To multiply one fraction by another. T53. 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, (T52,) 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 multıplying fraction, (1 49,) is 4 1, which, multiplied by 2, the numerator, (151,) is, Ans. The process, if carefully considered, will be found to consist in multiplying together the two numerators for a new numerator, and the two denominators for a new denominator. EXAMPLES FOR PRACTICE. 2. A man, having of a dollar, gave 3 of it for a dinner; what did the dinner cost him? 3. Multiply by . Multiply by 4. How much is of of off? Ans. dollar. Product, . 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. 168 480 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 × 8 = 40, for a new denominator, making 1=7, Ans. as before. 5. & of of of of off of = how much? Ans. 1. 6. What is the continual product of 7, 초, & of 3 and 31? Note. The integer 7 may be reduced to the form of an improper fraction by writing a unit under it for a denominator, thus, 1. 25 Ans. 211. 7. At of a dollar a yard, what will of a yard of cloth cost? 8. At 68 dollars per barrel for flour, what will 7 of a barrel cost? 6 = 51; then 51= =?=$21, Ans. 9. At & of a dollar per yard, what cost 7 yards? Ans. $611. 10. At $24 per yard, what cost 60 yards? Ans. $1439. 11. What is the continued product of 3, 4, 8 of 1, 2, and of for? 12 Ans.. 54. The RULE for the multiplication of fractions may now be presented at one view : I. To multiply a fraction by a whole number, or a whole number by a fraction,-Divide the denominator by the whole number, when it can be done without a remainder; otherwise, multiply the numerator by it, and under the product write the denominator, which may then be reduced to a whole or mixed number. II. To multiply a mixed number by a whole number,-Multiply the fraction and integers, separately, and add their products together. III. To multiply one fraction by another, Multiply together the numerators for a new numerator, and the denominators for a new denominator. Note. If either or both are mixed numbers, they may first be reduced to improper fractions. EXAMPLES FOR PRACTICE. 1. At$ per yard, what cost 4 yards of cloth ? -5 yards? 8 yards? 6 yards? 20 yards? Ans. to the last, $15. by 2 will it take to keep 3 horses the same time? by Last product, 44-4 10 3. If29 tons of hay keep 1 horse through the winter, how much 7 horses? Ans. to the last, 377 tons. 13 horses? 4. What will 8-7 barrels of cider come to, at $3 per barrel ? 12 5. At $144 per cwt. what will be the cost of 147 cwt.? 6. A owned 3 of a ticket; Bowned of the same; the ticket 5 15 was so lucky as to draw a prize of $1000; what was each one's share of the money? 11. Multiply & of 2 by & of 4. 13. Multiply 1000000 by §. Product, . Product, 152. Product, 2 Product, 1. Product, 3. 12, Multiply continually together 1 of 8,3 of 7, of 9, and 4 of 10. To divide a whole number by a fraction. 55. We have already shown (149) how to divide a fraction by a whole number; we now proceed to show how to divide a whole number by 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? 4 1 dollar is, and 9 dollars is 9 times as many, that is, 26; then is contained in 36 as many times as 3 is contained in 36. 4 Ans. 12 persons. 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 product by the numerator. 2. How many times is a contained in 87 8÷3= how many? OPERATION. 8 Dividend. Numerator, 3) 40 Quotient, 13 times, the Answer. To multiply by a fraction, we have seen, (1 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, whether the multiplicand be a whole number or a fraction, as has been already shown, (152,) we divide by the denominator of the multiplying fraction, and multiply the quotient by the numera tor. To divide by a fraction, whether the dividend be a whole number or a fraction, we multiply by the denominator of the dividing fraction, and divide the product 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 product is 9. In multiplication, the multiplier being less than unity, or 1, will require the product to be less than the multiplicand, (52,) to which it is only equal when the multiplier is 1, and greater when the multiplier is more than 1. 