5 4 45. Add together, 92, 124, 16, and 213. 46. What is the sum of+3+ 9 + 1 + 2 1 ? 47. What is the sum of 19 of 5 +2 15+ 48. What is the sum of 1 of 1937+63 +11 49. Find the sum of of a shilling and of a penny In this example, first reduce the of a shilling to pence, and the fraction of a penny. 50. Find the sum of of a gallon and of a gill. 51. What is the sum of 5 days and 525 minutes ? 52. What is the sum of of a cwt., 81⁄2 lb. and 3 oz.? SUBTRACTION OF FRACTIONS. As in addition of fractions we find the sum of their num ors, so in subtraction of fractions we find the diffe ence of their numerators. RULE. If either quantity be a compound fraction, reduce it to a simple fraction, and if the two fractions have different denominators, reduce them to a common denominator. Subtract the numerator of the subtrahend from the numerator of the minuend, and place the remainder over the common denominator. When the minuend is a mixed number, and the fraction in the subtrahend is greater than that in the minuend, subtract the numerator of the subtrahend from the denominator, and to the difference add the numerator of the minuend; and consider the integer of the minuend to be 1 less than it stands. It is not always obvious, which of two fractions expresses the greater quantity. In such case, the fractions are denoted with a character between them, thus, 1; and the greater is discovered by reducing them to a common denominator. 53. What is the difference between 24% and 26? 72 27 Here the fraction in the subtrahend is the greater, and we are obliged to convert a unit into seventy-seconds to obtain a quartity from which to subtract $9. 54. What is the difference between 4 and 19? 26 24 142 56 56 55. Perform subtraction on 1. 56. What will remain if 512 be taken from 843? 57. Subtract of 3 from 36. 58. What is the difference between 4 and 10? 59. What will remain if of be taken from a unit? 60. What is the difference between and ? 61. 4—of of is equal to what quantity? 39 MULTIPLICATION OF FRACTIONS. The following rules for multiplication of fractions, are based on the Propositions 1, and 11, stated in page 37. CASE I. To multiply a fraction by a whole number. RULE. Either multiply the numerator, or div the denominator by the whole number. CASE II. To multiply a whole number by a fraction. RULE. Multiply the whole number by the numerator, and divide the product by the denominator. CASE III. To multiply a fraction by a fraction. RULE. Multiply numerator by numerator, and denominator by denominator, for a new fraction. When both factors are mixed numbers, it is generally more convenient to reduce them to improper fractions and then proceed according to the rule under Case III. The effect of multiplying any quantity by a proper fraction is, to give in the product, such a part of the quantity multiplied as the fraction indicates. Thus the product must be less than the multiplicand. This effect of the operation will appear consistent with the principle of multiplication, when it is considered, that multiplying any number by 1, gives only the same number in the prodict; and, therefore, multiplying by less than 1, must give a product less than the number multiplied. 62. Multiply 23 by 9. 25X9-25X9 = 225-818 28 : 63 Multiply 49 by . (See rule under Case 11.) 64. Multiply by. (See rule under Case 111.) 65. Multiply 6 by 3. (Remark under Rule 111.) 66. What is the product of by 15? 67. What is the product of 9241 by? 68. What is the product of 3 10 by 23? 69. What is the product of 8 by 12? 70. Which is the most, X 65, or, 65 X 7? 71. What is the product of 29413 by 25? In this example, it will be most convenient to find the product of the whole numbers without regard to the fraction first; then find the product of the fraction in a separate operation, and, finally, add the two products together. 72. What is the product of 361 by 3415? 