= of what number? 9. 3 Solution. — If = 35 of some number, of that number must equal of, which may be expressed by making 35 a factor of the denominator, and 36, or the number, must be 36 times the last product, which may be expressed by making 36 a factor of the numerator. 'This gives the following written work: : Solution. Since 8372 = .07 of some number, .01 of that number must equal of 8372, which is 1196, and 188, or the number, must equal 100 times this result, which, found by removing the point two places towards the right, is 119600. Of what number — 19. Does 79.84.004? 18. Does 2.475 = .03? 20. Does .58674.0011': 146. Practical Problems. 1. If of a yard of cloth cost $3.74, how much will 1 yard cost? Solution.. If of a yard of cloth cost $3.74, of $3.74, which is $1.87, and †, or a yard, will which is $5.61 = Ans. of a yard will cost cost 3 times $1.87. The work may be written in either of the following forms: Second Form. 2) $3.74 $1.87 $5.61 Proof-If 1 yard of cloth costs $5.61, of a yard will cost of $5.61, which is $1.87, and of a yard will cost two times $1 87, which is $3.74. 2. If of an acre of land cost $8.54, what will an acre cost! of a vessel cost $17645, what will the vessel cost? of a cask of oil is 143 gallons, how many gallons are there in the cask? 3. If 4. If 5. If .09 of a lot is worth $594.72, how much is the lot worth? 6. If .13 of a cargo of coal is worth $213.46, how much is the cargo worth? 7. If of a pound of tea is worth of a dollar, what is a pound of tea worth? Solution. Since the answer is to be in dollars, we write of a dollar as the number to be operated upon. If of a pound of tea cost so much, of a pound will cost of this, which may be expressed by making 5 a factor of the denominator; and 7, or a pound, will cost 7 times this result, which may be expressed by making 7 a factor of the 8. If of a bale of cotton weighs 18 of a ton, how much will the bale weigh? 9. Bought of a gallon of oil for 3 of a dollar. How much would a gallon have cost at the same rate? 10. If of a yard of silk cost $1.572, what will of a yard cost? Solution.- Since the answer is to be in dollars, we first write $1.572 as the number to be operated upon. If of a yard cost $1.572, † of a yard will cost of $1.572, which may be expressed by writing 8 under $1.572 as a denominator, and, or a yard, will cost 9 times this result, expressed by making 9 a factor of the numerator. If 1 yard costs so much, of a yard will cost of this, found by making 20 a factor of the numerator, and 27 a factor of the denominator. This gives the following written work: 2 3 NOTE. Questions like the above can always be resolved into two or more simple ones. Thus If of a yard of silk costs $1.572, what will 1 yard cost? If 1 yard costs the last result, what will yard cost? of a 11. How much will of a cord of wood cost, if § of a cord cost $3.85 ? 12. If a ship sails 157 miles in of a day, how miles will she sail in 8 days? many 13. A man sold 84 tons of iron for $384, and afterwards sold 93 tons at the same rate. the last lot? Suggestion. How much did he receive for Reduce the mixed numbers to improper fractions. The question then will read, A man sold of a ton of iron for $384, and afterwards sold 5 of a ton, &c. 14. How much must be paid for 5 acres of land, when $868 are paid for 2 acres ? 15. If 6g barrels of flour cost $38.25, how much will 14ğ barrels cost? 16. If of a dollar will purchase 7 lb. of coffee, how many pounds can be purchased for $51? 80 Solution. Since the answer is to be in pounds, we write 71, or , as the number to be operated on. If of a dollar will purchase lb. of coffee, of a dollar will purchase of lb., and §, or a dollar, will purchase 8 times this result, and $54, or $41, will purchase of this last result. The work, then, would be written thus: 17. If 12 yards of calico are given for 17 yards of sheeting, how many yards of calico must be given for 25 yards of sheeting? Suggestion. — Observe that as the answer is to be yards of calico, the number of yards of calico is that to be operated on. 18. If 17 tons of hay cost $227, how much will 9 tons cost? 19. If 13 lb. coffee are worth as much as 3g lb. of tea, how many pounds of coffee are worth as much as 7 lb. of tea? 20. If 8 yards of silk are worth as much as 24 yards of gingham, how many yards of gingham can be obtained for 574 yards of silk? 21. When 34 yards of silk, worth $14 per yard, are given for 13 yards of broadcloth, how many dollars should be given for 1 yards of broadcloth? 147. Division by Fractions. (a.) Since 1=? ==ˋ =, &c., it follows that there must be two times as many halves, three times as many thirds, four times as many fourths, &c., as there are ones in any number. Or, which is the same thing, (b.) Since 12 × 1, = 3 × 1,= 4 × 1, &c., it follows that there must be twice as many times, three times as many times, four times as many times, five times as many times, &c., as there are times 1 in any number. (c.) But the quotient of a number divided by 1 equals the number itself; hence, the quotient of a number divided by must equal twice the number; divided by must equal three times the number; by 4, four times the number; &c. NOTE. This is but an application of the principle, that if one number contains another a certain number of times, it will contain half shat number twice as many times; of it three times as many times of it four times as many times; &c. 24 ÷ 12 = 2; 24 of 12, or 6, = 2 × 2, or 4. First Solution. 9 divided by 1 = 9; hence, divided by, it must equal 4 times 9, or 36. Second Solution. Since 9 contains 1, 9 times, it must contain 1, 4 times 9 times, or 36 times. ? 12. What is the quotient of First Solution. } divided by 1 = , and divided by must equal 6 times, which, by cancelling, = 3 times = 21, = 54. Second Solution. Since contains 1, = = times, it must contain 1, 6 times times, 3 times = 5 times. By either form of solution, the work may be written thus: — "4= 16. Of 18 18 ? 17. Of 187÷&? 18. Of 720? 1728 These questions are really equivalent to, "9 = how many fourths?" how many thirds?" (See 126, 16th Example.) |