and 6. 62 and 6. 162 and 6. 1062 and 6. 1066 and 2. 166 and 2. 66 and 2. 76 and 2. 6. How much is 3 + 4? 20 + 3 + 4? 30 + 3 + 4? 40 + 3 + 4? 3 + 40 + 4? 60 + 4 + 3? 4 + 3 + 70? 7. How much is 5 + 2? 50 + 5+2? 51+5 + 2? ADDITION. y 8. How many are 9 and 1 ? 89 and 1 ? 99 and 1 ? 109 and 1 ? 199 and 1 ? 209 and 1 ? 299 and 1 ? 9. How many are 6 and 4 ? 6 and 5 ? 16 and 4 ? 16 and 5 ? 26 and 5 ? 36 and 5 ? 56 and 5 ? 10. How many are 8 and 7 ? 18 and 7 ? 118 and 7 ? 128 and 7 ? 28 and 7 ? 38 and 7 ? 638 and 7 ? 11. What is 3 + 9 equal to? 4 + 9? 23 + 9? 24 + 9? 54 + 9? 55 + 9? 56 + 9? 52 + 9? 12. What is 2 + 5 + 7 equal to? 12 + 5 + 7? 22 + 5 + 7? 42 + 5 + 7? 72 + 5 + 7? 92 + 5 + 7? . 13. How many are 63 and 5 ? 79 and 2 ? 103 and 8? 47 and 6? 99 and 3 ? 102 and 7? 113 and 5? 14. How many are 10 and 10? 20 and 10? 50 and 10? 90 and 10? 100 and 10 ? 200 and 10 ? 15. 6 and 9 make how many ? 34 and 6 ? 43 and 9 ? 52 and 8 ? 58 and 2 ? 50 and 3 ? 79 and 6 ? 88 and 4 ? 103 and 3 ? Ill and 9 ? 124 and 4 ? 16. How many are 7 and 7 and 8? 21 and 9 and 6? 34 and 8 and 7 ? 41 and 10 and 9? 72 and 4 and 5 ? 99 and 10 and 1 ? 199 and 2 and 9 ? 17. One tree bears 34 apples, another 6, and a third 10 ; how many apples do all three bear? MODEL. If one tree bears 34 apples, another 6, and a third 10, all three together will bear the sum of 34, 6, and 10 apples, or 50 apples. Amwer, 50 apples. 18. A farmer set out 42 apple, 8 pear, and 9 plum trees ; how many trees did he set out altogether ? 19. If I travel 75 miles by boat, 7 by railroad, and 10 by stage, how many miles do I go in all ? 20. 103 English books, 20 French books, and 6 German books, make how many books in all ? 10 ADDITION. 21. James bought 20 cents' worth of bread, 10 cents' worth of cake, and 9 cents' worth of crackers ; how much did he have to pay the baker ? 22. How many miles will you travel in a day, if you go 21 miles in the morning, 30 miles in the after- noon, and 8 miles in the evening ? 23. In a field were 19 sheep, 8 cows, and 5 calves ; how many animals were in the field ? 24. John was eight years older than Charles. How old was John when Charles was thirteen ? How old was each, nine years afterward ? 25. How many windows in a factory, if there are 12 in front and 6 on each of the other three sides ? 26. In a certain school were 10 boys and 6 more girls than boys ; how many scholars altogether ? 27. Helen laid out 45 cents for muslin, 5 cents for thread, 9 cents for pins, and 6 cents for needles ; how many cents did she spend in all ? 28. A fisherman had sold all his stock, except 17 shad, 6 bass, and 9 eels ; how many fish had he left ? 29. Count by 2's, beginning 2, 4, 6, &c., up to 100. 30. Count by 2's, beginning 1, 3, 5, &c., up to 99. 31. Count by 3's, beginning 1, 4, 7, &c., up to 100. 32. Count by 4's, beginning 2, 6, 10, &c., up to 98. 33. Count by 5's, beginning 1, 6, 11, &c., up to 96. 34. Count by 6's, beginning 2, 8, 14, &c., up to 98. 35. Count by 7's, beginning 3, 10, 17, &c.,up to 94. 36. Count by 8's, beginning 5, 13, 21, &c., up to 93. 37. Count by 9's, beginning 1, 10, 19, &c., up to 100. [These exercises may be continued, and varied by commencing differently, till they are made perfectly familiar.] ADDITION. 1 1 SECTION 3. 1. How many are 43 and 26 ? MODEL. 43=3 units 4 tens; 26=6 units 2 tens. 6 units and 3 units are 9 units ; 2 tens and 4 tens are 6 tens. 6 tens 9 units are 69. Ans. 69. 2. What is the sum of 31 and 52 ? Of 74 and 1 5 ? 3. How many are 22 and 60? 17 + 81? 65 + 32? 4. How many are 14 and 54? 46 + 31? 80 + 19? 5. Add 123 and 876. 456 and 543. 739 and 40. 6. What is the sum of 267 and 431 ? SHORT FORM. 1 and 7 are 8 ; 3 and 6 are 9 ; 4 and 2 are 6. Ans. 698. 7. How many are 240 and 326 ? 579 and 120 ? 8. How many are 165 and 722 ? 410 and 378 ? 9. How many are 821 and 73 ? 26 and 933 ? 10. What is the sum of 59 and 63 ? SHORT FORM. 3 and 9 are 12, 2 units and 1 ten ; 1 and 6 and 5 are 12. Ans. 122. 11. How many are 14 and 18? 17 + 23? 21 + 16? 12. How many are 40 and 69? 53 + 29? 37 + 47? 13. How many are 99 and 12? 86 + 62? 29 + 59? 14. How many are 108 and 15? 221+49? 72 + 68? 15. How many are 29 and 29? 38 + 57? 93 + 67? 16. Count by tens, 10, 20, 30, &c., up to 100. 17. Count by elevens, 11, 22, 33, &c. 5 up to 132. 18. Count by twelves, 12, 24,36, e ra'wTHS. 4. Read each fraction ; name its terms ; mention its numerator and denominator ; tell what each shows. i* f- fi* TP !& W- Tlr- 44 FRACTIONS. 5. Write in figures : Two fifths. Three sevenths. Five sixths. Seven elevenths. Five nineteenths. 6. Write in figures : Seven tenths. Seven hun- dred ths. Seven thousandths. Fifty-one thousandths. Xineteen sixtieths. Four seventy-fifths. 7. How much is ^ of 18 ? How much is | of 18 ? o o MODEL. One third of 18 is 6, and two thirds are twice 6, or 12. Ans. 12. 8. How much is f of 12 ? of 10 ?. f of 20 ? 9. How much is f of 18 ? f of 21 ? -f o f 35 ? 10. How much is T 9 o f 100 ? f of 36 ? -f of 77 ? 11. How much is -fr of 121 ? of 40 ? | of 54 ? 12. How much is -& of 144 ? f of 60 ? f of 72 ? 13. How much is ^ of 160 ? f of 45 ? -f of 88 ? 14. How much is J- of 90 ? f of 50 ? f of 91 ? 15. How much is Jf of 40 ? T ^ of 96 ? T 6 T of 99 ? 16. How much greater is T 7 of 80 than f of 48 ? 17. How much less is f of 35 than T 5 T of 33 ? 18. Which is greater, -^ of 108 orf of 63 ? 19. A person having $1550 divided it into five equal parts, and gave one of these parts to each of his three children. What part of the whole did the chil- dren receive, and how much? 20. A road 160 miles long is composed of 10 equal sections. What fraction of the whole road are 6 of these sections, and how many miles do they contain? 21. Two brothers are equal heirs with 6 other per- sons to an estate of $8000. What part of the whole do the two brothers receive, and how much money? 22. If the rent of a house for 30 days is $90, how much will it be for 1 7 days ? FRACTIONS. 45 SECTION 20. 1. What is a Fraction ? 2. What does one third mean ? Ans. One of three equal parts into which a whole is divided. 3. W T hat does two thirds mean ? Ans. Two of three equal parts into which a whole is divided. 4. What does one fourth mean ? Two fourths ? f ? 5. How is this expression, 6 J, read ? Ans. Six and three fourths. 6. Of what does it consist ? Ans. Of a whole number (6), and a fraction (f). 7. What is a number that consists of a whole num- ber and a fraction called ? Ans. A Mixed Number. 8. How many fourths in 6 ? MODEL. In 1 there are 4 fourths, and in 6 six times 4 fourths, or 24 fourths. Ans. ^. 9. How many fourths in 6| ? MODEL. In 1 there are 4 fourths, and in 6 six times 4 fourths, or 24 fourths. 24 fourths and 3 fourths are 27 fourths. Ans. *f. 10. How many eighths in 7 ? In 7f ? In 9| ? 11. How many fifths in 11 ? In 11^? In 12|? 12. How many ninths in 12 ? In 12? In llf ? 13. How many sevenths in 4 ? In 4-f ? In 8-| ? 14. How many tenths in 10 ? In 10-^ ? In 4^ ? 15. How many twelfths in 6 ? In 5 T V ? In 8 T \? 16. 5 =how many halves? 5 J ? 6? 8? 17. How many thirds in 3 ? In 7 ? In 9 ? In 5 ? 18. 8=how many sixths? 9| ? 8| ? 4f ? 4 how many elevenths ? 4 T 2 T ? 7 T 3 T ? 9^ ? 6 T \ ? 19. How many halves in 50^ ? Twentieths in 5l? Hundredths in2 T i 7 ? Fortieths in 4- 4 % ? Thirds in 13^ ? Eighths in 14| ? Quarters in 7J ? 46 REDUCTION OF FRACTIONS. 20. What kind of fractions are | 5 , &c. ? Ans. Improper Fractions. 21. What is an Improper Fraction ? Ans. A Frac- tion whose numerator is equal to, or greater than, its denominator. 22. What is a fraction called, whose numerator is less than its denominator ? Ans. A Proper Fraction. SECTION 21. 1. How many thirds do 4j equal ? 2. When you say *- for 4J, do you change the form of the mixed number ? Do you change its value ? 3. What is the process of changing the form, with- out changing the value, called ? Ans. Reduction. 4. In the last Section, what kind of numbers did you reduce ? To what kind of fractions did you re- duce them ? 5. 5 units equal how many thirds ? 15 thirds, then, equal how many units ? 6. How many units in 16 thirds ? MODEL. Since 3 thirds make 1 unit, 16 thirds make as many units as 3 thirds are contained times in 16 thirds, or 5J-. Ans. 5^. 7. How many units in ^ ? In ^ ? In * ? 8. How many units in ^- ? In *- ? In -\ 5 ? 9. How many units in *$- ? In ^ ? In ^ ? 10. How many wholes in i ? In ^ ? In f ? 1 1. How many wholes in f| ? In fj ? In f| ? 12. Which is greater, |J- or fj? ^ or - 4 /? 13. Which is greater, ^ O r \ 3 - ? \f or ? 14. How many times 1 in - 4 / ? In 15. How many times 1 is REDUCTION OF FRACTIONS. 47 16. |=how many units ? f | ? -f f ? -^- ? 17. Which is greater, \ 5 - or - 5 / ? - 6 T 2 - or -%*- ? 18. How many dollars in ^ of a dollar? In 29 quarter-dollars ? In 29 half-dollars ? 19. In this Section, what kind of fractions have you reduced ? 20. To what have you reduced them ? SECTION 22, 1. What is a Fraction ? A Proper Fraction ? An Improper Fraction ? 2. What is a Mixed Number ? 3. What is meant by the Reduction of Fractions : 4. How many halves in ] 2 ? How many units in %- ? 5. How do you reduce units to halves ? How do you reduce halves to units ? 6. How do you reduce thirds to units ? Units to thirds ? 7. How do you reduce units to sixths ? How do you reduce sixths to units ? 8. How do you reduce ninths to units ? Units to ninths ? 9. Reduce 8 to twentieths. Reduce J-f- 2 - to units. 10. Reduce - to a mixed number. Reduce 2f to an improper fraction. 11. At a dinner, 47 plates of pie were called for; how many pies were used, if each plate contained quarter of a pie ? 12. A druggist is putting up soda in powders of \ of a dram each. If he uses 9|- drams of soda, how many powders does he put up ? 48 FRACTIONS. 13. How many bows, requiring ^ of a yard of ribbon each, can be made out of 7f yards ? 14. At a lecture 43 dollars, 24 half-dollars, and 32 quarter-dollars, were taken in. How much in all was received ? 1 5. A pint is one eighth of a gallon ; how many pints in 13f gallons ? How many gallons in 141 pints ? 16. If a boy can saw and split y 1 ^ of a cord of wood in a day, how many such boys will it take to saw and split 4 T \ cords in a day ? 17. How many cords would the boy just mentioned saw and split in 300 days ? 18. How many dollars will 53 tops cost, at the rate of -Q of a dollar apiece ? 19. How many quarter-pound weights will it take to balance 56 pounds ? How many half-pound weights ? 20. If an apple costs -^ of a dollar, how many apples can you buy for $1 ? For $2 ? For $5 ? 21. How many lots -^ of an acre in size can be laid out from 5 acres ? From 7 acres ? From 2 T 3 T acres ? 22. How many inches in ^J- of an inch ? 23. One day is % of a week; how many weeks in 50 days ? 24. How much is | of 18 ? 1 4 is how many times 2 ? 25. | of 18 is how many times 2 ? 26. | of 60 is how many times 9 ? 27. -| of 56 is how many times 8 ? 28. ^ of 50 is how many times 15 + 5? 29. | of 72 is how many times 178 ? 30. | of 84 is how many times ^ of 16 ? 31. i of 96 is how many times | of 12 ? FRACTIONS. 49 SECTION 23. 1. 6 is | of \vhat number ? MODEL. If 6 is one third of the required number, three thirds, or the whole, must be three times 6, or 18. Am. 6 is ^ of 18. 2. 4 is | of what number ? - 3. 7 is T V of what number ? 4. 9 is ^ of what number ? 5. 3 times 4 is | of what number ? 6. The sum of 5 and 9 is J of what number ? 