73. How many square inches of paper in a sheet that is 14 inches long, and 11 inches wide? DIVISION OF FRACTIONS. The rules for division of fractions, like those for multiplication, are based on Propositions I, and II. CASE I. To divide a fraction by a whole number. RULE. Either divide the numerator, or multiply the denominator, by the whole number. CASE II. To divide a whole number by a fraction.RULE. Multiply the whole number by the denominator, and divide the product by the numerator. CASE III. To divide a fraction by a fraction. RULE. Invert the divisor, and then proceed as in multiplying a fraction by a fraction. Observe, that the operation of this last rule is, to multiply the denominator of the dividend by the numerator of the divisor for a new denominator, and the numerator of the dividend by the denominator of the divisor for a new numerator. Compound fractions are to be reduced to simple ones, and mixed numbers to improper fractions, before the adoption of either of the above rules. 74. Divide by 8. 9 75. Divide 14 by. (See rule under Case 11.) 76. Divide by . (See rule under Case 111.) 77. Divide the compound fraction of by 6. 124 78. Divide 325 by the mixed number 5. 2 83. What is the quotient of 91 divided by 15? 84. What is the quotient of 2063 divided by 947 85. How many times is 24 contained in 319? 86. How many times is 193 contained in 99? 87. How many times of an inch in of a yard? First, reduce the of a yd. to the fraction of an inch 88. How many times of a gill in 3 barrels ? 89. Suppose a wheel to be 11 feet in circumference how many times will it roll round in going 39 rods ? MISCELLANEOUS EXAMPLES. In the following examples, all fractions which appear in the answers, must be reduced to their value in whole numbers of lower denominations, whenever there is opportunity for such reduction. 90. What distance will a car run in 9 hours, allowing its velocity to be 231⁄2 miles an hour? 91. Suppose a car wheel to be 8 feet 7 inches in circumference, how many times will it turn round in running 46 miles? 92. If 37cwt. of sugar be taken from a hogshead containing 14 cwt. 1 qr. 6 lb., how much will remain in the hogshead? 93. What is the sum of 16 cwt., 7cwt. 3qr. 81lb., 2T. 19cwt., 2cwt. 11qr., and 7 of a ton? 94. A farmer owning 132 acres of land, sold 46 A. 3R. 12r. How much land had he remaining? 95. What is the value of 36 acres of land, at $473 per acre? 96. What is the value of 15 barrels of flour, at $4.62 per barrel ? 97. What is the value of a load of wond, containing 6 feet, [ of a cord,] at $5.25 per cord? Or, what is of $5.25? Or, $5.25 X = ? 98. How much land is there in a square lot, measuring 354 rods on every side? (See page 28.) 99. What quantity of land in a lot, which is 65 rods long and 47 rods wide? 100. What quantity of wood is there in a pile, 142 feet long, 3 feet wide, and 62 feet high? 101. Suppose a lot of land to be 6 rods wide, how long must it be, to contain 1 acre? (See PROB. v, page 21. Consider that 1 acre contains 160 rods.) 102. What quantity of loaf sugar must be sold at 19 cents per pound, that the price shall amount to $524? 103. What cubical quantity of earth must be removed, in digging a pit, 13 feet deep, 12 feet long, and 93 feet wide? 104. What quantity of hewn timber is there in a stick that is 12 feet long, 24 feet deep, and 12 foot wide ? 105. Suppose a stick of timber to be 112 foot deep, and 8 inches wide; what must be the length of the stick, in order that its quantity shall be 1 ton of hewn timber? (See PROB. VIII, page 22. Consider a ton as the pro duct of three factors.) 106. Suppose wood to be piled on a base 18 feet long and 72 feet wide, what must be the height of the pile, to contain 9cords? 107. What quantity of molasses in 4 casks, containing severally, 55gal., 313 gal., 275 gal., and 581 gal.? 108. What is the cost of 486 bushels of corn, at 62 cents per bushel ? 109. Suppose 65 gallons to have leaked from a hogshead of wine, what is the value of the remainder of the wine, at 87 cents per gallon? 110. How many bottles, each holding 1 pint, are required for bottling 3 barrels of cider? 111. Suppose 4 gallons of cider to have evaporated from a barrel; what number of bottles, each holding 1 pt. 3gi., will be required to bottle the remainder? 112. What is the value of 142 tons of coal, at 73 dollars per ton? 113. What is the value of of a bushel of wheat, at the rate of of a dollar per bushel? [X=?] 1⁄2 |