IvIBRARY OF THE University of California. GTF"T OF" Accession No . 6 9 J' J C? • Class No.'^ * Received THE FRANKLIN ELEMENTARY ARITHMETIC EDWIN P. SEAVER, A. M. HEAD MASTER OF THE ENGLISH HIGH SCHOOL, BOSTON ; FORMERLY ASSISTANT PROFESSOR OF MATHEMATICS IN HARVARD COLLEGE GEO. A. WALTON, A. M. AUTHOR OF WALTONS' ARITHMETICS, ARITHMETICAL TABLES, ETC. BOSTON WILLIAM WARE AND COMPANY [Successors to Brewer and Tileston] 1879 6y t>- 3c Copyright By E. p. SEAVER and G. A. WALTON. 1878. University Press; John Wilson & Son, Cambridge. PREFACE. The Franklin Elementary Arithmetic, though designed to be an introduction to the Franklin Written Arithmetic, is, nevertheless, a complete arithmetic of its kind. It con- tains a short course in the elements of numbers, with such applications as are necessary in ordinary business transac- tions. With the oral teaching which should accompany the use of any book, this book contains enough to meet the wants of that large class of pupils who leave the schools at twelve or thirteen years of age, and of all who desire to master the essentials of arithmetic but have not time to study all that is found in the larger books. The method of the book is indicated by its title. It distinguishes arithmetical operations from the science of arithmetic. While it gives a systematic practice in the former, it leaves the latter to be learned after the mere operations have become familiar by practice. No attempt is made to establish general principles, but the pupil is led to operate by imitating processes illustrated by simple examples, generally in the concrete. General statements and formal rules are thus rendered unnecessary. The special features of the work are : — 1. The uniting of oral exercises with the written work, so that the same analysis answers for both processes. iv PREFACE, 2. The thorough manner of treating the four fundamental operations, especially multiplication and division. These opera- tions are so arranged and combined, that the oral and written processes can be learned in the time ordinarily given to learning the multiplication and division tables. 3. The introduction of United States money and denominate numbers into the same sections with simple numbers and frac- tions. In this way the reductions in United States money and compound numbers are made a part of the necessary practice in the fundamental operations, and thus much time is saved. 4. Only one method is given for each operation. The pupil is thus spared the perplexity which often results from a multi- plicity of methods. 5. The Appendix to the book. This contains full sets of tables, with other methods of operation, to be used instead of those given in the body of the book, if the teacher prefers. 6. Drill Tables, by which the pupil's work may be indefi- nitely extended, without the teacher being required to search in other books for examples to apply as tests in class exercises. 7. The adaptation of the exercises for use in the class-room, — all oral and illustrative examples being designated by letters of the alphabet, while the examples for the slate and blackboard are numbered consecutively through the section. Boston, August, 1878. TABLE OF CONTENTS. Reading and writing Num- bers TO Thousands 7 Addition 14 United States Money 20 Subtraction 23 Multiplication 34 Liquid and Dry Measures. . . 48 Division 51 Measures of Time 65 Weights and Numbers 66 Reading and writing Num- bers to Millions 72 Factors 78 Multiples 79 Common Fractions 81 Reduction 83 Addition 87 Subtraction 89 Multiplication 91 Long Measure 94 Division 95 To find the whole when a part is given 99 Decimal Fractions 104 Reduction 106 Addition 108 Subtraction 109 Multiplication 110 Division 112 Percentage 115 Application to Profit and Loss, etc 120 Interest 122 Bank Discount 126 Mensuration 130 Area of Rectangles 1 30 Square Measure 131 Triangles, etc 132 Volume of Rectangular Solids 133 Cubic Measure 134 Lumber and Boards 135 Prisms, etc. 135 Appendix 1 38 Tables of Weights and Measures 143 Multiplication Table 144 Drill Tables 32, 70, 76, 128 Exercises on Drill Tables 33, 71, 77, 129 Miscellaneous Oral Exercises ...29,48,64, 101 Miscellaneous Examples for Slate... 29, 49, 67, 74, 90, 102, 137 Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/franklinelementaOOseavrich fUiri7BR3 Elementary Arithmetic. SEOTIOI^ I. READING AND "V^TRITING NUMBERS. Numbers from One to Ten. One man. Two horses. Three dogs. Four boys. Five trees. Six cows. Seven sheep. Eight birds. Nine posts. Ten rails. One. Two. Three, Four. Five. Six. Seven Eight. 8 Nine. 9, Ten. 10, Article 1. How many men do you see in the picture ? How many horses ? boys ? dogs ? rails ? trees ? birds ? 8 READING AND WRITING In answering these questions, you have named some numbers. Name the numbers from one to ten ; from ten to one. 2. Copy on your slate the figures that stand for the numbers you have named. 10. Numbers from Ten to Twenty. 1 ^ CD Ten blocks and seven blocks ? 17 means ten and what ? iz. Ten blocks and eight blocks ? 18 means ten and what ? i. Ten blocks and nine blocks ? 19 means ten and what ? J. Ten blocks and ten blocks ? 20 means how many tens? SIMPLE NUMBERS, Exercises. 4. Count by ones from ten to twenty ; from twenty to ten. 6. Eead the following : k. 11. 222. 18. 0. 15. q. 19. s. 16. 1. 13. n. 17. p. 12. r. 14. t. 20. 6. Write in figures on your slate all the numbers from ten to twenty. /^ // /J /J /^ /S S ^/ Wi/f30 Numbers made up of Tens. BCD i^qz: E "~ ~ ~ ~ , 20. 30. 40. 50. 100. Twenty. Thirty. Forty. Fifty. A hundred. 7. From the collections of blocks represented above find a. How many tens in twenty ? d. How many tens in fifty ? 5. How many tens in thirty ? e. How many tens in a hun- c. How many tens in forty ? 8. Twenty is written 20. Thirty is written 30. Forty is written 40. Fifty is written 50. dred? Sixty is written 60. Seventy is written 70. Eighty is written 80. Ninety is written 90. A hundred is written 100. 10 READING AND WRITING 9. The figure is called zero, and stands for no numhei\ How many figures are required to write any number of tens to nine tens ? Which figure shows how many tens there are ? Exercises. 10. Count by tens from ten to one hundred; from one hundred to ten. 11. Eead the following : /. 20. h. 30. j. 90. L 60. n, 10. g. 40. i. 70. k. 50. m. 80. o. 100. 12. Write in figures the numbers made up of tens from ten to one hundred. Numbers made up of Tens and Ones. y 7 y' i 1 1 1 1 PL. 1 1 1 p m. 13. How many tens and how many ones are there in the collection marked A? in ±a H-j-H j y±rhi ^ ^ i^ c ^ I t P M I I P H I I iP Two tens and three are ^ ^ ^ twenty-three, written 23. Three tens and four are thirty- four, ivritten 34. Four tens and seven are forty-seven, written 47. Five tens and eight are fifty- eight, written 58. Nine tens and nine are ninety-nine, wTitten 99. How many figures are needed to write tens with ones ? What does the left-hand figure stand for? What does the right-hand figure stand for ? 14. Ones are called units, and the place at the right where the ones are written is the units' place. The sec- ond place from the right, where the tens are written, is the tens' place. SIMPLE NUMBERS. 11 e. From 50 to 70. /. From 70 to 100. P' 77. s. 95. Q' 84. t. 46. r. 69. u. 81. Exercises. 15. Count by ones a. From 20 to 30. c. From 30 to 50. b. From 30 to 20. d. From 50 to 30. 16. Read the following : g. 28. J. 49. 222. 50. h. 35. k. 31. 22. 63. 2. 42. 1. 27. o. 59. 17. Turn to page 32, and read all the numbers written in A. 18. E-ead again the numbers written in A, and tell how many tens and how many units there are in each number. Thus, " Fourteen : one ten and four units." " Twenty-six : two tens and six units," and so on. 19. Write in order all the numbers from fifty to ninety-nine. Numbers made up of Hundreds. 20. Count by hundreds from one hundred to ten hundred. One hundred is written 100. Two hundred is written 200. Three hundred is written 300. Nine hundred is written 900. Exercises. 21. Eead the following : a. 400. c. 700. e. 500. b. 300. d. 200. /. 600. How many figures are needed to write hundreds ? many of these figures are zeros ? g. 900. h. 800. How 12 READING AND WRITING 22. Write in figures the numbers made up of hundreds from one hundred to nine hundred. Numbers made up of Hundreds, Tens, and Units. 23. Three hundreds, two tens, and five units are three hundred twenty-five, writ- ten 325. ^ In which place from the right are the hundreds written ? the tens ? the units ? Exercises. 24. Eead the following : i. 123. k, 629. m. 934. o. 206. j. 304. 1. 580. 22. 468. p. 370. l>Fi J'^ivn fn page 32, and read the numbers written in B, 26. Write in figures the following : q. Seven hundred eighty-one. t. Kine hundred fifty. T. Five hundred fifteen. u. Eight hundred six. s. Three hundred forty-seven. v. Six hundred eleven. Let the teacher dictate other numbers made up of hundreds, tens, and units for the pupil to write. Numbers made up of Thousands. 27. Ten hundreds make a thousand, written 1000 or 1,000. Count by thousands from one thousand to ten thousand. Two thousand is written 2000 or 2,000. Three thousand is written 3000 or 3,000. One thousand. -^^^ SO on. SIMPLE NUMBERS. 13 Exercises. 28. Eead the following : a. 1,000. d. 5,000. b. 7,000. e. 8,000. c. 3,000. /. 6,000. g. 2,000. 2z. 4,000. i. 9,000. How many figures are needed to write thousands ? How many of these figures are zeros ? 29. Write in figures the numbers made up of thousands from one thousand to nine thousand. Numbers made up of Thousands, Hundreds, Tens, and Units. 30. One thousand three hundred twen- ty - five is written 1325. In whicji place from the right are the thousands written ? Exercises. 31. Read the following : j. 1,234 m, 2,304. p. 1,008. k. 5,067. 22. 5,372. q. 3,200. i. 8,009. o. 8,097. r. 5,970. 32. Turn to page 32, and read the numbers written in C. 33. Write in figures : S» One thousand, two hundred. t. One thousand, two hundred seventeen. u. Four thousand, six hundred forty-three. V. Nine thousand, nine hundred eighty. Let the teacher dictate other numbers made up of thousands, hundreds, tens, and units for the pupil to write. 14 ADDITION. SEOTIOE" II. ADDITION. Addition of I's, 2's, and 3's to other Numbers. ^ ^^1^^^^^^ ^^' How many apples ^ are in the basket ? If the apples in the hand and on the table be put with those in the bas- ket, how many will there be in the basket then ? To find the answer to this question, count the numbers together thus : three, five, six. 36. Counting numbers together is adding them. 36. The number found by adding is called the sum, a. Add 4 and 2 ; 4 and 3 ; 5 and 2 ; 5 and 3. b. What is the sum of 6 and 2 ? 6 and 3 ? 7 and 2? 7 and 3 ? c. What is the sum of 8 and 2 ? 8 and 3 ? 9 and 3 ? 9 and 2 ? Drill Exercises. 37. You w^ill need to do examples like the following over , « (.) (5) (« (5) (6) (?) (S) («) andoveragain d. Add 1 1 1 1 1 1 1 1 1 t^ll yo" can to 1 3 6 7 4 2 6 8 9 add at sight, — — — — — — — — — that is, till you e. Add 222222222 can name the to J J. J _6 _3 _5 _9 _7 _8 sum of a pair of numbers as /. Add 3 3 3 3 3 3 3 3 3 ^^^^^ ^^ to_2JJJJJ_8_6_9 seethe figures. ADDITION. 15 38. g. How many are 2 and 1 ? 12 and 1 ? 22 and 1 ? 42 and 1 ? h. How many are 4 and 2 ? 14 and 2 ? 34 and 2? 44 and 2 ? i. How many are 9 and 2 ? 19 and 2 ? 39 and 2 ? 59 and 2 ? j. How many are 8 and 3 ? 28 and 3 ? 48 and 3 ? 68 and 3 ? k. How many are 9 and 3 ? 19 and 3? 59 and 3? 79 and 3 ? 39. i. Add by 2's from 2 to 20, writing on your slate each sum as you find it ; thus, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. 122. Now look away from your slate and add by 2's without writing the sums. In the same way, 22. Add by 2's from 1 to 21. o. Add by 3's from 3 to 30 ; from 1 to 31 ; from 2 to 32. Examples for the Slate. 40. Copy and add the numbers in the following examples. In Example a, begin at the bottom and add thus : "1, 3, 5, 7, 9, 12, 15 ; sum, 15." Write this sum under the line as it is written in the book. To see if your work is right, begin at the top and add downwards, thus : ^'3, 6, 8, 10, 12, 14, 15; sum again, 15." In Example b, say "2, 3, 4, 6, 9, 10, 13; sum, 13 tens." Write this sum as it is Avritten in the book, putting a zero to fill the units' place. In the same way do the other examples. a. b. (1-) (2.) (3.) (4.) 3 30 2 20 30 300 3 10 3 30 30 100 2 30 2 10 20 300 2 20 1 30 20 200 2 10 3 20 30 100 2 10 2 10 10 300 1 20 2 20 20 300 16 130 16 ADDITION. (5.) (6.) (7.) (8.) (9.) ^(10.) 3 2 300 20 300 3 3 2 200 30 300 2 2 1 300 30 300 3 3 1 100 20 200 1 1 2 300 20 200 2 3 3 200 10 100 3 2 3 300 30 300 3 3 2 100 30 200 3 1 3 200 20 200 2 41. In the next examples, beginning with the units, add the units, the tens, and the hundreds separately. (11.) (12.) (13.) (14.) Ill 221 102 312* 201 101 30 112 322 230 323 310 203 13 112 33 132 323 232 231 42. In the following examples write the nuriibers to be added J units under units, tens under tens, and hundreds under hundreds. 15. Charles counted 201 bricks in one pile, 312 in another, 133 in another, and 123 in another. How many bricks did he count in all ? 16. Mrs. Otis has four strawberry-beds. In the first there are 321 plants, in the second 120, in the third 230, and in the fourth 203. How many plants has she in all ? 17. Mr. Otis bought a horse for 120 dollars, another horse for 230 dollars, another for 312 dollars, and a carriage for 330 dollars. How many dollars did he pay for all ? 18. Add together 123, 301, 22, 210, 300, 123, and 310. 19. Add together 201, 121, 330, 12, 13, 211, and 300. ^ADDITION. 17 Addition of 4's and 5's. 43. a. If you have 2 cents and earn 4 cents more, how many cents will you have then ? b. 2 and 4 are how many ? 3 and 4 ? 4 and 4 ? 4 and 3 ? c. Charles had 6 cents and Albert gave him 4 cents. How many cents did he then have ? d. How many are 6 and 5 ? 7 and 5 ? 7 and 4 ? 7 and 5 ? e. If you should make 8 snowballs and Dwight should make 4, how many would you both make ? /. 8 and 5 are how many ? 9 and 5 ? 9 and 4 ? 44. Drill Exercises. {1) (^) (3) iU) (5) (G) (?) {8) {9) (10) (11) (IS) g. Add 44444444432 3 to 132685974879 A. Add 555555555232 to 259 163748976 45. i. How many are 4 and 4 ? 14 and 4 ? 24 and 4 ? J. How many are 7 and 4 ? 17 and 4 ? 37 and 4 ? 47 and 4 ? A. How many are 9 and 4 ? 29 and 4 ? 49 and 4 ? 59 and 4 ? 1. How many are 7 and 5 ? 27 and 5 ? 47 and 5 ? 67 and 5 ? 222. How many are 9 and 5 ? 19 and 5 ? 39 and 5 ? 79 and 5 ? 22. How many are 8 and 5 ? 28 and 5 ? 58 and 5 ? 98 and 5 ? 46. o. Add by 4's from 4 to 40, writing down each sum as you find it. Add by 4's from 1 to 41 ; from 2 to 42 ; from 3 to 43. p. Add by 5's from 5 to 50 ; from 1 to 51 ; from 2 to 52 ; from 3 to 53 ; from 4 to 54. 47. Of. Name two numbers which added together make 3. T. Name a pair of numbers which added together make 4 j name another pair. s. Name a pair of numbers which added together make 5 ; name another pair; name another pair. 18 ADDITION. Examples for the Slate. 48. Copy and add the following : (20.) (21.) (22.) (23.) (24.) (25.) 2 2 3 30 300 500 3 4 2 40 500 300 3 5 4 20 100 500 5 4 1 50 400 200 3 2 5 30 200 400 4 5 2 40 500 300 1 4 5 50 ' 500 400 49. In addmg the units of Example a you have 12 units, which you have learned is the same as 1 ten and 2 units (Art. 13), so you may only write down the 2 units, and then add the 1 ten with the tens written in the next column. When adding the tens begin with the 1 ten ; thus, " 1, 4, 6, 8." a. 425 324 533 1282 In the same way do the following examples. {2Q>.) (27.) (28.) (29.) (30.) 301 406 304 421 330 404 313 225 312 445 423 105 514 204 352 524 432 233 134 35 32 624 22 225 110 414 314 415 542 643 31. Alfred sold 24 tomato-plants to one person, 35 to another, 14 to another, 42 to another, and 35 to another. How many tomato-plants did he sell ? 32. John picked up 35 nuts on Monday, 114 Tuesday, 135 Wednesday, 255 Thursday, 54 Friday, and 25 Saturday. How many did he pick up in all ? 33. Add together 204 trees, 45 trees, 543 trees, 404 trees, 350 trees, and Bb trees. ADDITION. 19 Addition of 6's and 7's. 60. a. Frank has set out 3 grape-vines, and his father has set out 6. How many grape-vines have both set out ? fe. How many are 3 and 6 ? 4 and 6 ? 3 and 7 ? 4 and 7 ? 51. 4 and 7 may be written 4 + 7. The sign + means and. This sign is read " plus." c. How many are 5 + 6? 5 + 7? Q^^Q^? 7 + 6? d. Mary knit 8 times around her mitten this morning and 6 times this afternoon. How many times did she knit around in all? e. 7 + 7 equals what number ? 8 + 7 equals what ? 62. The sign = means " equals," or " is equal to." /. 7 + 2 + 6 = what number ? 6 + 3 + 7 = what ? 63. Drill Exercises. (Jf) .(f) {3) (A) (5) (6) (7) {8) (9) (10) {11) {12) g. Add ^^QQ>Q>Q%QQ4.b4. to 4215 3 9687987 h. Add 777777777545 to 523164879987 64. i. 5 + ^-^"^ 15 + 6-? 25 + 6? Q>-\-Q,? 76 + 6? ^^ + Q? 5 + ^? m + Q,? 78 + 6? 6 + 7? 16 + 7? 46 + 7? 7 + 7? 27 + 7? 47 + 7? 8 + 7? 38 + 7? 88 + 7? 27 + 6? 85 + 7? 48 + 7? 66. o. Add by 6's from 6 to 60, writing the sums. p. Add by 6's from 1 to 61 ; from 2 to 62 ; from 3 to 63 ; from 4 to 64 ; from 5 to ^6. q. Add by 7's from 7 to 70 ; from 1 to 71 ; from 2 to 72 ; from 3 to 73 ; from 4 to 74; from 5 to 75; from 6 to 76. ^Q. r. Name each pair of numbers which added together make 6 ; which make 7 ; 8 ; 9 ; 10. J- 7 + 6-? 17 + 6=? 37 + 6? k. 9 + 6 = ? 29 + 6 = ? 49 + 6? 1. 4 + 7-? 34 + 7 = ? 54+7? m. ,5+7=? 45 + 7 = ? 75 + 7? 22. 9 + 7 = ? 19 + 7 = ? 69 + 7? 20 ADDITION, UNITED STATES MONEY. 67. Ten cents make a dime. Ten dimes make a dollar. One hundred cents make a dollar. The sign $ stands for dollars. The sign f stands for cents. 68. Examples for the Slate. What is the sum of $37, $502, $657, $462, 102 34. $735? 35. What is the sum of 35/, 72/, 45/, 9/, and 47/? 69. Dollars and cents are written together, twenty-five dollars and thirty-six cents are written (36.) $707 674 547 436 776 (37.) $1.67 2.34 7.06 6.57 7.65 (38.) $7.12 4.37 3.24 2.16 5.74 (39.) $ 12.42 17.32 62.45 77.23 54.36 \1^, and ^ 604/, Thus, $25.36. (40.) $ 54.07 67.35 44.72 52.47 77.64 60. In order to add the numbers in the following example easily, write them dollars under dollars and cents under cents. 41. M}^ sister went shopping, and bought a veil for 72 /, a pair of gloves for $1.25, some slippers for $2.25, some cambric for $ 3.57, and had $ 7.65 left. How much money had she at first ? ADDITION, 21 Addition of 8's and 9's. 61. a. Ira has 2 school-books and James has 8. How many school-books have both together ? b. How many are 2 + 8 ? 3 + 8? 3 + 9? 2 + 9? - c. Ira made 4 paper bags to-day and James made 8. How many paper bags did both make ? d. How many are 4 + 8 ? 4 + 9? 5 + 8? 5 + 9? e. If you have 6 pencils and buy 8 more, how many pencils will you then have ? /. Add together 7 pencils and 8 pencils ; 6 pencils and 9 pencils. ^. 2 + 5 + 9 = what? 2 + 6 + 8? 5 + 3 + 9? 5 + 4 + 9? 62. Drill Exercises. (1) {2) (5) (A) (5) (6) (7) (5) (9) (10) {11) {IS) h. Add 888888888676 to 685231479768 i. Add 9 9 9 9 9 9 9 9 9 7 6 7 to 3 6 2 1 5 4 8 7 9 8 9 7 63. j. 4 + 8? 14 + 8? 34 + 8? 6 + 8? 26 + 8? 66 + 8? k. 5 + 8? 25 + 8? 45 + 8? 8 + 8? 38 + 8? 68 + 8? 1. 9 + 8? 39 + 8? 69 + 8? 7 + 8? 17 + 8? 27 + 8? m. 4 + 9? 44 + 9? 94 + 9? 6 + 9? 16 + 9? 36 + 9? 22. 5 + 9? 25 + 9? 46 + 9? 7 + 9? 27 + 9? 57 + 9? 0. 8 + 9? 18 + 9? 48 + 9? 9 + 9? 29 + 9? 99 + 9? 64. p. Add by 8's from 8 to 80 ; from 1 to 81 ; from 2 to 82 ; from 3 to 83 ; from 4 to 84 ; from 5 to 85 ; from 6 to 86; from 7 to 87. q. Add by 9's from 9 to 90; from 1 to 91; from 2 to 92 ; from 3 to 93 ; from 4 to 94 ; from 5 to 95 ; from 6 to 96 ; from 7 to 97 ; from 8 to 98. 65. r. Name each pair of numbers which added together make 11; 12; 13; 14; 15; 16; 17; 18; 19. 22 ADDITION. QQ» Examples for the Slate. Copy and add the following : (42.) (43.) (44.) (46.) (46.) $3.28 $9.19 $0.36 $9.64 $8.19 32.35 8.20 7.21 9.65 7.28 5.92 7.39 8.92 8.70 6.37 8.67 6.48 6.79 2.78 5.00 7.98 5.57 5.43 7.87 6.67 0.83 4.66 2.19 9.86 7.68 6.77 3.75 0.87 6.93 8.79 0.06 2.84 6.54 2.39 9.80 47. I bought a hat for 13.75, a coat for $12, a pah- of gloves for § 1.75, some handkerchiefs for $ 3, and a pair of hoots for $ 7.50. What was the cost of all ? 48. How much money must you have that you may buy a dictionary for $1.25, a slate for 35,^, a ruler for 12/, a geog- raphy for $ 1.37, an arithmetic for 92/, a reader for 87/, and a writing-desk for $ 2.35 ? 49. My father owes $25.78 to one man, $79.48 to another, 97/ to a third, $75.94 to a fourth, and $5.48 to me. How much money does he owe in all ? 50. Charles bought, at a grocery-store, a barrel of flour for $9.35, butter for $17.36, beans for $3.50, fish for 92/, eggs for 58/, and $ 18.15 worth of sugar. What did tlie whole cost? 51. Mr. Smith spent during the winter months $ 18.88 for groceries, $ 15.91 for meat, $ 20 for fuel, $ 36 for rent, and $ 84 for clothing. What did he spend in all ? 62. What is the sum of $ 5.79, $87.68, $9.87, $87.38, $17, and $ 4.82 ? 53. What is the sum of $4.35, $25.98, $8.37, 13/, $38.49, and $1.67? 54. Add $9.98, $17.74, $0.49, $15, $23.05, and 6Q^f ? For other examples in Addition, see page 33. SUBTRACTION, 23 SECTION III. SUBTRACTION. Subtraction of I's, 2's,'and 3's from other Numbers. 67. What is the whole number of apples in the basket? If the two apples under the hand be taken away, how many will be left in the basket ? 68. Taking part of a num- ber away to find how many are left is subtracting. 69. The number, part of which is to be taken away, is the minuend. 70. The part of the minuend to be taken away is the subtrahend. 71. The part of the minuend left after a part has been taken away is the remainder. a. Subtract 2 from 5 ; 3 from 5 ; 2 from 6 ; 3 from 6. b. If you subtract 2 from 1, how many will be left ? c. What will be the remainder if 3 be subtracted from 4 ? from 7 ? from 8 ? from 11 ? from 12 ? Drill Exercises. 72. You will need to do examples like the following until you can subtract at sight. (7) (8) (9) m 8 7 9 10 1111 {!) (^) {3) ih) (-5) (6) d. Prom 2 3 6 5 1 4 subtract 1 1 1 1 1 1 24 I: SUBTRACTION. (1) (2) (S) (-4) (5) (6) (7) (5) (i?) (iO) e. From 4 11 5 3 8 6 9 7 2 10 subtract 2 2 2 2 2 2 2 2 2 2 /. From 4 3 T 5 10 12 8 11 6 9 subtract 3 3 3 3 3 3 3 3 3 3 78. g. How many are 11 less 2 ? 21 less 2 ? 31 less 2 ? A. 11 less 3 ? 21 less 3 ? 51 less 3 ? 71 less 3 ? 91 less 3 ? i. 12 less 3 ? 42 less 3 ? 62 less 3 ? 82 less 3 ? 92 less 3 ? 74. j. Subtract from 20 by 2's, writing down each remain- der as you find it ; thus, 20, 18, 16, 14, 12, and so on. k. Now subtract from 20 by 2's without writing the remain- ders. Subtract in the same way by 2's from 21. 1, Subtract by 3's from 30 ; from 31 ; from 32. Examples for the Slate. 75. Copy the following examples upon your slate and subtract, taking units from units, tens from tens, hundreds from hundreds, and writing the remainders as shown in Ex- amples a and h- To see if your work is right, Add the remainder to the subtrahend. The suni ought to equal the mimiend. 76. In Example c the 2 tens are taken from 11 tens. Subtract in the same way in the other examples. a. b. (1-) (2.) (3.) (4.) 33 43 56 125 468 359 12 33 20 13 121 202 21 10 c. (5.) (6.) (7.) (8.) (9.) L15 114 410 205 911 328 23 21 102 31 103 32 92 10. If you take 32 from 127, how man y will be left? 11. How many are 105 nuts less 22 nuts ? 112 less 31 ? SUBTRACTION. 25 Subtraction of 4's, 5's, and 6's. 77. a. Alvin had 5 melons and sold 4 of them. How many melons had he left ? b. How many are 6 less 4 ? 7 less 4 ? 7 less 5 ? 7 less 6 ? 78. 7 less 6 may be written 7-6. The sign -means less. This sign is read " minus " or " less." c. How many are 8 - 6 ? 8-4? 8-5? 9-5? d. Jane found 9 ripe peaches and Mary found 4. How many more peaches did Jane find than Mary ? e. How many more are 10 than 4 ? than 5 ? than 6 ? /. If you have 11 cents and spend 5 of them, how many cents will you have left ? g. If you have 11 and spend 6^ how many will you have left ? h, 12 -4 are how many? 12-5? 12-6? 13-6? 13-5? 79. DriU Exercises. (1) (3) (3) (4) (5) (6) (7) (8) (9) (10 i. From 13 5 8 4 7 11 9 10 6 12 subtract 4 4. A 4. 4c 4. 4 4 4 4 jf. From 5 14 7 8 11 6 9 13 10 12 subtract 5 5 6 5 5 5 5 5 5 5 k. From 10 8 11 14 7 13 12 15 6 9 subtract 6 6 6 6 6 6 6 6 6 6 80. i. How many more are 8 than 4? 18 than 4? 11 than 4 ? 21 than 4 ? 221. How many are 12 - 4 ? 22 - 4 ? 13 - 4 ? 53 - 4 ? n. 11-5? 21-5? 41-5? 12-5? 32-5? 42-5? 0. 13-5? 23-5? 53-5? 14-5? 44-5? 64-5? p. 11-6? 21-6? 31-6? 13-6? 23-6? 43-6? 