7. 2 x 8 is I of what number ? 8. | of 21 is | of what number? - 9. f of 21 is I of what number? ~ 10. ^ of 25 is T V of what number ? 11. | of 25 is 1 of what number? 12. -f of 45 is -J- of what number ? 13. 2 + 8 3 is of what number? 14. 10 is T 1 T of what number ? 10 is T ^ of what? 15. If a man can do -^ of a piece of work in 2 hours, how long will it take him to do the whole ? MODEL. If a man can do one tenth of a piece of work in 2 hours, to do ten tenths, or the whole, will require 10 times 2 hours, or 20 hours. Ans. 20 hours. 16. If a family consume T V of a barrel of flour in 11 days, how long will the barrel last them ? 17. If a boat is 3 hours in performing \ of its trip, how long, at that rate, will its whole trip take ? 18. If a locomotive can go six miles in ^ of an hour, how many miles can it go in one hour? 19. With $24 collected from one customer, $16 from another, and $11 from a third, a person paid ^ of his taxes ; what did his taxes amount to ? 20. 7 + 8 + 5 + 4 is of what number ? 5 r -/r- 50 FRACTIONS. 21. 6 is f of what number? MODEL. If 6 is fa>o thirds of the required number, one third of it is of 6, or 3 ; and three thirds, or the whole, are 3 times 3, or 9. An*. 6 is f of 9. /22. 4 is f of what number ? / 23. 12 is f of what number ? 24. 36 is T 6 T of what number? - 25. 48 is -f of what number? 26. 27 is -f$ of what number? 27. 54 is T 9 ^ of what number? 28. 33 is f of what number ? 29. 40 is -| of what number? rl$0. 3 times 10 is f of what number? 31. 12 7 is | of what number? 32. | of 72 is f of what number ? 33. f of 44 is ft of what number? J* 34. | of 105 is | of what number? - 35. 8 is -f- of what ? 28 is how many times 4 ? 36. 8 is -| of how many times 4 ? 37. 9 is T 3 Q- of how many times 5 ? MODEL. If 9 is three tenths of a certain number, one tenth of it is of 9, or 3 ; and ten tenths, or the whole, are 10 times 3, or 30. 30 is 6 times 5. Ans. 6 times. 38. 14 is T \- of how many times 8 ? 39. 64 is | of how many times 12 ? - 40. \ of 36 is f of how many times 2 ? ^ 41. Charles lost 18 marbles, which were -^ of all he had. How many had he at first ? 42. A boy, having received $5 apiece from 6 per- sons, had only T 5 F of what he needed to buy a pony. What was the price of the pony ? FRACTIONS. 5 1 43. 70 is I of how many times 5 ? / 44. 70 is -J of how many times of 10 ? 45. 35 is -| of how many times 3 ? 46. 35 is f of how many times of 12 ? 47. 12 is T 3 of how many times f of 12 ? 48.. 16 is -^ of how many times ^ of 18 ? 49. of 80 is how many times 5 ? 50. J of 80 is how many times -J- of 30 ? 51. 3* T of 44 is how many times ^ of 96 ? 52. -/Q- of 100 is how many times 6 ? 53. -J of 36 is how many times ^ of 84 ? 54. -| of 49 is how many times ^ of 27 ? 55. A is 80 years old, and T 5 ^- of his age is of B's ; Low old is B ? 56. From Buffalo to Syracuse is 150 miles, and | of this distance is ^ the distance from Buffalo to Verona; how far is it from Buffalo to Verona ? 57. Wilmington is 28 miles from Philadelphia, and J of this distance is -^ of the distance from Philadel- phia to Baltimore. Plow far is Philadelphia from Baltimore ? 58. From Boston to Newburyport is 36 miles, which is 8 miles more than |- the distance from Boston to Portsmouth. How far is Portsmouth from Boston ? 59. A man, having bought some corn for $80, sold it for % of its cost ; did he gain or lose, and how much ? 60. A man bought some corn for $80, which was -J of what he sold it for ; did he gain or lose, and how much ? 61. A firmer had 24 cows, and f as many sheep. Had he more sheep or cows, and how many more? 52 FRACTIONS. x x 2 SECTION 24. 1. What is meant by Reducing a fraction ? 2. If we divide a pie into two __ __ e( l ual parts, each part is called If we divide each half into two equal parts, we get four equal parts in all, and each is J of the whole. It is clear that two of these fourths equal one half, or that f may be reduced to |. 3. f =. What operation performed on f 2 ^ 2 _ } gives ? Am. Dividing its terms by 2. 4 -*-2 ~ T 4. i=f . What operation performed on \ gives f ? ^L?i5. Multiplying its terms by 2. 5. What principle may we lay down ? ^4n. The value of a fraction is not changed by dividing or mul- tiplying its terms by the same number. 6. When we divide both terms of a fraction by 2, show why we do not change its value. Am. We get only half as many parts as before, but each part is twice as large. 7. When we multiply both terms of a fraction by 2, show why we do not change its value. Am. We get twice as many parts as before, but each part is only half as large. 8. When is a fraction in its lowest terms ? Ans. When no number greater than 1 is exactly contained in both terms. \ is in its lowest terms; f is not, be- cause 2 is exactly contained in its numerator and de- nominator. 9. Is | in its lowest terms ? f ? f ? | ? REDUCTION OF FRACTIONS. 53 10. How is a fraction reduced to its lowest terms? Ans. B} r dividing its terms by whatever number or numbers, greater than 1, are exactly contained in both. 11. Reduce / T to its lowest terms. Ans. ^. 12. Reduce the following to their lowest terms : f. V*. ,V f if- M- M- A- M. 13. Reduce y^ to its lowest terms. MODEL. Dividing both terms by 5, we reduce the fraction to JJ. Again dividing both terms by 5, we get f . Ans. f . 14. Reduce the following to their lowest terms: *f tt- i 3 -- T 7 oV fi- itfr- 15. Reduce f-J to its lowest terms. |f. f 16. Reduce ff to its lowest terms, ff. | 1 7. How many halves in i ? In $- ? In 18. How many thirds in T | ? In J ? In 19. How many fourths in 3| ? How many halves ? 20. How many fifths in 7^ - ? In 3 T 6 o ? In 5yV ? 21. Reduce f| to a mixed number. NOTE. Always see that a fraction occurring in an answer is in its lowest terms. 22. Reduce -\/ to a mixed number. i. J-p. / 23. How much is || of 16? MODEL. J= }. ^ of 16 is 4, and f is 3 times 4, or 12. ^/w. 12. NOTE. It is often best to reduce a fraction to its lowest terms before operating with it. > ^/v 24. How much is |f of 6 ? II f 12 ? 25. How much is f f of 24 ? f of 20 ? 26. 10 is f of vi^at number? 27. I of 16 is f| of what number? 28. 24 is |4 of how many times 2 ? 54 FRACTIONS. SECTION 25. 1. What may we do to the terras of a fraction, without changing its value ? 2. How is a fraction reduced to its lowest terms ? 3. How may a fraction be reduced to higher terms? Ans. By multiplying its terms by the same number; *=i- 4. Explain why, when we multiply both terms of a fraction by 3, we do not change its value. 5. How many sixths in f ? MODEL. In 1 there are 6 sixths, and in f there are f of 6 sixths, or 4 sixths. Ans. -J. 6. How many ninths in ^ ? How many tenths in f ? 7. How many 24ths in % ? In f ? In T \ ? In f ? 8. How many 36ths in f ? In | ? In f ? In T 5 ^ ? $ 9. How many 18ths in f ? In f ? In 1^ ? In 1 f ? 10. Reduce 1^ to twentieths. To 30ths. To50ths. 11. Reduce 3| to twelfths. Reduce 2| to 63rds. 12. How many twelfths in J- ? In | ? In | ? In ? 13. What do you observe with respect to the frac- tions T 3 ^, T 6 , T 2 ^, and T % ? ^l??,s. They have a common denominator (12). 14. Reduce ^ and | to fractions that have a com- mon denominator. MODEL. 2x5 = 10; the common denominator is 10. ^equals -ro ; t equals -,% -4w. 1%, rV 15. Reduce to fractions with a common denomina- and |; f and | ; | and ^_ ; ^ and f. 16. Reduce to fractions with a common denomina- tor 2} (that is, f) and |; | and 1| (f) ; Ij and If 17. Reduce to fractions having a common denomi- nator f \, and i ; f, f, and }; ^,4, and f REDUCTION OP FRACTIONS. 55 18. If an ounce is T ^ of a pound, how many ounces in J of a pound ? In f- of a pound ? 19. One cent is one hundredth of a dollar; what fraction of a dollar is 10 cents? 25 cents? 75 cents? 20. How many inches, or 36ths of a yard, in 1 of a yard ? In | of a yard ? In | of a yard ? 21. How many eighths of a gallon in a gallon and a half? In 5 gallons ? In quarter of a gallon ? 22. Reduce J, f , and -J, to fractions with the least common denominator. MODEL. As 8, the third denominator, exactly contains the others (2 and 4), it is the least common denominator. equals | ; equals fi AvtR A fi 1 b . -ATM*, a, g , 8 . 23. Reduce to fractions with the least common de- nominator f and ^V |, J, and f. J, T V, and f. 24. Reduce -J, f , and |f, to fractions having the least common denominator. Reduce ^, f , , and ^. 25. Reduce , &, f, and f , to fractions with the least common denominator. Reduce T 5 ^, ^-, |, |. 26. Reduce T 5 , f, 1^ (that is, |), to fractions with the least common denominator. Reduce 1J, ^, If. 27. Reduce f and f to fractions having a com- mon denominator. 28. Reduce f and | to fractions having the least common denominator. MODEL. 2, being a factor of both denominators 4 and 6, may be rejected from their product. 4 x 6=24. 24-f-2=12, least common den. J equals -, a 2 - ; J equals |f . Ans. -ff, }$. 29. Reduce % and -f to fractions having the least common denominator. Reduce ^ and f. Reduce f and |. Reduce \% and -f. Reduce -$ arid f. 56 FRACTIONS. SECTION 26. 1. What is Addition ? By what sign is it denoted ? 2. What is the result of addition called ? 3. How much are 9 times 5 and ^ of 5 ? 4. Ho\v much are 6 times 7 and ^ of 42 ? 5. How much are 4 times 6 and ^ of 33 ? 6. How many are 2 apples and 3 apples ? 2 books and 3 books ? 2 ninths and 3 ninths ? 7. How many are 4 tops and 5 tops ? 4 tenths and 5 tenths ? 4 elevenths and 5 elevenths ? 8. How much are 5 thirds and 6 thirds ? f + | ? 9. How much are 7 sixths and 8 sixths ? -J + f ? 10. What is the sum of -| and -f ? T \ and ^ ? 11. What is the sum of \\ and T \ ? MODEL. 11 fourteenths and 5 fourteenths are jf, or l-ft-, equal to 14-. -4ns. ly. 12. What is the sum of and f ? Off and | ? 13. What is the sum of T \, -f$, and T 6 g ? 14. Add T V T V and IT- A < i I, and f. 15. How much is -f f -f f ? How much is -f -f-f ? 16. How much is J + f+J-f- J? 17. How much is 7 increased by ^- + ^ ? 18. How much is 3 added to the sura of ^ and ^ ? 19. Add 4 and |. Add 4 and ^+f. 20. Add 4, 2, and the sum of | and |. 21. How much are 4^- and 2-| ? 3^ and 4^ ? 22. How much are 5f and 3| ? If and 7 ? 23. What is the sum of ^ of 26 and of 18 ? 24. What is the sum of ^, 5^, and 2^ ? MODEL, -fa, -,Aj-, and -,%, are |f, equal to 1-ft, or l. 1 and 5 and 2 are 8. ^4ws. 8k ADDITION OF FRACTIONS. 57 25. How much is 2 + -h3 ? IfV + T 9 o+ 5 ? 26. How much are If, arid ^ of 19 ? 27. A person who owned T ^of a steamboat, bought from two other parties T % and T 5 ; what part of the boat had he then ? 28. A man, having a farm of 253 acres, divides it equally among his 5 children. If his eldest child al- ready had 25| acres, how much land has he now ? 29. Reduce and | to fractions having a common denominator. SECTION 27. 1. How much are f and f ? 2. How much are f and f ? MODEL. }=&; J=|J. A + j|=^, or 1-fr ^w. 1-ft. 3. What is the difference between Examples 1 and 2? 4. How do we add fractions that have a common denominator ? 5. What must we first do, when they have not a common denominator ? 6. What is the sum of and ? Of \ and J? 7. What is the sum of } and -j- ? Of ^ and 1 ? 8. What is the sum of J and \ ? Of J and | ? 9. AYhat is the sum of \ and \ ? Of J- and ? 10. How much is \ + \ ? How much is |-f^? 11. How much is f + ? How much is |-f J? 12. How much is f + f? How much is | + f ? 13. How much is f + f ? How much is f + -J? 14. ^- + | how much? ^--f = how much ? 