2. Harry had of a dollar, and Rufus ; what part of a dollar has Rufus more than Harry? How much does from leave? 3. How much does 18 from 12 leave ? 6. How much does 6 50 9 Too from leave? 72 100 From the foregoing examples, it appears that fractions may be subtracted by subtracting their numerators, as well as added, and for the same reason. 1. Bought 20 yards of cloth, and sold 15 yards; how much remained unsold? In this example, we cannot take 12 from 12, but, by borrowing 1 (unit), which is, we can proceed thus, and 12 are 12, from which taking Tz, or 9 parts from 20 parts, leaves 11 parts, that is, }; then, carrying 1 (unit, for that which I borrowed) to 15, 9 makes 16; then, 16 from 20 leaves 4, which, joined with, makes 41, Ans. 2. From nator, give take. and, reduced to a common denomi 5 and; then, from 2 leaves, Ans. From these illustrations we derive the following Q. What is the rule? RULE. A. Prepare the fractions as in addition, then the difference of the numerators written over the denomi nator, will give the difference required. ¶ XLVI. TO DIVIDE A WHOLE NUMBER BY A FRACTION Lest you may be surprised, sometimes, to find in the following examples a quotient very considerably larger than the dividend, it may here be remarked, by way of illustration, that 4 is contained in 12, 3 times, 2 in 12, 6 times, 1 in 12, 12 times; and a half (1) is evidently contained twice as many times as 1 whole, that is, 24 times. Hence, when the divisor is 1 (unit), the quotient will be the same as the dividend; when the divisor is more than 1 (unit), the quotient will be less than the dividend; and when the divisor is less than 1 (unit), the quotient will be more than the dividend. 1. At of a dollar a yard, how many yards of cloth can you buy for 6 dollars? 1 dollar is, and 6 dollars are 6 times, that is, 24; then, 3, or 3 parts, are contained in 24, or 24 parts, as many times as 3 is contained in 24, that is, 8 times. A. 8 yards. In the foregoing example, the 6 was first brought into 4ths, or quarters, by multiplying it by the denominator of the divisor, thereby reducing it to parts of equal size with the divisor; hence we derive the following RULE. Q. How do you proceed to divide a whole number by a fraction? A. Multiply the dividend by the denominator of the dividing fraction, and divide the product by the nume rator. Exercises for the Slate. 2. At of a dollar a bushel how many bushels of rve can I have for 80 dollars? 3. If a family consume of a quarter of flour in one week, how many weeks will 48 quarters last the same family? A. 128 weeks. 4. If you borrow of your neighbor of a bushel of meal at one time, how many times would it take you to borrow 96 bushels? A. 960 times. 5. How many yards of cloth, at of a dollar a yard, may be bought for 200 dollars? A. 1000 yards. contained in 720? A. 840. contained in 300? Reduce 8 to 6. How many times is 7. How many times is 8 an improper fraction. A. 36. 8. Divide 620 by 871. 9. Divide 84 by 28. 168 A. 753 10. Divide 92 by 4. 11. Divide 100 by 24. 12. Divide 86 by 157. 13. How many rods in 220 yards? 14. How many sq. rods in 1210 sq. yards? 15. How many barrels in 1260 gallons? ¶ XLVII. TO DIVIDE ONE FRACTION BY ANOTHER. 1. At of a cent an apple, how many apples may be bought for of a cent? How many times in? How many times in ? 2. William gave of a dollar for one orange, how many oranges, at that rate, can he buy for of a dollar? How many for of a dollar? For? For 24? For 27 ? For? Hence we see that fractions, having a common denominator may be divided by dividing their numerators, as well as subtracted and added, and for the same reason. 1. At of a dollar a yard, how many yards of cloth may be bought for of a dollar? In this example, as the common denominator is not used, it is plain that we need not find it, but only multiply the numerators by the same numbers as before. This will be found to consist in multiplying the numerator of the divisor into the denominator of the dividend, and the denominator of the divisor into the numerator of the dividend. But it will be found to be more convenient, in practice, to invert the divisor, then multiply the upper terms together for a numerator, and the lower terms for a denominator; thus, taking the last example, and, by inverting the divisor, become and; then, x=22 yards, as before, Ans. 3X3 PROOF., the quotient, multiplied by, the divisor; thus, 33, gives 12, the gives, divisor. From these illustrations we derive the following RULE. Q. How do you proceed to divide one fraction by another? A. I invert the divisor, then multiply the upper terms together for a new numerator, and the lower for a new denominator. Note.-Mixed numbers must be reduced to improper fractions, and com pound to simple terms. PROOF.-I would be well for the pupil to prove each result, as in Simple Multiplication, by multiplying the divisor and quotient together, to obtain the dividend. More Exercises for the Slate. 2. At of a dollar a peck, how many pecks of salt may be bought for of a dollar? A. 43 pecks. 3. Divide 3 by 4. A. 17=2. 14' 4. Divide by 2. A. 150=213. 7. How many times is contained in? 5 ? A. 215. 9. What number multiplied by will make? A. 2. It will be recollected, that in Reduction (¶ XXIX.) whole numbers were brought from higher to lower denominations by multiplication, and from lower to higher denominations by division; hence, fractions of one denomination may be reduced to another after the same manner, and by the same rules. * XLVIII. TO REDUCE WHOLE NUMBERS TO THE FRACTION OF A GREATER DENOMINATION. 1. What part of 2 miles is 1 mile? 2. What part of 4 miles is 1 mile? Is 2 miles? Is 3 miles? 10. What part of 1 hour is 11 minutes? Is 21 minutes? |