OPERATION. 48 and : these, joined to their respective whole numbers, give the following expressions, viz. By adding together all the 60ths, Cwt. Cwt. viz. 45, 12 and 40, we have 7=187; 19 1948 then writing the down, and carry201= 2013 ing the whole number, 1, to the 22 = 2248 amount of the column of whole num bers, makes 62, which, joined with Ans. 6287 cwt. 37, makes 6287, Ans. 2. How much is of , and , added together? } of =; then and , reduced to a common denominator, give it and 24, which, added together as before, give =124, Ans. From these illustrations we derive the following RULE. 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 ? More Exercises for the Slate. 3. What is the amount of 164 yards, 17} yds. and 3f yards ? A. 3717. 4. Add together and 6. A. 1795 A. 3072 SUBTRACTION OF FRACTIONS. STION 9 XLV. 1. William, having 1 of an orange, gave to Thomas; how much had he left? How much does : from leave ? 2. Harry had } of a dollar, and Rufus $; what part of a dollar has Rufus more than Harry? How much does from f leave ? 3. How much does 18 from 13 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 20 yards of cloth, and sold 158 yards; how much remained unsold ? OPERATION. In this example, we cannot and , reduced to a com take 12 from 1, but, by bormon denominator, make 12 rowing 1 (unit), which is li, and ; then, we can proceed thus, 14 and 20% = 20 are 12, from which taking 154 = 15 PS 12, or 9 parts from 20 parts, leaves 11 parts, that is, 12; 4 11 yards, Ans. then, carrying 1 (unit, for that which I borrowed) to 15, makes 16; then, 16 from 20 leaves 4, which, joined with 12, makes 412, Ans. 2. From take . and ţ, reduced to a common denomi- 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. 2. From 1 take a A. 3. From 11 take 11. A. For 4. From 1 take 4. 112. 5. From } take . 4. 18. A. 1. 6. From it take z. 66 520 A. = 65 DIVISION OF FRACTIONS. 9 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 (!) 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, **; then, t, or 3 parts, are contained in 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. A. Multiply the dividend by the denominator of the dividing fraction, and divide the product by the numerator. Exercises for the Slate. 2. At it of a dollar a bushel, how many bushels of rye can I have for 80 dollars ? 02 OPERATION. In this example, 80 dividend. we see more fully illustrated the fact 16 denominator. that division is the 480 opposite of multi plication ; for, to 80 multiply 80 by we should multiNumerator, 5) 1280 ply by the numeraQuotient, 256 bushels, Ans. tor, and divide by the denominator ; TXXXIX. 3. If a family consume f 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 Ty 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 36 contained in 720? A. 140. 7. How many times is 8 contained in 300 ? Reduce 85 to an improper fraction. A. 36. 8. Divide 620 by 811. A. 757. 9. Divide 84 by S. A. 160. 10. Divide 92 by 4:. A. 205. 11. Divide 100 by 2. A. 3641. 12. Divide 86 by 157. .4. 512 A. 40 rods. 53 to of th 1 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 I in 1? 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 g of a dollar ? For ? For 44 ? For 272 For $go ? 9 12 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. In this example, as Reducing the fractions } and to a the common denomi. common denominator, thus: nator is not used, it is plain that we need not } 录 find it, but only multiply the numerators by the same numbers Then, iis contained in 12 as many as before. This will times as 4 is contained in 9,=24 be found to consist in A. 24 yards. multiplying the nu merator 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 Proof. , the quotient, divisor, become į and if ; then, multiplied by }, the divisor, ix==2+ yards, as be- thus, 11, gives =, the fore, Ans. divisor. RULE. 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. 48 pecks. 3. Divide by 14. A. 41=2. |