15. |+| + }=howmuch? j + ^ + . = how much? 58 FRACTIONS. 16. + |-f=howmuch? | + J + T 5_ how much? 17. How much is 1+f + J-? How much is f -f 11? 18. How much is 1 + 2 + f + T 1 g-? ij + 2^? 1 9. How much is 4f -f 2J- ? How much is 5^ + 3^ ? 20. How much is 1& + 7&? 3 4 3 + 4 l ? 21. How much are ^ of 19 and of 18 ? 22. How much are J of 23 and -J- of 42 ? 23. How much are r V of 20, of 18, and | of. 10 ? 24. A market-woman sold -J of her eggs for 40 cents, T 1 T of them for 9 cents, and -f of them for 57 cents. What part of her eggs did she sell, and for how much ? 25. A can do of a piece of work in 1 day, B J-, and C |. How much can all three do in a day ? 26. If a boy who had $4^, earned $2f more, and had $1-3*5- given him, how many dollars had he then ? 27. A walked half a mile in ^ of an hour, f of a mile in ^ of an hour, and 1| miles in T 8 T of an hour. How many miles did he walk, and how long did it take him ? 28. If I buy some muslin for $3J, lace for 82^, calico for $2^, and J- dozen collars for $7 2 %, how much change should I receive for a twenty-dollar bill ? 29. If a person buys a razor for f of a dollar and a strop for f of a dollar, for how much must he sell them both in order to make half a dollar ? 30. Helen had 83g\, Louise $5, and Mary 811. They put their money together, and divided it equally among several poor persons, so that each received ^ of the sum. How many poor persons were there, and how much did each get ? SUBTRACTION OF FRACTIONS. 59 SECTION 28. 1. What is Subtraction ? By what sign is it denoted ? 2. What is the result of subtraction called ? 3. How much is 11 times 3-^ of 40? 4. How much is 9 times 8 f of 36 ? 5. How much is 7 times 4 f of twice 8 ? 6. 4 pins 3 pins =how many pins? 4 ninths 3 ninths = how many ninths? $$? 7. 10 knives 3 knives=how many knives? 10 elevenths 3 elevenths ? H ?V ? 8. How much is H""i^ ? Ho w much i 9. How much is f T V ? How much is 10. From the sum of -g% and ff take %. 11. From the sum of fa fj, and f$, take J-J-. 12. From the sum of -^ and J take H"~A' 13. -| from ^- leaves how much ? -| from 2|- ? 14. | from Jy*. leaves how much ? | from 2f ? 15. Reduce f and f to fractions having a common (i denominator. SECTION 29. 1. From | take |. 2. From | take f. MODEL. equals fj; f equals |J. fj |J=A- ^ w - A* 3. What is the difference between Ex. 1 and 2 ? 4. How do we subtract one fraction from another, when they have a common denominator ? 5. What must we first do, when they have not a common denominator? 6. From take J. Take | from . 7. From $ take f Take | from J. 60 FKACTIOXS. 8. How much is f J ? How much is J 1 ? 9. How much is f f ? How much is J f ? 10. How much is y\? How much is J ? 11. Howmuchislf- ^? How much is 2f| | ? /- f - 12. How much is 3^ ^ ? "How much is4J-| 1? 13. How much is j J- ? How much is 1^ J ? 14. How much is |f f ? How much is 1^f ? 15. How much is 2| | ? How much is 5f ? 16. How much is 4| J-| ? How much is 3J |? 17. How much is 6J f ? How much is 2 J ? 18. How much is |fr? 2| J ? 2f-li ? 19. How much is A-t? 4 rV-|? 4 A~3J? 20. How much is 5|| ? 4^-f ? 3^- T V ? 21. How much is 3f- 1? 5^-2^? 4| ^ ? 22. What remains, if we take ^ from 5 ? NOTE. We reduce one of the 5 units to thirteenths, and then subtract. 5=4 + 1, or 4}f . -ft from 4|| leaves 4 |J. .4^. 4H- 23. From 3 take J. Take f from 7. $ from 9. 24. From 5 take f. Take f from 6. f from 1. 25. From 8 take f. Take ^ from 7. ^ from 1. 26. From 6 take If. XOTE. = J r ; 4=H- As we can not take in fr m ?V, w e re- duce one of the 6 units to f R, and add it to - 2 A ^ making f J. l^g from 5f * leaves 4-^ -. Ans. 4- 2 9 (J . 27. From 2J take If. Subtract 3f from 5f. 28. From 3| take If. Subtract 4^ from 9f 29. From 9 take 2|. Subtract 5f from 20. 30. From 50 take 40^. Take 99^ from 100. 31. From the sum of \ and T 5 take f. 32. From of 28 take the sum of 1J and f. SUBTRACTION OF FRACTIONS. 61 SECTION 30.--1. A grocer, having a bushel of po- tatoes, sold J of it to one customer, to another, and T 3 to a third. What part remained unsold ? 2. From a farm of 100 acres were taken three fields, containing 2^-, 4^-, and 3 T ^- acres. Plow many acres were left ? 3. If the sum of ^ and T \ of a person's age is 14 years, how old is he ? 4. From two remnants of calico, containing respect- ively 4f and 8J yards, were cut 12fi- yards for a dress. How much did what was left lack of 1 yard ? 5. The difference between f- and J of F's age was 26 years ; how old was he ? 6. A stage, after making and -f of its trip, had 7 miles yet to go ; how long was its trip ? MODEL. i= 2 %; l=2 8 o; 2~o + 2%=io. The whole trip was f$ of itself; when the stage had made | of the trip, there remained IS iu> r JMT- 1^ 2^0" of the trip was 7 miles, -$ was 4- of 7 miles, or 1 mile ; and f g, or the whole trip, was 20 times 1 mile, or 20 miles. Ans. 20 miles. 7. A man performed \ of his journey in the morn- ^ n g> i f ft i n tne afternoon, and the rest (12 miles) in the evening. How long was the journey ? 8. A person divided $101 equally between his two daughters. The elder then spent $25 J, and gave away $15| ; how much of her share had she left? 9 8 Two thirds of a certain rod is blue, f of it red, and the rest white. If the white part is 3 inches long, how long is the whole rod ? 10. How old am I, if the difference between \ and \ of my age is 3 years ? 62 FRACTIONS. 11. Bought some paper for $12-^; sold it for $13 V- By how much did the cost exceed the profit ? 12. Three pans contained respectively 2J, 4-|, and 3} quarts of milk. If 1 quarts were spilled, how many quarts were left ? 13. John and Cyrus had 6 dozen eggs each ; John sold 2 dozen, Cyrus 3 dozen. How many dozen did Cyrus have left, and how many eggs less than John ? 14. A can do a piece of work in 12 days, and B in 8 days ; what part can each do in one day ? How much can both together do in one clay ? After they have worked one day, how much of the job will remain ? 15. C can do a piece of work in 4 days, and D in 6 days. How much can both, working together, do in one day, and how much will then remain to be done ? 16. E can do a certain job in 9 hours, and F in 6 hours. After they have worked together at it one hour, what part of the job remains to be done ? SECTION 31. 1. What is Multiplication? By what sign is it denoted ? 2. What is the result of multiplication called ? 3. 6 times 3 equals what ? 5 times 3 apples equal what ? 5 times 3 sevenths ? . 5 times 3 ninths ? 4. How much is 5 times f ? * 5 times f ? 5. How much is 4 times \ ? 6 times f ? 6. How much is 7 times f ? 3 times f ? 7. How much is 9 times f ? 8 times f ? * Reduce the products to whole or mixed numbers, and fractions in the answers to their lowest terms. MULTIPLICATION OF FRACTIONS. 63 8. Multiply f by 2. NOTE. To multiply J by 2, we may double the number of parts. Twice three fourths is six fourths. Ans. f, or f . Or, we may double the size of the parts. Halves are twice as great as fourths (as we found on p. 52). Hence, twice three fourtfix is three halves. Ans. f . The answers agree. In the first case, we multiply the numerator by 2 : j x2 =f. In the second case, we divide the denominator by 2 : J +8 =|. The second method is shorter, when it can be used, because it brings the answer in its lowest terms at once. 9. How much is -^ x 5 ? ^ x 2 ? f x 3 ? 10. Multiply -j 5 g- by 4. ^ by 8. -^ by 9. 11. Multiply ^ by 3. Ji by 10. $ by 8. 12. What is the product of \ and 6 ? ^and 12 ? 13. What is the product of fa and 5 ? -^ and 11 ? 14. What is the product of ^f and 9 ? f| and 8 ? 15. What is the product of 10 and f ? T 4 and 7 ? 16. What two ways are there of multiplying a frac- tion by a whole number ? 1 7. Which is preferable ? Why ? 18. What is 4 times |- ? 4 times f? 4 times f? 19. When you multiply a fraction by its own de- nominator, what do you get ? 20. How much is 5 times f ? 10 times T 7 ? 21. How much is 9 times f ? 19 times -f4 ? 22. How much is 100 times yfg- ? 25 times f|? 23. How much does 4 times fa lack of 10 ? 24. What must be added to 15 times ^f , to make 7 ? 25. Which is greater, 5 times T 2 T or 2 1|? 26. 7 times ^ is |- of what number? 27. 8 times -^ is of what number ? - 28. 3 times -^ T is f of what number? 64 FRACTIONS. SECTION 32. 1. How much is 5 times 3-| ? MODEL. 5 times f is 3, or 2 ; 5 times 3 is 15. 15 + 2?=17f-. /4?w. 17f. 2. How much is 6 times 2J? 4 times 8f ? 3. How much is 3 times 5| ? 7 times 1^ ? 4. How much is 9 times 4-J ? 5 times 2J ? 5. How much is 10 times 7|- ? 12 times 3f ? 6. Multiply 6f by 2. By 5. By 7. By 10. 7. What cost 8 vests, at $5f each ? 8. How much ale in 6 cans, holding 2f pints each ? 9. What will 9 hens weigh, averaging 2|- pounds ? 10. Multiply 6 by J. NOTE. 6 multiplied by % must be * as much as 6 multiplied by 1, that is, J of 6. Multiplying a number by *, i, &c., is therefore equivalent to taking *, , &c., of that number. 11. Multiply 24 by J. By f. By 1 By f. 12. Multiply 18 by f By f. By f. By f. 13. How much is of 2 ? MODEL. J of 1 is J, and J of 2 is twice , or f. -4ws. f . 14. How much is \ of 2? iof2? T Vof7? 15. How much is } of 3 ? J of 5 ? T V of 6 ? 16. Multiply 6 by ^. Multiply 4 by T V 7 by / f . 17. How much is f of 21 ? MODEL. One sixth of 21 is 3$, and five sixths are 5 times 3, or 17*. ^ws. 17*. 18. How much is f of 5? | of 9 ? f of 11 ? 19. How much is f of 8 ? f of 9 ? & of 14 ? 20. How much is of 2 ? f of 2 ? $ of 2 ? 21. How much is J of 3 ? ^of4? -^oflG? 22. How much is of 100 ? f of 100 ? f of 100 ? 23. How much is 7 times 2f- ? f of 4 ? -f of 11 ? MULTIPLICATION OF FRACTIONS. 65 24. How far will a locomotive, moving at the rate of 20 T ^ miles an hour, go in 5 hours ? In 8 hours ? 25. What cost two pieces of meat, weighing re- spectively 8-| and 6^-g- pounds, at 20^ cents a pound ? 26. At $8f an acre, what will be the cost of 3 fields containing 3f acres each ? 27. What is the weight of four cheeses, if two them weigh 9f pounds each, and the other two pounds each ? 28. If I divide $2 equally among 8 beggars, what part of $2 will each receive ? What part of $1 ? 29. A person bought 56 pounds of butter, and sold f of it ; how many pounds remained on hand ? 30. Emma is 3| years old ; Laura is 5 times as old as Emma. In how many years will Laura be 21 ? SECTION 33, 1. How much is 5f times 7? MODEL. 5 times 7 is 35 ; f of 7 is V~> or 4|; 35 + 4|=39|. Ans. 39J. 2. How much is 2 times 24 ? 3f times 14 ? 3. How much is 5f times 10 ? 4f times 15 ? 4. How much is 8f times 12? 6| times 16 ? 5. How much is 3J times $ of 18 ? 6. What cost ?! acres, at $10 an acre? 7. If f of a ton of hay cost $15, what will 1 ton cost ? What will of a ton cost ? 8. If | of a ton of hay cost $20, what will f of a ton cost ? What will 2f tons cost ? 9. What are 3^ gallons of petroleum worth, if | of a gallon is worth 21 cents ? 60 FRACTIONS. 10. |- of 4 equals how many? -|- of 4 apples ? of 4 ninths? Jof^? Jof-^? 11. ^ of 6 equals how many? ^ of 6 knives ? ^ of 6 sevenths ? of ? J- of ^ ? 12. How much is J of f ? of f J ? T V of f-? 13. What is multiplying by a fraction equivalent to ? Ans. To taking such a part as is denoted by the fraction. Multiplying by % is equivalent to taking -J-. 14. Multiply |f by f Multiply ff by T V Multi- ply ^ by |. Multiply -|f by ^. 