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 Mental Arithmetic 
 
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 - -umERfcAN • Book- Company 
 
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 IN MEMORIAM 
 FLOR1AN CAJOR1 
 
 M 
 
 K& 
 
 (y 
 

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 PEEFACE. 
 
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 In the solution of problems there are two distinct steps — the selection 
 of the operations, and their performance. Mental and written arith- 
 metic agree in that the choice of operations is determined in the same 
 manner ; they differ in that the operations are wholly mental in the one, 
 while external aids are used in the other. Mental arithmetic should, 
 therefore, embrace all cases in written arithmetic except those which 
 teach how to add, subtract, multiply, and divide large numbers. This 
 arithmetic is intended as a drill-book in which the principles of written 
 arithmetic, except those mentioned above, shall be concisely stated and 
 illustrated. The examples and problems are such as the average mind 
 should be able to solve readily without a pencil. 
 
 He who teaches by the printed page must use every artifice of arrange- 
 ment to make his statements clear and attractive. The placing of prin- 
 ciples and illustrations in parallel columns aids the student to grasp the 
 subject as a whole, since each column may be read independently, and 
 each conveys the same thought in a different manner. The beginning 
 of each subject at the top of a page, the systematic placing of explana- 
 tions and directions under exercises, and the continuous numbering of all 
 the examples in a chapter, aid the teacher to announce and the pupil to 
 understand the requirements. 
 
 In Addition, the combination method is made prominent. The number 
 of seconds which should be required for the solution of each example is 
 stated after each exercise. Since ninety per cent of all arithmetical 
 computation in the work-shop, farm, and counting-room is Addition, this 
 subject cannot be too zealously pressed. Many who have broken the 
 habit, in adding, of saying " 6 and 8 are 14 and 6 are 20," are still saying 
 in subtracting, " 6 from 10 leaves 4 " ; in multiplying, " 9 times 8 are 72, 
 and 4 are 76 " ; and in dividing, " 12 + 5 = 2 and 2 remaining." Special 
 
 3 
 
4 PREFACE. 
 
 stress is laid upon the importance, in performing operations, of dropping 
 all unnecessary words, since the mind reaches results much more rapidly 
 without them. 
 
 In factoring, the introduction of a new conception, that of numbers 
 severally prime to each other, will be appreciated by experts, and cannot 
 fail to benefit learners, because it obviates the cumbersome expression of 
 numbers by their prime factors. Those who, in dividing fractions, have 
 never practiced mentally the method largely used in Europe, will be 
 delighted with the ease by which results can be obtained. 
 
 Attention is called to the presentation of the Metric System. By 
 memorizing the table of submultiples and the table of units, the student 
 acquires the principles of the whole subject, and will only need practice 
 to master it. 
 
 Percentage is taught without rules or formulae, and without the use of 
 the terms base, amount, and difference, although one page is devoted to them 
 after the subject has been completed. The student comes to see clearly 
 that the various exercises in percentage do not need special rules, but are 
 familiar cases slightly modified since the symbol " % " is used instead of 
 hundredths. Interest is taught by the 6 % method and by the modification 
 of this method in general use among bankers. 
 
 The practical exercises " at the lumber yard," " at the carpet store," 
 etc., are to drill the student in methods daily used at such places. Men- 
 suration has been developed with a view of showing the necessity for the 
 existence of the various forms, their relations, and their limitations. 
 
 Few principles are presented, but these few are the keys to all depart- 
 ments of the science. Let it be remembered, that he who relies upon 
 thousands of special rules is but a pygmy beside the giant who can apply 
 a score of general principles to millions of particulars. 
 
 M. A. BAILEY. 
 
 State Normal School of Kansas. 
 
15 IS" 7 
 
 TABLE OF CONTENTS. 
 
 PAGE 
 
 Addition 7 
 
 Combinations — Two Figures 8 
 
 Combinations — Three Figures 14 
 
 Combinations — Four Figures 15 
 
 Addends 16 
 
 Problems 18 
 
 Subtraction 21 
 
 Problems . . 26 
 
 Multiplication 29 
 
 Problems 33 
 
 Division 36 
 
 Precedence of Signs 41 
 
 Parenthesis or Bar 42 
 
 Problems -. .45 
 
 Factoring 48 
 
 Multiplication and Division 52 
 
 Greatest Common Divisor 54 
 
 Least Common Multiple . . . * 55 
 
 Common Fractions 56 
 
 First Conception — An Expression of Division .... 56 
 
 Second Conception — One or More of the Equal Parts of a Unit . 57 
 
 Change of Form — To Higher Terms 58 
 
 Change of Form — To Lower Terms . 59 
 
 Change of Form — To a Whole or Mixed Number . . .60 
 
 Addition and Subtraction 61 
 
 Multiplication — Universal Case 62 
 
 Division — Universal Case 63 
 
 Problems 64 
 
 Decimals 72 
 
 Reduction — Common Fractions to Decimals 74 
 
 Reduction — Decimals to Common Fractions 75 
 
 Per Cent 76 
 
 Short Methods 77 
 
 Denominate Numbers — English Tables 81 
 
 Money, Weights 81 
 
 Long Measure 82 
 
 5 
 
CONTENTS. 
 
 PAGE 
 
 Denominate Numbers — English Tables (Continued). 
 
 Square and Cubic Measures 83 
 
 Capacity, Time 84 
 
 Circular Measure, Counting, Paper, Equivalents .... 86 
 
 Exercises in Tables 87 
 
 Reduction in Same Table 90 
 
 Reduction Table to Table . . 92 
 
 Denominate Numbers — Metric System . . . . . .94 
 
 Practical Questions . 98 
 
 Reduction — Metric 99 
 
 Reduction — English and Metric 100 
 
 Percentage 101 
 
 Reduction 101 
 
 The Operation Directly Stated 102 
 
 Operations to be Determined 104 
 
 Profit and Loss .• . 109 
 
 Commission 112 
 
 Interest 115 
 
 Simple Interest 116 
 
 Trade Discount 123 
 
 True Discount 124 
 
 Bank Discount 125 
 
 Stocks 126 
 
 Practical Exercises 131 
 
 At the Lumber Yard 131 
 
 Measurement of Logs 135 
 
 At the Carpet Store 136 
 
 With the Paper Hanger 137 
 
 Average 138 
 
 Involution and Evolution 139 
 
 Proportion 140 
 
 Mensuration . 141 
 
 One Dimension 141 
 
 Two Dimensions 142 
 
 Three Dimensions 147 
 
 Similarity ' 150 
 
 Miscellaneous 152 
 
 Arithmetical Progression . . . 152 
 
 Geometrical Progression 153 
 
 Specific Gravity 154 
 
 Zero and Infinity 155 
 
 General Review Exercises 156 
 
AMERICAN MENTAL ARITHMETIC. 
 
 ADDITION. 
 
 Addition is indicated by the 
 sign + . 
 
 The numbers to be united are 
 addends; the result, the sum or 
 amount. 
 
 The sign of equality is = . 
 
 The sum of two or more num- 
 bers may be found by counting. 
 
 Addition is a process shorter 
 than counting for finding the 
 sum of numbers. 
 
 A number may be written by 
 the decimal notation or by its 
 addends. 
 
 A number may be spelled by 
 naming its addends, just as a word 
 is spelled by naming its letters. 
 
 A number may be spelled in 
 several different ways. 
 
 Illustration. 
 
 6 + 4 = 10 
 
 read 
 
 6 plus 4 equals 10. 
 
 6 and 4, addends. 
 
 10, sum or amount. 
 
 To find the sum of 6 and 
 4 by counting. 
 
 Counting to 6 and mak- 
 ing a mark at each count, 
 llllll; counting to 4 and 
 making a mark at each 
 count, llllll III/; count- 
 ing the result, we have 10. 
 
 Ten may be written 
 
 in 5 6 7 8 9 
 
 10; or 5 , 4 , a , 2 , r 
 
 Ten, as written above, is 
 spelled five five, six four, seven 
 three, eight two, or nine one. 
 
AMERICAN MENTAL ARITHMETIC. 
 
 § 1. Combinations — Two Figures. 
 
 There are 45 different combinations of figures taking two 
 at a time, viz. : 
 
 999999999888888887777 77 
 9' 8' 7' 6' 5' 4' 3' 2' 1' 8' 7' 6' 5' 4' 3' 2' 1' f 6' 5' 4' 3' 2' 
 76666665555544443 33221 
 1' 6' 5' 4' 3' 2' 1' 5' 4' 3' 2' 1' 4' 3' 2' 1' 3' 2' 1' 2' 1' 1' 
 
 These combinations are shown in the following table, and 
 should be thoroughly memorized. 
 
 The 
 Number. 
 
 Combinations. 
 
 The 
 Number. 
 
 Combinations. 
 
 The 
 Number. 
 
 Combinations. 
 
 2 
 
 1 
 1 
 
 8 
 
 4 5 6 7 
 4' 3' 2' 1 
 
 14 
 
 7 8 9 
 T 6' 5 
 
 3 
 
 2 
 1 
 
 9 
 
 5 6 7 8 
 4' 3' 2' 1 
 
 15 
 
 8 9 
 
 7' 6 
 
 4 
 
 2 3 
 
 2' 1 
 
 10 
 
 5 6 7 8 9 
 5' 4' 3' 2' 1 
 
 16 
 
 8 9 
 
 8' 7 
 
 5 
 
 3 4 
 
 2' 1 
 
 11 
 
 6 7 8 9 
 5' 4' 3' 2 
 
 17 
 
 9 
 
 8 
 
 6 
 
 3 4 5 
 
 3' 2' 1 
 
 12 
 
 6 7 8 9 
 6' 5' 4' 3 
 
 18 
 
 9 
 9 
 
 7 
 
 4 5 6 
 3' 2' 1 
 
 13 
 
 7 8 9 
 6' 5' 4 
 
 
 
 1 2 2 3 
 
 Memorize thus : 2, -. (one one) ; 3, .. (two one) ; 4, „, n (iwo £w?o, or, ^ree 
 
 u»e) ; etc. 
 
 1. What are the combinations whose sum is 10 ? 
 
 2. What are the combinations whose sum is 12? 18? 9? 4? 
 
 3. What are the combinations whose sum is 17? 5? 11? 3? 
 
 4. What are the combinations whose sum is 2? 16? 6? 15? 
 
 5. What are the combinations whose sum is 7 ? 14? 8? 13 r' 
 
 State the answers without reading the questions aloud. 
 5 6 7 8 9 
 
 Ex. 1. 
 
 5' 4' 
 
 2' r 
 
ADDITION 9 
 
 Name the combinations whose sum is i 
 
 6. Two, three, four, five, six, seven, eight, nine, ten, 
 eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, 
 eighteen. 
 
 7. Eighteen, seventeen, sixteen, fifteen, fourteen, thirteen, 
 twelve, eleven, ten, nine, eight, seven, six, five, four, three, two. 
 
 8. Ten, twelve, fifteen, two, six, eleven, nine, sixteen, 
 three, thirteen, four, seventeen, five, eighteen, seven, fourteen, 
 eight. 
 
 State the answers without reading the questions aloud. 
 
 i? .7 9 9 8 9 8 9 . 
 ■ bx - 7> ; 8 ; 8' 7 ; 7' 6' etc ' 
 
 Read the sums : 
 
 Q 97847678632467 
 y - 9' 6' 5' 3' 2' 5' 8' 8' 4' f 8' 4' 9' 6' 
 
 _ n 56972484458377 
 10 " 5' 3' 2' 2' 6' 4' 8' 1' 3' 8' T 3' 4' 5* 
 
 ..44389612794624 
 **: 8' 6' 5' 9' 9' 8' 1' 1' 7' 5' 9' 7' 2' 2* 
 
 Ex. 9. 18, 13, 13, 7, 9, etc. Do not say 9 and 9 are 18. 
 
 Read the sums : 
 
 12. 9 + 9, 3 + 2, 6-}-6, 2 + 2, 5 + 5, 4 + 3, 7 + 6, 9 + 8, 6+5, 
 2 + 1, 8 + 8, 7 + 7, 1 + 1, 5 + 4, 8 + 7, 4 + 4, 3 + 3. 
 
 13. 9 + 7, 6 + 2, 9 + 4, 8 + 6, 9 + 6, 7 + 2, 6 + 4, 5 + 1, 9 + 5, 
 7 + 5,8 + 4,5 + 3,8 + 5, 5 + 2, 7 + 1, 8 + 3, 7 + 3, 8 + 2, 6 + 3, 
 
 6 + 1,4 + 2,9 + 3,7 + 4,4 + 1. 
 
 14. 8 + 9,7 + 6,4 + 5,2 + 3,7 + 8, 8 + 4, 7 + 9, 7 + 7, 8 + 4, 
 
 7 + 2, 8 + 7, 9 + 9, 8 + 8. 
 
 Do not say 9 plus 9 equals 18, but state sums directly. 
 Ex. 12. 18, 5, 12, etc. 
 
10 AMERICAN MENTAL ARITHMETIC. 
 
 To 1, 3, 6, 8, 9, 4, 7, 5, 2 
 
 15. Add 9. 18. 
 
 16. Add 8. 19. 
 
 17. Add 7. 20. 
 Ex. 15. 10, 12, 15, 17, 18, etc. 
 
 State the results : 
 
 Add 6. 
 
 21. Add 3. 
 
 Add 5. 
 
 22. Add 2, 
 
 Add 4. 
 
 23. Add 1, 
 
 24. 
 
 8 + 9 = 
 
 25. 
 
 7 + 8 = 
 
 26. 
 
 8 + 4 = 
 
 27. 
 
 Q + 5 = 
 
 28. 
 
 5 + 9= 
 
 29. 
 
 3 + 7 = 
 
 30. 
 
 6 + 3 = 
 
 31. 
 
 4 + 6 = 
 
 32. 6 + 9 = 
 
 40. 9 + 5 = 
 
 33. 4 + 7 = 
 
 41. 3 + 9 = 
 
 34. 6 + 8 = 
 
 42. 9 + 2 = 
 
 35. 9 + 9 = 
 
 43. 6 + 6 = 
 
 36. 6 + 7 = 
 
 44. 8 + 8 = 
 
 37. 7 + 5 = 
 
 45. 4 + 9 = 
 
 38. 8 + 5 = 
 
 46. 3 + 8 = 
 
 39. 7 + 9 = 
 
 47. 7 + 7 = 
 
 nd7? 6and5? 
 
 4 and 9? 3 and 7? 
 
 48. How many are 9 and 7 
 
 5 and 8? 7 and 2? 8 and 6? 
 
 49. How many are 7 and 6 ? 8and3? 9and5? 8and8? 
 
 6 and 2? 4 and 5? 5 and 3? 7 and 9? 8 and 8? 4 and 
 9? 4 and 3? 
 
 50. How many are 9 and 9? 6 and 4 ? 5 and 7 ? 8 and 9 ? 
 3 and 6 ? 5 and 2 ? 9 and 6 ? 8 and 7 ? 7 and 7 ? 6 and 
 6? 5 and 5? 
 
 51. Read as rapidly as possible 12, 16, 14, 13, 11, 18, 14, 
 16, 17. 
 
 52. Read as rapidly as possible 
 
 698759989 
 6' 7' 6' 6' 6' 9' 5' 8' 8' 
 
 The student has mastered these combinations when he can read numbers 
 as expressed in Ex. 52 as rapidly as he can read numbers as expressed in 
 Ex. 51. 
 
ADDITION. 11 
 
 § 2. In General. 
 
 Declare the sums: 
 
 — 8 28 38 58 78 88 _ 8 48 68 98 78 58 
 
 53 ' 9' 9' 9' 9' 9' 9* 56 ' 6' 6' 6' 6' 6' 6* 
 
 9 89 39 49 59 79 
 4' 4' 4' 4' 4' 4* 
 
 8 48 88 78 58 28 
 8' 8' 8' 8' 8' 8' 
 
 54. _, _ , „y -) - > „ • 57. 
 
 RR 9 89 29 79 39 69 _ Q 
 
 55 ' 9' 9' 9' 9' 9' 9* 58 ' 
 
 Ex. 53. 17, 37, 47, etc. 
 
 /What is the right hand figure of the sum, 
 
 59. When 9 is added to a number ending in 9? 8? 3? 7? 
 4? 2? 5? 6? 1? 
 
 60. When 8 is added to a number ending in 8? 2? 6? 3? 
 5? 9? 1? 4? 7? 
 
 61. When 7 is added to a number ending in 1? 3? 5? 7? 
 9? 2? 4? 6? 8? 
 
 62. When 6 is added to a number ending in 6? 3? 8? 2? 
 4? 1? 5? 7? 9? When 5 is added? 
 
 63. When 4 is added to a number ending in 2? 6? 4? 8? 
 1? 9? 5? 7? 3? When 3 is added? 
 
 64. When 2 is added to a number ending in 3? 5? 1? 7? 
 4? 9? 2? 6? 8? When 1 is added? 
 
 Ex. 59. 8, 7, 2, 6, 3, 1, 4, 5, 0. 
 
 Beginning with 1, count as rapidly as possible to about 100 : 
 
 65. By 9. 68. By 6. 71. By 3. 
 
 66. By 8. 69. By 5. 72. By 2. 
 
 67. By 7. 70. By 4. 73. By 1. 
 
 Ex. 65. 1, 10, 19, 28, 37, etc. 
 
12 AMERICAN MENTAL ARITHMETIC. 
 
 Add: 
 
 74. 3, 7, 6, 8, 9, 2, 7, 8, 5, 1, 2, 3, 4, 5, 6, 7, 8„9, 5, 9, 3. 
 
 75. 4, 9, 8, 1, 1, 3, 9, 9, 7, 3, 4, 5, 6, 7, 8, 9, 6, 8, 7, 8, 5. 
 
 76. 6, 8, 2, 4, 1, 3, 5, 7, 9, 1, 4, 7, 2, 6, 1, 6, 8, 9, 9, 8, 6. 
 
 77. 7, 8, 9, 9, 8, 7,' 6, 5, 4, 4, 5, 6, 3, 8, 7, 7, 8, 3, 7,- 4, 7. 
 
 78. 5, 3, 7, 6, 2, 4, 9, 8, 7, 4, 5, 9, 2, 9, 8, 7, 8, 6, 3, 8, 9. 
 
 79. 2, 7, 5, 3, 9, 6, 8, 8, 8, 5, 6, 2, 5, 6, 8, 3, 4, 7, 6, 6, 9. 
 so. 8, 7, 7, 6, 6, 5, 5, 4, 3, 2, 1, 6, 5, 4, 3, 3, 5, 7, 9, 8, 7. 
 
 81. 9, 6, 3, 4, 2, 4, 6, 8, 9, 8, 7, 6, 3, 1, 3, 5, 9, 9, 8, 4, 5. 
 
 82. 4, 3, 5, 8, 7, 6, 5, 1, 3, 2, 4, 5, 7, 8, 2, 4, 5, 7,6, 8, 8. 
 
 83. 3, 5, 7, 2, 2, 4, 6, 7, 8, 3, 6, 9, 9, 8, 7, 6, 3, 2, 5, 1, 7. 
 
 84. 8, 2, 4, 5, 6, 2, 7, 8, 6, 4, 5, 9, 8, 3, 4, 5, 3, 6, 9, 4, 2. 
 
 85. 9, 9, 8, 7, 3, 4, 6, 1, 5, 4, 2, 3, 3, 9, 1, 7, 8, 9, 7, 8, 2. 
 
 86. 8, 5, 6, 9, 4, 3, 3, 4, 6, 1, 4, 3, 1, 6, 9, 9, 6, 2, 3, 2, 4. 
 
 87. 8, 2, 4, 8, 1, 5, 1, 5, 8, 3, 3, 1, 9, 5, 4, 8, 6, 3, 5, 8, 9. 
 
 88. 1, 5, 8, 6, 3, 5, 1, 9, 3, 3, 7, 7, 2, 5, 9, 2, 8, 6, 6, 2, 2. 
 
 Ex. 74. 3, 10, 16, 24, 33, 35, etc. Allow 9 seconds for each example. 
 
 89. On the next page find the sum of the columns giving 
 the population of the U. S. in 1850. 
 
 90. In the same manner find the sum of the columns for 
 1860. 
 
 91. Find the sum of the columns for 1870. 
 
 92. Find the sum of the columns for 1880. 
 
 93. Find the sum of the columns for 1890. 
 
 Ex. 89. 9, 16, 20, 25, 31, . . . 166 ; 5, 13, 22, 24, 28, . . . 171 ; etc. 
 Allow 19 seconds for a column. 
 

 
 ADDITION. 
 
 
 13 
 
 U.S. 
 
 1850 
 
 1860 
 
 1870 
 
 1880 
 
 1890 
 
 N.Y. 
 
 3,097,394 
 
 3,880,735 
 
 4,382,759 
 
 5,082,871 
 
 5,997,853 
 
 Penn. 
 
 2,311,786 
 
 2,906,215 
 
 3,521,951 
 
 4,282,891 
 
 5,258,014 
 
 111. 
 
 851,470 
 
 1,711,951 
 
 2,539,891 
 
 3,077,871 
 
 3,826,351 
 
 Ohio 
 
 1,980,329 
 
 2,339,511 
 
 2,665,260 
 
 3,198,062 
 
 3,672,316 
 
 Mo. 
 
 682,044 
 
 . 1,182,012 
 
 1,721,295 
 
 2,168,380 
 
 2,679,184 
 
 Mass. 
 
 994,514 
 
 1,231,066 
 
 1,457,351 
 
 1,783,085 
 
 2,238,943 
 
 Tex. 
 
 212,592 
 
 604,215 
 
 818,579 
 
 1,591,749 
 
 2,235,523 
 
 Ind. 
 
 988,416 
 
 1,350,428 
 
 1,680,637 
 
 1,978,301 
 
 2,192,404 
 
 Mich. 
 
 397,654 
 
 749,113 
 
 1,184,059 
 
 1,636,937 
 
 2,093,889 
 
 Iowa 
 
 192,214 
 
 674,913 
 
 1,194,020 
 
 1,624,615 
 
 1,911,896 
 
 Ky. 
 
 982,405 
 
 1,155,684 
 
 1,321,011 
 
 1,648,690 
 
 1,858,635 
 
 Ga. 
 
 906,185 
 
 1,057,286 
 
 1,184,109 
 
 1,542,180 
 
 1,837,353 
 
 Tenn. 
 
 1,002,717 
 
 1,109,801 
 
 1,258,520 
 
 1,542,359 
 
 1,767,518 
 
 Wis. 
 
 305,391 
 
 775,881 
 
 1,054,670 
 
 1,315,491 
 
 1,686,880 
 
 Va. * 
 
 1,421,661 
 
 1,596,318 
 
 1,225,163 
 
 1,512,565 
 
 1,655,980 
 
 N.C. 
 
 869,039 
 
 992,622 
 
 1,071,361 
 
 1,399,750 
 
 1,617,947 
 
 Ala. 
 
 771,623 
 
 964,201 
 
 996,992 
 
 1,262,505 
 
 1,513,017 
 
 N.J. 
 
 489,555 
 
 672,035 
 
 906,096 
 
 1,131,116 
 
 1,444,933 
 
 Kans. 
 
 
 107,206 
 
 364,399 
 
 996,096 
 
 1,427,096 
 
 Minn. 
 
 6,077 
 
 172,023 
 
 439,706 
 
 780,773 
 
 1,301,826 
 
 Miss. 
 
 606,526 
 
 791,305 
 
 827,922 
 
 1,131,597 
 
 1,286,600 
 
 Cal. 
 
 92,597 
 
 379,994 
 
 560,247 
 
 864,694 
 
 1,208,130 
 
 S.C. 
 
 668,507 
 
 703,708 
 
 705,606 
 
 995,577 
 
 1,151,149 
 
 Ark. 
 
 209,897 
 
 435,450 
 
 484,471 
 
 802,525 
 
 1,128,179 
 
 La. 
 
 517,762 
 
 708,002 
 
 726,915 
 
 939,946 
 
 1,118,587 
 
 Nebr. 
 
 
 28,841 
 
 122,993 
 
 452,402 
 
 1,058,910 
 
 Md. 
 
 583,034 
 
 687,049 
 
 780,894 
 
 934,943 
 
 1,042,390 
 
 W.Va. 
 
 
 
 442,014 
 
 618,457 
 
 762,794 
 
 Conn. 
 
 370,792 
 
 460,147 
 
 537,454 
 
 622,700 
 
 746,258 
 
 Me. 
 
 583,169 
 
 628,279 
 
 626,915 
 
 648,936 
 
 661,086 
 
 Colo. 
 
 
 34,277 
 
 39,864 
 
 194,327 
 
 412,198 
 
 Fla. 
 
 87,445 
 
 140,424 
 
 187,748 
 
 269,493 
 
 391,422 
 
 N.H. 
 
 317,976 
 
 326,073 
 
 318,300 
 
 346,991 
 
 376,530 
 
 R.I. 
 
 147,545 
 
 174,620 
 
 217,353 
 
 276,531 
 
 345,506 
 
 Vt. 
 
 314,120 
 
 315,098 
 
 330,551 
 
 332,286 
 
 332,422 
 
 Oreg. 
 
 13,294 
 
 52,465 
 
 90,923 
 
 174,768 
 
 313,767 
 
 D.C, 
 
 51,687 
 
 75,080 
 
 131,700 
 
 177,624 
 
 230,392 
 
 Dei. 
 
 91,532 
 
 112,216 
 
 125,015 
 
 146,608 
 
 168,493 
 
 Nev. 
 
 
 6,857 
 
 42,491 
 
 62,266 
 
 45,761 
 
 Best 
 
 72,927 
 
 184,497 
 
 311,030 
 
 606,819 
 
 1,621,118 
 
14 AMERICAN MENTAL ARITHMETIC. 
 
 § 3. Combinations — Three Figures. 
 Declare the sums : 
 
 01888757461111374922 
 
 94. 9, 7, 8, 7, 3, 3, 4, 8, 4, 7, 1, 1, 5, 7, 8, 7, 9, 9, 6, 8. 
 
 29223139588999979999 
 
 95. 
 
 96. 
 
 97. 
 
 98. 
 
 99. 
 
 3 
 
 1 
 
 5 
 
 8 
 
 C» 
 
 3 
 
 2 
 
 1 
 
 1 
 
 2 
 
 5 
 
 6 
 
 8 
 
 8 
 
 8 
 
 5 
 
 9 
 
 1 
 
 1 
 
 1 
 
 7, 
 
 7, 
 
 r>. 
 
 ( A 
 
 6, 
 
 4, 
 
 2, 
 
 4, 
 
 6, 
 
 5, 
 
 7, 
 
 7, 
 
 8, 
 
 8, 
 
 8, 
 
 5, 
 
 9, 
 
 4, 
 
 8,9. 
 
 8 
 
 9 
 
 9 
 
 9 
 
 9 
 
 4 
 
 6 
 
 8 
 
 8 
 
 9 
 
 8 
 
 9 
 
 8 
 
 9 
 
 5 
 
 3 
 
 3 
 
 9 
 
 9 
 
 9 
 
 4 
 
 5 
 
 9 
 
 8 
 
 3 
 
 L> 
 
 2 
 
 2 
 
 2 
 
 3 
 
 3 
 
 3 
 
 4 
 
 4 
 
 6 
 
 3 
 
 5 
 
 4 
 
 6 
 
 5 
 
 8, 
 
 6, 
 
 6, 
 
 8, 
 
 5, 
 
 5, 
 
 2, 
 
 3, 
 
 5, 
 
 5, 
 
 3, 
 
 9, 
 
 4, 
 
 8, 
 
 8, 
 
 4, 
 
 9, 
 
 4, 
 
 9, 
 
 8. 
 
 <s 
 
 8 
 
 5 
 
 3 
 
 9 
 
 8 
 
 9 
 
 7 
 
 5 
 
 3 
 
 4 
 
 9 
 
 4 
 
 9 
 
 9 
 
 6 
 
 9 
 
 7 
 
 9 
 
 9 
 
 3 
 
 7 
 
 4 
 
 6 
 
 7 
 
 4 
 
 7 2 
 
 7 
 
 1 
 
 1 
 
 1 
 
 3 
 
 4 
 
 4 
 
 4 
 
 3 
 
 4 
 
 4 
 
 5 
 
 3, 
 
 8, 
 
 5, 
 
 8, 
 
 8, 
 
 4, 
 
 7, 
 
 4, 
 
 9, 
 
 6, 
 
 5, 
 
 4, 
 
 7, 
 
 5, 
 
 6, 
 
 6, 
 
 5, 
 
 7, 
 
 5, 
 
 7. 
 
 (5 
 
 7 
 
 5 
 
 8 
 
 8 
 
 6 
 
 9 
 
 9 
 
 9 
 
 6 
 
 6 
 
 5 
 
 9 
 
 9 
 
 9 
 
 8 
 
 8 
 
 8 
 
 8 
 
 7 
 
 3 
 
 1 
 
 6 
 
 6 
 
 1 
 
 6 
 
 2 
 
 3 
 
 1 
 
 1 
 
 1 
 
 4 
 
 1 
 
 1 
 
 3 
 
 4 
 
 5 
 
 2 
 
 2 
 
 1 
 
 6, 
 
 1, 
 
 6, 
 
 7, 
 
 1, 
 
 6, 
 
 3, 
 
 8, 
 
 1, 
 
 8, 
 
 2, 
 
 4, 
 
 1, 
 
 1, 
 
 3, 
 
 C>, 
 
 6, 
 
 2, 
 
 2, 
 
 3. 
 
 6 
 
 1 
 
 7 
 
 7 
 
 2 
 
 8 
 
 3 
 
 9 
 
 3 
 
 8 
 
 2 
 
 9 
 
 4 
 
 5 
 
 3 
 
 6 
 
 6 
 
 3 
 
 4 
 
 3 
 
 3 
 
 1 
 
 3 
 
 5 
 
 2 
 
 1 
 
 2 
 
 1 
 
 1 
 
 3 
 
 1 
 
 4 
 
 1 
 
 5 
 
 2 
 
 5 
 
 1 
 
 6 
 
 1 
 
 4 
 
 7, 
 
 3, 
 
 6, 
 
 6, 
 
 3, 
 
 2, 
 
 2, 
 
 2, 
 
 2, 
 
 6, 
 
 2, 
 
 6, 
 
 4, 
 
 5, 
 
 2, 
 
 5, 
 
 3, 
 
 6, 
 
 1. 
 
 7. 
 
 7 
 
 4 
 
 9 
 
 7 
 
 4 
 
 4 
 
 2 
 
 6 
 
 3 
 
 8 
 
 5 
 
 7 
 
 4 
 
 7 
 
 5 
 
 8 
 
 5 
 
 6 
 
 7 
 
 7 
 
 51432222111211222112 
 
 100. 5, 1, 5, 5, 6, 5, 4, 4, 2, 4, 3, 6, 2, 7, 4, 7, 2, 6, 3, 3. 
 66776664876 8^ 88778985 
 22117233214 2 3 3313114 
 
 101. 4, 6, 2, 3, 7, 5, 3, 3, 3, 5, 3, 3, 4, 6, 4, 6, 5, 4, 5, 5. 
 879917 979778 8 7 9 7 5 9 8 6 
 7464 5 978733 49 8 5 9 6 3 4 8 
 
 102. 7, 8, 6, 3, 2, 9, 6, 9, 4, 8, 4, 7, 2, 3, 4, 7, 6, 4, 5, 5. 
 8 9718431279814926347 
 16892585232796378799 
 
 103. 4, 7, 2, 8, 9, 4, 7, 4, 1, 4, 4, 4, 5, 4, 3, 1. 9, 4, 6, 5. 
 58177363932789892134 
 
 9 9 
 Ex. 94. 20, 17, 18, 17, 14, 11, 12, etc. Look upon q as 18 ; then 9 
 
 »> 
 
 18 
 appears 2 ; say 20. Speak no words except the sums. Allow 14 seconds 
 
 for each example. 
 
ADDITION. 15 
 
 § 4. Combinations — Four Figures. 
 
 Read the sums : 
 
 99999999999999999999 
 „^ 99999999999999999999 
 104 ' 9' 5' 6' 2' 5' V 9> 5' 2' 8' 8' 6' 8' 8' V 6' 4' 7' 8' 4' 
 
 94625621179223132454 
 
 
 9 9 9 9 9 9 
 
 9 9 
 
 9 9 
 
 9 9 9 
 
 9 9 9 9 9 9 9 
 
 
 8 8 8 8 8 8 
 
 8 8 
 
 8 8 
 
 8 8 8 
 
 8 8 8 8 8 8 8 
 
 105. 
 
 V 5' 9' 2' 7' 6' 
 
 4' 3' 
 
 V 6' 
 
 4' 7' 8' 
 
 6' 8> 5' 9' 8' V 9* 
 
 
 15 7 17 4 
 
 4 3 
 
 5 2 
 
 113 
 
 16 2 3 7 2 4 
 
 
 12 2 3 3 4 
 
 5 3 
 
 4 9 
 
 8 7 6 
 
 9 8 7 6 9 8 7 
 
 
 112 2 3 3 
 
 5 1 
 
 1 2 
 
 3 4 5 
 
 3 4 5 6 4 5 6 
 
 106. 
 
 9' 9' 8' 5' 8' 8' 
 
 6' 8' 
 
 7' 9' 
 
 4' 2' 4' 
 
 6' 6' 7' 5' 3' 8' 5* 
 
 
 9 8 8 5 8 7 
 
 6 3 
 
 5 1 
 
 12 3 
 
 4 12 2 113 
 
 
 9 8 9 8 9 9 
 
 9 8 
 
 7 6 
 
 5 8 7 
 
 6 5 7 6 5 4 6 
 
 
 6 7 7 8 8 9 
 
 1 2 
 
 3 4 
 
 5 12 
 
 3 4 12 3 4 1 
 
 107. 
 
 V 4' 3' 4' 9' 8' 
 
 1' 6' 
 
 V 8' 
 
 9' 8' 7' 
 
 6' 2' 6' 6' V 8' 8* 
 
 
 3 2 2 4 11 
 
 1 6 
 
 6 5 
 
 4 5 7 
 
 5 15 14 18 
 
 
 12 3 3 6 1 
 
 5 9 
 
 4 8 
 
 3 7 2 
 
 6 2 19 7 5 3 
 
 
 8 9 6 5 8 2 
 
 6 1 
 
 5 9 
 
 4 8 3 
 
 7 4 3 19 7 5 
 
 108. 
 
 5' 6' 7' 8' 9' 3' 
 
 V 2> 
 
 6' 1' 
 
 5' 9' 4> 
 
 8 ? 6' 5' 3' V 9' 7' 
 
 
 3 6 9 7 5 4 
 
 8 3 
 
 7 2 
 
 6 15 
 
 9 8 7 5 3 19 
 
 
 3 8 4 6 6 8 
 
 8 7 
 
 8 9 
 
 7 9 5 
 
 6 8 3 4 9 3 9 
 
 
 6 4 8 2 5 2 
 
 5 7 
 
 8 1 
 
 2 4 4 
 
 7 5 3 4 9 8 6 
 
 109. 
 
 2' 1' 2' 8' 4' 7' 
 
 2' 7' 
 
 8> 8' 
 
 8' V 8' 
 
 2' 9' 6' 8' 2' 7 ? 2' 
 
 
 5 7 6 4 7 3 
 
 5 7 
 
 8 3 
 
 3 6 3 
 
 12 6 8 15 1 
 
 
 7 7 7 7 7 7 
 
 7 7 
 
 7 7 
 
 7 7 7 
 
 7 7 7 7 7 7 7 
 
 
 6 6 5 5 6 6 
 
 5 4 
 
 3 2 
 
 16 6 
 
 5 5 4 3 2 18 
 
 110. 
 
 1' 2' 3' 4' 5' 6' 
 
 7> 8' 
 
 9' 5' 
 
 4' 3' 2' 
 
 V 9' 5' 6' V 8' 9" 
 
 
 9 8 7 6 5 4 
 
 3 2 
 
 1 9 
 
 8 7 6 
 
 5 4 3 2 19 8 
 
 Ex. 104. 36, 27, 30, 22, 28, 31, 29, etc. Look upon g as 18; then 
 
 9 
 
 g appears as :S ; say 36. Speak no word except the sums. Allow 18 
 
 9 
 
 seconds for each example. 
 
16 AMERICAN MENTAL ARITHMETIC. 
 
 § 5. Addends. 
 Add: 
 
 111. 18, 17, 11, 16, 15, 14, 13, 12, 19, 20, 17, 19, 18, 17, 19. 
 
 112. 19, 19, 18, 18, 17, 17, 16, 16, 15, 15, 14, 14, 13, 13, 12. 
 
 113. 21, 22, 23, 16, 15, 18, 11, 10, 19, 24, 16, 18, 12, 14, 15. 
 
 114. 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25. 
 lis. 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40. 
 
 116. 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25. 
 
 117. 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10. 
 
 118. 40, 11, 39, 12, 38, 13, 37, 14, 36, 15, 35, 16, 34, 17, 33. 
 
 Ex. 111. 35, 46, 62, 77, 91, etc. 
 
 A glance determines whether the sum of the units is more 
 
 than 9 or less than 10. If more than 9, we increase the sum 
 
 of the tens by 1 ; if less than 10, we take sum of the tens. 
 
 Ex. 111. 18 + 17 ; the sum of the units is more than 10 ; we increase 
 the sum of the tens by 1 and say 35 ; 35 + 11 ; the sum of the units is less 
 than 9 ; we take the sum of the tens and say 46, etc. 
 
 Find the sum : 
 
 119. 78 + 94. 122. 74 + 92. 125. 63 + 18. 
 
 120. 86 + 75. 123. 83 + 75. 126. 94 + 87. 
 
 121. 73 + 68. 124. 43 + 58. 127. 63 + 25. 
 
 Ex. 119. 172. 
 
 Beginning with 1 count to about 200 : 
 
 128. By 19. 131. By 16. 134. By 13. 
 
 129. By 18. 132. By 15. 135. By 12. 
 
 130. By 17. 133. By 14. 136. By 11. 
 Ex. 128. 1, 20, 39, 58, 77, 96, etc. 
 
ADDITION. 17 
 
 Add: 
 
 '' 2469477731 ... 7542468273 
 
 137. r f 5 , 5 , 2 , 2 , f 6 , g , 9 - 141. x , 5 , 4 , 6 , 4 , 9 , 4 , g, g , g . 
 
 , 00 3944969726 _._ 1111289986 
 
 138. 2 , g, 2 , 7 , g, g, g, 2 , g , 3. 142. g , g,. g , 9 , 4 , r g , 4 , 2 , 3- 
 
 1M 5 2 5 4 U 5 5 6 7 ... 4653745689 
 
 139 - 9' 6' t 9' 1* f 5' 5' 6' 3' 143 * 1' 2' 4' 5' 4' 3' 5' T 8' 6* 
 
 , An 1377392611 1AA 2722112468 
 
 140. 2 , ^ 4 , 7 , 3, 7 , g , 6 , g , g . 144. 6 , 2 , 8 , 4 , 1 , 7 , 3 , g , 7 , g . 
 
 6391462135 2417232157 
 
 145. 7, 8, 7, 9, 8, 3, 1, 1, 4, 8. 148. 8, 6, 2, 4, 3, 6, 8, 9, 2, 2. 
 754725 5 863 4428928649 
 
 1924538111 2548324554 
 
 146. 5, 6, 7, 7, 4, 2, 1, 8, 2, 1. 149. 6, 2, 6, 7, 5, 1, 2, 6, 7, 9. 
 9827492816 4548897294 
 
 3446937111 5725644127 
 
 147. 5, 8, 4, 1, 6, 1, 9, 5, 9, 5. 150. 2, 8, 7, 4, 4, 5, 1, 7, 1, 2. 
 1345221116 1766914895 
 
 2694923647 3132322133 
 
 ,-, 1417975952 __, 2122278868 
 
 151 ' 9' 5' 5' 1' 9* 6' 2' 5' 5' 8* 153 * 4' 3' 3' 8' 6' 9' 5' 8' 2' 2* 
 
 3519261793 1984865957 
 
 5892953941 4175357282 
 
 -„ 9786365882 ncA 8166755514 
 
 152 ' 7' 3' 4' 4' 7' 1' 6' 2' 7' 5* 154 ' 6' 6' 7' 2' 4' 6' 9' 9' 1 ? 7* 
 
 4399931257 9833767115 
 
 Ex. 137. 9, 20, 31, 45, 51, 60, 74, etc, Look at J, say 9 ; at 4 20 ; at ®, 
 31 ; at k, 45 ; etc. 4 17 
 
 Ex. 154. 27, 43, 66, 82, etc. Look at 5, say 27 ; at g, 43 ; at fj, 66 ; etc. 
 