81. q. Subtract by 4's from 40 ; from 41 ; from 42 ; from 43. r. Subtract by 5's from 50 ; from 51 ; from 52 ; from 53. s. Subtract by 6's from 60 ; from 61 ; from 62 ; from 63. 26 SUBTRACTION. 82. Examples for the Slate. (12.) (13.) (14.) (15.) (16.) (17.) From 247 125 ♦ 712 513 138 714 subtract 134 43 405 306 54 206 83. In Example a, as we cannot take 4 from 2, we change one of the 8 tens (leaving 7 tens) to units. This 1 ten equals (7){i2) 10 units. Ten units and 2 units are 12 units. We now 8 2 take 4 units away from 12 units and have 8 units left. 2 4 Next we take 2 tens from 7 tens, and have 5 tens left, r o and we have for the whole remainder 5 tens and 8 units, or 58. In subtracting, say " 4 from 12, 8 ; 2 from 7, 5. Ans. 58." In the same way you may do the following examples. (18.) (19.) (20.) (21.) {22.) (23.) 262 374 560 305 618 736 125 236 414 154 432 144 (24.) (26.) (26.) (27.) (28.) 1 21.40 $ 33.62 $42.34 180.91 1 73.85 15.36 26.16 14.05 25.44 65.16 29. A train of cars took 425 passengers into Buffalo and brought back only 216. How many more passengers were taken into Buffalo than were brought back? 30. In the Bates School are 527 pupils and in the Lincoln School are 455 pupils. How many more pupils are in the Bates School than in the Lincoln ? 31. In a certain town are 642 children, of whom only 526 attend school. How many do not attend school ? 32. Mrs. Eay bought a house for $2450, but paid only $1625 down. How much does she still owe? 33. James's father made 625 pounds of maple sugar last year, and has made 1242 pounds this year. How many more pounds has he made this year than he made last ? SUBTRACTION. Subtraction of 7's, 8's, and 9's. 84. a. Ned's father had 10 horses and sold 7 of them. How many horses had he left ? b. Seven from 10 leaves what ? 8 from 10 ? 9 from 10 ? c. Mary is 11 years old and Ethel is 6. What is the dif- ference of their ages ? d. Howmanyarell-8? 11-9? 12-9? 12-7? 12-8? e. John lives 13 miles from his cousin. After going 7 miles on the way, how many more miles has he to go ? /. What number added to 7 will make 13 ? 14 ? 15 ? g. What number added to 8 will make 13 ? 14 ? 15 ? h. Emma had 16 rabbits and gave away all but 7. How many did she give away ? i. What number taken from 16 will leave 7 ? 8 ? 9 ? 85. Drill Exercises. {1) {2) (3) iU) (5) (6') (7) {8) (9) {10) j. From 12 11 7 9 8 10 13 15 16 14 subtract 7 77-77 7 7 7 7 7 k. From 11 13 10 8 12 9 15 14 17 16 subtract 8 8 8 8 8 8 8 8 8 8 1. From 12 17 15 10 11 9 13 16 14 18 subtract 9 9 9 9 9 9 9 9 9 9 86. m. 11 - 7 ? 21 - 7 ? 41 - 7 ? 12 - 7 ? 62 - 7 ? 72 - 7 ? n, 14-7? 24-7? 54-7? 16-7? 46-7? 96-7? 0. 13-7? 43-7? 83-7? 15-7? 25-7? 75-7? p. 11-8? 31-8? 51-8? 13-8? 23-8? 53-8? q. 15-8? 25-8? 45-8? 17-8? 37-8? 87-8? r. 14-8? 44-8? 84-8? 13-8? 33-8? 73-8? -s-. 13-9? 33-9? 63-9? 15-9? 25-9? 85-9? t. 17-9? 67-9? 87-9? 16-9? 46-9? 96-9? 87. u. Subtract by 7^8 from 70; from 71; from 72; from 73; from 74; from 75; from 76. 28 SUBTRACTION. V. Subtract by 8's from 80 ; from 81 ; from 82 ; from 83 ; from 84 ; from 85 ; from 86 ; from 87. w. Subtract by 9's from 90 ; from 91 ; from 92 ; from 93 j from 94 ; from 95 ; from 96 ; from 97. Examples for the Slate. 88. a. Subtract 168 from 300. WRITTEN WORK. In this example, as we have no units to take the (2) (9) iio) 8 units from, and no tens to take the 6 tens from, 3 we change one of the hundreds (leaving 2 hundreds) 16 8 to 10 tens, and then change one of the tens (leaving 13 2 ^ tens) to ten units. We have now 2 hundreds, 9 tens, and 10 units, from which we can easily take the 1 hundred, 6 tens, and 8 units. In subtracting, say " 8 from 10, 2 ; 6 from 9, 3 ; 1 from 2, 1." (34.) (35.) (36.) (37.) (38.) (39.) 400 502 800 $ 17.28 1 98.76 $86.42 285 259 769 5.79 67.89 35.69 40. Mary took $5.10 to the store, and bought a pair of boots for $2.38. How much money had she left? 41. A man bought a horse for $ 378 and sold him for $ 417. How many dollars did he gain ? • 42. Sarah's father bought her a piano for $425, paying all but $287 of the money at the time he bought it. How much did he pay ? 43. A house and lot together were worth $4500, and the land alone was worth $ 1185. What was the house worth ? 44. Otis bought of Mr. Carter cloth to the amount of $ 29.80, and Mr. Carter took off $ 2.98 of the cost, for present payment. How much did Otis then have to pay ? 45. Mr. Diller raised 1280 baskets of peaches this year and 927 baskets last year. How many more baskets of peaches did he raise this year than last ? For other examples in Subtraction, see page 33. MISCELLANEOUS EXAMPLES, 29 89. Miscellaneous Oral Examples. a. James has 24 doves, and Walter has 9 more than James. How many doves has Walter ? b. If Burt has 26 chestnuts and gives 9 of them to Ida and 7 to you, how many will he have left ? c. Annie had 50/, and bought with it some paper for 9/, some ink for 10/, and an orange for 6 ^. How many cents had she left ? d. Add 9 to 32 and then subtract 7 from the sum. e. From 59 take the sum of 14 and 6. /. A boy threw a ball 7 feet beyond the fence. If the fence was 24 feet from the boy, how far must he go to get the ball ? How far to get the ball and return ? g. If John is 7 feet from you on the right and Alice 14 feet from you on the left, how far is John from Alice ? How far must John walk to go past you to Alice and then back to you again? h. The end of the school-room is 30 feet long. There are 2 doors in the end each 4 feet wide, and the rest of the distance is filled with blackboards. How many feet are filled with blackboards ? i. Another school-room is 33 feet long. Ellen is standing at one end, and Jane is standing opposite her at the other end. If Jane now walks 5 feet towards Ellen, and Ellen walks 7 feet towards Jane, how far apart will they be ? 90. Miscellaneous Examples for the Slate. 46. If a boy earns $ 9.50 and spends $ 2.35 for books, how much money will he have left ? 47. Grace has 1 29.63 and Charlotte has $ 7.85 more than Grace. How much money has Charlotte ? 48. Mr. Rolfe owned 435 acres of land and sold 42 acres to one person and 37 acres to another. How many acres had he left ? 49. The material for Kate's dress cost $5.84, the making cost $ 3.15 and the trimming $ 1.75. What did the whole dress cost? 30 SUBTRACTION. 50. A bookcase has 5 shelves. On the first are 24 books ; on the second, 37 ; on the third, 45 ; on the fourth, 48 ; and on the fifth, 52. If 4 books be taken from the first shelf, 8 from the second, and 17 from the third, how many books will remain in the case ? 51. Mrs. Bush lent a neighbor $ 20. He paid her at one time $ 6.50, at another time I b.S^, and at another time gave towards jDayment of the debt, a calf worth $ 7.50. How much of the debt did he then owe ? 52. Mr. Mason paid $ 121.80 for his passage to Liverpool, and $ 26.32 more for his jDassage back than for his passage out. How much did he pay for his passage back ? 53. A grocer paid $ 18 for a barrel of sugar and $ 0.75 for freight upon it. For how much must he sell it to gain $6.75? 54. Mr. Gove's income this year is $ 568 besides his salary of $ 1500. He pays $ 250 for rent, $ 67 for fuel, and a tax of $ 48. How much has he left for his other expenses ? ^^. Dora's mother gave her a 5-dollar bill, with which she bought some cambric for $1.55, some thread for 10/, buttons for 37^, and a sack pattern for 35/. How much change should she receive in return ? b^. Find the sum of $ 3684 and $ 2748, and then subtract from this sum $ 197. 57. From $ 30.07 take the sum of $ 5.87 and 62/. bS. A man who has $ 8600 owes $ 2107 to one person and $ 3648 to another. What will he have left after paying what he owes ? 59. A man owned a house that cost $ 5285 and a barn that cost $ 670 ; he was obliged to sell them both for $ 4500. How much money did he lose by the sale ? 60. John has 264/, which is 187/ less than Charles has. How many cents has Charles ? 61. Find the difference between 284 and 109, and then take that difference from 248 + 2964 4- 307. MISCELLANEOUS EXAMPLES. 31 62. A man bought a carriage for $465, spent $38 for re- pairing it, and then sold it for $ 485. Did he gain or lose, and how mnch ? 63. James shot an arrow 74 feet and Fred shot one 2S feet farther than James. How many feet must Fred go to pick up his arrow and return ? 64. Ralph lives 150 yards north of the school-house. One morning he went for Arthur, who lives 85 yards south of the school-house, and then went to school. How many yards did he travel ? Q5. Two girls start from the same place to roll hoops. One goes east 298 feet, and the other west 284 feet. How far apart are they then ? QQ>. If the two girls are 582 feet apart and should go towards each other, one 132 feet and the other 175 feet, how far apart would they he then ? 67. From a tank containing 412 gallons of water 148 gal- lons were drawn off, after which 327 gallons ran in. How many gallons were there in the tank then ? ^S. Here is a boy's account of what he received and w^hat he spent last June. How much has he left to begin his July account with ? June 1 17 June 3 " 6 " 8 " 15 On hand from last month Received from father For doing an errand For weeding for Mrs. Bates, 4 hours . Mother gave me Amount. Spent for cap, 75 f; for slate, S7 f Qave Mrs. Fox JfOf ; bought ball, 35 f Paid for square of glass, J^Of ; for paper, 5f\ Birthday's present for Susie Bought shoes, ^ 2.25 ; hoe, 50 f Amount Balance on hand June 30, B 42 00 10 50 25 25 32 DRILL TABLE. 92. DRILL TABLE No. 1. Simple Numbers. A B c D E F gh ijk I mno P'l rs t u vwx 14 752 4 505 25 324 1 231 26 720 5 604 48 747 7 038 46 237 6 162 37 421 5 370 32 471 7 390 16 908 6 325 26 623 8 919 84 162 2 591 45 305 7 052 98 143 4 028 21 750 7 644 32 681 3 237 33 626 5 381 18 645 9 804 42 638 2 486 47 970 8 275 45 242 1 827 54 352 5 129 31 327 9 413 81 147 2 624 34 428 3 796 72 638 4 140 15 514 4 468 45 902 1 561 23 762 8 574 32 540 7 018 35 644 7 631 29 412 5 381 44 371 3 628 17 915 4 973 41 427 2 376 41 109 9 462 36 755 6 705 95 484 4 092 45 237 8 892 82 723 8 407 24 642 4 534 34 528 1 295 32 768 2 347 66 892 3 Oil 45 374 3 926 19 717 2 814 42 461 9 319 78 158 5 390 26 175 9 235 39 243 4 321 15 780 1 527 46 905 j 6 804 DRILL EXERCISES. \ \\ 93. Explanation of the Use of the Drill Tables. The following illustration will show how the table on page 32 may be used for class drill so as to give each pupil a different example : 1. Let the members of the class number themselves 1, 2, 3, etc., to any given number up to 25 ; and let each member find his number in the left-hand margin of the table. 2. The teacher then gives a direction in this form : " Add A, D, and G." 3. In obedience to this direction, each pupil will add the numbers that he finds expressed under the letters A, D, and Gr, and in the line of his own number. Thus, pupil No. 1 will add 14, 25, and 23 ; No. 2 will add 26, 48, and 38 ; and so on. Thus a series of examples is given out at a single dictation, and the pupils are taught to work independently. The key contains answers to all these examples. 94. Exercises on Table No. 1. Addition. 1. Add A, D, and G. 2. Add A, B, and C. 3. Add D, E, F, and G. 4. Add B, C, D, and E. 5. Add A, B, 0, D, E, and 269. Subtraction. 6. From B take A. 7. From E take D. 8. From C take B. 9. Subtract E from F. 10. Find the difference between C andF. Oral Practice. 11. Add g and h ; add j; and q ; add y and z. 12. Add %y j, and Tc ; m, n, and o ; r, s, and t. IS. Add If m, n, o ; add u, v, w, x. 14. Add g, h, i, J, k, I, m, n, ; add p, q, r, s, t, u, V, u\ x, ij, z. 15. In each column g, h, i, etc., add all the numbers. How many are 16. U-g-ht 17. 57-g-h, etc., to H 18. S5- g-h, etc., tog'? 19. H-r-s-t, etc., to2;? 20. lOO-g'-A-i, etc., t0 2;? 34 MUL TIP Lie A TION. SECTION lY. MULTIPLICATION. Multiplication of 2's, 3's, and 4's. 95. How many apples are there in the hand? in the basket? on the table ? How many apples in ail ? a. How many are three 2's, or 3 times 2 ? h. How many are four 2's? five 2's? 6 times 2? In finding the answer to each of these questions, you united two or more equal numbers. 96. Uniting two or more equal numbers is multiplying, 97. One of the equal numbers to be united is the xnultiplicsLiid. 98. The number that tells how many equal numbers are to be united is the multiplier, 99. The result obtained by multiplication is the product. c. In the example " seven 2's are 14," which is the multi- plicand ? the multipher ? the product ? d. What is the product of 2 multiplied by 8, or 8 times 2 ? 100. 2 multiplied by 8 may be written 2x8. The sign X means '' multiplied by." e. What is the product of 2 x 9 ? of 2 x 10 ? /. Add by 2's from 2 to 24. The numbers you have named are all the products of 2 up to 24. (7) (8) (9) (iO) (H) 2 2 2 2 2 6 11 8 12 10 MULTIPLICATION. 35 Drill Exercises. 101. g- Write the products found by multiplying 2's from one 2 to twelve 2's, in the form given below. table * h. Kepeat from memory what you have q^^ 2 is 2 written. Two 2's are 4. i. Eepeat the table several times, for- Three 2's are 6. ward and back. Etc. Name the products of the pairs of numbers written below till you can give them in any order at sight. (!) (2) (5) (A) (5) (6) j. Multiply 2 2 2 2 2 2 by 2 4 3 7 5 9 k. Multiply these numbers again, and add 1 to each product. In this exercise, think what the products are, but name only the final results. Thus, in multiplying the first, think 4 and then say 5 ; in multiplying the second, think 8 and then say 9 ; and so on. 102. L Add by 3's from 3 to 36. m. Write the table of 3's from one 3 '^^^^^• to twelve 3's. ^"*^ 2, ^^ ^^ n. Repeat tlie table from memory for- ^ ward and back. {1) (2) (3) (A) (5) (6) (7) (5) (9) (10) (11) o. Multiply 33333333333 by 2 5 10 3 8 4 11 6 12 7 9 p. Multiply these numbers again and add 1 to each product. q. Multiply again and add 2 to each product. 103. r. Add by 4's from 4 to 48. TABLE. s. Write the table of 4's from one 4 to r^ a • * One 4 IS 4. twelve 4's. Two 4's are 8. t. E-epeat the table forward and back. Etc. * Sec Appendix, page 138. 3 6 MUL T I PLICA TION. (1) (2) (5) (^) (J) (6) (?) (5) (9) (10) (11) u. Multiply 44444444444 by 3 5 2 7 9 4 10 12 6 8 11 V. Multiply these numbers again and add 1 to each product. w. Multiply and add 2. x. Multiply and add 3. 104. Oral Examples. a. If a chair has 2 arms, how many arms have 3 chairs ? 4 chairs ? Solution. — If a chair has 2 arms, 3 chairs will have 3 times 2 arms, which is 6 arms. Aiis. 6 arms. (See Appendix, page 139.) b. "If a knife has 3 blades, how many blades have 2 knives? 3 knives ? c. If each of Harry's 6 playmates gave him 3 peaches, how many peaches did he receive ? d. A fork has 4 tines. How many tines have 2 forks ? 3 forks ? 7 forks ? e. At $ 4 a week, how many dollars will a person earn in 4 weeks ? in 5 weeks ? in 10 weeks ? /. What cost 6 mats at $ 1 each ? at $ 2 ? at $3 ? at $4 ? g. One dollar equals 2 half dollars. How many half dollars do 2 dollars equal ? 3 dollars ? 5 dollars ? iz. A square has 4 sides. How many sides have 3 squares ? i. A triangle has 3 angles. How many angles have 2 tri- angles ? 3 triangles ? 8 triangles ? J. If a pear is cut into 3 equal parts, each of the parts is 1 third. How many thirds in 4 pears ? in 5 pears ? 105. k. How many are three 2's ? two 3's ? Is there any difference in the answers ? 1. How many are 2 X 4 ? 4x2? four 3's ? three 4's? 222. How many are 3 x 2 ? 3 tens x 2 ? 3 hundreds x 2 ? 22. How many are 3x5? 3 tens x 5 ? 3 hundreds x 5 ? o. How many are 4 x 7 ? 4 tens x 7 ? 4 hundreds x 7 ? p. How many are 4x9? 4 tens x 9 ? 4 hundreds x 9 ? MUL TIPLICA TION. 37 Examples for the Slate. 106. Copy the following examples on your slate. Begin- ning with the units, midti2oly the Q. -Qo units, the tens, and the hitndreds scjp- 2 3 arately, writing the products as they — are written in Examples a and b. (1.) (2.) (3.) (4.) (5.) Multiply 313 423 304 402 404 by 3 3 4 5 6 c. d. In Example c, the product of Multiply 43 $2.40 the units is 18. 18 units are "j^y g 7 equal to 1 ten and 8 units. We o^ ^ A~Qn ^^'i^ite 8 under the line and keep the 1 ten to add to the product of the tens. The entire product is 258. In Example d, as the multiplicand is dollars and cents, so the product is dollars and cents. For the sake of rapid working, use as few words as possible. Thus, in Example c, say ''^eighteen, twenty-four, twenty-five^ While saying " eighteen,^^ write 8 ; and while saying " twenty-five," write 5 and 2. (6.) (7.) (8.) (9.) (10.) (11.) Multiply $413 $423 $ 1.42 $3.14 $3.24 $2.30 by 7 10 11 9 8 12 12. Mr. Granger has 115 sheep and Mr. Oaks has 3 times as many. How many sheep has Mr. Oaks ? 13. After travelhng 134 miles on a journey, Mr. Niles had 5 times that distance to go. How many miles had he to go ? 14. A man bought a boat for $ 132 and a horse for 4 times that sum. What did he pay for his horse ? 15. What will 8 hats cost at $ 2.40 apiece ? 16. What will 9 caps cost at $ 1.34 apiece ? 17. What will 12 chairs cost at % 4.42 apiece ? 38 MULTIPLICATION. Multiplication of 5's and G's. Drill Exercises. 107. a. Add by 5's from 5 to 60. b. Write the table of 5's from one 5 to twelve 5's in the same way that you wrote the tables of 2's, 3'S; and 4's. c. Eepeat the table forward and back. {!) (^) {3) U) (5) {6) (7) (S) {9) (10) {11) d. Multiply 555555 5555 5 by 3 8 7 2 9 4 10 12 6 11 5 e. Multiply these numbers again and add 1 to each product. /. Multiply and add 2. g. Multiply and add 3. h. Multiply and add 4. 108. i. Add by 6's from 6 to 72. J. Write the table of 6's from one 6 to twelve 6's. k. Eepeat the table forward and back. {1) (^) (5) L Multiply 6 6 6 by 2 7 10 222. Multiply these numbers again and add 1 to each product. 22. Multiply and add 2. p. Multiply and add 4. o. Multiply and add 3. q. Multiply and add 5. 109. Oral Examples. r. If there are 5 books on the table and 10 times as many on the shelf, how many are there on the shelf ? s. There are 6 working days in one week. How many are there in 2 weeks ? in 5 weeks ? in 10 weeks ? t. A cube has 6 faces. How many faces have 3 cubes ? 6 cubes ? 12 cubes ? u. If five men can do a piece of work in 2 d^ys, in what time can 1 man do it ? (^) (5) (6) (7) {8) (9) (10) ill) 6 6 6 6 6 6 6 6 3 8 11 4 9 12 5 6 MULTIPLICATION, 39 110. Examples for the Slate. (18.) (19.) (20.) (2i.) (22.) Multiply 456 564 645 450 536 by 2 ■ 3 4 5 6 (23.) (24.) (25.) (26.) (27.) (28.) 524 650 15.46 16.34 $4.65 $5.42 7 8 9 10 11 12 29. There are 365 days in a common year. How many days are there in 3 common years ? 30. There are 366 days in a leap year. How many days are there in 5 common years and 1 leap year ? 31. At $ 6.Q>b a week for board, what is the cost of 7 weeks' hoard ? 32. At $6.35 a day, what are Mr. Dole's wages for 6 days ? 33. Charles has $ 6.40 and his brother has 8 times as much. How much money has his brother ? 34. Mary bought 12 yards of silk at $ 1.65 a yard. What did it cost her ? 35. What must be paid for 11 acres of land at $2.56 an acre ? 36. By working 9 hours a day, a man can do a certain piece of work in 156 days. How many days will it take him if he works 1 hour a day ? 37. If 6 men can build a fence in 46 days, in how many days can 1 man build it ? 38. How many are 10 times 650 trees ? 12 times $ 536 ? 39. Multiply 456 by 3 and by 4 and add the products. 40. Multiply 564 by 5 and by 6 and add the products. 41. Multiply 654 by 7 and by 8 and add the products. 42. Multiply 346 by 9 and by 11 and add the products. 43. Multiply Q>QQ> by 4 and by 6 and add the products. 40 MUL T I FLIC A TION. Multiplication of 7's and 8's. Drill Exercises. 111. a. Add by 7's from. 7 to 84. b. Write tlie table of 7's from one 7 to twelve 7's. c. Eepeat the table forward and back. (i) (2) (3) {U) (v) (6) (7) {8) (9) {10) (11) d. Multiply 77777777777 by 3 6 4 2 7 5 11 9 12 10 8 e. Multiply these numbers again and add 2 to each product. /. Multiply and add 3. h. Multiply and add 5. g. Multiply and add 4. i. Multiply and add 6. 112. J. Add by 8's from 8 to 96. k. Write the table of 8's from one 8 to twelve 8's. 1. Eepeat the table forward and back. (i) (?) (.5) (A) (5) (6) (7) [8) (9) (10) (11) 222. Multiply 8,8888 88 88 88 by 7 9 2 8 3 10 4 11 5 12 6 22. Multiply these numbers again and add 3 to each product. o. Multiply and add 4. q. Multiply and add 6. p. Multiply and add 5. r. Multiply and add 7. 113. Oral Examples. s. At 7 cents a yard, what will 4 yards of ribbon cost ? t. There are 7 days in a week. How many days are there in 2 weeks ? in 3 weeks ? in 5 weeks ? in 9 w^eeks ? iz. There are 8 quarts in 1 peck. How many quarts are there in 2 pecks ? in 3 pecks ? in 4 pecks ? V. John is 8 years old, and his father is five times as old. How old is his father ? w. Annie has made a quilt with 8 rows of squares in it, 8 squares in each row. How many squares are there in the quilt ? MUL TIPLICA TION. 41 X. If 8 quires of paper can be bouglit for $ 1, how much can be bought for $6? for $7? y. If a piece of work can be done in 2 days by working 8 hours a day, in how many days can it be done by working 1 hour a day ? z. What is the product ofTxlO? 7x11? 8x7? 7x8? 114. Examples for the Slate. (44.) (45.) (46.) (47.) (48.) Multiply 678 875 748 837 728 by ^ 2 3 4 5 6 (49.) (50.) (61.) (52.) (53.) (54.) 187 780 878 787 684 768 7 8 9 10 11 12 bb. At $ 2 each, what will 187 pear-trees cost ? At $2 each, 187 pear-trees will cost 187 times $2. But 187 times $2 is the same as 2 times $ 187. (Art. 105.) Hence we multiply 187 by 2. Ans. $374. 6Q>. At 1 3 a day, what will a person receive for 278 days' work ? 57. At 6 cents a pound, what will 168 pounds of sweet potatoes cost ? 6^. At 11 cents a pound, what will 287 pounds of sugar cost? 59. What will 2 boxes of soap cost, each box contilining 74 pounds, at 9 cents a pound ? 60. How many days are there in 48 weeks and 1 day ? 61. At 8 cents a yard for muslin, what is the cost of 4 pieces, 47 yards in a piece ? 62. Multiply 278 by 6, and multiply the product by 5. 63. Multiply 426 by 3, and multiply the product by 7. 64. Multiply 177 by 4, and multiply the product by 9. 65. Multiply 387 by 2, and multiply the product by 11. 42 ' J/ UL T I FLIC A TION. Multiplication of 9's and lO's. Drill Exercises. 115. a. Add by 9's from 9 to 108. h. Write the table of 9's from one 9 to twelve 9's. c. Repeat the table forward and back. (i) (^) (5) U) (J) (6) (7) {S) {[)) (10) ill) d. Multiply 999999999 9 9 by 11 3 9 2 8 12 4 7 10 5 6 e. Multiply these numbers and add 4 to each product. /. Multiply and add 5. h. Multiply and add 7. g. Multiply and add 6. i. Multiply and add 8. 116. j. Write the table of lO's from one 10 to twelve lO's. k. Repeat the table forward and back. (i) (S) (5) iU) (5) (G) {7) {8) [9) {10) [11) L Multiply 10 10 10 10 10 10 10 10 10 10 10 by 3 10 12 2 11 9 4 7 5 8 6 222. Multiply these numbers and add 5 to each product. 22. Multiply and add 6. p. Multiply and add 8. o. Multiply and add 7. q. Multiply and add 9. 117. Write upon your slate the numbers from 1 to 9, as they are written below. (See Appendix, p. 139.) Beginning at the right, r. Multiply each number by 2 and add 1 to the product. s. Multiply by 3 and add 2. w. Multiply by 7 and add 6. t. Multiply by 4 and add 3. x. Multiply by 8 and add 7. u. Multiply by 5 and add 4. y. Multiply by 9 and add 8. V. Multiply by 6 and add 5. z. Multiply by 11 and add 9. MUL T I PLICA TION. 43 118. Oral Examples. a. If a man works 9 hours a day, how many hours does he work in 2 days ? in 3 days ? in 4 days ? b. Mary pays 10 cents a yard for braid. How much does she pay for 4 yards ? for 5 yards ? c. If you sleep 9 hours a day, how many hours do you sleep in 6 days ? in a w^eek ? d. At 10 cents apiece, what will 6 tops cost ? What will 7 tops cost? 8? 11? 12? e. How many dimes are there in $ 1 ? in $ 6 ? /. In one eagle there are $ 10. How many dimes are There ? g. What is the product of 9x8? 9x10? 9x12? h. What is the product of 10 x 11 ? 10 x 12 ? i. Is there any difference in the number of trees in two orchards, if one has 9 rows of 10 trees each, and the other 10 rows of 9 trees each ? Why ? j. How many tenths in one ? in two ? in ten ? 119. Examples for the Slate. Q>Q. Multiply 798 by 3 ; by 4 ; and add the products. 67. Multiply 897 by 5 ; by 6 ; and add the products. ^^. Multiply 590 by 7 ; by 8; and add the products. 69. Multiply 489 by 9 ; by 10 ; and add the products. 70. Multiply 397 by 11 ; by 12 ; and add the products. 71. Multiply 1399 by 2 ; by 3 ; and add the products. 72. Find the amount which must be paid for 48 cords of wood at $ 6 a cord, and 96 cords at $ 5 a cord ? 