15. How much is f of *- ? MODEL. One fourth of L f- is f , and three fourths are 3 times ^ or ^, equal to 1^. Ans. 1J. 16. How much is f of |-|? How much is f of ff-? 17. How much is J of fj ? How much is f of 1|| ? 18. How much is f of- 2 / ? How much is f of 2f ? 19. How much is f of -^ ? How much is of 1^ ? 20. How much is of f of -^-? * A 21. How much is f of If ? How much is f of f ? 22. How much is f of f of If ? 23. How much is f of f of 2f|? 24. If a person bought f of a ship, and then sold of his share, what part of the ship did he retain ? 25. When cheese is -f^ of a dollar a pound, how much will f of a pound cost ? 26. Jane's age is J of Clara's, and Clara's is f of Lucy's. If Lucy is 18, how many years old is Jane ? 27. If a cow eats -fa of a ton of hay in a week, what will her daily supply of hay cost when hay is $21 a ton? 4/s DIVISION OF FRACTIONS. ' 67 SECTION 34. 1. What is Division ? By what sign is it denoted ? 2. What is the result of division called ? 3. Dividing by 2 is equivalent to taking what part ? 4. Dividing by 3 is equivalent to what ? Dividing by 4? Dividing by 7 ? 5. How many times is 6 contained in 18 ? How much is i of 18 ? Divide 18 by 6. 6. How many times is 6 contained in 18 sevenths ? How much is % of if- ? Divide if- by 6. 7. Divide | by 2. NOTE. To divide f by 2, we may take of the number of parts ; of eight halves is four halves, or 2. Ans. 2. Or, we may make each part half as great. FourtJis are half as great as halves (as we found on p. 52). Hence, of 8 halves is 8 fourths, or 2. Ans. 2. The answers agree. In the first case, we divide the numerator by 2 : **== 2 In the second case, we multiply the denominator by 2 : x2 =i 2 The first method is preferable when the whole number is exactly contained in the numerator of the fraction. 8. What two ways are there of dividing a fraction by a whole number? Which is preferable? 9. When can the first method be used ? 10. How many times is 6 contained in f ? NOTE. As 6 is not exactly contained in 2, we have to use the second method: f x 6-&i or JM- ^ ns - ~2T- 11. Divide f by 9. Divide f by 6. Divide f by 8. 12. How many times is 10 contained in T 9 T ? In -i~| ? 13. How much is J of f ? $ of T V? -J- of & ? ' 14. How much is f of f ? MODEL. One fourth of f is / -, or -fa ; and three fourths are 3 tims -fa, or -fc Ans. -, 3 . 08 FRACTIONS. 15. How much is of f ? of f ? f of f ? 16. How much is of if ? i of If ? f of If ? 17. How much is | of f ? 1 of 1 ? f of 1| ? 18. How much is f of 2| ? f of 2| ? f of 3| ? 19. A, B, and C, own a ferry, having equal shares. B sells f of his share to C ; what part of the ferry does B own after this sale, and what part C ? How much more has C than B ? 20. If 1 pound of butter costs f of a dollar, how much will J of a pound cost ? 21. Mary has $1-^ and George $2f. They divide J of what they both have equally between two poor persons. What part of a dollar does each receive ? 22. By how much does f of T ^ exceed \ of f- ? 23. How much is 3| times 2J ? MODEL. 2}=-^. 3 times *=*, or 8 J. One third of ^ is f^ and two thirds are twice -J-J, or \* =lf. 8+ 1 J = 10f 2 -. Am. 10 -,V 24. How much is 2J- times 1 J ? If times 4 ? 25. How much is 4^ times 60J? 3| times 41|? 26. How much is 5| times 10 T V? 6 times 18 T V? SECTION 35. 1. How many halves are there in 1? In 2 ? In 5 ? 2. How many times is contained in 1 ? In 2 ? 3. Dividing by ^ is equivalent to what ? Ans. Dividing by 1 is multiplying by 2. 4. Multiplying by ^ is equivalent to what ? Ans. Multiplying by \ is taking ^, or dividing by 2. 5. If we divide 8 by -J, what is the quotient ? If we multiply 8 by , what is the product ? DIVISION OF FRACTIONS. 69 6. Dividing by -^ is equivalent to what? Am. Dividing by | is multiplying by 3. 7. Dividing by 4- is equivalent to what ? By \ ? 8. How many times is ^ contained 'in 3 ? In 9 ? 9. How many times is ^ contained in 2 ? In 3-^- ? In4 T V? In6f? In7|? 10. How many times is 3| ? 4-- ? 5f ? 6| ? 11. How many times % is f ? Divide f by . 12. How many times \ is |- ? Divide f by ^-. 13. How many times ^ is -f- ? 14. How many times T ^ is If? 15. How many times is -^ contained in 1^ ? 16. How many times is -^ contained in 2^ ? 17. How many times is $ contained in f ? MODEL. One sixth is contained in f , L f or f times ; and five sixths, being 5 times as great, is contained in it i of J times, or -, 9 - times. Ans. <$,- times. 18. How many times is f contained in f ? f in f ? 19. How many times is f contained in f ? | in -|? 20. f in ^- how many times? -| in If? ^ 21. -| in ^ how many times ? f in 1^- ? 22. ^ in 4| how many times ? -^ in 1^? 23. f in f how many times? If in f ? 24. J in | how many times ? If in 1? 25. if i n 2_2 now mari y times? If in 34-? 26. 3-^ in 2f how many times ? 4| in 5|? 27. How far will a person, walking at the rate of 2-| miles an hour, walk in 3^ hours ? 28. If the current of a river moves 2f miles in 3J hours, how far will it move in 1 hour ? 29. How much is 5 x 2f ? How much is 6^-7- 2f ? 70 FRACTIONS. SECTION 36. 1. How many times is 9 contained in lOOf ? MODEL. 9 is contained in 100, 11 times and 1 over. 9 is con- tained in If, or 2, i time. Ans. \\\ times. 2. How many times is 11 contained in 84^? 3. How many times is 7 contained in 42|^? 4. How many times is 12 contained in 39f? 5. How many times is 5 contained in 43f ? 6. How many times is 4 contained in 49f ? 7. How many times are 7 apples contained in 14 apples ? 7 cups in 14 cups ? 7 ninths in 14 ninths ? 8. How many times are 5 pins contained in 45 pins ? 5 eighths in 45 eighths ? -| in ^ ? ^ in *f- ? 9. How do we divide one fraction by another, when they have a common denominator? ^Lns. Divide nu- merator by numerator, rejecting the denominators. 1( >. - 4 <5M-f = now many? ^M-? If-v-f? 5-f-v- 119 J.4 I 2 9 q R -ISO 1 8' f" s " 1 f- r t5 T2-~ 1 T3 i 11. What cost | of a barrel of cider, at $8 a bar. ? 12. A person laid out $55J for fuel. He bought 5 tons of coal at 87 J a ton, and spent the rest for wood at $3 1- a load ; how many loads of wood did he buy ? 13. If a horse goes 2^ miles in an hour, what part of an hour will it take him to go one mile ? 14. A farmer sold f of his flock, and had 17 sheep left; how many sheep had he originally? 15. A farmer sold 14 sheep, and had f of his flock left ; how many sheep had he left ? 16. What will a pile of 16f bushels of potatoes bring, if put up in bags holding 2| bushels each, and sold for 8l T V per bag ? FRACTIONS. 71 17. 7 is f of what number ? MODEL. If 7 is two thirds of a certain number, one third of it is of 7, or I ; and three thirds, or the whole, are 3 times J, or ^ which equals 10. Ans. 10. 18. 9 is f of what number ? 19. 12 is f of what number ? 20. | of 20 is | of what number? 21. ^ of 30 is | of how many times 8 ? 22. -J of 32 is | of how many times |? 23. of 16 is | of how many times 10 ? 24. f of 25 is f of how many times 12 ? 25. f of 63 is -J- of how many times J of 24 ? 26. | of 40 is | of how many times 3| ? 27. | is f of what number? 28. | is f of what number? 29. |f is | of what number ? 30. f is f of what number? 31. If is |f of what number? 32. If is | of what number ? 33. f is |^ of how many times f? MODEL. If f is four thirds of a certain number, one third of it is of f , or - 2 3 y ; and three thirds, or the whole, are 3 times -/-$, or ro- 2 9 o is as many times f as f is contained times in f . One sixth is contained in 2% 6 times 2 %, or f , which equals f }, times ; and Jive sixths is contained in it -t of f times, or f . ^?w. f J. 34. |-| is f of how many times 1 ? 35. of li is | of how many times 1| ? 36. | of -fif is 4 times what number ? 37. f of l^j is |- of | of what number ? 38. | of 24- is | of J of how many times 2 ? 39. | of 3f is Y\F of how many times | ? 72 FEDERAL MONEY. 40. If a grocer buys cheese for 14| cents a pound, and sells it for 16 cents, how many pounds will he have to sell in order to make 7-J cents ? 41. Suppose that it takes 2-J yards of merino to make a sack, and 7J yards to make a dress. After 3 sacks have been cut from a piece containing 22 f yards, how many dresses can be cut from what remains ? CHAPTER SIXTH. FEDERAL MONEY. SECTION 37. 1. One hundred cents make $1 ; how many cents are there in $5 ? MODEL. Since there are 100 cents in $1, in $5 there are 5 times 100 cents, or 500 cents. Ans. 500 cents. 2. One hundred cents make $1 ; how many dollars in 500 cents ? MODEL. Since there are 100 cents in $1, in 500 cents there are as many dollars as 100 cents are contained times in 500 cents, or 5. Ans. $5. 3. When we change $5 to 500 cents, do we alter the value ? Do we alter the form ? What is this pro- cess called ? Ans. Reduction. 4. When we change dollars to cents, do we go to a higher or lower denomination ? What is this kind of Reduction called ? Ans. Reduction Descending. 5. When we change cents to dollars, do we go to a higher or lower denomination ? What is this kind of Reduction called ? Ans. Reduction Ascending. REDUCTION OF FEDERAL MONEY. 73 SECTION 38. What is Federal Money? Ans. Federal Money is the currency of the United States. TABLE OF FEDERAL MONEY. 10 mills (in.) make 1 cent, . . . . c., ct. 10 cents, 1 dime, . . . . di. 10 dimes, 1 dollar, . . . $. 10 dollars, 1 eagle, . . . . E. Accounts are kept in dollars and cents. Cents are written at the right of dollars, with a period between, and occupy two places. The first place is filled with a naught, if the cents are expressed by but one figure. Mills are written at the right of cents. Thus : Nine dollars, 5 cents, ..... $9.05 Nine dollars, fifty cents, ..... $9.50 Nine dollars, fifty cents, five mills, . . $9.505 Nine dollars, five mills, ..... $9.005 As 100 cents make a dollar, 1 cent is j^ of a dollar, 2 cents T^O> & c - Cents are sometimes written as hundredths of a dollar ; five dollars and twenty-five cents may be written 1. How many mills in 3 cents ? * In 7c. ? In 19c. ? 2. How many cents in 5 dimes ? In 11 dimes? 3. How many dimes in $2 ? In $8 ? In $10 ? 4. How many dollars in 4 eagles? In 12 eagles ? 5. How many dollars in 50 dimes?* In 170 dimes? 6. How many eagles in $60 ? In $90 ? In $20 ? 7. How many dimes in 50 cents ? In 90c. ? In40c. ? 8. How many cents in $5 ? MODEL. Since 10 cents make 1 dime, and 10 dimes make $1, in $1 there must be 10 times 10, or 100, cents ; and in $5, 5 times 100, or 500, cents. Am. 500 cents. * See MODELS, on page 72. 74 FEDERAL MONEY. 9. How many cents in $9 ? In 4 eagles ? How many dimes in 7 eagles ? How many mills in 3 dimes ? In $2 ? 10. How many cents in 6 dollars 35 cents ? MODEL. In $1 there are 100 cents, and in $6 six times 100 cents, or 600 cents. 600 cents + 35 cents =635 cents. Ans. 635c. In such cases, to reduce to cents we need only remove the pe- riod and dollar-mark : $6.35=635 cents 11. How many cents in $4.89? In $10.10? In ~$9.05 ? How many mills in $2.375 ? In 87c. 5m. ? 12. How many dollars in 635 cents ? MODEL. 100 cents make $1, and 635 cents will make as many dollars as 100 cents are contained times in 635 cents, or 6-^fj. Ans. :6-ft&, or $6.35. In such cases, we need only cut off the two right-hand figures for cents, and what remains on the left will be dollars : 635 cent3=,$6.35 13. How many dollars in 757 cents? In 843 cents? In 926 cents ? In 4270 mills? 14. How many eagles in $17 ? How many cents in 53 mills ? How many dimes in 47 cents? 15. How many cents in half a dime? In 1 9|- dimes ? In 5| dollars ? How many dollars in 5 half-eagles ? 