9. At $3 a box, what will 4 boxes of raisins cost? ANALYSIS. $3} = $15, and 4 times $15= $15. Or, 4 times $4 = $3, and 4 times $3 = $12; $12+ $3 = $15. 10. Multiply 6 by 4; by 5; 101⁄2 by 4; 94 by 6. 11. At $8 a barrel, what is the cost of 3 barrels of flour? 12. What is of 12 pounds? of 24 men? of $40? 13. Multiplying by 1, 1, 1, 1, 1, etc., is the same as dividing by what integers? When a fractional part of an integer, or of a fraction, is to be taken, the word of, and not times, should be used. 14. At $7 a ton, what will of a ton of coal cost? ANALYSIS. It will cost of $7, or 3 times of $7. of $7 is $13, and 3 times $1 are $51. 15. If an acre of land produce 40 bushels of corn, how much will of an acre produce?? j? $? ? 10 16. What is of $5? Of $7? of $16? # of $20 ? 17. Multiply 6 by ; 8 by ; # by 8; 15 by ; 24 by . Show that a fraction of an integer equals the product of the integer by the fraction. How is an integer multiplied by a fraction? 18. At $12 a ton, what will 5 tons of cheese cost? ANALYSIS.-It will cost 5 times $12. 5 times $12 are $60, and of $12 are $43, which added to $60 make $641. 19. If a man earn $4 a day, how much will he earn in 6 days? In 5 days? In 8 days? In 10 days? 20. What is of 6 tenths? of 9 twelfths? 21. Which is greater, of, or of? 22. If I own farm do I sell? of 8 ninths? of 4, or of ? of a farm, and sell of it, what part of the What part do I retain ? 23. A boy having $4, gave § of it for a knife. What part of a dollar did he pay for the knife? ANALYSIS.-He paid of $, or 5 times of $4. of $4 is $4, and 5 times $ are 14, or $5. 24. If a pound of coffee cost $3, what cost of a pound? 25. At $ a gallon, what will of a gallon of syrup cost? Fractions with the word of between them are sometimes called compound fractions. The word of is equivalent to the sign (×) of multiplication. Thus, { of } = { × }; of 9 x9, etc. = 27. What is x 8 ? 7 % × 8 ? 8 × 12? J× 14? 28. A man owning of a mill, sold of his share to his brother. What part of the mill did each then own? 29. At $84 a barrel, what will of a barrel of flour cost? ANALYSIS. —It will cost of $84, or 3 times of $83. of $81 is $21, and 3 times $21 are $63. 30. At $91 a case, what will of a case of slates cost? 31. At $5 a yard, what will ✯ of a yard of cloth cost? 32. At $ a yard, what will 61 yards of flannel cost? ANALYSIS.—It will cost 6 times $3. 6 times $3 are $4, and of $} is $, which added to $4 makes $43. 33. If a man hoe acres can he hoe in 5 of an acre of corn in a day, how many days? In 63 days? In 8 days? 34. What will 24 yards of silk cost, at $34 a yard? ANALYSIS. 21 = 2, and 31 = 10; and 10×2=15=$73. Reduce mixed numbers to improper fractions, and then proceed as in multiplying one fraction by another. 119. PRINCIPLES.-I. A fraction is multiplied by multiplying its numerator, or by dividing its denominator by any integer. II. The product of two or more factors, whether integral or fractional, is the same, in whatever order they are used. WRITTEN EXERCISES. 120. All operations in multiplication of fractions may be reduced to one uniform and simple process, and often much abbreviated by cancellation. Since the analyses and principles of the Inductive Exercises are equally applicable to similar written exercises, further explanations are unnecessary. All factors common to the numerators and the denominators should be cancelled before multiplying. It is obvious, that in multiplying a fraction by an integer, or an integer by a fraction, the result is the same. (PRIN. II.) RULE.-1. Change integers and mixed numbers, if any, to the form of fractions, and write the numerators on the right, and the denominators on the left of a vertical line. (74, N. 2). 2. Cancel equal factors, if any. Then the product of the remaining factors on the right will be the numerator, and the product of the remaining factors on the left the denominator of the required product. When the multiplier is a mixed number, and the multiplicand large, and an integer, we may multiply by the integer and fractional part separately, and add their products. Find the cost 37. Of of a yard of flannel, at 83 a yard. 39. Of 8 40. Of 21 42. Of 9 pounds of coffee, at 284 cents a pound. 43. Of 56 pounds of cotton, at 183 cents a pound. 44. Of of 5 pounds of wool, at 47 cents a pound. 45. Of of 9 pounds of honey, at of 182 cents a pound. 46. Of 15 cords of bark, at $4 a cord. 47. Of 24 pounds of tea, at $3 a pound. 48. Of 80 yards of cloth, at $4 a yard. 49. Of 213 bushels of corn, at $1o a bushel. 50. Of of 5 tons of hay, at $15 a ton. 51. Of 18 barrels of crude oil, at $74 a barrel. 52. Of of 18 yards of silk, at 4 of $5 a yard. 53. Of 126 pounds of beef, at 94 cents a pound. 54. Of 36 tons of railroad iron, at $62 a ton. 55. Of 35 horses, at $2053 each. 56. Of of 156 acres of land, at of $54% an acre. 57. Of 28 bushels of sweet potatoes, at $13 a bushel. 58. Of of a yard of satin, at $18 a yard. 59. Of of an acre of land, at $125 an acre. 60. Of tons of middlings, at $264 a ton. 61. Paid $365 for a horse, and sold him for of what he cost. What was the loss? 62. When peaches are worth $3 a basket, what are 1261 baskets worth? 63. What is the cost of 4 of 6 tons of coal, at of $7 per ton? 64. If a family consume 14 barrels of flour a month, how many barrels will 5 such families consume in 4 months? 65. Find the value of (129-76) × √3 of 121−2}+(21§ × 67). |