73. What must be paid for 2 barrels of sugar, the first con- taining 249 pounds at 8 cents a pound, the other 268 pounds at 11 cents a pound ? 74. What cost 498 tons of iron at $9 a ton, and 904 tons at $ 12 a ton ? For other examples in Multiplication by numbers no greater than 12, see page 71. 44 MUL T I PLICA TION. Multiplication of ll's and 12's. Drill Exercises. 120. a. Add by ll's from 11 to 132. b. Write the table of ll's from one 11 to twelve ll's. c. Eepeat the table forward and back. (i) {^) (3) (i) (5) (6) (7) {8) (9) {10) (11) d. Multiply 11 11 11 11 11 11 11 11 11 11 11 by 7 4 8 3 9 2 10 5 11 6 12 ^ — e. Multiply these numbers and add 5 to each product. /. Multiply and add 6. h. Multiply and add 8. g. Multiply and add 7. i. Multiply and add 9. 121. j. Add by 12's from 12 to 144. k. Write the table of 12's from one 12 to twelve 12's. 1. Repeat the table forward and back. (i) (S) {3) (i) (.5) {6) (7). {8) (9) {10) {11) m. Multiply 12 12 12 12 12 12 12 12 12 12 12 by 7 10 2 11 3 12 4 9 5 8 6 22. Multiply these numbers and add 5 to each product. o. Multiply and add 6. q. Multiply and add 8. p. Multiply and add 7. r. Multiply and add 9. 122. Oral Examples. s. If it taltes 11 yards to make a dress, how many yards will it take to make 2 dresses ? 3 dresses ? t. There are 4 cousins each 11 years old, what is the sum of their ages ? They have a teacher 5 times as old as either of them. How old is the teacher ? u. In one foot there are 12 inches. How many inches are there in 4 feet ? in 5 feet ? in 10 feet ? V. How many inches are there in a yard, which is 3 feet ? in 2 yards, or 6 feet ? w. How many eggs are there in 7 dozen ? in 10 dozen ? a. b. c. Multiply 9 9 9 by 10 100 1000 MULTIPLICATION. 45 123. Examples for the Slate. Ten nines equal 9 tens (Art. 105), or 90. So 100 nines equal 9 hundreds, or ~90 ~900 QQQQ 900 ; and 1000 nines equal 9 thousands, or 9000. Hence, to multiply by 10, annex a zero to the multiplicand; to mul- tiply by 100, annex two zeros; to multiply by 1000, annex three zeros. 75. Multiply 69 by 10 ; 69 by 100 ; 6 by 1000. 76. Multiply 47 by 10 ; by 100 ; and add the products. 77. Multiply 8 by 100 ; by 1000 ; and add the products. 78. Multiply 9 by 10 ; by 100 ; by 1000 ; and add the products. 79. Multiply 5 by 1000; by 100; by 10; and add the products. 80. Multiply 42 by 10 ; 268 by 10 ; 3 by 1000 ; and add the products. When the multiplicand and multi- 124. d. Multiply 30 p^-g^,^ ^^ ^^^^^^ ^^ ^j^^j^^ ^^^^^ ^g^^g ^^ ^y ^^ the right hand, disregard the zeros in 1200 multiplying, hut annex to the p^roduct as many zeros as were disregarded. (81.) (82.) (83.) (84.) (85.) (86.) Multiply 40 300 240 710 480 20 by 60 2 20 30 30 400 87. When hay is $ 20 a ton, what must be given for 10 tons ? for 100 tons? for 20 tons ? 88. There are 20 things in a score. How many things are there in 100 score ? 89. What is the cost of 20 rakes at 35 cents each ? 90. Mr. Oakes sold 140 bushels of potatoes at 60 cents a bushel. What did he receive for them ? ^--«==ss=::=:= 91. How many men working 1 hour would he^>^^^^^ ^o do as much as 80 men working 10 hours ? f^"^ ^^ 46 MULTIPLICATION. 92. Multiply 23 by 2 and by 30 and add the products. 93. Multiply 42 by 4 and by 50 and add the products. 94. Multiply bQ by 3 and by 20 and add the products. 95. Multiply 61 by 5 and by 50 and add the products. 96. Multiply 72 by 6 and by 50 and add the products. 97. Multiply 37 by 4 and by 70 and add the products. 98. Multiply 36 by 3 and by 40 and add the products. 99. Multiply 95 by 5 and by 80 and add the products. 100. Multiply 48 by 7 and by 60 and add the products. 101. Multiply 29 by 4 and by 30 and add the products. 126. e. Multiply 29 by 34 WRITTEN WORK. 29 To multiply by 34, multiply first by 4, 34 then by 30, and add the products. J 1 /( ^^ multiplying by 30, the product is 116 = product by 4. g^^ ^^^^ .^ .^ ^^^ necessary to write the 87 -product by 30. zero. 986 = product by 34. (102.) (103.) If you have made no mistake, the an- M Ifi-nlv S2 84 ^^^'srs to Examples 102 and 103 are equal. , QA ^9 Hence, to prove examples in multiplica- *^ tion, multiply the multiplier by the multi- plicand. The two products ought to he equal. Multiply and prove the following examples : (104.) (105.) (106.) (107.) (108.) (109.) 34 52 76 48' 39 83 x45 x37 x58 x64 x25 x98 110. How many are 67 x 79 ? (114.) 94 x 79 - what ? 111. How many are 67 X 36 ? (115.) 83 x 37 = what ? 112. How many are 52 x 45 ? (116.) 79 x 67 - what ? 113. How many are 68 x 25 ? (117.) 94 x 89 - what ? MULTIPLICATION. 47 126. /. Multiply 145 by 27 ; 205 by 27. PROOF. PROOF. 27 27 145 205 135 = product by 5. 135 = product by 5. 108 = product by 40. 54 - product by 200. 2J_ = product by 100. ^ ^ p^^^^^^ ^^ 205. 3915 = product by 145. 118. * Multiply 184 by 47. 122. How many are 583 x56? 119. Multiply 235 by 39. 123. How many are 28 x 470 ? 120. Multiply 605 by 15. 124. How many are 731 x 13 ? 121. Multiply 62 by 134. 125. How many are 33 x 206 ? For other examples in Multiplication, see page 71. 127. Applications. 126. At 11 cents a pounds what will a barrel of 269 pounds of sugar cost ? 127. If a train of cars runs 453 miles in a day, bow many miles will it run in 21 days ? 128. If a tailor makes 432 coats in a year, and puts 14 but- tons on each coat, how many buttons does he use ? 129. Mr. Frost's cow Lily yielded 154 pounds of butter in a year. The butter was sold at 35 cents a pound. How much money did Mr. Frost receive for the butter ? 130. A party of 23 persons hired a schooner, paying $ 3.75 apiece for the use of it. How much did all pay ? 131. What must be paid for 5 cheeses weighing 85 pounds each, at 16 cents a pound ? 132. Twelve dozen make a gross. Twelve gross make a great gross. How many pens are there in a great gross of pens ? 133. A family of 4 boarded in Plymouth for 9 weeks at the rate of $ 6.50 a week for each person. What was the cost of board for the whole time ? 48 MISCELLANEOUS EXAMPLES. 128. Miscellaneous Oral Examples. Eepeat the following tables : liiquid Measure. Dry Measure. 4 gills = 1 pint. 2 pints = 1 quart. 2 pints =1 quart. 8 quarts = 1 peck. 4 quarts = 1 gallon. 4 pecks = 1 bushel. a. How many quarts are there in one gallon ? how many pints ? how many gills ? b. What is the cost of a gallon of milk at 6 cents a quart ? c. If a gill of milk is required for a pint of coffee, how many gills are required for a quart ? for a gallon ? d. How many quarts are there in 2 gallons and 1 quart ? e. In a bushel there are how many pecks ? how many quarts ? how many pints ? /. At 10 cents a quart for tomatoes, how many cents will a man receive for 1 bushel and 2 pecks ? g. If I buy 3 bunches of asparagus at 8 cents a bunch, 2 quarts of new potatoes at 10 cents a quart, and some lettuce for 13 cents, how many cents should I pay for them ? h. If what I bought came to 57 cents, and I gave in pay- ment a 1-dollar bill, how much change should I receive in return ? Note. To take 57 cents from $ 1, or 100 cents, take away jfirst 50 cents, and then take 7 cents from the remainder. i. John was sent to the store with 60 cents to buy 3 pounds of beef at 11 cents a pound, 6 pounds of rhubarb at 2 cents a pound, and 2 bunches of radishes at 5 cents a bunch. How many cents should he bring back ? J. Edith bought 4 yards of cambric at 12 cents a yard and had 9 cents left. How many cents had she at first ? k. 12 inches make a foot and 3 feet make a yard. How many inches make a yard ? MISCELLANEOUS EXAMPLES. 49 L Measure your height in inches, also measure the distance you can reach with your arms extended, and find the difference. Notp:. In doing the following examples write the numbers on your slate, if necessary. m. A traveller bought for his breakfast, 2 sandwiches at 8 cents apiece, a cup of coffee at 10 cents, and 2 pears at 6 cents apiece. What did his breakfast cost him ? n. For his dinner he bought steak for 30 cents, 2 bananas at 8 cents apiece, some pie for 10 cents, and a cup of tea for 5 cents. What did his dinner cost him ? o. How much more did his dinner cost than his breakfast? p. How much did his breakfast and his dinner both cost ? q. How much change should he receive if he gave in pay- ment for his breakfast and dinner a 1-dollar bill ? T. 3 and 2, multiplied by 2, less 6, multiplied by 3, are how many ? s. 7 times 3, plus 4, plus 5, less 6, less 8, are how many ? t. Take 14, subtract 8, add 3, add 2, multiply by 2. u. 6, and 4, and 7, and 5, less 12, multiplied by 4 = what ? V. Think of any number less than 8, multiply that number by 5, subtract 3 times the number thought of, add 2, add 4, subtract twice the number thought of. What number have you ? 129. Examples for the Slate. 134. In a quire of paper there are 24 sheets. How many sheets are there in 20 quires or a ream ? 135. After a collection was taken in church, one box con- tained $ 25.34, another $ 10.17, another $ 14.21, and another $ 15.98. How much did all contain ? 136. In a basket of eggs I counted 175. S2 of them were turkey's eggs and the rest were hen's eggs. How many hen's eggs were there ? 137. Charles started to carry a basket containing 132 eggs, but he fell and broke 17 turkey's eggs and 29 hen's eggs. How many eggs remained whole ? 50 MISCELLANEOUS EXAMPLES 138. Add together all the numbers from 1 to 24. 139. There were landed from a ship 38 boxes of lemons. If each box contained 250 lemons, how many lemons were there in all ? 140. In an orchard of 125 trees, the owner set out 37 more, of which 13 died. How many remained alive ? 141. If 1 sparrow destroys 235 caterpillars in a day, how many will 6 sparrows destroy in a week ? 142. How many panes of glass are there in a house which has 45 windows of 6 panes each, and 4 cellar windows of 3 panes each ? 143. A hod-carrier went up and down a ladder 42 times in a day. If the ladder had 30 rounds, how many steps did he take upon it during the day ? 144. The children of a school can be arranged in 24 rows, 28 in a row. How many more children would be needed to make 27 rows, 27 in a row ? 145. A boy bought a writing-desk for $2.75 and a jack- knife for $ 1.37. He spoiled his jack-knife in mending his writing-desk and then sold the writing-desk for $ 3.86. Did he gain or lose by the whole transaction, and how much ? 146. A barrel of flour contains 196 pounds. If a pound of flour is put into each loaf of bread, how many loaves may be made from 25 barrels of flour ? 147. If 7 persons consume a barrel of flour in 13 weeks, how many persons will consume 20 barrels in 1 week ? 148. A man bought 4 bushels of potatoes for $ 3.50, and sold them at 80 cents a peck. How much did he receive for what he sold ? How much did he gain ? 149. A grocer paid $ 5.00 for a box containing 294 oranges ; 17 of the oranges were spoilt, and he sold the remainder at 5 cents apiece. Did he gain or lose, and how much ? 150. A dealer counted 4 boxes of oranges, and found in the first box 32 dozen and 4 : in the second, 35 dozen and 3 ; in the third, 28 dozen and 11 ; and in the fourth, 36 dozen and 7. How many oranges were there in all ? DIVISION. 51 SEOTIOIf T. DIVISION. Division by 2, 3, and 4. 130. How many apples are there on the table ? If you give these 6 ap- ples to some children, giv- ing them 2 apiece, to how many children will you give them ? a. How many 2's, or how many times 2, are there in 6 ? 131. Finding how many times one number is contained in another is dividing. 132. The number to be divided is the dividend. 133. The number by which we divide is the divisor. 134. The result obtained by dividing is the quotient. b. Divide 10 by 2, that is, find how many 2's there are in 10. c. What is the quotient of 18 divided by 2 ? of 20 divided by2? d. If you have 11 apples, to how many children can you give 2 apiece, and how many apples will be left ? 135. The part of the dividend left after the equal num- bers are taken away is the remainder. e. How many 2's are there in 13, and what is the remainder ? 52 DIVISION, f. What is the quotient of 2S divided by 4 ? 136. 28 divided by 4 may be written thus, 28 ~ 4. The sign -^ means '' divided by." ^. What is the quotient of 36 - 4 ? of 36 - 3 ? h. What is the quotient of 27 -- 3 ? of 22 - 2 ? i. What is the quotient of 13 ^ 3, and what remains ? 137. Drill Exercises. j\ Repeat the multiplication table of 2's. k. How many 2's in 2 ? 4 ? 6 ? 8 ? 10 ? 12 ? 14 ? 16 ? 18? 20? i. Write the division table in the form table.* given in the margin^ from 2 in 2 to 2's in 24. 222. Eepeat the division table forward and back. Divide by 2 the numbers written below, naming quotients and remainders at sight. (i) {2) {3) iU) {5) {6) (7) (8) (9) {10) {11) {12) n. 2)2 10 12 4 8 6 16 14 18 24 22 20 o. 2)3 11 13 5 9 7 17 15 19 25 23 21 138. p. Repeat the multiplication table of 3's. q. Write the division table of 3's from 3 in 3, to 3's in 36. r. Repeat the table forward and back. Divide by 3 the numbers written below : {!) {3) {3) (^) (.3) (6) (7) {S) {0) {10) {11) {12) s. 3) 3 15 9 27 6 18 12 24 36 21 33 30 t. 3) 4 17 11 2S 8 19 14 25 37 23 34 31 2 in 2 1. 2's in 4 2. 2's in 6 3. Etc. See Appendix, page 138. DIVISION. ^^ 139. K. Eepeat the multiplication table of 4's. V, Write the division table of 4's from 4 in 4 to 4's in 48. w. Eepeat the table forward and back. Divide by 4 the numbers written below : (i) (2) {3) {U) (5) (6) (7) (S) (9) {10) {11) {IS) X. 4)16 8 4 24 12 28 20 32 40 48 36 44 y, 4) 18 11 7 25 15 30 23 35 43 49 37 45 140. Oral Examples. a. How many tops at 2 cents each, can be bought for 6 cents ? for 10 cents ? h. At 2 cents each, how many tops can be bought for 21 cents, and how many cents will be left ? Sc/lution.^ — As many tops can be bought for 21 cents as there are 2's in 21; which is 10, and 1 remains. Ans. 10 tops; 1 cent remains. c. How many rows of chairs, 2 in a row, can you make with 4 chairs ? 12 chairs ? 19 chairs ? d. How many quarts are there in 10 pints ? in 14 pints ? in 24 pints ? e. In one yard there are 3 feet. How many yards are there in 6 feet ? in 9 feet ? in 10 feet ? /. At 3 cents an hour, how many hours must you work to earn 15 cents ? 24 cents ? 30 cents ? g. Four quarters make 1 dollar. How many dollars in 8 quarters ? in 12 quarters ? in 15 quarters ? h. How many gallons are there in 12 quarts ? in 14 quarts ? i. How many bushels are there in 4 pecks ? in 16 pecks ? in 21 pecks ? J. How many pints in 4 gills ? in 20 gills ? in 30 gills ? k. If a coat can be made from 4 yards of cloth, how many coats can be made from 24 yards ? from 29 yards ? * See Appendix, page 139. 54 DIVISION. Dividing Numbers into Equal Parts. 141. If 8 pears be divided equally between 2 boys, bow many pears will each boy have? Each boy wdll have one of the two equal parts into which 8 is divided. We have seen by multiplication that 2 fours are 8, so we know that each boy will have 4 pears. 142. If 1 pear be divided equally between two boys, _ %^ what will each boy have ? ^^m^m^i^' When any number or any thing is di- 'Llft ■I^^Ip vided into two equal parts, the parts are called halves. When it is divided into three equal parts, the parts are called thirds. When it is divided into four equal parts, the parts are called fourths; and so on. 143. Oral Exercises. a. What is 1 half of 8 apples ? of $ 10 ? of 14 ? of 18 ? b. What is 1 third of 6 pears ? of $ 9 ? of 15 ? of 27? c. What is 1 fourth of 8 cents ? of $ 16 ? of 12 ? of 20 ? d. If 12 figs be divided equally between two boys^ how many figs will each boy receive ? Solution. — Each boy will receive 1 half of 12 figs, which is 6 figs. Ans. 6 figs. e. How many figs w^ould each boy receive if 12 figs were divided equally among 3 boys ? among 4 boys ? /. How many fishes would each person receive if 24 fishes were divided equally among 3 jDersons ? among 4 persons ? g. By what do you divide to find 1 half of a number ? 1 third ? 1 fourth ? h. How many are 9 - 3 ? 9 tens - 3 ? 9 hundreds - 3 ? i. How many are 8^4? 8 tens -^ 4 ? 8 hundreds ^ 4 ? J. How many are 27 - 3 ? 27 tens - 3 ? 27 hundreds - 3 ? DIVISION. 55 Examples for the Slate. 144. Copy the following examples on your slate. Begin- iiinor with the hundreds, divide the a. ^ * Divisor ^) 306 Dividend, huudreds, the tens, and the units sep- ~77j^ n r « arately, writing the quotient as it is written in Example a. Ill doing Example a, say " 3's in 3, 1; in 0, none; in 6, 2." To prove your work, find the product of the divisor and quotient. This product ought to equal the dividend. (1.) (2.) (3.) (4.) (5.) 2) 842 2) 460 3) 366 3) 906 4) 480 b. c. In Example b, as 3 is larger than 1, begin 3) 1230 2) 1418 by dividing 12 hundreds. In Example c, ~410 /TAQ begin by dividing 14 hundreds ; and when you come to the tens, as 2 is larger than 1, put a zero in the tens' place Crt' the quotient and divide 18 units. (6.) (7.) (8.) . (9.) (10.) 3) 1569 3) 2118 2) 1612 4) 1620 4) 8360 "• In Example d, after dividing 26 by 4, there is a 4) 426 — J hem. remainder of 2, which is written at the right of the 1Q5 dividend. Note. In proving the work, the remainder must he added to the product of the divisor and quotient. (11.) (12.) (13.) (14.) (15.) 3) 2722 4) 2814 4) 3225 3) 2405 3) 3277 16. If 364 children are marching, 3 in a row, how many rows are there ? 17. How many rows would there be, if 364 were marching, 4 in a row ? 18. Albert bought 4 canary birds for $8.32. What was the price of each ? 19. A father left $ 8420 to his four children. What was the share of each child ? 56 DIVISION. Division by 5, 6, and 7. 145. Drill Exercises. a. Repeat the multiplication table of 5's. b. Write the division table of 5's from 5 in 5 to 5's in 60. c. Repeat the table forward and back. Divide by 5 the numbers written below : {!) (?) {3) (A) (5) {G) (7) (5) (9) {10) {11) {IS) d. 5) 10 30 50 25 5 35 m 15 40 60 20 45 e. 5) 13 32 54 26 7 39 57 16 43 62 24 48 146. /. Repeat the multiplication table of 6's. g. Write the division table of 6's from 6 in 6 to 6's in 72. h. Repeat the table forward and back. Divide by 6 the numbers written below : {1) {2) {3) {h) {5) {G) (7) {8) {9) {10) {11) {12 L 6) 12 36 60 72 6 42 66 18 48 24 54 30 j. 6) 15 ' 37 64 77 11 44 68 23 49 29 58 34 147. k. Repeat the multiplication table of 7's. 1. Write the division table of 7's from 7 in 7 to 7's in 84. m. Repeat the table forward and back. Divide by 7 the numbers written below : {1) p) {3) (/.) (5) {G) (7) {8) (.9) {10) {11) {12) n, 7) 28 49 7 63 77 14 m 70 21 84 35 42 o. 7) 31 54 11 69 82 17 62 72 23 %S 36 46 148. Oral Examples. p. If a sack can be made out of 5 yards of cloth, how many sacks can be made out of 25 yards ? 30 yards ? 34 yards ? q. 6 feet make a fathom. How many fathoms deep is it to a rock that is under water 24 feet ? 36 feet ? 48 feet ? DIVISION. 57 T. If an anchor sinks 59 feet, how many fathoms does it sink and how many feet besides ? s. How many weeks and how many days over are there in 25 days ? in 37 days ? in 60 days ? in 72 days ? t. What is 1 fifth of 2^ days ? 1 sixth of $ 36 ? 1 seventh of 84 cents ? 12. Charles bought a velocipede for $ 15, and paid 1 fifth of the price every month. Hai^tjnany dollars did he pay a month ? How many months did it take him to pay the whole ? 149. Examples for the Slate. (20.) ^ (21.) (22.) (23.) 5) 1520 6) 3541 6) 1809 7) 6329 ^' Ijli^ividing 17 (tens) by 5, we have 3 (tens) for the 5) ilo-6 quotient, and 2 tens remain. We write 3 under the 34 line in the tens' place, and unite the 2 tens with the 3 units of the dividend, making 23 units. 5's in 23, 4 and 3 remain, which we express as in the written work above. Ans. 34 and 3 remain. (24.) {25.) (26.) (27.) (28.) (29.) 5)184 ^5)283 6)728 7)1230 6)326 7)8142 30. How many sheep at $ 6 apiece can be bought for $ 159, and what will remain ? 31. In 365 daj^s how many weeks, and what remains ? 32. A milkman received $86.45 for milk which he sold at 7 cents a quart. How many quarts did he sell ? 33. How man}^ settees will be required to accommodate 1000 persons if 5 are seated on each settee ? 34. Five men hired a boat for $ 12.75, agreeing to divide the cost equally. What was the share of each ? 35. Six men shared equally a profit of $ 7464. What did each receive ? 36. What is 1 fifth of 1375 oxen ? 1 seventh of $ 4529 ? 58 DIVISION. Division by 8, 9, and 10. Drill Exercises. 150. a. Repeat the multiplication table of 8's. b. Write the division table of S's from 8 in 8 to 8's in 96. c. Kepeat the table forward and back. Divide by 8 the numbers written below : (I) (f) (3) (A) (5) {&)^fk («) (9) {10) (11) (11) d. 8) 16 40 64 80 8 48 88 24 56 96 32 72 e. 8) 22 45 66 87 13 54 89 31 .^58 99 36 75 151. /. Eepeat the multiplication table of^'s. g. Write the division table of 9's from 9 in 9 to 9's in 108. h. E-epeat the table forward and back. Divide by 9 the numbers written below i^*^ (1) (2) (3) (^) (5) (G) (7) (8) (9) (10) (11) 90 36 72 54 99 18. 63 81 27 108 i. 9) 45 J. 9) 51 92 42 76 61 102 21 65 "^P 32 112 152. k. Repeat the multiplication table oi 10^ 1. Write the division table of K|'s from 10 in 10 to iO's in 120. 222. Repeat the table forward and back. Divide by 10 the numbers written below : (1) (2) (3) (A) (5) (6) (7f (8) (9) (10) (11) 22. 10) 20 31 45 60 57 73 88 90 108 123 110 Oral Examples. 153. o. Divide 9 yards of ribbon equally among 8 girls. Divide 10 yards. If 9 yards are divided among 8 girls, each girl will receive 1 yard, and there will still be 1 yard to be divided. If 1 yard is divided amoncr 8 girls, each girl I I I I I I I I I -^ r> n 7 n will receive 1 eighth of a yard. So each girl 1 eighth. 'vvill have 1 and 1 eighth yards. DIVISION, 5^^ If 10 yards are divided among 8 girls, after giving 1 yard to each . ■..■■,,. girl, there will be 2 yards to be divided, which 1 eigjjth. will give 2 eighths of a yard apiece. So each I I 1 I I I I I I girl will have 1 and 2 eighths yards. ' ''^'^'^- p. 8 men bought a fishing-net for $ 21. What part of the price should each pay ? How many dollars ? q. What is 1 eighth of 20 ? of 23 ? of 34 ? T. Divide 41 rods of land into 8 equal parts ; divide 52 rods ; divide 78 rods. 5. If a party of 10 men catch ^Q pounds of fish, what is the share of each ? 154. Examples for the Slate. a. Divide $ 6530 by 8. in the second form of the written wPviTTEN WORK. work of this example, the last re- ^ 8) $ 6530 - $ 2. 8) $ 6530 mainder, $ 2, is divided, giving a ~r^7^ ^Qifa quotient of 2 eighths, which is ^ written thus, |. Ans. $816|. Divide in the same way the following examples : (37.) (38.) (39.) (40.) 8) 3297 8) 4542 8) 4125 8) 6321 (41.) (42.) (43.) (44.) 9) 7234 9) 1928 9) 1234 9) 7361 (45.) (46.) (47.) (48.) 8) 5187 9) 1904 -10) 8761 10) 3205 49. Mr. Eustis bought a house for $ 3824 and paid 1 eighth of it every year till it was paid for. How much did he pay a year ? 50. Mr. Clark teaches 9 months in the year at a salary of $ 1900. What is that a month ? 51. Miss Alvord teaches 10 months in the year at a salary of $ 1275. What is that a month ? 52. If it costs a family of 8 persons $ 1535 a year to live, what is the average cost for 1 person ? 60 DIVISION. Division by 11 and 12. Drill Exercises. 156. a. Repeat the multiplication table of ll's. b. Write the division table of ll's from 11 in 11 to ll's in 132. c. Eepeat the table forward and back. Divide by 11 the numbers written below : {1) (^) (5) (>i) (5) (6) (7) [8) (9) {10) [11) d. 11) 22 33 66 88 110 44 77 121 55 99 132 e. 11) 25 37 71 95 118 53 81 122 65 101 141 166. /. Eepeat the multiplication table of 12's. g. Write the division table of 12's from 12 in 12 to 12's in 144. h. Eepeat the table forward and back. Divide by 12 the numbers written below : I 12)36 iS) {3) 24 96 144 (•5) (6) 132 48 (7) [8) 120 60 (9) {10) {11) 108 84 72 J. 12) 42 31 101 152 141 52 122 71 113 93 73 167. Oral Examples. k. At 12 cents apiece, how many melons can be bought for 24 cents ? for 40 cents ? for 50 cents ? i. It takes Mary 60 hours to crochet a jacket. How many days will it take her if she works 12 hours a day ? 221. How many days will it take Mary to crochet the jacket if she works 11 hours a day ? 21. How many feet are there in 12 inches ? in 24 inches ? in 18 inches ? in 96 inches ? (See Example u, page 44.) o. When 10 yards of silk can be bought for $ 15, what is the price of 1 yard ? p. There are 12 months in a year. In 36 months how many years ? in 30 months ? in 42 months ? DIVISION. 61 q. What is 1 eleventh of $ 22 ? of $ 44 ? of $ 66 ? r. What is 1 eleventh of $ 27 ? of $ 51 ? of $ 72 ? s. What is 1 twelfth of 36 n of 60 / ? of 75 / ? t. What is 1 twelfth of 35 days ? of 100 days ? 