16. How many cents in half a dollar? In \ of a -dollar? In of 81? In $t ? In $? In $-1 ? 17. What part of a dollar is 50 cents ? MODEL. Since 100 cents make a dollar, 1 cent is TOTT of $1, ; and 50 cents are 50 times T foj, or -&% which equals . Am. i of .-a dollar. 18. What part of a dime is 3 cents ? What part of :an eagle is $4 ? What part of a cent is 6 mills ? REDUCTION. 75 19. What part of a dollar is 75 cents? 33^ cents? 10 cents? 12J cents? 25 cents ? 20 cents? 16| cents? 20. What cost 44 Grammars, at 75c. apiece ? MODEL. 75c. is f of $1. 44 Grammars, at $1 apiece, would cost $44, and at f of a dollar, they cost J of $44, or $33. Ans. $33. 21. What cost 66 rulers, at 12|- cents apiece ? 22. What cost 2 dozen Readers, at 33-|c. each ? 23. At 25 cents each, what cost 18 slates? 24. At 16f cents each, what cost 40 magazines ? 25. At 50c. each, how many books can I buy for $5 ? 26. At 33^c. each, how many balls can I buy for $9 ? CHAPTER SEVENTH. REDUCTION. SECTION 39, 1. What is Reduction ? Ans. Re- duction is changing a quantity from one denomina- tion to another, without altering its value. 2. What is Reduction Ascending ? Ans. Reduc- tion Ascending is changing a quantity from a lower denomination to a higher, without altering its value. 3. What operation do we use in Reduction Ascend- ing ? Ans. Division. 4. What is Reduction Descending ? Ans. Reduc- tion Descending is changing a quantity from a higher denomination to a lower, without altering its value. 5. What operation do we use in Reduction De- scending? Ans. Multiplication. 76 REDUCTION. SECTION 40. What is English or Sterling Money ? Ans. Tb^ currency of Great Britain. TABLE OF STEELING MONEY. 4 farthings (far., qr.), 1 penny, . . d. 12 pence, 1 shilling, . . s. 20 shillings, 1 pound, . . . 21 shillings, 1 guinea, . . guin. 1. How many pence in 1 5s. 6d. ? MODEL. In 1 are 20s. 20s. + 5s.=25s. In Is. are 12d., and in 25s. 25 times 12d., or 300d. 300d. + 6d.=306d. Am. 306d. 2. How many pounds, &c., in 306 pence ? MODEL. 12d. make Is. ; 306d. will therefore make as many shillings as 12 is contained times in 306, or 25s. and 6d. over. 20s. make 1 ; 25s. will therefore make as many pounds as 20 is con- tained tunes in 25, or 1 and 5s. over. Am. 1 5s. 6d. 3. How many shillings in 9 3s. ? In 4 guin. 7s. ? 4. How many farthings in 9s. 8d. ? In lid. 3qr. ? 5. How many pounds, &c., in 73s. ? In 251d. ? 6. How many pence in 5 ? In 6 lid. ? 7. How many shillings, &c., in 89 far. ? In 5 guin. ? 8. How many pence in 1 guin. 4s. ? In % of 1 ? 9. How many farthings in 3s. 9fd. ? In ll|d. ? 10. How many pence in 1 ? Farthings in Is. ? 11. What part of a pound is 5s. ? 4s. ? 10s. ? 12. What part of a shilling is 4d. ? 8d. ? 9d. ? 13. At 6d. a pound, what cost 40 pounds of meat? NOTE. Reduce the answers to the highest denomination possible, 14. What cost 8 knives, at 3s. each ? 15. What cost 6 dozen slates, at 8d. each? TROY WEIGHT. 77 SECTION 41. For what is Troy Weight used ? A/is. For weighing gold, silver, and precious stones. TABLE OF TROY WEIGHT. 24 grains (gr.) make 1 pennyweight, pwt. 20 pennyweights, 1 ounce, . . . oz. 12 ounces, 1 pound, . . To. 1. Reduce 86 oz. to pounds, &G. To pennyweights, 2. How many grains in 1 oz. ? In 5 pwt. 20 gr. ? 3. How many pounds in 250 pwt. ? In 100 oz. ? 4. How many grains in 4 oz. 3 pwt. ? 5. How many ounces in J of a Ib. ? In 5-f- Ib. ? 6. How many pennyweights in \ Ib. ? In 4|- oz. ? 7. How many grains in f of a pwt. ? In 2f pwt. ? 8. How many grains in \ of an ounce ? In f pwt. ? 9. What part of a pwt. is 18 grains ? 14 grains? 30. What part of an ounce is 15 pwt. ? 1 grain ? 11. What part of an ounce is -fa of a pound ? MODEL. 1 Ib. is 12 oz. ; and -^ r of a Ib. is -^ of 12 oz., or J of 1 oz. iJ=f. ^4n*. I oz. 1 2. What part of a pennyweight is -fa of an ounce ? 13. What part of a grain is -fa of a pennyweight? 1 14. What part of a pound is 120 pwt. ? 15. A jeweller bought 7 pwt. of pure gold, 9 pwt. of gold coin, and 16 pwt. of silver ; how many ounces, did he buy altogether ? 16. What is the weight in pounds, &c., of 12 spoons, of 30 pwt. each ? 17. How many spoons, weighing 35 pwt. each, canu be made out of 1 Ib. 2 oz. of silver ? 78 REDUCTION. SECTION 42. By whom is Apothecaries' Weight used ? Ans. By apothecaries, in mixing medicines, TABLE OF APOTHECARIES' WEIGHT. 20 grains (gr.) make 1 scruple, . sc. or 3. 3 scruples, 1 dram,. . dr. or 3. 8 drams, 1 ounce, . oz. or 3 . 12 ounces, 1 pound, . Ib. or ft. 1. How many scruples in 5 oz. ? In 1 Ib. ? 2. How many drams in 95 gr. ? In 4 Ib. 5 oz. ? 3. How many grains in 3 dr. 5 sc. ? In J Ib. ? 4. Ho\v many pounds in 113 3 ? In 113 ? 5. Reduce 500 gr. to ounces, &c. -| oz. to scruples. 6. What part of a scruple is 17 gr. ? 8 gr. ? 7. What part of an ounce is 3 drams? 2 sc. ? 8. What part of a pound is 5 oz. ? -J- dram ? 9. What fraction of a dram is -fa of an ounce ? 10. What part of a grain is -^ of a scruple ? 11. How much are f of a dram and ^ of an ounce? MODEL. In 1 oz. are 8 dr., and in of an ounce J of 8 drams, or 2| drams. 2| drams + J dr.=3 ,% drams. Ans. 3 ^ dr. 12. How much are f of a Ib. and ^ of an ounce ? 13. How much are f of a dram and 3J sc. ? 14. How much are f of an ounce and ^ of a dram ? 15. How many drams, &c., will it take for 20 pow- ders, each containing 20 grains ? 16. How many powders, of 1 J sc. each, can be put up from an ounce of soda ? 17. If a druggist charges 50c. for ten powders, con- taining 15 gr. each, at what rate is that per ounce? AVOIRDUPOIS WEIGHT. 79 SECTION 43. For what is Avoirdupois Weight used ? Ans. For weighing groceries, meat, coal, cot- ton, drugs when sold in quantities, and all articles except gold, silver and precious stones. TABLE OF AVOIRDUPOIS WEIGHT. 16 drams (dr.) make 1 ounce, . . . oz. 16 ounces, 1 pound, . . . . Ib. 25 pounds, 1 quarter, . . . qr. 4 quarters, 1 hundred- weight, cwt. 20 hundred-weight, 1 ton, . . . . T. 1. How many drams in a pound ? Ounces in a quarter? Pounds in 1 cwt. ? Pounds in a ton ? 2. How many pounds in 5 T. 3 cwt. 21 Ib ? In \ of a ton ? In 35 ounces ? In 4 T. ? 3. How many ounces in 5 Ib. ? In 83 dr. ? In 2 cwt. 10 Ib. 5 oz. ? In 3 qr. 15 Ib ? 4. In 12 bales of cotton, averaging 400 Ib. each, how many tons, &c. ? 5. HQW many tons in 9 hogsheads of sugar, con- taining an average of 1000 pounds each ? 6. How many four-ounce weights can be made out of 2 pounds of brass ? 7. How many seven-pound packages of flour can a grocer put up from 2 cwt. ? 8. What cost 2 cwt. of cheese, at 16jc. a pound ? 9. Bought 500 Ib. of straw, at 75c. per cwt., and 400 Ib. of hay, at $1.25 per cwt. ; what was the bill ? 10. Bought 1 cwt. of meat for $17.50; sold it at 22 cents a pound. What was the whole selling price ? What was the profit ? 80 SEDUCTION. SECTION 44. What articles are weighed by Tro y Weight ? By Avoirdupois Weight ? By Apotheca- ries' Weight ? What kind of pounds (Troy, Avoirdu- pois, or Apothecaries') are those in the following Table ? MISCELLANEOUS TABLE. 14 pounds, . . 1 stone of iron or lead. 60 pounds, . . 1 bushel of wheat. 100 pounds, . . 1 quintal of dried fish. 100 pounds, . . 1 cask of raisins. 196 pounds, . . 1 barrel of flour. 200 pounds, . . 1 bar. of beef, pork, or fish. 1. How many pounds in 5J- stone ? In 3^ quin- tals of cod-fish ? In a quarter of a barrel of flour ? 2. Which is greater, 7 stone or 1 cwt., and how much ? 3. How many bushels in 3000 Ib. of wheat ? 4. What cost 4 quintals of fish, at 6c. a pound ? 5. What part of 1 cwt. is 1 stone ? 6. What part of a ton is 5 cwt. ? 1200 Ib. ? 7. What part of an avoirdupois pound is 2 ounces ? What part of a Troy pound is 2 ounces? What part of an apothecaries' pound is 2 ounces ? 8. What fraction of an avoirdupois ounce is -fa of a pound ? What part of a Troy ounce ? 9. How do the ounce and pound of Troy Weight compare with those of Apothecaries' Weight ? Arts. They are the same. 10. How do the ounce and pound of Troy Weight compare with those of Avoirdupois Weight ? Ans. The Troy ounce is greater, the Troy pound less. LONG MEASUEE. 81 SECTION 45. For what is Long Measure used ? Ans. For measuring length or distance. 1 inch. TABLE OF LONG MEASURE. 12 inches (in.) make 1 foot, . . ft. 3 feet, 1 yard ) . . y<*. 5| : yards, 1 rod, . . . rd. 40 rods, 1 furlong, . . . fur. 8 furlongs, 1 mile, mi. 1. How many inches in t. JJft. ? In 2 yd. 9 2. How many inches in 4 yd. 1 ft. 5 in. ? 3. How many yards, &c., in 91 ft. ? In 80 inches? 4. How many rods in a mile ? How many yards ? 5. How many feet in a mile ? In half a mile ? 6. How many miles in 480 rods ? In 18 furlongs ? 7. How many feet in 2 rods ? In 99 inches ? 8. How long will it take a person, walking at the rate of 20 rods a minute, to go 2 miles ? 9. 4 inches make a hand. What is the height in feet of a horse 15 hands high? 10. What is used in measuring drygoods ? Ans. The yard of long measure, divided into halves, quar- ters, eighths, and sixteenths. 11. Three dress-patterns, 9, 10^, and 8-J yards long, were cut from a piece containing 36 yards ; how many yards were left ? What did the three dresses cost, at 60 cents a yard ? 12. How many half-yards of velvet can be cut from a piece 9|- yards long ? How many eighths ? 1 inch. 82 REDUCTION. SECTION 46. 1. For what is Square Measure used ? Ans. For measuring surfaces ; such as land, walls, floors, &c. 2. What is a Square? Ans. A A SQUARE INCH. Square is a figure that has four equal sides perpendicular one to another that is, leaning no more to one side than to the other. 3. What is a Square Inch? Ans. A square whose sides are each an inch long. TABLE OF SQUARE MEASURE. 144 square inches (sq. in.), 1 square foot, sq. ft. 9 square feet, 1 square yard^ sq. yd. 30J square yards, 1 square rod, sq. rd. 40 square rods, 1 rood, . . R. 4 roods, 1 acre, ... A. 640 acres, 1 square mile, sq. mi. 4. How many square rods in 2 A. 3 R. ? 5. What part of a square foot is 36 sq. in. ? 6. How many square feet in 7 sq. yd. ? 7. Reduce sq. rd. to square yards. 8. How many 80-acre farms will 1 sq. mi. make ? 9. Reduce 8 square rods to square yards. 10. What part of a rood is ^ of an acre ? 11. How many square rods in J- A. and f R. ? 12. A person, having 20 A. of land, sold it off in lots of 10 sq. rd. How many lots did it make? 13. What will it cost to plaster 288 sq. ft,, at 25 cents a square yard ? CUBIC MEASURE. 83 SECTION 47. 1. For what is Cubic Measure used ? Ans. For measuring bodies, which have length, breadth, and thickness ; such as timber, earth, boxes, &G. 2. What is a Cube? Ans. A Cube is a body bounded by six equal squares. 3. What is a Cubic Inch ? Ans. A cube, one inch long, one inch broad, and one inch thick. Each of its six sides is a square inch. 4. W h a t is a Cord ? Ans. A Cord is a pile of wood 8 ft. long, 4 ft. wide, and 4 ^ ft. high. One foot in length of such a pile is called a Cord Foot. TABLE OF CUBIC MEASURE. 1728 cubic inches (cti. in.), 1 cubic foot, cu. ft. 27 cubic feet, 1 cubic yard, cu. yd. 40 cu. ft. of round, or ) , n ,, , . -, > 1 ton or load, T. 50 cu. ft. ol hewn timber, ) 16 cubic feet, 1 cord foot, cd. ft. 8 cord feet, 1 cord, . . Cd. 5. How many cords in a pile of wood, 24 feet long, 4 feet wide, and 4 feet high ? MODEL. 1 cord being 8 ft. long, 4 ft. wide, and 4 ft. high, there are as many cords as 8 x 4 x 4 is contained times in 24 x 4 x 4, or 3. Ans. 3 Cd. 6. How many cords in a pile of wood, 36 feet long, 4 feet wide, and 8 feet high ? 