 9 8 3 
 
 By reading several figures at a glance and combining as 
 in this section, the columns on page 13 may be added in 5 
 seconds each. 
 
 AM. MENT. AR. — 2 
 
18 AMERICAN MENTAL ARITHMETIC. 
 
 § 6. Problems. 
 
 Declare the answer to each as quickly as possible with- 
 out reading the problem aloud and before explaining. 
 
 If required to explain, avoid repetitions and unneces- 
 sary words. 
 
 155. A paid 16^ for a book, 15? Ans , 36 ^ He pa id in 
 for a slate, and 5^ for a pencil ; how all the sum of 16^, 15^, and 
 much did he pay in all ? Explain. ^ or 36 ^ 
 
 156. Jane bought some apples for 10^, some peaches for 
 18^, some plums for 20^, and an orange for 8^; how much 
 did she pay for all ? 
 
 157. John has 28 marbles in one bag, 16 in another, 14 in 
 another, 35 in another, and 19 in another; how many has 
 he in all ? 
 
 158. There are 9 birds in one flock and 27 in another ; 
 how many are there in both ? Explain. 
 
 159. One day I walked 5 miles and the next day 14 miles ; 
 how far did I walk in both days ? 
 
 160. A man has horses in 3 pastures : in the first 9, in the 
 second 11, and in the third 13; how many has he in all? 
 Explain. 
 
 161. A baker sold 36 loaves of bread on Monday, 30 on 
 Tuesday, 27 on Wednesday, 34 on Thursday, 25 on Friday, 
 and 40 on Saturday ; how many loaves did he sell during the 
 week? 
 
 162. A man has 5 baskets of eggs : in the first basket there 
 are 24 eggs, in the second 36, in the third 18, in the foarth 
 12, and in the fifth 16 ; how many has he in the five bas- 
 kets? 
 
ADDITION. 19 
 
 163. 38 years, 29 years, 10 years, 19 years, 23 years, and 
 14 years are how many years in all ? 
 
 164. Walter saw three flocks of prairie chickens ; the first 
 contained 28 chickens, the second 19, and the third 33; 
 how many chickens did he see ? 
 
 165. I bought on account : a cabbage for 10 cents, a dozen 
 eggs for 24 cents, a peck of apples for 15 cents, a bushel of 
 potatoes for 65 cents, and a quart of beans for 20 cents ; how 
 much do I owe the merchant for these ? 
 
 166. How many books are there in the Bible if the Old 
 Testament contains 39 and the New Testament 27 books ? 
 
 167. In a factory there are 15 men, 12 women, 17 girls, 
 and 21 boys at work ; how many persons are employed in 
 the factory? 
 
 168. Alfred earned 44 cents one day, 50 cents the next 
 day, and found 35 cents ; how many cents did he then have ? 
 
 169. Clay studied 25 minutes one afternoon, 55 minutes 
 the next afternoon, and 12 minutes the next afternoon; 
 how many minutes did he study in all? 
 
 170. A farmer raised 36 bushels of wheat, 18 bushels of 
 oats, 27 bushels of rye, and 19 bushels of corn ; how many 
 bushels of grain did he raise ? 
 
 171. A lady canned, during the summer, 12 quarts of 
 peaches, 9 quarts of cherries, 26 quarts of strawberries, 
 17 quarts of blackberries, 13 quarts of raspberries, and 19 
 quarts of gooseberries ; how many quarts of fruit has she 
 for winter use? 
 
 172. Find the number of days in the first six months of 
 the year when January has 31 days, February 29 days, 
 March 31 days, April 30 days, May 31 days, and June 
 30 days. 
 
20 AMERICAN MENTAL ARITHMETIC. 
 
 173. Find the sum of 62, 15, 9, 32, 27, 18, 8, 6, and 4. 
 
 174. A carpenter used 7 bunches of lath for the kitchen, 
 16 bunches for the dining-room, 12 bunches for the parlor, 
 and 11 bunches for a bed-room ; how many bunches did he 
 use for the four rooms? 
 
 175. Bought berries for 17 cents, cherries for 19 cents, 
 and apples for 13 cents ; what was the cost of all? 
 
 176. A lady bought a dress for $18, a muff for $16, a 
 shawl for $17, and other articles for $19; what was the 
 whole bill? 
 
 177. A merchant sold 18 barrels of flour one week, 16 the 
 next week, 12 the next, 13 the next, and 14 the next; how 
 many barrels did he sell during the five weeks ? 
 
 178. Mary had 56 oranges, and Susan had 19 more than 
 Mary ; how many had Susan ? 
 
 179. A man is 48 years old, and his wife is 36 years old : 
 what is the sum of their ages ? 
 
 180. James had 59 cents and found 48 cents ; how many 
 cents had he then ? 
 
 181. Mary gave 58 cents to her brother and 96 cents to 
 her sister ; how many cents did she give away ? 
 
 182. A merchant sold rice for $198, sugar for $18, oil for 
 $17, candy for $13, molasses for $16, and salt for $12; how 
 much did he receive for all? 
 
 183. A girl made 15 red pin-wheels, 16 blue ones, and 17 
 blue and white. How many pin- wheels did she have ? 
 
 184. It is 38 miles from A to B, 19 miles from B to C, 17 
 miles from C to D, 18 miles from D to E ; how many miles 
 does a man travel who goes from A to E, passing through 
 B, C, andD? 
 
SUBTRACTION. 
 
 Subtraction is indicated by the 
 sign -. 
 
 The number to be subtracted is 
 the subtrahend. 
 
 The number from which to sub- 
 tract is the minuend. 
 
 The result is the difference or 
 remainder. 
 
 The difference between any two 
 numbers may be found by counting. 
 
 Subtraction is a process shorter 
 than counting for finding the 
 difference between numbers. 
 
 Read the remainders as rapidly as 
 1. 10 10 10 10 10 10 10 10 
 
 Illustration. 
 8-3 = 5 
 read 
 8 minus 3 equals 5. 
 8, minuend. 
 3, subtrahend. 
 5, difference or remainder. 
 
 To find the difference be- 
 tween 8 and 3 by counting. 
 
 Counting to 8 and making a 
 mark at each count, ////////; 
 counting to 3 and crossing a 
 mark at each count, XXX/////; 
 counting what is left, we 
 have 5. 
 
 possible : 
 
 10 11 11 11 11 11 11 
 
 9 5 3 
 
 7 
 
 4 18 
 
 2 6 9 
 
 6 
 
 2 
 
 4 8 3 
 
 2. 11 11 12 
 
 12 
 
 12 12 12 
 
 12 12 13 
 
 13 
 
 13 
 
 13 13 13 
 
 5 7 9 
 
 5 
 
 8 4 6 
 
 3 7 9 
 
 5 
 
 7 
 
 4 8 6 
 
 3. 14 14 14 
 
 14 
 
 14 15 15 
 
 15 15 16 
 
 16 
 
 16 
 
 17 17 18 
 
 9 7 5 
 
 8 
 
 6 9 7 
 
 8 6 9 
 
 7 
 
 8 
 
 9 8 9 
 
 Ex. 1. 1, 5, 7, 3, 6, 9, etc. Do not say 9 from 10 leaves 1. 
 
 21 
 
22 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 What must be added to : 
 
 4. 9 to make 18 ? 7. 8, 9, 7, 6 to make 15 ? 
 
 5. 9, 8 to make 17 ? 8. 7, 9, 5, 6, 8 to make 14 ? 
 
 6. 7, 9, 8 to make 16 ? 9. 5, 8, 7, 9, 6, 4 to make 13 ? 
 
 io. 6, 7, 5,4, 8, 3, 9 to make 12? 
 
 11. 5, 8, 7, 4, 3, 6, 2, 9 to make 11 ? 
 
 12. 9, 2, 3, 5, 4, 7, 6, 1, 8 to make 10? 
 
 Ex. 8. 7, 5, 9, 8, 6. 
 
 Beginning with 100 count backwards : 
 
 13. By 9. 16. By 6. 19. By 3. 
 
 14. By 8. 17. By 5. 20. By 2. 
 
 15. By 7. 18. By 4. 21. By 1. 
 
 Ex. 13. 100, 91, 82, etc. 
 
 Begin at the right and read the remainder : 
 
 22. 
 
 98736 
 86224 
 
 26. 
 
 85672 
 43461 
 
 23. 
 
 5897638 
 2634521 
 
 27. 
 
 5463267 
 3252145 
 
 24. 
 
 12345678 
 1343527 
 
 28. 
 
 76924711 
 56412300 
 
 25. 
 
 98768975 
 82345723 
 
 29. 
 
 64723108 
 23412106 
 
 Ex. 22. Say 2, 1, 5, 2, 1. The habit of saying "4 from 6 leaves 2 ; 2 
 from 3 leaves 1 ; 2 from 7 leaves 5," etc., should be broken up. While the 
 four words, " 4 from 6 leaves, 1 ' are being formed, no progress can be made in 
 subtracting. There is no reason why the student should not call off the 
 figures of the remainder as rapidly as he can talk. 
 
SUBTRACTION. 
 Begin at the right and read the remainder 
 
 23 
 
 30. 31. 
 
 
 32 
 
 33. 
 
 36854 3000205 
 29876 1864783 
 
 
 43000005 568120001 
 17652436 497203854 
 
 34. 
 
 35. 
 
 
 36. 
 
 503784325 
 
 282958298 
 
 6234567890 
 3929802958 
 
 3050702003 
 
 1234567898 
 
 37. 
 
 
 
 38. 
 
 286540302567200 
 192830605088739 
 
 
 
 806304205102030 
 507080610503028 
 
 39. 
 
 
 
 40. 
 
 736904521300671 
 497000457369712 
 
 
 
 192470030060091 
 141398765432189 
 
 41. 
 
 
 
 42. 
 
 862300100439610 
 
 765489012786937 
 
 
 
 572000000700123 
 324986574309165 
 
 43. 
 
 
 
 44. 
 
 432176090135200 
 168349827563425 
 
 
 
 678230004000500 
 437654321234567 
 
 Ex. 30. Say 8, 7, 9, 6, and no other words. Practice will enable the 
 student to read the figures of the remainder almost as rapidly as he can talk. 
 
24 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 45. To make 100, what must be added to 56? 48? 32? 
 74? 83? 76? 25? 73? 44? 86? 92? 38? 53? 27? 49? 33? 
 18? 58? 67? 77? 
 
 46. To make 1000, what must be added to 72? 102? 148? 
 156? 63? 179? 185? 196? 144? 156? 175? 183? 122? 104? 
 157? 163? 177? 192? 
 
 47. To make 1000, what must be added to 676? 687? 575? 
 762? 349? 534? 296? 105? 428? 777? 388? 499? 
 
 48. To make 1000, what must be added to 375 ? 804? 783? 
 926? 439? 604? 593? 355? 707? 599? 
 
 49. To make 10000, what must be added to 3608? 5732? 
 4963? 6078? 7095? 2801? 5678? 4209? 
 
 50. From 1000000, subtract 886097, 407864, 360835, 
 479632, 582769, 380803, 760967, 320457, 978654. 
 
 51. From 100000000, subtract 23456789, 92037405, 
 50640720, 40009265, 70904055, 66090207. 
 
 52. 
 
 1000000000 
 372840625 
 
 53. 
 
 1000000000 
 102030458 
 
 54. 
 
 1000000000 
 289076430 
 
 55. 
 
 1000000000 
 203572763 
 
 56. 
 
 1000000000 
 123456789 
 
 57. 
 
 1000000000 
 246813579 
 
 58. 
 
 1000000000 
 764031246 
 
 59. 
 
 1000000000 
 837964012 
 
 60. 
 
 1000000000 
 543212345 
 
 Ex. 48. 625, 196, 217, etc. In these examples, it is best to begin at the 
 left and call ont what must be added to each figure of the subtrahend except 
 the last to make 0, but what must be added to the last figure to make 10. 
 The student should read results as fast as he can talk. 
 
SUBTRACTION. 25 
 
 61. From 276 take 189. 72. From 506 take 489. 
 
 62. From 364 take 278. 73. From 287 take 198. 
 
 63. From 467 take 389. 74. From 802 take 746. 
 
 64. From 123 take 74. 75. Take 123 from 210. 
 
 65. From 106 take 98. 76. Take 299 from 323. 
 
 66. From 207 take 199. 77. Take 145 from 223. 
 
 67. From 111 take 46. 78. Take 345 from 421. 
 
 68. From 203 take 145. 79. Take 258 from 324. 
 
 69. From 209 take 167. 80. Take 456 from 531. 
 
 70. From 245 take 169. 81. Take 489 from 503. 
 
 71. From 456 take 389. 82. Take 286 from 345. 
 
 Ex. 61. 87. To make 200, 11 must be added to 189 ; 76 + 11 = 87. Do 
 not say, "9 from 16 leaves 7 ; 9 from 17 leaves 8." 
 
 83. To make 922, what must be added to 648? 396? 479? 
 553? 764? 875? 283? 697? 785? 892? 189? 527? 819? 
 634? 311? 439? 510? 609? 
 
 84. To make 816, what must be added to 378? 496? 785? 
 396? 519? 439? 382? 786? 758? 729? 715? 638? 525? 
 444? 775? 314? 678? 248? 
 
 85. To make 725, what must be added to 648 ? 639? 675? 
 686? 695? 683? 681? 649? 663? 671? 535? 598? 419? 
 307? 212? 199? 25? 63? 
 
 86. To make 513, what must be added to 416? 438? 269? 
 183? 68? 75? 233? 175? 254? 285? 54? 19? 153? 240? 
 369? 452? 387? 299? 
 
 Ex. 83. 274, 526, 443, etc. Consider what must be added to each to 
 make 900; then add 22; e.g. to make 900, 252 must be added to 648; 
 252 + 22 = 274 ; to make 900, 504 must be added to 396 ; 504 + 22 = 526, 
 etc. 
 
26 AMERICAN MENTAL ARITHMETIC. 
 
 § 7. Problems. 
 
 Declare the answers to each as quickly as possible with* 
 oat reading the problem aloud and before explaining. 
 
 If required to explain, avoid repetitions and unneces- 
 sary words. 
 
 87. From a box containing Ans. 8 marbles. There remained 
 37 marbles 29 were lost ; how tlie difference between 37 marbles 
 many remained ? Explain. and 29 marbles ' or 8 marbles - 
 
 88. Henry had 75^ and gave 45^ for a knife ; how many 
 had he left? 
 
 89. A man had $ 145 in the bank and drew out $ 89; how 
 many dollars had he left in the bank ? Explain. 
 
 90. A merchant bought 124 barrels of apples, and sold 98 
 barrels ; how many had he left ? 
 
 91. A farmer had 115 sheep and after selling some he had 
 86 left ; how many did he sell ? Explain. 
 
 92. In an orchard there are 100 peach trees and 57 plum 
 trees ; how many more peach trees than plums trees are there ? 
 
 93. A receives a salary of $125 per month, and after pay- 
 ing his necessary expenses he has $57 left; what are his 
 expenses ? Explain. 
 
 94. B borrowed $105 and paid $ 79 of the debt ; how much 
 did he still owe ? 
 
 95. The sum of two numbers is 122, one of the numbers is 
 39; what is the other? Explain. 
 
 96. A pays $130 for a horse and a saddle. He pays $13 
 for the saddle ; how much does he pay for the horse ? 
 
 97. From a bin containing 112 bushels of corn, 76 bushels 
 were sold ; how many bushels remained ? 
 
SUBTRACTION. 27 
 
 98. From a school of 109 pupils, 18 were absent ; how many 
 were present? Explain. 
 
 99. Ray's father gave him 75^, his mother gave him 45^, 
 his sister gave him 20^, and his brother gave him 15^. He 
 spent 92^ ; how much had he left ? Explain. 
 
 100. The sum of three numbers is 145 ; the first is 24, the 
 second is 48 ; what is the third ? 
 
 101. A merchant had on hand 36 pounds of butter ; he 
 bought 95 pounds more and then sold 57 pounds ; how 
 much had he left ? 
 
 102. What is the difference between 69 + 82 and 73 + 49? 
 
 103. What is the difference between 132 - 29 and 110 - 61 ? 
 
 104. What is the difference between 101 + 48 + 24 and 
 111-19? 
 
 105. What number subtracted from 98 + 23 will leave 73? 
 
 106. Ethel had 45^ ; she spent 29^, after which she earned 
 57^ ; how many cents had she then ? 
 
 107. A farmer raised 150 bushels of wheat ; he sold at one 
 time 35 bushels, at another time 49 bushels and kept the 
 remainder ; how many bushels did he keep ? 
 
 108. A earns $ 100 per month, and pays $ 15 for board and 
 $46 for other expenses ; how much does he save each month? 
 
 109. From a piece of carpet containing 76 yards, two 
 pieces were cut and 29 yards remained ; the first piece cut off 
 contained 13 yards; how many yards did the second piece 
 contain ? Explain. 
 
 110. Subtract 87 from 104, add 45 to the remainder, and 
 declare the result. 
 
 ill. On Monday a gentleman deposited in a bank $53, 
 on Tuesday he deposited $85, on Wednesday he drew out 
 how much did he leave in the bank ? 
 
28 AMERICAN MENTAL ARITHMETIC. 
 
 112. A owes me 76^; I owe him 91^; how may we settle 
 the account ? Explain. 
 
 113. From the sum of 65 and 88 subtract the sum of 37 
 and 49. 
 
 114. James bought 100 oranges and sold 68 of them ; how 
 many had he left ? 
 
 115. John is 77 years old, and Joseph is 38 ; John is how 
 many years older than Joseph ? 
 
 116. In a school of 66 pupils 29 are present; how many 
 are absent ? 
 
 117. Of 1000 men 128 were sick ; how many were well ? 
 
 118. From a herd of 256 cattle 189 were sold ; how many 
 remained? 
 
 119. A man sold 48 cows, then bought 19, and then had 
 65 ; how many had he at first ? 
 
 120. Prove that your answer to the 119th is correct. 
 
 121. A farmer exchanged eggs costing 35^, butter 19^, 
 potatoes $1.05, for cloth costing 36^, sugar 50^, starch 12^; 
 how much was due him ? 
 
 122. A man bought a horse for $57, received for his use 
 $19, and paid for his keeping $12 ; he sold him for $65 ; how 
 much did he gain ? 
 
 123. Paid $37 for sugar, $29 for molasses ; how much did 
 both cost? How much more did the sugar cost than the 
 molasses ? 
 
 124. John bought 35 apples at one store and 48 at another ; 
 he sold 29 of them ; how many remained ? 
 
 125. I bought a horse for $ 65 ; for how much must I sell 
 him to gain $38? 
 
 126. A farmer sold a cow for $38, which was $19 more 
 than she cost him ; how much did he pay for her ? 
 
MULTIPLICATION. 
 
 Multiplication is indicated by the 
 sign x . 
 
 The number to be multiplied is the 
 multiplicand. 
 
 The number by which to multiply 
 is the multiplier. 
 
 The result is the product. 
 
 Any number of times a given num- 
 ber may be found by adding. 
 
 Multiplication is a process shorter 
 than adding for finding the sum 
 of equal addends. 
 
 Illustration. 
 
 6 x 4 = 24 
 
 read 
 
 6 multiplied by 4 equals 24 
 
 or 
 
 4 times 6 equals 24. 
 
 6, multiplicand. 
 
 4, multiplier. 
 
 24, product. 
 
 To multiply 6 by 4 by 
 adding. 
 
 6x4=6+6+6+6 
 or 
 6 taken 4 times as an ad- 
 dend. 
 .-. 6 x 4 = 24. 
 
 X 
 
 1 
 
 2 | 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 12 
 
 12 
 
 24 36 
 
 48 
 
 60 
 
 72 
 
 84 
 
 96 
 
 108 
 
 120 
 
 132 
 
 144 
 
 11 
 
 11 
 
 22 
 
 33 
 
 44 
 
 55 
 
 66 
 
 77 
 
 88 
 
 99 
 90 
 
 110 
 100 
 
 121 
 110 
 
 132 
 120 
 
 10 
 
 10 
 
 20 
 
 30 
 
 40 
 
 50 
 
 60 
 
 70 
 
 80 
 
 9 
 
 9 
 
 18 
 
 27 
 
 36 
 
 45 
 
 54 
 
 63 
 
 72 
 
 81 
 
 90 
 
 99 
 
 108 
 
 8 
 
 8 
 
 16 
 
 24 
 
 32 
 
 40 
 
 48 
 
 56 
 
 64 
 
 72 
 
 80 
 
 88 
 
 96 
 
 7 
 6 
 
 7 
 6 
 
 14 
 12 
 
 21 
 18 
 
 28 
 24 
 
 35 
 30 
 
 42 
 36 
 
 49 
 42 
 
 56 
 
 48 
 
 63 
 54 
 
 70 
 60 
 
 77 
 66 
 
 84 
 
 72 
 
 13 x 13 = 169 
 
 18 x 18 = 324 
 
 24 x 24 = 576 
 
 5 x 5 x 5 = 125 
 
 14 x 14 = 196 
 
 19 x 19 = 361 
 
 25 x 25 = 625 
 
 6 x 6 x 6 = 216 
 
 15 x 15 = 225 
 
 21 x 21 = 441 
 
 2x2x2=8 
 
 7 x 7 x 7 = 343 
 
 16 x 16 = 256 
 
 22 x 22 = 484 
 
 3 x 3 x 3 = 27 
 
 8 x 8 x 8 = 512 
 
 17 x 17 = 289 
 
 23 x 23 = 529 
 
 4 x 4 x 4 = 64 
 
 9 x 9 x 9 = 729 
 
 29 
 
30 AMERICAN MENTAL ARITHMETIC. 
 
 When a number is multiplied by itself, as 13 x 13, 14 x 14, 
 etc., the product is called a square. When the square of a 
 number is multiplied by the number, as 2x2x2, 3x3x3, 
 etc., the product is called a cube. 
 
 Declare the products of : 
 
 1. 12,11,10, 9, 8, 7, 6, 5, 4, 3, 2, by 9. 
 
 2. 9, 6, 2, 5,11, 4, 2, 7, 3, 8, 10, by 8. 
 
 3. 2, 10, 6, 9, 4, 12, 8, 11, 7, 5, 3, by 7. 
 
 4. 10, 4, 9, 3, 8, 12, 2, 7, 11, 6, 5, by 6. 
 
 5. 6, 8, 4,10, 2, 9,12, 3,11, 7, 5, by 5. 
 
 6. 6, 9, 2, 8, 3, 7,10,12, 4,11, 5, by 4. 
 
 7. 12, 4,11, 2,10, 6, 9, 2, 8, 5, 7, by 11. 
 
 8. 11, 2, 7, 3, 2, 4, 6, 5, 8,12, 9, by 12. 
 
 Ex. 2. 72, 48, 16, 40, etc. Do not say, "8 times 9 are 72." 
 
 Declare the products of : 
 
 9. 13 by 1, 2, 3, 4, 5, 6, 7. 15. 19 by 1, 2, 3, 4. 
 
 10. 14 by 1, 2, 3, 4, 5, 6, 7. 16. 21 by 1, 2, 3, 4. 
 
 n. 15 by 1, 2, 3, 4, 5, 6. 17. 22 by 1, 2, 3, 4. 
 
 12. 16 by 1, 2, 3, 4, 5, 6. 18. 23 by 1, 2, 3, 4. 
 
 13. 17 by 1, 2, 3, 4, 5. 19. 24 by 1, 2, 3, 4. 
 
 14. 18 by 1, 2, 3, 4, 5. 20. 25 by 1, 2, 3, 4. 
 
 21. 3 by 33, 27, 31, 28, 26, 29, 30, 32. 
 
 22. 2 by 49, 30, 39, 31, 42, 32, 47, 48, 28, 38, 29, 46, 27, 36, 
 26, 41, 33, 45, 43, 37, 34, 40, 35. 
 
 23. 13 x 13, 2x2x2, 14 x 14, 8 x 8 x 8. 
 
 24. 14x14, 9x9x9, 25x25, 4x4x4, 21x21, 7x7x7, 
 17 x 17, 5 x 5 x 5, 19 x 19, 6 x 6 x 6, 15 x 15, 3 x 3 x 3, 16 x 16, 
 24 x 24. 
 
 Ex. 9. 13, 26, 39, 52, etc. Do not say, " 13 times 1 are 13." 
 
MULTIPLICATION. 
 
 31 
 
 Give the multiplication table : 
 
 25. '13 times' to 13 x T. 
 
 26. « 14 times ' to 14 x 7. 
 
 27. * 15 fo'w08 ' to 15 x 6. 
 
 28. ' 16 times ' to 16 x 6. 
 
 29. '17 times' to 17x5. 
 
 30. '18 times' to 18x5. 
 
 31. '19 times' to 19x5. 
 
 32. '21 ^'raes ' to 21 x 4. 
 
 33. ' 22 times ' to 22 x 4. 
 
 34. ' 23 times ' to 23 x 4. 
 
 35. ' 24 times ' to 24 x 4. 
 
 36. ' 25 times ' to 25 x 4. 
 
 Ex. 25. 13 x 1 are 13 ; 13 x 2 are 26 ; 13 x 3 are 39 ; etc. 
 
 Declare the results rapidly : 
 
 37. 19x5, 17x4, 16x4, 14x4, 18x5, 19x4, 16x6, 
 13x3, 5x16, 17x5, 5x13, 17x3, 4x13, 16x3, 19x3. 
 
 38. 19x2, 3x18, 17x2, 2x16, 15x6, 5x14, 15x5, 
 14x3, 13x2, 4x18, 15x3, 2x14, 13x6, 7x14, 2x18. 
 
 39. 4x15, 6x16, 7x13, 9x5x2, 2x3x7, 8x13, 
 4 x 14, 6 x 15, 6 x 13, 2 x 6 x 8, 5 x 6 x 2, 6 x 14, 18 x 4, 14 x 6. 
 
 40. 16x5, 15x2, 14x2, 15x4, 14x5, 14x7, 3x4x5, 
 2x4x6, 5x4x2, 3x5x6. 
 
 State rapidly: 
 
 41. The- squares of the integers from 1 to 25. 
 
 42. The cubes of the integers from 1 to 9. 
 
 43. The square of 25, 23, 24, 21, 19, 16, 17, 15, 13, 22, 
 
 20, 18. 
 
 44. The square of 12, 14, 16, 18, 20, 22, 24, 13, 15, 17, 
 19, 21. 
 
 45. The cube of 9, 6, 3, 1, 4, 7, 8, 5, 2, 5, 7, 9, 4, 6, 8. 
 
 46. The cube of 4, 3, 5, 7, 9, 2, 4, 6, 8, 10, 9, 2, 8, 6, 7. 
 
32 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 Multiply : 
 
 
 
 47. 
 
 48. 
 
 49. 
 
 2030876514 
 9 
 
 5708392943 
 
 8 
 
 3886546312 
 
 7 
 
 50. 
 
 51. 
 
 52. 
 
 4571263972 
 4 
 
 7360925168 
 12 
 
 54. 
 
 2784235879 
 11 
 
 53. 
 
 55. 
 
 1203045678 
 
 8 
 
 6543712345 
 
 6 
 
 6789098765 
 5 
 
 56. 
 
 57. 
 
 58. 
 
 4630902034 
 11 
 
 5762198345 
 
 7 
 
 60. 
 
 6290311269 
 
 8 
 
 59. 
 
 61. 
 
 3862048001 
 9 
 
 3572607983 
 3 
 
 6238496785 
 12 
 
 62. 
 
 63. 
 
 64. 
 
 4837254912 
 4 
 
 5789564328 
 6 
 
 7912765046 
 11 
 
 Ex. 47. 36, 12, 40, 58, 68, etc. Do not say, "9 times 4 are 36 ; 9 times 1 
 are 9 and 3 are 12 ; 9 times 5 are 45 and 1 are 46," but declare the results 
 only. 
 
MULTIPLICATION. 33 
 
 § 8. Problems. 
 
 Declare the answers to each as quickly as possible with- 
 out reading the problem aloud and before explaining. 
 
 If required to explain, avoid repetitions and unneces- 
 sary ivords. 
 
 65. At 13^ each what will Ans - 156 ^ Since x basket costs 
 
 ^ ^ , , . -^ , . 13^, 12 baskets will cost 12 times 
 
 12 baskets cost? Explain. 13^ r 156^. 
 
 66. Is the following a cor- 
 rect explanation of Ex. 65? 
 
 Since 1 basket costs W, 12 10 N °; Because the denomination of 
 
 , 12 is baskets and not cents. 
 baskets will cost 13 times VLr, t 
 
 or 156£ 
 
 67. At $3 per barrel what will 19 barrels of flour cost? 
 
 68. At 19 per head what will 21 sheep cost? 
 
 69. If a man travels 5 miles an hour, how far will he 
 travel in 16 hours? Explain. 
 
 70. A train runs 29 miles an hour ; how far will it run 
 in 8 hours? 
 
 71. If a man earns $6 per week, how much does he earn 
 in 12 weeks? 
 
 72. How much will 4 acres of land cost at 1 57 an acre? 
 
 73. If 34 men can do a piece of work in 11 days, how long 
 will it take one man to do it? Explain. 1 
 
 74. If 3 pipes fill a cistern in 19 hours, how long will it 
 take one pipe to fill it ? 
 
 75. Henry is 12 years old, and his father is 5 times as old ; 
 how old is his father ? 
 
 76. If a ship sails 9 miles an hour, how far will it sai] 
 in 9 hours ? 
 
 AM. MENT. AR. 3 
 
34 AMERICAN MENTAL ARITHMETIC. 
 
 77. A farmer bought 19 An». 2Zf. If l yard cost Sf, 19 
 
 yards of cloth at 3/ a yard, ^ ards cost 19 *?"*■ " 57 ^ ; if X 
 
 ^ . ^ dozen eggs sold for 16^, 5 dozen sold 
 
 and gave in exchange 5 dozen f or ^ . if the cloth cost 67 , and the 
 eggs at 16/ a dozen; how eggs brought 80^, the farmer's due 
 much was due him ? Explain. was the difference, or 23^. 
 
 When the scholar has thoroughly mastered this form, he should be required 
 to abbreviate the explanation. Thus, the cloth cost 57^ ; the eggs brought 
 SOf ; therefore 23^ was due the farmer. 
 
 78. A man bought 7 yards of cloth at 16/ a yard, and 5 
 yards of cloth at 12/ a yard; what was the entire cost? 
 
 79. If the income from one cow is 1 16 per year, and from 
 one sheep $2, what is the income from 6 cows and 8 sheep? 
 
 80. I paid 18/ each for 12 chickens, and 2/ each for 13 
 eggs ; what was the entire cost ? 
 
 81. What will be the cost of 5 pictures at 19/ each, and 7 
 hooks at 3/ each ? 
 
 82. A lady bought 6 dozen buttons at 12/ per dozen, and gave 
 in payment one dollar ; how much change should she receive ? 
 
 83. A man bought 6 barrels of apples at $3 per barrel, 
 and gave in exchange 12 sacks of flour at 1 2 per sack; how 
 much was due him? 
 
 84. James bought 8 marbles for 5/ each, 6 pencils for 10/ 
 each, and a book for 25/ ; he gave in payment 150/ ; how 
 much change should he receive ? 
 
 85. Harvey bought 9 oranges for 7/ each, and 11 lemons 
 for 5/ each ; he gave in exchange 9 pounds of butter at 15/ 
 per pound ; how much was due him ? 
 
 86. John traded 8 marbles worth 8/ each, for a jack-knife 
 worth 50/ and some money; how much money should he 
 receive ? 
 
MULTIPLICATION. 35 
 
 87. What is the profit in Arts. $30. It is best to find the 
 
 buying 6 cows at $20 each, P rofit on °f ' Th f ■ «" Profit on 
 , ° ,,. ~ ~- , owe cow is $ 5 ; on 6 cows, $30. 
 
 and selling at $25 eacnr 
 
 88. What is the profit in buying 10 shares of stock at $ 99 
 each, and selling them at $103 each? 
 
 89. What is received from the sale of 16 cows at $ 21 each, 
 if $ 1 each is paid the agent for selling them ? 
 
 90. What is received from the sale of 16 shares of stock 
 at $101 each, if $1 each is paid the broker for selling them ? 
 
 91. How much does a man gain by buying 6 cows at $21 
 each, paying an agent $1 each for purchasing, and selling 
 them at $ 27 each, paying an agent $ 2 each for selling them ? 
 
 92. Does a man gain or lose and how much by buying 6 
 shares of stock at $101 each, paying a broker $1 each for 
 purchasing, and selling at $103 each, paying a broker $2 
 each for selling them? 
 
 93. When beef is b$ a pound, and pork 6^ a pound, how 
 much more will 17 pounds of beef cost than 14 pounds of 
 pork? 
 
 94. Which costs the more, the keeping of 16 horses 9 
 weeks at $1 a week each, or the keeping of 12 cows 12 
 weeks at 50^ a week each ? How much more ? 
 
 95. Two persons start from the same point and travel in 
 opposite directions: one travels 5 miles an hour, and the 
 other 7 miles an hour ; how far apart are they at the end of 
 13 hours ? 
 
 96. How far apart are they at the end of 13 hours if they 
 travel in the same direction ? 
 
 97. Which is cheaper and how much per dozen, to buy 
 eggs at 25^ a dozen or at 3^ each ? 
 
DIVISION. 
 
 Division is indicated by the sign -i-. 
 
 The number to be divided is the 
 dividend. 
 
 The number by which to divide is 
 the divisor. 
 
 The result is the quotient. 
 
 That which remains when the divi- 
 sion is not exact is the remainder. 
 
 The number of times one number 
 is contained in another may be found 
 by subtracting. 
 
 Division is a process shorter than 
 subtracting for finding how many 
 times one number is contained in 
 another. 
 
 Illustration. 
 
 20 -*■ 7 = 2f 
 read 
 20 divided by 7 = 2 and 
 6 remainder. 
 
 20, dividend. 
 7, divisor. 
 2, quotient. 
 6, remainder. 
 To divide 9 by 4, by sub- 
 tracting. 
 
 9-4-4 calls for the num- 
 ber of times that 4 may be 
 subtracted from 9. 
 
 9-4-4 = 1; 
 that is, 
 9-4-4 = 2^ 
 
 Declare the quotients of: 
 
 1. 144, 96, 36, 60, 72, 48, 24, 84, 108, 132, 120, divided by 12. 
 
 2. 88, 44, 77, 132, 99, 121, 66, 22, 55, 33, 110, divided by 11. 
 
 3. 72, 108, 81, 54, 63, 18, 45, 90, 27, 108, 36, divided by 9. 
 
 4. 64, 56, 96, 24, 32, 16, 72, 40, 80, 48, divided by 8. By 4. 
 
 5. 63, 14, 35, 56, 21, 70, 84, 28, 42, 35, 49, 70, divided by 7. 
 
 6. 54, 36, 18, 72, 54, 12, 24, 48, 42, 60, divided by 6. By 3. 
 
 7. 25, 40, 30, 15, 50, 35, 55, 20, 60, 45, 10, 65, divided by 5. 
 
 8. 10, 18, 6, 16, 22, 14, 20, 8, 12, 4, 24, 28, 32, divided by 2. 
 
 Ex. 1. 12, 8, 3, 6, 6, etc. Do not say, " 144 -*- 12 are 12." 
 3d 
 
DIVISION. 37 
 
 Declare the quotient and remainder of : 
 
 9. 119, 111, 113, 81, 117, 86, 118, 77, 116, 90, 87, -f-12. 
 
 10. 109, 71, 106, 80, 102, 79, 105, 60, 107, 59, 86, +11. 
 
 n. 89, 60, 76, 83, 52, 63, 50, 73, 62, 55, 80, 25, -*- 9. 
 
 12. 79, 70, 61, 62, 73, 71, 67, 60, 78, 68, 63, 65, -f- 8. 
 
 13. 69, 59, 62, 54, 58, 23, 67, 57, 64, 55, 60, 53, + 7. 
 
 14. 59, 50, 41, 55, 51, 27, 57, 47, 46, 53, 45, 56, -*- 6. 
 is. 49, 41, 44, 38, 36, 22, 46, 39, 48, 34, 42, 37, -j- 5. 
 
 16. 39, 30, 33, 37, 26, 22, 34, 23, 31, 25, 29, 35, -*- 4. 
 
 17. 29, 19, 17, 23, 11, 16, 8, 28, 20, 22, 10, 13, + 3. 
 Ex. 9. 9, 11 ; 9, 3 j 9, 5 ; 6, 9 ; etc. Do not say, " 119 -=- 12 are 9 and 11 
 
 remaining." 
 
 Declare the quotient and remainder of : 
 
 18. 12 contained in 119, 17, 111, 14, 113, 13, 117, 18, 118, 
 15, 116, 16, 114, 19, 112, 28, 110, 24, 115, 20, 109, 25, 106, 26, 
 102, 21, 105, 23, 107, 22, 101, 27, 108, 29, 104, 34, 100, 30, 103, 
 38, 99, 35, 97, 37, 90, 32, 93, 33, 98, 31, 96, 36. 
 
 19. 12 contained in 91, 39, 94, 40, 95, 48, 92, 44, 89, 41, 
 85, 47, 80, 43, 88, 46, 84, 42, 81, 45, 86, 49, 82, 87, 53, 83, 57, 
 79, 52, 73, 55, 77, 51, 74, 58, 78, 50, 76, 56, 70, 54, 75, 59, 71, 
 64, 69, 68, 66, 62, 63, 60, 67, 65, 61, 72. 
 
 20. 11 contained in 109, 65, 106, 11, 102, 12, 105, 17, 107, 
 14, 101, 13, 108, 18, 104, 15, 100, 16, 103, 19, 99, 28, 97, 24, 
 90, 20, 93, 25, 98, 26, 96, 21, 91, 23, 94, 22, 95, 27, 92, 29, 89, 
 34, 85, 30, 80, 38, 88, 77, 35, 71, 37, 66, 32, 69, 33, 79, 31, 54. 
 
 21. 9 contained in 89, 10, 80, 12, 84, 14, 86, 18, 87, 16, 
 79, 28, 77, 20, 78, 26, 70, 22, 71, 29, 66, 30, 67, 35, 65, 32, 62, 
 31, 64, 39, 54, 48, 50, 41, 51, 43, 52, 42, 53, 49, 45, 57, 46, 55, 
 47, 58, 44, 56, 40, 59, 36, 68, 33, 60, 37, 61, 38, 63, 34, 69, 27. 
 
 Ex. 18. 9, 11 ; 1, 5 ; 9, 3 ; 1, 2 ; etc. 
 
38 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 Read the quotient : 
 
 
 22. 
 
 23. 
 
 12)119076324 
 
 11)207806035 
 
 25. 
 
 26. 
 
 7)369246810 
 
 6)121416181 
 
 28. 
 
 4)245678900 
 
 31. 
 
 3)902637052 
 
 34. 
 
 29. 
 
 5)312760030 
 
 32. 
 
 7)510000900 
 
 8)507603295473286023 
 
 36. 
 
 6)102003040507623086 
 
 38. 
 
 4)816540092367813362 
 
 40. 
 
 9)923634373532146874 
 
 42. 
 
 12)823476298345151718 
 
 44. 
 
 5)345678987654321235 
 
 46. 
 
 9)876678543345210038 
 
 24. 
 
 9 )803706256 
 
 27. 
 
 3 )920212233 
 
 30. 
 
 8 )405001762 
 
 33. 
 
 12 )634310271 
 
 35. 
 
 7)123456780924572568 
 
 37. 
 
 5)102305067052342125 
 
 39. 
 
 3 )276300009823145911 
 
 41. 
 
 11)382900768419582123 
 
 43. 
 
 6)102030456783961048 
 
 45. 
 
 4)953872641219382116 
 
 47. 
 
 11)398476521830057621 
 
 over." 
 
 Ex. 22. 9, 9, 2, 3, 0, 2, 7. Speak no words except the quotient figures. 
 i not say, "12 into 119, 9 times and 11 over; 12 into 110, 9 times and 2 
 
DIVISION. 39 
 
 State results rapidly: 
 
 48. 7 times 14 are how many times 2? 13? 12? 
 
 49. 9 times 11 are how many times 6? 12? 8? 
 
 50. 5 times 15 are how many times 3 ? 16 ? 9 ? 
 
 51. 4 times 17 are how many times 2 ? 15 ? 8 ? 
 
 52. 2 times 33 are how many times 3 ? 11 ? 22 ? 
 
 53. 9 times 10 are how many times 5 ? 18 ? 15 ? 
 
 54. 6 times 16 are how many times 3? 32? 6? 12? 24? 
 
 55. 8 times 9 are how many times 2? 18? 4? 6? 8? 
 12? 24? 
 
 56. 5 times 12 are how many times 2? 3? 4? 6? 10? 151 
 
 State results rapidly : 
 
 9x 5, +15 are how many times 12? 15? 4? 7? 
 14 x 6, - 7 are how many times 11 ? 7 ? 18 ? 6 ? 
 