158. Examples for the Slate. Find how many times $ 11 is contained in the sums of money written below : (530 (54.) {66.) (56.) $ 11) $ 5602 1 11) $ 6281 $ 11) $ 3406 $ 11) $ 4987 Find 1 twelfth of the sums of money written below : (57.) (58.) (59.) (60.) 12) $ 7681 12) $ 3908 12) $ 85.29 12) $ 50.03 For other examples in Division by numbers no greater than 12, see page 71. 159. a. Divide 1698 by 32. WRITTEN WORK. In this example, since 32 is a larger number 32) 1698 (53 A *^^^^ ^^^ ^^ ^^'^^ divide 169 (tens) by 32. -j^^Q There are about as many 32's in 169 as there are 3's in 16, which is 5. We write 5 for the tens of the quotient, at the right of the curved ^ line. Multiplying 32 by 5 (tens), we have 160 2 (tens). Subtracting 160 from 169, we find 9 tens remain. Uniting with these 9 tens the 8 units of the dividend, we have 98 units. 32's in 98, 3 and 2 units remain. We write 3 for the units of the quotient. Dividing the remaining 2 by 32, we have 2 thirty-seconds, and the entire quotient is 53^^. Divide in the same way the following examples : (61.) (62.) (63.) (64.) _ 21) 1164 ( 42) 1984 ( 44) 3668 ( 51) 4897 ( k(65.) (66.) (67.) (68.) 3) 2998 ( 62) 4528 ( 67) 2363 ( 41) 3784 ( 62 DIVISION. 69. How many are 394 - 34 ? 73. Divide 3949 by 84. 70. How many are 538 - 47 ? 74. Divide 7009 by 51. 71. How many are 784 - 26 ? 75. Divide 7958 by 68. 72. How many are 892 - 75 ? 76. Divide 9876 by 92. 160. b. Divide 1792 by 24. WRITTEN WORK. j^ Example b, when we say that there are about 24) 1792 (74if as many 24's in 179 as 2's in 17, we find 8 for the 168 first term of the quotient; but on multiplying we ~Tio get 192, a number too large to subtract from 179. Therefore 8 is too large for the first term of the quotient, and we try 7, as shown in the work 16 above. 77. How many are 1443 - 74 ? 81. Divide 9278 by 87. 78. How many are 2938 - 77 ? S2. Divide 2876 by 43. 79. How many are 8642-46 ? 83. Divide 6444 by 96. 80. How many are 4672 - 58 ? 84. Divide 5105 by 64. To divide by 10, 100, or 1000. 161. c. In 1456;, how many tens and what remains ? Ans. 145 tens and 6 remain. d. In 1456j how many hundreds and what remains ? how many thousands and Avhat remains ? e. Divide 1456 by 10 ; by 100 ; by 1000. To divide by 10, put a point in tlie 1456 -f- 10 =145.6 dividend one place to the left; to divide by 1456-^100 =14.56 100, put a point two places to the left; to 1456 -=- 1000 = 1.456 divide by 1000, put a point three ptlaces to the left. In Example e, the first answer is 145 and 6 tenths ; the second is 14 and 56 hundredths ; and the third is 1 and 456 thousandths. 85. Divide 3456 by 100. 88. Divide $ 8100 by 10 ; by 100. 86. Divide 252 by 100. 89. Divide $1649 by 100; by 10. 87. Divide $ 3000 by 10. 90. Divide $ 4930 by 1000. DIVISION. 63 To divide United States Money, carrying the Division to Cents. 162. /. Divide $ 34.87 by 32. g. Divide $45 by 32. WRITTEN WORK. In Example /, /. s- the dividend has 2 32) $ 34.87 ($ 1.08§i 32) i >45 ($ 1.40§J places for cents, so 32 32 in the quotient we 287 1300 mark off 2 places 256 128 for cents. 31 20 In Example g. after the dollars are divided there is a remainder of $ 13, which equals 1300 cents. 1300 cents divided by 32 gives 40|^ cents, and the whole quotient is$1.40ff In doing the following examples, whenever a remainder is dollarSj annex two zeros and continue the division to cents. 91. Divide $ 48.40 by 15. 97. Divide $ 48 by 14. 92. Divide $ 82.36 by 32. 98. Divide $ 28 by 32. 93. Divide $ 97.48 by 63. 99. Divide $ 42 by 34. 94. Divide $ 99.19 by 42. 100. Divide $ 84 by 41. 95. Divide $ 49.32 by 54. 101. Divide $ 4 by 25. 96. Divide $ 85.12 by 81. 102. Divide $ 8 by 27. 163. Applications. 103. How many piles of 20 half-dollars each, can you make with 348 half-dollars ? 104. How many piles of 40 quarter-dollars each, can you make wdth 583 quarters ? 105. Mr. Goss planted 1296 hills of corn in rows of 27 hills each. How many row^s did he plant ? 106. If Mr. Goss had planted the same number of hills in 16 rows, how many hills would he have had in a row ? 107. Mr. Green earned $ 28.68 in a month and spent 1 twelfth of what he earned for books. How much did he spend for books ? 64 DIVISION. 108. A man bought a house for $2175, agreeing to pay 1 tenth of the price every year till it was paid for. How much did he agree to pay each year. How many years would it take him to pay the debt ? 109. How many barrels can be filled with 2000 pounds of flour, each barrel containing 196 pounds ? 110. A load of potatoes weighing 742 pounds was put into bags of 2 bushels each. If each bushel weighed 60 pounds, how many full bags were there, and how many pounds re- mained ? 111. How many dollars are there in 2762 cents ? in 5648 cents ? 112. If a man walks 25 miles a day, how many days will it take him to walk 1000 miles ? 113. There are 5280 feet in a mile. How many times must a wagon-wheel turn in going a mile if it is 14 feet round the rim ? For other examples in Division, see page 71. Miscellaneous Oral Examples. 164. a. Ira sells cherries at 10 cents a quart. How many quarts must he sell to receive 50 cents ? 90 cents ? $ 1 ? b. Carl sells berries at 9 cents a quart. How many quarts must he sell to get enough money to buy a hat worth 70 cents and a slate worth 11 cents ? c. Jane makes paper-bags at 8 cents an hour. How much will she earn in half an hour ? in 4 and 1 half hours ? d. How many peck baskets will it take to hold 8 quarts of plums ? 32 quarts ? 28 quarts ? e. How many times must a quart measure be filled to meas- ure out a bushel and a half of cranberries ? /. How many quart cans will it take to hold 8 and 1 half gallons of milk ? g. How many times must you pick a pint dish full of ber- ries to fill a basket that holds a j)eck ? that holds a peck and 2 quarts ? MISCELLANEOUS EXAMPLES, 65 h. How many pounds of beef at 12 cents a pound will be required to pay for 9 quarts of cherries at 8 cents a quart ? i. If $81 is paid for 9 weeks' labor, what is paid for 4 weeks' labor ? for 7 weeks' ? for 12 weeks' ? j. If 6 yards of muslin cost 60/, what will 7 yards cost ? 10 yards? 12 yards ? k. If 7 persons consume a barrel of sugar in 9 months, how long will it last 1 person ? How long will it last 8 per- sons ? 1. If a family of 3 persons consume a barrel of flour in 6 months, how long will it last if 2 persons are added to the family ? 165. Repeat the following table of Time. 60 seconds = 1 minute. 60 minutes =1 hour. 24 hours = 1 day. 7 days =1 week. 52 weeks and 1 day \ or 365 days ; 100 years =1 century. a. How many seconds are there in half a minute ? b. How many minutes are there in half an hour ? in a quarter of an hour ? in an hour and a quarter ? in an hour and a half ? The word ''day" as used in the table denotes the interval of time be- tween one midnight and the next. As used in the following example, it denotes the interval of time between sunrise and sunset. c. When the sun rises ^t six o'clock and sets at six o'clock, how long is the day ? how long is the night ? d. When the days are 11 hours long, how long are the nights ? e. If a person is 84 years old, how many years longer must he live to be a century old ? = 1 year. 66 DIVISION. There are 12 months in the year, with days as follows : ** Thirty days hath September, April, June, and November ; All the rest have thirty- one. Except the second month alone, To which we twenty-eight assign, Till leap-year gives it twenty-nine." /. Name the months that have 30 days. g. What month has 28 days ? How many days has it in leap-year ? h. How many days has each of the other months ? i. The winter months are December^ January, and Feh- ruary. How many days are there in winter ? J. The spring months are March, April, and May. How many days are there in spring ? k. The summer months are June, July, and August. How many days are there in summer ? 1. The autumn months are September, October, and No- vember. How many days are there in autumn ? 166. Repeat the following tables : Numbers. 16 ounces =1 pound. 12 dozen = 1 gross. 2000 pounds = 1 ton. 20 units =1 score. 12 gross = 1 great gross. 222. If an ounce of chocolate is used at each meal, for how many meals will 2 pounds of chocolate serve ? 22. If 200 pounds of beef are put into a barrel, how many barrels will it take to hold a ton ? o. How many dozen buttons are there on a card that con- tains half a gross ? How many buttons on the card ? p. The allotted age of man being "threescore years and ten," how old is a man who has lived 17 years beyond the allotted age ? MISCELLANEOUS EXAMPLES. 67 167. Miscellaneous Examples for the Slate. 114. A farmer has 45 sheep and 18 lambs. How many more sheep has he than lambs ? How many of both has he ? 115. A flour-dealer sold 28 barrels of flour to one man, 33 to another, 17 to a third, and then had 115 barrels left. How many barrels had he at first ? 116. If 6 sheets of paper are required to make a book of 192 pages, how many pages are printed on one sheet ? 117. At 4/ a dozen for hooks, what will it cost for hooks for 4 closets having 18 hooks in each, and 2 closets having 30 hooks in each. 118. Belle gave a half-dollar, a dime, and 3 cents for a fan, and had ^2.bS left. How much had she at first ? How many cents did the fan cost ? 119. A newsboy sold 16 Journals at 3 ^ apiece, 23 Heralds at 2f apiece, 13 Advertisers at 4/^ apiece, and 18 Transcripts at 4/ apiece. What did he receive for all ? 120. At $ 2 a pound for worsted, what will 1 ounce cost ? 121. Kate bought a pound and a quarter of worsted at $ 2 a pound, a thimble for 85 cents, and a crochet-needle for 7 cents. How much did she pay for all ? 122. Mr. Hubbard bought 6 gross of lead-pencils at $1.80 a gross. What did he pay for the lot ? 123. If 1 gross of pencils can be bought for $ 1.80, what is that per dozen ? 124. Here are the charges for charges. Olive's bonnet. What did it cost ? ^^^^ ^ ^-^^ 125. Olive's bonnet cost $1.89 ^^^^^^ ^-^^ more than Mary's. What did Mary's RMon 63 cost? Lace 38 126. What was the cost of both Frame and making 1.60 bonnets, and 2 scarfs at $1.62 Amount i apiece ? 127. If 196 pounds of flour cost $ 9.80, what cost 50 pounds ? 128. At $ 3.20 a bushel for cranberries, what cost 9 quarts ? 68 MISCELLANEOUS EXAMPLES. 129. I sent a message of 34 words by telegraph, paying 25 cents for the first 10 words, and for each remaining word 2 cents. What did the message cost ? 130. If a horse-car makes 12 trips a day, and takes on the average 24 passengers at 5 cents apiece, how much money does the conductor take in a day ? 131. Dr. Lamb caught 13 and a half pounds of trout, worth 36 cents a pound. What was the lot worth ? 132. When oysters are 38 cents a quart and shad 12 cents a pound, what shall I pay for a pint of oysters and 3 and a half pounds of shad ? 133. If a horse-shoer puts 8 nails in each shoe and has used up 1376 nails, how many shoes has he set ? How many horses has he shod, each horse being shod all round ? 134. If a watch loses 7 seconds a day, how many seconds will it lose in a year ? How many minutes ? 135. At 4 cents a dozen for clothes-pins, what will 300 clothes-pins cost ? 136. If you buy 6 papers containing 25 needles each, for 25 cents, how many needles do you get for 1 cent ? 137. At 62 cents a bushel, what will 5 and a half bushels of potatoes cost ? 138. A clothier bought 12 coats for $ 13.92, and sold them for $ 1.50 apiece. What did he gain on the lot ? 139. If two gallons of lemonade are made with 14 lemons costing 36/ a dozen, and 2 pounds of sugar costing 11/ a pound, what is the cost of a gallon of lemonade ? of a quart ? of one glass holding half a pint ? 140. If a family consume 2 quarts of milk a day at 7 cents a quart from October 1 to April 1, and 3 quarts a day at 6 cents a quart from April 1 to October 1, what is the cost of milk for the year ? 141. A man bought 350 cords of wood for $ 962.50. If he sold 100 cords at $2.65 a cord, 175 cords at $3.20 a cord, and the rest at $ 3.33 a cord, did he gain or lose, and how much ? MISCELLANEOUS EXAMPLES. 69 142. How many pins are there in a paper having 12 rows, and 30 pins in a row ? If you can buy the pins for 18 cents, how many pins do you get for 1 cent ? 143. Which is cheaper, to buy 360 pins for 18 cents, or a paper containing 12 rows of 20 pins each, for 10 cents ? 144. As Mark rode to his uncle's, he watched one of the wagon-wheels and found it turned 550 times. The wheel measured 13 feet round. How many feet did he ride ? How many miles ? 145. Alfred can walk around a certain pond in 9 minutes, and Bertram can walk around in 8 minutes. How many times can Alfred walk around while Bertram walks around 36 times ? 146. How many feet is it from the first floor of a house to the attic, which is reached by 2 flights of stairs, the first having 16 steps, and the second 13 steps, each step being 8 inches high, and 12 inches making 1 foot ? 147. Here is a bill of goods bought by Mr. James Springer of R. H. White & Co. Find what ali the goods cost, and make a copy of the bill upon paper. Boston, Oct. 7, 1878. G^^l. Ja??zed (^/iu'??.^e^ ""^Otigllt Of E. H. WHITE & 00. S yu'U^ z^eAe^ @ ^ S.^{? / ^ 4 '' cadd77tei.e " .^6^ £ - ac'/d - S.(y/9 § -^mae^ . '' . S§ / /^/, ^4.^^; 1/ cap?,e, //.i^<^ / c/o-iieTt /lac^ aocu^ 7^ / Received 'payment, h M. s^if. ^m. i(-i^' \ ^o. 70 DRILL TABLE. 168. DRILL TABLE No. 2. simple Numbers. A B c D E F M 41- N 5 tu O viu xy 4 16 25 39 67 66 28 987 6909 7 19 32 43 61 67 49 996 2985 2 22 29 38 55 61 24 872 4360 6 14 35 46 49 66 54 327 2943 3 20 20, 37 64 68 36 436 2616 8 13 34 44 56 62 72 216 1728 5 17 30 40 60 70 46 543 1086 7 23 27 45 58 63 56 842 5894 11 15 33 41 52 69 88 471 1884 9 21 31 47 59 64 27 174 1044 10 18 28 42 53 71 66 862 3448 12 24 36 48 60 72 96 854 1708 G H 1 J K L 81 534 3738 74 90 99 115 124 135 16 291 2619 80 93 97 110' '121 139 39 487 3896 73 86 98 114 126 134 68 390 1950 82 95 103 118 130 143 94 876 2628 75 85 101 109 128 138 63 912 5472 79 91 108 116 122 133 42 396 2772 81 87 107 113 131 142 33 483 1932 76 89 100 111 125 136 96 236 1880 78 94 106 117 123 141 57 387 1161 83 88 104 112 127 137 18 897 4485 77 92 102 119 129 140 48 624 5616 ' 84 ■96 108 120 132 144 57 246 1476 DRILL EXERCISES. 71 169. Exercises Examples for the Slate. Multiply N 21. By 2. 26. By 7. 22. By 3. 27. By 8. 23. By 4. 28. By 9. 24. By 5. ^a By 11. ^J. By 6. 30. By 12. Multiply M 31. By 23. .?^. By 76. ^£ By 54. 34. By 98. Divide N 35. By 2. 4^. By 7. ,56. By 3. 4i. By 8. 57. By 4. ^^. By 9. 38. By 5. ^. By 11. ,5R By 6. 44. By 12. Divide 45- By 2. 50. By 7. ^5. By 3. 51. By 8. 47. By 4. 52. By 9. 4^. By 5. 53. By 11. 49. By 6. 54. By 12. 55. Divide ty P. 56. Divide ty M. 57. Divide by N. Miscellaneous. 58. What will P yards of cloth cost at M cents a yard ? 59. If a person buys goods for N cents, and gives in payment a upon the Table. 10-dollar bill, what should he receive in return ? 60. At P cents each, how many pears can you buy for N cents ? 61. Divide by P and add N to the quotient. 62. If 7 men can do a piece of work in M days, in how many days can 8 men do it ? Oral Practice. 63. Multiply qhy r and add s to the product. 64. Multiply r by s and add t to the product. 65. Multiply shj t and add ii, to the product. 66. Divide A and B each by 2. 67. Divide A, B, and C each by 3. 68. Divide B to D each by 4. 69. Divide B to E each by 5. 70. Divide B to F each by 6. 71. Divide B to F each by 7; by 8; by 9; by 10; by 11; by 12. 72. Divide G by 7. 73. Divide G and H each by 8. 74. Divide G, H, and I each by 9. 75. Divide G to J each by 10. 76. Divide G to K each by 11. 77. Divide G to L each by 12. 78. Multiply g by r and divide the quotient by s. 72 READING AND WRITING. SEOTIO]^ VI. Numbers from Ten Thousand to Millions. 170. Count by tens from ten to a hundred. Count by ten-thousands from ten thousand to a hundred thousand. Ten thousand is written 10000 or 10,000. Twenty thousand is written 20000 or 20,000. And so on. How many figures are needed to write ten-thousands ? Which figure shows how many ten-thousands there are? 171. Read the following : a. 10,000. c. 75,000. e. 98,000. g. 48,007. h. 40,000. d. 17,000. /. 36,549. h. 93,080. Turn to page 77, and read the numbers written in column C. 172. Write in figures the numbers made up of ten-thousands from ten thousand to ninety thousand. Write in figures the following : 1. Thirty-four thousand, two hundred seventy-five. 2. Eighty-nine thousand, thirty-six. 3. Fifty-four thousand, five hundred two. 4. Fifteen thousand, twelve. Let the teacher dictate other numhers for the pupil to write. 173. Count by hundred-thousands from one hundred thou- sand to ten hundred thousand. One hundred thousand is written 100000 or 100,000. Two hundred thousand is written 200000 or 200,000. And so on. How many figures are needed to write hundred-thousands ? Which figure shows how many hundred-thousands there are ? SIMPLE NUMBERS. 73 174. Eead the following : i. 400,000. k. 527,810. m. 324,517. o. 407,086. j\ 438,261. 1. 906,224. n. 872,239. p. 370,805. Turn to page 77, and read the numbers written in column B, 175. Write in figures : 5. Five hundred sixty-eight thousand, four hundred. 6. Four hundred twenty thousand, twenty-six. 7. Nine hundred nine thousand, nine. 8. Eight hundred forty-one thousand, live hundred three. Let the teacher dictate other numbers for the pupil to write. 176. The figures in the fourth, fifth, and sixth places taken together form a group called the thousands' group, while the figures in the first, second, and third places form a group called the units' group. The figures in the units' group express units, tens, and hundreds of units, while the figures in the thousands' group express units, tens, and hundreds of thousands. In writing numbers, separate the groups of figures hy a comma, as shown in Articles 173 and 174. 177. Beyond the second group are higher groups. A third group of figures, expressing units, tens, and hundreds of millions, is shown in the following TABLE. Sfl53 Sflo SS-^ ^ ^ ^ ^ ^ .d ^ ^ -^ HJ-4J-t-5 ^J^J.^J rS'^OT OOOt^ OOrt< COCNrH 7 2 5, , 3 2 8 , 7 8 3^ Figures. 3d group. 2d group. ist group. Millions. Thousands. Units. How many figures are needed to write millions? ten-mil- lions ? hundred-millions ? 74 SIMPLE NUMBERS. Which figure shows how many millions there are ? which shows how many ten-millions there are ? which shows how many hundred-millions there are ? The number written in the table is read " Seven hundred twenty-five million, three hundred twenty- eight thousand, seven hundred eighty-three." 178. Kepeat the following table : Ten ones (or units) make a ten. Ten tens make a hundred. Ten hundreds make a thousand. Ten thousands make a ten-thousand. Ten ten-thousands make a hundred-thousand. Ten hundred-thousands make a million. 179. Eead the following : q. 2,684,500. t. 8,109,019. w, 1,326,709. r. 9,275,405. u. 4,414,8.93. x, 5,070,890. s. 3,118,184. V. 6,005,928. y. 7,654,321. 180. Turn to page 76, and write in figures the numbers given in column A. Let the teacher dictate other numbers to millions for the pupil to write. 181. Miscellaneous Examples for the Slate. 9. How much must be paid for 625 acres of land, at $ 175 an acre ? 10. It is about 25,000 miles round the earth. How many days would it take a person to travel this distance if he trav- elled 40 miles a day ? 11. Mr. Gaines owned real estate valued at $56,000. Of this he sold land worth $5,000 and a house worth $6,750. What was the value of what he had left ? 12. At 7 cents a pound, how many cents will a dealer receive MISCELLANEOUS EXAMPLES. 75 for 50 boxes of soap each containing 72 pounds ? How many dollars will he receive ? 13. How many square miles are there in the United States, the Northern Lake region containing 484,339 square miles; the Atlantic Slope, 304,530 square miles; the Gulf region, 1,683,303 square miles ; and the Pacific Slope, 854,314 square miles ? 14. In 1865 Massachusetts had 1,267,031 inhabitants, and in 1875 she had 1,651,912 inhabitants. What was the increase in ten years ? 15. How many seconds are there in the month of January ? 16. When Charles is 11 years old, how many hours has he iived, if in that time there have been 3 leap-years ? 17. The great bell at Moscow weighs 448,000 pounds. How many tons does it weigh ? 18. In one part of her orbit the moon is 224,000 miles from us. If a cannon-ball moves at the rate of 16 miles a minute, in how many minutes will it move through this distance ? 19. In 2,468 cocks of hay, averaging 78 pounds each, how many pounds ? How many tons ? 20. A man bought 42 bushels of potatoes for $7.50, and sold them for 80 cents a peck. How much did he receive for them ? • How much did he gain ? 21. At 4 cents apiece, what will it cost for slats to fence a lot of land having 4 sides each 250 feet long, 5 slats being required for each foot of length ? 22. What will be the cost of ties at 35 cents each, to build a mile of railroad, there being 6 ties to a rod and 320 rods in a mile ? 23. A farmer bought 5,960 feet of boards at % 0.03 a foot, and gave in payment cheese at $ 0.12 a pound. How many pounds did it take ? 24. How many are 3,687,543 + 245,871 + 3,684 + 932,185 ? 25. From 4,357,859 + 248,946 take 2,783,947. 26. Multiply 7,285 by 394 and by 207. 27. Divide 4,893,683 by 14 and by 25. 76 DRILL TABLE. 182. DRILIi TABLE No. 3. Simple Numbers. A Five million^ eighty-one thousand, two hundred six. One million, five hundred seven thousand, forty-one. Eight million, three hundred twenty-one thousand, forty. Five million, twenty-nine thousand, three hundred. Four million, seven hundred thousand, four hundred four. Three million, forty thousand, two hundred thirty-four. Eight million, six hundred ten thousand, ninety-one. Nine million, fLYe> hundred thirty thousand, seven. Six million, one hundred sixteen thousand, six hundred. Two million, seventy-nine thousand, one hundred eighty. Five million, eighty-four thousand, four hundred nine. Three million, nine hundred thousand, one hundred two. Eight million, seven hundred six thousand, fifty-nine. One million, forty-eight thousand, two hundred eight. Nine million, eight thousand, three hundred twenty. Four million, twenty-three thousand, five. Five million, seven hundred sixty-eight. Eight million, seventy-two thousand, eighty-nine. Three million, five hundred seven thousand. Two million, three hundred forty-eight thousand, eight. Nine million, twelve thousand, six hundred ninety. Seven million, seven thousand, seven hundred. Six million, fifty thousand, two hundred ninety-one. Four million, five hundred eighty-eight. Seven million, seven hundred thousand, four hundred six. DRILL EXEEGISES, 77 DRILL TABLE No. 3 {continued). B 987 449 905 788 679 435 369 663 153 967 ^^% 455 806 399 917 517 927 931 953 205 667 871 728 431 845 219 144 SQ>S 225 189 199 598 941 529 795 721 734 256 323 583 824 174 769 416 108 824 872 892 444 764 c_ 2 5~3i7 4 5 995 3 7 872 8 8 324 7 5 436 5 7 216 9 4 503 6 6 842 19 471 6 7 174 2 6 862 9 6 354 2 9 534 16 111 5 4 197 4 5 490 8 876 3 4 902 7 6 396 5 6 484 4 9 235 7 4 387 6 2 897 3 8 624 8 7 476 183. Exercises upon the Tables Addition. 79. Write A in figures. 80. Add A, B, and C. 81. Add in A from 1 to 5; 2 to 6, etc. 82. Add in B from 1 to 8; 2 to 9, etc. 8S. Add in C from 1 to 11 ; 2 to 12, etc. Subtraction. 84. From A take B. 85. From B take C. 86. From A take C. 87. From 1,001,001 take B. 88. From 9,900,600 take A. Multiplication. Multiply B 89. By 2. 92. By 5. 95. By 8. 90. By 3. 93. By 6. 96. By 9. 91. By 4. 94. By 7. 97. By 10. Multiply D 98. By 12. 101. By 67. IO4. By 168. 99. By 34. 102. By 78. 105. By 312. 100. By 57. 103. By 87. 106. By 970. Division. Divide A 107. By 2. 110. By 5. 113. By 8. 108. By 3. 111. By 6. II4. By 9. 109. By 4. 112. By 7. ii<5. By 11 Divide B 116. By 17. 119. By 904. i27. By 45. i^(?. By 686. 118. By 198. ' i^i. ByC. 122. Divide A by B. 2;^^. Divide A by C. "^S FACTORS. SEOTIOIsr YII. FACTORS. 184. What numbers multiplied together will make 6 ? Ans. 2 and 3, also 1 and 6 ; thus, 2x3 = 6, and 1x6 = 6. A number that may be used as multiplicand or multi- plier to make another number is a factor of that number. Thus, 2 and 3 are factors of 6 ; so also are 1 and 6. a. Name two factors of 4 ; of 9 ; of 12 : of 15. 185. iSTame some factors of 8 besides 8 and 1. A number that has other factors besides itself and 1 is a composite number. Thus, 8 is a composite number. 186. Name the factors of 5. A number that has no other factors besides itself and 1 is a prime number. Thus, 5 is a prime number. 