84 REDUCTION. SECTION 48, For what is Liquid or Wine Measure used ? Ans. For measuring liquids generally. TABLE OF LIQUID MEASURE. 4 gills (gi.) make 1 pint, . . . . pt. 2 pints, 1 quart, . . . . qt. 4 quarts, 1 gallon, . . . gal. 3 1|- gallons, 1 barrel, . . . . bar. 2 barrels (63 gal.), 1 hogshead, . . hhd. 2 hogsheads, 1 pipe, . . . .pi. 2 pipes, 1 tun, .... tun. 1. How many gallons in 45 pt. ? In 150 gi. ? 2. How many quarts in 101 gi. ? In 2 bar. ? 3. Reduce 5 gal. 1 pt. 1 gi. to gills. 4. Reduce f of a gallon to pints. To gills. 5. Reduce 3 qt. 1 pt. to gills. 6. What part of a gallon is 1 qt. ? 1 pt. ? 7. Add gal., % qt., and | pt. 8. How many gallons in 5 hhd. ? In 8 bar. ? 9. If a tumbler holds half a pint, how many times will 1 gallon of water fill it ? 10. Bought 5 gal. of oil for $10, what is the price per quart ? 11. Bought a pint of milk for 4 cents; at what rate is that per gallon ? 12. What cost 18 quarts of wine, at 16s. per gal.? 13. 3 pt. of molasses having been sold out of 2 gal., what is the remainder worth at 80c. a gal. ? 14. If 5 bottles hold a gallon of wine, what part of a quart does each hold ? DRY MEASURE. 85 SECTION 49. For what is Dry Measure used ? Ans. For measuring grain, vegetables, salt, coal, and other articles not liquid. TABLE OF DRY MEASURE. 2 pints (pt.) make 1 quart, . . . . qt. 8 quarts, 1 peck, .... pk. 4 pecks, 1 bushel, . . . bu. 36 bushels, 1 chaldron, . . . chal. 1. How many bushels in 165 qt. ? In 191 pt. ? 2. Reduce 1 bu. 5 qt. to quarts. 3. How many bushels in 5 chaldrons ? 4. How many pints in 3 bu. 3 pk. 3 qt. ? 5. Reduce -f^ of a peck to lower denominations. MODEL. 1 pk. = 8 qt. Hence, -f- - of a peck is -, 9 - of 8 qt., or ^ of 1 qt, which equals 7-J- qt. 1 qt.=2 pt. Hence, J of a quart is of 2 pt., or | of 1 pt. Ans. 7 qt. f pt. 6. Reduce -^ of a bushel to lower denominations. 7. Reduce f of a peck to lower denominations. 8. How many bushels in 5f chaldrons ? 9. If a chaldron of coal costs $12, what is the price per bushel ? 10. How many baskets, holding 2^ pecks each, will 5 bushels of peaches fill ? 11. What part of a bushel is half a peck? 12. What part of a quart is fa of a peck ? 13. 6 quarts are -| of how many pecks ? 14. If a horse is fed 6 qt. of oats a day, how long will it take him to consume 3 bushels ? 15. If a family consume 4 bushels of coal a day, how long will it take them to use 3^ chaldrons ? 86 REDUCTION. SECTION 50. 1. What are the natural divisions of time ? Am. The Year, in which the Earth revolves round the Sun ; and the Day, in which it turns on its axis. 2. How is the year divided ? Ans. Into twelve calendar months, differing in length. 3. How is the day divided ? Ans. Into hours, minutes, and seconds. TABLE OF TIME MEASURE. 60 seconds (sec.) make 1 minute, . . min. 60 minutes, 1 hour, . . . h. 24 hours, 1 day, . . . da. 7 days, 1 week, . . . wk. 365 days or ) _ < _ 12 calendar months, ) 366 days, 1 leap year. 100 years, 1 century, . . cen. 4. Learn the names of the calendar months, and the number of days they contain : DATS. /st month, January, 31. 2d month, February, 28. 3d month, March, 31. 4th month, April, 30. 5th month, May, 31. 6th month, June, 3Q. DATS. 7th month, July, 31. 8th month, August, 31. 9th month, September, 30. 10th month, October, 31. llth month, November, 30. 12th month, December, 31. 5. How many hours in the month of April ? 6. How many minutes in a day ? In 2 days 3 h. ? 7. How many seconds in 2 h. ? In 1 h. 10 min. ? 8. How many weeks, &c., in 240 hours? PAPER MEASURE. 8 7 SECTION 51. TABLE OP PAPER MEASURE. 24 sheets make 1 quire. 20 quires, 1 ream. 2 reams, 1 bundle. 5 bundles, 1 bale. TABLE OF COLLECTIONS OF UNITS. 12 units make 1 dozen, doz. 12 dozen, 1 gross. 12 gross, 1 great gross. 20 units, 1 score. 1. How many sheets in a ream of paper ? 2. How many sheets in 8f quires ? In f of a ream ? 3. How many quires will 240 sheets make? 4. How many bundles will 100 quires make ? 5. What part of a ream is 37 sheets ? 15 quires ? 6. What part of a gross is 5 dozen ? 1 score ? 7. How many units in a great gross ? In 11 gross ? 8. If a box of pens holds a gross, how many pens are there in 5 boxes ? At ic. a pen, what will each box cost ? 9. If a person buys a ream of paper for $3, and re- tails it at 1 cent a sheet, how much profit will he make ? 10. If 5 dozen buttons are sold out of a gross that cost $2.40, how much are what remain worth ? 11. Bought 7 quires of foolscap, 12 sheets of letter- paper, and 2|- quires of note-paper ; what was it worth, at the rate of $3.20 a ream ? 12. How many units in 4 great gross? REDUCTION. SECTION 52. 1. Which is the shortest month ? 2. How many days has February in leap-year? Ans. 29. 3. Which years are leap-years ? Ans. Those that can be exactly divided by 4, except such of the even hundreds as can not be exactly divided by 400. 4. Name the leap-years between 1868 and 1901. 5. From January 1st to February 1st how many months ? From Jan. 3d to Feb. 3d ? From Jan. 7th to Oct. 7th? 6. From Jan. 5th to March 5th how many months ? From Feb. 1st to Aug. 1st ? From June 2d to Nov. 2d ? 7. From April 6th to Oct. 6th how many months 9 From Aug. 9th to Nov. 9th ? 8. What part of a year is 2 months ? 6 months ? 8 months ? 3 months ? 9 months ? 5 months ? 9. How many months in f of a year ? In -% of a year ? In T \ of a year ? In f of a year ? 10. In business calculations, how many days are generally allowed to a month ? Ans. 30 clays. 11. In business calculations, what part of a month would we call 1 day ? 2 days ? 20 days ? 15 days ? 5 days ? 24 days ? 1 days ? 12. In business calculations, what part of a year is a day considered ? Ans. -^ of T 1 g-, or of a year. 13. What part of a year is 5 days considered? 14. What part of a year is a day really? What part is five days ? 15. How many calendar months in a century ? 16. Which of the months have 31 days? How many days has the 5th month ? The llth month ? THE METKIC SYSTEM. 89 CHAPTER EIGHTH. THE METRIC SYSTEM. SECTION 53. 1. What is the Metric System? Am. A system of weights and measures in which it takes 10 of a lower denomination to make 1 of the next higher. 2. Where is the Metric System used? Am. In France, Belgium, and other countries of Europe. Its use is also authorized by law in the United States. 3. What is the unit of length ? Am. The Metre, from which the Metric System has its name. 4. How long is a Metre ? Am. About 39^^ inches. 5. How are lower denominations formed? Am. From the metre other denominations, -^ y^-, and ToW as g reat j are formed with the prefixes deci (pro- nounced des'e), centi, and milli. 6. How are higher denominations formed ? Ans. Denominations 10, 100, 1000, and ten thousand times as great as the metre, are formed with the prefixes deca, Recto, kilo, and myria. MEASURES OF LENGTH. iu mu iimetres r naK( ) l cen timetre TOT incn - 10 centimetres a 1 dec'imetre = 3 1 9 ^ s - inches. 10 decimetres u 1 ME'TKE = 39 T Vu- inches. 10 metres u 1 dec'ametre = 32 ft, 9 T V in. 10 decametres a 1 hec'tometre = 328 ft. 1 in. 10 hectometres u 1 kil'ometre = 3280 ft. 10 in. 10 kilometres a 1 myr'iametre = 6 T 2 o^ miles. * The equivalents given in these Tables are not exact, hut nearly so. 00 THE METRIC SYSTEM. 7. How many centimetres in a metre? In half a metre ? In 5 metres ? In 17 metres ? In a decimetre ? 8. How many kilometres in 4000 metres ? 9. How many kilometres in 500 decametres? 10. What part of a decametre is a decimetre ? A metre ? 5 metres ? 8 metres ? 11. What part of a metre is a decimetre ? A centi- metre ? What part of a hectometre is a decametre ? 12. How many metres in 15 decimetres ? In of a decametre? In 250 centimetres ? In J of a hectometre ? 13. Is a metre more or less than a yard ? How many inches more ? 14. About how many metres equal 1 rod ? 15. What cost 4 metres of cloth, at $3 a metre ? 16. If a person sells 7 decimetres of velvet from a piece containing 3 metres, how much is what remains worth, at $10 a metre ? 17. How many kilometres of fence will be required, to surround a square field 500 metres on each side ? 18. At 75c. a metre, what will it cost to fence a square field, 30 metres on each side ? SECTION 54, 1. What is the unit of surface ? Ans. The Are (pronounced a/r), a square whose side is 10 metres, and which equals 119| square yards. MEASURES OF SURFACE. The cen'tiare is 1 square metre, or 1550 square inches. 100 centiares make 1 ARE = 119f sq. yd. 100 ares " 1 hec'tare = 2-^ acres. THE METRIC SYSTEM. 91 2. What is the unit of capacity ? Ans. The Litre (pronounced le'tur), a cube equal to about T 9 of a quart of dry measure. 3. How are other denominations formed? Ans. As before, by means of the prefixes deci, centi, and milli for lower denominations, and deca, hecto, and kilo for higher ones. MEASUEES OF CAPACITY. Dry Measure. Liquid Measure. 10 nnTlilitres, 1 cen'tilitre = f cu. in. = J- fluid oz. 10 centilitres, 1 dec'ilitre =6 T V cu. in. = f gill. 10 decilitres, 1 LI'TRE = T 9 7 quart = 1^ quart. 10 litres, 1 dec'alitre =ly peck = 2f gallons. 10 decalitres, 1 hec'tolitre =2f bushels = 26f gallons. 10 hectolitres, 1 kil'olitre =1 A cu. yd. =264^ gallons. The kilolitre (1 cubic metre), when used in measuring wood, is called the STERE, and equals about J^ of a cord. 10 steres make 1 dec'astere = about 2| cords. 4. How many ares in 550 centiares ? In 5 hectares ? 5. Are 2 hectares more or less than 5 acres, and how much ? 6. What part of an are is 65 centiares ? 7. What part of a litre is 5 centilitres ? 7 decili- tres ? 2 millilitres ? 8. How many litres in 1 7 J decalitres ? In fa of a hectolitre ? In 70 decilitres ? 9. What is the cost of 1^ hectolitres of molasses, at 12 cents a litre? 10. How many litres of molasses, at 11 cents a litre, should be given in exchange for 3 decalitres and 3 litres of vinegar, at 8 cents a litre ? 92 THE METRIC SYSTEM. 11. If a person bought 2 hectolitres of molasses, and sold 10 decalitres, what was the rest worth, at 12J cents a litre ? 12. If one stere equals J-J- of a cord, how many steres make a cord ? MODEL. If 1 stere equals -Ji of a cord, it will take as many steres to make a whole cord, or JJ, as -J-^ is contained times in f ; that is (rejecting the denominators), as 11 is contained times in 40, or 3-^-. Ans. 3 fr steres. 13. If 1 litre equals T 9 F of a quart, dry measure, how many litres make a quart ? 14. Bought a decastere of wood, and sold 4 steres of it for $8 ; how much was that a stere ? How much was what remained worth at the same rate ? 15. At $1.75 a stere, what is the cost of 2 decasteres of wood ? 16. A man bought 3 hectolitres of potatoes for 89. He sold them for 40 cents a decalitre. Did he gain or lose, and how much ? 17. Reduce 1 litre 1 decilitre 1 centilitre to centi- litres. 18. The kilolitre being a cubic metre, how long in inches is each side of the cube it represents ? 19. If 1 decastere equals 2| cords, how many cords do 5 decasteres equal ? 5^ decasteres ? 20. How many decasteres do 19 cords equal? 21. If a litre equals T \ of a quart, dry measure, how many litres are there in 45 quarts ? 22. What prefixes are used with the metre, times 5 is 50. Am. 50 rods. 4. How far will the thief mentioned in the last Example run before he is caught ? MISCELLANEOUS EXAMPLES. 