16. 12 divided by, the quotient is In division, the divisor being. less than unity, or 1, will be contained a greater number of times; consequently will require the quotient to be greater than the dividend, to which it will be equal when the divisor is 1, and less when the divisor is more than 1. EXAMPLES FOR PRACTICE. 1. How many times is contained in 7? 7 how many? 2. How many times can I draw 4 of a gallon of wine out of a cask containing 26 gallons? 3. Divide 3 by 4. 6 by 3. 10 by . 4. If a man drink 9 of a quart of rum a day, how long will 3 gallons last him? 16 5. If 24 bushels of oats sow an acre, how many acres will 22 bushels sow? 222= how many times? Note. Reduce the mixed number to an improper fraction, 2 = 4. Ans. 8 acres. 6. At $42 a yard, how many yards of cloth may be bought for $37? 7. How many times is 96 contained in 84? 103 8. How many times is 36 contained in 6? 9. How many times is 8 & contained in 53? Ans. 822 yards. Ans. & of 1 time. Ans. 615 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. 57. 1. At of a dollar per bushel, how much rye may be bought for 3 of a dollar? is contained in 3 how many times? Had the rye been 2 whole dollars per bushel, instead of of a dollar, it is evident, that of a dollar must have been divided by 2, and the quotient would have been Tổ; 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 3 is 3 times toc 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 3 = bushel, Ans. 9 TO To The process is that already described, T 55 and 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, 3X3 9 in the above example, -, as before, 2X5 10 EXAMPLES FOR PRACTICE. 2. At of a dollar per bushel for apples, how many bushels may be bought for of a dollar? How many times is & contained in ? Ans. 31⁄2 bushels. 3. If of a yard of cloth cost of a dollar, what is that per yard? It will be recollected, (T 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 yard. PROOF. If the work be right, ( 16, "Proof,") the product of the quotient into the divisor will be equal to the dividend; thus, =. This, it will be perceived, is multiplying the price of one yard (3) by the quantity (7) to find the cost (3;) and is, in fact, reversing the question, thus, If the price of 1 yard be of a dollar, what will & of a yard cost? Ans, 3 of a dollar. Note. Let the pupil be required to reverse and prove the succeeding examples in the same manner. 4. How many bushels of apples, at of a dollar per bushel, may be bought for of a dollar? Ans. 43 bushels. 5. If 48 pounds of butter serve a family 1 week, how many weeks will 363 pounds serve them? The mixed numbers, it will be recollected, may be reduced to improper fractions. Ans. 88 weeks. 36 8. Divide 24 by 1t. Quot. 1. Divide 103 by 24. Quot. 41. 9. How many times is contained in ?? 10. How many times is 3 contained in 47 ? 11. Divide of by & of 4. Ans. 4 times. Ans. 113 times. 58. The RULE for division of fractions may now be presented at one view: I. To divide a fraction by a whole number, Divide the numerator by the whole number, when it can be done without a remainder, and under the quotient write the denominator; otherwise, multiply the denominator by it, and over the product write the numerator. II. To divide a whole number by a fraction, - Multiply the dividend by the denominator of the fraction, and divide the product by the numerator.. III. To divide one fraction by another, --Invert the divisor, and multiply together the two upper ternis for a numerator, and the two lower terms for a denominator. -Note. If either or both are mixed numbers, they may be reduced to improper fractions. EXAMPLES FOR PRACTICE. 1. If 7 lb. of sugar cost 63 of a dollar, what is it per pound? 637= how much? To이 1 of 63 is how much? 100 100 2. At for of a barrel of cider, what is that per barrel? 3. If 4 pounds of tobacco cost & of a dollar, what does 1 pound cost? 4. If of a yard cost $4, what is the price per yard? 5. If 143 yards cost $75, what is the price per yard? Ans. 525 barrels of cider, what is that per barrel ? 6. At 31 dollars for 10 Ans. $3. Ans. 1989. |