187. Exercises. b. Which of the following numbers are prime and which are composite: 6? 7? 8? 10? 11? 12? 13? 14? c. Write the composite numbers from 1 to 30 ; from 30 to 50. d. Write the prime numbers from 1 to 30 ; from 30 to 50. 188. e. Name any factors of 12 that are prime numbers. Name any that are not prime. A factor that is a prime number is a prime factor. Thus, 2 and 3 are prime factors of 12. 189. A composite number equals the product of all its prime factors. Thus, 12 = 2x2x3. FACTORS. 79 190. Exercises. Separate the following numbers into their prime factors and write the results, thus : 6 = 2x3; 8 = 2x2x2, and so on. /. 6. h. 10. j. 18. 1, 2S. 22. 40. g. 8. i. 16. k. 20. 222. 30. ^ o. 45. 191. p. Write all the prime factors of 18 and of 24, and name the numbers that are factors of both. . ( The prime factors of 18 are 2, 3, and 3« ( The prime factors of 24 are 2, 2, 2, and 3, The factors of both are 2, 3, and their ]3roduct 6. 192. A number that is a factor of two or more numbers is a common factor of those numbers. Thus, 2, 3, and 6 are common factors of 18 and 24. q. Which of the numbers 2, 3, and 6 is the greatest com- mon factor of 18 and 24 ? 193. Exercises. r. Name a common factor of 8 and 12 ; of 10 and 15. s, Name all the common factors of 18 and 12. t. ISfame the greatest comm.on factor oi 18 and 12. What is the greatest common factor H. Of 9 and 12 ? w. Of 8 and 12 ? j^. Of 6 and 18 ? V. Of 12 and 15 ? x. Of 18 and 20 ? z. Of 24 and 36 ? Multiples. 194. a. Name some numbers which are made by using 2 as a factor. Ans. 2, 4, 6, 8, etc. A number made by using another number as a factor is a multiple of the number so used. Thus, tlie numbers 2, 4, 6, and 8 are multiples of 2. 80 FACTORS. 195. b. Write the multiples of 3 and of 4 to 24. ^^^^ f Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24. '^^* t Multiples of 4 are 4, 8, 12, 16, 20, 24. c. Which of these numbers are multiples of both 3 and 4 ? A number that is a multiple of two or more numbers is a common multiple of those numbers. Thus, 12 is a common multiple of 3 and 4 ; so also is 24. d. Which of these numbers, 12 and 24, is the least common multiple of 3 and 4 ? 196. Exercises. e. Write all the multiples of 2 and of 3 to 30. /. Which of the numbers 3'ou have written are common multiples of 2 and 3 ? g. Which is the least common multiple of 2 and 3 ? h. Write all the multiples of 4 and of 5 to 40. i. Which of the numbers you have written are common multiples of 4 and 5 ? j. Kame the least common multiple of 4 and 5. Name any common multiple k. Of 3 and 4 ; of 3 and 9 ; of 2, 3, and 4. ^ i. Of 2 and 5 ; of 4 and 9 ; of 5, 10, and 15. m. Of 5 and 7 ; of 8 and 12 ; of 5, 8, and 10. Name the least common multiple 22. Of 3 and 5. s. Of 4, 6, and 8. 0. Of 6 and 7. t. Of 4, 8, and 10. p. Of 4 and 6. u. Of 4, 5, and 6. (?. Of 6 and 10. v. Of 3, 8, and 10. r. Of 9 and 12. w. Of 6, 9, and 12. For fuller treatment of Greatest Common Factor and Least Common Multiple, see Appendix, x^ages 139, 140. COMMON FRACTIONS. SECTION YIII. COMMON FRACTIONS. 197. Eichard cut a melon into two equal parts and gave his brother one of the parts. What part of the melon did each then have ? (Art. 142.) a. How many halves equal a whole thing ? 198. Three boys divided a pineapple equally among themselves. What part of the pineapple did each boy have ? Z). How many thirds equal a whole thing ? c. If any single : thing or unit is divided into four equal parts, what is each of the parts called ? What are two of the parts called ? three of the parts ? d. If a unit is divided into five equal parts, what is each of the parts called ? two of the parts ? three of the parts ? four of the parts ? e. If a unit is divided into six equal parts, what is each of the parts called ? two of the parts ? three of the parts ? /. In one how many fourths ? thirds ? fifths ? sixths ? 199. The equal parts of a unit, as halves, thirds, fifths, etc., are fractions. Numbers that are not fractions are .called whole numbers or integers. 82 COMMON FRACTIONS, g. To obtain two thirds, into how many equal parts is the unit divided ? How man}^ parts are taken ? 200. The number of equal parts into which a unit is divided is the denominator of the fraction. Thus, in the fraction two thirds, the denominator is three. 201. The number of equal parts taken is the numerator of the fraction. Thus, in the fraction two thirds, the numerator is two. 202. The numerator and denominator are called the terms of the fraction. h. What is the denominator of the fraction three fifths ? i. What is the numerator ? Writing Common Fractions. 203. Two thirds is written as in the margin, the Numerator 3 nuuicrator two abovc the line, and the de- Denominator..^ nominator th^e below. Exercises. 204. Write in figures the following : a. One half. d. Three fourths. b. One third. e. Four fifths ; five fifths. c. Two thirds. /. Seven twelfths ; seven fifths. g. Write any fraction you please having for a denominator six ; seven ; nine ; eleven. h. Write any fraction you please having for a numerator four ; six ; seven ; nine ; twelve. 205. Kead the following, and tell how many parts the unit is divided into, and how many parts are taken in each case : i. fapp^e. k. tday. m. $ A- o. f. q. H- J. f pear. 1. J quart. n. $ |. p. f . r. -j- J. COMMON FRACTIONS. 83 To change a Fraction to smaller or larger Terms. 206. Mary cut an apple into halves, and then cut one of the halves into two equal parts. What part of the whole apple was one of these smaller parts ? If Mary should put these two fourths to- gether, what pg.rt of the whole apple would they make? a. One half eqiials how many fourths ? b. Two fourths equal how many halves ? c. If a piece of paper one inch long be cut into thirds, and each of these thirds be cut into two equal parts, how many equal parts will the whole piece be cut into ? d. What part of an inch will one of these smaller parts be ? e. One third equals how many sixths ? /. Two sixths equal how many thirds ? 207. Compare the terms of the equal fractions \ and |. The terms of the fraction ^ are one half as large as the i i i terms of the fraction |. And the terms of the fraction | are twice as large as the terms of the fraction ^. So, 208. If both terms of a fraction he multiplied hy the same number, the value of the fraction will not be changed. And, If both terms of a fraction be divided by the same num- ber, the value of the fraction will not be changed. g. By what will you multiply the terms of the fraction J to change it to fourths ? to sixths ? to eighths ? S4 COMMON FRACTIONS, h. By what will you divide the terms of the fraction | to change it to J ? i. By what will you divide the terms of the following frac- tions to change them to halves : f ? j\? ^^ ? ^^ ? 209. Oral Exercises. First do the following examples, writing the answers upon the slate; then practise doing them mentally till you can name the results at sight. a. Change to smaller terms : I ; f ; f ; f; f ; f ; A ; ^ ; b. Change to smaller terms: ^^', ^%', ^^', -j:'^; ig; ^s^ ^ c. Change to smaller terms: if 5 M; II; A; U 5 M; •f*; i§; if. M; H; If; M- c?. Change ^ to fourths ; to sixths ; to tenths ; to twelfths. e. Change -^ to sixths ; to ninths ; to twelfths ; to fifteenths. /. Change f to sixths ; to ninths ; to twelfths ; to fifteenths. g. Change to twenty-fourths : i; J-; §; J; f; J; f; J; I ; t ; i ; tV ; tV ; t\* iz. Change to thirtieths : J; J; f; J; f; J; f; f; ^i^ ; To ; tV ; A" ; t^s ; A ; A* To change a Fraction to its smallest Terms. 210. To change a fraction to its smallest terms, Divide hj all the factors that are common to numerator and denomi- nator ; or divide hoth terms hy their greatest common factor. 211. Examples for the Slate. Change the following to equivalent fractions of smallest terms : a. Do the work mentally, and write the result as it is if ~ f • written in a. (1.) M- (4.) li- • (7.) if. (2.) if. (5.) n. (8.) U- (3.) ^. (6.) H. (9.) If. For other examples, see j)age 129. (10.) |§. (13.) n- (11.) fg. (14.) IS. (12.) U- (15.) T*/^, COMMON FRACTIONS. 85 To change Improper Fractions to Whole or to Mixed Numbers. 212. A fraction whose numerator equals or exceeds its denominator is called an improper fraction. Thus, I, j, and | are called improper fractions, a. 4 half dollars equal how many dollars? Z). J apples ,1^ ^^^ equal how many whole :^^^ apples ? (&.) Since 2 halves make 2) 7 a whole one, |- apples 3I will equal as many whole apples as there are 2's in 7, which is 3, and 1 half remains. ^^5. 3^ apples. 213. The number 3J consists of a whole number and a fraction. A number consist- ing of a whole number and a fraction is a mixed number. 214. Oral Exercises. c. Alvin picked berries into a dish that held J- of a quart j if he filled the dish 8 times, how many quarts did he pick ? d. How many quarts are there in f quarts ? in f ? in -y- ? e. How many dollars are there in $ -\0- ? in $ V" ^ in $ -^ ^ /. By what do you divide to change halves to units ? to change thirds ? fourths ? fifths ? sevenths ? tenths ? ^. Change to whole numbers : | ; | ; -V" j t j V" ? § 5 ¥-• iz. Change to mixed numbers : J ; | ; -V^- ; J ; J ; J^- ; ^- ; i. Change to whole or mixed numbers: |; -^; f ; -^ ; ^-] ¥; ¥; -V-; ¥; V-; ¥-; ¥-; ¥; -¥; ¥; ¥-5 M- 215. Examples for the Slate. Change the following fractions to whole or mixed numbers, writing the results as in a and b, below. a. V- = 16|. (16.)1F. (18.)iF. (20.)iF. (22.) W- Z>. ^ = 19J. (17.) ^|2. (19.) ^s.. (21.) if 1. (23.) W- 86 COMMON FRACTIONS. To change a "Whole or a Mixed Number to an Improper Fraction. 216. a. Two dollars equal how many half dollars ? ^^ h. How many half apples are there in 3 apples ? in 4 ? in 5 ? in 7 ? c. By what do you mul- tiply to change units to halves ? d. How many half apples are there in 3 J apples ? Since in 1 there are 2 halves, {d.) m 3 there are 3 times 2 halves, or 6 halves, which with ^i = 2- 1 half added are 7 halves. Ans. ^ apples. 217. Oral Exercises. e. Ada had 2 yards of ribbon, which she made into knots of 1 third of a yard each. How many knots did she make ? /. How many thirds are there in 2 ? in 3 ? in 5 ? in 7 ? g*. By what do you multiply to change a number to thirds ? h. How many thirds are there in 4§? in 5J ? in 6f ? i. If it takes 1 fourth of a 3^ard of cloth to make a cap, how many caps can be made from 5 j^ards ? from 5 J yards ? J. By what do you multiply to change a number to fourths ? fifths? sixths? eighths? ninths? tenths? k. Change 3 to fourths ; 4 to fifths ; 5 to eighths ; 8 to tenths. I. Change to improper fractions: IJ; 8J; 2J; 4J; 7 J ; 3J. m. Change to improper fractions : If ; 4J ; 8| ; 3f ; S^q ; 8|; 6|; 8i; 5^; 7J ; 12^; 6^ ; of; 2J; 4f; 3f; Sf; 7|. 218. Examples for the Slate. Change the following numbers to improper fractions, writing the result as in a, below. a. 16f=Aj4. (26.) 33J. (29.) 32f. (32.) 1211. (24.) 37^. (27.) 66$. (30.) 272i. (33.) 271 f. (25.) 17i. (28.) 301. (31.) 187f (34.) 368|. ADDITION OF FRACTIONS. 87 ADDITION OF FRACTIONS. To add Fractions having a Common Denominator. 219. a. In J of a day, | of a day, and f of a day, how many fourths ? How many days ? The fractions \, |, and | have the same denominator, 4. Fractions which liave the same denominator are said to have a common denominator. h. In adding \, |, and f, which terms did you add ? 220. Oral Examples. c. Charles picked ^ of a peck of berries, William picked f of a peck, and Alfred picked ^ of a peck. How many eighths did all pick ? How many pecks ? d. How many sevenths are f , f , and \ ? e. How many sixths are J, f , and f ? /. How many elevenths are ^j, -j^, and y\ ? g. Add t\, /_ and \\, i. Add ^, ^^r, and /^. iz. Add ^0? hh <^nd ^J. j. Add ^^, ^%, and ^. To add Fractions not having a Common Denominator. 221. a. James worked | of a day for Mr. Smith and J- of a day for Mr. Clark. How many fourths of a day did he work for both ? How many days ? b. How many eighths in ^ plus J ? in J plus | ? c. In I plus f how many ninths ? How many units ? cf. In f plus y^^ how many tenths ? How many units ? 222. e. Add J, |, and #. WRITTEN WORK. To add these fractions, they must be changed i = ^ to equivalent fractions having a common de- f - "ft" nominator. I = ^^ The most convenient denominator is the TT ^ - - least common multiple of all the denomi- ^ ^ ^ -* nators. 88 COMMON FRACTIONS. The least common multiple of 3 and 4 is 3 times 4, or 12, which is also a multiple of 6 * To change \ to twelfths, the denominator 3 must be multiplied by 4. Hence the numerator must be multiplied by .4. Thus, \ is found to equal ■^. In a similar way | is found to equal ^%, and | to equal l^. Adding these fractions, we have ||, or 1^-^, for the sum. Ans. l\^. 223. Oral Examples. /. Add i and \ ; ^ and J ; J- and J ; \ and \ ; J and ^^ ; J and ^ ; ^ and ^J^j ; \ and ^ ; | and ■^. g. Add 4^ and ^ ; J and i ; J and i ; ^ and J ; J- and ^. A. Add ^ and f ; J and f ; f and f ; f and J ; f and f . i. Add J, 1, and J ; i J, and i ; ^, \, and -jio ; J. h and T^; i? i, and ■^^. j. Add I, J, and t ; if, and J ; |, f , and t\ ; |_^ ..|^ and T^; f , f , and^y^. k. Miss Prencli cut from a piece of ribbon § and | of a yard, and then had | of a yard left. How long was the piece ? 224. Examples for the Slate. In doing the following examples, write the work as in e. Art. 222. 35. Add f , /o, and ^%. 39. Add ig, J, and yV 36. Acjd ^^, T^, and ^^, 40. Add -^, 2%, and f . 37. Add tV, f,t and f . 41. Add ^^2, M. and ^V 38. Add ^\, f, and \. 42. Add yV, %, and /^. In doing the following examples, add the wdiole numbers and the fractions separately. 43. Add 5i, 4f, and 7^. 44. Add 4|, 9§, and 18f . * If in any example the least common multiple is not readily seen, the pupil may take for a common denominator any common multiple of the denominators. The product of all the denominators is frequently the most convenient. t What should first be done to this fraction ? SUBTRACTION OF FRACTIONS. 89 SUBTRACTION OF FRACTIONS. To subtract Fractions having a Common Denominator. 225. Oral Examples. a. Jolin owned J of a football, and sold f to Burt. JEIow many eighths did he own then ? Z). I - § - what ? d. From -^^ take ^^, c. f - 1 - what ? e. From \^ take yV How do you subtract when the minuend and subtrahend have a common denominator ? /. Charles had a melon, and gave § of it away. What part had he left ? 5-. l-§? 1-1? 1-A? 1-A? 1-*? 2-J? h. 4-2- f? 4-2f? 12-3t7^? 20 -6§? 100 -12^? To subtract Fractions not having a Common Denominator. 226. Oral Examples. J. Silas could have the use of a boat for half an hour. After he had used it a quarter of an hour^ how much longer could he use it ? j. If 1^ of a yard of muslin be cut from | of a yard, how many eighths will be left? k. J less y'^j are how many tenths ? How many fifths ? 1. ^ less ^ are how many ninths ? 227. m. From f take -|. WRITTEN rm 1 T . o t ^ P ' WORK. That the subtraction may be performed, these fractionfi 4 = 7^2" must be changed to equivalent fractions having a com- ■| = -^^ mon denominator. The least common denominator is 12. «• \-V i-i? i-J? l-i? i-i? o- 5-i? f-f? l-f? f-l? «-F 90 COMMON FRACTIONS. 228. Examples for the Slate. In the following examples, write the work as in 122, Art. 227. (45.) t- A- what? (49.) 17 -2J- what? (46.) t-f = what? (50.) 1/^-1 = what? (^^') A-TV = what? (51.) Ij-J = what? (48.) ^T-f-what? (52.) 121 -4f = what? 229. Examples in Addition and Subtraction. 53. What are the contents of three remnants of carpeting containing as follows : 24^ yards, S^ yards, and 18 J yards ? 54. Clifford had 2J miles to row. After rowing If miles, how much farther had he to row ? 55. A man sold 4J bushels of cranberries at one time, 3| bushels at another, and 1^2 ^^ another. How many bushels did he sell in all ? b%. If it took 12f hours to plough a field, and 60j hours to plant it, how many more hours did it take to plant than to plough it ? 57. A farm of 23 acres contained a peach-orchard of 4J acres, a wheat-field of 5^ acres, a corn-field of 8^ acres ; | of an acre was planted with peas, Ij acres with strawberries, and the rest was a garden. How much was a garden ? ^S. How many more acres of the above farm were there in the corn-field than in the wheat-field ? 59. In making some boxes a carpenter used 4J pounds of board-nails, 1^ pounds of shingle-nails, and 6f pounds of lath- nails. How many pounds did he use in all ? 60. From a chest of tea containing 42f pounds, 17^^^ pounds were sold. How many pounds were left ? 61. A boy once paid his fare of $ 2^^^ from Springfield to Chatham, but, falling asleep, was carried on to Albany, where he paid % Ij for his lodging and breakfast, and $ J g- ^^ return to Chatham. What was the entire expense of his trip ? For other examples in Addition and Subtraction, see page 129. MULTIPLICATION OF FRACTIONS, 91 MULTIPLICATION. To multiply a Fraction by an Integer. 230. If James earns f of a dollar in 1 day, how much will he earn in 2 days ? Solution. — If he earns | of a dollar in 1 day, in 2 days he will earn 2 times f of a dollar, which equals |, or \^ dollars. Ans. l\ dollars. In multiplying f hy 2, which term of the fraction is multi- plied? 231. Oral Examples. a. If it takes Dana J of an hour to walk a mile^ how long will it take him to walk 2 miles ? 3 miles ? 6 miles ? 8 miles ? b. How many dollars will it take to give ^ of a dollar to each of 4 boys ? 5 boys ? 8 boys ? 9 boys ? c. How many are 4 times \? 1 times J ? 7 times \ ? 7 times f ? 7 times | ? 5 times f ? 4 times f ? cZ. How many are 6 times | ? 8 times f ? 9 times If ? Note. To multiply 1|, multiply the integer and the fraction separately. e. Multiply 4f by 6; 7f by 8 ; 8fby9; 9| by 8. 232. Examples for the Slate. /. Multiply f by 25. WRITTEN WORK In doing the following examples, write the 3 X 25 75 o r > = — = 15. work as it is written in Example /. If there are common factors in the numbers written „ ^„ above and below the line, strike them out as is or 3 X 2^ ^ t; ' = lo. done m the second form of the written work. 233. Striking out equal factors in the dividend and divisor is cancelling. 62. Multiply f by 16. 64. Multiply ^7_ by 15. 63. Multiply I by 63. 65. Multiply {}^ by 14. 92 COMMON FRACTIONS. 234. g. Multiply 16J by 11. WRITTEN "ifii Multiplying -^ by 11, we have J^, which equals 5-|. 2 Multiplying 16 by 11, we have 176, which, with 5^, gives 181^. Ans, ]8U. 1I6_ 66. Multiply 5J by 7. 68. Multiply 161 by 10. 181J 67. Multiply 12§ by 12. 69. Multiply Q>^ by 25. 70. Simon can walk a mile in 14 J minutes. How long will it take him to walk 3 miles ? 71. If a lot of cloth can be woven in 12f\ days on 4 looms, in how many days can it be woven on 1 loom ? 72. Eight persons bought a tent together, each paying as his share $ 15f . What was the price of the tent ? To multiply an Integer by a Fraction. 235. What is J of 2 yards ? fH — I — I — I — \ — I — I — I If -g- of each of 2 yards be taken, w^e shall Teighth. have f of a yard. Ans. | of a yard. I— fH — h- H — H — I (See illustration ; also page 58, Example o.) 1 eighth. iof2 = f. 236. Oral Examples. a. Whatisiof2? ^ofS? ^of3? i^oi%%? ^ of 8 days ? b. If 1 yard of tape is worth 3 cents, what is f of a yard worth ? Solution. — If 1 yard is worth 3 cents, f of a yard is worth f of 3 cents. ■| of 3 is f , and -| of 3 is 2 times |, which is |, or \\. Ans. l\ cents. c. What is ^ of 4 ? | of 4 ? f of 14 ? f of 17 ? f of 19 ? d. What is i of 7 ? | of 7 ? J^ of 13 ? f of 13 ? | of 17 ? 237. Finding the fractional part of a number is called multiplying by a fraction. Thus, in finding f of 17 we are said to multiply 17 by f . MULTIPLICATION OF FRACTIONS. 93 238. Examples for the Slate. e. Multiply 17 by J. /. Multiply 24 by /^. WRITTEN WORK. To multiply 17 by I is to take | of 17. 17 X 3 _ 51 _ ^ /x , To find -| of 17 we may either divide ^ '^ 5 5 ~ ^* ^^^' by 5 and then multiply the result by 3, 4 or we may first multiply by 3 and then .^^x_5_20^g2 divide the result by 5. The latter method J^ 3 is generally easier, as in Examples e ^ and /. 73. What is I of 82? 76. Multiply 18 by t^^. 74. What is f of 324 ? 77. Multiply 30 by ^. 75. What is T^ of 581 ? 78. Multiply 35 by A. 79. If a barrel of flour costs $ 8.75, what will f of a barrel cost? 80. If a wheel turns 480 times in going a mile, how many times will it turn in going f of a mile ? 81. If a clerk can copy 50 pages in a day, how many pages can he copy in 2^ days ? in 3 J days ? 82. What cost 14^ yards of cambric at 42 cents a yard ? 83. What cost 16| yards of silk at $ 2.75 a yard ? To multiply a Fraction by a Fraction. 239. What is J of i ? \ \ \ To find \ of ^, the -J- must be divided into , , I I . 2 equal parts. . Since the entire unit will contain 3 times ^ 1 of 4 = 1. 2 or 6 such parts, one of the parts will be \ of the unit. Ans. \. In finding J of J, how was the new denominator obtained ? 240. Oral Examples. a. What is i of i ? i of i ? i of J ? i of ^ ? ^ of i ? 5. What is i of J ? i of J ? J of J ? i of ^ ? i of J ? 94 COMMON FRACTIONS. 241. c. What is f oft? WRITTEN WORK. J of | is ^, (-^) ; then ^ of I must he ^ X f = A- A' (5^)' ^^^^ I o* f ^^^st ^^ 2 times as much, In finding § of f , how was the new numerator ohtained ? the new denominator ? To multiply a fraction by a fraction, Find the product of the nimierators for the new mLmerator, and the product of the denominators for the new denominator. d. Find § of i; ioff; ^off; fofj-; f of |. e, Find i oft; f of§; f of f ; |of§; f of |. /. If a pound of butter costs $ f^, what part of a dollar will ^ of a pound cost ? What will J of a pound cost ? g. A paper-hanger had | of a barrel of flour, and used f of it for paste. What part of the barrel did he use ? 242. Examples for the Slate. 84. What is | of ^% ? 86. What is | of ^\ ? S5. What is | of /_ ? 87. What is | of ^ ? In the following examples change all mixed numbers to im- proper fractions before multiplying. 88. Find f of f. 90. Find f of 18^. (92.) 3} x 5J ? 89. Find f of 14f . 91. Find ^V of 3J. (93.) 6^ x 4§ ? 94. What cost 3^ barrels of peas at $ 6| a barrel ? 95. Dora is 15f years old, and Oscar is f as old as Dora. How old is Oscar ? 243. Eepeat the following table : liong: Measure. 12 inches = 1 foot. 5i yards or I64 feet = l rod. 3 feet =1 yard. 320 rods or 5280 feet = l mile. 96. How many inches are there in 16^ feet ? 97. How many feet are there in 33J yards ? 98. Change 160 rods to yards. 99. Change 3J rods to feet. For other examples in Multiplication of Fractions, see page 129. DIVISION OF FRACTIONS. 95 DIVISION. To divide a Fraction by an Integer. 244. If J of a melon be di- vided equally between 2 boys, what part of the melon will each boy have ? If 1^ of a melon be divided equally between 2 boys, each boy will have 1 half of -f, or f of a melon. 1 half of I = |. Ans. f of a melon. In dividing J by 2, what was done to the numerator ? 245. If f of a melon be divided equally between two boys, what part of the melon wiK each boy have ? As the number of parts, 3, can- not be divided by 2 without a re- mainder, each one of the parts, fifths, may be divided into two equal parts ; these parts will be tV tenths of the whole melon. There llialfof|=yV will then be 6 tenths, of which each boy's part will be 3 tenths. In dividing f by 2, what was done to the denominator ? 246. To divide any fraction by an integer, Divide the imnierator or multi;ply the denominator hy the integer. 247. Oral Examples. a. Divide|by2; f by 2 ; f by 3 ; |by4; \^hj ^, b. Divide§by3; f by 2 ; f by3; ^by4; t^^ by 5. c. If 2 girls divide f of a yard of ribbon equally between themselves, what part of a yard will each have ? d. If 9 shrubs cost $ 2^, what is the cost of each ? 1 ^ First change the mixed number to an \ ') t-f- 4x^""4* improper fraction. 96 COMMON FRACTIONS. e. If $ 5 was paid for 2^ feet of wood, what was the cost of one foot ? /. In a boarding-house 7 breakfast tickets were sold for $ 1 J, 7 dinner tickets for $ 1|, and 7 supper tickets for $ 1 J. What was the cost apiece of each kind ? g. A family put up 8^- bushels of potatoes for the 3 winter months. What was that a month ? /p. \ 3\ 31 i of 8 is 2, with a remainder of 2. -- 2 units equal f , which, with ^, are J. ^ of J is J. ^ ^Tis. 2|- bushels. iz. If the railroad fares of 5 persons are $ 31 J, what is the fare for each person ? i. What must be the width of matting of which it takes 5 breadths to reach 8| yards ? j. What must be the width of paper of wliich it requires 10 breadths to reach 24 J feet ? k. Divide If by 6; 2f by 4 ; 4i by 3 ; 8| by 11. L Divide ISJ by 4 ; by 5 ; by 6 ; by 7 ; by 8 ; by 9. 