140. When the multiplicand is a whole, and the multiplier a mixed number. 7. What is the product of 48 by 8? NOTE. First multiply 48 by, which gives 8; then by 8, which gives 384, and the sum 892 is the product: Hence, Rule. OPERATION. 48 × 1/ = 8 48 × 8 = 384 392 Multiply first by the fraction, and then by the whole number, and add the products. 8. Multiply 67 by 91. 9. Multiply 9 by 123. 10. Multiply 108 by 12. 11. Multiply 59 by 31. 12. What is the product of 6, 27, and of 12? 13. What will 24 yards of cloth cost, at 33 dollars a yard? 14. What will 63 bushels of wheat cost, at 3 dollars a bushel? 3 15. A horse eats of 3 of 12 tons of hay in three months: how much did he consume? 16. If of of a dollar buy a bushel of corn, what will of of a bushel cost? 6 17. What is the cost of 5 gallons of molasses, at 961 cents a gallon? 18. What will 711 dozen candles cost, at. of a dollar per dozen? 19. What must be paid for 175 barrels of flour, at 7} dollars a barrel? 20. If of of 2 yards of cloth can be bought for one dollar, how much can be bought for 7 of 13 dollars? 21. What is the cost of 153 cords of wood, at 39 dollars a cord? 140. How may you multiply, when the multiplicand is a whole, and the multiplier a mixed number? DIVISION. 141. DIVISION OF FRACTIONS is the operation of finding how many times one number is contained in another, when one or both, are fractional. 1. What is the quotient of 5 divided by ? ANALYSIS.-One-sixth is contained in 1, 6 times, because there are 6 sixths in 1: one-sixth is contained in 5, 5 times as many times as in 1: hence, is contained in 5, 30 times. OPERATION. 5 ÷ 2 = 5×6 = 30. 142. To divide a fraction by a whole number. 1. If 4 bushels of apples cost bushel cost? ANALYSIS. Since 4 bushels cost dollar, 1 bushel will cost of a dollar, what will 1 of of a dollar. OPERATION. of a 90 8÷ 4 = 3. Dividing the numerator of the fraction by 4, we have (Art. 117). Multiplying the denominator by 4, will produce the same result (Art. 118): Hence, Rule. Divide the numerator, or multiply the denominator, by the divisor. 141. What is Division of Fractions? What is the quotient of 8 divided by ? 142. What is Case I.? What is the rule? 9. If 6 horses eat of a ton of hay in 1 month, how much will one horse eat? 10. If 9 yards of ribbon cost of a dollar, what will 1 yard cost? 11. If 1 yard of cloth cost 4 dollars, how much can be bought for of a dollar? 12. If 5 pounds of coffee cost 15 of a dollar, what will 1 pound cost? 13. At $6 a barrel, what part of a barrel of flour can be bought for of a dollar? 14. If 10 bushels of barley cost 3 dollars, what will 1 bushel cost? NOTE.-Reduce the mixed number to an improper fraction, and divide as in the case of a simple fraction. 15. If 21 pounds of raisins cost 4 dollars, what will 1 pound cost? 16. If 12 men consume 63 pounds of meat in a day, how much does 1 man consume? CASE II. 143. To divide a whole number by a fraction. 1. At of a dollar apiece, how many hats can be bought for 6 dollars? ANALYSIS.-As many as of a dollar is contained times in 6 dollars. x 4. To divide 6 by, is to divide it by, and then the quotient by 4 (Art. 76). OPERATION. 66×4 =6x=307 hats. 6 ÷ 3 = 6 × 5 = 30; then, dividing by 4, we have 307 for the answer. Hence, Rule. Invert the terms of the divisor, and multiply the whole number by the new fraction. Examples. 1. Divide 14 by 7. 3. Divide 63 by 19. 2. Divide 212 by 18 4. Divide 420 by 5. At of a dollar a yard, how many yards of cloth can be bought for 9 dollars? 6. If a man travel of a mile in 1 hour, how long will it take him to travel 10 miles? 7. If of a ton of hay is worth 9 dollars, what is a ton worth? CASE III. 144. To divide one fraction by another. 1. At of a dollar a gallon, how much molasses can be bought for of a dollar? ANALYSIS.-As many times as of a dollar is contained times 7÷}=}÷} × 2 in of a dollar: OPERATION. == × 2; = = 35 ÷ 2 = 35 = 2,3% gall. hence, to divide by 3, is to divide by and 2.35; then dividing by 2, we have, 2 gallons. Hence, Rule. I. Invert the terms of the divisor: 3 = II. Then multiply the numerators together for the numerator of the quotient, and the denominators together for the denominator of the quotient. NOTES.-1. Cancel all common factors. 2. If the dividend and divisor have a common denominator, they will cancel, and the answer will be the quotient of their numerators. 3. When the dividend or divisor contains a whole or mixed number, or compound fractions, reduce to the form of simple fractions, before dividing. 7. At of a dollar a pound, how much butter can be bought for of a dollar? 8. If 1 man consume 1 pounds of meat in a day, how many men would 83 pounds supply? 9. If 6 pounds of tea cost 4 dollars, what does it cost a pound? 10. At of a dollar a basket, how many baskets of peaches can be bought for 113} dollars? 11. If of a ton of coal cost 63 dollars, what will 1 ton cost, at the same rate? 12. How much cheese can be bought for of a dollar, at of a dollar a pound? 13. A man divided 2 dollars among his children, giving them of a dollar apiece; how many children had he? 14. How many times will of a gallon of beer fill a vessel holding of of a gallon? 15. How many times is of of 27 contained in of of 42 ? 16. If 5 bushels of potatoes cost 23 dollars, how much do they cost a bushel? 17. If John can walk 21 miles in of a day, how far can he walk in 1 day? 18. If a turkey cost 13 dollars, how many can be bought for 12 dollars? 19. At of of a dollar a yard, how many yards of ribbon can be bought for 2 of a dollar? 143. What is Case II.? What is the rule? |