45. Add together, 9, 12 11, 1o, i and 21. 46. What is the sum of lots+ + ta? 47. What is the sum of 191 tj of +21+1? 48. What is the sum of į of : + 37'+6tú 49. Find the sum of of a shilling and 5 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 fi of a gallon and I of a gill. 51. What is the sum of 5 days and 525 minutes ? 52. What is the sum of 4 of a cwt., 83 lb. and 3 oz.? As in addition of fractions we find the sum of their numerators, so in subtraction of fractions we find the difference 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, an it; and the greater is discovered by reducing them to a common denominator. 53. What is the difference between 24 and 263 ? 72 | Here the fraction in the sub trahend is the greater, and we 247 56 are obliged to convert a unit into seventy-seconds to obtain a quan 1432 43 tity from which to subtract 54. What is the difference between from 1? 263 55. Perform subtraction on an 11. 56. What will remain if 511 be taken from 34ş? 57. Subtract of from 36% 58. What is the difference between 4 ) and 107? 59. What will remain if şof be taken from a unit? 60. What is the difference between 4 and ? 61. 43-of of is equal to what quantity? MULTIPLICATION OF FRACTIONS. The following rules for multiplication of fractions, are based on the Propositions I, and 11, stated in page 37. CASE I. To multiply a fraction by a whole number. · RULE. Either multiply the numerator, or divide 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 111. 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 product; and, therefore, multiplying by less than 1, must give a product less than the number multiplied. 62. Multiply 2h by 9. 3X9= 2539=2=80 63. Multiply 49 by 5. (See rule under Case 11.). 64. Multiply 1 by Ž. (See rule under Case 111.) 65. Multiply 6 % by 34. (Remark under Rule iii.) 66. What is the product of 4 by 15? 67. What is the product of 9241 by ?? 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 34 15 ? 73. How many square inches of paper in a sheet that is 14 inches long, and 11inches wide ? DIVISION OF FRACTIONS. The rules for division of fractions, like those for multiplication, are based on Propositions 1, and 11. 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 x by 8. +8=Taxe=”. Ans. 75. Divide 14 by 7s. (See rule under Case 11.) 76. Divided by <. (See rule under Case ini.) 77. Divide the compound fraction of io by 6. 78. Divide 325 by the mixed number 52. 79. What is the quotient of divided by 13? 80. What is the quotient of 57 divided by ✓? 81. What is the quotient of 1s divided by is? 82. Divides of lo by of of ź. 83. What is the quotient of 91 divided by 15 ? 84. What is the quotient of 206 divided by 94? 85. How many times is 21 contained in 319 ? 86. How many times is 194 contained in 991 ? 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 4 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 23 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 3cwt. of sugar be taken from a hogshead containing 14 cwt. 1 qt. 6 lb., how much will remain in the hogshead ? 93. What is the sum of 16 zcwt., 7cwt. 3qr. 811b., 2T. 19cwt., 2 cwt. 1 gr., and of a ton ? 94. A farmer owning 1325 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 $47 per acre ? 96. What is the value of 154 barrels of flour, at $4.62} per barrel ? 97. What is the value of a load of wood, containing 6 feet, [e 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 47rods wide ? 100. What quantity of wood is there in a pile, 14 % feet long, 31 feet wide, and 6 feet high? 101. Suppose a lot of land to be 64 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 3 cents per pound, that the price shall amount to $524? 103. What cubical quantity of earth must be removed, in digging a pit, 132 feet deep, 121 feet long, and 93 feet wide ? 104. What quantity of hewn timber is there in a stick that is 12 feet long, 2 feet deep, and 1foot wide ? 105. Suppose a stick of timber to be 11 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 product of three factors.) 106. Suppose wood to be piled on a base 18 feet long and 7 feet wide, what must be the height of the pile, to contain 91 cords ? : 107. What quantity of molasses in 4 casks, containing severally, 557 gal., 31; gal., 27 gal., and 58 11 gal.? 108. What is the cost of 436 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 872 cents per gallon? 110. How many bottles, each holding 11 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. 31 gl., will be required to bottle the remainder ? 112. What is the value of 142 tons of coal, at 7 dollars per ton ? 113. What is the value of of a bushel of wheat, at the rate of of a dollar per bushel ? [áx =?] |