248. Examples for the Slate, 100. Mrs. Grant gave $ 10 J for 27 Christmas wreaths. How much did she give apiece ? 101. A gentleman paid $ 67 J^ for board for 120 days. What was the price per day ? 102. If 1 needle-woman can do a piece of work in 11^ days, in what time can 12 do it ? In what time can 10 do it ? 103. If a certain field can be reaped in 15f days, in what time could a field that is -^^ as large be reaped ? - 104. If 119J yards of cambric are required to make 9 dresses, how much is required for 1 dress ? 105. If a railroad car runs 122/ij miles in 5 hours, what is the average rate an hour ? 106. Divide ^Vt by 25. (109.) 363.-11. 107. Divide II by 26. (110.) 16f-f-20. 108. Divide tl by 110. (111.) 2724-18. DIVISION OF FRACTIONS. 97 To divide an Integer or a Fraction by a Fraction. 249. Oral Examples. a. In one there are how many halves ? thirds ? fourths ? b. In two there are how many halves ? thirds ? fourths ? c. Julian had 5 cents, which he spent for pears at J cent apiece. How many pears did he buy ? d. How many pears could he buy at J of a cent apiece ? at J of a cent apiece ? e. How many mugs at $ ^ each can you buy for $ 3 ? for $ 5 ? 250. /. Helen made 4 knots of worsted into tassels, putting § of a knot into each tassel. How many tassels did she make ? She must have made as many tassels as WRITTEN WORK. ^, ^. o - a there are times f m 4. (/.) 4 == \^- We first change 4 to thirds, making J^. 1 2. ^ 2 _ ][2 -f- 2 = 6. There are as many times f in ^ as there are 2's in 12, or 6. ^ns. 6 tassels. Note. For a different analysis of this example, see Appendix, page 141. g. Miss Breck made 6 yards of crash into towels, putting J of a yard in each towel. How many did she make ? h. Divide 6 by f; by f ; by ,^ ; by | • by #. i. Divide 8 by ^ ; by f ; by 1^ or | ; by 2^ or J. J. How many hats at $ 1 J each can be bought for $ 12 ? How do you change an integer to divide it by a fraction ? How do you then divide ? How do you divide by a mixed number ? 251. k. Divide f by f . o. 2f are how many times f ? i. Divide | by f . i?. 4f are how many times f ? 122. Divide f by f . g. 3| are how many times 1\? n. Divide 3 J by g. -^- ^i a-re how many times 2 J ? s*. In a farm-school 5\ acres of land were given to some boys to plant, each boy having \ of an acre. To how many boys was the land given ? When fractions have a common denominator, how do you divide ? 98 COMMON FRACTIONS, 252, t. How ma,ny feet of rope can be made from 1^ pounds of hemp, there being J of a pound to a foot ? WRITTEN wo UK. As many feet can be made as there (t.) 13 = I = fi ; i - tV. are times J- in If or |. 2 1 _^ 2 = 21 - 2 = 101- i ^^^ 6 changed to fractions liaving ^ ^* a common denominator are 2^ and ^2^. f^ divided by ^ equals 21 divided by 2, or 10^. Ans, 10^ feet. When fractions have different denominators, how do you prepare them to divide ? How do you then divide ? 253. To divide an integer or a fraction by a fraction, Change the dividend and divisor to fractions having a common denominator, and then divide the numerator of the dividend hy the numerator of the divisor. (See Appendix page 141.) 12. How many times is f contained in f ? f in § ? | in f ? V, How many times is -^^ contained in f ? f in J ? J in 1 J ? w. I have a dish which holds § of a quart. How many times must it be filled to measure 2^ quarts ? X. How many house-lots each containing \ of an acre can be made from a lot containing 2 J acres ? 254. Examples for the Slate. 112. What is 36-^^? 117. Divide ^ by t^. 113. What is 48 -IJ? 118. Divide f by IJ. 114 What is 52 -3i? 119. Divide 33 i by 4 J. 115. What is 100-61? 120. Divide If by 7f. 116. What is 4^ -1|? 121. Divide 8 J by 4J-. 122. How many lengths of fencing f of a rod long will it take to enclose a square lot of land, each side 40 rods long ? 123. The rations of a ship's company were Ij pounds of meat a day to each person, but after losing a part of their pro- visions in a storm, they were allowed only -f^ as much. What were their rations then ? For other examples in Division, see page 129. COMMON FRACTIONS. 99 To find the "Whole when a Part is given. OrcQ Eszercises. 255. a. If J pound of yarn costs 12 cents, what will a pound cost at the same rate ? b. In \ of a ream of paper there are 5 quires. How many- quires are there in a ream ? c. Mr. Fitch sold a boat and gained $ 8, which was \ of what it cost him. How much did it cost him ? d. 8 is \ of what number ? e. A man sold a wagon and lost $ 11, which was ^ of what it cost him. What did it cost him ? /. If ^ of a yard of silk cost $ 1^, what will 1 yard cost ? g. From a cask of oil 15 gallons leaked out. If this was -fV of the contents of the cask, what did the cask contain at first? h. Frank has a sister whose age equals f of his own age. If the difference between their ages is 2 years, what is Frank's age ? Suggestion. Frank's age is -| of itself. The difference between -f and f , which is \, is 2 years» i. After losing J of his money, a man had $ 12 left. How much money had he at first ? j. A man traded off a cow and got | of what she cost him. If he gained $ 5, what did she cost him ? k. The difference between ^ and ^ of a bushel is 8 quarts. How many quarts are there in a whole bushel ? L The difference between f and y% of a number is 5. What is that number ? m. The difference between J and ^ of a number is Ij. What is that number ? n. $ 5 is .^ of how many dollars ? 0. 1 J is J of what number ? \ of what number ? p. 65 is J of what number ? J of what number ? 100 COMMON FRACTIONS. 256. a. If f of a piece of work can be performed in 4 days, in how many days can ^ of the work he performed ? In how many days can the whole work be performed ? b. If f of a number is 4, what is the whole number ? Solution. — If f of a number is 4, \ of the number is \ of 4, which is 2, and \ of the number is 5 times 2, or 10. Ans. 10. c. A man performed J of a journey in 15 days. In how many days could he perform \ of the journey ? In how many days could he perform the whole journey ? d. If f of a number is 15, what is the whole number ? e. If f of a quire of paper cost 10 cents, what did the whole quire cost ? /. f of a certain number is 10. What is that number ? g. If there are 21 lines in ^ of a page, how many lines are there in the whole page ? h, 21 is J of what number ? i. A grocer sold eggs at 20 /' a dozen, which was | of what they cost him. What did they cost him ? J. A man gained $ 15 by selling candy for J as much money as it cost him. What did it cost ? k. 15 is I of what number ? i. Edgar sold a kite for f of what it cost him, and thereby lost 8 cents. What did the kite cost him ? 222. 8 is f of what number ? 22. By selling beef at 6J ^ a pound, a market-man gets but g of what it cost him. How much did it cost him ? o. 6J is f of what number ? p. After I of a certain month is past there are 24 days re- maining. How many days are there in the month ? q. A farmer finds after selling f of his pears that he has 8 J barrrels left. How many barrels had he at first ? r. 8J is f of what number ? iz. J is f of what ? s. 1-^y is f of what number ? t^. ^ is f of what ? t. 4J is j% of what number ? w. ^ is f of what ? MISCELLANEOUS EXAMPLES. 101 257. Miscellaneous Oral Examples. a. At $ J a day for board, how many daj^s' board may be obtained for $ 7 J ? b. If } of a cord of wood costs $ 6, what will 1 cord cost ? c. If 5 men can do a piece of work in f of a week, in what time can 1 man do it ? d. A man bought a quantity of drugs for $53, and was obliged to sell them for -f^ of what they cost him. How much did he receive for them ? e. A miller paid 90 cents a bushel for wheat, and sold it so as to lose § as much as it cost. What did he receive for it ? /. If f of a bale of cotton cost % 30, what will 1 bale cost ? How many yards of cloth at $ 5 a yard will pay for a bale of cotton ? g. If 9 bottles of ink cost $ 2\, what will 5 bottles cost ? h. If 6 pounds of lead cost $ 1 J, what will 7 pounds cost ? i. How many packages of f of a pound each can be made from 9 pounds of starch ? j. If ^ of a barrel of oil cost $ 36, what cost J of a barrel ? k. If $ I will buy 6 gallons of oil, how much will $ | buy ? 1. If 20 cents are paid for f of a basket of peaches, what is the cost of f of the basket ? 122. If § of a box of raisins is worth $ 4, how many clocks at $ 2 apiece will pay for a whole box ? 22. Mr. Wing sold 6 pigs for $ 24, which was f of what he paid for them. What did he pay apiece for them ? 0. If a man by travelling 8 hours a day can perform a jour- ney in If days, in how many days can he perform it by travel- ling 10 hours a day ? p. I sold a horse for $ 48, which was f of what he cost me. When I bought the horse I paid for him with hay at $ 10 a ton. How many tons did it take ? q. If a man can walk at the rate of | of a mile in 7J min- utes, in What time can he walk 1 mile ? 5 miles ? 102 MISCELLANEOUS EXAMPLES. 258. Miscellaneous Examples for the Slate. 124. Mr. Sage lias 4 pieces of velvet ; the first contains 7| yards, the second 8^ yards, the third 5/(j yards, and the fourth 8]^ yards. How many yards are there in all ? 125. At $ 5 J a ream, what cost ^ of a ream of paper ? 126. Mr. Gray bought 18^ acres of land, and gave 1 half of it to his son and 1 lialf of the remainder to his nephew. How much did he give to his nephew ? 127. Emma's dress cost $11|, and Mary's g as much. What did Mary's cost ? 128. Mr. Brown had a huilding-lot that measured 12^ rods m front. What did it measure in yards ? in feet ? 129. How many rods of fencing on both sides of a road that is 1 J- miles long ? 130. What is the cost of fencing a road which is 2 of a mile in length, at $ 14i- a rod ? 131. Olive had 111'^. If she spent all but $5^^ of it for cambric at $ |^ a yard, how many yards did she buy ? 132. Winthrop received $ 18.10 for work, and after j)aying a debt of $ 4.80, had money enough to pay for 3 J weeks' board. What was the price of board per week ? 133. Mr. Channing bought land for $ 1 If an acre, and sold it for $ 19 J an acre. What did he gain on 1 acre ? on 3 acres ? 134. Mr. Howes sold 12 dozen cabbages at 8J/ apiece, and received in payment 46 pounds of beef at 9 >^ a pound, and the rest in money. What did he receive in money ? 135. How many dominos can be cut from a piece of ivory 7J inches long, allowing Ij inches for each domino ? 136. How many checkers can be made from a block 15 inches long, if f of an inch is allowed for each checker ? 137. A man owned 23f acres of meadow, 11\ acres of pas- ture, and 5J acres of woodland. How many acres did he own in all ? 138. What will 25^^ yards of carpeting cost at $1| per yard ? MISCELLANEOUS EXAMPLES. 103 139. At $4J per week, wliat is the cost of board for 3^ weeks ? 140. How man}'- rations of Ij pounds each can be furnished from a barrel of beef weigliing 200 pounds ? 141. Three boys started together to skate down a river. One skated a mile in 8 J- minutes, another in 10 J minutes, and the third in 9J- minutes. How much longer did it take the second than the first ? than the third ? 142. Tliese boys started together to return, and found that it took them 11 J minutes, 12 § minutes, and 13 J^ minutes re- spectively, to skate the mile. How many minutes had the first to rest before the second came up ? How many before the third came up ? 143. When 1 pound of cotton is put into 3| yards of cloth, how much is put into half a yard ? into 12^ yards ? 144. If 30 yards of cloth 1 yard wide weigh 12% pounds, what will 30 yards of the same kind of cloth weigh if it is J of a yard wide ? if it is | of a yard wide ? 145. After going J of his journey, Mr. Otis had 32| miles to go. What was the length of his journey ? 146. Homer's saddle cost $25, which was -^^ of what his horse cost. What did his horse cost ? 147. A man sold a watch for $ 18^, which was f of what he gave for it. What did he give for it ? 148. Mr. Bracket sold a coat and gained $ 1.32, which was \ of what it cost him. What did it cost him ? 149. If 5 yards of cloth cost $ 2.25, what will 7 J yards cost ? 150. If 14 cedar posts cost $8|, what will 10 cost ? 151. If ^ pounds of lead cost $ 1.70, what will 5J pounds cost? 152. A man sold a iirkin of butter for $ 16.80, and gained J of what it cost him. What did it cost him? 153. On counting his money, Mr. Gould found he had $ 36.72. If this was f as much as he had spent, how much had he spent ? How much had he at first ? 104 DECIMAL FRACTIONS. SEOTION^X, < < a DECIMAL FRACTIONS. 259. a. If a block be divided into ten equal parts, what is each of the parts called ? b. If one of these tenths be divided into ten equal parts, what will each of these parts be called ? What is^Joof 1^0? c. If one of these hundredths be divided into ten equal parts, what will each of these parts be called ? What is -^ of -^^-q ? d. How many tenths make a unit ? e. How many hundredths make a tenth ? /. How many thousandths make a hundredth ? g. What part of a tenth is a hundredth ? h. What part of a hundredth is a thousandth ? One tenth of a thousandth is a ten-thousandth; one tenth of a ten-thousandth is a hundred-thousandth ; one tenth of a hundred-thousandth is a millionth; and so on. 260. The fractions, tenths, hundredtJis, thousandths , etc., are decimal fractions, or decimals. To read and write Decimal Fractions. 261. Decimal fractions may be written like common fractions, with denominators. But they are usually written in the same Avay that integers are written. To distinguish DECIMAL FRACTIONS. 105 them from integers, a point (.), called a decimal point, is put at the left of the tenths' place. Thus we write One tenth 0.1 One hundredth 0.01 One thousandth 0.001 Three tenths 0.3 Two hundredths 0.02 Five thousandths 0.005 Three tenths and two hundredths, or thirty-two hundredths.... 0.32 Two hundredths and five thousandths, or twenty-five thousandths 0.025 Three tenths, two hun- dredths, and five thousandths, or three hundred twenty-five thousandths 0.325 One and three hundred twenty-five thou- sandths 1.325 It will be seen that in this method of writing decimals it is the numerator only that is written. How is the denominator indicated ? Ans. The denominator is indicated hy the place of the right- hand figure, counting from the decimal point. 262. The method of writing decimals is more fully shown by the following table, which is merely an extension of the table given on page 73. .1 $ .9 . I 4 I I 9 I In which place from the right of the decimal point are ten- thousandths written ? hundred-thousandths ? millionths ? 106 DECIMAL FRACTIONS. 263. Exercises. Eead the following : a. b. 0.4 0.04 6. d. 0.001 e. 41.5 0.9 0.36 0.5 0.668 0.06 4.6 8.88 3.21 3.704 60.209 3.2 7.59 6.13 20.032 0.006 7.7 3.40 25.07 6.006 13.013 0.0001 6.2724 h. 68.5 i. 1324. 0.0045 0.3662 0.007 132.4 0.0182 0.3079 2.904 13.24 0.3247 17.3761 0.8' 764 1.324 26.0026 42.6305 279.0017 0.1324 Turn to page 129, and read the numbers written in H and I. 264. Examples for the Slate. Write in figures the following numbers : 1. Eight tenths. (6.) 4 units and 5 hundredths. 2. Seven hundredths. (7.) 274 units and 8 tenths. 3. Twelve thousandths. (8.) 906 hundred-thousandths. 4. Thirty-four hundredths. (9.) 4 units and 7 ten-thousandths. 5. Eight ten-thousandths. (10.) 2148 hundred-thousandths. Let the teacher dictate other decimals for the pupils to write. REDUCTION OF DECIMALS. To change a Decimal to lo"wer Denominations. 265. a. Change 3 to tenths, b. 0.5 to hundredths; to thousandths. WRITTEN WORK. rpo Gxpress a decimal 3 = 3.0 read 30 tenths. fraction in any lower de- 0.5 = 0.50 read 50 hundredths. nomination, Annex zeros 0.5 = 0.500 read 500 thousandths. ^^ ^^^ ^.^^.^ .a:^r...^.^ until the place of the required denomination is reached. REDUCTION OF DECIMALS. 107 266. Examples for the Slate. 11. Change 7 to tenths ; to hundredths ; to thousandths. 12. Change 5 to thousandths ; 0.25 to ten-thousandths ; To change Decimals to Common Fractions. 267. c. Change 0.5 to a common fraction in its simplest form. wrjTTEN WORK. To changG a decimal to a common (c.) 0.5 = /w = o* f Inaction, First ivrite the fraction luith its denominator; then change the common fraction to its smallest terms. 268. Examples for the Slate. Change the following to common fractions in their simplest forms : (13.) 0.2; 0.4; 0.5; 0.6; 0.7; 0.05; 0.15; 0.25; 0.75. (14.) 0.125; 0.375; 0.625; 0.875. To change Common Fractions to Decimal Fractions. 269. d. Change | to a decimal fraction. WRITTEN WORK. The fraction | is the same as \ of 3, or | of (d.) 8 ) 3.000 3.000, which is found hy dividing 3.000 by 8 in 0.375 the usual w^ay. To change a common fraction to a decimal, Express the nnmerator in tenths, hundredths^ thousandthsy etc., hy an-, nexing as many zeros as may he reqitired, and then divide it hy the denomincctor. ^. 270. Examples for the Slate. Change (15.) ^ to tenths. (19.) J to hundredths. (23.) i to thousandths. (16.) i to tenths. (20.) f to hundredths. (24.) f to thousandths. (17.) i to tenths. (21.) ^7^ to hundredths. (25.) J to thousandths. (18.) I to tenths. (22.) ^ to hundredths. (26.) ^ to thousandths. For other examples, see page 129. 108 ADDITION OF DECIMALS. ADDITION OF DECIMALS. 271. B»^mples for the Slate. Write and add the following ; WRITTEN WORK. 5.7 13.8 9.14 6.27 a. 5 and 7 tenths, 13 and 8 tenths, 9 and 14 hundredths, 6 and 27 hundredths. Ans. 34.91 (27.) Add 3 and 7 tenths, 75 and 8 tenths, 6 and 43 hundredths, 16 and 9 hundredths. (28.) Add ^S and 72 hundredths, 3 and 9 tenths, 43 and 46 hundredths, 7 and 7 hundredths. 200 and 6 hundredths. (29.) Add 8 and 19 hundredths, b^ and 9 tenths, b^ hundredths, 24 and 8 hundredths, 9 tenths. (30.) Add 5 and 126 thousandths, 14 and 374 thousandths, 276 and 11 thousandths, 489 thousandths, 6 and 108 thousandths. In adding decimals, Add as in integers, fixing the decimal point in the answer as soon as the tenths are added. (31.) Add 1 and 926 thousandths, 754, 37 and 47 hundredths, 8 and 78 thousandths. (32.) Add 5 and 4763 ten-thousandths, 21 and 2131 ten-thousandths, 3245 ten-thousandths, 9 and 875 ten-thousandths, 47 and 26 thousandths. (33.) Add 14 and 3748 ten-thousandths, 4 and 93 hundredths, 7 and 872 thousandths, 5 and 8 tenths, 9875 ten-thousandths. (34.) Add 7 and 1 hundredth, 6^25 ten-thousandths, 875 thousandths, 46 and 97 ten-thousandths, 8 and 1008 ten-thousandths. SUBTRACTION OF DECIMALS. 109 35. Add twelve and three tenths ; seventy and thirty-five hundredths ; two thousand three hundred forty-seven ten- thousandths ; and seventy-eight thousandths. 36. Add forty-seven and three hundred seventy-eight thou- sandths ; twenty-six ten-thousandths ; four and nineteen hun- dredths; and sixty-six thousand six hundred and sixty-six hundred-thousandths. 37. Turn to page 106, and add the numbers written in Exercises a to i. For other examples in Addition, see page 129. SUBTRACTION OF DECIMALS. 272. Examples for the Slate. a. From 2 and 75 hundredths take 928 thousandths. WRITTEN WORK. In subtiacting decimals, Subtract as in 2.75 integers, fixing the decimal point in the Q'^^^ remainder as soon as the tenths are sub- Ans. 1.822 tr acted. 38. Prom 14 and 5 tenths take 75 hundredths. 39. From 7 and 43 hundredths take 647 thousandths. 40. From 675 thousandths take 497 thousandths. 41. Subtract 445 thousandths from 2 and 4 tenths. 42. Subtract 6^ thousandths from 6 and 6 tenths. 43. Subtract 9 ten-thousandths from 9 hundredths. Find the difference of the following : (44.) (45.) (46.) (47.) (48.) (49.) 4.075 75.09 10.066 0.202 27.9384 1.11 2.439 7.446 0.0066 0.0202 3.42853 0.9999 273. Miscellaneous Examples. 50. A dozen lemons cost % 0.38 ; four pounds of sugar, $ 0.36 ; and three pounds of crackers, $ 0.25. What did all cost ? 51. One eighth of a dollar is $ 0.125 ; five eighths is $ 0.625. What is the sum expressed decimally ? What is the difference ? 110 MULTIPLICATION OF DECIMALS. 52. If from a cask containing 15.4 gallons of ink 7.875 was drawn out, how much remained ? 53. If the ages o^JPbhn's grandfathers are 71.125 and 72.5 years, and of his grandmothers 69.33 and 70.25 years, what is the sum of all their ages ? 54. What is the difference between the ages of John's grand- fathers ? between the ages of his grandmothers ? For other examples in Subtraction, see page 129. MULTIPLICATION OF DECIMALS. To multiply a Decimal by an Integer. 274. a. Multiply 0.3 by 4 ; 0.03 by 4; 0.003 by 4. WRITTEN WORK. Three tenths multiplied by 4 is 12 0.3 0.03 0.003 tenths ; three hundredths multiplied by 4 4 4 4 is 12 hundredths ; three thousandths 1.2 0.12 0.012 multiplied by 4 is 12 thousandths. In multiplying a decimal by an integer, how many places for decimals do you point off in the product ? 275. Examples for the Slate. m. Multiply 0.4 by 9 ; 0.04 by 9 ; 0.004 by 9. m. Multiply 1.2 by 12 ; 0.12 by 12 ; 0.024 by 12. 57. Multiply 7.5 by 15 ; 6.25 by 5; 87.5 by 20. 58. What is the cost of 5 pounds of butter at $0.3 per pound ? at $ 0.4 ? at % 0.35 ? at S 0.40 ? 59. There are 16.5 feet in one rod. How many feet in a mile, or 320 rods ? To multiply an Integer by a Decimal. 276. b. What is tV of 7? yV^f^? yio of ^ ? ^l^oi25? c. Multiply 11 by 0.1 ; by 0.01 ; by 0.12. WRITTEN WORK. To multiply 11 by 0.1 is to take 1 tenth 11 11 11 of it, which we express by placing the 0.1 0.01 0.12 decimal point so that the figures 11 may T~l Q 11 1 32 express tenths, thus, 1.1. MULTIPLICATION OF DECIMALS. Ill To multiply 11 \>y 0.01 is to take 1 hundredtli of it, which we express by placing the decimal point so that the figures may express hundredths; thus, 0.11. To multiply 11 by 0.12 is to take 12 hundredths of it. One liundredth of 11 is 0.11, and 12 hundredths is 12 times 0.11, which equals 1.32. Aiu. 1.32. In multiplying an integer by a decimal, how many places for decimals do you point off in the product ? 277. Examples for the Slate. 60. Multiply 9 by 0.2; by 0.02 ; by 0.05. 61. Multiply 12 by 0.6; by 0.08; by 0.008. 62. Multiply 25 by 0.12 ; by 0.012; by 1.2. 63. At $8 a ton for coal, what cost 0.9 of a ton? 0.09 of ton? 5.4 tons? 7.23 tons ? To multiply a Decimal by a Decimal. 278. d. What is J^ of tV ^ tV of t\ ? ^js of A^ A of ^7^? e. What is jV of jgi) ? 0.1 of 0.05 ? 0.4 of 0.06 ? /. What is tJo of tV ^ 0.02 of 0.7 ? 0.06 of 0.9 ? g. Multiply 0.5 by 0.9. h. Multiply 0.5 by 0.09. WRITTEN WORK. {g.) To multiply 0.5 by 0,9 is to take 9 tenths {g.) (h.) of 5 tenths. One tenth of 5 tenths is 5 hun- 0.5 0.5 dredths, and 9 tenths is 9 times 5 hundredths, 0.9 0.09 which is 45 hundredths, 0.45. "TTT^ 0~04^ (^^•) ^^^ ^^^ same way we find that 0.5 multi- plied by 9 hundredths gives 45 thousandths. As it requires three decimal places to write thousandths, and as in 45 there are but 2, we supply the deficiency by prefixing a zero before placing the decimal point, thus, 0.045. In multiplying a decimal by a decimal, how many decimal places do you point off in the product ? 279. In multiplying by decimals, Multiply as in in- tegers, and point off as many j^^^f^es for decimals in the product as there are decimal p)laces in the midtiplicand and multiplier counted together. 112 DIVISION OF DECIMALS. ^ 280. Examples for the Slate. 64. Multiply 11.6 by 4. 71. 30.04x0.105 = 9 ^^, Multiply 1.16 by 7. 72. 920.8x706.1- 9 m. Multiply 17.07 by 11. 73. 3.007 X 0.005 = 9 67. Multiply 106 by 0.3. 74. 5005. X 0.001 = 9 68. Multiply 20 by 1.71. 75. 88.04 X 36 = ? 69. Multiply 3.21 by 28. 76. 0.325 X 0.018 = 9 70. Multiply 30.2 by 1.4. 77. 0.0101 X 4.16 - 9 78. At $0.62^ ($0,625) each for hats, what is the cost of 10 hats ? of 2 dozen ? 79. At $ 0.20 a square foot for land, what is the cost of 1000 square feet? of 2500 ? 80. What cost 124^ yards of cloth at 1 1.875 a yard ? 81. What must be paid for 100 feet of land at $ 0.06^ a foot ? 82. There are 30.25 square yards in a square rod. How many square yards are there in 160 square rods or an acre ? DIVISION. To divide a Decimal by an Integer. 281. a. What is 1 third of 0.6 ? 1 fourth of 0.8 ? of 1.2 ? b. What is 1 half of 0.04 ? 1 fifth of 0.25 ? 1 eighth of 0.56 ? c. Divide 13.75 by 5. d. Divide 10.272 by 4. WRITTEN WORK. Ill Example c, the dividend has two (c.) (cf.) places for decimals, so in the quotient 5) 13.75 4) 10.272 we mark off two places for decimals. ~2T5 ~~2 568 ^^ Example d, the dividend has three places for decimals, so in the quotient we mark off three places for decimals. In dividing a decimal by an integer, how many places do you point off for decimals in the quotient ? 282. Examples for the Slate. 83. Divide 6.24 by 12. SQ. Divide 1.576 by 8. 