EXAMPLES FOR PRACTICE. 2. What will 40 bushels of corn come to at of a dollar per bushel? 40 × 2 how much? 3. What will 24 yards of cloth cost at of a dollar per yard? 24 X = how much? ? 4. How much is of 90? of 369 ? of 45? 5. Multiply 45 by 7. Multiply 20 by 1. To multiply one fraction by another. 53. 1. A man, owning of a ticket, sold of his share; what part of the whole ticket did he sell?of is how much? We have just seen, (T 52,) that, to multiply by a fraction, is to divide the multiplicand by the denominator, and to multiply the quotient by the numerator. divided by 3, the denominator of the multiplying fraction, (49,) is 15, which, multiplied by 2, the numerator, (T 51,) is, Ans. The process, if carefully considered, will be found to consist in multiplying together the two numerators for a new nur merator, and the two denominators for a new denominator. EXAMPLES FOR PRACTICE. 2. A man, having of a dollar, gave of it for a dinner; what did the dinner cost him? Ans.doilar. Product, 3. Multiply by . Multiply by 4. 4. How much is 4 of 3 of 7 of ? Note. Fractions like the above, connected by the word of, are sometimes called compound fractions. The word or implies their continual multiplication into each other. Ans. 188=20. When there are several fractions to be multiplied continually together, as the several numerators are factors of the new numerator, and the several denominators are factors of the new denominator, the operation may be shortened by dropping those factors which are the same in both terms, on the principle explained in T 46. Thus, in the last example, , ,,, we find a 4 and a 3 both among the numerators and among the denominators; therefore we drop them, multiplying together only the remaining numerators, 2 × 7=14, for a new numerator, and the remaining denominators, 5 x 8 = 40, for a new denominator, making ʊ, Ans. as before. K* $ 8 5. of t of of & of off of how much? Note. The integer 7 may be reduced to the form of an improper fraction by writing a unit under it for a denominator, thus, 7. Ans. 2H Ans. f. and 33? 7. At of a dollar a yard, what will of a yard of cloth cost? 8. At 63 dollars per barrel for flour, what will of a barrel cost? 681; then 4×7 == $2191, Ans. 9. At of a dollar per yard, what cost 7 yards? 2285 Ans. $611 10. At $2 per yard, what cost 63 yards? Ans. $1433, 11. What is the continued product of 3, 4, § of 2, 24, and of of t? Ans. 208. ¶ 64. The RULE for the multiplication of fractions may Row be presented at one view : I. To multiply a fraction by a whole number or a whole number by a fraction,-Divide the denominator by the whole number, when it can be done without a remainder; otherwise, multiply the numerator by it, and under the product write the denominator, which may then be reduced to a whole or mixed number. II. To multiply a mixed number by a whole number,-Multiply the fraction and integers, separately, and add their products together. III. To multiply one fraction by another, Multiply together the numerators for a new numerator, and the denominators for a new denominator. Note. If either or both are mixed numbers, they may first be reduced to improper fractions. EXAMPLES FOR PRACTICE. 1. At $ per yard, what cost 4 yds. ? 6 yds.? 8 yds.? 2. Multiply 148 by by f. 3. If 2 tons of hay keep 1 horse yards. of cloth? .5 20 yds. ? Ans. to the last, $15. • by ਲੰਨ by f Last product, 44. through the winter, how much will it take to keep 3 horses the same time? 7 horses? 13 horses? 4. What will 8 barrel ? Ans. to last, 377 tons barrels of cider come to, at $3 per 5. At $14 per cwt., what will be the 6. A owned of a ticket; B owned ticket was so lucky as to draw a prize of each one's share of the money? 7. Multiply of by of t 8. Multiply 7 by 2. 9. Multiply by 24. 10. Multiply of 6 by 3. 11. Multiply of 2 by of 4. 12. Multiply continually together and 4 of 10. 13. Multiply 1000000 ʊy §. cost of 147 cwt.? of the same; the $1000; what was To divide a whole number by a fraction. T55. We have already shown (T 49) how to divide a fraction by a whole number; we now proceed to show how to divide a whole number oy a fraction. 1. A man divided $9 among some poor people, giving them of a dollar each; how many were the persons who received the money? 