 5? 
 
 7? 
 
 9? 
 
 6? 
 
 8? 
 
 7? 
 
 3? 
 
 6? 
 
 2? 
 
 8? 
 
 6? 
 
 3? 
 
 57. 
 58. 
 59. 
 60. 
 
 63. 
 64. 
 65. 
 
 L4 x 6, - 7 are how many times 11 ? 7 ? 18 ? 6 ? 
 7x12, - 8 are how many times 19? 13? 14? 15? 
 6 x 8, + -4 are how many times 13? 6? 9? 7? 
 
 61. 10 x 4,+ 2 are how many times 6? 8? 9? 12? 
 
 62. 8x 9, +16 are how many times 8? 12? 16? 9? 
 14 x 4, + 8 are how many times 16? 8? 4? 32? 
 18 x 3, + 6 are how many times 15? 12? 5? 10? 
 17 x 5, + 5 are how many times 18? 10? 16? 9? 
 
 State results rapidly : 
 
 66. How many times 21 are 14 x 6 ? 7x9? 3 x 28 ? 
 
 67. How many times 24 are 6x16? 8x6? 6x12? 
 
 68. How many times 13 are 39 x 2?39x 3?26x 2? 
 
 69. How many times 12 are 15 x 4? 6x16? 54 x 2? 
 
 70. How many times 11 are 33 x 3?22x 4?44x 2? 
 
 71. How many times 9 are 12x12? 6x12? 5x18? 
 
 72. How many times 14 are 7x8? 2x49? 7x10? 
 
40 AMERICAN MENTAL ARITHMETIC. 
 
 Declare the results rapidly : 
 
 73. 24-12; 25 + 5; 26 + 13; 27 + 9; 28-1-7; 30 + 15 
 32-16; 33 + 11; 34 + 17; 35 + 5; 36 + 18; 38 + 19. 
 
 74. 39 + 13; 40 + 8; 42 + 14; 44+4; 45 + 15; 48 + 16 
 49 + 7; 50 + 10; 51 + 3; 52 + 4; 54 + 3; 55 + 11. 
 
 75. 56 + 4; 57 + 3; 60 + 4; 64 + 4; 65 + 5; 66 + 6; 68 + 4 
 70 + 7; 72 + 18; 75 + 5; 76+4; 77 + 11. 
 
 76. 78 + 6; 80 + 5; 81 + 9; 84 + 6; 85 + 5; 88 + 11 
 90 + 18; 91 + 13; 95 + 19; 96 + 16; 98 + 7; 99 + 11. 
 
 Declare the results rapidly : 
 
 77. 96 + 16, 12, 24, 3, 8, 32, 48, 4, 6; 98 + 2, 7, 49, 14. 
 
 78. 99 + 11, 33, 9, 3; 94 + 2, 47; 93 + 3, 31; 92 + 2, 23, 46, 
 4; 91 + 7, 13; 90 + 9, 5, 18, 3, 45, 30, 2, 15, 6. 
 
 79. 88 + 11, 2, 22, 44, 4, 8; 87 + 3, 29; 86 + 2, 43. 
 
 80. 85 + 5, 17; 84 + 7, 21, 12, 14, 4, 6, 42, 2, 3, 28; 81 + 9, 
 27,3; 80 + 16,8,10,5,40, 20. 
 
 81. 78 + 39, 2, 13, 6, 3; 76 + 38, 2, 19, 4; 74 + 37, 2; 72 + 
 18, 12, 24, 6, 3, 4, 36, 2, 4, 8, 9. 
 
 82. 70 + 35, 7, 5, 14, 2; 64 + 8, 4, 16, 32; 63 + 9, 3, 7, 21. 
 
 83. 54 + 6, 3, 9, 27 ; 48 + 8, 4, 6, 2, 24, 3, 16. 
 
 Name sets of two numbers each, whose product is: 
 
 84. 99, 98, 96, 95, 94, 93, 92, 91, 90, 88, 87, 86. 
 
 85. 85, 84, 82, 81, 80, 78, 77, 76, 75, 74, 72, 70. 
 
 86. 69, 68, 66, 65, 64, 63, 62, 60, 58, 57, 56, 55. 
 
 87. 54, 52, 51, 50, 49, 48, 46, 45, 44, 42, 40, 39. 
 
 88. 38, 36, 35, 34, 33, 32, 30, 28, 27, 26, 25, 24. 
 
 89. 22, 21, 20, 18, 16, 15, 14, 12, 10, 9, 8, 6, 4. 
 
 Ex. 84. 33 and 3 ; 9 and 11. 7 and 14 ; 49 and 2. 6 and 16 ; 8 and 12 ; 
 4 and 24 ; 3 and 32 ; 2 and 48; etc. 
 
DIVISION. 
 
 41 
 
 § 9. Precedence of Signs 
 
 What is the value of 6 + 4 x 5 ? 
 
 Mathematicians have agreed 
 to use the sign 'x' before the 
 sign '+.' or '-.' 
 
 What is the value of 6 - 4 - 2 ? 
 
 Mathematicians have agreed 
 to use the sign '+' before the 
 sign '+' or '-.' 
 
 What is the value of 24 -=-4 
 x2? 
 
 There is no agreement as to 
 which sign shall be used first. 
 It is best to avoid such ex- 
 pressions. 
 
 What is the value of 6 - 4 + 8 ? 
 
 It makes no difference in 
 what order the signs '+' and 
 '-' are used. 
 
 If the signs are used as they 
 occur, the answer is 50 ; if the 
 sign ' x ' is used first, 26. 
 
 If the signs are used as they 
 occur, the answer is 1 ; if the 
 sign ' -7- ' is used first, the answer 
 is 4. 
 
 If the signs are used as they 
 occur, the answer is 12 ; if the 
 sign ' x ' is used first, the answer 
 is 3. 
 
 If the signs are used as they 
 occur, the answer is 10 ; if the 
 sign ' + ' is used first, the answer 
 is 10. 
 
 Find the value of : 
 
 90. 6 + 8-5-2-3 + 2. 98. 
 
 91. 72-6-64-*-8-3. 99. 
 
 92. 6x8 — 12 + 4. loo. 
 
 93. 96-5-16 + 72-24-8. 101. 
 
 94. 99-11-81-5-9 + 25. 102. 
 
 95. 7 + 8-5-4 + 9x2-12-4. 103. 
 
 96. 9+16-5-8-18-5-3 + 2x5. 104. 
 
 97. 25 + 10-f-5-27-5-3. 105. 
 
 8 + 4-7 + 6-9-5-3. 
 18 + 15-3 + 72-24. 
 30 + 5^5-36-3-8. 
 92 -j- 23 + 87 -s-3 + 49-7. 
 98-J-7-42-6 + 18-9. 
 33-3-10-5-2-8-5-4. 
 64x2-5-32 + 88-5-8-11. 
 96-f-32 + 84-*-12-70-*-7. 
 
 Ex. 90. 9. Say 6, 10, 7, 9. 
 
42 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 10. Parenthesis or Bar. 
 
 To indicate that several 
 quantities are to be subjected 
 to the same operation, they 
 are written within curved 
 lines or brackets, or under or 
 over a straight line. 
 
 Commas are sometimes 
 used to indicate that the signs 
 are to be used in the order of 
 their occurrence. 
 
 Illustration. 
 (6 + 3) x 5, 
 
 or 
 
 6 + 3 x 5, 
 
 means, 
 
 the sum is to be multiplied by 5. 
 
 read, 
 the expression 6 plus 3 times 5. 
 
 6, + 8, + 2, x 7 
 means, 
 to 6, add 8, divide by 2, multiply 
 by 7. 
 
 Find the value of : 
 
 106. [(9x8 + 9) + 9 + 5] + 2 + 8. 
 
 107. (6 + 8)-r-2 + (5-3)x2. 
 
 108. (7x5-3x8) x 8 + 4. 
 
 109. (8x12-18x4)-h(9x7-19x3). 
 
 Ex. 106. 15. 
 
 Find the value of : 
 
 no. 9, x8, + 6, -=-13, x5, -4-2, x4, +4, -8, +9, x5. 
 ill. 90, +9, -nil, x2, +6, -8, x7, +4, -5, x8, +2, 
 -7, x8, +1, -7. 
 
 112. 19, x4, + 5, -i-9, x5, + 6, -17, xl8, +7, +3, -;-J6. 
 
 113. 18, x5, + 6, h-8, xl2, -44, +8, +-12, x9, +7. 
 
 114. 17, x 4, +5, +8, +6, +3, +2, +5, +14, -10, x8, 
 + 9, +8, -1, +7. 
 
 Ex. 110. 7. Say 9, 72, 78, 6, 30, 15, 60, 64, 8, 17, 85. 
 
DIVISION. 43 
 
 Find the value of : 
 
 115. 6, +7, x5, + 5, + 2, +18, x4, +4, +5, +3, x6, 
 + 2, -11, X 8, +1, -i-8, x7, +8. 
 
 lie. 98, +7, +7, +8, +7, +12, x29, +3, +12, +30, 
 h-12, xll, x3, +11, +12, -5, +1. 
 
 117. 16, +17, -18, x6, +1, h-13, xl4, +2, -10, +25, 
 + 18, +22, -15, xl8, +9, +9, -9. 
 
 lis. 45, -15, x3, x5, x2, -6, x4, -5, xl2, -16, x8, 
 -9, x5, +10, x7, x2, +7. 
 
 119. 2, +19, -16, +15, +18, -13, -12, +9, +8, -17, 
 + 25, -13, +18, -6, -9, -5, +4. 
 
 120. 18, +17, -19, -12, +23, +48, -16, -19, +13, 
 -11, +12, -9, +16, -8, +17, -18, -19. 
 
 121. 13, +12, -11, +15, -14, +22, -19, -3, +8, +9, 
 + 17, +13, -6, -9, -12, -25, +18. 
 
 122. 7, x7, +1, +5, x8, +5, +17, x6, +2, +16, x49, 
 *-7, x6, +6, +18, +6, x8, +4, +23, +6. 
 
 123. 19, +11, +13, +2, +16, +5, +14, +18, +4, +7, 
 f 13, +10, +9, +12, +14, +16, +6, +8, +11. 
 
 124. 245-18-13-15-16-12-11-10-9-8-7-6-5 
 _4_3_2-l-ll_14-17-25-8. 
 
 125. 19, +8,-7,-6, +13, -14, +18, -17, +16, +12, 
 -14, -15, +9, -8, +7, -6, -11. 
 
 126. 4, x5, +1, +7, x5, +2, +6, -3, +4, x6, +2, 
 + 8, x4, +4, +2, x8, +1, +9, x6, +2, +7. 
 
 127. 5, xl2, +3, +7, x8, -5, -3, +8, x6, +1, +7, x6, 
 + 2, +11, x4, x4, +7, +8, -11, -^17. 
 
 128. 6, x7, +9, -3, +8, x6, +4, +5, x3, +1, +5, x6, 
 + 6, +18, xlO, +1, +7, x6, +4, +11. 
 
 129. 8, x5, +2, +7, x6, +4, +10, xl4, +6, +9, -8, +7, 
 x2, +4, +2, xll, +11, +12. 
 
44 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 24 
 
 48 
 
 4 = 6. 
 
 8 = 6. 
 
 24-4 
 
 12-2 
 
 0, 
 
 24 - 4 = 6. 
 48 - 4 = 12. 
 
 24 
 12 
 
 4 = 6. 
 4 = 3. 
 
 24-4 
 
 24-8 
 
 24 
 24 
 
 4 = 6. 
 2 = 12. 
 
 § 11. Principles 
 
 Multiplying both dividend 
 and divisor by the same number 
 does not affect the quotient. 
 
 Dividing both dividend and 
 divisor by the same number does 
 not affect the quotient. 
 
 Multiplying the dividend 
 multiplies the quotient. 
 
 Dividing the dividend divides 
 the quotient. 
 
 Multiplying the divisor di- 
 vides the quotient. 
 
 Dividing the divisor multi- 
 plies the quotient. 
 
 Division is expressed in four 
 ways : 
 
 By writing the dividend above 
 and the divisor below a hori- 
 zontal line. 
 
 By writing the sign 4 -f-' be- 
 tween the terms. 
 
 By writing the sign ' : ' be- 
 tween the terms. 
 
 By writing the divisor at the 
 
 left, and the dividend at the 
 
 right, of a curved line. 
 
 The first method was originally used. Later, to get both terms on the 
 same horizontal line the dividend was written, then the line ' — ' with a dot 
 over it for the dividend and a dot below for the divisor, then the divisor. 
 Later the line was omitted. 
 
 Eight divided by three is ex- 
 pressed : 
 
 , fractional method. 
 
 8 — 3, common method. 
 
 8:3, ratio method. 
 
 3)8, working method. 
 
DIVISION. 45 
 
 § 12. Problems. 
 
 Declare the answer to each as quickly as possible without 
 reading the problem aloud and before explaining. 
 
 If required to explain, avoid repetitions and unnecessary 
 words. 
 
 130. If 14 apples cost 28**, 
 
 what will 1 apple cost ? Explain. Ans ' ^ If u & ™ les cost 28 ?> 
 
 1 apple will cost as many cents as 
 Ans. 2f. If 14 apples cost 28^, 1 14 fe contained times in 28, or 2?. 
 apple will cost T \ of 28^, or If. 
 
 131. What is the cost of 1 yard of cloth when 16 yards 
 cost 96^? 
 
 132. A man divided $ 200 among 20 persons ; how much 
 did each receive ? 
 
 133. If 18 yards of cloth cost 1 54, for how much must it 
 be sold per yard to gain $ 36 ? 
 
 134. A farmer gave 18 barrels of flour, worth 1 4 a barrel, 
 for 12 yards of cloth; how much was the cloth a yard? 
 
 135. If 24 hours equal 360 degrees, how many degrees 
 equal 1 hour? 
 
 136. If 360 degrees equal 24 hours, how many hours 
 equal 1 degree ? 
 
 137. Eleven cows were sold for $ 220 ; what was the sell- 
 ing price of each ? 
 
 138. Eleven shares of stock were sold for $ 1111 ; what 
 was the selling price of each ? 
 
 139. Eleven cows were sold for $231, and $1 per cow was 
 paid to the agent for selling them ; how much did the owner 
 receive for each cow? 
 
 140. If 9 tables cost 1 108, what will 12 tables cost? 
 
46 AMERICAN MENTAL ARITHMETIC. 
 
 141. At 3^ each, how many pears Ans. 13 pears. Since 1 
 
 can be bought for 39^ ? P ear costs ^ 39 ^ wiU bu y 
 
 TTr , , . . . . , . . as many pears as 3 is con- 
 
 142. Would this explanation be tained times in 39} or 13 
 
 correct ? "If one pear costs 3^, 39^ pea rs. 
 
 will buy as many pears as 3^ U con- Yes> Because S f is con- 
 
 tained times in 39^, or 13 pears" tained times in 39^. 
 
 143. Would this explanation be 
 
 correct? « Since 1 pear costs 3£ 39^ co^i^t^ in'sfc " 
 will buy as many pears as 3^ is con- 
 tained times in 39." 
 
 144. Would it be right to say, "If 1 pear costs 3^, 39^ will 
 buy as many pears as 3 is contained times in 39^ " ? 
 
 145. At 4^ each, how many lemons can I buy for 72^? 
 
 146. If 1 cow costs % 15, how many can be bought for 175 ? 
 
 147. At $19 each, how many sheep can be bought for $95? 
 
 148. A and B started together in the same direction from 
 the same point, A at the rate of 5 miles an hour, and B at 
 the rate of 3 miles an hour ; in how many hours will A be 
 14 miles ahead of B? 
 
 149. Traveling as before, B has 20 miles the start ; in how 
 many hours will A overtake B ? 
 
 150. A and B started at the same time from the same 
 point in opposite directions, with the same rate as before ; 
 how far apart will they be in 10 hours ? 
 
 151. After traveling 10 hours they turn around towards 
 home. Who will reach home first? How far will A have 
 traveled? How far will B have traveled? 
 
 152. If 16 oranges are worth 32 pears, and 3 pears are 
 worth 6 apples, and apples are worth 2$ each, how many 
 oranges can be bought for 40^? 
 
DIVISION. 47 
 
 153. If 6 quarts of berries cost 18*, what will 12 quarts 
 cost? 
 
 Ans. 36^. 1 quart will cost £ of Ans. 36^. 12 quarts are 2 times 
 
 18^, or 3^ ; 12 quarts will cost 12 6 quarts ; 12 quarts will cost 2 times 
 times 3f, or 36f. 18^, or 36^. 
 
 154. If 12 pounds of cheese cost 108*, what will 36 pounds 
 cost? 
 
 155. How many pounds of butter, at 15 cents per pound, 
 must be given for 18 pounds of sugar at 5 cents a pound? 
 
 156. If 8 sheep cost $ 80, how much will 24 sheep cost ? 
 
 157. If 5 men can do a piece of work in 20 days, in how 
 many days can 4 men do it? 
 
 158. How many barrels of flour can be bought for $40, if 
 5 barrels cost $ 50 ? 
 
 159. How long will it take Paul to earn 99 cents, if he 
 earns 18 cents in 2 weeks ? 
 
 160. How many years will it take to pay a debt of $ 1080, 
 if $ 720 are paid in 6 years? 
 
 161. How much will 24 barrels of apples cost, if 6 barrels 
 cost $24? 
 
 162. At 24* for 12 apples, what will 72 apples cost ? 
 
 163. At 18* for 3 dozen clothes-pins, how many clothes- 
 pins can be bought for 30* ? 
 
 164. If 19 apples cost 57/, what will 14 apples cost ? 
 
 165. If 19 apples cost 57*, how many apples can be bought 
 for 51* ? 
 
 166. If 17 books cost $153, what will 22 such books cost? 
 
 167. At $185 for 5 cloaks, what will 7 cloaks cost? 
 
 168. If 23 cows sell for $ 920, at the same rate what wil) 
 30 cows bring ? 
 
FACTORING. 
 
 A number may exactly contain an- 
 other; the container is a multiple; 
 the contained, & factor. 
 
 A number may have other factors 
 besides itself and one; a composite 
 number. 
 
 A number may have no other 
 factors besides itself and one ; a prime 
 number. 
 
 Several numbers may have no 
 common factor greater than one; 
 numbers prime to each other. 
 
 Each of several numbers may be 
 prime to each of the others ; numbers 
 severally prime. 
 
 Name: 
 
 1. All the composite numbers from 1 to 100. 
 
 2. All the prime numbers from 1 to 100. 
 
 3. Two composite numbers prime to each other. 
 
 4. Three numbers prime to each other. 
 
 5. Three numbers severally prime. 
 
 Define : 
 
 6. A multiple of a number. 9. A prime number. 
 
 7. & factor of a number. 10. Numbers prime to each other. 
 
 8. A composite number. 11. Numbers severally prime. 
 
 48 
 
 Illustration. 
 
 12 contains 6, 2 times. 
 
 12, a multiple of 6. 
 6, a factor of '12. 
 
 12, a composite number. 
 Its factors are, 1, 2, 3, 4, 
 6, 12. 
 
 7, a prime number. It 
 has no factors except 7 
 and 1. 
 
 8, 12, 25, are prime to 
 each other. 
 
 8, 9, 25, 49, are severally- 
 prime . 
 
FACTORING. 
 
 49 
 
 A number is divisible : 
 
 By 2, when its last digit is 
 divisible by 2. 
 
 By 5, when the number de- 
 noted by its last digit is divisi- 
 ble by 5. 
 
 By 4, when the number de- 
 noted by its last two digits is 
 divisible by 4. 
 
 By 8, when the number de- 
 noted by its last three digits is 
 divisible by 8. 
 
 By 3, when the sum of its 
 digits is divisible by 3. 
 
 By 9, when the sum of its 
 digits is divisible by 9. 
 
 By 11, when the difference 
 between the sum of its digits in 
 the odd places and the sum of 
 its digits in the even places is 
 divisible by 11. 
 
 By the product of any num- 
 ber of its factors which are sev- 
 erally prime to each other. 
 
 Illustration. 
 
 3960 is divisible by 2, because 
 is divisible by 2. 
 
 3960 is divisible by 5, because 
 is divisible by 5. 
 
 3960 is divisible by 4, because 
 60 is divisible by 4. 
 
 3960 is divisible by 8, because 
 960 is divisible by 8. 
 
 3960 is divisible by 3, because 
 18, the sum of its digits, is divisi- 
 ble by 3. 
 
 3960 is divisible by 9, because 
 18, the sum of its digits, is divisi- 
 ble by 9. 
 
 3960 is divisible by 11, because 
 0, the difference between 9 (the 
 sum of its digits in the odd places) 
 and 9 (the sum of its digits in the 
 even places) is divisible by 11. 
 
 3960 is divisible by the product 
 of 3 x 4 x 5 x 11, or 660, because 
 3, 4, 5, and 11 are factors of 3960, 
 and are severally prime. 
 
 Which of the numbers 2, 3, 4, 5, 8, 9, 11, are factors of : 
 
 12. 27720? 15. 48532? 18. 72754? 
 
 13. 3960? 16. 9768? 19. 3675? 
 
 14. 6732? 17. 19998? 20. 14175? 
 
 Ex. 12. 2, 3, 4, 5, 8, 9, 11. 
 
 AM. MENT. AS. 4 
 
50 AMERICAN MENTAL ARITHMETIC. 
 
 Name all the following numbers that are factors of 360360 
 
 21. 2, \ \\ 7, 8, \ 11, 13, 17, 19, 23. 
 
 22. 2x3,2x4,2x5,2x7,2x8,2x9. 
 
 23. 2x11,2x13,3x4,3x5,3x7x2. 
 24., 3 x 9, 3 x 11, 3 x 13, 4 x 5, 4 x 7 x 3. 
 
 25. 4 x 9, 4 x 11, 4 x 13, 5 x 7, 5 x 8 x 3. 
 
 26. 5 x 11, 5 x 13, 7 x 8, 7 x 9, 7 x 11 x 2. 
 
 27. 8x9,8x11,8x13,9x11,9x13. 
 
 28. 2 x 9, 3 x 8, 4 x 8, 5 x 9, 7 x 13, 6 x 7. 
 
 29. 2x3x8,2x4x5,2x5x9,6x9. 
 
 30. 2 x 7 x 9, 2 x 8 x 11, 3 x 4 x 5 x 8. 
 
 31. 3 x 4 x 11, 3 x 5 x 9, 3 x 7 x 9, 11 x 12. 
 
 32. 9x11x13,7x11x13,5x11x13. 
 
 33. 3 x 8 x 11, 3 x 11 x 13, 4 x 5 x 11 x 2. 
 
 34. 4x7x11,4x9x11,5x7x9x4. 
 
 35. 2 x 3 x 4, 2 x 3 x 5, 2 x 3 x 7, 20 x 7. 
 
 36. 2 x 3 x 9, 2 x 3 x 11, 2 x 3 x 13 x 5. 
 
 37. 2 x 4 x 7, 2 x 5 x 7, 2 x 5 x 8, 12 x 15. 
 
 38. 2x5x11,2x5x13,2x7x8x5. 
 
 39. 2 x 7 x 11, 2 x 7 x 13, 2 x 8 x 9 x 7. 
 
 40. 2 x 9 x 11, 2 x 9 x 13, 2 x 11 x 13. 
 
 41. 3x4x7,3x4x8,3x4x9x7x11. 
 
 42. 3x4x13, 3x5x7,3x5x8x11x2. 
 
 43. 3 x 5 x 11, 3 x 5 x 13, 3 x 7 x 8 x 9. 
 
 44. 3 x 7 x 11, 3 x 7 x 13, 3 x 8 x 9 x 2. 
 
 45. 3 x 8 x 13, 3 x 9 x 11, 3 x 9 x 13 x 5. 
 
 46. 3 x 5 x 11 x 13, 4 x 8 x 9 x 11 x 13. 
 
 47. 5x7x8x13,5x7x8x11x9x13. 
 
 48. 5 x 8 x 9 x 11, 5 x 8 x 9 x 13 x 2 x 3. 
 
 49. 3x5x9x11,3x4x7x9x11x13. 
 
 Ex. 29. 2x5x9, because 2, 5, 9 are severally prime. 
 
FACTORING. 51 
 
 Of the following, some of the factors are given. Find 
 two or three more for each, by taking the product of factors 
 severally prime : 
 
 50. Number 360 ; factors 4, 9, 8. 
 
 51. Number 1155 ; factors 3, 7, 5, 11. 
 
 52. Number 1260 ; factors 12, 15, 7. 
 
 53. Number 600 ; factors 3, 4, 10, 5. 
 
 54. Number 210 ; factors 15, 14, 7, 2. 
 
 55. Number 660 ; factors 20, 33, 3. 
 
 56. Number 2520 ; factors 5, 7, 8, 9, 4, 3, 6. 
 
 57. Number 2431 ; factors 11, 13, 17. 
 
 Ex. 53. 12 or 3 x 4, 30 or 3 x 10, 15 or 3 x 5, etc. 
 
 By inspection tell why : 
 
 58. 1224 is divisible by 72. 63. 2034 is divisible by 18. 
 
 59. 3465 is divisible by 55. 64. 1463 is divisible by 77. 
 
 60. 2394 is divisible by 63. 65. 3144 is divisible by 24. 
 
 61. 10208 is divisible by 88. 66. 980 is divisible by 35. 
 
 62. 10197 is divisible by 99. 67. 1728 is divisible by 72. 
 
 Ex. 58. 1224 is divisible by 8 and by 9 ; hence by 8 x 9 because 8, 9 are 
 severally prime. 
 
 By inspection determine a common factor of : 
 
 68. 36, 48, 72. 73. 300, 250, 400. 
 
 69. 77, 88, 121, 22. 74. 3260, 84, 96. 
 
 70. 96, 56. 75. 395, .95, 625. 
 
 71. 360, 144, 9872. 76.. 88, 84, 90. 
 
 72. 235, 25. 77. 378, 117, 234. 
 

 Illustration. 
 
 
 4x6x8 
 
 2 
 
 4x6x8 
 2 
 
 4x6x8 
 
 2 
 
 4 x 6 x 16 
 
 4 x 12 x 8 
 
 8x6x8 
 
 2)4x6x8 
 2x6x8 
 
 2)4x6x8 
 4x3x8 
 
 2)4x6x8 
 4x6x4 
 
 52 AMERICAN MENTAL ARITHMETIC. 
 
 § 13. Multiplication and Division. 
 
 A number expressed by its 
 factors will be multiplied by 
 a number, if any one of its 
 factors is multiplied by that 
 number. 
 
 A number expressed by its 
 factors will be divided by a 
 number, if any one of its 
 factors is divided by that 
 number. 
 
 Multiply : 
 
 78. 2x3x4 by 6. 83. 2x3x4 by 5 x 6. 
 
 79. 7 x 5 x 8 by 7. 84. 3 x 4 x 5 by 2 x 3. 
 
 80. 9 x 7 x 3 by 4. 85. 6 x 5 x 7 by 4 x 5. 
 
 81. 8 x 2 x 5 by 3. 86. 8 x 3 x 9 by 2 x 6. 
 
 82. 9x6x7 by 2. 87. 7 x 9x8 by 2x3. 
 
 Ex. 78. 2 x 3 x 24, 2 x 18 x 4, or 12 x 3 x 4. 
 
 Ex. 83. 10 x 18 x 4, or 2 x 15 x 24, or 10 x 3 x 24, etc. 
 
 Divide : 
 
 88. 9x18x6 by 3. 93. 54x64x18 by 18 x8. 
 
 89. 12x10x8 by 5. 94. 72x96 x 48 by 24x8. 
 
 90. 17x3x6 by 17. 95. 81 x 72x44 by 9x11. 
 
 91. 14x18x12 by 6. 96. 76x24x34 by 19x17. 
 
 92. 18 x 24 x 36 by 12. 97. 48 x 36x24 by 16x12. 
 Ex. 88. 3 x 18 x 6, or 9 x" 6 x 6, or 9 x 18 x 2. 
 
 Ex. 93. 3 X 8 x 18, or 54 x 8 x 1. 
 

 FACTORING. 
 
 Divide : 
 
 98. 
 
 85x6x7 by 17. 104. 64x7x9 by 56 
 
 99. 
 
 95x8x3 by 19. 105. 70x2x6 by 14 
 
 100. 
 
 95x8x3 by 24. 106. 90x3x7 by 54 
 
 101. 
 
 80x9x7x11 by 88. 107. 78x3x7 by 39 
 
 102. 
 
 96x12x7x11 by 112. 108. 75x4x8 by 50 
 
 103. 
 
 72 x 8 x 7 by 168. 109. 52 x 8 x 7 by 91 
 
 
 110. 9x12x14 by 36; 42; 108. 
 
 
 ill. 56 x 8 x 9 by 64 ; 72 ; 168. 
 
 
 112. 26 x 8x12 by 24; 32; 104. 
 
 
 lis. 75 x 84 x 16 by 25 ; 16 ; 175. 
 
 
 114. 96x35x17 by 48; 14; 672. 
 
 
 lis. 84 x 20 x 16 by 28 ; 40 ; 480. 
 
 
 116. 19 x 18x14 by 28; 36; 126. 
 
 
 117. 75 x 42 x 28 by 56 ; 42 ; 100. 
 
 
 lis. 30x70x20 by 14; 24; 210. 
 
 
 119. 17 x 19x18 by 34; 38; 153. 
 
 
 120. 20 x 21 x 22 by 28 ; 56 ; 154. 
 
 
 121. 23 x 24x25 by 92; 30; 115. 
 
 
 122. 26 x 27 x 28 by 13 ; 63 ; 117. 
 
 
 123. 29 x 30x31 by 58; 62; 186. 
 
 
 124. 32 x 33 x 34 by 88 ; 44 ; 136. 
 
 
 125. 44x45x46 by 55; 22; 460. 
 
 
 126. 47x48x49 by 47; 21; 112. 
 
 
 127. 62 x 63x64 by 93; 16; 288. 
 
 
 128. Q5xQ6x 67 by 26 ; 67 ; 143. 
 
 
 129. 68x69x70 by 69; 23; 115. 
 
 53 
 
 Ex. 98. 5x6x7. 85 + 17 = 5. 
 
 Ex. 101. 10 x 9 x 7. Factors 88 are 8 and 11 ; 80 -*- 8 = 10 ; 11 -r- 11 = 1. 
 
 Ex. 127. 2 x 21 x 64. Factors 93 are 31 and 3 ; 62 -r- 31 = 2 ; 
 63 -j- 3 = 21. 
 
54 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 14. Greatest Common Divisor. 
 
 Illustration. 
 
 2 
 
 72 
 
 144 
 
 108 
 
 3 
 
 6 
 
 12 
 
 9 
 
 
 2 
 
 4 
 
 3 
 
 12 x 3 
 
 32)70(2 
 64 
 6 
 
 25)75 
 3 
 
 G. C. D. 
 
 G. C. D. 32, 70 is 2. 
 G. C. D. 32, 6 is 2. 
 
 G. C. D. 75, 36 is 3. 
 G. C. D. 3, 36 is 3. 
 
 The G. C. D. of two or more 
 numbers is the product of all 
 the common factors which may 
 be used as successive divisors 
 until the quotients are prime to 
 each other. 
 
 The G. C. D. of two numbers 
 is the G. CD. of the smaller and 
 of the remainder found by divid- 
 ing the greater by the smaller. 
 
 One of the numbers may be 
 divided by a number prime to 
 one of the others without affect- 
 ing the G. C. D. 
 
 By the second principle, find the G. C. D. of : 
 
 130. 64, 96. 134. 35, 75. 138. 46, 69. 
 
 131. 56, 84. 135. 44, 90. 139. 
 
 132. 72, 108. 136. 27, 84. 140. 
 
 133. 24, 76. 137. 36, 75. 141. 
 
 Ex. 130. 32. The G. C. D. of 64 and 96 is the G. C. D. of 64 and 32 (the 
 remainder), or 32. 
 
 By the third principle, find the G. C. D. of : 
 
 142. 75, 96. 146. 36, 44. 150. 35, 91. 
 
 143. 98, 72. 147. 22, 36. i5i 
 
 144. 46, 68. 148. 77, 91. 152 
 
 145. 51, 72. 149. 80, 64 
 
 Ex. 142. 3. 75 + 25 = 3. 
 96, or 3. 
 
 40, 60. 
 32, 48. 
 38, 57. 
 
 72, 56. 
 
 33, 75. 
 
 153. 45, 95. 
 
 G. C. D. of 75 and 96 is the G. C. D. of 3 and 
 
FACTORING. 
 
 55 
 
 § 15. Least Common Multiple. 
 
 To find the L. C. M. of two 
 numbers, divide one of them by 
 their G. C. D., and multiply the 
 quotient by the other. 
 
 To find the L. C. M. of more 
 than two numbers, find the 
 L. C. M. of two of them, then of 
 the result and a third, and so 
 on. 
 
 If one of the numbers exactly 
 contains another, the smaller 
 may be neglected. 
 
 Illustration. 
 L. C. M. 10 and 12 is 60. 
 
 12 - 2 = 6. 
 10 x 6 = 60. 
 
 L. C. M. 10, 12, 15 is 60. 
 L. C. M. 10, 12 is 60. 
 L. C. M. 60, 15 is 60. 
 
 L. C. M. 12, 24 is 24. 
 
 12 may be neglected. 
 
 By the first principle, find the L. C. M. of 
 
 154. 12,14. 
 
 155. 15, 12. 
 
 156. 16, 20. 
 
 157. 24, 32. 
 
 158. 40, 50. 
 
 159. 60, 80. 
 
 160. 30, 26. 
 
 161. 14, 21. 
 
 162. 18, 20. 
 
 163. 32, 36. 
 
 164. 36, 40. 
 
 165. 60, 72. 
 
 166. 25, 30. 
 
 167. 30, 40. 
 
 168. 24, 27. 
 
 169. 96, 84. 
 
 170. 49, 63. 
 
 171. 72, 96. 
 
 172. 48, 52. 
 
 173. 75, 80. 
 
 174. 60, 72. 
 
 175. 35, 42. 
 
 176. 78, 52. 
 
 177. 68, 85. 
 
 178. 95,57. 
 
 179. 90, 72. 
 
 180. 80, 96. 
 
 181. 42, 63. 
 
 182. 84, 63. 
 
 183. 56, 42. 
 
 184. 76, 72. 
 
 185. 50, 75. 
 
 186. 65, 26. 
 
 Ex. 154. 84. The G. C. D. of 12, 14 is 2 ; 14 +- 2 = 7 ; 12 x 7 = 84. 
 
COMMON FRACTIONS. 
 
 § 16. First Conception — An Expression of Division. 
 
 Division may be expressed 
 by writing the dividend 
 above, and the divisor below, 
 a line. Such an expression 
 is a common fraction ; the di- 
 vidend is the numerator ; the 
 divisor, the denominator. 
 
 The numerator, or the de- 
 nominator, or both, may con- 
 tain fractions ; such an expres- 
 sion is a complex fraction. 
 
 We sometimes speak of a 
 fraction of a fraction ; a com- 
 pound fraction. 
 
 An integer plus a fraction 
 is a mixed number. The plus 
 sign is usually omitted. 
 2. 
 
 Analyze -|. 5. 
 
 Define by first conception : 
 
 1. A common fraction. 
 
 2. A complex fraction. 
 
 3. A compound fraction. 
 
 66 
 
 Illustration. 
 
 f , common fraction. 
 
 It means 4-^-5. 
 
 4, numerator. 
 
 5, denominator. 
 read, 4 -r- 5. 
 
 4 f t 9 i 
 
 complex fractions. 
 
 f of f , a compound fraction. 
 
 6f , a mixed number. 
 
 f ; 3 is the numerator ; 4, the de- 
 nominator ; it means 3-^-4. 
 
 f is the numerator ; f , the denom- 
 inator ; it means | -t- 1 . 
 
 4. A mixed number. 
 
 5. The numerator. 
 
 6. The denominator. 
 
COMMON FRACTIONS. 
 
 57 
 
 § 17. Second Conception — One or More of the 
 Equal Parts of a Unit. 
 
 A unit may be divided 
 into two or more equal parts, 
 and one or more of those 
 parts may be taken. 
 
 The number denoting into 
 how many parts the unit is 
 divided is written below a 
 horizontal line, and is the 
 denominator. 
 
 The number showing how 
 many parts are taken is 
 written above the line, and 
 is the numerator. 
 
 The whole expression is a 
 common fraction. 
 
 According to this concep- 
 tion, is | a fraction? No. 
 It is called an improper frac- 
 tion, i.e. not properly a frac- 
 tion. 
 
 According to this concep- 
 
 2 
 
 tion, is | a fraction ? 
 I 
 
 Illustration. 
 
 A C B 
 
 AB is divided into 8 equal parts ; 
 AG contains 5 of them; AG = 5 
 eighths of AB; expressed, AG — f of 
 AB. 
 
 f, common fraction. 
 5, numerator. 
 8, denominator. 
 read, 5 eighths. 
 
 It means that a unit is divided into 
 8 equal parts and 5 of those parts 
 are taken. 
 
 }, read one half; f, read 3 quar- 
 ters, or 3 fourths. 
 
 It is impossible to divide a unit 
 into 5 equal parts and then take 8 of 
 them. 
 
 No. It is impossible to divide a 
 unit into £ equal parts. 
 
 Define by second conception 
 
 7. A common fraction. 
 
 8. The denominator. 
 
 9. The numerator. 
 10. An improper fraction. 
 
58 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 18. Change of Form — To Higher Terms. 
 
 Multiplying both numera- 
 tor and denominator by the 
 same number does not change 
 the value of a fraction. 
 
 This is to prepare frac- 
 tions for addition and sub- 
 traction. 
 
 Illustration. 
 
 A B C 
 
 I ' I ' I ' I 
 
 AB = l orf, of AC. 
 
 1 = 1 
 
 (multiplying both terms by 2). 
 
 Change : 
 
 11. f to 16ths. 
 
 12. I to 12ths. 
 
 13. I to 40ths. 
 
 14. T 9 g to 48ths. 
 
 15. f to 35ths. 
 
 16. 4 to 36ths. 
 
 to 24ths. 
 
 17. 
 
 12 
 
 18. ^3 to 39ths. 
 
 19. -fj to 77ths. 
 
 20. T 5 ¥ to 28ths. 
 
 21. 
 
 T 9 T to 51sts. 
 
 22. 11 to 56ths. 
 
 23. 
 
 if to 60ths. 
 
 24. if to 84ths. 
 
 25. 
 
 if to 90ths. 
 
 Ex. 11. jf. To make the denominator 16, we must multiply it by 4 ; 
 multiplying both terms by 4, § = if. 
 
 Reduce to equivalent fractions having their least common 
 denominator : 
 
 26. 
 
 1 3 5 
 3' 4' 6* 
 
 31. 
 
 1 1 JL 
 
 5' 7' 3 0" 
 
 36. 
 
 5 3 4 
 l¥' 5 6' y 
 
 27. 
 
 f ' i 3 4' 2 8 r 
 
 32. 
 
 1 _5_ _7 
 ¥' 12' 2¥* 
 
 37. 
 
 4 6 7_ 
 33' 66' 11' 
 
 28. 
 
 g > 1 1 jr. 
 
 9' 18' 36* 
 
 33. 
 
 9' 3 6' Y2- 
 
 38. 
 
 Y' 2 8' T* 
 
 29. 
 
 1 6 3' 3 9' 52* 
 
 34. 
 
 4 7_ 11 
 
 2 5' 50' 10 0* 
 
 39. 
 
 16' 18' 8' 
 
 30. 
 
 1 1 1 
 
 2' 3' 6* 
 
 35. 
 
 4 X - 5 - 
 Y' Y7' 11* 
 
 40. 
 
 i\' f' eV 
 
 Ex. 26. T V, T \, if. Multiplying both terms of § by 4 ; of |, by 3 ; of £, 
 by 2. 
 
COMMON FRACTIONS. 
 
 59 
 
 § 19. Change of Form — To Lower, Terms. 
 Dividing both numerator and Illustration. 
 
 denominator by the same number 
 does not change the value of a 
 fraction. 
 
 This is to reduce fractions to 
 their simplest forms. 
 
 Which fraction is the more 
 readily comprehended, ||-, or f ? 
 Why? 
 
 J B 
 
 p-n 
 
 m 
 
 AB = %, or I, oiAC. 
 
 (dividing both terms by 2). 
 
 smaller. 
 
 Reduce to lowest terms 
 
 «• » A'flA' A- 
 
 42 _8_ 10. 16 11 18 
 
 **• 16' 20' 32' 33' 36* 
 
 4.0 19 11 16 12 21 
 * d * ¥¥' 4?' 4 8"' ¥¥' 5 0' 
 
 44 14 18 2 9 11 18 
 
 ™ 52' 5¥' 58' 68' T2 ' 
 
 45 11 15. 12 65 13 
 * 3, 22' 39' 65' 18* 91* 
 
 *«• ft ft ft' il- tt • 
 
 47. 
 