84. Divide 288.9 by 9. 87. Divide 112.84 by 11. ^B. Divide 91.05 by 15. ^%. Divide 14.607 by 27. DIVISION OF DECIMALS. 113 To divide Integers, carrying the Division to Decimals. 283. e. Find 1 fourth of 13. wiaxTEx "WGiiK. In Example e, after the integer is divided there 4) 13.00 is a remainder of 1. This we change to 10 tenths. 3 25 One fourth of 10 tenths is 2 tenths, and 2 tenths re- main, which are equal to 20 hundredths. One fourth of 20 hundredths is 5 hundredths. The entire quotient is 3.25. Ill tlio following examples carry the division to decimals : 89. Find 1 fifth of 157. 91. Find 1 fourth of $ 927. 90. Find 1 sixth of 879. 92. Find 1 eighth of § 721. 93. If 8 hamniochs cost $ 7, wdiat is the cost of 1 liammock ? 94. If 4 dozen hats cost $ 78, what is the cost of 1 hat ? 284. /. Divide 348 by 71. WKiTTEN woKK. We divide as in Example e till we have 11) 348 f4 901 thousandths in the quotient and there is still noj^ a remainder. We might keep on dividing, and thus come to greater accuracy in the ^'^O quotient. But it is not generally necessary C39 to carry the division beyond thousandths. -^ In the following examples carry the division to thousandths, and if there is ^^ a remainder then, indicate that there is one by dots, as in the answ^er to Example /. 95. How many are 4321-9? 98. Divide 876.1 by 8. 96. How many are 3214-12? 99. Divide 13.47 by 17. 97. How many are 9486-19? 100. Divide 9841 by 21. To divide an Integer or a Decimal by a Decimal. 285. g. How many times are 0.3 contained in 0.6 ? 0.4 in 0.8? 0.6 in 1.2? 0.9 in 2.7 ? 72. How many times are 0.03 contained in 0.06? 0.08 m ^.16? 0.12 in 1.44? ^^'"'7'^- i. Divide 0.006 by 0.002 ; 0.018 by 0.003; flfti^ty 0.O12.- Uiri7E 114 DIVISION OF DECIMALS. 286. J. Divide 35 by 0.7. k. 1.44 by 0.8. WRITTEN WORK. Before dividing by a fraction, the (j.) {k.) dividend must be expressed in the 0.7) 35.0a 0.8) 1.4a4 same denomination as the divisor. In ^ H o Example j, the divisor is a number of tenths ; the dividend expressed in tenths is 35.0 (350 tenths). 350 tenths divided by 7 tenths gives the same quotient as 350 divided by 7, which is 50. Ans. 50. In Example k, the divisor is, a number of tenths ; the dividend expressed in tenths is 14.4 tenths (the denomination may be indicated, as in the written work, by a caret). 14.4 tenths divided by 8 tenths gives the same quotient as 14.4 divided by 8, which is 1.8. Ans. 1.8. In Example j, bow did you prepare the dividend to divide ? How did' you then divide ? In Example k, bow did you prepare the dividend to divide ? How did you indicate the denomination tenths in the dividend? 287. To divide by decimals : 1. Uxpress the dividend in the same denomination as the divisor hy putting a mark as many places to the right of the decimal point as there are decimal places in the divisor. 2. Divide as if the divisor were an integer y fixing the decimal point of the qiiotient when the terms of the divi- dend have been used as far as the mark. 288. Examples for the Slate. 101. Divide 315 by 0.7. 106. Divide 1000 by 0.001. 102. Divide 86.1 by 0.08. 107. Divide 90.09 by 0.071. 103. Divide 70.32 by 0.38. 108. Divide 8.64 by 71.6. 104. Divide 0.172 by 0.12. 109. Divide 538.1 by 4.001. 105. Divide 0.5307 by 0.1. 110. Divide 3.027 by 21.1. 111. How many tops at $ 0.02 apiece can be bought for $ 2 ? 112. How many rods, each 16.5 feet, are there in 100 feet ? 113. At $ 0.125 a quart for cherries, how many quarts can he bought for $ 0.25 ? for $0,875? for $1.25? for $3? For other examples in Multiplication and Division, see page 129. — J PERCENTAGE. 115 SEOTIOS" X. PERCENTAGE. Oral Examples. 289. a. One is what part of 2 ? of 3? of 4? of 5? of 10? b. One is wha^ part of 20 ? of 30 ? of 50 ? of 100 ? c. What part of 100 is 1 ? is 2? 3? 4? 5? 10? 20? Suggestion. One is -^^ of a hundred; 2 is yfg-; 3 is yf^; and so on. 290. Any number of hundredths of a thing or number is a per cent of that thing or number. Thus, y§^ of a number is 2 per cent of it. d. What per cent of a number is y §^ of it ? y§^ ? -j%% ? \%% ? e. To take 20 per cent of a number is to take how many hundredths of the number ? /. How many hundredths of a number is 10 per cent of it ? 25 per cent of it ? To express a given Per Cent. 291. The sign % is used for the words '' per cent." Thus, 3 % means 3 per cent. 292. Any per cent may be expressed as a common frac- tion, as a decimal, or with the sign for per cent, %. Thus, 1 per cent may be expressed y^, 0.01, orl%. 5 percent " " y|^, 0.05, or 5%. 8J per cent " " ^, 0.08^, or 8^%. 100 percent " " 1^, 1.00, or 100%. 125 percent " " |||, 1.25, or 125%. 4 per cent " " y^, 0.00^, ori%. 293. Examples for the Slate. Express the following in the three forms given above : (1.) 2 per cent. (3.) b\ per cent. (5.) 145 per cent. (2.) 8 per cent. (4.) 100 per cent. (6.) ^ per cent. 116 PERCENTAGE. To change a Common Fraction to a Per Cent. 294, a. What per cent of a number is J^ of it ? &. ^ of it ? "vvniTTEN woKK. (^a.) Since any number equals ICO per (a.) {b.) cent of itself, \ of the number must equal 2) 100 % 3) 100 % \ of 100 % , or 50 % . Ans. 50 % . 50% 331% (ft.) -J of the number must equal \ of 100 % J 01^ 33 J % . Ans. 33 J % . 295. Oral Examples. You will need to practise on these exercises till you can give the answers at sight. c. What per cent of a number is -J of it ? ^ ? ^^ ? t^^? d. What per cent of a number is 5 of it ? f ? f ? J ? | ? A 9 3 9 49 7 9 3 9 7 9 9 IT • 8 • ^ • 8 • T(J • To •, 1(7 • e. What per cent of a number is ^^ of it ? /^ ? ^f^ ? ^5 ? ^^ 9 ^T_ 9 7 9 293. From what you have learned in the preceding exercises, you may tell what part of a number the following per cents are, giving the parts in their smallest terms, thus, "10%, T^t,; 20V., l? etc. f. 10%; 20%; 30%; 50%; 25%; 75%; 80%; 90%; m%; 37i%; 621%; 87^%; 33i%; 660o', 16S%. To find a Number that is a given Per Cent of another Number. Oral Examples. 297. g. What number is 7% of S800 ? Solution. — 7 % of $ 800 is ^^ of § 800, or $ 56. Ans. $56. A number found by taking a number of hundredths of another number is a percentage of that number. Thus, in the example above, S 50 is a percentage of S 800. h. What is 4% of 300 men ? of S 500 ? of 1000 ? i. What is 10% of 8 50? of $25? of $ 80 ? WniTTEX WORK. (a.) (ft.) 946 87.G0 0.15 0.08^ PERCENTAGE. 117 J. What is 1% of 2000 ? 3% ? 5% ? 6]% ? A:. What is 1% of $1200 ? i% ? 1% ? i% ? 2% ? 2^% ? i. What is 1% of $800? 100% ? 101%? 102% ? 110% ? 122. A farmer raised 400 bushels of corn, and sold 50% of it. How many bushels did he sell ? n. In a school of boys and girls containing 300 pupils, 70 % were girls. How many were boys ? 298. Examples for the Slate. a. What is 15% of $ 946 ? b. SJ % of $ 87.60 ? By what do you divide a number to findl% of it? Having found 1%, how do you find "4730 29 vn' ^^^ number of times 1 % ? 946 70080 Does it make any difference which 141 90 7.3000 operation you perform first ? (7.) What is 6% of $469? (14.) J% of 120 acres ? (8 ) 8% of 975 ? (15.) §% of 784 pounds "" (9.) 9% of 1800 ? (16.) ^5% of 1600 days ? (10.) 16% of 465? (17.) t% of 750 words? (11.) 25% of 6800 men ? (18.) 40% of 7.45 ? (12.) 42% of 876 miles ? (19.) 60% of 10.5 tons ? (13.) 1^1^^% of 873 bushels. (20.) 18^% of 960 barrels ? 21. Of a flock of 175 sheep 4% were killed by dogs. How many were killed ? 22. A stable-keeper bought two carriages ; for the first he paid $275, and for the second 75% as much as for the first. What did he pay for both ? 23. A landlord has lowered his rents 20%. What will he now charge for tenements that have rented for $ 200 a year ? for $ 425 ? for $ 12.50 a month ? 24. What does a man receive for a house-lot which cost $1524 by selling it for 81% more than it cost ? 2o. What prices must be charged for four kinds of tea cost- ing 36/, 48/, 54/, and 63/ per pound to gain 16§% ? 118 PERCENTAGE. To find a Number when a Per Cent of it is given. Oral Examples. 299. a. 20% of a number is $40. What is the number ? Solution. — Since 20 % of the number sought is $ 40, 1 % of the number sought is -^^ of $40, which is $2, and 100% of the number sought is 100 times $2, which is $200. Arts. $200. h. 60 pounds is 6 % of how many pounds ? c. $24 is 8% of how many dollars ? $ 30 is 12% of how many dollars ? d. 90 is 10 % of what number ? 220 is 11 % of what number ? e. 120 bushels is 60% of what number? $80 is 40% of what number ? /. John has 12/, which is 6% of what Charles has. How much has Charles ? g. A man gained $ 20 by selling his watch for 10 % more than it cost him. How much did it cost ? h. Mary spelt 98 % of the words given out to her class and missed the rest, which was 6 words. What per cent did she miss ? How many words were given out ? 300. Examples for the Slate. a. $ 250.50 is 25% of what number ? WRITTEN WORK. How do you find 1% of a number r}Jl^2i -.oA when a number of per cents are given? — 1UU.U. When you have found 1 % of a num- Ans $ 1002. ^^^y ^^^ ^^ y^^ ^^^ ^^^ whole number ? (26.) 74 is 37% of what ? (30.) $ 75 is 37|% of what ? (27) 150.50 is 5% of what ? (31.) 1 94.50 is 6J% of what ? (28.) $ 4.86 is 10 % of what ? (32.) $31.25 is 125 % of what ? (29.) $ 649.33 is 11 % of what ? (33.) $ 87.50 is 62 J % of what ? 34. A man expends 70 % of his income and saves $ 369. What is his income ? 35. I drew from a bank $ 1485, which was 45% of what was left. What w^as the sum in at first ? PERCENTAGE. 119 To find what Per Cent one Number is of another. Oral Examples. 301. a. What per cent of 50 is 7 ? Solution. — 7 is -^-^ of 50. -^^j- equals ^^q, or 14%. Am. 14%. b. What per cent of 20 is 1 ? is 3 ? is 9 ? c. What per cent of 10 is 1 ? is 3 ? is 7? d. What per cent of 25 is 2 ? is 3 ? is 6 ? e. What per cent of 80 is 20 ? Solution. — 20 is i- of 80. J equals ^, or 25 % . Ans. 25 % . /. What per cent of 40 is 8 ? of 16 is 4 ? of 12 is 6 ? of 25 is 5? g. A boy had 20 plums, and gave away 7 of them. What per cent did he give away ? What per cent did he keep ? h. A man hired $ 50, and paid $ 3 for the use of it. What per cent did he pay ? i. A boy who weighed 80 pounds lost 10 pounds by sickness. What per cent did he lose ? J. A watch which cost $ 20 was sold for $ 15. How many dollars were lost ? What per cent was lost ? 302. Examples for the Slate. k. What per cent of $ 50 is $ 3.25 ? WRITTEN WORK. To find what per cent one mim- 50) 3.25 (0.06f j-6i%. ber is of another, Divide the per- ^ ^^ centage hy the number it is a per- 25 centage of, carrying the division to hundredths. 36. What per cent of $ 480 is $ 40 ? 37. Of 16 days is 14 days ? 39. Of 15 tons is 2 tons ? 38. Of 35 cents is 8 cents ? 40. Of 1 score is 1 dozen ? 41. If I sell a horse for $ 100 which cost $ 150, what per cent do I lose ? 42. Flour which was bought at $ 4.80 per barrel is sold at $ 6.00. What is the per cent of gain ? 120 PROFIT AND LOSS. PEOriT AND LOSS. Oral Examples. 303. a. How much money is gained hy selling a cow for 10% above cost, the cost being $ 60 ? $80 ? b. How much money is lost on a horse which cost 8 200 by selling liim at a loss of 10% ? of 20% ? of 50%? c. At what price must books which cost $4 apiece bo sold to gain 50%? 25%? 20%? 10%? d. A man bought butter at 20 cents a pound. At what price must he sell it to lose 25% ? 20% ? 30% ? e. A man sold a cow for § GG, and gained 10%. What did she cost him ? Solufion. — Since he gained 10%, $66 must be 110% of the cost. If S 66 is 1 J^ or \^ of the cost, J-^ of the cost must be ^^ of $ 66, or $6, and the whole cost must be 10 times $6, or §60. Ans. $ 60. /. What must have been paid a bushel for wheat, if by sell- ing it for §1.50 there was a gain of 25% ? of 50% ? g. A man sold a wagon for $48, and lost 20%. What was the cost of the wagon ? Suggestion. Since he lost 20%, §48 is 80%, or |- of the cost. h. What was the cost of knives which sold for 27 cents each at a loss of 10%? 25%? 2. Wliat per cent would be gained by selling a plough for $10 which cost $8? Suggestion. § 2 was gained on § 8, which was |- or \- of 8. ^ = 25 %. j. WHiat per cent is gained if spades costing 80 cents are sold for $ 1 00 ? for § 1.20 ? for $ 1.30 ? 304. The difference between the cost of goods and the price at which they are sold is a profit or loss. 305. Examples for the Slate. 43. Cloth Avhich was bought at $ 7 a ^^ard was sold at a gain of 12%. W^liat was received for it ? 44. Por wliat must I sell 150 cords of wood which cost $ 3.50 a cord to gain 8% ? COMMISSION, INSURANCE, TAXES, ETC. 121 45. By selling corn at 25 % profit a merchant gained $462.50. Wliat did the corn cost him? 46. From a barrel of kerosene 15 1^ gallons leaked out. If this was 30% of the contents of the barrel, how many gallons did the barrel contain at first ? 47. Tlie expenses of a family in 1877 were %B^^. In 1878 they were 7?i% less. Wliat were their expenses in 1878 ? 48. A shoe-dealer sold boots at 8 4.50 a pair at a gain of 20% on wliat they cost him. What did tliey cost? 49. If 120% of a number is $ 4.50, wliat is the number ? 50. A man sold coal at $5.50, and lost 23% of the cost. What did his coal cost ? 51. What per cent is gained by selling nutmegs at 56 cents a pound which cost 48 cents a pound ? 52. A, having failed, pays B $ 280 instead of $ 480, which he owed him. What per cent does B lose ? 306. OOMMISSIGl^, lETSUEANOE, TAXES, ETO. Note. The tenns Commission, Insurance, Taxes, etc., should be ex- plained by tlie teacher. A full treatment of these subjects will be found in the Fi-anklin Written Arithmetic. 53. An agent sold a house for $ 2500. If he received a com- mission of 4% for selling, how much did he receive ? 54. A lawyer received 5% commission for collecting a bill of $ 375. What was his commission ? bo. An architect received a commission of § 140.30 for plan- ning and superintending the building of a house. If his com- mission was 3%, what was the cost of the house? 56. A man had his liousc insured for §2000. If he paid 1J% of this sum for the insurance, what did he pay ? 57. A man is taxed for property amounting to $ 4760. If his tax is 1] % of tin's amount, what is liis tax ? lj?>. A broker bought for a man 5 shares of railroad stock wortli 8 100 a share, and charged J % connnission. What was Ilia charge ? 122 INTEREST. SECTION XI. INTEREST. 307. A man lent me $200 for 1 year. At the end of the year I paid him back $ 200, with 5 % of % 200 for the use of the money. How much did I pay for the use of the money? How much in all ? Ans. $ 10 for the use ; $ 210 in all. Money paid for the use of money is interest. The money for the use of which interest is paid is the principal. The sum of the principal and interest is the amount. Thus, in the above example $ 10 is the interest, $ 200 is the principal, and $ 210 is the amount. 308. To find the interest, a certain per cent of the prin- cipal is taken for each year. This per cent is called the rate per cent, or simply the rate. Note. In reckoning interest, it is customary to consider a year to be 12 months, and a month 30 days. 309. a. What is the interest of $200 for 2 years at 6% ? WKITTEN WORK. 200 6% is ^fo, or 0.06. 0.06 of $200 is $12, the 0.06 interest for 1 year. 12.00 The interest for 2 years is 2 times $ 12, or $ 24. 2 Ans. $24. 24.00 310. Oral Examples. What is the interest b. Of $100 at 6% for 2 years ? 3 years ? c. Of $200 for 2 years at 3% ? 4% ? 5% ? 7%? 12%? d. Of $40 at 10% for 1 year? for 3 years 6 months ? SOLUTION. 1 40 X 10% = $ 4. Int. for 1 y. (year). 3 years 6 months $4 X 3i - $ 14. " " 3 y. 6 mo. (months), equals 3^ years. INTEREST. 123 What is the interest of 160 at 10% e. For 1 y. 6 mo. ? g. For 1 y. 9 mo. ? /. For 1 y. 3 mo. ? h. For 2 y. 8 mo. ? What is the interest of $120 at 10% i. For4y. 2mo. ? -6:. For 2 y. 5 mo. ? 7. For 4 y. 1 mo. ? L For 6 y. 7 mo. ? The Six Per Cent Method. A. general method of computing interest will be found on p. \42 of the Appendix. 311. Oral Examples. a. At 6%, how many hundredths of the principal is the in- terest for 1 y. ? for 2 y. ? for 3 y, ? Arts. 0.06 ; 0.12 ; 0.18. At 6%, how many hundredths of the principal is the interest b. For 6 y. ? for 8 y. ? 10 y. ? 11 y. ? c. For 6 mo. ? for 2 mo. ? Suggestion. Since the interest is 6 % for 1 year, for 6 months, which is 1 half of a year, it is 3 % , or 0.03 ; for 2 months, which is 1 sixth ot a year, it is 1 %, or 0.01 ; and so on. At 6%, how many hundredths of the principal is the interest d. For 2 mo. ? 4 mo. ? 8 mo. ? 10 mo. ? e. For 1 y. 6 mo. ? 2 y. 6 mo. ? 3 y. 2 mo. ? /. For 4 y. 2 mo. ? 3 y. 4 mo. ? 5 y. 8 mo. ? g. For 1 mo. ? 3 mo. ? 5 mo. ? Suggestion. Since the interest is 1 % for 2 months, for 1 month it is •^%, or 0.005,- for 3 months it is 1^%, or 0.015 ; and so on. At 6%, what part of the principal is the interest h. For 3 mo. ? 5 mo. ? 7 mo. ? 9 mo. ? 11 mo. ? i. For 1 y. 1 mo. ? 2 y. 1 mo. ? 2 y. 3 mo. ? 2 y. 2 mo. ? j. For 3 y. 1 mo. ? 4 y. 5 mo. ? 3 y. 6 mo. ? 3 y. 7 mo. ? k. For 5 y. 10 mo. ? 6 y. 9 mo. ? 6 y. 8 mo. ? 10 y. 11 mo.? 1. For 6 days ? for 1 day ? for 5 days ? for 7 days ? Suggestion. Since the interest is 0.005 of the principal for 1 month or 30 days, for 6 days, whf-^h is 1 fifth of a month, it is 0.001 ; for 1 day it is 0.000^ ; for 5 days, O.OOOf ; for 7 days, 0.001^; and so on. 124 INTEREST, At 6%, what part of the principal is the interest m. For 6 d. (days) ? 12 d. ? 18 d. ? 24 d. ? 1 mo. (30 d.) ? n. For Id.? 3d.? 9d. ? 10 d. ? lid.? 13 d.? o. For 15 d.? 19 d.? 25 d. ? 17 d.? 21 d.? 27 d. ? 312. Examples for the Slate. a. At 6%, what part of the principal is the interest for 2 y. 5 mo. 14 d. ? WRITTEN WORK. To fiiid wtat part of the 0.12 for 2 years. principal the interest is, r«Z;e 0.025 " 5 months. ^ ^^-^^^^^ ^^^ ^^ hundredths as 0.002 \ " 14 days. ,7 . . ,. ^^ ^ ^ ^ '^^ ^^ ^ ^^ ^ ^ ^ m(j?'e a?'^ ?/cars, 1 hcdj as many hundredths as there are months, 0.147i " 2 y. 5 mo. 14 d. and 1 sixth as many thousandths as there a?r, days. At 6%, what part of the principal is the interest 1. For 1 y. 1 mo. Id.? 6. For 4 y. 5 mo. 25 d. ? 2. For 1 y. 3 mo. 5 d. ? 7. For 8 y. 4 mo. ? 3. For 2 y. 6 mo. 9 d. ? 8. For 6 mo. tO d. ? 4. For 3 y. 8 mo. 23 d. ? 9. For 2 y. 10 d, ? 5. For 5 y. 7 mo. 7 d. ? 10. For 3 y. 5 /no. 14 d. ? 313. b. Find the interest of $120 at 6% for 3 y. 5 mo. 14 d. WEiTTEN WORK. At 6 % , the interest for 3 years 5 monUis 14 days 120 IS 0.207^ of the principal. 0.207-^ of $ 120 is § 24.88. 2071 Ans. $24.88. 7^ To compute the interest on any sum at 840 ^ % '- (I) Find the decimal that expresses the 240 pcirt ivhich the iyitercst is of the ^9?v!7irzpaZ ; 24.880 (2) iy t^^is decimal midtiply the lorinci'pal. Note. In doing the following examjoles, get the answers to the nearest cent. At G%, what is the interest 11. Of $ 800 for 1 y. 4 mo. ? 13. Of $ 350 for 3 y. G mo. 9 d. ? 12. Of $ 124 for 2 y. 7 mo. ? 14. Of $ 488 for 5 y. 7 mo. 14 d. ? INTEREST. 125 314. 15. At 6%, T\'liat is the interest of $500 from Jan. 1, 1877, to March 7, 1878 ? Suggestion. From Jan. 1, 1877, to Jan. 1, 1878, is 1 year ; from Jan. 1 to March 1 is 2 months ; and from March 1 to March 7 is 6 (lays. The whole time is 1 y. 2 mo. G d. At G%, ^vhat is tlie interest 16. Of § 268.90 from July 15, 1877, to Sept. 25, 1878 ? 17. Of § 1185 from May 3, 1878, to August 18, 1879 ? 18. Of $ 2000 from Oct. 25, 1878, to Jan. 29, 1879 ? Sitggsstion. The whole time is 3 months 4 days. At 6%, what is the interest 19. Of S 87.50 from Sept. 14, 1877, to Feb. 20, 1878 ? 20. Of 8380.70 from Sept. 17, 1877, to Marcli 29, 1879 ? 21. Of S 200 from Dec. 16, 1877, to May 10, 1878 ? Suggestion. From Dec. 16, 1877, to Ajoril IG, 1878, is 4 months. From April 16 to April 30 is 14 days, and to May 10 is 10 days more, or 24 days. The whole time is 4 months 24 days. At 6%, w^hat is the interest 22. Of $ 95 from Nov. 26, 1878, to May 15, 1879 ? 23. Of §284.40 from Nov. 27, 1877, to Oct. 15, 1879? 24. At 6%, what is the amount of $ 7500 from Feb. 15, 1878, to Dec. 29, 1879 ? Suggestion. To find the amount, add the principal to the interest. At 6%, wdiat is the amount 25. Of $ 496 from June 17, 1877, to Dec. 1, 1878 ? 26. Of $ 630.20 from Feb. 18, 1877, to July 5, 1878 ? 315. To find the Interest at any Per Cent other than 6%. What is the interest of 8 2S5, for 3 y. 4 mo. 27. At7%? 28. At8%? 29. At 5% ? Suggestion. First find the interest at 6% ; then to find the interest at 7%, add 1 of this interest to itself, to find the interest at 8%, arZr/ | or J. To find the interest ai 5%, subtract J, and so on. 126 INTEREST. What is the interest of $ 156.60 for 3 y. 6 mo. 18 d. 30. At 4% ? 31. At 8% ? 32. At 10% ? What is the interest 33. Of $ 474 for 2 y. 11 mo. at 7 % ? 34. Of $2500 from Oct. 15, 1878, to July 27, 1879, at 10% ? What is the interest and amount 35. Of $ 318 for 11 mo. 90 d. at 5% ? 36. Of 1 268 for 3 y. 8 mo. 27 d. at 8 % ? 37. Of $ 98.25 for 1 y. 5 mo. 1 d. at 7% ? What is the amount 38. Of $ 805.50 from Sept. 10, 1878, to Nov. 4, 1878, at 6^% ? 39. Of $ 962.25 from Apr. 8, 1875, to Oct. 1, 1875, at 7% ? 40. Of $ 500 from Jan. 1, 1878, to Jan. 15, 1879, at 7^% ? 41. A note for $400 was on interest at 6%, from June 5, 1875, to June 5, 1876, when a payment of $ 124 was made to pay the interest due and part pay the note. How much then remained due ? 42. If $ 300 was due on a note June 5, 1876, and was kept on interest till December 5, 1876, what was then required to pay the note and the interest due ? BANK DISCOUNT. 316. On the 7th of June, 1878, Mr. M. J. Oliver bought a horse of Mr. John Day, and agreed to pay $ 225 for it in 60 days, giving him the following written promise, called a PROMISSORY NOTE. ^ ^2^ X«icoZ?i, June 7, 18118, (25^^ c/ayd /i^7ri c/ci^ of^ ^'toTncae '^ ^uiy ^oupi. .^czy. If Mr. Day wants the money before the 60 days are over, he can take the note to a bank and get the money by pay- BANK DISCOUNT. 127 ing the bank the interest on it for the time to elapse before it is due, and for 3 days more. The note is then said to be discounted. 317. The interest paid to the bank is bank discount. 318. The money received from the bank is called the proceeds. 319. The three days for which interest is taken beyond the time named in the note are called days of grace. 320. a. Find the discount on the above note July 7, 1878, the discount being 6%. Find the proceeds. WRITTEN WORK. Bank discount is interest for the specified time $ 225 ^^d of 3 days' grace. 0.0055 '^^^ ^^^^^ horn July 7 to the end of 60 days and —7^ grace is 33 days. -1I9K The interest of 1 225 for 33 days at 6% is $ 1.24, 1125 which is the discount. $225 less $1.24 equals $ 1.2375 $ 223.76, the proceeds of the note. $ 225 - $ 1.24=$ 223.76. ^^s- $ 1-24, discount ; $ 223.76, proceeds. 321. Examples for the Slate. 43. What is the hank discount of a note for $ 500, payable in 30 days, discount 6 % . What are the proceeds ? Note. Add 3 days of grace to the specified time in all instances. Find the bank discount and proceeds of a note 44. For $300, payable in 60 days, discount'6%. 45. For 175.80, payable in 90 days, discount 7%. 46. For $1300, payable in 4 months, discount 5%. 47. For $486.50, payable in 45 days, discount 8%. 48. For $275.48, payable in 3 months, discount 10%. 49. For $80, due July 15, and discounted June 15, at 6%. 50. For $700, due Aug. 29, and discounted Aug. 2, at 5%. 51. For $650, dated Aug. 25, payable in 2 months, a^nd discounted Sept. 25, at 6%. 52. For $2000, dated Oct. 1, payable in 1 month, and dis- counted Oct. 16, at 9%. 128 DRILL TABLE. 322. DRILL TABLE No. 4. Common Fractions. — Decimals. — Percentage. A B c i 4 U 4 § U i tV u \% rAr 7 Ha T^3 1% i A is f tV iff H A H li 3V fs f A «• § # H ifV 40- T^d /a i-V * A J$ ft A -2S 1 tfj 21 5 A ja i?" ^ e -^^ 5 tf f A u H ^4- i^/k 3 in f§ flT ttV T^A ^ i§ §* ? A il D E F 3? 5 19J 50 IG 12s 125 13 ICtV 8} 19 9j n . 7 11§ 12§ 12 155- Gf 9 8i 14* 18 I'i n 8 12i iiA 21 17i 52^. 14 8J 9J 8 15J Cff 17 123- 8* 9 14^ 42 15 cj 7J 24 llj 5t\ 11 13j\ 12$ 13 15-^ 13i 16 20i 11 sV 7 11^ 162- 22 19? 7i 10 n lOi 12 lltV 8.^ 25 9* n 13 13i DRILL EXERCISES. 129 DRILL TABLE No. 4 {continued). Examples H I 1, 2.905 3.178 2. 3.061 50.817 0.052 0.703 38.04 209.63 200.1 54.68 0.05 3.0721 708.1 0.1001 362 3 0.0732 52.12 9.246 1.805 43.287 100.2 0.016 0.086 9.03 702.4 3.0072 0.7 0.245 5 099 3.18 80.06 5.4061 0.004 10.18 581.9 3287 600.7 540.91 7.101 10.6 71.01 4.8001 19.06 13.704 0.83 97 3001 2^.5 2.007 16.035 191.06 Exercises upon the Table. 323. Common Fractions. 124. Change C to smallest terms. 125. Change D to improper fractions. 