8 and these, joined to their respective whole numbers, give the following expressions, viz. OPERATION. Cwt. Cwt. 1921948 20 2018 223=2248 Ans. 62 cwt. By adding together all the 60ths, viz. 45, 12 and 40, we have 7=187; then writing the down, and carrying the whole number, 1, to the amount of the column of whole numbers, makes 62, which, joined with 87, makes 6287, Ans. 9 2. How much is of, and, added together?of=; then and , reduced to a common denominator, give 4 and 1, which, added together as before, give 24=124, Ans. From these illustrations we derive the following RULE. Q. How do you prepare fractions to add them? A. Reduce compound fractions to simple ones, then all the fractions to a common or least common denominator. Q. How do you proceed to add? A. Add their numerators. More Exercises for the Slate. 3. What is the amount of 16 yards, 17 yds. and 34 yards? 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? 4. How much does 19 from 12 leave? 5. How much does from leave? 50 6. How much does from 7 leave? 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 203 yards of cloth, and sold 15 yards; how much remained unsold? OPERATION. and 3, reduced to a com mon denominator, make 8 12 and; then, 204 20 = 12 1515 411 yards, Ans. In this example, we cannot take from 12, but, by borrowing 1 (unit), which is 1, we can proceed thus, 13 and are, from which taking , or 9 parts from 20 parts, leaves 11 parts, that is, 1; then, carrying 1 (unit, for that which I borrowed) to 15, makes 16; then, 16 from 20 leaves 4, which, joined with makes 411, Ans. 2. From take and , reduced to a common denominator, give and; then, from 12 leaves Jo, 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 denominator, will give the difference required. More Exercises for the Slate. 1 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 4, and 6 dollars are 6 times, that is, 24; then, 4, 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 numerator. Exercises for the Slate. 2. At of a dollar a bushel, how many bushels of rye can ave for 80 dollars? In this example, we see more fully illustrated the fact that division is the opposite of multiplication; for, to multiply 80 by 6' we should multiply by the numerator, and divide by the denominator; T XXXIX. flour in one week, 3. If a family consume of a quarter of 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. . How many times is 26 contained in 720 ? A. 140. 7. How many times is 8 contained in 300 ? an improper fraction. A. 36. 8. Divide 620 by 811. 9. Divide 84 by 198. 10. Divide 92 by 4. 11. Divide 100 by 22. 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? Reduce 8 to A. 757. Ꭿ. 160. A. 20. A. 36. .9.55 A. 40 rods. A. 40 sq. rods. I 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 2? How many times in §? of a dollar? How 2. William gave of a dollar for one orange; how many oranges, at that rate, can he buy for many for of a dollar? For 2? For? For 24? For 27 ? 130 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? OPERATION. Reducing the fractions and to a common denominator, thus: 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, 1-2 yards, as be fore, Ans. PROOF., the quotient, multiplied by, the divisor, thus,, gives, the 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 compound to simple terms. PROOF. It 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. 4 pecks. A. 솔루=2. A. 150-213. |