From the preceding demonstration it follows that, § 120. In fractional, as in integral, multiplication, the product remains the same, when the multiplicand and multiplier are taken, the one for the other. Thus of = of §. 11. What should be paid for of a yard of linen, at the rate of of a dollar per yard? It is plain that of a yard will cost of of a dollar. of , the same as X, is a compound fraction, to be reduced to a simple fraction. (§ 118). Ans. of a dollar. 12. What should be paid for of a barrel of apples, if the whole barrel be worth 1 of a dollar? Ans. of a dollar. 13. A person owning of a tract of land, sells of his share to A; what part of the whole tract does A purchase? Ans. 6 of the whole. 14. What should be paid for off of a pound of tea, at the rate of of a dollar per pound? Ans. of a dollar. 15. A gentleman owning of a ship, sells of his share to A, and the rest of it to B. What part of the whole ship does he sell to each? Ans. To A, and to B of the whole. 16. Bought of an acre of ground at the rate of 18 dollars per acre; required the sum to be paid for it. Ans. 114 dollars. 17. Sold 25 bushels of clover seed at 74 dollars per bushel, and 3 bushels at 7 dollars per bushel; what did the whole amount to? Ans. 205 dollars. 18. Find the entire cost of of a pound of pepper at dollar a pound, of a hundred weight of flour at 21⁄2 dollars a hundred, and 21 yards of cloth at 7 dollars a yard. Ans. 1615 dollars. of a 19. Find the value of 3 pieces of cloth, each containing 251 yards, at 63 dollars per yard. Ans. 511 dollars. 20. Find the distance traveled by a person in 61⁄2 days, allowing him to proceed for 3 days at the rate of 30 miles per day, and the rest of the time at the rate of 27 miles per day. Ans. 185 miles. 21. What would be the profit on 75 barrels of flour, purchased at 33 dollars a barrel, and sold at 41⁄2 dollars a barrel? Ans. 56 dollars. 22. What would be the profit or loss on 35 yards of silk, purchased at of a dollar per yard, if 16 yards of it be sold at 14 dollars a yard, and the remainder at § of a dollar a yard? Ans. Profit 5 dollars. of a load of pork, weighing 2460 and to B of his purchase; how each obtain? 23. A merchant bought pounds. He sold to A many pounds did A and B Ans. A 5274, and B 615 pounds. 24. Having purchased 350 bushels of wheat at dollar a bushel, and sold of the quantity at 1 dollar a bushel, what would I gain or lose on the whole by selling the remainder at dollar a bushel? Ans. Gain 14 dollars. 25. In how many days ought one man to accomplish a work equivalent to what 12 men performed in 74 days? Ans. 87 days. 26. In how many days ought one man to accomplish an undertaking which 17 men could perform in 133 days? Ans. 2273 days. 27. A farmer bought of a tract of land which contained 735 acres, and sold to his neighbor of his purchase. What part of the whole tract, and how many acres, did he sell? Ans. of the tract; 294 acres. 28. Bought of A 3 cords of wood, at 2 dollars per cord; of B 7 cords, at 1 dollars per cord; and of C 10 cords, at 2 dollars per cord. What was the whole quantity of wood purchased, and the whole sum paid for it? Ans. 207 cords; 4115 dollars. 29. If 3 masons can build a wall of a certain length, height, and thickness, in 13 days, in what time ought one mason to build a wall of the same height and thickness, but 24 times as long? Ans. 91 days. 30. A speculator bought of A 189 acres of land at 10 dollars an acre, and of B 250 acres at 13 dollars an acre. He sold of the first tract at 18 dollars, and of the second at 19 dollars an acre; what would he make, on the whole, by selling the remainder of both tracts at 20 dollars an acre? Ans. 335218 dollars. DIVISION OF FRACTIONS. $ 121. Dividing by a fraction, as well as by an integer, consists in finding how many times the dividend contains the divisor, or what part the dividend is of the divisor. Thus divided by gives the quotient 3, because contains 3 times; and divided by gives the quotient, because isof. How many times dues contain, or what is the quotient of divided by? Of divided by? Of af divided by f? What part is of 2, or what is the quotient of ‡ divided by 2? Of divided by ? Of divided by &? Of $÷? Of ÷10? § 122. In fractional, as in integral, division, the dividend is a product given, and the divisor one of its factors given, to find the other factor. is equal to 4, because is the product of X, expressing the part that of; the quotient is of the divisor . $123. The reciprocal of a fraction is the fraction inverted; and is equal to a unit divided by the fraction. (§ 50). Thus the reciprocal of is; equal to 1 or 7 divided by 4. What is the reciprocal of ? Of ? of? Of it? The reciprocal of a mixed number is that of its equivalent improper fraction. Thus the reciprocal of 5= What is the reciprocal of 24? Of 73? Of 10? Complex or Mixed Fractions. is. Of 98%? § 124. When the dividend or divisor is a fraction or a mixed number, the dividend placed over the divisor, with a line between, forms a complex or mixed fraction. Thus, ÷÷5, 2÷÷3, may be expressed, respectively, by the 2 complex fractions and These complex fractions may be read-numerator, denominator 5; numerator 2, denominator 31. A complex, as well as a simple, fraction is equal to its numerator divided by its denominator. (§ 92). RULE XXIII. § 125. To divide a fraction by a fraction. 1. Divide the numerator of the dividend by that of the divisor, and the denominator by the denominator; or 2. Multiply the dividend by the reciprocal of the divisor. 3. A fraction is divided by an integer, by dividing the numerator by the integer, or multiplying the denominator by the integer. 4. An integer is divided by a fraction, by multiplying the integer by the reciprocal of the fraction. 5. A mixed number may be used in division under the form of an improper fraction. A mixed number may also be divided by an integer, by dividing the integer and fraction in the mixed number, separately. 1. To divide 8 by . EXAMPLES. Dividing numerator by numerator, and denominator by denominator, 18-2 we have 18÷=20÷5—2—24. Here we cannot divide numerator by numerator, and denominator by denominator, without remainders. Multiplying, therefore, the dividend by the reciprocal of the divisor, we have÷==&=#• 3. To divide 18 by 5. Dividing the numerator of the fraction by the integer, 10÷÷5 we have 5= ̈ ̄ -= Or, multiplying the denominator by the integer, 10 we have 1÷5=13X5=1's, as before. 4. To divide 123 by . Multiplying the integer by the reciprocal of the fraction, we have 123÷=123X="15153. 5. To divide 163 by 5. Reducing the mixed number to an improper fraction, we have 163-5-435-42=3211. Otherwise. Dividing the integer and fraction in the mixed number, separately, 5) 163 3211 We say 5 in 16, 3 times and 1 over; 5 in 13, twice and 3 over: this 3 and the make 33, equal to ; then ÷5, that is, of, is 1. Dividing a fraction by a fraction is the reverse of multiplying a fraction by a fraction. (§122). When the terms of the dividend cannot be divided by those of the divisor without a remainder, suppose each term of the former by both terms of the latter; this will not alter the value of the dividend. ($93). Then the resulting numerator the numerator of the divisor, and the denominator by the denominator, gives, for the quotient, the dividend X the reciprocal of the divisor. Thus, in the second example, 9 X2X3 9 ×2×3 =35X2X3; and 9 X3 35X2X3=35X2=3X=the quotient. From the Rule, thus demonstrated, it follows that, § 126. In fractional, as in integral, division, the quotient is always such a part of the dividend as is expressed by the reciprocal of the divisor. For the quotient is found by multiplying the dividend by said reciprocal; and multiplying by a fraction finds such a part of the multiplicand as is expressed by the multiplier. |