20. When the dividend and the divisor are not similar fractions, how may the quotient be found? 21. How often is contained in 1? How often in ? in 1? in? in? in? in ? 22. How often is contained in ? 5 3 is con of 3 207. Analysis. Since 1 is contained in times, tained in 3 times times, or times; and is contained in times, or times. Hence, etc. 23. Find the quotient of; 3 ÷ 8; 9 ÷ 8; 17%. 24. If of a yard of cloth cost $, what will 1 yard cost? 25. Find the cost of 1 quart of wine at $1 for 3 of a quart. 26. If a mason can do of a piece of work in one day, in what time can he do of it? of it? of it? of a mile in 3 of an 27. If the current of a river moves hour, how far does it move in 1 hour? If it moves 4 of a mile? 28. If the quotient of is, how may the numerator of the quotient be found from the terms of the fraction? How may the denominator of the quotient be found? 29. Show in what two ways the quotient of a fraction divided by a dissimilar fraction may be found. 30. At $ a pound, how many pounds of tea can be bought for $23? 208. Analysis. — Since 1 pound of tea costs $1, for $23 there can be bought as many pounds as the number of times $3 are contained in $23. $23 equal $, and $ equal $§; $§ are contained in $19 as often as 6 is contained in 19, which is 33 times. Hence, etc. Or, (207). 31. What is the quotient of 33 days divided by § of a day? 32. How long will it take to earn $43 at $3 a day? 10 33. At $9% a yard, how many yards of cassimere must a merchant sell to receive $5? To receive $63? 34. If 7 years is of Harry's age, how old is he? 35. A teamster hauled 83 cords of wood in of a day. How many cords a day could he haul at that rate? 36. If a canal-boat averages 21⁄2 miles an hour, how long will it be in running 83 miles? 209. Analysis. — Since a canal-boat runs 23 miles an hour, to run 8 miles will require as many hours as the number of times 2 miles are contained in 8 miles. 23 miles equal 5 of a mile, and 83 miles equal 25 of a mile. Also, of a mile equal 15 of a mile, and 25 of a mile equal 50 of a mile; and of a mile are contained in 50 of a mile, etc. Hence, etc. Or, (207). 37. If a man can cut 1 cords of wood in a day, in how many days can he cut 4 cords? 6 cords? 10 cords? 38. How many weeks will 84 bushels of oats last a horse that eats 2 bushels a week? How long will 12,1 bushels last? 6 39. At $31 a pair, how many pair of shoes will $191⁄2 buy? How many pair can be bought for $223? For $321⁄2? 40. A farmer cut 12 tons of hay from a field that yielded 1 tons to the acre. How many acres were in the field? 41. If 63 bushels of wheat cost $104, what is the price of 1 bushel? If 7 bushels cost $114? 42. Explain how a fraction is divided by a mixed number. A mixed number by a mixed number. Find the quotient of | 43. divided by 2. 44. divided by 31. 45. 17 divided by 44. How often are 47. 31 feet contained in 63 feet? 48. 41 days contained in 21 days? 49. 63 hours contained in 10 hours? 50. 83 dollars contained in 4 dollars? 51. What number divided by gives a quotient 21 less than 41? A quotient 33 more than 51? 46. 24 divided by 55. 52. If the quotient is 7, the remainder is 1, and the divisor is 61, what is the dividend? If the divisor is 7? 53. If the dividend is 403, the quotient 7, and the remainder 5, what is the divisor? If the quotient is 8? 54. If the dividend is 523, the divisor 63, and the quotient 7, what is the remainder? If the divisor is 74? 55. The product of two factors is 333, and one of the factors is 61. Find the other factor. of the other factor. 56. If the product of two factors is 261, and one of the factors is of 33, what is of the other factor? 2338 57. The product of three factors is 50%, and two of the factors are 43 and 24. What is the third factor? 58. A newsboy bought 10 papers at 3 cents each, and sold them for 48 cents. What did he gain on each? 59. If a man earns $21 a day, and his expenses average $14, how many days must he work to save $20% ? 60. A farmer having $25, gave of his money for cloverseed at $33 a bushel. How many bushels did he buy? 210. Principle. Multiplying the dividend multiplies the quotient, and dividing the dividend divides the quotient. What is the quotient of 211. Model. Since Exercises. divided by §? 1 is contained in times, is contained in 6 times, or 42; and is contained in of 42 times, or times, or times, or 1 times (207). Or, by changing the fractions to similar fractions (206). Section VII. RELATIONS OF NUMBERS. 1. To Find what Part One Number is of Another. 1. 2 is how many times 1? 3 is how many times 1? 2. How much is 1 divided by 2? 1 divided by 3? 1÷4? 3. 1 is what part of 2? What part of 3? Of 4? Of 5? Of 6 ? 4. 2 is what part of 3? 4, of 7? 212. Analysis. Since 1 is of 3, 2 is 2 times, or, of 3. 5. 3 is what part of 5? 5, of 8? 6, of 9? 6. What part of 7 is 5? Of 9 is 7? Of 10 is 6? Of 12 is 8? 7. 6 tons are what part of 8 tons? 8 pounds of 14 pounds? 10 cents of 25 cents? 9 eggs of a dozen? $20 of $50? 8. What part of 12 feet are 6 feet? Of 18 yards are 9 yards? Of 25 miles are 18 miles? Of 40 bushels are 25 bushels? 9. If an acre of land can be bought for $50, what part of an acre can be bought for $5? For $10? For $25? 10. 12 inches are a foot, and 3 feet are a yard. What part, then, of a yard are 9 inches? 18 inches? 24 inches? 11. What part of 5 is ?? 213. Analysis. Since 1 is of 5, is of of 5, or 2 of 5; and is 3 times 2 of 5, or 2 of 5. Hence, etc. 5 12. is what part of 6? %, of 7?, of 8? 7, of 10? 13. What part of 12 is? Of 13 is ? Of 15 is 1% ? 14. 2 is what part of 6? 3, of 10? 11, of 15? 61, of 25? 15. $2 are what part of $3? 2 feet, of 12 feet? 41 miles, of 15 miles? 63 miles, of 30 miles? 71⁄2 miles, of 45 miles? 16. What part of 12 dozen are 2 dozen? 31 yards? Of 25 gallons are 8 gallons? Of 20 yards are Of $30 are $91? 17. Frank has 12 dollars, and Henry has $93. Henry's money is what part of Frank's? 20. What part of is? Of is ? Of is ? 21. of an ounce are what part of 2 ounces? ğ of a mile of 3 miles? of a ton, of 51⁄2 tons? 22. What part of 61 cents are 33 cents? Of $7 are $21⁄2? Of 163 cents are 64 cents? Of $183 are $12!? 23. What part of 15 days are of 15 days? of 51 days? of 10 days? of 12 days? of 93 days? 24. If 5 yards are one rod, what part of a rod are 23 yards? What part are 33 yards? 215. The relation of one number to another is the value of one number compared with the value of another; or, It is the number of times one of two numbers contains the other, or the part one number is of the other. 216. General Principle. Only similar numbers, or numbers of the same kind, can be compared with each other. Exercises. 1. What is the relation of 12 to &? 217. Model. - The relation of 12 to is the quotient of 12 divided by ; 12 divided by is 48, and 12 divided by is of 48, or 16. 2. of is what part of 13? 218. Model. divided by 1 is of 1. by 1, etc. of is such part of 13 as the quotient of of of equals, and 14 equals ; divided |