 84"' 8 4' "84' 8 4"' 8"4* 
 
 12 32 64 48 72 
 96' 96' 96' 96' 96* 
 
 P il 11 11 5 
 T2' 96' 64' F0' ¥5* 
 
 .30 41 11 30 
 60' 60' 90' f 5' 
 
 48. 
 49. 
 50. 
 51. 
 52. 
 53. 
 
 24 
 
 T8"' 98"' 3 0' 84' 
 
 1£ 4 1 14 13 2 _H_ 
 39' 92' 96' 14¥' 121* 
 
 112 14 4 1^5 _6_26_ 
 336' 1T2^' 625' 1250* 
 
 JUL 28. 11 _6_3_ XX - 3 1 
 133' 98' 98' 112' $4' 63* 
 
 169 116. 2 21 2 51 2 8 9. 
 13 0' 140' 150' 16 0' llO* 
 
 _46 _30_ _21_ JUL JUL 
 138"' 150' 200' 152' 253* 
 
 _3 9_ _7_6 56, 1-0 5 _9_2_ 
 351' 532' 385' 525' 368* 
 
 » ¥%' iVV S%% A 5 5' AV 
 
 54. 
 55. 
 56. 
 57. 
 
 59. 
 60. 
 61. 
 62. 
 63. 
 64. 
 
 .2 4. 3 61 
 
 4 41 41 
 
 6 0' 5 7 0^' 8 0' 8¥"0"' 8F0* 
 
 5 21 5 11 121 8 21 
 690' 720' 750' ¥0' 30* 
 
 6 4 12_5. 211 3 43. 512 
 80' 500' 360' ¥20' ¥70* 
 
 129 24 31 41 60 .71 
 9 00' 3 0' ¥2' 12' 6 6' 91* 
 
 7 2 14. 93 5 6 77 __8_5_ 
 18' 90' 102' 63' 84' 102* 
 
 2T' 6¥' l¥§"' 2 3 1 6 6' 3¥3' f ' 
 
 65 J5JL 8 1_ 4 21 111 12.1 
 ° 3, 512' T29' 850' 930' BTS' 
 
 66 2 21 777 101 245 _8_25_ 
 DO * 669' 630' 161' 735' 1650* 
 
 Ex. 41. |, f, |, \, i. Divide both terms of if by 6 ; of / T , by 9 ; etc. 
 
60 
 
 AMERICAN MENTAL ARITHMETIC, 
 
 § 20. Change of Form — To a Whole or Mixed 
 
 Number. 
 
 A fraction is an expression of divi- 
 sion. It means that the numerator 
 is to be divided by the denominator. 
 
 A mixed number is an integer plus 
 a fraction. If the integer is re- 
 duced to an equivalent fraction hav- 
 ing the denominator of the fraction, 
 the two parts may be united. 
 
 To reduce a mixed number to an 
 improper fraction is a case in addi- 
 tion of fractions. 
 
 Illustration. 
 
 8 92 
 
 £—3 
 
 a t». 
 
 2f = 2 + f . 
 
 2 = |. 
 + t=f. 
 
 therefore 
 2§ = f. 
 
 Reduce to a whole or mixed number 
 
 e?. yȴ- 
 
 
 "• ft 4f 
 
 75. 
 
 5 5 68. 
 13' 15' 
 
 68. ff, f f . 
 
 
 72. fl, |f. 
 
 76. 
 
 15 8.5_ 
 16' 13' 
 
 «»• if. ft- 
 
 
 73. |f |f. 
 
 77. 
 
 M 81 
 1^' 13" 
 
 70 - ihii- 
 
 
 74. |f, f|. 
 
 78. 
 
 .99 .9.6. 
 1^' 15" 
 
 Ex. 67. 13f Performing 
 
 the indicated division, 96 h 
 
 r- 7 = 135. 
 
 Reduce to an 
 
 improper fraction : 
 
 
 
 79. 5, 4^. 
 
 
 83. 5i|, 6 ^. 
 
 87. 
 
 Hi- 
 
 so. 71 8|. 
 
 
 84. 311 4 jy. 
 
 88. 
 
 m- 
 
 81. 9f, 6|. 
 
 
 85. 5i|, 4&. 
 
 89. 
 
 Hi- 
 
 82. 8|,8f. 
 
 
 86. 511 6 if . 
 
 90. 
 
 5if. 
 
 Ex. 79. f , |f . 4 = f f ; f f + T 5 T = |f . This example is often explained i 
 since there are || in 1, in 4, there are 4 times \% t or f$- ; f f + T 5 T = T f. 
 
COMMON FRACTIONS. 
 
 61 
 
 § 21. Addition and Subtraction. 
 
 Before fractions can be added or 
 subtracted, they must be reduced to 
 equivalent fractions having a common 
 denominator. 
 
 The least common denominator 
 should always be found. 
 
 Illustration. 
 
 A+A=tt- 
 
 Find the value of: 
 
 91. 
 
 2+3* 
 
 98. 
 
 A 4- 1 
 6^9* 
 
 105. 
 
 6* + 7f 
 
 92. 
 
 !+!• 
 
 99. 
 
 ll4_ A 
 13^ 6* 
 
 106. 
 
 8|+7f. 
 
 93. 
 
 t+fr 
 
 100. 
 
 tt+ f 
 
 107. 
 
 9f+8J. 
 
 94. 
 
 t+f 
 
 101. 
 
 8*+2f 
 
 108. 
 
 12f + 8f. 
 
 95. 
 
 HI- 
 
 102. 
 
 2*+8* 
 
 109. 
 
 10|+9f. 
 
 96. 
 
 *+*• 
 
 103. 
 
 3i+4f 
 
 110. 
 
 18f+3f. 
 
 97. 
 
 ♦+* 
 
 104. 
 
 5|+6f 
 
 111. 
 
 25f+6i 
 
 :. 91 
 
 • i+*=t+f«* 
 
 
 
 
 Find the value of : 
 
 112. i— J. 
 
 113. f- T V 
 
 115. 
 116. 
 
 117 - !-tV 
 
 118. f- |. 
 Ex. 125. 31 - 
 
 12' 
 ♦ "*■ 
 
 2_ A 
 
 119. 
 
 2f 
 
 126. 8J-2J. 
 
 120. 
 
 4|-lf 
 
 127. 
 
 n-2f 
 
 121. 
 
 8J-2i- 
 
 128. 
 
 8f-7|. 
 
 122. 
 
 H-H- 
 
 129. 
 
 Tf-6f. 
 
 123. 
 
 5|-3|. 
 
 130. 
 
 H-5f- 
 
 124. 
 
 2-l«. 
 
 131. 
 
 8J-6|. 
 
 125. 
 
 H'- *• 
 
 132. 
 
 9 f-4|- 
 
 l = 3xV-A = 2if- T \ = 2H. 
 
62 AMERICAN MENTAL ARITHMETIC. 
 
 § 22. Multiplication — Universal Case. 
 
 Multiply the numerators Illustration. 
 
 for a new numerator and the A D C B 
 
 denominators for a new de- I ' ' ' ' 
 
 nominator, canceling when 3 of AB = AC. 
 
 possible. % of AC = AD = ±oi AB. 
 
 .-. f of f of AB = I of AB. 
 Mixed numbers should 2 3 1 
 
 be reduced to improper £ x J ~ 2' 
 
 fractions. . 2 
 
 Find the value of : 
 
 133. T 3 ¥ x^-; ^xf; &xf. 139. f of 18; fx91; |x65. 
 
 134. §x6; 8x|; 9xf. 140. | of 25; f of 24; f of 16. 
 
 135. fx-&; 7xf; 9x^. 141. f of 21;^ of 44; T V of 65. 
 136. -J x 16; 18xf; 72 x|. 142. f of 20 ; ^ of 91 ; ^ of 55. 
 
 137. 84x^; £x96; |x48. 143. fxf; fxff; |x T V 
 
 138. 96x T \: i|x84; T 9_ X 64. 144. fof&j £of£; &ofl6. 
 
 145. T 9 eOfM;iiofi|;-lfof84. 
 
 146. 3|x6f ; 4f X5V; 51x12. 
 
 147. 2f X3J; 41x11; 6f x^- 
 
 148. 5l X ll; 21x21; 3ix2|. 
 
 149. ljxli; lfxli; 2|x2i 
 
 150. f of 16+§ of 9 + 11x8. 
 
 151. 1 Of 1 + 1 Off. 153. I X If -f Off. 
 
 152. f Of 1-1 Of 1 154. 31 X f-fx^. 
 
 Ex. 133. - 2 7 4 . Divide both 16 and 14 by 2. Speak no words except the 
 result. 
 
 Ex.152. T V; fof * = tf ; iof i = i; tf - * = A. 
 
COMMON FRACTIONS. 63 
 
 § 23. Division — Universal Case. 
 
 Divide the numerators for a new 
 numerator and the denominators for Illustration. 
 
 a new denominator, changing to equiva- 8 « = 4 
 
 lent fractions ivith their least common 
 denominator, if necessary. 
 
 f.+ l=A + A = 
 
 Mixed numbers should be re- 
 duced to improper fractions. 
 
 Find the value of : 
 
 155. 6-^f. 168. J.+f 181. 5-S-3J. 
 
 156. 18-^f. 169. £-s-f 182. 21-1-1. 
 
 157. 27 -*-£. 170. $•*-£, 183. 21-61 
 
 158. 84^-J- 171. f + f. 184. 7|^81 
 
 159. 63 -f 172. £-*■£. 185. 71 -h8f. 
 
 160. 48 -r-^g. 173. i 6 ^-- 1 /. 186. 9|-f-10f. 
 
 161. 54-^V ^4. l^lf 187. 6f^81 
 
 162. 77-f-^. 175. 21-3. 188. 9f-6f, 
 
 163. 20 -|, 176. 2f-=-4. 189. 4f-6i. 
 
 164. 32-^. 177. 3-21. 190 . 8f-f-6l 
 
 165. 16 - T \. 178. 4^f. 191. 9f+3f 
 
 166. 25-f-f 179. 1|-1 192. 7f-t-2f 
 
 167. 49 -f 180. 5-31 193. 8§-r-41 
 
 Ex. 155. 9. 6 -=- § = y -4- 1 = 9. If the dividend is an integer, it is 
 
 easier to divide the numerator and multiply the quotient by the denomi- 
 nator. Thus, 6 + § j 6-2 = 3; 3x3 = 9. 
 
 Ex.186. 11. 9J-*-10| = Y- ¥ = ¥■-*-¥ = *!• 
 
 Ex.188, ff. 9* + 6* = ¥ + ¥ = **• 
 
64 AMERICAN MENTAL ARITHMETIC. 
 
 § 24. Problems. 
 
 194. At \i each, what will 18 ap- Am. 12ft If 1 apple 
 pies cost ? costs ^ 18 apples will cost 
 
 195. If 1 yard costs 18* what will 18 tlmeS * ° r "* 
 
 | of a yard cost ? , ^ n f f • H ^ costs 
 
 • «r 18ft J of a yard will cost 
 
 .4ns. 12ft If 1 yard costs 18ft § of a yard i f 18ft or 6^ ; § of a yard 
 
 will cost § times 18ft or 12ft will cos t 2 times 6ft or 12ft 
 
 196. If 1 yard COSts 18£ what will Ans. 99ft If 1 yard costs 
 5| yards COSt ? 18ft 5 yards will cost 90f ; 
 
 4iw. 99ft If 1 yard costs 18ft 5 J yards will * of a ^ ard wiU cost ^ \ 
 cost 5£ times 18ft or 99ft 90^ + 9f = 99ft 
 
 The right-hand explanation of the last two examples is objectionable 
 because it explains the process of multiplication. The explanation should 
 simply point out the operation to be employed. 
 
 197. At %f each, what will 16 oranges cost ? 
 
 198. If 1 yard of cloth costs 25^, what will f of a yard cost ? 
 
 199. At 6^ a yard, what will 6| yards of cloth cost ? What 
 will 13 J yards cost ? 
 
 200. At 6 \t a yard, what will 6 yards of cloth cost ? 
 
 201. If a man earns f of a dollar per day, how much will 
 he earn in 20 days ? 
 
 202. If 1 quart pears costs 14^, what will ^ of a quart cost ? 
 
 203. If 1 apple costs |^, what will f of an apple cost ? 
 
 204. If 1 orange costs 2j^, what will 12 oranges cost? 
 
 205. At \f each, what will 20 pears cost? 
 
 206. At 6§^ a yard, what will 1\ yards of braid cost? 
 
 207. If 1 shawl costs I12 1 -, what will 15 shawls cost? 
 
 208. At 5J^ each, what will 16 cocoanuts cost ? 
 
 209. If a boy reads b\ pages of a book an hour, how much 
 will he read in 2 J hours ? 
 
COMMON FRACTIONS. 65 
 
 210. At \t each, how many apples can be bought for 18/ ? 
 
 Aiis. 24 apples. If 1 apple costs Ans. 24 apples. If 1 apple costs 
 
 f ?, \%? will buy as many apples as f f f, 1 cent will buy f apples ; 18^ will 
 is contained times in 18, or 24 apples. buy 18 times f apples, or 24 apples. 
 
 211. If -| of a yard of cloth costs 18/, what will 1 yard cost ? 
 
 Ans. 24?. If ^ of a yard costs 18^, If 5 yards cost 15?, what will 1 
 
 1 yard will cost § of 18^, or 24/-. yard cost ? 
 
 Ans. 24 ?. If f of a yard costs If 5 yards cost 15?, 1 yard will cost 
 
 \ I of a yard will cost £ of 18ft 
 
 or 6? ; f , or one yard, will cost 4 verting 5 ; in the same way we may 
 times Oft or 24ft invert f in the example above. 
 
 212. At J/ each, how many apples can be bought for 20/? 
 
 213. If | of an apple costs 2/, what will 1 apple cost ? 
 
 214. At §/ each, how many apples can be bought for 18/? 
 
 215. If | of an apple costs 4/, what does 1 apple cost ? 
 
 216. If 1 apple costs §/, how many apples can be bought 
 forf/? 
 
 217. If | of an apple costs |/, what does 1 apple cost? 
 
 218. If | of a melon cost 20/, what was the cost of the 
 whole melon? 
 
 219. If | of a house cost $ 1200, what did the whole house 
 cost? 
 
 220. If | of a ship is worth $ 15,000, what is the value of 
 the whole ship ? 
 
 221. If | of a farm cost $4200, what did the whole farm 
 cost? 
 
 222. If | of a store is valued at $6300, at what is the 
 whole store valued? 
 
 223. If ^ of a garden cost $120, what did the whole 
 cost ? 
 
 AM. MENT. AR. 5 
 
66 AMERICAN MENTAL ARITHMETIC. 
 
 224. 12 is | of what num* Ans. 18. If 12 is § of some num- 
 ber ? ber, i of the number is \ of 12, or 6 ; 
 
 Ans. 18. If 12 is § of some num- h or the whole number, is 3 times 
 
 ber, that number is 12 + §, or 18. °> or 18 ' 
 
 225. I of 20 is f of what number? f of what? 
 
 226. I of 40 is I of what number ? -^ of what ? 
 
 227. -J of 24 is -J of what number ? | of what ? 
 
 228. ^ of 52 is I of what number ? i| of what ? 
 
 229. T 8 g of 76 is I of what number ? |£ of what ? 
 
 230. T 5 T of 85 is || of what number ? f of what ? 
 
 231. T 3 6 - of 48 is f of what number? ^ of what? 
 
 232. I of 45 is f of what number ? - 2 ^ of \7hat ? 
 
 233. 2 8 t of 84 is if of what number? f of what? 
 
 234. I of 56 is I of what number ? |- of what ? 
 
 235. ^ °f 84 i s £ °f what number? § of what? 
 
 236. T 9 g of 80 is T 9 g of what number? jf of what? 
 
 237. -| of 50 is £ of what number ? -| of what ? 
 
 238. I of the scholars in a school, or 30, are girls; how 
 many are boys ? 
 
 239. A boy lost J of his kite-string, and gave away -J- of 
 the remainder; he then had 400 feet; how long was the 
 string at first? 
 
 240. Joseph is 12 years old; f of Joseph's age is f of 
 John's age ; how old is John ? 
 
 241. The head of a fish is 15 inches long;f| of the length 
 of the heac^ is T 3 T of the length of the rest of the body ; what 
 is the length of the fish ? 
 
 242. In a school, J of the students study algebra ; ^ of the 
 remainder, geometry ; and the rest, or 12, trigonometry ; how 
 many scholars are there ? 
 
COMMON FRACTIONS. 67 
 
 243. At 3 for 5^, how-many apples can be bought for 20^? 
 
 Ans. 12 apples. If 3 apples cost . ' , _. _.. . 
 
 „. ., . ... . . . Ans. 12 apples. Since 200 are 4 
 
 50, 1 apple will cost 40 ; as many orvrf ... .J 
 
 , ^ , , .* - ' _ , times 50, 200 will buy 4 times 3 
 
 apples can be bought for 200 as % is , , 
 
 . . ~« .J . apples, or 12 apples, 
 contained times in 20, or 12 apples. 
 
 244. If 6 quarts of berries cost 12^, what will 17 quarts 
 cost? 
 
 ^4ws. 34^. If 6 quarts cost 120, Ans. 340. Since 17 quarts are V- 
 
 1 quart will cost \ of 120, or 20 ; 17 times 6 quarts, 17 quarts will cost - 1 / 
 quarts will cost 17 times 20, or 340. times 120, or 340. 
 
 245. If 27 cans of tomatoes cost $ 2.70, what will 9 cans cost? 
 
 246. At 4 for 5^, how many apples can be bought for 20^? 
 
 247. At 2 for 3^, how many apples can be bought for 30^ ? 
 
 248. At 3 for 2^, how many apples can be bought for 30^ ? 
 
 249. If 4 quarts of berries cost 12J^, what will 6 quarts cost? 
 
 250. If 6 quarts of berries cost 12^, what will 18 quarts 
 cost? 
 
 251. If J of a pound of prunes costs 10^, what will f of a 
 pound cost? 
 
 252. If | of an apple costs f ^, what will f of an apple cost ? 
 
 253. If 5 cans of tomatoes cost 60^, what will 13 cans cost? 
 
 254. At 35^ a dozen, what will 3 oranges cost ? 
 
 255. At 20^ a dozen, what will 9 eggs cost ? 
 
 256. At 5^ a score, what will 60 clothes-pins cost ? 
 
 257. At $ 36 a dozen, what will eight pairs of shoes cost ? 
 
 258. If 16 men can earn $ 32, how much can 75 men earn ? 
 
 259. If | of a ship's cargo is worth $ 2400, what is § of the 
 cargo worth? 
 
 260. If 15 sheets of paper cost 10 cents, what will 24 sheets, 
 or one quire, cost? 
 
68 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 261. If A can do a piece 
 of work in 2 days, and B 
 can do the same work in 3 
 days, how long will it take 
 them working together? 
 
 262. If A and B together 
 can do a piece of work in 5 
 days, and A alone in 8 
 days, in how many days 
 can B alone do the work ? 
 
 Ans. 1 \ days. In 1 day, A can do \ 
 of it ; B can do \ of it ; they can both 
 do the sum of \ and i, or £ of it. If 
 they can do £ in 1 day, it will take as 
 many days to do f , or the whole, as $ is 
 contained times in |, or li days. 
 
 Ans. 13i days. In 1 day both can 
 do i of it ; A can do \, and Bi-|, or ¥ % 
 of it ; it will take B as many days to do 
 the whole as ^ 
 ££, or 131 days. 
 
 263. One pipe will fill a cistern in 4 hours; a second 
 pipe will fill it in 5 hours ; how long will it take both io 
 fill it? 
 
 264. Two pipes together fill a cistern in 6 hours ; the first 
 can fill it in 10 hours ; how long will it take the second to 
 fill it? 
 
 265. Two pipes carry water into a tank, and a third carries 
 water from it. The first pipe will fill it in 2 hours, the second 
 in 3 hours ; the third will empty the tank in 1 J hours ; if the 
 tank is empty and all 3 pipes are used, in what time will the 
 tank be full ? 
 
 266. A cistern holding 70 gallons has a pipe by which 15 
 gallons will run into the cistern in 1 hour, and another that 
 will discharge 10 gallons an hour; when both are running, 
 what part of the cistern will be filled in 3 hours? 
 
 267. A can do a piece of work in 3 days ; B can do the 
 same work in 4 days ; if A earns $ 2 a day, what does B 
 earn per day? 
 
 268. John is 16 years old, and James is f as old ; how old 
 is James? 
 
COMMON FRACTIONS. 69 
 
 269. In a school there are 27 girls, and f as many boys ; 
 how many scholars are there in the school ? 
 
 270. A boy, having 80 marbles, lost -f and sold T 3 B of them ; 
 how many had he left? 
 
 271. A man said that | of his money was 4 times his 
 week's wages ; he had $ 100 ; what were his week's wages ? 
 
 272. If a bushel of wheat costs $1-|, and a bushel of corn 
 $f, what is the difference in the cost of 5 bushels of each? 
 
 273. John gave away | of his marbles, and lost f of the 
 remainder ; how many had he left if he had 60 at first? 
 
 274. In traveling 72 miles a man went f of the distance 
 the first day, J of the distance the second day, and the 
 remainder the third day; how far did he travel the third 
 day? 
 
 275. Of a flock of sheep -J are in one field, J in a second, 
 ^ of the remainder, or 14, in the third ; how many sheep are 
 there in the flock? 
 
 276. After spending £ of my money, and losing J of the 
 remainder, I have $ 30 left ; how much had I at first ? 
 
 277. By selling a watch at a loss of $ 36, I lost |- of its 
 value ; what was its val»e ? C * j 
 
 278. By selling a watch for $36, I lost £ of its value; 
 what was its value ? 
 
 279. By selling a watch for $ 36, I gained -J of its value ; 
 what was its value ? 
 
 280. A man bought 6 gallons of vinegar at 12-|^ a gallon, 
 and paid for it in oranges at 36^ a dozen ; how many oranges 
 did it take ? 
 
 281. Walter is -| as old as his father, and f as old as his 
 mother; if he is 18 years old, how old are his father and 
 mother ? 
 
70 AMERICAN MENTAL ARITHMETIC. 
 
 282. I bought 100 pounds of sugar at 4f ^ a pound, and paid 
 for it with codfish at 12|/ a pound; how many pounds of 
 codfish did it take ? 
 
 283. How many barrels of flour, at $6-J a barrel, must be 
 given in exchange for 25 barrels of apples at $ 3 a barrel ? 
 
 284. How many dozen eggs, at 12|^ a dozen, will pay for 5 
 pounds of candy at 10^ a pound ? Qj 
 
 285. A person owning | of a ship sold I ©T his share for 
 jgfiffi ; ?? wliat was the value of the ship? * 
 
 286. If a man can do a piece of work in 12J days, work- 
 ing 8 hours per day, how many days will it take, working 10 
 hours a day ? 
 
 287. If 6 cakes cost 15/, what will 7 cakes cost? 
 
 288. If |- of a melon costs 12^, how many apples at 2^ each 
 will buy the melon ? 
 
 289. I sold a cow for $ 32, which was | of her cost ; what 
 was the cost? 
 
 290. A man sold a cow at a loss of $16; the loss was | of 
 her value ; what was her value ? 
 
 291. If | of a sum of money is T ^- of the value of a horse, 
 ^Rind \ of the value of the horse is $ 20, what is the sum of 
 
 money ? 
 
 292. At a selling price of $18 for sheep, \ the cost of the 
 sheep was lost ; what was the cost ? 
 
 293. | of | of 16 is | of H of how many times 5 ? 
 
 294. A lady bought 10 pounds of raisins at 12-^ a pound, 
 and paid for them with currants at 5^ a pound ; how many 
 pounds of currants did it take ? 
 
 295. If 8 pounds of soda cost 9|^, what will 5 pounds cost ? 
 
 296. How many pounds of coffee, at 33|^ a pound, must be 
 given in exchange for 8 pounds of tea at 66|^ a pound ? 
 
COMMON FRACTIONS. 71 
 
 297. A has $1|, and B $2|; they divide what they both 
 have equally between two persons; how much does each 
 receive ? 
 
 298. If a river flows 2| miles in 3£ hours, how far will it 
 flow in 1 hour? 
 
 299. -| is |- of what number ? 
 
 300. | is ^ of how many times -fa ? 
 
 301. | of 3|- is fa of how many times ^ ? 
 
 302. When cheese is -fa of a dollar a pound, what will § 
 of a pound cost ? 
 
 303. If I buy turkeys at the rate of 5 for $ 3, and sell at 
 the rate of 8 for $7, how much will I gain on 40 turkeys? 
 
 304. How many pigs can I buy for $ 75, at the rate of 3 
 for $7, and have $5 left? 
 
 305. If 4 men can dig a ditch in 16 days, what part of it 
 will three men dig in 7 days ? 
 
 306. If a man were twice as old, \ of his age would be 20 
 years; how old is he? 
 
 ^307. B gave f of all his money for a cow; he paid $12 
 for hens, which was f of all the money he had left ; how 
 much had he at first ? 
 
 308. I bought stock at $800 ; f of this is f of f of 2 times 
 the present value of the stock ; what is its present value ? 
 
 309. ^_ of 85\s if of how many times ff of -| of 50 T\ 
 
 310. -^ is | of what number? 
 
 311. hjfa of 7 T 6 ^)is | of what number ? 
 
 312. Frank is 16 years old ; if 4 years were added to his age, 
 he would be f as old as his brother; how old is his brother? 
 
 313. f of f of 2 x 28 is f of | of what number? 
 
 314. | the sum of two equal numbers is 20 ; what are the 
 numbers? 
 
DECIMALS. 
 
 The principles of the decimal nota- 
 tion may be extended to certain frac- 
 tions by placing a period, called a 
 decimal point, after units 1 place, and 
 writing the fraction to the right as 
 lower orders in the decimal scale. 
 
 The orders to the right of the 
 decimal point are tenths, hundredths, 
 thousandths, ten-thousandths, hundred- 
 thousandths, millionths, etc., and only 
 those fractions which have these de- 
 nominators can be written as decimals. 
 
 The part to the left of the decimal 
 point is an integer; the part to the 
 right, a decimal fraction. 
 
 The fractional part is not read on 
 the same plan as the integral part. 
 
 We read the whole as an integer 
 for the numerator, and then declare 
 the denomination of its last digit for 
 the denominator. 
 
 The first reading represents a frac- 
 tion plus a fraction ; the second, their 
 sum. 
 
 Illustration. 
 
 345.6789 
 
 5, in units' place. 
 
 . , decimal point. 
 
 1 unit = 10 tenths (jg) ; 
 tenth = 10 hundredths 
 
 ( T Vk); 1 hundredth 
 
 10 
 
 thousandths ( T £§o) 5 etc * 
 
 345.6789 
 
 345, integer. 
 
 .6789, decimal fraction. 
 
 .234567, on the plan of 
 reading integers, would be 
 read 234 thousandths, 567 
 millionths. 
 
 It is actually read, 234 
 thousand 567 millionths. 
 
 First. 
 
 Second. 
 
 rWff + Ttfro oinr — iVo 4 o 5 oV< 
 
 ITS' 
 
 72 
 
DECIMALS. 
 
 73 
 
 To write a fraction whose denomi- 
 nator is 10, 100, 1000, etc. (that is, 
 a decimal fraction), we write the 
 numerator in the usual manner, and 
 indicate the denominator by the aid 
 of a decimal point. 
 
 . 3 _ - .003 
 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 D 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ■• 
 
 
 
 z 
 
 
 
 
 
 
 
 
 
 
 
 CO 
 
 
 
 
 
 
 < 
 
 
 
 
 
 
 
 
 
 
 
 X 
 
 
 
 
 
 
 CO 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 H 
 
 
 
 CO 
 
 
 
 3 
 
 Q 
 
 
 
 
 
 
 
 
 
 
 Q 
 
 
 
 X 
 
 
 
 o 
 
 z 
 
 
 
 
 
 
 
 
 
 CO 
 
 Z 
 
 
 
 h 
 
 
 
 I 
 
 < 
 
 
 
 
 
 ^ 
 
 
 
 
 I 
 
 < 
 
 
 
 z 
 
 
 
 h 
 
 IL 
 
 o 
 
 CO 
 
 CO 
 
 3 
 
 o 
 
 x 
 
 CO 
 Q 
 
 CO 
 
 
 
 h 
 
 z 
 
 2 
 
 
 CO 
 
 X 
 
 h 
 
 to 
 
 X 
 
 g 
 
 h 
 
 Q 
 
 z 
 < 
 
 CO 
 
 CO 
 
 3 
 O 
 
 X 
 h 
 
 CO 
 
 X 
 
 CO 
 
 I 
 h 
 z 
 
 
 
 o 
 
 -1 
 
 I 
 
 CO 
 
 X 
 
 CO 
 
 s 
 
 u. 
 
 z 
 
 Q 
 
 
 
 _i 
 
 
 o 
 
 z 
 
 Q 
 
 Q 
 
 h 
 
 Zi 
 
 Q 
 
 h 
 
 z 
 
 w 
 
 
 
 < 
 
 UJ 
 
 
 
 < 
 
 (f) 
 
 Ul 
 
 < 
 
 
 Ul 
 
 z 
 
 -1 
 
 Ul 
 
 z 
 
 o 
 
 _l 
 
 cc 
 Q 
 
 z 
 
 CO 
 
 z 
 
 CO 
 
 3 
 O 
 
 cc 
 Q 
 
 z 
 
 CO 
 
 z 
 
 £ 
 
 5 
 O 
 
 I 
 fc 
 
 z 
 
 cc 
 
 a 
 
 z 
 
 CO 
 
 3 
 
 o 
 
 z 
 
 D 
 
 z 
 
 o 
 
 -1 
 
 I 
 z 
 
 cc 
 o 
 
 z 
 
 o 
 
 -1 
 
 
 3 
 
 Ul 
 
 X 
 
 3 
 
 UJ 
 
 z 
 
 
 Ul 
 
 3 
 
 X 
 
 Ul 
 
 3 
 
 
 Ul 
 
 3 
 
 
 1 
 
 I 
 
 1- 
 
 h 
 
 I 
 
 H 
 
 3 
 
 •— 
 
 h 
 
 X 
 
 H 
 
 h 
 
 X 
 
 I 
 
 h 
 
 X 
 
 CD 
 
 2 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 3 
 
 • 
 
 6 
 
 7 
 
 8 
 
 4 
 
 5 
 
 2 
 
 1 
 
 7 
 
 8 
 
 
 
 INTEGER. 
 
 
 
 
 
 
 
 DECIMAL. 
 
 
 
 
 Beginning with the decimal point, numerate : 
 
 1. .23456789254. 4. .365; .4685; .032; .007. 
 
 2. .000240506782. 5. .001; .00024; .007058. 
 
 3. .0000123045123. 6. .20304; .506; .8075. 
 
 Read: 
 
 7. Ex. 4. 
 
 8. Ex. 5. 
 
 9. Ex. 6. 
 
 io. .0000001. 13. 800.3065. 16. 49675.35. 
 n. .234|. 14. 40.087. 17. 3.000003. 
 12. 60.005. 15. 2683.3. 18. 5.604002. 
 
 Ex. 11. 234| thousandths. A common fraction is of the same denomina- 
 tion as the order which it follows. 
 
 Ex. 13. 800 and 3 thousand 65 ten-thousandths. In reading mixed num- 
 bers, and is used only for the decimal point. 
 
74 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 25. Reduction — Common Fractions to Decimals. 
 
 | means 5 ~- 8 ; performing the 
 indicated operation, we obtain .625. 
 
 The factors of 10 are 5 and 2 ; 
 hence, if a fraction has any factor 
 other than 2 and 5 in its denomina- 
 tor, the division will not be exact. 
 
 In such cases, the usual plan is to 
 
 carry out the division a few places 
 
 and to write the remainder, or to write 
 
 the sign «+' instead of the remainder. 
 
 As a curiosity, the division may be carried out until the quotient begins 
 
 to repeat, and dots may be placed over the first and last of the figures which 
 
 repeat, as in 2d result. The part which repeats is then called a repetend. 
 
 19. Reduce J to a 
 
 decimal. 
 
 8)5.000 
 
 
 .625 
 
 
 20. Reduce f to a 
 
 decimal. 
 
 1st result. 
 
 
 7)2.000 
 
 
 .285^, 01 
 
 • .285+ 
 
 2d result. 
 
 
 7)2.000000 
 
 
 .285714 
 
 
 To be memorized : 
 .50 
 
 1 
 
 2 
 
 I = .20 
 | = .831 
 
 f=.40 
 
 | = .75 
 .80 J=.16f 
 | = .621 j = . 87 j 
 
 Give the decimal equivalents rapidly : 
 
 a- h h h h h % h h h h 23- Vt> I* f> h h t. h h f • 
 
 421783211 | 
 & 3' P & 6' S> 5' 2' 5> 5* 
 
 22. |, h h b b b b b h !•■ 24 
 
 etc. Do not look at above table. 
 
 Reduce to decimals: 
 25. f f, f, t, f, 1, I £. 27. £, fa £, ft, 1J, J 3 , fl 
 
 26 - I. f f. ^> ft> A. A. A- 2 «- tV iV fe A- if > tV A 
 
 Ex. 25. 1st. .14f ; } = 1 + 7 =.14J. 
 
DECIMALS. 75 
 
 §26. Reduction — Decimals to Common Fkactions. 
 Give the equivalents in common fractions rapidly : 
 
 29. .50; .33 J ; .871. 33. .871; .66| ; .40. 
 
 30. .12|; .16}; .331 3 4. .75; .20; .60. 
 
 31. .25; .20; .371. 35. .88}; .62}j,87}. 
 
 32. .80; .831; .621 3 6. .75; .331 ;.66§. 
 
 Ex. 29. |, I , |. Having memorized the table in the previous section, 
 the student should read these at a glance. 
 
 Reduce to improper fractions : 
 
 37. 3.50; 2.75; 3.40. 45. 2.6; 5.8; 4.66|. 
 
 38. 1.331; 3.66f ; 2.121 4 6. 3.621 ; 5.5 . 6.25. 
 
 39. 3.371 ; 2.831; 1.16}. 47. 7.2; 8.75; 9.6. 
 
 40. 4.25; 6.871; 9.2. 48. 5.66}; 7.831; 6.871 
 
 41. 7.50 ; 9.871 ; 4.621 49. 8.50 ; 5.621 ; 6.831 
 
 42. 9.75; 5.831; 6.80. 50. 9.7; 8.621; 7.331. 
 
 43. 3.16f ; 5.331 ; 8.80. 51. 6.66f ; 7.90; 5.16f. 
 
 44. 3.121; 1.621; 9.30. 52. 8.80; 7.33J; 4.831 
 Ex.39. -V-. 3.37J-- 3| = V-. 
 
 Reduce to common fractions : 
 
 53. .48; .25; .375. 57. .58; .265; .735. 
 
 54. .960;. .225; .144. 58. .875; .3125; .4375. 
 
 55. .625; .200; .16. 59. .821; .721; .6|. 
 
 56. .516; .750; .90. 60. .7J ; .201; .181 
 
76 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 27. 
 
 Per Cent. 
 
 When the denominator is 100, there 
 are three ways of expressing the 
 fraction. 
 
 1st. As a common fraction. 
 
 2d. As a decimal fraction. 
 
 3d. By the per cent symbol. 
 
 Per cent and hundredths are inter- 
 changeable. 
 
 Change to % : 
 
 61. .06; .08; .09. 
 
 62. 500 hundredths. 
 
 64. .16; .85; .96. 
 
 65. 4000 hundredths. 
 
 66. 8365 hundredths. 
 Ex.62. 500%. Ex.66. 8365%. 
 
 Give the equivalents as % 
 
 Illustration. 
 Six hundredths may be 
 written, 
 T § 7 , a common fraction, 
 .06, a decimal fraction, 
 6 %, read 6 per cent. 
 8% =.08. 
 
 Change to decimals : 
 
 67. 5%; 8%; 23%. 
 
 68. 233% ; 4025%. 
 
 69. 86%; 75%. 
 
 70. i%; f%. 
 
 71. .000f % ; 2%. 
 
 72. 200% ; 65%. 
 Ex. 68. 2.88: 40.25. 
 
 73. h h h h h h h h h h h h h h f 
 
 74 1 1 JL JL i i J- 4 X i 4 
 '** ¥' 6' 16' 12' f' 6' 16' 12' 12' ¥' 6* 
 
 75. 11 2|, 3f , 4|, 5f , 6£, 8i, 9|. 
 
 Ex. 73. 50 %, 33J %, etc. See p. 74. 
 
 Give the equivalents as fractions : 
 
 76. 871%, 66$%, 80%, 50%, 621%, 16|%, 20%, 60%. 
 
 77. 16|%, 371%, 87-|-%, 831%, 40%, 75%, 25%, 331%. 
 
 78. 1831%, 280%, 1371%, 125%, 2331%, 175%, 225%. 
 
 79. 1161%, 220%, 140%, 375%, 4331%, 2 12£%, 416| %. 
 Ex. 76. 1, I, f , etc. See p. 74. 
 
DECIMALS. 77 
 
 § 28. Short Methods. 
 
 To multiply when the sum of the Illustration. 
 
 fractional parts is one, and the in- 6 I 
 
 tegers are the same. 
 
 »i 
 
 To the product of the integer 
 
 6x1 
 and th^ integer increased by one, 6 x a 
 
 anr^x the product of the fractions. 6x6 
 
 79. What is 6f x6^? 6 x 7 + | x f = 42ft. 
 
 Find the value of : 
 
 so. 6Jx6J. 96. ljxlf. 112. 8.3x8.7. 
 
 81. TJxTf 97. 9^x9J. 113. 9.2x9.8. 
 
 82. 8Jx8J. 98. 6|x6f. 114. 7.2x7.8. 
 
 83. 4£x4J. 99. 9Jx9f. lis. 2.2x2.8. 
 
 84. 6|x6f ioo. 8lx8|. 116. 2.3x2.7. 
 
 85. 5l*5i- ioi. 5fx5|. 117. 85x85. 
 
 86. 9fx9f io2. 8fx8f lis. 74x76. 
 
 87. 3fx3i 103. 7-^x7^. 119. 83x87. 
 
 88. 8|x8f 104. 8.5x8.5. 120. 92x98. 
 
 89. 3^x3 J. 105. 7.4x7.6. 121. 27x23. 
 
 90. 5-i- x 51 106. 9.3x9.7. 122. 38x32. 
 
 91. 101x101 107. 4.8x4.2. 123. 54x56. 
 
 92. llixlli 108. 11.9x11.1. 124. 31x39. 
 
 93. 991x991 109. 5.7x5.3. 125. 63x67. 
 
 94. 10 T Vxl0ii 110. 4.6x4.4. 126. 58x52. 
 
 95. 12^3x1211. 111. 10.9x10.1. 127. 4.75x4.25. 
 
 Ex. 80. 42.25. 6 x (6 + 1) = 42 ; .5 x .5 = .25. 
 
 Ex. 117. 7225. 85 x 85 = 8.5 x 8.5 x 100 = 7225. 
 
16. 
 
 78 AMERICAN MENTAL ARITHMETIC. 
 
 To multiply or divide when one 
 number can be readily reduced to a Illustration 
 
 simple fraction. 
 
 Reduce to the fraction and mul- 
 tiply. 
 
 Multiply : 
 
 128. 24 by .50. 134. 56 by .871 140. 23 by .331 
 
 129. 36 by .75. 135. 36 by .831 141. 26 by .66f . 
 
 130. 84 by .831 i 36 . 25 by .60. 142. 27 by .25. 
 
 131. 72 by .87-|. 137. 75 by .331 143. 29 by .50. 
 
 132. 33 by .331 i 38 . 21 by .66f . 144. 30 by .121. 
 
 133. 24by.66f. 139. 48 by .371 145. 31 by .371 
 
 Ex. 129. 27. .75= f; 36 x f = 27. 
 
 What is: 
 
 146. .50 of 16? .871 of 40? %50. .40 of 32? .60 of 13? 
 
 147. .331 of 60? .621 of 64? 151. .16| of 25? .331 f 19? 
 
 148. .66 J of 36 ? .371 of 56 ? 152. .20 of 53 ? .25 of 35 ? 
 
 149. .25 of 52? .121 of 72? 153. .37} of 22? 1.121 f 16? 
 
 154. 20% of 50? 75% of 16? 
 
 155. 66f% of 72? 25% of 28? 
 
 156. 16|%of42? 33^% of 18? 
 
 157. 40% of 60? 371% .of 16? 
 
