3. How many times is 9. How many times is 10. How many times is 11. How many times is contained in 13? Ans. 38. contained in 17? Ans. 24. contained in 13 ? of contained in 4 of 21? 12. What is the quotient of of 4, divided by 10 of 31? 13. What is the quotient of of 7 of 36 divided by Ans. 31. 7 times? 31 SOLUTION. This example is only another form for expressing division of fractions; it is sometimes called a com plex fraction, and the process of performing the division is called reducing a complex fraction to a simple one. We simply reduce the upper number, or dividend, to an improper fraction, and the lower number, or divisor, to an improper fraction, and then divide as before. 43 14. What is the value of 7. = 7 OPERATION. 35 8 35 = Ans. Ans. 8. Ans. 20. Ans. 25 242 Ans. 19. If a horse eats of a bushel of oats in a day, in how many days will he eat 51 bushels? Ans. 14 days. 20. How many times will 43 gallons of oil fill a vessel that holds of of 1 gallon? Ans. 10 times. 21. If 14 acres of meadow land produce 32 tons of hay, how many tons will 5 acres produce? Ans. 113 tons. 13. What part of 24 is 13? OPERATION. 24 = 44; 13 = §. } = √14 of 14. 1 or 2=5×14 or 5 of 14. SOLUTION. - We first reduce the mixed numbers to improper fractions. 2 and 13. The problem now reads: What part of is? Proceeding as in Example = § = § of 54 or 25 of 112, we find the answer to be. 14 Or, 24=13; 13=28. 23 = 25 of 43. Or, changing both numbers to fractions with a common denominator, we may find what part the smaller numerator is of the greater. .. 133 of 23. RULE. found by the part. What part of: 14. is ? Ans. . 16. 15. § Divide the number of which the part is to be = 142. To find a number when a fractional part of it is given. 1. 180 is of what number? is? ‡ = OPERATION. the number = 1 of 180 = 60. 4 × 60 = 240, Ans. Or, the number of 180. 60 the number of 180-240, Ans. 3 whole number, will be 4 times RULE.-Multiply the number by the fraction with its terms inverted; or, divide the number by the fraction. 18. 2 is 11? 19. 7 is 14? Find the number of which: 2. 75 is . 4. 15 is 2. Since SOLUTION. 180 is of a certain number, of the number will be of 180, which is 60; and the number will be 4 times 60, which is 240. Hence, 180 is of 240. Or, the number will be of 180, and, or the of 180, which is 240. 6. 392 is 4. 7. 1 is . 8. 625 is f. 9. 450 is §. PROMISCUOUS EXAMPLES. 9 6 1. Reduce 1, §, 3, and 4 to equivalent fractions whose denominators will be 24. Ans. 12, 3, 24, 24. 2. Change to an equivalent fraction having 91 for its denominator. Ans. 249 49 3. Find the least common denominator of 4, 13, 1 of 1, 2, of of 15%. 4. Add 4, 7, ‡ of 11, 3, and 1. 5. Find the difference between 3 of 67 and 5 of 4,8%. Ans. 1138. 6. The less of two numbers is 47564, and their difference is 1283. What is the greater number? Ans. 4885 7. What is the difference between the continued products of 3, 3, 3, 4, and 33, 3, 4, f? Ans. 317. 4 21/ 8. Reduce the fractions and 11 form. 9. What number multiplied by will produce 18257? 10. A farmer had of his sheep in one pasture, in another, and the remainder, which were 77, in a third pasture. How many sheep had he? Ans. 140 sheep. 11. What will 74 cords of wood cost at of $9 per cord? Ans. $2411. 12. At of a dollar per bushel, how many bushels of apples can be bought for $57? 13. I paid $1837 for 7350 much was that per bushel? 14. If 235 acres of land cost $4725g, what will 628 acres cost? Ans. $12601. 15. A man, owning of an iron foundry, sold of his share for $5403. What was the value of the foundry? Ans. $40558. to their simplest bushels of oats. How Ans. of a dollar. 16. 142 less of 83 140 off of what number? Ans. 27. 17. A merchant bought 43 cords of wood at $31 per cord, and paid for it in cloth at § of a dollar per yard. How many yards were required to pay for the wood? of a yard wide, will Ans. 34 yards. is 18. How many yards of cloth, line 20 yards, 14 yards wide ? 19. If the dividend is 3, and the quotient, what is the divisor? 20. If the sum of two fractions is §, and one of them is, what is the other? Ans. 21. If the smaller of two fractions is 4, and their difference is, what is the greater? Ans. 22. If 3 pounds of sugar cost 33 cents, what must be paid for 65 pounds? 23. A certain sum of money is to be divided among 5 persons; A is to have 4, B, C, D, and E the remainder, which is $20. What is the whole sum to be divided? Ans. $50. 24. What number, diminished by the difference between and of itself, leaves a remainder of 34? Ans. 40. 25. If 324 bushels of barley can be had for 259 bushels of corn, how much barley can be had for 2000 bushels of corn? Ans. 2500 bushels. 26. If of a farm is valued at $1728, what is the value of the whole? 27. I bought 320 sheep at $2 per head; I afterwards bought 435 at $13 per head; then sold of the whole number at $13 per head, and the remainder at $21. Did I gain or lose, and how much? Ans. Lost $441. 28. If 5 is added to both terms of the fraction 7, will its value be increased or diminished? Ans. Increased 29. If 5 is added to both terms of the fraction, will its value be increased or diminished? Ans. Diminished |