126. Add A and B. 127. Add A, C, and D. 128. Add D and F. 129. A-B. 136. AxD. 130. G-F. 137. DxF. 131. D-C. 138. A-^E. 132. F-D. 139. E-^B. 133. AxE. 140. A-^B. 134. GxB. 141. D-^A. 135. AxC. 142. F^D. 324. Decimals. 143. Change A to a decimal (3 places). 144' Change B to a decimal (4 places). 145. Add H, I, and 2.9784. 146. From 1000 take H. 147. Find the difference of H and I. 148. Multiply I by 7. 149. Multiply 20 by H. 150. Multiply H by I. 151. Divide H by 4. 152. Divide 57 by H (3 places). 153. Divide I by H (4 places). 325* Percentage. 154. What is 5 % of E dollars ? 155. What is 7% of G dollars? 156. What is 8 % of H dollars ? 157. Find the interest of G dollars for 3 y. 6 mo. at 6 % . 158. Find the interest of E hundred dol- lars for 4 y. 9 mo. 1 5 d. at 6 % . 159. Find the interest of G hundred dol- lars for 2 y. 3 mo. 14 d. at 5 %. 130 MENSURATION, SECTION XII. MENSURATION. Note. For those who may study no liigher Arithmetic, brief direction -. are here given for finding the areas of some surfaces and the vohnne of some solids. Further exi)lanation3 and illustrations of these surfaces and solids should le given by the teacher, who is referred to the Franklin Written Arithmetic, pages 137 to 139 ; also pages 277 to 288. SURFACES. 326. A square each of whose sides is 1 inch long is a square inch; a square each of whose sides is 1 foot long is a square foot; etc. 327. The area of a surface is its con- tents reckoned in square units. A Square. To find the Area of a Rectangle. 328. A rectangle is a four-sided figure which has all its corners square. a. If a rectangle is 4 inches long and 1 inch ^Yide, how many square inches does it contain ? How many will it contain if it is 4 inches long and 2 inches wide ? 3 inches wide ? b. What numbers will you multiply together to find the number of square units in any rectangle ? 329. The area of a rectangle is found by midtiplyincj the niivibcr of units in the length hj the member of like units in the hreadth. This is expressed for brevity as multiplying the length hy the hreadth. A Rectangle. MENSURATION. 131 330. Oral Exercises. c. How many square inches are there in the surface of a slate that is 9 inches long and 8 inches wide ? that is 12 inches (1 foot) long and 12 inches (1 foot) wide ? Then how many square inches are there in a square foot ? d. How many square feet are there in a rectangle that is 5 feet long and 3 feet wide ? How many square feet are there in a square whose sides are each 3 feet (1 yard) long ? 331. Eepeat the following tahle : Square Measure. 144 square inches =1 square foot. 9 square feet =1 square yard. m square yards or ) ^ ^ ^^^^ ^^^^ 272i square feet ) 160 square rods =1 acre. 640 acres =1 square mile. 332, Examples for the Slate. 1. In a walk 20 feet long and 6 feet wide, how many square feet ? how many square yards ? 2. How many square yards of carpeting will it take to cover a floor that is 15 feet long and 14 feet wide ? 3. In a piece of ground 200 feet long and 50 feet wide, how man}^ square feet ? how many square rods ? 4. In a lot 250 feet long and 120 feet wide, how many square rods ? 5. How many square feet of land in a lot 4 rods wide and 90 feet deep ? 6. At 20/ a square foot for land, what must be paid for a lot 100 feet deep and 3 J rods on the front ? 7. In a square field 60 rods on each side, how many acres ? 8. What is the difference in the size of two lots, one con- taining %b square rods, the other being (j5 rods square, that is, Qb rods long and Q6 rods wide ? 132: MENSURATION. TRIANGLES, POLYGONS, AND CIRCLES. 333. A triangle is half a rectangle of the same base and height. Hence, To find the area of a triangle, Multiply the A Triangle, j^^^g ly fj^^ height and divide the product hij ^. 334. To find the area of a polygon, Divide it into triangles and find the sicm of their areas. 335. When the diameter of a circle is given, To find the circumference, Multiply the diameter hij S.lJplS. 336. A circle may be said to consist of triangles whose bases form the circumfer- ence, the height of the triangles being equal to the radius of the circle. Hence, To find the area of a circle. Multiply the circumference hy the radius and divide the product hy 2. A Polygon. A Circle. 337. Examples for the Slate. 8. I have a piece of woodland shaped like a triangle, the base of which is 100 rods long and the height 62 rods. How many acres does it contain ? 9. How many square yards are there in a roof that is made up of 4 triangles, the base of each being 20 feet and the height 15 feet ? 10. I have a tin-pail that measures 8 inches across the top. What is the distance round it ? How many square inches must there be in a cover that will fit it ? 11. How many square feet of land in a circular flower-bed that measures 10 feet across ? MENSURATION. 133 SOLIDS. 338. A rectangular solid is a solid bounded by six rectangles. A brick is a rectangular solid body. 339. A cube is a rectangular solid bounded by six equal squares. A cube each of whose edges is 1 inch long is a cubic inch. A cube each of whose edges is 1 foot long is a cubic foot, etc. 340. The volume of a solid is its contents reckoned in cubic units. A Cube. ^ y A ^ ' y y y y y- y y y y y y To find the Volume of a Rectangular Solid. 341. a. If a rectangular solid is 4 inches long and 2 inches wide, how many square inches must its base contain ? (Art. 329.) "b. If the base of the solid contains 4 times 2, or 8 square inches, and it is 1 inch thick, how many solid or cubic inches must it contain ? If it is 2 inches thick, how many cubic inches must it contain ? If it is 3 inches thick, how many cubic inches must it contain ? What numbers have you multiplied together to find the vol- ume of the rectangular solid 4 inches long, 2 inches wide, and 3 inches high ? 342. The volume of any rectangular solid is found by multiplying the member of units in the length hy the number of like units in the breadth ^ and this product by the number of like units in the thickness. This is expressed, for brevity, as multiplying together the length, breadth, and thickness. 134 MENSURATION 343. Oral Exercises. c. How many cubic inches in a rectangular solid 6 inches long, 5 inches wide and 3 inches thick ? 4 inches long, 2 inches wide and 7 inches thick ? 12 inches long, 12 inches wide and 1 inch thick ? d. How many cubic feet in a cube 3 feet long, 3 feet wide and 3 feet thick, or in 1 cubic yard ? 344. Eepeat the following table : Cubic Measure. 1723 cubic inches = 1 cubic foot. 27 cubic feet =1 cubic yard. 128 cubic feet = 1 cord (iised in measuring wood). Wood is generally cut for the market into sticks 4 feet long, and laid in piles, so that ^ p^P the length of the sticks becomes the width of the pile. A pile 4 feet wide, 4 feet high, and 8 feet long, contains 1 cord. One eighth of a cord is called 1 cord foot. 1 cord foot Ai»J»^*^^. ' contains 16 cubic feet. (See illustration above.) 345. Examples for the Slate. 12. How many cubic inches will a box contain that is 11 inches long, 7 inches wide and 6 inches high, inside measure ? 13. How many cubic inches in a beam 5 feet long, 3 inches thick and 4 inches wide ? 14. A cubic foot of water weighs 62 J^ pounds. What is the weight of water that a cistern may contain which measures on the inside 4 feet in length, 3 feet in width and 4 feet in depth ? 15. How many cords of wood in a pile 20 feet long, 4 feet wide and 4 feet high ? 16. How many cords in a pile 30 feet long, 4 feet wide and 3i feet high ? MENSURATION. 135 Lumber and Boards. 346. Sawed timber and boards, when 1 inch or less in thickness, are generally reckoned by the square foot of sur- face measure. When more than 1 inch in thickness, they' are reckoned in proportion to their thickness. Thus, 2000 square feet, 1 inch or less thick = 2000 feet, board measure. 2000 square feet, 1^ inches thick = 3000 feet, board measure.' 2000 square feet, 2 inches thick = 4O0O feet, board measure. And so on. 17. How many feet of boards, 1 inch thick, in a tight board fence, 2 rode long and 4 feet high ? 18. How many feet board measure, in a plank 7 feet long, 2^- inches thick, 14 inches wide at one end and 10 inches wide at the other ? Note. First find the average width which equals one half the sum of the widths at the ends. 19. How many feet, board measure, in a pile of 12 boards, each board being J of an inch thick, 14 feet long, 14 inches wide at one end and 11 inches wide at the other ? PRISMS, CYLIJSTDERS, PYRAMIDS, CONES, AND SPHERES. 347, To find the volume of a prism or a cylinder, Mtdtijjhj the area of the base hj the height. Thus, if the area of the base of a prism is 15 square inches and the height 9 inches, a the vol- A Prism. A Cylinder. ume is 15 X 9, or 135 cubic inches. 348. A pyramid is 1 third of a prism of the same base and height, and a cone is 1 third of a cylinder X J^ of the same base and height, a Pyramid, a Cone. 136 MENSURATION. Hence, to find the volume of a pyramid or a cone, Multijphj the area of the base by the height and divide the product by 3. 349. The surface of a sphere is equivalent to four great circles of the sphere. Hence, ^ when the diameter is given. To find the surface of a sphere, Find the ' area of a great circle of the s'phere and multi- ply it by 4. ^ sp^®^®- 350. A sphere may be regarded as made up of pyramids whose bases taken together form the surface of the sphere and whose height is the radius. Hence, To find the volume of a sphere, Multiply the surface by the radius and divide the loroduct by 3. 351. Examples for the Slate. 20. The base of a prism contains 4 square feet, and its height is 3 feet 6 inches. How many cubic feet does it con- tain ? 21. How many cubic feet of water will a circular tub con- tain, that measures across the top and bottom 2 J- feet, and whose depth is li feet ? Note. First find the area of the bottom. 22. I have a paper weight made of flint glass in the shape of a pyramid, measuring 4 inches square on the base and 5 inches in height. A cubic inch of the glass weighs lf| ounces, what is the weight of the whole pyramid ? 23. How many cubic feet in a conical hay-stack that meas- ures across the base 16 feet, the height being 18 feet ? 24. How many square inches are there in the surface of a foot-ball that is 8 inches in diameter ? How many cubic inches does it contain ? 25. How many cubic feet in a globe 2 feet in diameter ? MISCELLANEOUS EXAMPLES. 137 352. Miscellaneous Examples for the Slate. 26. What will be the cost of a turkey weighing O-j^^- pounds at 20/' a pound? 27. A man owns 240 acres of Western land, which is J of a township. How many acres are there in the township ? 28. A shipmaster has bought f of J of a ship for $3000. What is the value of the ship at the same rate ? 29. After 11 § yards of a piece of linen were sold, there re- mained l^jr yards. How many yards were there at first ? 30. The old shilling of New England was J of a dollar, and the old shilling of New York was J of a dollar. How many New York shillings equalled 3 New England shillings ? 31. A dealer sold 60 bundles of asparagus at the rate of 3 for 25 cents. What did lie receive for them ? 32. By working 10 hours a day a person can do a piece of work in 2j days. In how many days can he do it working 7J hours a day ? 33. How many breadths of carpeting f of a yard wide will reach across a room 18 feet wide ? 34. A steamer going at the rate of 242 miles a day started from New York on Wednesday at noon. On the following Friday at noon another steamer going at the rate of 264 miles a day started on the same course. How far apart will the steamers be at the end of 10 days from Friday noon ? In how many days will they be together ? 35. I bougiit a barrel of sugar for 12/ a pound, and I can now buy the same kind of sugar for 9 J / a pound. What per cent do I lose on the sugar I have left ? 36. If § of a barrel of beef lasts a family 3| weeks, how long will 5 of a barrel last them ? 37. A man put $ 576 in a savings bank October 10, 1877. If it gained 2 J % of interest every six months, what was due April 10, 1878> 38. A lot of paper which cost $ 748.60, being damaged in the store, was sold at 12\ % less than it cost. What did it sell for? APPENDIX. Multiplication and Division Tables (Arts. 101, 137). 1. The pupil sliould be taught to write upon the slate the multiplication and division tables. Either of the following forms is shorter than that given in the book, and may on that account be preferred : 2. Multiplication of 2's and 3's (Art. 101). First Form. Second Form. 2's. 3's. . 2's, 3's, 1 2 2 1 3 3 2x1=: 2 3x1=. 3 2 2 4 2 3 6 2x2= 4 3x2= 6 3 2 6 3 3 9 2x3= 6 3x3= 9 4 2 8 1 4 3 12 2x4= 8 3x4--. 12 6 2 '10 6 3 15 2 X 5 = 10 3x5 = 15 Etc Etc Etc. Etc. The reading in the first form should be, " one two is two " ; " two two's are four," etc. In the second it should be, " two multiplied by one is two''; " two multiplied by two is four," etc. Or the reading in either case may be, "once two is two"; "twice two are four," etc. 3. Division by 2 and 3 (Art. 137). First Form. Second Form. 3. S. 3. 3. 2 2 1 3 3 1 2-2 = 1 ; 3-3=1 2 4 2 3 6 2 4-2 = 2 6-3 = 2 2 6 3 ' 3 9 3 6-2 = 3 9-3 = 3 2 8 4 3 12 4 84-2 = 4 12-3 = 4 2 10 5 : 3 15 5 10 - 2 = 5 15-3 = 5 Etc ] Etc. Etc. Etc. The reading in the first form should be, " two in two, one"; " two's in four, two," etc. In the second it should be, " two divided by two is one"; "four divided by two is two," etc. APPENDIX, 139 Solutions (Art. 104, 140). 4. There are two sorts of practice, both of which the pupil needs to follow at different times during his course in Aritli- metic. One consists in giving answers quickly, as soon as ho hears or reads the problem, without expressing the reason- ing at all ; the other in giving the analysis or reasoning pro- cess according to some approved model. The former secures promptness, the latter accuracy. The teacher will best decide at what time and to what extent his class may need either sort of practice. It may be remarked, however, that set forms of reasoning should not be so rigidly insisted on as to repress originality ; for oftentimes the less formal methods that come from real thinking are better than the more regular processes that come from mere imitation. (Art. 140.) 5. The following form of solution may be preferred to the one given in Art. 140. Solution. — Since 2 cents will buy 1 top, 1 cent will buy 1 half of a top, and 21 cents will buy 21 halves, which equal 10 tops and a half. Ans. 10 tops and a half (or, 10 tops and half enough money to buy another top). (Art. 117.) 6. The pupil learns by this exercise to combine adding with multiplying in a way similar to that required in multipli- cation where there are carryings. The following shows the form of the exercise with 2 for the multiplier : 4 ^ S 7 ^ 1/ 6 ^ § ^ ~i 7 7i 7^ ~s 7^ ?7 Tj Greatest Common Factor (page 80). 7. We have seen in Art. 191 that 6, the greatest common factor of 18 and 24, is the product of 2 and 3, the only prime 140 APPENDIX, factors common to 18 and 24. The greatest common factor of any two or more numbers is the prodicct of all the prime factors which are common to those numbers. Note. The letters g. c. f. are used for greatest common factor. 8. a. What is the greatest common factor of 12, 36 and 42 ? WRITTEN WORK. The prime factors of 12 are 2, 2, and 3. The 12 = 2 X 2 X 3 product of such of these as are common to 36 and ffcf=2x3 = 6 ^^ must be the g. c. f. required. We find that 2 is a factor of both 36 and 42. We find that but one 2 is a factor of 42 ; therefore only one 2 is used as a factor of the g. c. f. We find that 3 is a factor of both 36 and 42 ; therefore 3 is a factor of the g. c. f. Thus the g. c. f. sought is 2x3, equal to 6. To find the greatest common factor of two or more num- bers : Separate one of the numbers into its prime factors, and find the product of such of them as are common to the other numbers. Least Common Multiple (page 80). 9. As any number contains all its prime factors, a multiple of any number must contain all the prime factors of that number. A common multiple of two or more numbers must contain all the prime factors of those numbers, and The least common multiple of two or more numbers is the least number which contains all the prime factors of those numbers. Note. The letters 1. c. m. are used for least common multiple. 10. b. What is the least common multiple of 6, 12 and 15 ? WRITTEN WORK. The least multiple of 6 is 6, which 6 = 2x3 ^*^y he expressed in the form 2x3. 12 = 2 x 2 X 3 ^^^ ^^^^^ multiple of 12 is 12, which I P^ _ o K may be expressed in the form 2x2x3. But in 6 we have already two of the 1. c. m. = 2 X 3 X 2 x 5 = 60 factors (2 and 3) of 12; hence, if we put APPENDIX. 141 ■with the prime factors of 6 the remaining factor (2) of 12, we shall have 2x3x2, which are all the factors necessary to produce the 1. c. m. of 6 and 12. The least iiiultiple of 15 is 15, which may be expressed in the form 3x5. In the 1. c. m. of 6 and 12 we have one of the prime factors (3) of 15 ; hence if we put with the prime factors of 6 and 12 the re- maining factor (5) of 15, we shall have 2x3x2x5, which are all the prime factors necessary to produce tlie 1. c. m. of 6, 12 and 15. The product of these factors is 60, which is the L c m. sought. Note. In finding the least common multiple, the factors. of the given numbers seldom need to be expressed, and the written work may be greatly reduced. Thus, in this example the written work may be simply 1. c. m. = 2x3x2x5= 60. To find the least common multiple of two or more num- bers : Take the pri??ie factors of one of the numbers / with these take such prime factors of each of the other numbers in succession as are not contained in any preceding number, and find the product of all these prime factors. (Art. 250.) 11. /. One of the following forms of solution may be pre- ferred to that given in the body of the book : (1.) If she had put 1 knot into a tassel, she could have made 4 tassels. If she had put \ of a knot into a tassel, she could have made 3 times 4, or 12 tassels ; but since she put f of a knot into a tassel, she could make but \ of 12 tassels, which is 6 tassels. Ans. 6 tassels. (2.) She must have made as many tassels as \ is contained times in 4. WRITTEN WORK. 1 Is contained in 4, 4 times ; \ is contained in 4 ^^g three times 4, or 12 times, and | is contained in 4 -s— = 6. -^ of 12 times, which is 6 times. Ans. 6 tassels. 12. From either form of solution may be derived the follow- ing general rule, which may be preferred to that in Art. 253. To divide by a fraction : Multiply the dividend by the de- 142 APPENDIX. nominator of the cUinsor, and divide the result hy the numera- tor ; or, as more frequently stated, Invert the divisor and jproceed as in multiplication of fractions,' Note. This rule can be derived from the process given in Art. 250, thus ; If, in the i^lace of 12 in the written work, we put the factors which formed 12, we shall have (4x3)-^2, or l|a, in which the expression for the divisor is inverted. 13. General Method of computing Interest (page 123). The teacher is advised to have the pupils use but one method of com- puting interest. Some teachers may prefer the following method to that given in the book. a. What is the interest of % 750 for 2 years 3 months 21 days at 8% ? WRITTEN WORK. The interest of % 750 for 1 year at 8 % SJ^O is g 750 X 0.08, which is $ GO. The inter- est for 2 years is 2 times % 60, or § 120. 0.08 % 60.00 X 2 = $ 120.00 3 months 21 days equal 111 days. The 37 interest of % 750 for 1 year being % 60, the * (^0 X ITt interest for 1 day is -j-J-q of §60, and for '^^ ^ -= % 18.50 111 days it is \\\ of $60, or §18.50, 1138.50 which, added to §120, makes $138.50, 2 the entire interest. Ans. $ 138.50. By this method : 1. To find the interest at any per cent for any number of years^ Midtiply the iirinci'pal hy the rate for 1 year^ and that product hy the numher of years. 2. To find the interest for months and days, Change the months to days, and take as many 360^/^5 of a yearns interest as there are days in the given time. Note. By making the divisor 365 instead of 360, accurate interest can be obtained. APPENDIX. 143 14. "Weights and Measures. United States Money. = 1 cent. = 1 dime. = 1 dollar. = 1 eagle. 10 mills 10 cents 10 dimes ori 100 cents j 10 dollars liiquid Measure. 4 gills = 1 pint. 2 pints = 1 quart. 4 quarts = 1 gallon. Dry Measure. 2 jnnts = 1 quart. 8 quarts = 1 peck. 4 pecks = 1 bushel. Avoirdupois Weight. IG ounces = 1 pound. ^2000 pounds = 1 ton. Also sometimes used. 28 pounds = 1 quarter. 4 quarters = 1 hundi'cdweight. 20 hundredweight = 1 ton (called long ton). Troy Weight. 24 grains = 1 pennyweight. 20 pennyweights = 1 ounce. 12 ounces = 1 pound. As used in 7nixing medicine. 20 grains = 1 scruple ( 3 ). 3 scruples = 1 dram ( 3 ). 8 drams = 1 ounce ( 3 ). 12 ounces = 1 pound. liong Measure. 12 inches = 1 foot. 3 feet = 1 3'ard. 5J yards or IC^ feet = 1 rod. 320 rods or 5280 feet = 1 niilq. Square Measure. 144 square inches = 1 square foot 9 s(piare feet = 1 square yard 301 s(iuare yards or ") , 272i square feet j = 1 square rod. 160 square rods = 1 acre. 640 acres = 1 square mile. Cubic Measure. 1728 cubic inches = 1 cubic foot. 27 cubic feet = 1 cubic yard. 128 cubic feet = 1 co]-d. Time. = 1 minute. = 1 hour. = 1 day. = 1 week. 60 seconds 60 minutes 24 hours 7 days 52 weeks 1 day or ) = j common year, 365 days ) 366 days = 1 leap-year. 100 years = 1 century. Circular or Angular Measure. 60 seconds = 1 minute. 60 minutes = 1 degree. 360 degrees = 1 circumference. Surveyors* Ijong Measure. 7.92 inches = 1 link. 100 links = 1 chain (= 4 rods). 80 chains == 1 mile. Surveyors' Square Measure. 10000 square links = 1 square chain. 10 square chains = 1 acre. [tion. 640 acres = 1 square mile orsec- 36 sections = 1 township. Numbers. 12 units = 1 dozen. 12 dozen = 1 gross. 12 gross = 1 great gross. 20 units = 1 score. 144 APPENDIX. 15. Multiplication Table. 1 1 is 1 2 is 2 3 is 3 4 is 4 2 3 I's I's are 2 " 3 2's 2's are 4 " 6 3's 3's are 6 " 9 4's 4's are 8 " 12 4 I's " 4 2's " 8 3's " 12 4's " 16. 5 I's " 6 2's " 10 3's " 15 4's " 20 6 rs " 6 2's " 12 3's " 18 4's " 24 7 rs " 7 2's " 14 3's " 21 4's " 28 8 I's " 8 2's " 16 3's " 24 4's " 32 9 rs " 9 2's " 18 3's " 27 4's " 36 10 I's " 10 2's " 20 3's " 30 4's " 40 11 I's " 11 2's " 22 3's " 33 4's " 44 12 I's " 12 2's " 24 3's " 36 4's " 48 1 5 is 5 6 is 6 7 is 7 8 is 8 2 5's are 10 6's are 12 7's are 14 8's are 16 3 5's " 15 6's " 18 7's " 21 8's " 24 4 5's " 20 6's " 24 7's " 28 8's " 32 5 5's " 25 6's " 30 7's " 35 8's " 40 6 5's " 30 6's " 36 7's " 42 8's " 48 7 5's " 35 6's " 42 7's " 49 8's " 56 8 5's " 40 6's " 48 7's " 56 8's " 64 9 5's " 45 6's " 54 7's " 63 8's " 72 10 5's " 50 6's " 60 7's " 70 8's " 80 11 5's *•' 55 6's " 66 7's " 77 8's " 88 12 5's " 60 6's " 72 7's " 84 8's " 96 1 9 is 9 1 10 is 10 11 is 11 12 is 12 2 9's are 18 lO's are 20 1 I's are 22 12'sare24 3 9's " 27 lO's " 30 ll's " 33 12's " 36 4 9's " 36 lO's " 40 ll's " 44 12's " 48 5 9's " 45 lO's " 50 ll's " 55 12's " 60 6 9's " 54 lO's " 60 ll's " m 12's " 72 7 9's '' 63 lO's " 70 ll's " 77 12's " 84 8 9's " 72 lO's " 80 ll's " 88 12's " 96 9 9's " 81 lO's " 90 ll's " 99 12's "108 10 9's 9's 9's " 90 " 99 "108 lO's lO's "100 -ll:s "110 12's 12's ^ 12's "120 11 12 "132 "144 ^711711^ iIT7 Id I m^ I 'QAi I UNIVERSITY OF CALIFORNIA LIBRARY \ JUST PULLLHED! WORCESTER'S New Primary Spelling- Book. 96 pjy* Beautifully Illustrated, WORCESTER S New Pronouncing Spelling-Book. 170 pp. The following are some of the distinguishing features of these books : — 1. The lessons are 3>\iort and carr fully graded. 2. The words have i)een juc'iciuusly grouped in classes for phonic drill, and to enable the learner 'o overcome the difficulties more easily. 3. The lessons are varied in kind. 4. Review lesions are introduce'^ -xX convenient intervals. 5. The Primary S pel ling-Book is beautifully illustrated. Only com- mon words are found in the lessons. Short sentences in script are given to be copied. We would also call attention to Adamses Spelling-Book FOB ADVANCEB CLASSES. By WILLIAM T. ADAMS, Formerly Principal of the 'Bowditch School, Boston. Intended for scholars already familiar with the principles of pronouncing and syllabication, and designed to follow the ordinary spelling-book as a practical application of the pupil' knowledge in spelling, dividing, and |>ronouncing the more difficult words in common use. Liberal terms will be made for introduction and exchange. Address the Publishers, WILLIAM WARE & CO., Successors to Brewer & Tileston, 47 Franklin St., Boston.