how much will it take to keep 3 horses the same time? 7 horses? 13 horses? Ans. to last, 37 tons. 4. What will 8 barrels of cider come to, at $3 per barrel ? 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 2 of 4. 8. Multiply 7 by 21. 9. Multiply by 23. 10. Multiply of 6 by 3. 11. Multiply of 2 by of 4. 12. Multiply continually together and of 10. 13. Multiply 1000000 vy §. cost of 147 cwt.? of the same; the $1000; what was Product, . Product, 15. Product, 24. Product, 1. Product, 3. of 8, of 7, f of 9, Product, 20. Product, 555555§. To divide a whole number by a fraction. ¶ 55. 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? how many? 9 1 dollar is 4, 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. 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 contained in 8? many? 8÷= how OPERATION. 8 Dividend. 5 Denominator. Numerator, 3)40 Quotient, 134 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( 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 product 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 ? 2. How many times can I draw of a cask containing 26 gallons? 3. Divide 3 by 2. 4. If a man drink will 3 gallons last him? 6 by 3. 7÷= how of a gallon of wine out 10 by 2. of a quart of rum a day, how long 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 fraction, 22 = 4. 6. At $4 a yard, how bought for $37? 7. How many times is Ans. 8 acres. many yards of cloth may be Ans. 8 yards. 8. How many times is 36 contained in 6 ? Ans. of 1 time. 9. How many times is 83 contained in 53 ? Ans. 645 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 of a dollar? is contained in g 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 ; 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, ¶ 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, in the above example, 3 X 3 as before. 2 X 5 == 9 10' EXAMPLES FOR PRACTICE. 2. At of a dollar per bushel for apples, how many bushels may be bought for of a dollar? contained in ? How many times is Ans. 3 bushels. of a dollar, what is that 3. If of a yard of cloth cost 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 yard. Ans. of a dollar per vard, 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 (;) and is, in fact, reversing the question, thus, If the price of 1 yard be of a dollar, what will yard cost? of a Ans. g 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. 4 bushels. 15. If 43 pounds of butter serve a family 1 week, how many weeks will 367 pounds serve them? The mixed numbers, it will be recollected, may be reduced to improper fractions. Ans. 8 weeks. T58. 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 terms 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 actions. EXAMPLES FOR PRACTICE. 1. If 7 lb. of sugar cost of a dollar, what is it per pound? ÷ 7 = how much? of is how much? 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 14 yards cost $75, what is the price per yard? Ans. 5 6. At 31 dollars for 10 barrels of cider, what is that per barrel? 7. How many times is contained in 746 ? Ans. $3. Ans. 1989. π 59. 1. A boy gave to one of his companions of an orange, to another, to another; what part of an orange did he give to all? + 1 + 1 how much? Ans.. of a ton of hay; a of a ton; and a pair 2. A cow consumes, in 1 month, than the horse? ++= how much? 3. 1+3+1= how much? how much the oxen *-= -- how much? -&= 4.26+20 +2%+1+2%= how much? how much? 5. A boy, having of an apple, gave of it to his sister; what part of the apple had he left? how much? |