9÷how many? 1 dollar ist, and 9 dollars is 9 times as many, that is, 36; then is contained in 36 as many times as 3 is contained in 36. Ans. 12 persons. OPERATION. Product, Product, 1. of 8, 3 of 7, § of 9, That is,-Multiply the dividend by the denominator of the dividing fraction, (thereby reducing the dividend to parts of the same magnitude as the divisor,) and divide the produc by the numerator. 2. How many times is contained in 8? 8÷4 = how many? 8 Dividend. 5 Denominator. Numerator, 3)40 Quotient, 13 times, the Answer. To multiply by a fraction, we have seen, (T-52,) implies two operations a division and a multiplication; so, also, to divide by a fraction implies two operations-a multiplication and a division. 56. Division is the reverse of multiplication. To multiply by a fraction, To divide by a fraction, whether the multiplicand be whether the dividend be a a whole number or a fraction, whole number or a fraction, as has been already shown, we multiply by the denomina (T 52,) we divide by the de- tor of the dividing fraction, nominator of the multiplying and divide the product by the fraction, and multiply the quo- numerator. tient by the numerator. Note. In either case, it is matter of indifference, as it respects the result, which of these operations precedes the other; but in practice it will frequently be more convenient, that the multiplication precede the division. 12 multiplied by, the pro duct is 9. 12 divided by, the quotient is 16. In multiplication, the mul- In division, the divisor betiplier being less than unity, ing less than unity, or 1, will or 1, will require the product be contained a greater number to be less than the multipli- of times; consequently will recand, ( 52,) to which it is quire the quotient to be greatonly equal when the multi-er than the dividend, to which plier is 1, and greater when it will be equal when the dithe multiplier is more than 1. visor is 1, and less when the divisor is more than 1. EXAMPLES FOR PRACTICE. 1. How many times is contained in 7? 7÷1 many? } 2. How many times can I draw of a gallon of wine out of a cask containing 26 gallons? 6 by &. 3. Divide 3 by 2. 10 by 3. 4. If a man drink of a quart of rum a day, how long will 3 gallons last him? 5. If 23 bushels of oats sow an acre, how many acres will 22 bushels sow? 22-22 how many times? Note. Reduce the mixed number to an improper frac tion, 22 = . Ans. 8 acres. many yards of cloth may be 6. At $4 a yard, how bought for $37? yards. 7. How many times is contained in r = how Ans. 8 84 ? Ans. 90 times. & How many times is 36 contained in 6 ? 9. How many times is 8§ contained in 53 ? Ans. § of 1 time. Ans. 644 times. 10. At of a dollar for building 1 rod of stone wall, how many rods may be built for $87? 87÷how many times? To divide one fraction by another. TT 57. 1. At of a dollar per bushel, how much rye may be bought for g of a dollar? is contained in how many times? as before. Had the rye been 2 whole dollars per bushel, instead of of a dollar, it is evident, that g of a dollar must have been divided by 2, and the quotient would have been ; but the divisor is 3ds, and 3ds will be contained 3 times where a like number of whole ones are contained 1 time; consequently the quotient is 3 times too small, and must therefore, in order to give the true answer, be multiplied by 3, that is, by the denominator of the divisor;3 times bush. Ans. = = The process is that already described, T 55 and T 56. If carefully considered, it will be perceived, that the numerator of the divisor is multiplied into the denominator of the dividend, and the denominator of the divisor into the numerator of the dividend; wherefore, in practice, it will be more convenient to invert the divisor; thus, inverted becomes ; then multiply together the two upper terms for a numerator, and the two lower terms for a denominator, as in the multiplication of one fraction by another. Thus, in the above example, 3 x 3 9 2 X 5 10' EXAMPLES FOR PRACTICE. + 2. At of a dollar per bushel for apples, how many busǹmay be bought for of a dollar? How many times is contained in ?? Ans. 3 bushels. els 3. If of a yard of cloth cost of a dollar, what is that per yard? It will be recollected, ( 24,) that when the cost of any quantity is given to find the price of a unit, we divide the cost by the quantity. Thus, (the cost) divided by (the quantity) will give the price of 1 yard. Ans. of a dollar per vard. |