1. What is the quotient of 7 ; } ? SOLUTION. Since the quotient of 7 = 1 is 7, the quotient of 7 + f must be 3 times 7, or 21. 2. What is the quotient of $ • f? Solution. — Since the quotient of f+ 1 is f, the quotient of f + must be 6 times , or 25. (b.) From these examples, it is evident that the quotient of a number divided by } 5 times the number; divided by t=7 times the number; and, generally, that (c.) The quotient of a number divided by a fraction having 1 for its numerator equals the product of that number multiplied by the denominator of the fraction. (a.) What is the quotient of 3. 3 = ? 6. $ • ¢? 9. 8 ; .1? 5. 9 = $? 8. f? 11. 46 = .001 ? 12. What is the quotient of 7 $? Solution. Since the quotient of 7 divided by } is 5 times 7, the quotient of 7 divided by 4 must equal 4 of 5 times 7, or of 7 = 8%. 13. What is the quotient of f = f? Solution. Since the quotient of f divided by is 3 times ,, the quotient of & divided by z must equal of 3 times f- of f = 1f. (e.) What is the quotient of 14. j ; $? 17. If • } 20. 1= 24? 21. 41 : 512 16. = ? 19. t = 3? 22. 34 : 17? (f.) From the foregoing, it appears that the quotient of a number divided by f=of 3 times the number, or 3 of the number; that the quotient of a number divided by f of 5 times the number, or of the number; and, generally, that (g.) The quotient of a number divided by a fraction equals the product of that number multiplied by the fraction inverted. (h.) Hence, to find the quotient of a number divided by a fraction, we multiply that number by the fraction inverted. 23. What is the quotient of 4 = ? SOLUTION.—The quotient of $ = = the product of 4 xf = f g = 13. 24. What is the quotient of 3= of tš? Solution. — The quotient of 34 or of & divided by tf must equal the product of 7 multiplied by {}, and the quotient of of 14 must equal & of this last result, wbich gives the written work below: (i.) Perform the following examples : 25. } ; f? 28. 53 - 22 31. 69 : 877 26. 2} = 3} ? 29. 8} = 2% ? 32. 4 = 518? 30. 54 : 14? 33. 21% = 24? how many times ? how many times 6 ? 40. of 4 = 12 of 11 how many times 9 ? 41. A man owned 44 acres of land which he divided into houselots, each containing of an acre. How many house lots did he obtain ? 42. When of a dollar will buy 1 yard of cloth, how many yards will $44 buy? 43. When 21 yards of cloth cost $1, how many dollars will 8} yards cost? 44. How many bushels of grain at of a dollar per bushel can be bought for 7 quarter-eagles ? 45. A man exchanged 74 barrels of apples worth $2; per barrel for wheat worth $11per bushel. How many bushels of wheat did he receive ? 46. A man purchased 16 baskets of peaches each containing ? of a bushel, and sold them at the rate of 5 of a bushel for a dollar. How much did he receive for them ? 47. A man who had $50 bought 17 bushels of salt at the of a dollar per bushel, and paid the rest of his money for apples at $2} per bbl. How many barrels of apples did he buy? 48. A man who had $100 paid of his money for broadcloth at $2} per yard. How many yards did he buy? 49. I own a lot of land 48} rods long and 16f rods wide. How many dollars will it cost to build a fence completely around it, provided that the fence costs of a dollar per yard ? 96. To divide by a Decimal Fraction. (a.) Since dividing by any fraction is equivalent to multiplying by that fraction inverted, it follows that to divide by .023 is the same thing as to multiply by '99', which may be done by removing the point in the dividend three places to the right and then dividing by 23, or by dividing by 23 and then removing the point in the quotient three places to the right. As a similar thing is true of any other decimal fraction, it follows that (b.) To divide by a decimal fraction, we may perform the division as though the divisor were a whole number, and then remove the point in the quotient as many places to the right as there are decimal places in the divisor. The quotient will thus contain as many places of decimal fractions as there are in the dividend more than in the divisor. NOTE. — In case the dividend does not contain as many places of decimal fractions as the divisor, it will be well to annex ciphers enough to make as many, before beginning to divide. 1. What is the quotient of 21.201 3.7 ? SOLUTION. — The quotient of 21.201 divided by 3.7 equals the product of 21.201 multiplied by $4, which, found by dividing by 37 and removing the decimal point one place to the right, is 5.73. (c.) What is the quotient of — 2. 25 • 5? 10. .25 = 5? .25 · .05? 18. 48 = 1.6? 26. .6 of .24 of 4.8 • .012 of 3.6 of .002? 27. .5 of .05 of .005 • .005 of .05 of .5? 28. 1.7 of 62.5 of .07 = 5.1 of .125 of 4.9 ? 29. 7.2 of 1.47 of .0064 of .003 • 1800 of .0049 of .128 of 9000? 30. .02 of 1.5 of .075 of 8.7 = .3 of 1.25 of .0029 ? 97. Complex Fractions. (a.) A COMPLEX Fraction is one which has a fraction in one or hth its terms. (b.) Complex fractions may be regarded as indicating that the numerator is to be divided by the denominator. 39 1. Reduce 57 to a simple fraction. 4. (c.) In the same manner reduce the following: 2} .32 .44 2. 6. 8. .0016 5 9. .0058 (d.) Complex fractions may also be reduced by multiplying both terms by such a number as will give a whole number in place of each. 23 10. Reduce 38 to a simple fraction. SOLUTION. If 24 be multiplied by 4 or any multiple of 4, and 3g be multiplied by 6 or any multiple of 6, the result in each case will be a whole number. Hence, if both terms of the given fraction be multiplied by some multiple of both 4 and 6, the resulting fraction will be a simple one. Mul 23 33 tiplying by 12, the least common multiple of 4 and 6 gives 3 35 46 (e.) In the same manner reduce the following: |