 158. 60% of 20? 621% of 24? 
 
 159. 1831% of 30? 116|%of36? 
 
 160. 2121% of 40? 325% of 16? 
 
 161. 16fxl8? 25x120? 163. 66fx48? 83^x28? 
 
 162. 75x96? 371x64? 164. 75x40? 25x80? 
 
 Ex. 159. 55. 183|% = y J V of 30 = 55. 
 
 Ex. 161. 300. 16f x 18 = .16f x IB x 100 = 300. 
 
DECIMALS. 79 
 
 What is the cost of : 
 
 165. 120 yards of cloth at 50^ a yard? 33^? 25^? 
 
 166. 12 yards of cloth at $1.16§ a yard? $2.75? 13.25? 
 
 167. 25 yards of cloth at 25^ a yard? 12^? 87^? 
 
 168. 72 dolls at 25^ each? 33^? 37^? 50/? 62£tf? 
 
 169. 72 chairs at 83^ each? 87^? 20/? 40/? 60/? 
 
 170. 24 lamps at $1.37* a dozen? |2.75 a dozen ? 
 
 171. 60 chimneys at 40/ a dozen ? 37^/ a dozen ? 
 
 172. 9 vests at $ 8.75 a dozen ? $3.25 a dozen ? 
 
 173. 24 spools of thread at $2.75 per 100? $4.50 a gross? 
 
 174. 16 pounds of ham at 8J/ per pound? 12^/? 16§/? 
 
 175. 3 dozen lemons at 6 for 12|^? 8 for a quarter? 
 
 176. 48 lamp chimneys at 33^/ per dozen ? 
 
 177. 40 tables at $1.25 each? $1.75? $1.66f ? $1.87£? 
 
 178. 18 table-cloths at $9 a dozen? $12.50 a dozen? 
 
 179. 64 hammocks at $1.25 each? $ .1.50? $1,831? $1.75? 
 
 180. 28 oranges at 25/^5Pdozen? 50/ per dozen? 
 
 181. 45 bananas at 75/ per dozen ? 33^/ per dozen ? 
 
 182. 15 plums at 20/ per dozen? 12 J/ per dozen? 
 
 183. 30 knives at 60/ each? 75/ each? 80/ each? 
 
 184. 20 bottles of ink at 60/ per dozen? 30/ per dozen? 
 
 185. 12 dozen pens at 16f/ per dozen ? 25/ per dozen ? 
 
 186. 100 cloaks at $12J each? $50? $75? $20? 
 
 187. 21 pair of shoes at $2.75 a pair? $3.25? $5? $2.50? 
 
 188. 4 dozen plates at $1.25 per dozen? at 121/ e ach? 
 
 189. 8 dozen cups at $4.37* per dozen? $3.40 per dozen? 
 
 190. 60 marbles at 10/ per dozen ? 25/ per dozen ? 
 
 191. 80 chains at 75/ each? 50/? $2.25? $3.66f ? $4.40? 
 
 192. 50 glasses at 25/ each? 331/ ? 871/? 371/ ? 
 
 193. 75 baskets at 12 J/ each? 16|/? 20/? 25/? 75/? 
 
 Ex. 165. $ 60 ; $ 40 ; etc. 
 
80 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 Find the value of: 
 
 194. 24 -j- .66§. 
 
 195. 24-66f%. 
 
 196. 24 -66§. 
 
 197. 210 -j- .50. 
 
 Ex. 194. 3G 
 Ex. 196. A 
 
 198. 210 -.331 
 
 199. 210-66f%. 
 
 200. 210 -=-75. 
 
 201. 210-871 
 
 Ex.195. 30. 
 
 202. 210 -.25. 
 
 203. 210 + 25%. 
 
 204. 210-25. 
 
 205. 210-831 
 
 What is : 
 
 206. 21-60%? By 371%? 214. 
 
 207. 21 -=-50% ? By 75% ? 215. 
 
 208. 21 -=-871%? By 16|%? 216. 
 
 209. 20-66|%? By 33l%? 2 17. 
 
 210. 20 + 40%? By 621% ? 2 i8. 
 
 211. 20 + 831%? By 50%? 219. 
 
 212. 25 + 16f%? By 331%? 220. 
 
 221. 
 
 213. 25-5-831% ? By 121%? 
 
 450 
 
 = 4900. 
 
 33 + 150%? By 1371%: 
 42-140%? By 175%? 
 66-1831%? By 75%? 
 By 331? 
 
 By 50%? 
 
 By 66f % ? 
 
 By 16f % ? 
 
 By 121%? 
 
 122£ + 25%? 
 
 213 + 25% 
 
 472 
 
 230 + 2^7 
 
 25% 
 
 How many yards can be bought: 
 
 222. For $12 at 25^? 75^? 16f^? 66f^? 331^? 37^? 
 
 223. For $12 at 60^? 87^? 831^? 12^? 50^? &2±ft 
 
 224. For $120 at $1.25? $1,331? $1.66§? $1.50? 
 
 225. For$120at$1.87^? $3.75? $2.50? $1.60? 
 
 226. For$150at$1.20'? $1.25? $3,331? $.75? 
 
 227. For $150 at $2.50? $1,871? $1.66f? $1,331? 
 
 228. For $150 at $.121? $3.16f? $.331? $1.25? 
 
 229. For $140 at $1.60? $1.20? $121? |16|? 
 
 230. For $140 at $25? $371? $1,871? $1.75? 
 Ex. 224. 90 yards. 120 -*- 1.25 = 120 *. f = 90. 
 
DENOMINATE NUMBERS — ENGLISH 
 TABLES. 
 
 MONEY. 
 
 10 mills (m.) = l cent (f) 
 10 4 =1 dime (d.) 
 
 10 d. =1 dollar ($) 
 
 110 =1 eagle (E.) 
 
 ENGLISH MONEY. 
 4 farthings (far. or qr.) = 1 penny (t?.) 
 12 d. =1 shilling O) 
 
 20 s. =1 pound (<£) 
 
 21 s. =1 guinea (G.) 
 
 The table of U. S. money, like the tables of the Metric system, has the 
 submultiples : Latin mille, 1000, mill ; Latin centum, 100, cent ; Latin 
 decern, 10, dime. Ten of any denomination always make one of the next 
 higher. 
 
 A sovereign = £1. 
 
 In English money, the symbols £, s., d., qr., are the initials of the Latin 
 equivalents, libra, solidus, denarius, and quadrans. 
 
 TROY WEIGHT. 
 
 24 grains (gr.) = 1 pennyweight (pwt.) 
 
 20 pwt. =1 ounce (oz.) 
 
 12 Oz. =1 pound (lb.) 
 
 AVOIRDUPOIS WEIGHT. 
 
 16 ounces (oz.) =1 pound (lb.) 
 
 100 lb. =1 hundredweight (cwt.) 
 
 20 cwt. =lton(T.) 
 
 AM. MENT. AB. — 6 81 
 
STTfL 
 
 mod 
 
 > APOTHECARIES WEIGHT. 
 
 3^XA36vu«i 20 grains (gr.) = l scruple (3) 
 (J 3 3 = 1 draft (3) 
 
 83 =1 ounce ( 3 ) 
 
 12 5 =1 pound (lb) 
 
 ■U* l^ (^rn^S^W^TLvm measurT^ 
 60 minims (^l) = 1 fluid drachm (/ 3) 
 8/3 =1 fluid ounce (/J) 
 
 16 /S - = lpint(0.) 
 
 8 0. =1 gallon (Cong.) 
 
 Troy weight is used in weighing gold, silver, and precious stones ; avoirdu- 
 pois weight, in weighing nearly everything else ; apothecaries' weight, in retail- 
 ing dry medicines ; apothecaries' fluid measure, in retailing liquid medicines. 
 
 1 long ton =2240 lb. ; 1 carat =4 gr. Carat is also used to denote the num- 
 ber of parts in 24 that are pure gold. E.g. * 12 carats fine' means \\ gold. 
 
 Grain is from grain of wheat ; 7000 gr. = 1 lb. avoirdupois weight, 5760 
 gr. = 1 lb. troy or apothecaries' weight. Pennyweight was the weight of the 
 English penny in silver ; ounce, Latin uncia, twelfth ; pound, Latin pe?ido, 
 weigh ; scruple, Latin scrupulus, a little stone ; dram , Greek, drachma. Cong. 
 is from the Latin congius, a gallon ; O is the abbreviation for Latin octavus, 
 one eighth, the pint being one eighth of a gallon. The origin of symbols of 
 apothecaries' weight is unknown. 
 
 LONG MEASURE. 
 
 12 inches (in.) =1 foot (ft.) 
 
 3 ft. =1 yard (yd.) 
 
 51 yd. or 16* ft. =1 rod (rd.) 
 
 320 rd. =1 mile (mi.) 
 
 surveyors' long measure. 
 7.92 inches (in.) =1 link (li.) 
 100 li. =1 chain (ch.) 
 
 66 ft. =1 chain. 
 
 80 ch. =1 mile (mi.) 
 
 SUbo 0^ frtJrv " 1 C*^W\ixX& 
 
DENOMINATE NUMBERS — ENGLISH TABLES. 83 
 
 Long measure is used in measuring lengths and distances. For short 
 lengths, the foot rule or yard stick is commonly used ; for long distances, 
 a chain 4 rods long. The surveyor'' s chain was made 4 rods in order that 
 10 square chains might equal 1 acre. It is divided for convenience into 100 
 links ; 4 rd. = 66 ft. = 792 in. ; hence, 7.92 in. = 1 li. The engineer's chain 
 is 100 feet long. 
 
 Inch is from Latin uncia, twelfth ; foot, human foot ; yard, a twig ; mile, 
 Latin mille passuum, 1000 paces. 
 
 A line = j 1 ^ in. ; a furlong = 40 rd. ; a fathom = 6 ft. 
 
 SQUARE MEASURE. 
 144 square inches (sq. in.) = 1 sq. ft. 
 9 sq. ft. =1 sq. yd. 
 
 30£ sq. yd. or 272$ sq. ft. =1 sq. rd. 
 160 sq. rd. =1 acre (A.) 
 
 surveyors' square measure. 
 10000 sq. li. =lsq. ch. 
 10 sq. ch. =1A. 
 640 A. =lsq. mi. 
 
 36 sq. mi. = 1 township (Tp.) 
 
 Square measure is used in stating areas anc surfaces; it is formed by 
 squaring long measure ; 12 in. = 1 ft. ; squaring, 144 sq. in. = 1 sq. ft. ; 3 ft. 
 = 1 yd., squaring, 9 sq. ft. = 1 sq. yd. ; etc. 
 
 A perch = 1 sq. rd. ; a rood = 40 sq. rd. 
 
 CUBIC MEASURE. 
 
 1728 cubic inches (cu. in.) = 1 cu. ft. 
 27 cu. ft. . = lcu. yd. 
 
 WOOD MEASURET ' <->c/o 
 16 cu. ft. = 1 cord foot (cd. ft.) 
 8 cd. ft. =1 cord (cd.) 
 
 Cubic measure, is used in stating volumes or contents;' it is formed by 
 cubing long measure ; 12 in. = 1 ft., cubing, 1728 cu. in. = 1 cu. ft. ; 3 ft. 
 1 yd., cubing, 27 cu. ft. = 1 cu. yd. 
 
 )icn 
 
84 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 CAPACITY. 
 
 LIQUID MEASURE 
 
 DRY MEASURE. 
 
 2pt. =1 qt. 
 
 8 qt. =1 peck (pk.) 
 4 pk. = l bushel (bu.) 
 
 4 gills (gi.) = lpint(pt.) 
 2 pt. =1 quart ( 
 
 4qtf H =lgalk)U (gal'.) 
 
 ^Li^bi^Jiieasu^ i| us$idi$(\^&asuring liquids ; dry measure, for measuring 
 dry substances. 
 
 Gill is from Latin gilla, a drinking- glass ; pint, Spanish pinta, a mark ; 
 quart, Latin quartus, a fourth ; gallon, derivation unknown. 
 
 For convenience, the number of bushels of grain and seed is determined 
 by weighing instead of by using a bushel measure. Many of the states have 
 fixed by statute the number of pounds to be reckoned a bushel. See p. 86. 
 
 The barrel does jiot co ntain any uniform number of gallons ; the number 
 of gallons is usually" written on OUB Uf lEsheads/ 31£ gallons is the standard. \ 
 
 TIME. 
 
 60 seconds (sec.) = l minute (min.) 
 
 60 min. 
 
 = 1 hour (h.) 
 
 24 h. 
 
 = 1 day (da.) 
 
 7 da. 
 
 = 1 week (wk.) N 
 
 4 wk. 
 
 = 1 month (mo.) 
 
 12 mo. 
 
 = 1 year (yr.) ^ 
 
 365 da. 
 
 = 1 common yr. 
 
 366 da. 
 
 = 1 leap yr. 
 
 100 yr. 
 
 = 1 century (cen.) 
 
 Thirty days hath September, 
 April, June, and November ; 
 All the rest have thirty-one, 
 Except the second month alone, 
 Which has but twenty-eight, in fine, 
 Till leap year gives it twenty-nine." 
 
DENOMINATE NUMBERS — ENGLISH TABLES. 85 
 
 Every year exactly divisible bv 4, cen\ 18 9 2 > a lea P year, 
 temiial years excepted, is a leap year, k 1893 ' a common y ear ' 
 
 Ivery centennial year exactly divis- \ 2000, a leap year, 
 ible by 400 is a leap year ; the others I 1900 ' a common year, 
 are common years. / 
 
 How many days are there in Mar.? Nov.? June? Aug.? 
 Sept.? Feb.? Apr.? Jan.? May? July? Oct.? Dec? 
 
 What month numerically is Aug. ? June ? Nov. ? Mar. ? 
 Dec? Oct.? July? May? Jan.? Apr.? Feb.? Sept.? 
 
 A day is the time of the revolution of the earth upon its axis/^a month k 
 the time of the revolution of th e moon around the earthy The months are ■. 
 January (31 days), February (28 or 29), March (81), April (30), May (31), 
 June (30), July (31), August (31), September (30), October (31), November 
 (30), December (31). The months are also designated by the ordinal 
 numerals, January being the 1st month. r -*v 
 /in business, 30 days are counted as a month.) 
 
 itryear is the time required for the earth toTevolve around the sun. It is 
 365 days 5 h. 48 min. 49.7 sec, or very nearly 365 \ days. Instead of reckon- 
 ing this part of a day each year, it is disregarded, and an addition is made 
 when it amounts to one day, which occurs about every fourth year. This 
 addition of one day is made to the month of February. Since the part of a 
 day that is disregarded when 365 days are considered as a year, is a little 
 less than one quarter of a day, the addition of one day every fourth year is a 
 little too much, and, to correct this excess, addition is made to only every 
 fourth centennial year. This is why every year divisible by 4 except cen- 
 tennial years is a leap year. 
 
 Formerly time was reckoned very inaccurately. In 46 b.c. Julius Caesar 
 reformed the calendar and made the year consist of 365J days, but even this 
 was not absolutely accurate, and in 1582 the error in the calendar established 
 by him (called the Julian calendar) had increased to 10 days ; that is, too 
 much time had been reckoned as a year, so that the civil year was 10 days 
 behind the solar year. To correct this error, Pope Gregory XIII. ordered 
 10 days to be stricken from the calendar. The day after the 3d of October, 
 1582, was called the 14th, and thereafter only those centennial years which 
 were divisible by 400 were considered as leap years. 
 
86 AMERICAN MENTAL ARITHMETIC. 
 
 CIRCULAR MEASURE. 
 
 60 seconds (") = 1 minute (') 
 60' =1 degree (°) 
 
 30° = lsign(S.) 
 
 90° =1 right angle. 
 
 12 S., or 360° =1 circumference (C.) 
 Circular measure is used iu measuring angles. -The division of a circum- 
 ference into 360 parts may have been suggested by the days in a year. 
 
 COUNTING. PAPER. 
 
 12 units =1 dozen. 24 sheets =1 quire. 
 
 12 dozen = 1 gross. 20 quires =1 ream. 
 
 12 gross =1 great gross. 480 sheets = 1 ream. 
 20 units = 1 score. 
 
 EQUIVALENTS. 
 
 231 cu. in. =lgal. 
 
 2150.4 cu. in. =1 bu. 
 
 4 bu. =5 cu. ft. (nearly). 
 
 4 heaped bu. =5 struck bu. 
 
 7J gal. =1 cu. ft. (nearly). 
 
 621 lb. =wt. 1 cu. ft. water. 
 (7 ft.) 3 to (8 ft.) 3 =1 ton hay. 
 7 cu. ft. corn in ear =3 bu. shelled corn. 
 
 24 h. =360°. 
 
 ^ 6$ K = 1 teaspoonful. 
 
 5760 gr. =1 lb. troy. 
 
 *V>5 5760 gr. = 1 lb. apothecaries'. 
 
 ^ 7000 gr. =1 lb. avoirdupois. 
 
 ** In most states : 
 
 48 lb. = l bu. barley. 60 lb. = l bu. potatoes. 
 
 56 lb. = l bu. corn. 56 lb. = l bu. rye. 
 
 ~* O 32 lb. = l bu. oats. 60 lb. = l bu. wheat 
 
 ^ 
 
DENOMINATE NUMBERS — ENGLISH TABLES. 87 
 
 § 29. Exercises in English Tables. 
 
 U.S. MONEY. 
 
 How many : How many .• 
 
 1. m. make $ 1 ? 6. m. make 1 d. ? 
 
 2. d. make |1? 7. m. make IE.? 
 
 3. t make $1? ^ 8. m. make 1 il 
 
 4. i make 1 E. ? t 9. p make 1 d. ? 
 
 5. d. make IE.? 10. $ make IE.? 
 
 ENGLISH MONEY. 
 
 How many : How many : 
 
 11. far. make X 1 ? 16. d. make Is.? 
 
 12. s. make XI? 17. d. make 1 G. ? 
 
 13. s. make 1 G. ? 18. d. make XI? 
 
 14. far. make 1 s. ? 19. far. make Id.? 
 
 15. far. make 1 #. ? 20. X make 1 G. ? 
 
 TROY AND AVOIRDUPOIS WEIGHTS. 
 How many : How many : 
 
 21. gr. make 1 lb. (troy) ? 26. gr. make 1 lb. (apoth.) ? 
 
 22. gr. make 1 oz. (troy) ? 27. oz. make 1 lb. (apoth.) ? 
 
 23. oz. make 1 lb. (troy) ? 28. cwt. make IT.? 
 
 24. pwt. make 1 lb. ? 29. lb. make IT.? 
 
 25. pwt. make 1 oz. ? 30. gr. make IT.? 
 
88 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 apothecaries' weight and apothecaries' fluid 
 
 MEASURE. 
 
 How many : 
 
 36. gr. make 1ft? 
 
 37. gr. make 1 5 ?&!/ 
 
 38. 3 make 1ft? 
 
 39. 3 make 1ft? 
 35f /3 make 1 Cong. ? 40. 3 make IS? 
 
 LONG AND SURVEYORS' LONG MEASURE. 
 
 How many : 
 
 41. in. make 1 yd. ? 
 
 42. in. make 1 rd. ? 
 
 43. ft. make 1 rd. ? 
 
 44. ft. make 1 mi.? 
 
 45. rd. make 1 mi. ? 
 
 How many : / 
 
 46. li. make J ch. ? 
 
 47. li. Vnake 1 rd. ? 
 
 48. ch. make 1 rd. ? 
 
 49. ch. maKKl mi.? 
 
 50. in. niake In.? 
 
 SQUARE AND SURVEYORS SQUARE MEASURE. 
 
 How many : 
 
 56. sq. ch. make 1 A.? 
 
 57. sq. n\mal?e 1 sq. ch. ? 
 
 58. sq. li. nmke 1 sq. rd. ? 
 
 59. A. maj^e 1 sq. mi. ? 
 
 60. sq. ml. make 1 Tp.? 
 
 How many : 
 
 51. sq. in. make 1 sq. ft. ? 
 
 52. sq. in. make 1 sq. yd. ? 
 
 53. sq. ft. make 1 sq. rd. ? 
 
 54. sq. rd. make 1 A. ? 
 
 55. sq. ft. make 1 sq. yd.? 
 
DENOMINATE NUMBERS — ENGLISH TABLES. 
 
 89 
 
 CUBIC AND aSOJOB^MEASURES. 
 
 How many : 
 
 61. cu. in. make 1 cu. ft.? 
 
 62. cu. in. make 1 cu. yd. 
 
 63. cu. ft. make 1 cu. yd. ?| 
 
 64. cu. in. make 1 cd. ft. ? 
 
 65. cu. in. make 3 cu. ft. ? 
 
 How many : 
 
 66. cu. ft. make 1 cd. ? 
 
 67. cu. ft. make 1 cd. ft. ? 
 
 68. cu. ft. make 3 cu. yd. ? 
 
 69. cu. ft. make 2 cu. yd. ? 
 
 70. cu. ft. make 3 cd. ? 
 
 LIQUID AND DRY MEASURES. 
 
 How many : 
 
 71. pt. make 1 qt. (dry)^ 
 
 72. qt. make 1 pk. ? 
 
 73. pt. make 1 bu. ? 
 
 74. pt. make 1 pk.? 
 
 75. pk. make 1 bu. ? 
 
 How many : 
 
 76. pt. make 1 qt. (liquid) ? 
 
 77. qt. make 1 gal. ? 
 
 78. qt. make 1 bbl. of 31 J gal. 
 
 79. pt. make 1 bbl. of 45 gal. ? 
 
 80. qt. make 1 bbl. of 40 gal. ? 
 
 MISCELLANEOUS. 
 
 How many : 
 
 81. da. make 1 leap year ? 
 
 82. — etegrees-imke-4--Gr? 
 
 83. cu. in. make 1 gal. ? 
 
 84. ^n^in^-make-4-ter? 
 
 85. v_degrees~m&ke IS.? 
 
 How many : 
 
 86. units make 1 dozen? 
 
 87. units make 1 score ? 
 
 88. sheets make 1 quire?' 
 
 89. sheets make 1 ream? 
 
 90. units make 1 gross? 
 
 ijw 
 
90 AMERICAN MENTAL ARITHMETIC. 
 
 § 30. Reduction — In the Same Table. 
 Reduce : 
 
 91. <£3 to d. 96. 48 c?. to s. 
 
 92. 14 s. to far. 97. 300 far. to s. 
 
 93. 5 G. to s. 98. 3000 far. to £. 
 
 94. £2 3 s. 4 d. to far. 99. f *. to d. 
 
 95. 3 G. 5 s. 6 d. to d. loo. Xf to d. 
 
 Ex. 95. 822 d. 3 G. = 63s. ; 68 s. = 816 d.\ 816 cf. + 6 d. = 822 d. 
 
 Reduce : 
 
 101. 2 lb. 3 oz. 2 pwt. (troy) to pwt. 
 
 102. 960 gr. (troy) to lb. 
 
 103. -| lb. (apoth.) to lower integers. 
 
 104. J lb. (avoir.) to lower integers. 
 105. | T. (avoir.) to oz. 108. 1440 3 to lb. 
 106.- 8 Cong, to O: - 109. - 1 Con grtrr"t , 
 107. ^Cong. SO. 2/5 to/5-. no. 3000 hi lu/5, 
 
 Ex.103. 43 63 134gr.;fib=4i3; f J = Gf37|3^1 ±-t>; ft>=4gt. 
 
 Reduce : 
 
 111. 3 yd. 2 ft. 3 in. to in. 117. | rd. to lower integers. 
 
 112. 3 yd. 2 ft. 3 in. to ft. 118. 5 sq. rd. to sq. ft. 
 
 113. 3 yd. 2 ft. 3 in. to yd. 119. 1 A. to sq. yd. 
 
 114. etc 10 li. lu II. 120. 3500 cu. in. to higher 
 
 115. 160 cu. ft. to cd. integers. 
 
 116. 1728 in. to rd. 121. 5184 cu. in. to cu. ft. 
 Ex. 120. 2 cu. ft. 44 cu. in. 
 
DENOMINATE NUMBERS — ENGLISH TABLES. 
 
 91 
 
 Reduce : 
 
 122. .6 bu. to lower in- 
 tegers. 
 
 123. 96 qt. to bu. 
 
 124. 2 bu. 2 pk. 1 qt. to 
 lower integers. 
 
 125. 252 gal. to pt. 
 
 126. 1J gal. to qt. 
 
 Ex. 122. 2 pk. 3 qt. .4 pt. ; .6 bu. 
 
 Reduce : 
 
 131. 2 mo. 16 da. to da. 
 
 132. 1 leap yr. 2 da. to wk. 
 
 133. 2500 min. to higher 
 integers. 
 
 134. 2 h. 2 min. 2 sec. to 
 
 127. 2000 pt. to higher in- 
 tegers. 
 
 128. |- gal. to lower in- 
 tegers. 
 
 129. 1728 pt. to higher in- 
 tegers. 
 
 130. 1728 pt. to gal. 
 
 :2.4 pk. ; .4 pk. = 3.2 qt. ; .2 qt. = .4 pt. 
 
 sec. 
 
 135. | h. to lower inte- 
 
 136. 5 score to units. 
 
 137. 6 score to dozen. 
 
 138. 5 gross to dozen. 
 
 139. 36 gross to great gross. 
 
 140. 960 sheets to reams. 
 
 141. 1200 sheets to reams. 
 
 142. 1200 quires to reams. 
 
 143. 3° 4' 5" to'. 
 
 gers. 
 
 Ex. 143. 184 T V; 3° = 180' ; 5" = T y ; 180' +f-+ T V = 184 T V 
 
 Reduce : 
 
 144. £1 Is. Id. 1 far. to 
 far. 
 
 145. |lb to lower integers. 
 
 146. 7 lb. (troy) to lower 
 integers. 
 
 147. 1925 oz. (avoir.) to 
 higher integers. 
 
 148. 480 ttl to 3. 
 
 Ex. 151. T £o mi. ; 33 ft. = 2 rd 
 
 149. 1 bu. 1 pk. 1 qt. 1 pt. 
 to pk. 
 
 150. 63 gal. to bbl. ' 
 
 151. 33 ft. to a fraction of 
 a mi. 
 
 152. 1 A. to sq. yd. 
 
 153. 1 cu. yd. to a fraction 
 of a cd. 
 
 = 3^ mi. = T ^ mi 
 
92 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 31. Reduction — Table to Table. 
 
 For most computations, we 
 may consider 4 bu. equal to 5 
 cu. ft. 
 
 For most computations, we 
 may consider 1\ gal. equal to 
 1 cu. ft. 
 
 4 bu. = 2150.4 x 4 = 8601.6 cu. in. 
 
 5 cu. ft. = 1728 x 5 = 8640 cu. in. 
 
 7\ gal. = 231 x7£ 
 1 cu. ft. = 1728 cu. in. 
 
 Reduce : 
 
 154. 20 cu. ft. to bu. 
 
 155. 20 bu. to cu. ft. 
 
 156. 400 bu. to cu. ft. 
 
 157. 400 cu. ft. to bu. 
 
 158. 32 bu. to cu. ft. 
 
 159. 100 cu. ft. to heaped bu. 
 
 160. 80 heaped bu. to cu. ft. 
 
 161. 64 heaped bu. to cu. ft. 
 
 162. 75 cu. ft. to heaped bu. 
 
 163. 8 ft. x 8 ft. x 3 ft. to bu. 
 
 164. 7 ft. x 10 ft. x 2 ft. to bu. 
 
 165. 4300.8 cu. in. to bu. 
 
 166. 6451.2 cu. in. to bu. 
 
 167. 4 «bu. to cu. ft. 
 
 168. 1 qt. (dry) to cu. in. 
 
 169. 1 pt. (dry) to cu. in. 
 
 170. 60 cu. ft. to gal. 
 
 171. 60 gal. to cu. ft. 
 
 172. 15 cu. ft. to gal. 
 
 173. 15 gal. to cu. ft. 
 
 174. 105 cu. ft. to gal. 
 
 175. 105 gal. to cu. ft. 
 
 176. 90 cu. ft. to gal. 
 
 177. 90 gal. to cu. ft. 
 
 178. 25 cu. ft. to gal. 
 
 179. 3 ft. x 5 ft. x 6 ft. to gal. 
 
 180. 6 ft. x 10 ft. x 4 ft. to gal. 
 
 181. 462 cu. in. to gal. 
 
 182. 1155 cu. in. to gal. 
 
 183. 4 gal. to cu. in. 
 
 184. 1 qt. (liquid) to cu. in, 
 
 185. 1 pt. (liquid) to cu. in. 
 
 Ex. 154. 16 bu. ; 1 cu. ft. = $ bu. ; 20 cu. ft. = 20 x f bu. = 16 bu. 
 Ex. 155. 25 cu. ft. ; 1 bu. = f cu. ft. ; 20 bu. = 20 x f cu. ft. = 25 cu. ft. 
 Ex. 160. 125 cu. ft. ; 1 h. bu. = f bu. ; 1 h. bu. = ff cu. ft. ; 80 h. bu. 
 s 80 X ft cu. ft. = 125 cu. ft. 
 
DENOMINATE NUMBERS — ENGLISH TABLES. 93 
 
 Reduce : 
 
 186. 5000 gal. water to cu. ft. 191. 144 lb. barley to bu. 
 
 187. 10 cu. ft. water to lb. 192. 504 lb. rye to bu. 
 
 188. 125 lb. water to cu. ft. 193. 336 lb. corn to bu. 
 
 189. 6 cu. ft. water to lb. 194. 480 lb. potatoes to bu. 
 
 190. 720 lb. wheat to bu. 195. 640 lb. oats to bu. 
 
 Reduce : 
 
 196. 686 cu. ft. hay to T. (7 3 ). 
 
 197. 1024 cu. ft. hay to T. (8 3 ). 
 
 198. 3 T. hay to cu. ft. (7 3 ). 
 
 199. 3 T. hay to cu. ft. (8 3 ). 
 
 200. 21 cu. ft. corn in ear to bu. shelled. 
 
 201. 28 cu. ft. corn in ear to bu. shelled. 
 
 202. 50 cu. ft. corn in ear to bu. shelled. 
 
 203. 4 bu. shelled corn to cu. ft. corn in ear. 
 
 204. 3 bu. shelled corn to cu. ft. corn in ear. 
 
 205. 16 bu. shelled corn to cu. ft. corn in ear. 
 
 206. 3 ft. x 7 ft. x 2 ft. corn in ear to bu. shelled. 
 
 207. 40 cu. ft. corn in ear to bu. shelled. 
 
 208. 4 ft. x 6 ft. x 2 ft. corn in ear to bu. shelled. 
 
 209. 75 cu. ft. corn in ear to bu. shelled. 
 
 210. 1.8 bu. shelled corn to cu. ft. corn in ear. 
 
 Reduce : 
 
 
 
 211. 
 
 4 h. to degrees. 
 
 217. 
 
 14 h. to arc. 
 
 212. 
 
 4° to h. 
 
 218. 
 
 11 h. to arc. 
 
 213. 
 
 3° to time. 
 
 219. 
 
 6° to time. 
 
 214. 
 
 6 h. to arc. 
 
 220. 
 
 15° to time. 
 
 215. 
 
 60° to time. 
 
 221. 
 
 60° to time. 
 
 216. 
 
 360° to time. 
 
 222. 
 
 4 min. to arc 
 
DENOMINATE NUMBERS --METRIC 
 SYSTEM. 
 
 The Metric system is a decimal system in which all the 
 denominations are decimal submultiples or multiples of a 
 unit. Ten of any denomination make one of the next higher ; 
 that is, the multiple is 10. 
 
 The submultiples are : Latin, mille (1000), centi (100), deci (10). The mul- 
 tiples are: Greek, Deka (10), Hecto (100), Kilo (1000), My via (10000); the 
 abbreviation in each case is the first letter ; small, if Latin ; capital, if Greek. 
 
 ^uomtdtiplesScOA/fi Multiples. v ~\:Tl) 
 
 10 milli = 1 centi. 10 (units) = 1 Deka. 
 
 iti sa 1 deci. f" rt | a 
 
 10 centi 
 10 deci 
 
 A, 
 
 
 
 10 Deka 
 10 Hekto 
 10 Kilo 
 
 1 Hekto. 
 1 Kilo. 
 1 Myria. 
 
 Units of the Different Tables. 
 
 Name. 
 Long 
 
 Land 
 
 Area 
 
 Weight 
 
 Capacity 
 
 Wood 
 
 Unit. 
 
 meter 
 
 square 
 meter 
 
 gram 
 
 liter 
 
 stere 
 
 94 
 
 Abbre- 
 viation, 
 
 m. 
 
 sq. m. 
 
 How obtained. 
 
 ,0000001 distance 
 equator to pole. 
 
 10 m. x 10 m. 
 
 1 m. x 1 m. 
 
 wt. 1 cu. cm. 
 water. 
 
 1 cu. dm. 
 
 1 cu. m. 
 
 Eng. Eqniv. 
 
 1.37 in. 
 
 ? V acre 
 nearly 
 
 10 sq. ft. + 
 
 15.432 gr. 
 
 .908 qt. dry 
 1.05 qt. liquid 
 
 i cord 
 nearly 
 
 1 sq. Km. = 
 .3861 sq. mi. 
 
 1 Kg. = 2\ lb. 
 
 1 cu. m. water 
 weighs 1 T. 
 
DElffdMIMTK/ NUMBERS — 
 
 METRIC SYSTEM. 
 
 95 
 
 By placing the various units in the table of submultiples, 
 the following tables are formed. Square and cubic measure, 
 as in English, are formed by squaring and cubing long 
 
 measure. ru 
 Long Measure^ 
 
 sb 
 
 \o 
 
 m 
 
 & 
 
 Capacity. 
 
 Weight. 
 
 10 mm. = 1 cm. 
 
 10 ml. = 1 cl. 
 
 "^10 mg. = 1 eg. 
 
 10 cm. = 1 dm. 
 
 10 cl. =ldl. 
 
 10 eg. =^ldg. 
 
 10 dm. = 1 .m/CGbr 
 
 10 dl. = l\hXuT\ 
 
 IL. 10 dg. = 1 g. 
 ^10 g. = 1 Dg. 
 
 10 m. = 1 Dm. 
 
 
 10 1. = 1'D1. 
 
 10 Dm. = 1 Hm. 
 
 
 10* Dl. = 1 HI. 
 
 10Dg, = lHg. 
 
 10 Hm. = 1 Km. 
 
 
 10 Hi. - 1TC1. 
 
 10 Hg.'= 1 Kg.^ * 
 
 10 Km. = 1 Mm. 
 
 
 10 Kl. = 1 Ml. 
 
 , 10 Kg. = 1 Mg. 
 
 10 Mg.= 1 Quintal (Q.). 
 10 Q. =1 Tonneau (T.). 
 
 Land. 
 
 
 Square. . 
 
 Cubic. G> /C- 
 
 10 ma. = 1 ca. 
 
 100 
 
 sq. mm. = 1 sq. cm. 
 
 1000 cu. mm. = 1 cu. cm. i 
 
 10 ca. = 1 da. 
 
 100 
 
 sq. cm. — 1 sq. dm. 
 
 1000 cu. cm. = 1 cu. dm. 
 
 10 da. = 1 a., etc. 100 
 
 sq. dm. = 1 sq. in. 
 
 1000 cu. dm. = 1 cu. m. 
 
 Wood. 
 
 100 
 
 sq. m. =1 sq. Dm. 
 
 1000 cu. m/ = 1 cu. Dm. 
 
 10 ms. = 1 cs. 
 
 100 
 
 sq. Dm. = 1 sq. Hm. 
 
 1000 cu. Dm. = 1 cu. Hm. 
 
 10 cs. = 1 ds. 
 
 100 
 
 sq. Hm. = 1 sq. Km. 
 
 1000 cu. Hm. = 1 cu. Km. 
 
 10 ds. = 1 s., etc. 100 
 
 sq. Km. = 1 sq. Mm. 
 
 1000 cu. Km. = 1 cu. Mm. 
 
 Land measure is usually given 100 ca. = 1 a. ; 100 a. = l Ha., 
 
 — the other denominations being omitted. 
 Wood measure is usually given 10 ds. = l s. 
 
 The measure of weight has two extra denominations : the 
 quintal and tonneau. 
 
 The following denominations in italics are used in the 
 metric system when the denominations immediately preced- 
 ing them would be used in the English system. Foot, yard, 
 rod — meter ; mile — kilometer ; square rod, acre — are ; cord 
 
 — stere ; pound — kilogram ; pint, quart, gallon, peck, bushel 
 
 — liter ; ton — tonneau 
 
96 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 = — OD 
 
 00 
 
 = — K 
 
 CO 
 
 The metric system of weights and measures originated in 
 France in 1795. It has been adopted in Austria, Belgium, 
 Brazil, Denmark, Germany, Greece, Holland, 
 Sweden, and Switzerland. Its use has been 
 authorized by the Congress of the United 
 States and by the Parliament of Great Britain. 
 It is destined to displace the English system, 
 and is already exclusively used in scientific 
 research. 
 
 To illustrate long measure, prepare a strip 
 of paper 39| in. long ; divide it into 10 equal 
 parts (dm.) ; divide each dm. into 10 equal 
 parts (cm.) ; divide the first cm. into 10 equal 
 parts (mm.). 
 
 To illustrate land measure, stake out on 
 the playground a square 10 m. xlO m.; this 
 square will mark an are. 
 
 To illustrate wood measure, lay off on the 
 playground a square 1 m.xl m. ; at each 
 corner drive a stake leaving 1 m. above 
 ground; these stakes will hold a stere of 
 wood. 
 
 To illustrate weights, procure a piece of tin- 
 foil that weighs as much as a nickel 5^ piece ; 
 cut the foil into five equal parts ; each part 
 will weigh 1 g. By law, the nickel must 
 weigh 5 g. Procure a stone that weighs 2 
 lb. 3 oz. ; it will weigh a kilogram (nearly}. 
 
 To illustrate measures of capacity, draw 
 on paper, rectangles as ABOB (see p. 97), having the 
 dimensions given in the table; complete AEFD, and cut 
 
 -Lfl 
 
 =— *? 
 
 to 
 
 =^-c\i 
 
DENOMINATE NUMBERS — METRIC SYSTEM. 
 
 97 
 
 it out; roll over AD to BC, and paste BUFC, forming cylin- 
 ders ; these cylinders will hold the amounts given in the table. 
 
 B E 
 
 Name. 
 
 Length. 
 
 Breadth. 
 
 1 liter 
 
 29. 16 cm. 
 
 14.78 cm. 
 
 5 deciliters 
 
 23.15 cm. 
 
 11.73 cm. 
 
 2 deciliters 
 
 17.10 cm. 
 
 8.64 cm. 
 
 1 deciliter 
 
 13.54 cm. 
 
 (5.80 cm. 
 
 5 centiliters 
 
 10.74 cm. 
 
 5.44 cm. 
 
 2 centiliters 
 
 7.92 cm. 
 
 4.01 cm. 
 
 1 centiliter 
 
 6.28 cm. 
 
 3.18 cm. 
 
 C F 
 
 1. State the unit of long measure, how it was obtained, 
 and its English equivalent. 
 
 2. Describe in full the unit of weight (see p. 94). 
 
 3. Describe in full the unit of land measure. 
 
 4. Describe in full the unit of capacity. 
 
 *""5. Describe in full the unit of wood measure. 
 is 6. Name the Latin submultiples, and give their values. 
 
 7. Name the Greek -s«bmultiples, and give their values. 
 
 8. What is the multiple in all the measures except square 
 and cubic measures ?/ $What is the multiple in square measure ? 
 
 9. What is the multiple in cubic measure ? 
 
 10. Give the table of multiples and subnrultiples. 
 
 11. Give the table of long measure. Of square measure. 
 Of cubic measure. Of capacity. 
 
 12. Give the table of land measure in full. Give the table 
 of land measure as abbreviated. Is the multiple 10 or 100? 
 
 13. Give the table of wood measure in full. 
 
 14. Give the table of wood measure as abbreviated. 
 
 15. Give the table of weight. What are the two extra 
 denominations found in the table of weight? 
 
 AM. MENT. AR. — 7 
 
98 AMERICAN MENTAL ARITHMETIC. 
 
 § 32. Practical Questions. 
 
 16. How tall are you ? How long is your arm ? What is 
 the length of your forefinger ? What is the thickness of your 
 thumb-nail ? What is the width of your thumb-nail ? 
 
 17. What is the width of this room ? What is the length 
 of this room ? What is the height of this room ? What is 
 the diameter of a nickel ? What is the thickness of a nickel ? 
 
 18. How far is it to the P.O. ? How far is it to the nearest 
 city ? How fast does a passenger train run ? How fast can a 
 horse pace ? How fast can he walk ? How fast can he trot ? 
 