115 5. A hare is 25 leaps in advance of a hound, and takes 4 leaps to the hound's 3, but 1 leap of the hound equals 1 leaps of the hare. How many leaps will each take before the hare is caught ? 6. A train of cars is 10 miles behind a stage. If the cars go 5 times as fast as the stage, how many miles, &c., will the stage go before it is overtaken ? 7. At what time between 2 and 3 o'clock are the minute and hour hand of a watch together? MODEL. The minute and hour hand are together at 12 o'clock. In the course of 12 hours, the minute hand will overtake the hour hand 11 times; to overtake it once, therefore, will require ft of 12 hours, or l-ft hours. When the hands are together between 2 and 3, the minute hand will have overtaken the hour hand twice since 1 2 o'clock, which will require twice 1ft hours, or 2ft hours. Ans. 2ft hours past 12, or 10 min. 54ft sec. past 2. 8. At what time between 3 and 4 will the hour and minute hand of a clock be together ? 9. At what time between 7 and 8 will the hands be together ? At what time between 9 and 10 ? 10. A person received from his employer $74, under an agreement that he was to board with the lat- ter 40 days, and receive $2.50 a day when he worked, and lose 75c. a day when he was idle. How many days was he idle ? MODEL. His pay being $2.50 a day, if he had worked the whole 40 days, he would have received 40 times $2.50, or $100 ; hence he lost by being idle the difference between $100 and $74, or $26. Each day he was idle he failed to make $2.50, and forfeited 75c. besides, thus losing in all $3. Hence, if he lost $26, he must have been idle as many days as $3 is contained times in $26, or 8. Am. 8 days. 116 MISCELLANEOUS EXAMPLES. SECTION 70. 1. A contracted to work 36 days for B, on condition that he should receive $2 for every day he worked, and forfeit 50c. each day he was idle. If he received from B $47, how many days was he idle ? 2. How many square yards in an oblong surface 7 yards long and 4 yards wide? MODEL. A surface 1 yd. long and 1 yd. wide contains 1 sq. yd. ; a surface 7 yd. long and 1 yd. wide, would therefore contain 7 times 1, or 7, sq. yd. ; and a surface 7 yd. long and 4 yd. wide contains 4 times 7, or 28, sq. yd. Ans. 28 sq. yd. 3. How many square rods in an oblong field 20 rods by 15 rods ? 4. How many sq. yd. in a floor 12 ft. square? 5. At 30c. a square yard, what will it cost to plas- ter a wall 10 ft. high and 18 ft. long ? 6. What will it cost to paint a wall 9 yd. long and 4 yd. high, at 5c. per square foot ? 7. How many marble blocks 12 in. square will be needed to pave a walk 3 ft. wide and 20 ft. long ? 8. How much yard-wide carpeting will be required to cover a room 18 ft. by 15 ft. ? 9. How much carpeting f of a yard wide is required for a room 9 yd. long by 14 ft. wide? 10. How many cubic feet in a box 4 ft. long, 3 ft. wide, and 2 ft. high ? MODEL. A box 1 ft. long, 1 ft. wide, and 1 ft. high, would con- tain 1 cu. ft. ; a box, therefore, 4 ft. long, 1 ft. wide, and 1 ft. high, would contain 4 times 1, or 4, cu. ft.; a box 4 ft. long, 3 ft. wide, and 1 ft. high, would contain 3 times 4, or 12, cu. ft. ; and a box 4 ft. long, 3 ft. wide, and 2 ft. high, contains twice 12, or 24, cu. ft. Ans. 24 cu. ft. PERCENTAGE. 11 7 11. How many cubic yards in a bin 5 yd. long, 2 yd. wide, and 3 yd. high ? 12. What is the freight on a box 3 ft. long, 3 ft. wide, and 3 ft. high, at 50c. a .cubic foot ? 18. What will it cost to dig a cellar 12 feet long, wide, and high, at 60c. a cubic yard ? CHAPTER ELEVENTH. PERCENTAGE. SECTION 71. 1. What do the words Per cent, mean ? Ans. By or on the hundred. 2. What does 1 Per cent, mean, and how may it be written ? Ans. I Per cent, means one on every hun- dred^ or 1 hundredth, and may be written 1 %. 3. What does 2 Per cent, mean, and how may it be written ? Ans. 2 Per cent, means two hundredths (y| ), and may be written 2 . If the premium is $9, and the value of the store is 200 % of the value of the contents, how much is the store worth ? What is the value of the contents ? TAXES. 125 SECTION 77, 1. What is a Tax? Ans. A Tax is a sura assessed on the person, property, or income of an individual, for the support of government. 2. What is a Poll-tax ? Ans. A tax on the person, generally a uniform sum on each male citizen except such as are exempted by law. 3. Ho\v is a Property-tax reckoned? Ans. At a certain . 19. Find the interest of $240, for 1 yr., at ^f c . 20. What must I pay, to take up a note for $100. with interest at 5 % for 12 months ? 21. A person lends $1000 for 1 year, at 7 . With another $1000 he buys a house, which at the end of a year he sells for $1100. From which investment does he make the more money, and how much more ? SECTION 80. 1. What is the interest of $200, at 7 , for 2 years ? MODEL. The interest on $200, at 7 #, for 1 year, is $14 ; and for 2 years it is twice $14, or $28. Ans. $28. 2. Find the interest of $120, for 3 yr., at 5 f c . 3. Find the interest of $400, for 6 yr., at 7 f c . 4. Find the amount of $700, for 3 yr., at 6 $>. 5. Find the amount of $40, for 5 yr., at 4#. 6. Find the interest of $600, for 4 yr., at 8 . 7. Find the amount of $350, for 2 yr., at 7$. 8. What is the interest of $8000, for 10 yr., at 10 % ? Of $300, for 20 yr., at 5 % ? What is the in- terest on any principal, when the product of the rate and years is 100 ? 9. Find the interest of $80, for 3 mo., at 7 #. MODEL. The interest of $80, at 7 fr for 1 yr., is $5.60; and for S mo., which is of a year, it is of $5.60, or $1.40. Ans. $1.40. INTEREST. 129 10. At 8$, what is the interest of 81000, for 2 months ? For 5 months ? 11. At 7#, what is the interest of $1600, for 3 months ? For 9 months ? For 1 year 9 months ? 12. At 5$, what is the amount of $300, for 2 'months ? For 1 yr. 2 mo. ? For 2 yr. 2 mo. ? 13. At 4j-#, what is the amount of $200, for 10 months ? For 6 mo. ? For 3 yr. 6 mo. ? 14. At 7 $, what is the interest of $60, for 4 mo. ? For 1 yr. 4 mo. ? For 11 mo. ? 15. At 4#, what is the interest of $1000, for 1 mo. ? For 5 mo. ? For 8 mo. ? 16. At 5 J- #, what is the interest of $1200, for 7 mo. ? For 1 yr. 11 mo. ? 17. At 7 #, what is the interest of $2400, from Oc- tober 3d to December 3d of the same year ? 18. What is the interest of $750, from February 15th to August 15th of the same year, at 6 % ? 19. What is the amount of $600, from November 1st to June 1st of the next year, at 5 % ? 20. What must I pay for the use of $175, from May 8th, 1866, to January 8th, 1868, at 8 % ? 21. February 1st A lent B $100, and on March 1st $200 more. Interest being allowed at 6 $, how much should B pay A on settling up, September 1st of the same year ? 22. A person depositing $200 in a savings-bank, June 1st, 1867, gets 5f interest. Had he waited till July 1st, he might have invested it differently at 7 % ; in this case, how much better off would he have been on the 1st of June, 1868 ? 130 INTEREST. SECTION 81. 1. What is the interest of $600, for 10 days, at 5 ^ '? NOTE. In calculating interest, 30 days are allowed to the month MODEL. At 5 #, the interest of $600, for 1 year, is 830, for 1 month, -fa of 30, or $2.50, and for 10 days, which is of 1 month, ^ of $2.50, or 83^c. Arts. SS^c. We reject fractions of a cent less than J, and call or more an additional cent ; which makes the above answer 83c. 2. What is the interest of $950, at 6 fa for 3 days ? For 10 days ? 3. What is the amount of 8800, at 7 fa for 12 days ? For 15 days ? 4. What is the amount of $2000, at tyfa for 8 days ? For 20 days ? 5. What is the interest of $1200, at 4 fa for 5 days ? At 7 fa for 25 days ? 6. What is the interest of $900, at 5 fa for 4 days ? At 6 fa for 18 days? 7. What is the amount of $450, at 8 fa for 27 days ? At 7 #, for 7 days ? 8. What is the interest of $7500, at 5 fa for 22 days ? For 28 days ? 9. What is the interest of $85.50, at 8 fa for 15 days? For 18 days? 10. What is the interest of $100, at 7 fa from May 1st, 1868, to May 22d, 1868 ? 11. What is the amount of $250, at 6$, from Jan. 1st, 1868, to Jan. 31st, 1868? 12. Borrowed, May 28th, $200, and June 3d, $300. How much money will be needed to pay off these debts, with interest at 7 fa June 12th of the same year? INTEREST. 131 SECTION 82. 1. What is the interest of $300, at 7 fa for 1 yr. 6 mo. 6 da. ? MODEL. At 7 %, the interest of $300, for 1 year, is $21. For 1 month, the interest is -^ of $21, or $1.75, for 6 months, 6 times $1.75, or $10.50, and for 6 days, which is \ of a month, \ of $1.75, or 35c. For 1 year 6 months 6 days, therefore, the interest is $21 -f $10.50 + 35c., or $31.85. Ans. $31.85. 2. What is the interest of $900, at 7 fa for 1 year 4 months 10 days ? 3. What is the amount of $480, at 6 fa for 9 mo. 5 da. ? At 8 fa for the same time ? 4. At fa what is the amount of $1020 for 1 yr. 8 mo. 9 da. ? For 2 yr. 8 mo. 9 da. ? 5. At 5-J-^, what is the interest of $1000, for 1 year 3 days ? For 3 yr. 3 mo. ? 6. At 5 fa what is the interest of $3200 for 2 yr. 3 mo. 18 da. ? For 3 yr. 9 mo. 1 day ? 7. At 1 % a month, what is the interest of $5000, for 21 days ? For 5 mo. 2 da. ? For 2 mo. 27 da. ? 8. At J of 1 % a month, what is the interest of $800 for 3 mo. 28 da. ? For 1 mo. 15 da. ? 9. What is the amount of $625, at 10 fa from Oc- toher 1st, 1866, to April 4th, 1868? 10. A person bought some property, Jan. 1st, 1868, for $1000, borrowing the money to pay for it, at 6 fa For how much must he sell it July 1st, 1868, to make 10 f on the cost and interest ? 11. A house was bought for $8000, $2000 cash, and the balance to be paid in two equal semi-annual instalments, with interest at 7 fa What was the whole amount paid ? 132 INTEREST. SECTION 83. 1. What is meant by the Legal Rate of interest ? Ans. A rate fixed by law, for cases in which no rate is specified. 2. In what States is the legal rate 7 % ? Ans. In New York, New Jersey, Michigan, Wisconsin, Min- nesota, Kansas, South Carolina, and Georgia. 3. What is the legal rate in most of the United States ? Ans. In most of the United States, including all the New England States, the legal rate is 6$. Hence it is important to know the shortest methods of computing interest at 6 fa 4. At 6 fa what is the interest of $1, for 2 months ? At 6 fa the interest for 2 months is what part of the principal ? 5. Find the interest of $75, at 6 fa for 2 mo. MODEL. At 6 #, the interest for 2 months is T Jo of the princi- pal. 7-^0 of $75 is 75c. Ana. 75c. 6. Find the interest of $8300, at 6 fa for 2 mo. 7. Find the interest of $302.50, for 2 mo., at 6 fa NOTE. Dividing a sum consisting of dollars and cents by 100 is done by simply moving the point two places to the left. T J^ of 8302.50 is $3.025. Ana. $3.03. 8. Find the amount of $60.25, for 2 mo., at 6 fa 9. Find the interest of $999, for 2 mo., at 6 fa 10. At 6 fa what is the interest of $130.75, for 2 nonths ? The amount of $8497, for 60 days ? 11. Find the interest of $2450, for 16 mo., at 6 fa MODEL. The interest for 2 months is -j-foy of the principal, or $24.50; and for 16 months it is as many times $24.50 as 2 mo. is contained times in 16 mo., or 8. 8 times $24.50 is $196. Ana. $198. INTEEEST. 133 12. At 6$, what is the. interest of $88.90, for 8 months ? For 18 months ? For 1 year 6 months ? 13. At 6$, what is the amount of $9000, for 10 months ? For 1 year 2 months ? 14. At 6#, what is the interest of $4800, for 11 months ? For 7 months ? For 2 yr. 8 mo. ? 