230. When both factors are fractional numbers. ORAL EXERCISES. 1. A boy having of a melon, gave of it to his sister. What part of the melon did she receive? 5. If I own of an acre of land, and sell of it, what part of an acre do I sell? What part do I retain ? 6. If a yard of silk is worth $3, what is of a yard worth? 7. A boy having $2 gave of it for a knife. What part of a dollar did he pay for the knife? ANALYSIS. He paid of $4, or 5 times of $1. of $ is $4, and 5 times $ are 15, or $3. 24 8. 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 = × 9, etc. 10. What is ×? 11. A man owning of a mill, sold of his share to his brother. What part of the mill did each then own? 12. At $8 a barrel, what will of a barrel of flour cost? ANALYSIS.—It will cost of $83, or 3 times of $83. † of $8 is $21, and 3 times $25 are $63. Or, $8 equal $44, and of $44 = $33 = $63. 13. At $94 a case, what will of a case of slates cost? 14. At $5 a yard, what will 4 of a yard of cloth cost? 15. How much is & of 7 miles? of 244 pounds? 16. If a man travel 26 miles in a day, how far does he travel in of a day? In ? In &? In ? 17. At a yard, what will 64 yards of flannel cost? ANALYSIS.-It will cost 6 times $3. 6 times $3 are $4, and of $ is $, which added to $4, makes $4}. Or, 6125, and 25 of $3 = $25 = $41. 18. At $ a pound, what will 8 pounds of tea cost? 19. What will 93 pounds of beef cost, at $ a pound? 20. If a man hoe of an acre of corn in a day, how many acres can he hoe in 5 days? In 64 days? In 8 days? 21. What will 24 yards of silk cost, at $3 a yard ? ANALYSIS.—2}=}, and 3}=10; and 10 × 2=15=$7}. Reduce mixed numbers to improper fractions, and then proceed as in multiplying one fraction by another. 22. What cost 5 yards of alpaca, at $1 a yard? 23. Multiply 2 by 31; 71 by 12; 5 by 2§. 231. PRIN.-The product of a fraction by a fraction is such a part of either factor as the other is of a unit. WRITTEN EXERCISES. 232. 1. Multiply by 3. ANALYSIS. To multiply by } is to find of, which is 7 times of, and is found by multiplying the denominator (200, 1.) Hence of is, and of is 7 times, or is 15, the required product. In like manner multiply = 8. Find the product of, 1, 5, and 2. by 18. ANALYSIS.-Change the mixed number 5 to an improper fraction, and the integer 2 to the form of a fraction, then multiply as in Ex. 1. Find the value of the following expressions: Or, 316 2 op From the preceding principles and operations is derived the following general RULE.—I. Reduce all integers and mixed numbers to improper fractions. II. Multiply together the numerators, for the numerator; and the denominators, for the denominator of the product. Or, I. Multiply by the numerator of the fractional multiplier and divide by the denominator. II. When the multiplier is a mixed number, multiply by the fractional and integral parts separately and add their products. Cancel all factors common to numerators and denominators before multiplying, thus shortening the operation, and obtaining the answer in its lowest terms. Find the cost a yard. a bushel. 17. Of 15 cords of bark, at $4§ a cord. 27. Of of 156 acres of land, at § of $54% an acre. 28. Of 28 bushels of sweet potatoes, at $1 a bushel. 29. Of of a yard of satin, at $13 a yard. 30. Of 31. Of p of an acre of land, at $125 an acre. tons of middlings, at $264 a ton? 32. Paid $365 for a horse, and sold him for of what he cost. What was the loss? 33. When peaches are worth $3 a basket, what are 126 baskets worth? 34. Find the value of (129 — 76§) × 141⁄2 of 121 — 2} + 211 × 63. DIVISION. 233. When the divisor is an integral number. ORAL EXERCISES. 1. If of an acre of land is divided into 3 equal lots, what part of an acre does each lot contain? 2. of is what part of 1 ? of? of? of f1⁄2? 3. Dividing by 3, 4, and 5 is the same as multiplying by what fractions? 4. How much is ×? ÷3? $×1? &÷4? 5. If 4 slates cost $8, what will 1 slate cost? 15 6. If 5 boxes of figs cost $4, what will 1 box cost? 7. Divide of a barrel of flour equally among 3 families. What part of a barrel will each family receive? 8. What is of? divided by 8? 9. What is the quotient of divided by 2? by 3? 10. Show that dividing the numerator of by 3 divides the fraction by 3. 11. Show that multiplying the denominator of § by 3 divides the fraction by 3. How many ways to divide a fraction by an integer? by 7; § by 9. 12. Divide by 4; 1 by 5; 13. If 6 pounds of sugar cost $3, what will 1 pound cost? 14. Divide by of 20; 1 by % of 15; by of 40. 15. At $4 a yard, how many yards of silk can be bought for $214? ANALYSIS.—As many yards as $4 is contained times in $215, or of 215 = 4 of 152, or 38 = 5% times. == Or, 4 is contained in 215, 5 times and 15 or 12 remainder; of 2 is, which added to 5 makes 53. Hence, etc. |