 19. What is your weight? How much beefsteak would 
 you buy for breakfast for three? What is the weight of a 
 nickel ? How heavy a letter of the first class will go for 2^ ? 
 What does an average horse weigh? How much hay will 
 winter a cow? How much coal do you expect to burn in 
 your kitchen stove this winter ? 
 
 20. How much milk do you need for a cup of coffee? 
 How much milk per day would you use for a family of six ? 
 How much will a teacup hold ? How much will a tablespoon 
 hold? What is a good yield of potatoes to the acre? What 
 is a good yield for wheat? How large a cistern have you? 
 
 21. How much land do you need for a flower garden? 
 What is the area of this floor ? How much land is needed 
 for a good farm ? 
 
 22. What is the contents of this room in cu. m.? How 
 much earth will make a good load for two horses? How 
 many cu. m. are there in an ordinary rick of hay? 
 
 23. How much wood will an ordinary stove burn in winter ? 
 How much wood will produce as much heat as a ton of coal ? 
 
 Note. — Give answers in the metric system. 
 
DENOMINATE NUMBERS — METRIC SYSTEM. 
 
 99 
 
 § 33. Reduction — Metric. 
 
 24. Tell whether Latin or Greek, and give meaning : m, 
 D., c, M., K., d., H. 
 
 25. Give the Greek for : 100, 1000, 10, 10000. 
 
 26. Give the Latin for: 100, 1000, 10. 
 
 27. Read 102 mm. ; 468 cm. ; 3284 Kg. 
 
 28. Read 368 Kl. ; 5600 g. ; 123 s. ; 160 a. ; 238 Ha. 
 
 29. Read 397 Mg. ; 353 cu. m. ; 189 sq. mm. 
 
 30. Read 495 1. ; 672 Dl. ; 801 HI. ; 911 Kg. 
 
 31. Read 916 a. ; 210 cl. ; 717 dm. ; 111 Mg. 
 
 32. Read 495 Q.; 266 cu. dm. ; 889 sq. Dm. 
 
 How many : 
 
 33. mm. make 1 Dm. ? 45. 
 
 34. cm. make 1 Mm.? 46. 
 
 35. dm. make 1 Km. ? 47. 
 
 36. m. make 1 Hm. ? 48. 
 
 37. mm. make 1 Mm.? 49. 
 
 38. mm. make 1 dm. ? 50. 
 
 39. mm. make 1 m. ? 51. 
 
 40. mg. make 1 Mg. ? 52. 
 
 41. eg. make 1 Hg. ? 53. 
 
 42. Mg. make IT.? 54. 
 
 43. Kg. make 1 T.? 55. 
 
 44. Dg. make 1 Q.? 56. 
 
 sq. mm. make 1 sq. Mm. ? 
 sq. cm. make 1 sq. Dm. ? 
 sq. mm. make 1 sq. dm.? 
 sq. m. make 1 sq. Hm. ? 
 sq. Dm. make 1 sq. Km. ? 
 cu. m. make 1 cu. Mm.? 
 cu. mm. make 1 cu. Km.? 
 cu. Dm. make 1 cu. Mm. ? 
 dl. make 1 KL? 
 cl. make 1 Ml. ? 
 HI. make 1 ML? 
 a. make 1 Ha.? 
 
 Ex. 33. 10000 ; mille = 1000 ; Deka = 10 ; 1000 x 10 = 10000. 
 
 Ex. 38. 100 ; mille = 1000 ; deci = 10 ; 1000 -- 10 = 100. 
 
 Ex. 45. 1 with 14 ciphers ; mille = 1000 ; Myria = 10000 ; 1000 x 10000 
 =' 10,000,000, or 1 with 7 ciphers ; 1 with 7 ciphers squared = 1 with 14 
 ciphers. 
 
100 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 34. Reduction — English and Metric. 
 
 Reduce approximately : 
 
 * 57. 16^Km. to miles. / b 
 
 \ 58. $) miles to Km. J £ 
 ^59. 3 m. to yards. 3 -^ ^ 
 t^ 60. 4 yards to m. 
 
 ^61. 600 m. to fee 
 T^STMrnTtomTS OQ7. I 
 
 L- 63. 320 in. to m. 
 
 Ex. 57. 10 mi. ; 1 Km. = § mi. 
 
 Reduce approximately : 
 
 V 71. 5 gallons to 1. 
 
 \y 72. 30 1. to gallons. 
 
 ,/ 73. 60 gallons to 1. 
 
 i/ 74. 360 gallons to 1. 
 
 / 75. 126 1. to gallons. 
 
 76. 1 bushel to 1. 
 
 77. 256 1. to bu. 
 
 Ex. 71. 20 1. ; 1 1. = 1 qt. 
 
 f 64. 600 feet to m. — 
 
 65. 2000 sq. mi. to sq. Km. 
 
 66. 10,000 sq. Km. to sq. m. 
 
 67. 16 acres to a. 
 
 68. 400 a. to acres. 
 
 69. 24 cords to s. 
 
 70. 72 s. to cords. 
 
 Ex. 64. 200 m. ; 1 m. = 3 ft. ; 600 ft. 
 = 200 m. 
 
 78. 64 1. to bushels. 
 
 79. 75 1. to quarts (dry). 
 
 80. 75 1. to quarts (liquid). 
 
 81. 10 1. to cu. in. 
 
 82. 25 1. to cu. in. 
 
 83. 40 1. to gallons. 
 
 84. 100 1. to gallons. 
 
 y* 
 
 _ . Reduce approximately : 
 
 85. 10 cu. m. of water to tons. 
 
 86. 50 cu. cm. of water to 
 grains. 
 
 87. 10 tons of water to cu. m. 
 
 88. 1 cu. dm. of water to 
 pounds. 
 
 89. 1 Kg. to pounds. 
 
 Ex. 78. 2 bu. ; 1 1. = 1 qt. 
 
 90. 3 Hg. to pounds. 
 
 91. 1 Hg. to ounces. 
 
 92. 30 grains to g. 
 
 93. 10 g. to grains. 
 94. 
 95. 
 96. 
 
 6 tons of water to cu. m, 
 5 bushels to 1. 
 18 Km. to miles. 
 
 ■I 
 
 h 
 
 
PERCENTAGE. 
 
 The symbol % means hundredths. 
 Per cent and hundredths are inter- 
 changeable. 
 
 Illustration. 
 
 6% =.06 
 
 read 
 
 per cent = 6 hundredths. 
 
 §35. 
 
 Change to % : 
 
 1. .07, .09, .12. 
 
 2. .16, .25, .96. 
 
 3. 325 hundredths. 
 
 4 * TOO' TOO' TOO"* 
 
 5. 9723 hundredths. 
 
 6. .001 .00f, .0001 ^ 
 
 Ex.6. i%, etc. 
 
 Change to % : 
 
 13 - h h I' i* 
 "• I' h h t- 
 
 1«5 4 1 £ 1 
 
 15 * 5' 6' 6' "8"' 
 
 !«• f't'4'f- 
 
 17 2. 3 £ 3 
 - L/ - 3' ¥' 5' ¥' 
 
 18 5. A 2 1 
 •"*• 8' 6' 3' T* 
 
 Ex.13. 50%, 331%, etc. 
 
 Reduction. 
 
 Change to hundredths : 
 
 7. 17%, 13%, 14%. 
 
 8. 70%, 90%, 77%. 
 
 9. 466%, 100%, 325%. 
 
 io. \% \% \% 
 
 11. .0i%, .001%, .000J%. 
 
 12. 200%, 4000%, 7283%. 
 Ex. 11. ,0|- hundredths. 
 
 Change to common fractions 
 
 19. 621%, 87J%, 37J%. 
 
 20. 66f %, 331%, 16}%. 
 
 21. 621%, 25%, 40%. 
 
 22. 75%, 60%, 25%. 
 
 23. 20%, 80%, 331%. 
 
 24. 2871%, 2331%, 116|%. 
 
 Ex. 19. & |, etc. 
 
 101 
 
102 AMERICAN MENTAL ARITHMETIC. 
 
 § 36. The Operation Directly Stated. 
 What is : 
 
 25. 6 times 50? 42. 1331% of 24? 
 
 26. | of 50? 43. 266f% of 18? 
 
 27. .06 of 50? ^44. 325% of 160? 
 
 28. 6% of 50? 45. 187!% of 40? 
 
 29. 121% of 40? 46. 1121% of 64? 
 
 30. 331% f 60? 47. 331% of 16? 
 
 31. 16|%of36? 48. 2871% of 16? 
 
 32. 371% of 16? 49. 125% of 16? 
 
 33. 831% of 24? 50. 25% of 34? 
 
 34. 66f% of 72? jSi! 871% of 17? 
 
 35. 621% of 32? W 831% of 25? 
 
 36. 80% of 20? 53. 66|%of49? 
 
 37. 871% of 48? 54. 1121% of 33? 
 
 38. 871% of 40? 55. 2331% of 12? 
 
 39. 75% of 144? 56. 16|%of36? 
 
 40. 371% of 64? 57. 121% of 80? 
 
 41. 25% of 400? 58. 83J% of 30? 
 
 In these examples, since the operation is directly stated, 
 no explanation is required. 
 
 Thus, 6 times 50 = 300 ; f of 50 = 30 ; .00 of 50 = 3 ; 6% of 50 = 3 ; etc. 
 To explain the 26th : I of 50 = 10, £ of 50 = 3 x 10 = 30, is unnecessary, 
 because we have learned how to multiply 50 by f in a previous case, and we 
 should not at this place explain that process. 
 
 Ex. 31. 10| % of 36 is £ of 36, or 6. 
 
 Ex. 48. 287$ % of 16 is - 2 ¥ 3 of 16, or 46. 
 
PERCENTAGE. 103 
 
 59. If a rope 200 ft. long 
 
 _ . . r . F . . 6 ^Lws. 190 ft. It shrinks 5% of 200 
 
 shrinks 5% when wet, how ft ., or io ft. ; 200 ft. - 10 ft. = 190 ft. 
 long is it when wet? 
 
 60. A shepherd having 240 sheep, lost 16 1% of them in a 
 storm ; how many had he left ? 
 
 61. A had 8200 and gave 40% of his money to B; how 
 much did A retain ? 
 
 62. A mine produces 2000 tons of ore; 25% of the ore is 
 metal; 2% of the metal is silver; how many pounds of silver 
 does it produce ? 
 
 63. From a hogshead containing 480 lb. of sugar, 66 J % 
 was sold at one time ; 50% of the remainder, at another time ; 
 how many pounds remained ? 
 
 64. A dry article weighing 60 lb. gains 10% in weight 
 when soaked in water ; how much water does it absorb ? 
 
 65. The population of a town in 1890 was 16J% more than 
 1500 ; what was the population ? 
 
 66. If gunpowder contains 75% of saltpetre, 10% of 
 sulphur, and 15% of charcoal, how much of each is there in 
 40 lb. of powder? 
 
 67. Of a regiment of 1000 men, 2% are killed, 7% are 
 prisoners ; how many are left in the regiment ? 
 
 68. Of 70 children in a school 14^% are boys ; how many 
 girls are there in the school ? 
 
 69. A man who worked for $24 a week had his salary 
 diminished by 12^% ; what was it after the deduction? 
 
 70. A farmer raising 300 bu. of wheat, sold 66|% of it, 
 and fed the rest to his stock ; how much did he feed ? 
 
 71. A man was hired to work 80 days, but he lost 20% of 
 his time ; how many days did he work ? 
 
104 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 
 § 37. Operation to 
 
 BE 
 
 Determined. 
 
 72. 
 
 12 is liow many times 6 ? 
 
 83. 
 
 18 is how many hun- 
 
 73. 
 
 12 is what part of 24? 
 
 
 dredths of 72 ? 
 
 74. 
 
 12 is how many fifths of 
 
 84. 
 
 18 is what % of 72? 
 
 
 24? 
 
 85. 
 
 4 is what % of 2? 
 
 75. 
 
 12 is how many hun- 
 
 86. 
 
 12 is 6 times what num- 
 
 
 dredths of 24? 
 
 
 ber? 
 
 76. 
 
 12 is what % of 24? 
 
 87. 
 
 12 is 1 of what? 
 
 77. 
 
 15 is how many times 5 ? 
 
 88. 
 
 12 is .06 of what? 
 
 78. 
 
 15 is what part of 45 ? 
 
 89. 
 
 12 is 6% of what? 
 
 79. 
 
 15 is how many sevenths 
 
 90. 
 
 15 is 3 times what? 
 
 
 of 45? 
 
 91. 
 
 15 is 1 of what ? 
 
 80. 
 
 15 is how many hun- 
 
 92. 
 
 15 is .03 of what? 
 
 
 dredths of 45 ? 
 
 93. 
 
 15 is 3% of what? 
 
 81. 
 
 15 is what % of 45? 
 
 94. 
 
 24 is | of what? 
 
 82. 
 
 18 is how many sixths of 
 
 95. 
 
 24 is .06 of what? 
 
 
 72? . 
 
 96. 
 
 24 is 6% of what? 
 
 Ex. 72. Since 12 is some number times 6, the number is 12 -~ 6, or 2. 
 
 Ex. 73. Since 12 is some number times 24, the number is 12 -4- 24, or \. 
 
 Ex. 74. Since 12 is some number times 24, the number is 12 -4- 24, or £ 
 $ = 2} fifths. 
 
 Ex. 75. Since 12 is some number times 24, the number is 12 -4- 24, or £ 
 | = .50. 
 
 Ex. 76. Since 12 is some number times 24, the number is 12 -4- 24, or \ 
 £ = 50%. 
 
 Ex. 86. Since 12 is 6 times some number, the number is 12 -4- 6, or 2. 
 
 Ex. 87. Since 12 is § times some number, the number is 12 -=- f , or 18. 
 
 Ex. 88. Since 12 is .06 times some number, the number is 12 -4-. 06, or 200. 
 
 Ex. 89. Since 12 is 6 % times some number, the number is 12 -4- .06, or 200. 
 
 Note. — In these examples, the operation must be determined by reason- 
 ing. Each admits of being placed in the form of an equation, and the oper- 
 ation at once appears. Thus, Ex. 72, 12 = some no. x 6. Therefore, the no. 
 = 12-4-6, since the product divided by either factor equals the other. 
 
PERCENTAGE. 10£ 
 
 97. A boy having 10^, Ans. 50%. Since bf is some num. 
 lost bt\ what % of his money ber of times 10 ?> the number is 
 did ho lose? * + **«»* 
 
 98. A spent 60% of his Ans. $120. He spent 85 % of his 
 
 £ -L nzri t „ money and had 15% left ; since 15% 
 
 money for a horse, 2o% for a / . * > /0 
 
 at o / t_ °^ money is $ 30, his money must 
 
 saddle, and had 1 30 left; what be #80 + .16, or $200; 60% of $200 
 
 did he pay for the horse ? = $ 120. 
 
 99. Mary had 12^ and gave 3^ to Henry; what % of her 
 money remained? 
 
 100. From a cask containing 96 gallons of oil, 32 gallons 
 were drawn ; what % of the whole remained in the cask ? 
 
 101. A teacher whose salary is $2400, spends $2000 annu- 
 ally ; what °Jo of his salary does he save ? 
 
 102. The standard of gold and silver coin in the U. S. is 
 9 parts pure gold or silver and 1 part alloy ; what % is alloy ? 
 
 103. The population of a town in 1892 was 1600 ; in 1893 
 it was 2400 ; what was the °Jo of increase ? 
 
 104. A clerk spends $1200 a year, or 66|% of his salary; 
 what is his salary ? 
 
 105. A man drew from the hank $575, or 25% of his 
 deposit ; what was his deposit ? 
 
 106. A regiment of 800 men lost 160 men in battle ; what 
 (f -of the regiment remained? 
 
 107. After a battle 80% of the regiment, or 640 men, were 
 left; how many men were there in the regiment at first? 
 
 108. A spent 60% of his money for a horse, 25% for a 
 carriage, and had $60 left; how much did he pay for the 
 horse ? For the carriage ? 
 
 109. The master of a ship threw overboard 800 bbl. of 
 flour, or 16 J% of its cargo; what was its cargo at first? 
 
106 AMERICAN MENTAL ARITHMETIC. 
 
 110. What number increased by 5 times itself becomes 30 ? 
 
 ill. What number increased by § of itself becomes 30? 
 
 112. What number increased by .06 of itself becomes 212? 
 
 113. What number increased by 6% of itself becomes 212? 
 
 114. What number increased by 7 times itself becomes 40 ? 
 
 115. What number increased by | of itself becomes 24 ? 
 
 116. What number Increased by 16|% of itself becomes 42? 
 
 117. What number increased by 83J% of itself becomes 22? 
 
 118. What number diminished by | of itself becomes 30 ? 
 
 119. What number diminished by .06 of itself becomes 188? 
 
 120. What number diminished by 6% of itself becomes 188? 
 
 121. What number diminished by 16| -% of itself becomes 40? 
 
 122. What nu mber diminished by 33 J% of itself becomes 20 ? 
 
 123. What number diminished by 66| % of itself becomes 20 ? 
 
 124. What number diminished by 87^ % of itself becomes 20 ? 
 
 125. What number diminished by 8% of itself becomes 184? 
 
 Ex. 110. A number increased by 5 times itself becomes 6 times itself ; 
 since 6 times the number is 30, the number is 30 -f- 6, or 5. 
 
 Ex. 111. A number increased by § of itself becomes f times itself ; since 
 \ times the number is 30, the number is 30 -r- f , or 18. 
 
 Ex. 113. A number increased by 6% of itself becomes 106% times itself ; 
 since 106% times the number is 212, the number is 212 -f- 1.06, or 200. 
 
 Ex. 120. A number diminished by 6 % of itself becomes 94 % times itself ; 
 since-94% times the number is 188, the number is 188 -^ .94, or 200. 
 
 In these examples, what the % is of is not given ; it is therefore necessary 
 to assume something as a base. As in the former case, the equation enables 
 us to determine what operation to perform. Thus, Ex. 114, 8 x no. = 40 ; 
 the no. = 40 -^ 8. 
 
PERCENTAGE. 107 
 
 126. A merchant sells calico 
 
 for 10% more this year than Ans. 10?. This year's price is 
 
 last year ; this year he sells for \ 10 % of last y ear ' s ; since 110 % 
 
 . / J * of last year's price is 11^, last 
 
 11? a yard; what was last year's price is 11^- 1.10, or Hy. 
 year's price ? 
 
 127. A merchant sells calico Ans. IOj*. This year's price 
 
 for 10% less this year than last is 90 % of last y ear ' s P rice J since 
 
 year ; this year he sells for 9^ a 90 %, of last year ' s p ™ e is 9 /' last 
 J t , , o year's price is 9? -f- .90, or 10^. 
 
 yard ; what was last year s price ? 
 
 128. By running 15% faster than usual, a locomotive runs 
 690 miles a day ; what is the usual distance it runs per day ? 
 
 129. v A field having increased in productiveness 22% over 
 the preceding year, yielded 488 bushels of potatoes; what 
 was the yield the previous year? 
 
 130. ' A farmer sold 1800 pounds of wool, which was 12*-% 
 more than he sold the previous year ; how many pounds did 
 he sell the previous year? 
 
 131. * A man sold 20 cows for $400, which was 20% less 
 than they cost ; what did they cost ? 
 
 132. After paying 30% of his debts, a merchant found that 
 $ 210 would pay the remainder ; what did he owe at first ? 
 
 133. My salary this year is $75 per month, or 25% more 
 than last year ; what was my salary last year ? 
 
 134. A dealer has two kinds of apples : the first kind he 
 sells for 66|% more than the second ; he sells the first for 
 $1.50 a barrel; for how much does he sell the second kind? 
 
 135. A sells a cow for $49 which is 12*-% less than he 
 gets for his horse ; what does he get for the horse ? 
 
 136. I pay $12 per week for board this year, which is 
 20% less than I paid last year; what did I pay last year ? 
 
108 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 Names are sometimes given 
 to the terms used in percentage. 
 
 What the % is of, is the base ; 
 the base x the %, the percent- 
 age; base + percentage, amount; 
 base — percentage, difference ; 
 the %, the rate. 
 
 To find percentage : 
 
 137. Base 60; rate 7%. 
 
 138. Amount 80 ; base 60. 
 
 139. Difference 70; base 90. 
 
 140. Difference 40 ; rate 20 % . 
 
 141. Amount 63; rate 12-|-%. 
 
 To find difference : 
 
 146. Base 32; rate 25%. 
 
 147. Percentage 24 ; base 30. 
 
 148. Base 72; rate 33^%. 
 
 149. Percentage 25 ; rate 
 
 10%. 
 
 150. Percentage 30; rate 
 
 15%. 
 
 Illustration. 
 
 0% of 200 = 12. 
 200 + 12 = 212. 
 200 - 12 = 188. 
 200, base. 
 
 12, percentage. 
 212, amount. 
 188, difference. 
 C*%,rate%. 
 
 To find amount : 
 
 142. Base 60; rate 16f %. 
 
 143. Percentage 30 ; base 60. 
 
 144. Percentage 25 ; rate 5 % . 
 
 145. Percentage 400 ; base 
 4000. 
 
 To find rate % : 
 
 151. Base 60; percentage 40. 
 
 152. Amount 70 ; percentage 
 30. 
 
 153. Difference 60 ; percent- 
 age, 20. 
 
 154. Amount 75 ; base 60. 
 
 155. Difference 70 ; base 80. 
 
 To find base : 
 
 156. Rate 6 % ; percentage 12. 
 
 157. Rate 6% ; amount 212. 
 
 158. Rate 6%; difference 188. 
 
 159. Amount 90 ; percentage 
 
 10. 
 
 160. Rate 8% ; percentage 80. 
 
 To find base : 
 
 161. Rate 8% ; amount 324. 
 
 162. Rate 8% ; difference 368. 
 
 163. Difference 70 ; percent- 
 
 age 30. 
 
 164. Rate 10%; percentage 
 
 70. 
 
PERCENTAGE. 109 
 
 § 38. Profit and Loss. 
 
 What is the selling price? 
 
 165. Cost $10; gain $2. 170. Cost $50; loss $5. 
 
 166. Cost $36; gain831%. 171. Cost $12; loss 15%. 
 
 167. Cost $24; gain 121%. 172. Cost $20; loss 66f%. 
 
 168. Cost $25; gain 40%. 173. Loss $16; loss 8%. 
 
 169. Cost $54; gain 331%. 174. Gain $ 9; gain 75%?. 
 
 What is the cost ? 
 
 175. Selling price 4^; gain 331%. 
 
 176. Selling price $21; gain $6. 
 
 177. Selling price $30; gain 871%. 
 
 178. Selling price $20; gain 25%. 
 
 179. Selling price $1.90; loss 5%. 
 
 180. Selling price $25; loss $5. 
 
 181. Selling price 4^; loss 66|%. 
 
 182. Loss 5^; loss 831%,. 
 
 183. Gain $9; gain 5%. 
 
 What is the gain or loss % ? 
 
 184. Selling price $12; cost $10. 
 
 185. Selling price $12; gain $3. 
 
 186. Selling price $12; loss $3. 
 
 187. Cost $12; gain $3. 
 
 188. Cost $12; loss $3. 
 
 189. Cost $ 25 ; selling price $30. 
 
 The gain or loss is always regarded as some per cent of 
 the cost. 
 
 Ex. 166. The gain is § of $36, or $30 ; the selling price is §6G. 
 
 Ex. 175. 100% of C = cost ; 33£% of C = gain ; 133J% of C, or f of cost 
 = \f ; cost = \.<f> -4- f , or 3ft. 
 
 Ex. 184. The gain is $2 ; $2 = no. x $10 j no. = 2 * 10, or 20%. 
 
(S) 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 190. By selling a horse for Ana. $200. 100% of C=cost; 25% 
 8250 I gained 25% ; what did of C=gain ; 125 % of c = $250 > cost 
 
 * . 6 . o = # 250 -*- 1-25 = $ 200. 
 
 the horse cost r 
 
 191. By selling a horse for Ant $300. 100 % of C= cost; 10% 
 1270 I lost 10%; What did of C = loss; 90%ofC = $270; cost- 
 , , , , 9 $ 270 -T- .90 = $ 300. 
 
 the horse cost f 
 
 192. What must be/fche selling price of tea that cost 
 pound, to gain 20% ^ fi^G 
 
 193. What % is gained on an article bought at $ 4.50 and 
 sold for $6? 'bVgc/lt 
 
 194. A grocer sells corn at a profit of 12^ a bushel and 
 thereby gains 20% ; what is the cost? pQ ^ 
 
 195. Papers were sold for 5fl each at a gain of 25% ; 
 what was the gain on 4 papers ? fCy^ 
 
 196. By selling books at $ 1.88 there was a loss of 6% ; what 
 was the cost ? ^QJ)^ 
 
 197. Two horses were sold for $99 each ; on the first there 
 was a loss of 10% ; on^ie j^cond a .gain of 10% ; what was 
 paid for both horses ? ^ 
 
 198. Do I gain or lose or/ thf? sale of both horses in ex- 
 ample 197 ? How much ?fff * ^J^Q^J. 
 
 199. A merchant by selling silks for $12 mote than they 
 cost gained 66%% ; what was the selling price fjflJ 
 
 200. Find the v^ofit on land that cost $200 and was sold 
 at a gain of 12%/%/ 7 
 
 201. Find the selling price of grain that cost $ 200 and 
 was sold at a loss of 9%. 
 
 202. Find the % gained on oil bought at 12^ and sold at 14^. 
 
 203. A watch that cost $100 was sold at a gain of 10% ; 
 what was the gain ? What was the selling price ? 
 
u 
 
 PERCENTAGE. HI 
 
 204. What is gained per cent by selling coal at 1 6 a ton 
 that cost 1 5 a ton ? 
 
 205. A horse was sold at a loss of $50, which was 10% of 
 the cost ; what was the cost ? 
 
 206. A watch was sold for $ 240, at a gain of 20 % ; wh^t 
 was the cost ? 
 
 207. By selling a* cow for $24 more than she cost, a farmer 
 gained 37|% ; what was the selling price? 
 
 208. What must be the selling price of tea that cost 30^ a 
 lb., in order to gain 33^% ? 
 
 209. A boy, by selling newspapers at h$ each, gains 66%</ ; 
 what do they cost him ? 
 
 210. A boy, by selling newspapers at 5^ each, gains 60% on 
 the selling price ; what do they cost him ? 
 
 211. An article was bought for $4 and sold for $6 ; what 
 was the gain per cent ? What per cent of the selling price 
 was gained? f J 
 
 212. How shall I mark goods that cost $ 12, so as to gain 
 16}*? If 
 
 213. How shall I mark goods that cost $/5, so that I can 
 deduct 10% from the marked price, and yet make 8% on the 
 cost? 
 
 214. What % is gained on goods sold at double the cost ? ]Q 
 
 215. A carpet which cost $12 was sold for $9; what was 
 the loss % ? % iT 
 
 216. How must goods that cost $15 be marked so as to 
 gain 121% ? \\f 
 
 217. How shall I mark goods that cost $6, so that I can 
 deduct 10% from the marked price, and make 5% on the cost? 
 
 218. How must I sell goods that cost $56 so as to gain 
 37 J% ? 17 
 
112 AMERICAN MENTAL ARITHMETIC. 
 
 § 39. Commission. 
 A person may be employed to buy or sell for another ; the 
 employer is the principal; the other, the agent; the price 
 paid for the service, the commission ; the amount returned to 
 the principal, the net proceeds. 
 
 Illustration. — A farmer takes to a grocer 1(3 barrels of apples to be 
 sold at $2 a barrel, agreeing to pay him 10% commission for selling them. 
 It is just that the grocer should keep 10% of the entire sales. Hence, 
 
 If an agent sells, his commission is some per cent of 
 the sales. 
 
 Illustration. — A woolen manufacturer sends his agent $1050 with 
 'nstructions to buy wool after deducting his commission of 5%. If the agent 
 took 5% of $ 1050, he would take 5% of what he pays for the wool, and also 
 5% of what he keeps. It is not just that he should receive 5% of what he 
 keeps, because he performs no labor for it, but he is entitled to 5% of what 
 he pays for wool. Hence, 
 
 If an agent buys, his commission is some per cent of 
 the purchase. 
 
 219. Find the amount of sales when the principal receives 
 $40, at a commission of 20%. 
 
 Ans. $ 50. The commission is 20% of the sales ; 100% of sales = the sales ; 
 the sales — the commission = the net proceeds ; 80% of sales = $40 ; sales = 
 $ 40 h- .80 = $ 50. Proof. 20% of $ 50 = $ 10 ; $ 50 - $ 10 = $40. 
 
 220. An agent sells 10 bbl. of apples at $2 a bbl. ; what 
 is his commission at 10%. 
 
 Ans. $2. His commission is 10% of the selling price, or 10% of $20, or $2. 
 
 221. Find the rate of commission when $ 2 is paid for a 
 sale of $8. 
 
 Ans. 25%. The commission is some per cent of the sales ; if $ 2 is some 
 number times $8, the number is 2 h- 8, or 25%. Proof. 25% of $8 = $2. 
 
PERCENTAGE. 113 
 
 V?> 
 
 222. A woolen manufacturer sends his agent $ 1050 to 
 invest in wool after deducting 5% commission; what is the 
 purchase price and what is the commission? 
 
 Ans. $1000; $50. 5% of purchase is commission; 100% of purchase is 
 the purchase; 105% of purchase = $ 1050 ; purchase is $1050 -=- 1.05, or 
 $ 1000 ; the commission is 5% of $ 1000, or $ 50. Proof. 5% of $ 1000=$ 50 ; 
 $1000 + $50 = $1050. 
 
 223. Find the amount of sales when an agent receives $4 
 
 from a 2% commission. 
 
 Ans. $ 200. The commission is some per cent of the sales ; 2% of the sales 
 is $4 ; the sales are $4 -4- .02, or $200. Proof. 2% of $200 = $4. 
 
 224. Find the commission on the sale of a farm for $1000, 
 at 3%. 
 
 225. Find the commission on the purchase of a mill for 
 $1000, at 3%. 
 
 226. Find the commission when an agent receives $ 220 to 
 be invested in goods after deducting his commission of 10%. 
 
 227. How many pounds of sugar, at 8^ a pound, can an 
 agent buy for $40.80 after deducting his commission of 2% ? 
 
 228. Find the rate of commission when $2 is paid for a 
 sale of $10. 
 
 229. Find the amount of sales when a commission of 2% 
 pays the agent $8. 
 
 230. Find the commission on the sale of a house for 
 $20,000, at 5%. 
 
 231. Find the rate of commission when an agent receives 
 $5 for a sale of $200. 
 
 232. Find the commission when an agent receives $ 3360 to 
 be invested in goods after deducting his commission of 12%. 
 
 233. Find the amount of sales when a commission of 6% 
 pays the agent $ 600. 
 
 AM. MENT. AS. — 8 
 
114 AMERICAN MENTAL ARITHMETIC. 
 
 234. Find the net proceeds from the sale of 20 barrels of 
 sugar at $4, commission 10%. 
 
 235. Find the amount of the purchase, when an agent 
 invests $440 in sugar after deducting his commission of 10%. 
 
 236. A lawyer, having a debt of $ 7000 to collect, settles 
 for 60% ; his commission is 1|% ; how much does he remit? 
 
 237. A lawyer collects a debt, takes 2% for his fee, and 
 remits the balance, or $490 ; what is his fee? 
 
 238. An agent receives 15050 with which to purchase 
 goods, after deducting his commission of 1 % ; what was the 
 cost of the goods ? 
 
 239. A buys corn at 1J% commission, and 2J% for guar- 
 anteeing payment; if the whole cost, including commission 
 and guaranty is 1 416, what was the first cost of the corn? 
 
 240. My agent in Paris has bought for me 16 bales of 
 calico, each bale containing 50 pieces of 30 meters each, at 
 20^ a meter; his commission is 1% ; how much must I send 
 him? 
 
 241. In buying shoes at a commission of 2J%, an agent's 
 commission was $ 25 ; how much did he invest ? 
 
 242. An agent sells 10000 lb. of sugar at 8^ per pound, 
 and receives 1-|% commission; he pays $10 for freight; 
 find his commission and the net proceeds. 
 
 243. An agent receives $1010 with which to buy shoes 
 and pay his commission of 1% ; what does he pay for the 
 shoes? What is his commission? ' 
 
 244. D bought a horse for $250, paying 2% of the cost 
 for commission, and 2% of cost for traveling expenses; he 
 sold him at an advance of 10% on the entire cost, including 
 commission and expenses ; how much did he gain ? 
 
INTEREST. 
 
 Money paid for the use of 
 money is interest ; the money 
 loaned is the principal; the 
 sum of the principal and in- 
 terest is the amount. 
 
 There are three conceptions 
 of interest : 
 
 That the principal alone 
 bears interest, simple interest. 
 
 That the principal and the 
 interest on the principal at 
 the end of each year bear in- 
 terest, annual interest. 
 
 That the principal and the 
 interest on the principal at 
 the end of each year, and all 
 other unpaid interest at the 
 end of each year bear inter- 
 est, compound interest. 
 
 Unless otherwise stated, 
 simple interest is always un- 
 derstood. The rate of interest 
 is regulated by law; when- 
 ever interest is in excess of the 
 legal rate, it is called usury. 
 
 If $ 6 is paid for the use of $ 100, 
 $100 is the principal; $6, interest; 
 $ 106, amount. 
 
 Suppose $ 100 is loaned for 3 yr., 
 at 6 %, and the interest remains un- 
 paid until the end of this period. 
 
 At the end of the first year, $6 
 interest is due by each of the three 
 conceptions. 
 
 At the end of the second year, by 
 the first conception, another $6, or 
 $12 in all, is due; by each of the 
 second and third, in addition, the 
 interest of the first $6 for 1 yr. 
 (36^), or $ 12.36 in all. 
 
 At the end of the third year, by the 
 first conception, another $ 6 is due, or 
 $18 in all; by the second, in addi- 
 tion, the interest of the first $ 6 for 
 2 yr. (72^), and the interest of the 
 second $6 for 1 yr. (36^), or $19.08 
 in all ; by the third, in addition, the 
 interest of the first 36^ for 1 yr. 
 (2.16^), or $19.1016 in all. 
 
 For 1 yr. the simple, annual, and 
 compound interest are the same ; for 
 2 yr., the annual and compound in- 
 terest are the same, and greater than 
 the simple ; for 3 yr. or more, each 
 is different, the order of magnitude 
 being compound, annual, simple. 
 
 115 
 
116 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 40. Simple Interest. 
 
 1. What is the interest of 
 |1 for lyr. at 6%? 
 
 2. What is the interest of 
 llfor 1 mo. at 6%? 
 
 3. What is the interest of 
 II fori da. at 6%? 
 
 The interest of $ 1 for 1 yr. at 6% 
 is .06 of $ 1, or Of. 
 
 Since the interest of $ 1 for 12 mo. 
 is G^, for 1 mo. it is yV of 6fi, or 5 m. 
 
 Since the interest of $ 1 for 30 da. 
 is 5 m., for 1 da. it is -fa of 5 m., or 
 £ of a m. 
 
 To be memorized. 
 
 The interest of $ 1 for 1 yr. at 6% is 6? ; for 1 mo., 
 for 1 da., \ of a mill. 
 
 What is the interest of : 
 
 4. llfor 2 yr. at 6%? 
 
 5. $2 for 3 yr. at 6%? 
 
 6. $4 for 5yr. at 6%? 
 
 7. |1 for 2 mo. at 6%? 
 
 8. |1 for 3 mo. at 6%? 
 
 9. |1 for 5 mo. at 6%? 
 
 10. |2 for 7 mo. at I 
 
 11. 1 3 for 9 mo. at i 
 
 12. 1 5 for 7 mo. at l 
 
 13. $1 for 4 da. at ( 
 
 14. |1 for 10 da. at 
 
 15. |3 for 6 da. at ( 
 ■? 
 
 re? 
 
 1c? 
 
 16. 14 for 8 da. at 6% 
 
 17. |1 for 2 yr. 2 mo. 2 da. at 6% ? 
 
 18. 1 10 for 3 yr. 18 da. at 8% ? 
 
 19. |30 for 4 yr. 8 mo. at 6% ? 
 
 20. 1100 for 63 da. at 10%? 
 
 21. |1 for 5 yr. 5 mo. 5 da. at 10% ? 
 
 22. |1 for 33 da. at 12%? 
 
 Ex. 18. $2.44. The interest of $1 for 3 yr. at 6% is $.18; for 18 da., 
 % .003 ; for the whole time, $ .183 ; of $ 10, 10 times $.183, or $ 1.83 ; at 8%, 
 \ more, or $2.44. 
 
 Ex. 20. $ 1.75. The interest of $ 1 for 63 da. at 6% is $.0105 ; of $ 100, 
 $ 1.05 ; at 10%, f more, or $ 1.75. 
 
INTEREST. 117 
 
 What is the interest of $1: 
 
 23. For 3 yr. at 4% ? 32. For 8 mo. at 12% ? 
 
 24. For 5yr. at 7%? 33. For 1 da. at 7% ? 
 
 25. For 8 yr. at 9%? 34. For 1 da. at 8% ? 
 
 26. For 2 yr. at 5}% ? 35. For 3 da. at 9% ? 
 
 27. For 7 yr. at 2%? 36. For 7 da. at 12% ? 
 
 28. For 5 mo. at 4%? 37. For 5 da. at 2% ? 
 
 29. For 7 mo. at 7% ? 38. For 2 yr. 5 mo. at 7% ? 
 
 30. For 9 mo. at 4%? 39. For 7 yr. 7 mo. at 8%? 
 
 31. For 4 mo. at 9% ? 40. For 8 yr. 5 mo. at 6% ? 
 
 What is the amount of $ 1 : 
 
 41. For 2yr. 6 mo. at 8% ? 
 
 42. For 3 yr. 4 mo. at 6% ? 
 
 43. For 5 yr. 8 mo. 6 da. at 6% ? 
 
 44. For 1 yr. 10 mo. 24 da. at 5% ? 
 
 45. For 9 mo. 18 da. at 9% ? 
 
 46. For 3 mo. 12 da. at 6% ? 
 
 47. For 7 yr. at 7%? 
 
 48. For 4 yr. 10 mo. at 6% ? 
 
 49. For 8 mo. 12 da. at 3%? 
 
 50. For 7 mo. 6 da. at 4% ? 
 
 51. For 6 mo. 9 da. at 6%? 
 
 52. For 4 mo. 24 da. at 8% ? 
 
 53. For 1 yr. 6 mo. at 7%? 
 
 54. For 5 yr. 9 mo. at 6% ? 
 
 55. For 10 yr. 1 mo. 6 da. at 2% ? 
 
 56. For 12 yr. 10 mo. 3 da. at 6% ? 
 
 Ex. 23. 12^. When years alone, months alone, or days alone are given, 
 it may be simpler to find the interest directly without the 6 % method. Thus, 
 the interest of $ 1 for 1 yr. at 4 % is 4 ^, for 3 yr. 12 f. 
 
 Ex. 41. % 1.20. The amount is the interest plus the principal. 
 
69. |40 in 4 mo. at uy 
 
 70. |60 in 3 yr. at 5% 
 
 71. |15 in 9 mo. at 10% 
 
 118 AMERICAN MENTAL ARITHMETIC. 
 
 What principal will gain : 
 
 57. $24 in 4 yr. at 6% ? 61. $63 in 3 yr. at 3% ? 
 
 58. $36 in 3 yr. at 2% ? 62. $72 in 9 yr. at 4%? 
 
 59. $48 in 6 yr. at 4% ? 63. $20 in 2 yr. at 5% ? 
 
 60. $56 in 2 yr. at 7% ? 64. $ 30 in 3 mo. at 6% ? 
 
 65. $32 in 4 mo. 24 da. at 8% ? 
 
 66. $330 in 6 mo. 18 da. at 6% ? 
 
 67. $100 in 5 mo. 30 da. at 5% ? 
 
 68. $120 in 10 mo. at 12%? 
 
 ? 
 
 ? . 
 ? 
 
 72. $30 in 10 mo. at 6%? 
 
 73. $80 in 1 yr. 4 mo. at 12% ? 
 
 74. $50 in 2 yr. 6 mo. at 10% ? 
 
 What principal will amount to : 
 
 75. $496 in 4 yr. at 6% ? 84. $396 in 4 yr. at 8% ? 
 
 76. $236 in 2 yr. at 9% ? 85. $516 in 8 yr. at 9% ? 
 
 77. $ 600 in 2 yr. at 10% ? 86. $412 in 4 mo. at 9% ? 
 
 78. $ 620- in 3 yr. at 8% ? 87. $ 205 in 5 mo. at 6% ? 
 
 79. $ 226 in 2 yr. 2 mo. at 6% ? 88. $254 in 9 yr. at 3% ? 
 