15. At 6 $, what is the amount of $145, for 20 mo. ? 16. At 6 $, what was the interest of $700, from July 7th, 1866, to October 7th, 1867 ? 17. How much was to pay on taking up two notes, for $200 each, both due March 3d, 1868, and bearing interest at 6 % ; one being dated Nov. 3d, 1867, and the other Feb. 3d, 1867 ? SECTION 84.- 1. What is the interest of $620, for SO days, at 6 % ? MODEL. The interest for 60 days is jfa of the principal, or $6.20 ; and for SO days (which is of 60 days) it is of $6.20, or $3.10. Ans. $3.10. 2. Find the interest of $355, for 30 days, at 6 %. 3. Find the interest of $80, for 30 days, at 6 %. 4. Find the interest of $800, for 3 days, at 6 #. 5. What is the interest of $650, at 6 #, for 30 days ? For 3 days ? For 33 days ? 6. What is the interest of $2040, at 6$, for 90 days ? For 3 days ? For 93 days ? 7. What is the amount of $18.60, at 6$, for 60 days ? For 3 days ? For 63 days ? 8. What is the interest of $1100, at 6 #, for 33 days ? For 63 days ? For 93 days ? 134 INTEREST. 9. What is the interest of $1200, for 1 year 10 months 18 days, at 6 % ? MODEL. The interest for 2 months is T ^Q of the principal, or 812, for 1 yr. 10 mo., or 22 months, it is 11 times $12, or $132, and for 18 days (which is \% of 2 months) it is J, or -,%, of 812, which is $3.60. Hence for 1 yr. 10 mo. 18 da. it is $132 + $3.80, or $135.60. Am. $135.60. 10. At 6 fo y what is the interest of $336, for 2 years 12 days ? For 1 yr. 7 mo. 10 da. ? 11. At 6& what is the amount of $1500, for 10 months 15 days ? For 1 yr. 11 mo. 5 da. ? 12. At 6#, what is the interest of 1800, for 4 months 5 days ? For 1 yr. 3 mo. 25 da. ? 1 3. At 6 $, what is the interest of $2000, for 8 months 21 days ? For 3 yr. 4 mo. 8 da. ? 14. At 6 f c , what is the interest of $2400 for 6 days ? At 6 %, the interest for 6 days is what part of the principal ? 15. At 6$, what is the interest of $8880 for 6 days ? For 12 days ? For 3 days ? For 9 days ? 16. Find the interest of $540, for 2 mo., at 7 %. MODEL. At .6 % it would be $5.40 ; at 1 #, \ of $5.40, or 90c. ; and at 7 #, it is 7 tunes 90c., or $6.30. Ans. $6.30. Solve this Example according to the method shown in Section 80, and see whether the answers agree. Solve the following accord- ing to both methods. 17. Find the interest of $600, for 10 mo., at 7 $. 18. Find the interest of $9000, for 9 mo., at 5 . 19. Find the amount of $420, for 18 mo., at 4 . 20. Find the interest of $6500, for 14 mo., at 8 %. 21. Find the amount of $836, for 8 mo., at 7#. INTEREST. 135 SECTION 85. 1. What part of a year's interest is 1 month's interest ? 2. Allowing 30 days to a month, what part of a year's interest is 1 day's interest ? 3. How many days are there in a year? What part of 1 year's interest, then, would 1 day's interest exactly be ? 4. By taking 1 d"ay's interest as -g-J-^ of 1 year's in- terest (in stead of -5^), do we take more or less than the exact amount ? Is the difference great ? 5. Why, do you suppose, are 30 $ays allowed to the month in computing interest ? 6. Required the interest, by the shortest method, Of $70.50, at 6 fa for 60 days. Of $1040, at 6 fa for 90 days. Of $88.00, at 8 fa for 30 days. Of $3750, at 4 fa for 63 days. Of $225, at 6 fa for 2 yr. 6 mo. Of $22.50, at 5 fa for 1 yr. 6 mo. 7. Required the amount, by the shortest method, Of $3.47, at 10 fa for 10 years. Of $1400, at 8 fa for 12 yr. 6 mo. Of $219.90, at 6 fa for 16 yr. 8 mo. Of $2000, at 6 fa for 1 yr. 1 mo. 1 da. Of $860, at 7 fa for 1 yr. 8 mo. 9 da. Of $900, at 7 fa for 7 mo. 7 da. Of $400, at 7 fa for 3 mo. 10 da. 8. Jan. 1st, A sells for B 6 bales of cotton, aver- aging 350 Ib. each, at 25c. a pound, tfan. 21st, he remits to B $125. How much is due to B, Feb. 21st, interest being allowed at 6 $ from the day of sale ? 136 INTEREST. SECTION 86. 1. At what rate will 8200 yield 89 interest in 1 year ? MODEL. At 1 , 8200 will yield $2 interest in 1 year ; if it yields 9, the rate must be as many times 1 % as $2 is contained times in $9, or4. Ans. 4\%. 2. At w T hat rate must $650 be invested, to yield 839 interest annually ? To yield 845.50 ? 3. At what rate of interest will 83000 produce 8210 a year ? At what rate will it produce 8120 ? 4. If 82500 produces 8200 interest in 12 months, what is the rat ? 5. A person lends 81000, and at the end of a year receives for principal and interest 81065. What is the interest ? What rate of interest does he receive ? 6. If a person receives 8481.50 for a loan of 8450 for 1 year, what rate of interest does he get? 7. At what rate must 2000 be invested, to amount to 2060 in lyear? 8. At what rate will 9000 yield 8675 interest in 1 year 6 months ? MODEL. 1 yr. 6 mo.=l| or I J r - If tne interest for f years is 8675, for \ year it is \ of $675, or $225 ; and for 1 yr. it is twice 8225, or $450. At 1 & $9000 will yield $90 in 1 year ; if it yields $450, the rate must be as many times 1 % as $90 is contained times in $450, or 5. Am. 5 %. 9. At what % will 8800 produce 860 in 1 yr. 3 mo. ? 10. At what % will 8900 give 8126 interest in 2 yr. ? 11.. At what rate will 8240 produce 822.40 interest in 1 year 4 iftonths ? 12. At what rate will 8606 produce $60.60 interest in 60 days ? INTEREST. 137 13. At what rate must $1000 be invested, to yield $50 interest in 10 mo. ? To yield $200 in 4 yr. ? 14. At what rate is $1100 invested, if it yields $38.50 interest semi-annually ? 15. At what rate must $2500 be invested, to amount to $2850 in 2 years ? 16. If the amount of $6700, for 1 yr. 8 mo., is $7370, what is the rate? 17. At what rate must $490 be put at interest, to amount to $627.20 in 3 yr. 6 mo. ? 18. At what rate will the interest on any sum equal the principal in 10 years ? MODEL. The interest will equal the principal in 1 year at 100 <, and in 10 years at -^ of 100 #, or 10 %. Am. 10 %. 19. At what rate will $100 produce $100 interest in 4 years ? In 6 years ? In 20 years ? 20. At what rate will the interest on a certain sum equal the principal in 12 years ? In 5 years ? 21. At what rate will any principal double itself in 2 years ? In 8 years ? In 15 years ? SECTION 87. 1. How long will it take $200, at 5 discount ? 3. If a franc is worth 19^c., what is the value of 1000 francs in U. S. currency? 4. The pound sterling of Great Britain is worth $4.86-^, and the Canada dollar is worth $1, in U. S. gold. How many Canada dollars are equivalent to 500 pounds sterling? 5. How many Canada dollars are 100 eagles worth ? 6. A grocer mixes 100 Ib. of coffee, at 24c., with 20 Ib., at 15c. ; what is the mixture worth per Ib. ? MODEL. 100 Ib., at 24c. a pound, are worth $24. 20 Ib., at 15c., are worth $3. The whole, 120 Ib., is therefore worth $24 + $3, or $27 ; and 1 Ib. is worth fa of $27, or 22|c. Ans. 22^c. 7. If 10 gall, of brandy, at $4 a gallon, were mixed with 5 gall., at $10 a gallon, how much a gallon was the mixture worth ? 8. If a train of cars goes 20 miles an hour for 2 hours, and then 30 miles an hour for 3 hours, what is its average rate for the whole time ? o 9. If a boat runs 30 miles in 2 h., and then 15 miles in 1^ h., what is its average speed ? 10. A grocer mixes 20 Ib. of tea that cost 70c. a Ib. with 20 Ib. that cost $1.10 a Ib. At what price per Ib. must he sell the mixture, to make 20$ ? MISCELLANEOUS EXAMPLES. 151 11. Two thousand bushels of wheat, worth $2.50 a bushel, were mixed with 1000 bu., worth $1.90. How much a bushel was the mixture worth ? 12. An equal number of geese, chickens, and tur- keys, were sold for $15. Each goose brought 90c., each chicken 50c., and each turkey -^ of the cost of a goose and chicken ; how many of each were sold ? 13. A person bought some property for $2000 on the 10th of Jan., and on the 10th of May following sold it for $2200. What r. T. HARRIS, LL. D., Supt. of Schools, St. Louis, Mo.; ANDREW J. EICKOFF, A. M., Supt. of Instruction, Cleveland, O. ; and MARK BAILEY, A.M., Instructor in Elocution, YaleCollege. Appletons' First Reader 90 pages. Appletons' Second Header 142 " Appletons' Third Reader 214 " Appletons' Fourth Reader 248 " Appletons' Fifth Reader 460 " These Readers, while avoiding extremes and one-sided ten- dencies, combine into one harmonious whole the several results desirable to be attained in a series of school reading-books. These include good pictorial illustrations, a combination of the word and phonic methods, careful grading, drill on the peculiar combinations of letters that represent vowel-sounds, correct spelling, exercises well arranged for the pupil's preparation by himself (so that he shall learn the great lessons of self-help, self-dependence, the habit of application), exercises that develop a practical command of correct forms of expression, good lit- erary taste, close critical power of thought, and ability to in- terpret the entire meaning of the language of others. The high rank which the authors have attained in the edu- cational field and their long and successful experience in prac- tical school-work especially fit them for the preparation of text-books that embody all the best elements of modern educa- tive ideas. In the schools of St. Louis and Cleveland, over which two of them have long presided, the subject of reading lias received more than usual attention, and with results that have established for them a wide reputation for superior elo- cutionary discipline and accomplishments. Of Prof. Bailey, Instructor of Elocution in Yale College, it is needless to speak, for he is known throughout the Union as being without a peer in his profession. His methods make nat- ural, not mechanical readers. D, APPLETON & CO., Publishers, 549 & 551 Broadway, New York. YA 02420 / ARITHMETICAL *SERIES, By G-. P. QUACKENBOS, LL. D. Cpen the tais of the tt erk8 tf GEORGE B. PEBFJKS, IX. D. I. A PRIMARY ARITHMETIC. Beautifully Illustrated ; carries the beginner through the first four Rules uJ the simple Tables, combining mental examples with sum* for the slate. 16mo, 108 pages. 22 cents. H. AH ELEMENTARY ARITHMETIC. Reviews the sub- jects of the Primary in a style adapted to somewhat maturer minds; also em- braces Fractions. Fedoral Money, Reduction, and the Compound Rule*. 12mo, 144 pages. 40 cents. m. A PRACTICAL ARITHMETIC. FulU clear, condensed. Prepared with nco to the wants of Common Schools, and designed to prepare- pupils thoro^hly for the business of life. Its methods are those actually used by busies men. 12mo, 336 pages. 80 cents. IV. A HIGKffER ARITHMETIC. This work carries the subject further, an* o. thorough mastery of tb theory and pra< It is really whr.i its name imports | P a S V. A'ME?-*.L/*L ARITHBtBTIC. For imparting read ; mental ealcuh. new and beautiful processes. 16mo, j 35 cent*. The ditf mber? of this Series are perfectly 1 ! graded. i'rom step to step is inductive a gradual ; i anticipated, nothing required to be supplied by the teacher; The definitions are simple, the rules brief, the analyses unencumbered with verbiage, the examples drawn from the practical matters of life, the arrangement the most natural, the methods taught the: shortest possible. Every device is resorted to, to prevent i the mere mechanical solving of examples ; the^ pupil's I mind is constantly kept on the alert, and his Arithmetic letson is thus made an invaluable mental discipline. Taohert that want the best books should examine the abore,