 80. $318 in 8 mo. at 9% ? 89. $312 in 6 mo. at 8% ? 
 
 81. $ 540 in 3 yr. 6 mo. at 10%? 90. $416 in 4 mo. at 12% ? 
 
 82. $256 in 4 yr. at 7% ? 91. $321 in 7 mo. at 12% ? 
 
 83. $345 in 3 yr. at 5% ? 92. $436 in 18 mo. at 6% ? 
 
 Ex. 57. $ 100. Assume $ 1. $ 1 in 4 yr. at 6 % will gain 24^ ; it will take 
 as many dollars to gain $24 as 24^ is contained times in $24, or -$100. 
 
 Ex. 75. $400. Assume $1. $ 1 in 4 yr. at 6% will amount to $1.24; 
 it will take as many dollars to amount to $496, as $1.24 is contained times 
 in $ 496, or $ 400. 
 
INTEREST. 119 
 
 In what time will : 
 
 93. flOO gain $24 at 6% ? 99. $125 gain $20 at 2% ? 
 
 94. $200 gain $30 at 5% ? ioo. $100 gain $90 at 9% ? 
 
 95. $150 gain $36 at 4% ? 101. $250 gain $50 at 10% ? 
 
 96. $300 gain $42 at 7% ? 102. $425 gain $51 at 12% ? 
 
 97. $400 gain $64 at 8% ? 103. $325 gain $39 at 3% ? 
 
 98. $108 gain $27 at 5% ? 104. $500 gain $60 at 6% ? 
 
 105. A sum gain J of itself at 6 % ? 
 
 106. A sum gain i of itself at 8% ? 
 
 107. A sum gain | of itself at 10% ? 
 
 108. A suln gain itself at 8% ? 
 
 109. A sum gain ^ of itself at 12% ? 
 
 110. A sum gain 3 times itself at 4% ? 
 ill. In what time will $100 amount to $124 at 6% ? 
 
 112. $50 to $52 at 4% ? 119. $40 to $88 at 12% ? 
 
 113. $25 to $40 at 6% ? 120. $50 to $75 at 5% ? 
 
 114. $40 to $68 at 7% ? 121. $50 to $53 at 6% ? 
 
 115. $40 to $72 at 8% ? 122. $30 to $36 at 10% ? 
 
 116. $80 to $84 at 5% ? 123. $40 to $76 at 9% ? 
 
 117. $70 to $84 at 2%? 124. $50 to $62 at 6%? 
 
 118. $30 to $33 at 10%? 125. $60 to $84 at 4% ? 
 
 126. In what time will a sum amount to twice itself at 6% ? 
 
 127. In what time will a sum quadruple at 10% ? 
 
 128. In what time will a sum double at 7% ? 
 
 Ex. 93. 4 yr. Assume 1 yr. $ 100 in 1 yr. at 6 % will gam $6 ; it will 
 take as many years to gain $ 24 as $ 6 is contained times in $ 24, or 4 years. 
 
 Ex. 105. 8 yr. 4 mo. This means, ' in what time will $ 12 (any princi- 
 pal) gain $6 at 6%? ' Assume 1 yr., etc. 
 
 Ex. 111. 4 yr. This means, • in what time will $ 100 gain $ 24 at 6 % ? ' 
 Assume 1 yr., etc. 
 
 Ex. 127. 30 yr. This means, 'in what time will $10 (any principal) 
 gain $ 30 at 10 % ? ' Assume 1 yr., etc. 
 
120 AMERICAN MENTAL ARITHMETIC. 
 
 At what % will : 
 
 129. $ 100 gain $24 in 4 yr.? 135. $ 175 gain $70 in 2 yr.? 
 
 130. $ 200 gain $98 in 7 yr.? 136. $ 800 gain $96 in 8 yr.? 
 
 131. $ 300 gain $ 81 in 9 yr.? 137. $ 600 gain $ 60 in 5 yr.? 
 
 132. $ 500 gain $ 75 in 5 yr.? 138. $ 190 gain $76 in 4 yr.? 
 
 133. $ 400 gain $84 in 3 yr.? 139. $ 250 gain $35 in 7 yr.? 
 
 134. $ 100 gain $36 in 6 yr.? 140. $100 gain $81in9yr.? 
 
 141. A sum gain J of itself in 8 yr.? 
 
 142. A sum gain ^ of itself in 6 yr.? 
 
 143. A sum gain | of itself in 7 yr.? 
 
 144. A sum gain itself in 8 yr*. ? 
 
 145. A sum gain two times itself in 10 yr.? 
 
 146. A sum gain three times itself in 5 yr.? 
 
 147. At what % will $100 amount to $124 in 4 yr.? 
 
 148. $40 to $72 in 8 yr.? 154. $40 to $84 in 10 yr.? 
 
 149. $80 to $88 in 5 yr.? 155. $30 to $39 in 6 yr.? 
 
 150. $20 to $38 in 9 yr.? 156. $20 to $42 in 11 yr.? 
 
 151. $20 to $48 in 7 yr.? 157. $30 to $48 in 12 yr.? 
 
 152. $80 to $96 in 2 yr.? 158. $20 to $27 in 7 yr.? 
 
 153. $60 to $96 in 3 yr.? 159. $20 to $29 in 9 yr.? 
 
 160. At what % will a sum amount to twice itself in 10 yr. ? 
 
 161. At what % will a sum triple in 20 yr. ? 
 
 162. At what % will a sum double in 10 yr. ? 
 
 163. At what % will a sum quadruple in 12 yr.? 
 
 Ex. 129. 6 %. Assume 1 %. $ 100 in 4 yr. at 1% will gain $ 4 ; it will 
 take as many % to gain $ 24 as $ 4 is contained times in $> 24, or 6 %. 
 
 Ex. 141. 6J %. This means, ■ at what % will $ 12 (any principal) gain $ 6 
 in 8 yr. ? ' Assume 1%, etc. 
 
 Ex. 147. This means, ' at what % will $ 100 gain $ 24 in 4 yr. ? ' Assume 
 1 %, etc. 
 
 Ex. 161. 10%. This means, ' at what % will $12 (any principal) gain 
 $ 24 in 20 yr. ? ' Assume 1 %, etc. 
 
INTEREST. 121 
 
 164. Is it proper to reason 
 
 thus: "Since $1 amounts to „„ Mkm .„ 
 
 r * es - Because $5 will amount to 
 
 11.06, $5 will amount to 5 6 times as much as * 1. 
 times 11.06, or $5.30"? 
 Why? 
 
 165. Is it proper to reason _ . , 
 
 i o • dn No * Tne am0llnt ls in every case 
 
 thus: "Since $1 amounts to once the principal plus the interest< 
 
 $ 1.06 in 1 yr., in 2 yr. it will By reasoning as at the left, we make 
 amount to 2 times $1.06, or the amount twice the principal plus 
 $2.12"? Why? the interest. 
 
 166. If $60 amounts to $70 in 1 yr., what will it amount 
 to in 2 yr. ? 
 
 167. If $40 amounts to $70 in 5 yr., what will it amount 
 to in 10 yr.? 
 
 168. If $60 amounts to $100 in 4 yr., what will it amount 
 to in 8 yr. ? 
 
 169. At what % will $200 gain $56 in 4 yr.? 
 
 170. At what % will a sum triple itself in 40 yr. ? 
 
 171. At what °Jo will a sum gain 3 times itself in 30 yr. ? 
 
 172. What principal will gain $200 in 3 yr. at 10% ? 
 
 173. What principal will amount to $ 224 in 2 yr. at 6 % ? 
 
 174. In what time will $200 gain $160 at 8% ? 
 
 175. In what time will $200 amount to $256 at 7% ? 
 
 176. In what time will a sum gain 3 times itself at 10% ? 
 
 177. In what time will a sum quadruple itself at 2% ? 
 
 178. What is the interest of $100 for 2 yr. 6 mo. at 8% ? 
 
 179. How many dollars in 6 yr. at 3% will gain the interest 
 of $100 for 4 yr. at 6%? 
 
 180. If $30 amounts to $60 in 3 yr., what is the rate of 
 interest ? The amount in 4 yr. ? The interest each year ? 
 
122 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 When the time does not 
 exceed 123 days, a modifica- 
 tion of the 6% method is in 
 general use. 
 
 Moving the decimal point 
 two places to the left in the 
 principal, gives tlve interest 
 for 60 days at 6 %. 
 
 Since the interest of $ 1 for 60 da. 
 at 6% is If, and \f is .01 of f 1, 
 dividing the principal by 100, that 
 is, moving the decimal point two 
 places to the left, will give the in- 
 terest for 60 days at 6%. 
 
 What is the interest of : 
 
 181. |125 for 60 da. at 6% ? 
 
 182. 1250 for 60 da. at 6%? 
 
 183. $313 for 60 da. at 6%? 
 
 184. |400 for 63 da. at 6% ? 
 
 185. $100 for 93 da. at 6%? 
 
 186. $200 for 33 da. at 6% ? 
 
 187. $100 for 63 da. at 10% ? 
 
 194. $100 for 33 da. at 12% ? 
 
 195. $100 for 93 da. at 12% ? 
 
 196. $200 for 93 da. at 7%? 
 
 197. $100 for 63 da. at 
 
 198. $200 for 33 da. at 
 
 199. $500 for 63 da. at 
 
 200. $800 for 33 da. at 
 
 7%? 
 7%? 
 
 188. $100 for 93 da. at 10% ? 201. $200 for 93 da. at 
 
 189. $100 for 33 da. at 10% ? 
 
 190. $100 for 63 da. at 8%? 
 
 191. $100 for 93 da. at 8% ? 
 
 192. $100 for 33 da. at 8%? 
 
 193. $100 for 63 da. at 12% ? 
 
 202. $300 for 33 da. at 10% ? 
 
 203. $200 for 93 da. at 10% ? 
 
 204. $400 for 63 da. at 10% ? 
 
 205. $400 for 33 da. at 10% ? 
 
 206. $800 for 63 da. at 8% ? 
 
 Ex. 184. $4.20. $4 (moving the decimal point two places to the left) is 
 the interest for 60 days ; ^ of $4, or 20^, the interest for 3 days. 
 
 Ex. 191. $2.07. $1 is the interest for 60 days at 6%; \ of $1, or 50^, 
 the interest for 30 days ; ^ of 50/', or 5^, the interest for 3 days, $1.55, the 
 interest at 6 % ; $1.55+1 of $ 1.55, or $2.07, the interest at 8 %. 
 
 Ex.202. $2.75. $3 is the interest for 60 days at 6%; $1.50, for 30 
 days; Yof for 3 days; $1.65, the interest at 6%; 10% is- 1 / f 6%; $1.65 
 x 10 x J = $2.75, the interest at 10%. The last step is taken by moving the 
 decimal point one place to the right, and dividing by 6. 
 
INTEREST. 
 
 123 
 
 § 41. Trade Discount. 
 
 Merchants and manufac- 
 turers usually have fixed price 
 lists of their goods, and when 
 the market varies they change 
 the rate of discount instead of 
 changing the fixed price. The 
 fixed price is the list price; 
 the deduction is the trade dis- 
 count, the amount paid, the 
 net price. 
 
 They announce their terms 
 upon their bill heads thus, 
 "Terms 30 days less 5%," 
 etc. In addition to this, they 
 frequently offer an additional 
 discount for cash. 
 
 Sold a bill of goods, list 
 price |20, on 4 mo. at 5% 
 discount, and deducted 10% 
 for cash; what was the net 
 price ? 
 
 Ans. $17.10. This means that 
 payment was not due for 4 mos.; 
 that 5% of list price was to be de- 
 ducted because of the condition of 
 the market ; and that an additional 
 discount of 10%, after the first dis- 
 count had been subtracted, was made 
 for cash. 
 
 $20 less 5% of $20 = $19; $19 
 less 10% of $19 = $17.10. 
 
 207. Sold a bill of goods, list price 1 20, on 3 mo. at 10% 
 discount, and deducted 5% for cash; what was the net price? 
 
 208. Sold a bill of goods, list price $ 20, on 3 mo. at 5% 
 discount, and deducted 10% for cash ; what was the net price ? 
 
 209. Compare examples 20T and 208. Does it make any 
 difference with the net price whether the discount is 10% 
 off and 5% for cash, or 5% off and 10% for cash? 
 
 210. Which is the better for the purchaser, 10% off and 
 5% for cash, or 5% off and 10% for cash? 
 
 211. Compare examples 207 and 208 and decide whether 
 any account should be taken of the time before the bill is due 
 in computing the net price. 
 
124 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 42. True Discount. 
 
 The true present worth of 
 
 a long-time note, or of a sum . „ . ., dbftA _ ,. 
 
 * ' Am. Present worth $ 200 ; dis- 
 
 of money due a long time in count $ 12 . The present worth is 
 
 that sum which put at interest to- 
 day will amount to $212 in 1 yr. 
 Assume $1. $1 in 1 yr. at 6% 
 amounts to $ 1.06 ; it will take as 
 many dollars to amount to $212, as 
 $ 1.06 is contained times in $212, or 
 $200. The discount is the debt 
 minus the present worth, or $ 12. 
 
 advance, is that sum which, 
 put at interest now, will 
 amount to the debt at the 
 expiration of the time. 
 
 John Smith owes me $212 
 a year from to-day; what is 
 the present worth, money at 
 6%? What is the discount ? 
 
 Find the present worth of : 
 
 212. |412 due in 6 mo., int. 6%. 
 
 213. $324 due in 8 mo., int. 12%. 
 
 214. $321 due in 1 yr., int. 7%. 
 
 215. $430 due in 1 yr. 6 mo., int. 5%. 
 
 216. $520 due in 8 mo., int. 6%. 
 
 217. $340 due in 1 yr. 4 mo., int. 10%. 
 
 218. $210 due in 10 mo., int. 6%. 
 
 219. $590 due in 2 yr., int. 9%. 
 
 220. $336 due in 3 yr., int. 4%. 
 
 221. $644 due in 5 yr., int. 8%. 
 
 222. $427 due in 9 mo., int. 9%. 
 
 223. $324 due in 1 yr. 4 mo., int. 6%. 
 
 224. $230 due in 1 yr. 8 mo., int. 9%. 
 
 225. $300 due in 4 yr. 2 mo., int. 12%. 
 
 226. $266 due in 5 yr. 6 mo., int. 6%. 
 
 Ex. 212. $ 400. $ 1 will amount to $ 1.03 in 6 mo. at 6 % ; it will take as 
 many dollars to amount to $412 as $ 1.03 is contained times in $412, or $400. 
 
INTEREST. 125 
 
 § 43. Bank Discount. 
 
 The true method of finding the present worth of a short- 
 time note is to find that sum which will amount to the face 
 of the note in the given time at the given rate. But since 
 the interest on the face for the given time at the given rate 
 is nearly the same, more easily found, and to the advantage 
 of the lender, the latter method, known as Bank Discount, is 
 employed. 
 
 The payer is allowed, on all notes, three days, called days 
 of grace, for payment, after the note becomes due. Hence, 
 
 In finding bank discount, three days are always added 
 to the time. 
 
 In this set the days of grace are included. 
 
 What is the bank discount of : 
 
 227. 1360 for 33 da. at 6% ? 239. $240 for 63 da. at 12% ? 
 
 228. $240 for 63 da. at 6% ? 240. $ 240 for 93 da. at 12% ? 
 
 229. $360 for 33 da. at 10% ? 241. |160 for 63 da. at 8% ? 
 
 230. $360 for 63 da. at 10% ? 242. $370 for 63 da. at 4% ? 
 
 231. $430 for 33 da. at 6% ? 243. $280 for 33 da. at 5% ? 
 
 232. $520 for 63 da. at 9% ? 244. $410 for 93 da. at 6% ? 
 
 233. $640 for 93 da. at 6% ? 245. $190 for 63 da. at 6% ? 
 
 234. $150 for 33 da. at 6% ? 246. $200 for 33 da. at 6% ? 
 
 235. $260 for 63 da. at 6% ? 247. $900 for 33 da. at 10% ? 
 
 236. $830 for 93 da. at 6% ? 248. $720 for 63 da. at 8% ? 
 
 237. $240 for 93 da. at 6% ? 249. $650 for 93 da. at 6% ? 
 
 238. $240 for 33 da. at 12%? 250. $875 for 33 da. at 4% ? 
 
 Ex. 227. % 1.98. The interest of $ 360 for 60 da. at 6 %, is $ 3.60 j for 30 
 da., $ 1.80 ; for 3 da., $ .18 ; for 33 da., $ 1.98. 
 
126 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 44. Stocks. 
 
 To raise money for the pros- 
 ecution of business enterprises, 
 stock companies are often formed. 
 Shares are issued with a face value 
 (par value) of $100, but some- 
 times this is made $50, or $25. 
 
 Shares do not often sell for their 
 par value, but for more (above par), 
 or for less (below par), according 
 to the success of the enterprise. 
 
 Earnings (dividends) are paid 
 at certain periods. 
 
 Brokers buy and sell stock for 
 their principals, charging some- 
 thing (brokerage) both for buy- 
 ing and selling. 
 
 Illustration. 
 On a mining claim in Mexico, 
 gold and silver were found in 
 such abundance that a stock 
 company was organized. They 
 issued 100000 shares with a face 
 value of $ 100 a share, but sold 
 them all for $ 2 a share. The 
 earnings at the end of the first 
 year were $660000, and a divi- 
 dend of 6% was declared. After 
 the dividend, the stock sold for 
 $110 a share. At this time 
 James Lyman bought of John 
 Fluker, through a broker, 50 
 shares, paying \% brokerage, 
 and received the certificate rep- 
 resented. He afterwards sold 
 the shares at $75, paying \% 
 brokerage. 
 
INTEREST. 
 
 127 
 
 On this certificate $100 is 
 the par value of each share ; 
 |2, 1110, 175 were the mar- 
 ket values at different times. 
 
 The par value is $100 
 -unless otherwise stated. 
 
 The brokerage and divi- 
 dend are some % of the par 
 value. 
 
 251. What was the par 
 value of the 50 shares when 
 bought by John Fluker? 
 What was the market value ? 
 
 252. What was the par 
 value of the 50 shares when 
 bought by James Lyman? 
 What was the market value ? 
 
 253. How much stock did 
 John Fluker own? 
 
 254. How much stock did 
 James Lyman own? 
 
 255. What was the income 
 on one share at the time of 
 first dividend? 
 
 256. How much brokerage 
 did Lyman pay on one share 
 when he bought ? When he 
 sold? 
 
 257. How much did James 
 Lyman pay per share ?i How 
 much did he receive ? 
 
 This understanding saves confu- 
 sion, and makes it unnecessary to 
 call attention to the par value. 
 
 The brokerage and dividend must 
 be some per cent either of the market 
 value or of the par value. The par 
 value never changes, the market value 
 is constantly changing ; hence the 
 former is selected. 
 
 Ans. Par value, $ 5000 ; market 
 value, $ 100. 
 
 Ans. Par value, $5000; market 
 value, $ 5500. 
 
 Ans. $5000 stock. 
 
 Ans. $ 5000 stock. 
 
 Ans. $6. The income on one 
 share was 6 % of $ 100, or $ 6. 
 
 Ans. $ \ when he bought, $ \ 
 when he sold. The brokerage was 
 \% and £% of $100. 
 
 Ans. He paid $110^. $110+$|- 
 brokerage. He received $ 74|. $ 75 
 — $ I brokerage. 
 
128 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 258. Does the market value 
 affect the dividend ? 
 
 259. Does the dividend or 
 the market value in any way 
 affect the brokerage ? 
 
 260. What is the dividend 
 on one share of 6% stock 
 bought at 50 and sold at 60 ? 
 
 261. What is the market 
 value of 16000 5% stock? 
 
 How many shares in : 
 
 262. $500 5% stock? 
 
 263. 18000 3% stock? 
 
 264. $1000 7% stock? 
 
 265. 11200 10% stock? 
 
 266. $1100 6% stock? 
 
 Find the cost of : 
 
 272. $800 stock, at 80, brokerage J? 
 
 273. $400 stock, at 90, brokerage J? 
 
 274. $500 stock, at 60, brokerage J? 
 
 275. $1200 stock, at 50, brokerage -J? 
 
 276. $1000 stock, at 70, brokerage -| ? 
 
 277. $8100 stock, at 80, brokerage J? 
 
 278. $7200 stock, at 50, brokerage |? 
 
 279. $400 stock, at 40, brokerage f ? 
 
 280. $600 stock, at 90, brokerage J? 
 
 281. $700 stock, at 60, brokerage j? 
 
 Ex. 262. 5 shares. The par value of one share is $ 100 ; $ 500 is the par 
 value of as many shares as $ 100 is contained times in $ 500, or 5 shares. 
 Ex. 272. $ 641. The entire cost of one share is $ 801 ; the cost of 8 shares 
 
 Ans. No. The dividend is some 
 % of the par valio'. 
 
 Ans. No. The brokerage is some 
 % of the par value. 
 
 Ans. $ 6. The dividend is some % 
 of the par value. 
 
 Ans. We have no means of know- 
 ing. 
 
 267. $15000 8% stock? 
 
 268. $2000 9% stock? 
 
 269. $1400 12% stock? 
 
 270. $9900 3% stock? 
 
 271. $2500 4% stock? 
 
INTEREST. 129 
 
 Find the net proceeds of : 
 
 282. $800 stock sold at 80J-, brokerage J. 
 
 283. $400 stock sold at 90, brokerage £. 
 
 284. $ 500 stock sold at 50^, brokerage \. 
 
 285. $600 stock sold at 70, brokerage J. 
 
 286. $ 1200 stock sold at 80, brokerage f . 
 
 287. $ 1000 stock sold at 40, brokerage |. 
 
 288. $900 stock sold at 90|, brokerage |. 
 
 289. $1200 stock sold at 60 J, brokerage \. 
 
 Ex. 282. $ 640. The net proceeds on one share is $ 80 ; on 8 shares, 8 
 times $80, or $640. 
 
 Find the dividend on : 
 
 290. $800 4% stock bought at 90. 
 
 291. $400 2% stock sold at 80. 
 
 292. $500 6% stock sold at 70 {. 
 
 293. $600 7% stock bought at 50. 
 
 294. $1200 3% stock sold at 80 J. 
 
 295. $1500 5% stock sold at 60. 
 
 296. $2000 9% stock sold at 89. 
 
 Ex. 290. $32. The dividend on one share is $4 ; on 8 shares, 8 times 
 
 $4, or $32. 
 
 How many shares may be bought : 
 
 297. For $800 at 79|, brokerage |? 
 
 298. For $1000 at 49f, brokerage J? 
 
 299. For $1200 at 59|, brokerage -J? 
 
 300. For $1400 at 69^, brokerage f ? 
 
 301. For $700 at 19|, brokerage |? 
 
 302. For $2000 at 39|, brokerage |? 
 
 303. For $3000 at 59|, brokerage £? 
 
 Ex. 297. 10 shares. The cost of one share is $ 80 ; $ 800 will buy as 
 many shares as $ 80 is contained times in $ 800, or 10 shares. 
 
 AM. MENT. AR. 9 
 
130 AMERICAN MENTAL ARITHMETIC. 
 
 How much stock gives an income : 
 
 304. Of $200; stock 4% ? 309. Of 18000; stock 4% ? 
 
 305. Of $400; stock 5% ? 310. Of $5000; stock 5% ? 
 
 306. Of $120; stock 6%? 311. Of $1800; stock 9% ? 
 
 307. Of $100; stock 2%? 312. Of $2000; stock 10%? 
 
 308. Of $900; stock 3% ? 313. Of $4800; stock 12% ? 
 
 Ex. 304. $ 5000 stock. The income on one share is $ 4 ; it will take as 
 many shares to yield $ 200 as $ 4 is contained times in $ 200, or 50 shares. 
 50 shares = $ 5000 stock. 
 
 What % will I realize on my investment : 
 
 314. When 6% stock is bought at 80? 
 
 315. When 5% stock is bought at 50? 
 
 316. When 8% stock is bought at 40? 
 
 317. When 9% stock is bought at 70? 
 
 318. When 4% stock is bought at 20? 
 
 319. When 3% stock is bought at 60? 
 
 320. When 7% stock is bought at 70? 
 
 Ex. 314. 7 $ %. One share costs $ 80 and gains $ 6 ; the gain % is $ 6 -=- $ 80, 
 
 or7i%. 
 
 Find the price of a 4% stock : 
 
 321. To equal a 6% stock at 50. 
 
 322. To equal a 5% stock at 20. 
 
 323. To equal a 7% stock at 60. 
 
 324. To equal an 8% stock at 80. 
 
 325. To equal a 5% stock at 40. 
 
 326. To equal a 12% stock at 80. 
 
 327. To equal a 10% stock at 50. 
 
 Ex. 321. $33^. One share costs $50 and gains $6 ; $1 gains -fa of $6, 
 or 12^ ; if $ 1 gains 12^, it will take as many dollars to gain $ 4 as 12^ is 
 contained times in $4, or $33k 
 
PRACTICAL EXERCISES. 
 
 § 45. At the Lumber Yard. 
 
 The same piece of lumber may be called by different names 
 since the classification is not exact. The following may be 
 helpful : — 
 Lumber — wooden building material. 
 
 I. Boards — 1 in. thick — less if specified. 
 
 1. Stock boards — boards of uniform width — 12 in. 
 
 wide. 
 
 2. Fencing — 6 in. wide. 
 
 3. Flooring — matched boards. 
 
 4. Siding or clapboards — Jin. thick — thicker at 
 
 one edge. 
 II. Dimension Stuff. 
 
 1. Scantling — 2 in. to 4 in. thick — 3 in. to 4 in. 
 
 wide. 
 
 2. Joist — 2 in. thick — any width. 
 
 3. Plank — 2 in. thick — wider than 4 in. 
 
 4. Timber — thicker than 2 in. — wider than 4 in. 
 
 III. Foot Stuff — sold by linear foot. 
 
 1. Battens — for covering cracks. 
 
 2. Moulding — for finishing. 
 
 IV. Laths — 4 ft. long, 1 J in. wide — 50 to a bunch. 
 
 V. Shingles — 4 in. wide — 250 to the bunch. They are 
 not of uniform width, but every 4 in. is reckoned as 
 one shingle. 
 
 131 ' 
 
132 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 Lumber is sold by the board 
 foot. 
 
 A board foot is the equivalent ^dzwft+iiJJt^ 
 of 1 ft. long, 1 ft. wide, and 1 in. ^mmmM^kz^ 
 thick. 1 board ft. 
 
 Inch lumber 12 ft. long con- 
 tains as many board ft. as there 
 are inches in its width. 
 
 For 12 ft. x t^- ft. x 1 in. 
 1 ft. x 1 ft. x 1 in. 
 
 How many feet of lumber are there in 
 
 1. 18 in., 12 ft. board? 
 
 2. 18 in., 10 ft. board? 
 
 3. 18 in., 6 ft. board? 
 
 4. 18 in., 14 ft. board? 
 
 5. 18 in., 18 ft. board? 
 
 6. 3 6 in., 10 ft. boards? 
 
 7. 4 7 in., 16 ft. boards? 
 
 8. 1 joist 2 x 4, 12? 
 
 9. 1 joist 2 x 4, 16 ? 
 
 10. 1 scantling 3 x 4, 12 ? 
 
 n. 1 plank 2 x 10, 12? 
 
 12. 1 plank 2 x 10, 16 ? 
 
 13. 4 timbers 4 x 4, 20 ? 
 
 14. 3 pieces 8 x 8, 24? 
 
 15. 5 pieces 2 x 4, 12? 
 
 16. 8 3 in., 12 ft. boards? 
 
 17. 6 14 ft. fencing? 
 
 18. 6 pieces 4 x 6, 20 ? 
 
 19. 10 6 in., 12 ft. siding? 
 
 20. 10 scantling 3 x 4, 16 ? 
 
 Ex. 2. 7. An 8 in. 12 ft. board would contain 8 ft, ; 10 = 12 - \ of 12 ; 
 8 — |of8 = 6|; taking the nearest whole number, 7. 
 
 Ex. 7. 37. 4 7 in. boards = 1 28 in. board. A 28 in. 12 ft. board would 
 contain 28 ft. ; 16 =12+* \ of 12 \ 28 + \ of 28 = 87$ ; taking the nearest 
 whole number, 37. 
 
 Ex. 9. 11. This is read, "1 joist 2 by 4, 16." The joist is 2 in. thick, 
 4 in. wide, 16 ft. long. 
 
 Ex. 13. 107. 4, 4 x 4 pieces =1 64 in. board. A 64 in. 12 ft. board 
 would contain 64 ft. ; 20 = 12 + § of 12 ; 64 + 1 of 64 = 106§ ; taking the 
 nearest whole number, 107. 
 
 * Unless otherwise specified, stuff less than an inch thick is counted as an 
 inch thick. 
 
PRACTICAL EXERCISES. 
 
 133 
 
 Dealers sometimes use a card like the following, carried out 
 for a great variety of lengths and dimensions. Usually, as in 
 this table, fractions of a foot are neglected. 
 
 Lumber Table. 
 
 SIZE. 
 
 12 
 
 14 i 16 
 
 18 | 20 | 22 
 
 24 
 
 26 28 
 
 30 | 32 | 34 
 
 36 
 
 38 
 
 40 
 
 2x6 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 26 
 
 28 
 
 30 
 
 32 
 
 34 
 
 36 
 
 38 
 
 40 
 
 2x8 
 
 16 
 
 19 
 
 21 
 
 24 
 
 27 
 
 29 
 
 32 
 
 35 
 
 37 
 
 40 
 
 43 
 
 45 
 
 48 
 
 51 
 
 53 
 
 3x4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4x6 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2x4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2x3 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 21. Verify the results in the 1st horizontal line. 
 
 22. Verify the results in the 2d horizontal line. 
 
 23. Declare the results for the 3d horizontal line. 
 
 24. Declare the results for the 4th horizontal line. 
 
 25. Declare the results for the 5th horizontal line. 
 
 26. Declare the results for the 6th horizontal line. 
 
 How many board feet are there in 
 
 27. 4plk., 2x4, 12? 
 
 28. 2 timbers, 4x6, 40? 
 
 29. 9 joists, 2 x 3, 16 ? 
 
 30. 12bds., 1x12, 16? 
 
 31. 24 fencing 1 x 6, 16 ? 
 
 32. 4 piece stuff, 3x3, 12? 
 
 33. 6 scantling, 2 x 4, 16 ? 
 
 34. 3 timbers, 6x6, 40? 
 
 35. 2 timbers, 8x8, 60? 
 
 36. 3 joists, 2x8, 12? 
 *37. 6plk., Ijxl2, 12? 
 
 38. 6plk., Ifxl2, 12? 
 
 39. 6plk., I|xl2, 12? 
 
 40. 6plk., I|xl2, 12? 
 
 * Count lumber less than 1 in. thick as 1 in. ; from 1£ to 1\, as 1| ; from 
 If to 1£, as 1£ ; from If to 2, as 2. 
 
134 AMERICAN MENTAL ARITHMETIC. 
 
 Laths. 
 
 Laths are nailed upon joists ! of an inch apart to receive 
 tlie plaster. 
 
 41. How many laths are there in a bunch ? See p. 131. 
 
 42. What are the dimensions of a lath ? 
 
 43. How many sq. in., including the space between two 
 laths, does 1 lath cover ? 
 
 44. How many sq. in. will 1 bunch cover? 
 
 45. How many sq. in. are there in 3 sq. yd. ? 
 
 46. How do the results in Exs. 44 and 45 agree with the 
 contractor's rule, "One bunch of laths will cover 3 sq. yd."? 
 
 47. Using contractor's rule, how many bunches of laths 
 will be required for the ceiling of a room 18 ft. x 18 ft.? 
 
 Shingles. 
 A hunch of shingles is 20 in. wide and contains 25 
 courses on each side; a shingle averages J^ in. wide. 
 
 48. How many shingles are there in a bunch ? 
 
 49. If a shingle is laid 4 in. to the weather, how many sq. 
 in. will one shingle cover? 
 
 50. How many sq. in. will 1000 shingles cover? 
 
 51. How many sq. in. are there in 100 sq. ft. ? 
 
 52. How do results in Exs. 50 and 51 agree with the 
 carpenter's rule, " One thousand shingles laid 4 in. to the 
 weather will cover 100 sq.ft. "? 
 
 53. How many bunches of shingles laid 4 in. to the weather 
 will be required for a roof 20 ft. x 40 ft. ? Use contractor's 
 rule. 
 
 54. How many bunches of shingles laid 4 in. to the weather 
 will be required for a roof 18 ft. x 12 ft. ? 
 
PRACTICAL EXERCISES. 
 
 135 
 
 § 46. Measurement of Logs. 
 
 By the number of 
 board ft. in a log is 
 meant the number 
 of board ft. in the 
 largest piece of tim- 
 ber that can be sawed 
 from the log. 
 
 A piece 1 in. x 1 in. x 12 ft. 
 (usually written 1 x 1, 12) contains 
 1 board ft. 
 
 12. feet 
 
 4 Board ft. 
 
 The area of the greatest 
 inscribed square is AB . 
 AB 1 + AC 2 = BC 2 . 
 (sq. of hyp. = sum of sqs.) 
 
 2AB =BC\ 
 
 AB> = ^- 
 2 
 
 A log 12 ft. long contains as 
 many board ft. as there are sq. 
 in. on its squared end, or as many 
 board ft. as there are sq. in. in 
 half the square of its diameter. 
 
 If the log is to be sawed into 
 boards, deduct \ for waste. 
 
 How many feet of lumber are there in a log : 
 
 55. Length 12 ft. ; D. 8 in. ? 58. Length 24 ft. ; D. 10 in. ? 
 
 56. Length 16 ft. ; D. 10 in.? 59. Length 32 ft. ; D. 18 in. ? 
 
 57. Length 20 ft. ; D. 12 in.? 60. Length 16 ft. ; D. 10 in. ? 
 
 Ex. 56. 67 ft. of lumber. 10 2 = 100 ; \ of 100 = 50 ; if the log were 12 ft. 
 long, there would he 50 hoard ft.; 50 + \ of 50 = 67. 
 
 How many feet of boards are there in a log : 
 
 61. Length 12 ft. ; D. 8 in. ? 64. Length 20 ft. ; D. 6 in. ? 
 
 62. Length 18 ft. ; D. 10 in.? 65. Length 36 ft. ; D. 10 in. ? 
 
 63. Length 24 ft. ; D. 12 in. ? 66. Length 40 ft. ; D. 16 in. ? 
 Ex. 61. 26 ft. There would he 32 ft. of lumber ; 32 - \ of 32 = 26. 
 
13G AMERICAN MENTAL ARITHMETIC. 
 
 § 47. At the Carpet Store. 
 
 Carpeting is usually a yard wide, and sold by the lineal 
 yard. It is cut up into breadths, and these breadths are 
 matched and sewed together. As the same figure is repeated 
 at intervals varying from 1 inch to 6 feet according to the 
 pattern, few carpets can be matched without loss. 
 
 Find the number of breadths and the length of each 
 breadth. 
 
 A room 14 ft. by 13 ft. is to be carpeted. 
 
 67. How many breadths will be required if they run 
 lengthwise? How many if they run crosswise? Ans. 5. 
 
 68. If there is no loss in matching, how many yards, 1 yd, 
 wide, should be purchased, the breadths running lengthwise ? 
 How many if the breadths run crosswise ? 
 
 69. How many yards are turned under in each case in 
 Ex. 68? 
 
 70. If the same figure is repeated at intervals of 9 in., 
 how many linear inches are wasted on each strip running 
 lengthwise ? Explain. 
 
 71. If the same figure is repeated at intervals of 9 in., how 
 many linear inches are wasted on each strip running cross- 
 wise ? Explain. 
 
 72. If there are five breadths, on how many is there waste 
 in matching ? Explain. Ans. 4. 
 
 73. What is the cost, at $2 a yard, breadths running 
 lengthwise, the same figures 1 ft. apart, carpet | yd. wide ? 
 What is the cost, breadths running crosswise? 
 
 74. 14 ft.xl3 ft. = 182 sq. ft.; 182 sq. ft. = 20f sq. yd. 
 Will 20| yd. of carpet 1 yd. wide be sufficient? Why? 
 
PRACTICAL EXERCISES. 137 
 
 § 48. With the Paper Hanger. 
 
 Wall paper is sold by the double roll, 48 ft. x 1| ft., or by 
 the single roll, 24 ft. x 1|- ft. It is cut up into strips, matched, 
 and pasted upon the walls or ceiling. 
 
 From the distance around the room in feet, deduct 3 ft. 
 for each opening {door or window). The remainder -*- 1 will 
 give the number of strips required for the walls. 
 
 The walls and ceiling of an 8 ft. room 20 ft. x 16 ft. are to 
 be papered ; there are four windows and a door. 
 
 75. How many strips for the walls will a double roll 
 supply? Explain. 
 
 Ans. 6 strips. There is a loss in matching, but this need not be consid- 
 ered, because the paper does not extend to the floor, on account of the base 
 board, nor to the ceiling, on account of the border. 
 
 76. By the rule, how many double rolls will be needed for 
 the wall? Explain. Ans. 7. 
 
 77. If the strips run lengthwise, how many strips for the 
 ceiling will a double roll supply? Explain. Ans. 2. 
 
 78. If the strips run crosswise, how many strips for the 
 ceiling will a double roll supply ? Explain. Ans. 3. 
 
 79. How many double rolls y will be required for the ceil- 
 ing, if the strips run lengthwise ? 
 
 80. How many double rolls will be required for the ceil- 
 ing, if the strips run crosswise ? 
 
 81. Do we ever in practice find the exact number of sq. 
 ft. on the walls, deduct for the doors and the windows, and 
 then divide by the number of sq. ft. in a double roll ? Why 
 not? 
 
138 AMERICAN MENTAL ARITHMETIC. 
 
 § 49. Average. 
 What is the average of 80, Ana. 78f . Their sum is 630 ; 
 
 85, 90, 70, 75, 100, 60, 70 ? 630 + 8 = 78 *' 
 
 A better way is to begin, with the first, find the difference between it and 
 each of the others, and unite these differences as we proceed. Thus, 
 (85 - 80 = 5), 5 ; (90 - 80 = 10 ; add to 5, because 90 is more than 80), 15 ; 
 (80 - 70 = 10 ; subtract 10, because 70 is less than 80), 5 ; (80 - 75 = 5 ; 
 subtract), 0; (100-80 = 20; add), 20; (80-60 = 20; subtract), 0; 
 (80 — 70 = 10 ; subtract), — 10. Since the total of the differences for 8 num- 
 bers is — 10, the average difference must be | of —10, or— 1J ; 80— 1J = 78|. 
 
 In practice, we say 5, 15, 5, 0, 20, 0, — 10, — 1£, 78£. 
 
 Find the average of : 
 
 82. 80, 70, 70, 60, 90, 84, 85, 87, 90, 93. 
 
 83. 85, 86, 87, 88, 90, 92, 93, 94, 95, 96. 
 
 84. 70, 75, 80, 85, 90, 95, 100, 90, 80, 70. 
 
 85. 75, 80, 82, 84, 78, 85, 71, 60, 54, 96. 
 
 86. 50, 80, 85, 70, 75, 100, 99, 86, 87. 
 
 87. 76, 68, 54, 47, 97, 98, 96, 97, 95. 
 
 88. 84, 83, 80, 87, 89, 88, 82, 81, 85. 
 
 89. 66, 67, 52, 53, 47, 84, 87, 91, 90. 
 
 90. 100, 100, 90, 95, 98, 99, 97, 100. 
 
 91. 75, 76, 73, 50, 62, 64, 70, 80, 89. 
 
 92. 91, 89, 94, 93, 91, 99, 100, 50, 60. 
 
 93. 89, 89, 88, 87, 86, 93, 95, 96, 97. 
 
 94. 45, 37, 29, 50, 49, 61, 62, 59, 48. 
 
 95. 84, 64, 94, 74, 54, 100, 90, 80, 70. 
 
 96. 65, 70, 80, 85, 90, 94, 86, 85, 72, 84. 
 
 97. 90, 92, 92, 93, 93, 94, 94, 94, 96, 98. 
 
 Ex. 83. 90.6. It is best to take as the base some number ending in a 
 cipher ; in this take 90. - 5, - 9, - 12, - 14, - 12, - 9, - 5, 0, 6 ; 90.6. 
 
INVOLUTION AND EVOLUTION. 
 
 A number written to the right of 
 another, a little above, shows how many 
 times the latter is used as a multiplier. 
 
 The number used as a multiplier is 
 the base; the number showing how many 
 times the base is used, the exponent; the 
 result, the power ; the process, involution. 
 
 A number written to the left of another 
 in the symbol •*/, or the denominator of 
 a fractional exponent, calls for the base 
 which taken this number of times as a 
 multiplier will produce the latter. 
 
 The result is the root; the process, 
 evolution. 
 
 When the second root, usually called 
 the square root, is required, the figure 2 
 is not written in the -^/. 
 
 The third root is usually called the 
 cube root. 
 
 Illustration. 
 
 2 4 = 16 ; 
 
 read, 
 
 2 to the 4th power = 16. 
 
 2, base; 
 
 4, exponent; 
 
 16, power ; 
 
 means 
 
 2x2x2x2 = 16. 
 
 Vl6 = 2, or 16* = 2. 
 4 calls for the base which 
 taken 4 times as a multi- 
 plier will produce 16 ; 
 read, 
 the 4th root o/16 = 2. 
 Vl6 = 4; 
 read, 
 the square root of 16 = 4. 
 
 V8 = 2; 
 read, 
 
 the cube root of 8 = 2. 
 
 Declare the value of : Declare the value of : 
 
 1. 2*; 3*; 6*; 7*. 6 . Vl6; V27; V32; VM. 
 
 2. 8 3 ; 9 3 ; 5 3 ; 4 3 . Ifis- 27^- 32^- 81* 
 
 3. 25 2 ; 23 2 ; 22 2 ; 21*. ?> *,' » j . 
 
 4. 17 2 ; 18 2 ; 19 2 ; 16 2 . 8 27 > 164 5 64 *5 A 
 
 5. 3 4 ; 2 6 ; 3 5 ; 2 3 . 9. 125^; 64*; 36*; 27*. 
 
 Ex. 8., Ans. 9. This means extract the cube root of 27, and square the 
 result. 
 
PROPORTION. 
 
 Division may be expressed by writing 
 the dividend before and the divisor after 
 a colon. Such an expression is a ratio. 
 See p. 44. 
 
 The dividend is the antecedent; the 
 divisor, the consequent. 
 
 Two ratios may be equal. An equality 
 of two ratios is a proportion. 
 
 The sign of equality is often abbrevi- 
 ated by writing only the extremities of 
 the sign ' =,' making : : 
 
 The first and last terms of a propor- 
 tion are extremes; the second and third, 
 means. 
 
 In a proportion, the product of the 
 extremes must equal the product of 
 the means. 
 
 If three terms of a proportion are 
 given, the other may be found. 
 
 The mean proportional of two quanti- 
 ties is the square root of their product. 
 
 1. What is the value of the ratio 27 : 81 ? 
 
 Illustration. 
 
 3 : 4, ratio ; 
 
 read, 
 
 3 is to 4 ; 
 
 means, 3-^-4. 
 
 3, antecedent. 
 
 4, consequent. 
 3:4 = 0:8, proportion; 
 
 read, 
 3 is to 4 equals 6 is to 8. 
 3 : 4 : : 6 : 8, proportion ; 
 
 read, 
 3 is to 4t as 6 is to 8. 
 
 3 and 8, extremes. 
 
 4 and 6, means. 
 3 : 4 : : 6 : 8. 
 
 3x8 = 4x6. 
 
 3 : ( ) : : 6 : 8. 
 
 3x8 
 
 () = 
 
 
 
 4. 
 
 12:()::():3. 
 () = Vl2 x 3 
 
 = 6. 
 36 : 72 ? 48 : 144 ? 
 
 2. What is the difference between 24 -r- 3 and 27 : 9 ? 
 
 3. Which is the greatest, f , 2 -f- 3, or 1 : 2 ? 
 
 4. Find the missing term in the proportion 9 : 18 : : 4 : ( ) ; in 
 4 : 12 : : ( ) : 2; in 5 : ( ) : : 7 : 14 ; in ( ) : 6 : : 8 : 12. 
 
 5. What is the mean proportional of 9 and 4 ? 20 and 5 ? 
 
 140 
 
MENSURATION. 
 
 § 50. One Dimension. 
 
 That which has one dimension is a 
 line. 
 
 A line may extend in the same direc- 
 tion, a straight line ; or it may constantly 
 change its direction, a curved line. 
 
 If two straight lines in a plane are ex- 
 tended, they will meet, or they will not 
 meet. If they do not meet, they are par- 
 allel; if they meet, they form angles. 
 
 If two lines meet, the angles will be 
 equal, right angles, or not equal, oblique 
 angles ; the larger is obtuse, the smaller, 
 acute. 
 
 A straight line may be parallel to the 
 horizon, a horizontal line ; perpendicular 
 to the horizon, a vertical line ; or neither 
 parallel nor perpendicular to the hori- 
 zon, an oblique line. 
 
 1. Straight. 
 
 2. Curved. 
 
 Parallel Lines. 
 
 Angles. 
 1&2. Right. 
 3 & 4. Oblique. 
 
 3. Obtuse. 
 
 4. Acute. 
 
 / 
 
 1. Horizontal Line. 
 
 2. Vertical Line. 
 
 3. Oblique Line. 
 
 Define : 
 A line. 
 
 A straight line. 
 A curved line. 
 Parallel lines. 
 
 An angle. 
 A right angle. 
 An obtuse angle. 
 An acute angle. 
 
 9. Oblique angles. 
 
 10. A horizontal line. 
 
 11. A vertical line. 
 
 12. An oblique line. 
 
 141 
 
142 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 51. Two Dimensions. 
 
 That which has two dimensions is a 
 surface. 
 
 . Straight lines may inclose a plane 
 surface, a polygon. 
 
 The least number of straight lines 
 which can inclose a plane is three, a 
 triangle (1). The three lines may be 
 equal, an equilateral triangle (2) ; two 
 of them may be equal, isosceles tri- 
 angle (3) ; or no two of them equal, a 
 scalene triangle (4). 
 
 A triangle may have one right angle, 
 a right-angled triangle (6) ; one obtuse 
 angle, an obtuse-angled triangle (7) ; or 
 three acute angles, an acute-angled tri- 
 angle (8). 
 
 The next number of straight lines 
 which can inclose a plane is four, a 
 quadrilateral (9). The quadrilateral 
 may have both pairs of its opposite 
 sides parallel, a parallelogram (10) ; one 
 pair parallel, a trapezoid (11); or neither 
 pair parallel, a trapezium (12). 
 
 The parallelogram may have its angles 
 right angles, a rectangle (13) ; or not 
 right angles, a rhomboid (14). 
 
 The rectangle may have its sides all 
 equal, a square (15); the rhomboid may 
 have its sides all equal, a rhombus (16). 
 
 Illustration. 
 
 The face of this page is a plane 
 surface, or a plane. 
 
 All the figures on this page are 
 polygons. 
 
 % 
 
 ^ 
 K 
 
 Triangles. 
 
 2. Equilateral. 
 
 3. Isosceles. 
 
 4. Scalene. 
 
 kd 
 
 Triangles. 
 
 6. Right-angled. 
 
 7. Obtuse-angled. 
 
 8. Acute-angled. 
 
 / \i«r\ r fi3-]--[i5i 
 
 Quadrilaterals. 
 
 10. Parallelogram. 
 
 11. Trapezoid. 
 
 12. Trapezium. 
 
 13. Rectangle. 
 
 14. Rhomboid. 
 
 15. Square. 
 
 16. Rhombus. 
 
MENSURATION. 
 
 143 
 
 If the angles of a polygon are equal, 
 it is a regular polygon. 
 
 A regular polygon of five sides is a 
 regular pentagon (17); six sides, a regu- 
 lar hexagon (18) ; seven sides, a regular 
 heptagon, etc. ; infinite number of sides, 
 a circle. 
 
 Illustration. 
 
 Regular pentagon. 
 Regular hexagon. 
 Circle. 
 
 Define : 
 
 13. A surface. 
 
 14. A polygon. 
 
 15. A triangle. 
 
 16. An equilateral triangle. 
 
 17. An isosceles triangle. 
 
 18. A scalene triangle. 
 
 19. A right-angled triangle. 
 
 20. An obtuse-angled triangle. 
 
 21. An acute-angled triangle. 
 
 22. A regular polygon. 
 
 23. A regular pentagon. 
 
 24. A regular hexagon. 
 
 25. A regular heptagon. 
 
 26. A circle. . 
 
 27. Beginning with plane surface (see note), define parallelo- 
 gram, rectangle, rhomboid, rhombus, square. 
 
 28. Beginning with quadrilateral (see note), define parallelo- 
 gram, rectangle, square, rhombus. 
 
 29. Beginning with parallelogram (see note), define square, 
 rhombus. 
 
 30. Give as short a definition as possible of square, rhombus. 
 
 Note. — A definition may begin with different terms, e.g. : 
 
 A square is a plane surface bounded by two pairs of opposite sides, having 
 each pair parallel, having its angles all right angles, and having its sides all 
 equal. Or, 
 
 A square is a quadrilateral having its opposite sides parallel, having its 
 angles all right angles, and having its sides all equal. Or, 
 
 A square is & parallelogram having its angles all right angles and having 
 its sides all equal. Or, 
 
 A square is a rectangle having its sides all equal. 
 
 That definition is the best which is the shortest, provided it begins with a 
 term which is understood by the person for whom the definition is given. 
 
144 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 B 
 BC, Base. 
 AB + BC+AC, Perimeter. 
 
 AB and BC, Legs. 
 AC, Hypotenuse. 
 
 D 
 CD, Radius. 
 AB, Diameter. 
 
 § 52. Parts of a Polygon. 
 
 That side of a polygon on which 
 it is supposed to rest is its base; 
 the distance around a polygon, 
 its perimeter; the perimeter of 
 a circle, its circumference. 
 
 In a right-angled triangle, the 
 side opposite the right angle is the 
 hypotenuse; the other sides, legs. 
 
 In a circle, a line from the 
 center to the circumference is a 
 radius; a line passing through 
 the center and bounded at both 
 extremities by the circumference 
 is a diameter. 
 
 The altitude of a triangle, par- 
 allelogram, or trapezoid, is a per- 
 pendicular to the base, from the 
 vertex opposite the base. 
 
 AB is the altitude in each of 
 these figures. Observe that the 
 base must sometimes be extended. 
 
 31. On how many sides may a triangle be supposed to rest ? 
 
 32. How many bases may it have ? 
 
 33. How many altitudes may it have ? 
 
 34. Define the base of a polygon. 
 
 35. Define the perimeter of a polygon ; circumference of a circle. 
 
 36. Define the radius of a circle ; the diameter. 
 
 37. Compare the following with the definition of a circle de- 
 veloped on p. 143 : A circle is a plane surface bounded by a curved 
 line, every point of which is equally distant from a point within 
 called the center. 
 
 u 
 
 c 
 
 D C 
 
 D B E D 
 AB, Altitude. 
 
MENSURATION. 
 
 145 
 
 § 53. Rules. 
 
 The area of a parallelogram 
 is the product of its base and 
 altitude. 
 
 The area of a trapezoid is half 
 the product of its altitude and 
 the sum of its parallel sides. 
 
 The area of a trapezium is 
 half the product of its diagonal 
 and the sum of the perpendicu- 
 lars from the vertices to the 
 diagonal. 
 
 The area of a triangle is half 
 the product of its base and alti- 
 tude. 
 
 The area of a triangle is the 
 square root of the continued 
 product of the half sum of its 
 sides and the remainders found 
 by subtracting each side from 
 the half sum separately. 
 
 The square of the hypotenuse 
 of a right-angled triangle is 
 equal to the sum of the squares 
 of the other two sides. 
 
 The circumference of a circle 
 is twice the radius times 3.1416. 
 
 The area of a circle is the 
 square of the radius times 
 3.1416. 
 
 AM. MENT. AR. — 10 
 
 Illustration. 
 Area = 6 x 10 = 60 sq. in. 
 
 m 
 
 O I If) 
 
 Area = — i — x 6 = GO sq. in. 
 
 2 
 
 Area = ^-±_- x 10 = 60 sq. in. 
 
 2 H 
 
 Area = 
 
 6 x 10 OA 
 
 = 30 sq. in. 
 
 2 
 
 
 6 + 8 +10 _ ■ 10 
 2 
 12- 6 = 6. 
 12 - 8=4. 
 
 XX 
 
 
 12 - 10 = 2. 
 
 Area = Vl2 x 6 x 4 x 2 = 24 sq. in. 
 
 52 = 32 + 42 or, 25 = 9 + 16. 
 
 Circum. = 2 x 10 x 3.1416 
 = 62.832 in. 
 Area = 10 2 x 3.1416 = 314.16 sq. in. 
 
146 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 Applications. 
 
 Find the circumference of : 
 48. A circle, R 2 in. 
 
 § 54 
 Find the area of : 
 
 38. A rectangle, B 6 in. ; A 4 in. 
 
 39. A rhomboid, B12in.*; A 9 in. 
 
 40. A square, B 13 in. 
 
 41. A parallelogram, B 14 in. ; 
 
 A 7 in. 
 
 42. A trapezoid, II sides 8, 10 
 
 in. ; A 6 in. 
 
 43. A trapezium, D 12 in. ; J§ 
 
 6, 8 in. 
 
 44. A triangle, sides 3, 4, 5 in. 
 
 45. A triangle, B 6 in. ; A 8 in. 
 
 46. A rhombus, B 12 in. ; A 9 in. 
 
 47. A circle, R 5 in. 
 
 Find the altitude of : 
 
 53. A tri., S 16 sq. in. ; B 2 in. 
 
 54. A rect., S 40 sq. in. ; B 5 in. 
 
 55. A trapezoid, S 60 sq. in. ; 
 
 II sides 4, 8 in. 
 
 In the following figures find the part indicated. 
 
 59. 60. 61. 62 
 
 Find the hypotenuse of : 
 
 49. A rt. tri., sides 9, 12 in. 
 
 50. A rt. tri., sides 7, 24 in. 
 
 Find the other leg of : 
 
 51. A rt. tri., hyp. 25 in., A lb in. 
 
 52. Art. tri., hyp. 13 in., B 12 in. 
 
 Note. — B, base; A, altitude; II, 
 parallel ; D, diagonal ; ±, perpendicu- 
 lar; R, radius; rt. tri., right-angled 
 triangle ; rect., rectangle ; S, area. 
 
 Find the base of : 
 
 56. A tri., S 24 -sq. in. ; A3 in. 
 
 57. A parallelogram, S 32 sq. 
 in. ; A 4 in. 
 
 58. A square, S 64 sq. in. 
 
 R = ? 
 
 R = ? 
 
 S of circle = ? 
 
 Area ring 
 
MENSURATION. 
 
 147 
 
 § 55. Three Dimensions. 
 
 Illustrations. 
 
 That which has three dimensions is 
 a solid. 
 
 That part on which a solid rests is 
 its base; its other surfaces are faces; 
 the union of two faces, an edge; the 
 union of three or more edges, a vertex. 
 
 A solid may have two bases equal 
 and parallel polygons, and its faces 
 rectangles, a prism. If its bases are 
 triangles, triangular prism; squares, 
 square prism; . . . circles, circular prism 
 or cylinder. 
 
 A solid may have two bases parallel 
 polygons, and its faces trapezoids, frus- 
 tum of a pyramid. If its bases are 
 triangles, frustum of triangular pyra- 
 mid; . . . circles, frustum of circular 
 pyramid or frustum of cone. 
 
 A solid may have one base and its 
 faces triangles, a pyramid. If its 
 base is a triangle, triangular pyramid ; 
 square, square pyramid; . . . circle, 
 circular pyramid or cone. 
 
 A solid may have all of its surfaces 
 equal and regular polygons ; four tri- 
 angles, tetrahedron; eight triangles, 
 octahedron; twenty triangles, icosahe- 
 dron; six squares, cube ; twelve penta- 
 gons, dodecahedron ; an infinite number 
 of infinitely small polygons, sphere. 
 
 A- BCD, Solid. 
 BCD, Base. 
 ACD, Face. 
 AC, Edge. 
 A, Vertex. 
 
 A, Pentagonal prism. 
 
 B, Cylinder. 
 
 A, Frustum of hexagonal pyramid 
 
 B, Frustum of cone. 
 
 A, Pentagonal pyramid. 
 
 B, Cone. 
 
 A, Tetrahedron. 
 C, Icosahedron. 
 
148 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 56. Rules. 
 
 The convex surface of a 
 prism or cylinder is the prod- 
 uct of the perimeter of its 
 base, by its altitude. 
 
 The volume of a prism or 
 cylinder is the product of the 
 area of its base, by its altitude. 
 
 The convex surface of a 
 pyramid or cone is half the 
 product of the perimeter of 
 its base, by its slant height. 
 
 The volume of a pyramid 
 or cone is one third the prod- 
 uct of the area of its base, by 
 its altitude. 
 
 The convex surface of the 
 frustum of a pyramid or cone 
 is half the product of the 
 sum of the perimeters of its 
 two bases, by its slant height. 
 
 The volume of the frustum 
 of a pyramid or cone is one 
 third the product of the sum 
 of the areas of its upper base, 
 lower base, and mean propor- 
 tional base, by its altitude. 
 
 The surface of a sphere is 
 four times the square of its 
 radius times 3.1416. 
 
 The volume of a sphere is 
 four thirds times the cube of 
 its radius times 3.1416. 
 
 G 
 V 4 
 
 s=2(3+4) x6. 
 v=3x4x6. 
 
 Illustration. 
 
 ICE* 
 
 s=2x2 
 
 x 3. 1416x6. 
 t?=2 2 x 3. 1416x6. 
 
 s=i(16 + 16 + 16 s= 1(2x8x3.1416) 
 + 16)xl0. xlO. 
 
 v=U16xlQ)x6. tf = K 8 x 8 
 
 x 3.1416) x 6. 
 
 s=i(8 + 8+8+8 s=K2x4x3.1416 
 
 + 16 + 16+16 +2x8x3.1416)x5. 
 + 16)x5. 
 
 (8 2 +16 2 
 
 v= i(4 2 x 3.1416 
 
 + V8 2 xl6 2 ) x 3. +8 2 x3.1416 
 
 + \/4 2 x8 2 x3.1416 2 ) 
 x3. 
 
 s=4x5 2 x 3.1416. 
 v=fx5 3 x 3.1416. 
 

 MENSURATION. 149 
 
 Define : 
 
 
 
 67. Solid. 
 
 
 75. Regular octahedron. 
 
 68. Prism. 
 
 
 76. Regular dodecahedron. 
 
 69. Cylinder. 
 
 
 77. Regular icosahedron. 
 
 70. Pyramid. 
 
 
 78. Sphere. 
 
 71. Cone. 
 
 
 79. Triangular prism. 
 
 72. Frustum of 
 
 pyramid. 
 
 80. Pentagonal pyramid. 
 
 73. Frustum of 
 
 cone. 
 
 81. Frustum of hexagonal pyra- 
 
 74. Regular tetrahedron. 
 
 mid. 
 
 State the rule f oi 
 
 f convex s 
 
 surface of : 
 
 82. A prism. 
 
 83. A cylinder. 
 
 84. A pyramid. 
 
 85. A cone. 
 
 State the rule for volume of: 
 
 89. A prism. 
 
 90. A cylinder. 
 
 91. A pyramid. 
 
 92. A cone. 
 
 Find the convex surface of : 
 
 96. A triangular prism, each 
 side of base 3 in., altitude 12 in. 
 
 97. A pentagonal pyramid, 
 each side of base 2 in., slant 
 height 8 in. 
 
 Find the volume of : 
 
 100. A cylinder, radius of 
 base 5 in., altitude 10 in. 
 
 101. A hexagonal pyramid, 
 area of base 36 sq. in., alti- 
 tude 12 in. 
 
 86. A frustum of a pyramid. 
 
 87. A frustum of a cone. 
 
 88. A sphere. 
 
 93. A frustum of a pyramid. 
 
 94. A frustum of a cone. 
 
 95. A sphere. 
 
 98. Frustum of a square 
 pyramid, one side of upper base 
 5 in., one side of lower base 10 
 in., slant height 12 in. 
 
 99. A sphere, radius 6 in. 
 
 102. Frustum of a cone, radi- 
 us of upper base 5 in., of lower 
 base 10 in., altitude 12 in. 
 
 103. A sphere, radius 3 in. 
 
150 
 
 AMERICAN MENTAL ARITHMETIC. 
 
 § 57. Similarity. 
 
 Similar figures must fulfill two 
 conditions : 
 
 1st. For every angle of the 
 one there must be an equal 
 angle in the other. 
 
 2d. The sides about the 
 equal angles must be in pro- 
 portion. 
 
 In similar figures : 
 
 Linear parts are to each 
 other as homologous linear 
 parts. 
 
 Surfaces are to each other 
 as the squares of homologous 
 linear parts. 
 
 Volumes are to each other 
 as the cubes of homologous 
 linear parts. 
 
 104. The radii of two spheres 
 are 4 in. and 2 in. What is the 
 ratio of their circumferences ? 
 
 105. What is the ratio of their 
 surfaces ? 
 
 106. What is the ratio of their 
 volumes ? 
 
 Illustration. 
 
 Oo 
 
 Similar figures. 
 CF:cf::DC:dc. 
 12 : 6 : : 4 : 2. 
 
 Area DF : Area df; : CF 2 : c/ 2 . 
 Area DF: Area df: : 12 2 : 6 2 . 
 
 Vol. AF: Vol. af: : CF* : c/ 3 . 
 Vol. AF: Vol. af: : 12 3 : 6 3 . 
 
 C x : C 2 : : 4 : 2. 
 2 s 1, Arts. 
 
 S X :S 2 ::&: 2 2 . 
 4 : 1, Ans. 
 
 V x : V 2 : : 4 3 : 2 3 . 
 8:1, Ans. 
 
 107. The circumference of a lead pipe is 6 in. ; what is the 
 circumference of a pipe whose diameter is half the diameter of 
 the first ? 
 
MENSURATION. 151 
 
 108. The area of a circle is 10 sq. in.; what is the area of a 
 circle whose diameter is twice the diameter of the first ? 
 
 109. Two lead pipes are 1 in. and 2 in. in diameter. The area 
 of a horizontal section of the one is how many times a similar 
 section of the other ? 
 
 110. How many lead pipes 1 in. in diameter will discharge as 
 much water as one pipe 4 in. in diameter ? 
 
 111. A cannon ball weighs 32 lb.; what is the weight of a 
 similar ball whose diameter is half the diameter of the first ? 
 
 112. What is the ratio of the surfaces of the two balls in Ex. Ill ? 
 
 113. A is 6 ft. tall ; his bronze statue is 12 ft. tall ; if the length 
 of A's little finger is 2\ in., what is the length of the little finger 
 of the statue ? 
 
 114. If it costs $ 1 to paint a statue of A's size, what will it cost 
 to paint the statue in Ex. 113 ? 
 
 115. If a statue of A's size weighs 500 lb., what will the statue 
 in Ex. 113 weigh ? 
 
 116. If a bin 6 ft. deep holds 60 bu., what is the contents of 
 a similar bin 12 ft. deep ? 
 
 117. If it costs $10 to make an excavation 6 ft. deep, what is 
 the approximate cost of a similar excavation 24 ft. deep ? 
 
 118. If it costs $1200 to build a house 20 ft. by 30 ft., what 
 will be the approximate cost of a similar house 30 ft. by 45 ft. ? 
 
 119. If it costs $ 16 for material and labor to lay a floor 16 ft. 
 by 20 ft., what will it cost approximately to lay a similar floor 
 20 ft. by 25 ft. ? 
 
 120. If it costs $ 40 to paint a house 30 ft. by 40 ft., what will 
 it cost approximately to paint a similar house 45 ft. by 60 ft. ? 
 
 121. Four pipes each 2 in. in diameter empty into a tank ; what 
 must be the diameter of a single pipe to carry away all of the water? 
 
 122. A and B bought a ball of twine 8 in. in diameter for $ 1 ; 
 A wound from the outside until the diameter of the ball that was 
 left was 4 in. ; what should each pay ? 
 
MISCELLANEOUS. 
 
 § 58. Arithmetical Progression. 
 
 A series of numbers may increase or 
 decrease by a common difference, an 
 arithmetical progression. 
 
 The first term is written, a; the 
 last term, I', the number of terms, n\ 
 the common difference, d ; and the sum 
 of the terms, s. 
 
 Every problem may be solved by the 
 formulae : 
 
 Z=a + (tt-l)d (1) 
 
 -=(«+<> 
 
 (2) 
 
 Illustration. 
 3, 5, 7, 9, 11 
 14, 11, 8, 5, .3 
 arithmetical progressions. 
 Formula (1). The last 
 term equals the first term, 
 plus the number of terms 
 less one times the common 
 difference. 
 
 Formula (2). The sum 
 of the terms equals half the 
 number of terms times the 
 sum of the first and last 
 terms. 
 
 Ans. 19; 1= 7+4x3 = 19. 
 Ans. 65; g=§ (7 + 19) =05. 
 
 The arithmetical mean of two 
 numbers is half their sum. 
 
 1. Find I when a = 7, n = 5, d = 3. 
 
 2. Find s when n = 5, a = 7, / = 19. 
 
 3. State the series in Ex. 1. Prove the answer to Ex. 2. 
 
 4. Find the arithmetical mean between 7 and 19. 
 
 5. Find the sum of the numbers 1 to 25 inclusive. 
 
 6. Find the sum of the numbers 1 to 99 inclusive. 
 
 7. Translate each formula for arithmetical progression. 
 
 8. How far can a man walk in 10 days, going 10 miles the 
 first day and increasing the rate 5 miles per day ? 
 
 152 
 
MISCELLANEOUS. 
 
 153 
 
 § 59. Geometrical Progression. 
 
 A series of numbers may increase or 
 decrease by a common ratio, a geometri- 
 cal progression. 
 
 The first term is written, a; the 
 last term, I; the number of terms, n; 
 the ratio, r ; and the sum of the terms, s. 
 
 Every problem may be solved by the 
 
 formulae : 
 
 I =ar n - 1 ; (1) 
 s = rl-a (2) 
 r — 1 
 The geometrical mean of two 
 numbers is the square root of their 
 product. 
 
 9. Find I when a = 2, r = 5, n =3. 
 
 10. Find s when r = 5, I = 50, a = 2. 
 
 Illustration. 
 
 2, 6, 18, 54 
 
 64, 32, 16, 8 
 
 geometrical progressions. 
 
 Formula (1). The last 
 
 term equals the first term, 
 
 times the ratio raised to 
 
 the power denoted by the 
 
 number of terms less one. 
 
 Formula (2). The sum 
 
 of the terms equals the 
 
 quotient, whose dividend 
 
 is the ratio times the last 
 
 term, less the first term; 
 
 and whose divisor is the 
 
 ratio less one. 
 
 1= 2x5 2 =50. 
 5x50-2 
 
 Ans. 50 
 Arts. 62 
 
 -=62. 
 
 •5-1 
 
 11. State the series in Ex. 9. Prove the answer to Ex. 10. 
 
 12. The extremes are 2 and 250 ; the ratio is 5 ; find s. 
 
 13. A man bought 6 yards of cloth, giving 2 4 for the first yard, 
 6 $ for the second, 18 4 for the third, and so on ; what did he pay 
 for the last yard ? What did he pay for all ? 
 
 14. What is the geometrical mean between 4 and 25 ? 
 
 15. State the two formulse for geometrical progression ; trans- 
 late each. 
 
 16. A man sold a pair of horses, receiving % 1 for the first shoe, 
 % 2 for the second, % 4 for the third, and so on ; the horses being 
 fully shod, how much did he receive ? 
 
 17. A man bought a pair of oxen, paying 1 f for the first shoe, 
 2^ for the second, 4^ for the third, and so on; how much did he 
 pay for the last shoe, the oxen being fully shod ? 
 
154 AMERICAN MENTAL ARITHMETIC. 
 
 § 60. Specific Gravity. 
 
 The weight of a substance divided Illustration. 
 
 by the weight of an equal volume of a vol. lead weighs ... 22 lb. 
 
 water is its specific gravity. Same vol. water weighs 2 lb. 
 
 Approx. Table, S. G. s# g# lead _ 22 1b. _ n 
 Gold, 19 Glass, 3 Water, 1 2 lb - 
 Lead, 11 Stone, 3 Acid, 1.8 Gold is 19 times as heavy- 
 Silver, 10 Oak, .7 Oil, .9 as water. 
 Iron, 7 Cork, .2 Air, .001 Oak wood is .7 as heavy as 
 
 A pint is a pound the world 
 round. 
 
 A cubic foot of water weighs a pint of water weighs a 
 
 62.5 lb. pound (approx.). 
 
 water. 
 
 Air is .001 as heavy as 
 water. 
 
 18. What is the weight of a pint of gold? Of lead? Of air? 
 
 19. What is the weight of a cubic foot of cork ? 
 
 20. What is the weight of a gallon of water ? Of oil ? Of acid ? 
 
 21. What is the weight of a bushel of cork ? 
 
 22. What is the weight of a cu. ft. of oak wood ? 
 
 23. Two volumes of lead and 3 of the same volumes of water 
 weigh 25 oz. ; what is the specific gravity of lead ? Explain. 
 
 24. Of a mixture, \ in volume is oil and -§ water ; what is the 
 S. G. of the mixture ? 
 
 25. What is the S. Gr. of lead and silver compounded of equal 
 volumes ? 
 
 26. How many cu. ft. of cork will weigh as much as a cu. ft. 
 of lead? 
 
 27. What is the difference in lb. between the weight of a 
 cu. ft. of lead and a cu. ft. of silver ? 
 
 28. A woman who had learned " A pint is a pound," gave a 
 pint of shot for a lb. ; how much did she lose, if shot is 15^ a lb. ? 
 
 29. How many gallons of air will weigh one pound ? 
 
MISCELLANEOUS. 
 
 155 
 
 § 61. Zero and Infinity. 
 
 Numbers may be regarded as existing 
 in three realms. 
 
 1. Where their values can be ex- 
 pressed by the decimal notation, the 
 finite. 
 
 2. Where their values are too great to 
 be expressed by the decimal notation, 
 the infinite. 
 
 3. Where their values are too small 
 to be expressed by the decimal notation, 
 the infinitesimal. 
 
 Every number in the greatest realm 
 is expressed by the character oc. 
 
 This does not stand for a single 
 number, but for any one of the 
 countless numbers in this realm. 
 
 Every number in the smallest realm 
 is expressed by the character 0. 
 
 This does not stand for a single 
 fraction, but for any one of the 
 countless fractions in this realm. 
 
 Illustration. 
 
 The number of ft. in a 
 .mi. can be expressed by the 
 decimal notation. 
 
 The number of cu. in. in 
 space is too great to be ex- 
 pressed by the decimal no- 
 tation. 
 
 The difference between 
 2 and 1.999-.., where 9 is 
 repeated without limit, is 
 too small to be expressed 
 by the decimal notation. 
 
 One cc (infinity) may be 
 2, 3, or any other number 
 greater or less than another 
 oc ; or twice three times, or 
 any number of times as 
 great. 
 
 One (infinitesimal) may 
 be twice, three times, or any 
 number of times as great as 
 another 0. 
 
 30. What is the value of - ? 
 
 
 
 31. What is the value of « ? 
 
 cc 
 
 32. What is the value of - ? 
 
 6 
 
 33. What is the value of |? 
 
 34. What is the value of 5. ? 
 
 QC 
 
 Ans. Any finite no. as 2, 1000. 
 Ans. Any no. as 2, 100, oc. 
 Ans. 0. 
 
 Ans. cc. The smaller the divisor 
 the greater the quotient. 
 
 Ans. 0. The larger the divisor 
 the smaller the quotient. 
 
GENERAL REVIEW EXERCISES. 
 
 1. The sum of eight numbers is 95; the sum of seven of them 
 is 87 ; what is the eighth number ? 
 
 2. The addends are 6, 8, 3, 9, 7, 4, 5, 6, 8 ; what is the sum ? 
 
 3. The subtrahend is 986; the minuend 1000; what is the 
 remainder ? 
 
 4. The minuend is 36 ; the rem. 12 ; what is the subtrahend ? 
 
 5. The multiplier is 12; the multiplicand 13; what is the 
 product ? 
 
 6. The multiplicand is 11 ; the product 132 ; what is the 
 multiplier ? 
 
 7. The dividend is 144; the divisor 18; what is the quotient? 
 
 8. The dividend is 119 ; the quotient 9 ; the remainder 11 ; 
 what is the divisor ? 
 
 9. The dividend is 125 ; the divisor 16 ; what is the remainder ? 
 
 10. The divisor is 9 ; the quotient 13 ; the remainder 1 ; what 
 is the dividend ? 
 
 11. What number multiplied by 13, with 7 added to the 
 product, will give 85 ? 
 
 12. By what number must 11 be multiplied so that when 4 is 
 taken from the result the remainder will be 128 ? 
 
 13. What is the result when 13 is taken 7 times as an addend ? 
 
 14. How many times must 12 be taken as an addend to pro- 
 duce 108 ? 
 
 15. Define a prime number ; numbers 'prime to each other ; 
 numbers severally prime. 
 
 16. Name three composite numbers prime to each other but 
 not severally prime. 
 
GENERAL REVIEW EXERCISES. 157 
 
 17. Give the rule for the divisibility of a number by 2 ; by 3 ; 
 by 4 ; by 5 ; by 8 j by 9 ; by 11 ; in general. 
 
 18. Name 20 factors of 180180. 
 
 19. 77 and 91 are factors of 360360; is their product a 
 factor ? Why ? 
 
 20. Why is 231 exactly contained in 360360 ? 
 
 21. Multiply 5x6x8 by 7, and express the result by its 
 factors. 
 
 22. Divide 27 x 18 x 9 by 3, and express the result by its 
 factors. 
 
 23. How many times is 6x8x4x3 contained in 48 x 36 ? 
 
 24. How many times is 17 x 6 contained in 51 x 2 x 3 ? 
 
 25. By an illustration, show that the remainder found by 
 dividing a number by 9, is the same as the remainder found by 
 dividing the sum of its digits by 9. 
 
 26. By an illustration, show that the remainder found by sub- 
 tracting the sum of its digits from a number is divisible by 3. 
 
 27. Show that a number is equal to its digit in unit's place, 
 plus ten times its digit in ten's place, plus one hundred times its 
 digit in hundred's place, and so on. 
 
 28. State three principles for finding the G. C. D. 
 
 29. By the second principle, how can you tell that 4 must be 
 the G. C. D. of 64 and 68 ? 
 
 30. By the third principle, how can you tell that 1 is the 
 G. C. D. of 625 and 1728 ? 
 
 31. State three principles for finding the least common multiple. 
 
 32. Find the L. C. M. of 20 and 30 ; 24 and 36 ; 30 and 35. 
 
 33. Find the L. C. M. of 3, 8, 12, 24, 48, 72. Did you use the 
 third principle ? 
 
 34. Analyze and explain the meaning of -f by the first concep- 
 tion ; by the second. 
 
 35. Which of these two methods was first used to indicate that 
 5 is to be divided by 8, 5 -r- 8, or | ? 
 
158 AMERICAN MENTAL ARITHMETIC. 
 
 36. 14 -*- 3 = 4|. Which is the more natural conception, that 
 14 divided by 3 equals 4 units and f of a unit, or that 14 divided 
 by 3 equals 4, with 2 which is yet to be divided by 3 ? 
 
 37. Change 6} to an improper fraction ; why is this an exam- 
 ple in addition of fractions ? 
 
 38. Divide 17£ by 2\. See p. 63. 
 
 39. Divide 11\ by 2\ by inverting the divisor and proceeding 
 as in multiplication. Is this process as easy for mental work as 
 dividing the numerators ? 
 
 40. What is the difference between f of 24 and the number of 
 which 9 is f ? 
 
 41. What is -| and J- of ^ of 6^ ? Did you find the sum of ^ 
 and i of ^ before you multiplied by 6J ? 
 
 42. Reduce ^ 2 - to a mixed decimal, and express the result in 
 two ways. 
 
 43. State the numeration table for decimals. 
 
 44. Express 6 -=- 100 in three ways ; what are they ? 
 
 l 
 
 45. Is there any difference among -£-, .00^, and \°l ? 
 
 46. Eeduce 100 to %. 
 
 47. What month of the year is January ? Oct. ? Dec. ? Aug. ? 
 
 48. How many days in the first six months of a common year ? 
 
 49. State the number of cu. in. in a gal. ; cu. in. in a bu. ; rela- 
 tion between cu. ft. and bu. ; relation between cu. ft. and gal. 
 
 50. How many cu. ft. of hay make a ton ? State the relation 
 between cu. ft. corn in the ear and bu. shelled corn. 
 
 51. How many drops make 1 teaspoonful ? gr. make 1 lb. troy ? 
 gr. make 1 lb. apothecaries' ? gr. make 1 lb. avoirdupois ? 
 
 52. How many lb. make 1 bu. oats ? 1 bu. corn ? 1 bu. pota- 
 toes ? 1 bu. wheat ? 
 
 53. In the metric system, state tabulated facts about the unit 
 of long measure ; unit of land measure ; unit of weight ; unit of 
 capacity ; unit of wood measure. 
 
GENERAL REVIEW EXERCISES. 159 
 
 54. How many 1. in 1 Ml. ? How many 1. make 1 qt. ? 
 
 55. How many pints in 3 HI. ? How many 1. make 1 cu. m. ? 
 
 56. How many Ha. in 2 acres ? 
 
 57. How many cords of wood in 40 steres ? 
 
 58. Which is the cheaper, to buy meat at 10^ a lb. or at 20^ a 
 Kg. ? By how much a lb. ? 
 
 59. What is the sum in lb. of a common English ton, a long 
 English ton, and a metric ton ? 
 
 60. How much is made per quart by buying chestnuts at 
 $ 1.60 a bu. and selling at 5^ a half-pint ? 
 
 61. How much is gained per lb. by buying salt at $ 20 a ton 
 and selling at \$ an oz. ? 
 
 62. What is gained on 6 dozen eggs by buying 3 for 2^ and 
 selling 2 for 3^ ? 
 
 63. By buying apples at 2 for a cent, and the same number at 
 3 for a cent, and selling all at 5 for 2^, I lost 2^ ; how many 
 apples did I buy ? 
 
 64. A man 45 years has a daughter 5 years old. In how 
 many years will she be \ as old as he ? \ as old ? \ as old ? Of 
 the same age ? 
 
 65. One hunter shot 24 pigeons, another shot 0. The first 
 shot how many times as many as the second ? 
 
 66. If 8 men will eat a quantity of flour in 15 days, how long 
 will it last if 4 men join them ? 
 
 67. A loaned $10 and B $ 15 for the same time and rate; 
 together they received $2 interest; what was the share of 
 each? 
 
 68. Three men hired a pasture for $9; A put in 4 horses, 
 B 6, and C 8 ; what ought each to pay ? 
 
 69. ^ of a cargo was lost ; A, who owned \ of the whole, lost 
 % 100 ; what was the value of the part that remained ? 
 
 70. If a man can do f of a piece of work in 15 days, how long 
 will it take him to do i of it ? 
 
160 AMERICAN MENTAL ARITHMETIC. 
 
 71. If a hen and a half lay an egg and a half in a day and 2 
 half, how many eggs will 4 hens lay in 3 days ? 
 
 72. If 3 cats catch 3 rats in 3 minutes, how many cats will b( 
 required to catch 100 rats in 100 minutes ? 
 
 73. A thief bought a pair of boots for $ 5, and gave in pay 
 ment a $ 50 counterfeit bill. The merchant having' no money a1 
 all, changed the bill at a bank and gave the thief $ 45 in gooc 
 money. After the merchant had paid $ 50 in good money for th( 
 bill, what was his entire loss ? 
 
 74. The freezing and boiling points in the Centigrade ther 
 mometer are 0° and 100°; in Fahrenheit's, 32° and 212°. Ho^ 
 many degrees C. equal 45° F. ? 
 
 75. How many degrees F. equal 45° C. ? 
 
 76. When F. reads 59°, what is the reading of C. ? 
 
 77. When F. reads 23°, what is the reading of C. ? 
 
 78. When C. reads 25°, what is the reading, of F. ? 
 
 79. When C. reads 10° below zero, what is the reading of F. \ 
 
 80. By selling a horse for $ 36 a man gained \ of the cost 
 what was the cost ? State this as an example in percentage. 
 
 81. By selling a horse for $ 60 a man lost \ of the cost ; whal 
 was the cost ? State this as an example in percentage. 
 
 82. On an article which cost $24 a merchant gained 33^% : 
 what would have been the selling price if he had gained half as 
 much? 
 
 83. What is the value of 9. ? 
 
 oc 
 
 84. What is the value of ^? 
 
 85. What is the value of x <x ? 
 
 86. May 5 = 1? May 5 = 2? May 5 = 100000 ? Why? 
 
 87. When the, diameter of a circle is 0, what is its circumfer- 
 ence ? What is the ratio of its circumference to its diameter ? 
 What is the value of this ratio ? 
 
%pJU^> a cJi^f***-*! 
 
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