265. To find any per cent. of a number. 1. What is 5 per cent. of 365 bushels? OPERATION. 365 Ans. 18.25 bu. Since 5 % is .05, 5 % of 365 bu. is .05 of 365 bu., or 18.25 bu. Therefore, etc. RULE. Multiply the given number by the rate per cent., expressed decimally. 10. Find 4 per cent. of 3 pk. 2 qt. SOLUTION. 3 pk. qt. =26 qt.; 26 qt. X .04 1.04 qt., Ans. 11. Find 45 % of 12 cwt. 2 qr. 20 lb. Ans. 5 cwt. 2 qr. 211⁄2 lb. 266. When the number of the per cent. is an aliquot part of 100, to find the percentage, we may Take the same part of the base that the rate per cent. is of 100. Explain the operation. What the Rule? When the number per cent. is an aliquot part of 100, how may the percentage be found? 12. What is 33% of 2880 hogsheads? 331 SOLUTION. 33% of 2880 hhd.= or, of 2880 = 960 hhd., Ans. 13. What is 25 % of $968.56? Ans. $242.14. Ans. 2 h. 12 m. 36 sec, 14. What is 50 % of 4 h. 25 m. 12 sec.? APPLICATIONS. 1. A farmer raised 3674 bushels of grain, and has sold 65 %, of it; how much has he left? SOLUTION. If he has sold 65% of it, he has left the whole, or 100%, less 65%, which is 35%; and 35% of 3674 bushels is 3674 × .35 : 1285.9 bushels, Ans. = 2. If cloth will shrink in sponging 5 per cent. of the length, what will be the shrinkage of a piece of cloth containing 43 yards before sponging? 3. If a man's salary is $2250 a year, and his expenses 87 per cent. of that sum, how much can he save yearly? Ans. $281.25. 4. A speculator bought 3160 barrels of produce, but on examining them, found 15% of the whole worthless; of the remainder he reserved for his own use 5 %, and sold what was left at $3 per barrel; how much did he get for what he sold? Ans. $7610.07. CASE II. 267. To find what per cent. one number is of another. 1. What per cent. is 18.25 of 365? 18.25 365 OPERATION. = = .05, or 5 %, Ans. and .05 = 5%, 18.25 is 5% of 365. RULE. Divide the number denoting the percentage by that denoting the base, and extend the division to hundredths. Explain the operation. Repeat the Rule. Examples. What per cent. is 2. 15 lb. of 50 lb.? Ans. 30 %. | 7. $28.47 of $657? 3. 57 men of 600 men? 4. $7.80 of $1040? Ans. %. 5. $782.80 of $760? 6. 90.848 of 801.6? Ans.11%. 8. 1994 rd. of 2850 rd.? Ans. 7 %. 9. $235.50 of $1500? Ans. 15% % 10. Find what per cent. 1.04 qt. is of 3 pk. 2 qt. SOLUTION. 3 pk. 2 qt. =26 qt.; 1.0426.04 4%. Ans. 11. Find what per cent. 5 cwt. 2 qr. 213 lb. is of 12 cwt. 2 qr. 20 lb. 12. Find what per cent. is of .625? Ans. 20 %. APPLICATIONS. 1. On a plantation containing 4000 trees, 20 trees died; what per cent. is that of the whole? Ans.% 2. If 43 yards of cloth shrink in length, by sponging, 2.365 yards, what is the shrinkage per cent.? 3. A farmer had last year 320 sheep, and this year 344; what is the increase per cent.? SOLUTION. If he had last year 320 sheep, and this year 344, the increase is 24 sheep; and 24 is 2 = .071=73% of what he had last year. 4. The population of a certain town was once 5600, but is now 4802; what per cent. is the decrease? Ans. 14%. 5. Mr. Wilson has paid $110 of a demand for $235; what per cent. remains unpaid? Ans. 537 %. 6. A farmer gave his son Thomas 100 acres of land, and his son Arthur the same quantity; but Thomas decreased his 50 %, and Arthur increased his 50 %; what per cent. then was Thomas' land of that of Arthur? Ans. 333 %. REVIEW QUESTIONS. What is meant by per cent.? (262) What is Percentage? (263) Percentage of a number? (263) The Base of percentage? (263) The Rate per cent.? (269) 268. To find a number when any per cent. of it is given. 1. $450 is 15 per cent. of what number of dollars? RULE. Divide the given percentage by the rate per cent. expressed decimally. Examples. 2. Of what number is 96 bushels, 6 %? Ans. 1600 bushels. 3. Of what number is 57 men, 9% ? 4. $375 is 30 % of what number? 5. 1.04 qt. is 4 % of what number? 6. $235.50 is 15% % of what number? 7. $7.80 is % of what number? 8. 7 lb. 11.2 oz. is 5 % of what number? Ans. $1250. Ans. 3 pk. 2 qt. Ans. $1040. Ans. 1 cwt. 2 qr. 4 lb. 269. When the number of the per cent. is an aliquot part of 100, to find the base, we may Consider the percentage the same fraction of the base that the rate per cent. is of 100. 9. 960 hogsheads is 333 per cent. of what number? SOLUTION. If 331%, or, of some number is 960 hogsheads, f, or that number, must equal 3 times 960, or 2880, hogsheads. 10. $242.14 is 25 % of what number? 11. 2 h. 12 m. 36 sec. is 50 % of what number? Ans. 4 h. 25 m. 12 sec. Explain the operation. What is the Rule? How may the base be found when the rate per cent. is an aliquot part of 100? APPLICATIONS. 1. From a cask of molasses there leaked out 10.08 gallons, which was 16 per cent. of what it contained; how much did it contain at first? Ans. 63 gallons. 2. The tax on a house is $8, which is % of its valuation; what is its valuation? Ans. $2000. 3. If a man, by saving 37% of his income, lays up $281.25 a year, what is his income? 4. If 17 bu. 2 pk. is 7% of a farmer's crop of grain, how much is the whole of it? Ans. 233 bu. 13 pk. 5. In a battle an army had 75 men killed, 93 wounded, and 112 taken prisoners, and the entire loss was 17%; what was the number at first? Ans. 1600 men. 6. A man took from a bank $393, which was 13% of what he had deposited; how much then remained? CASE IV. Ans. $2607. 270. To find a number, when a given number is a given per cent. more or less than the required number. 1. What number, increased by 10 per cent. of itself, is equal to 385 ? OPERATION. 10 385 × 100 350, Ans. 110 Or, 3851.10350, Ans. must be equal to 1.7 of 385 = A number increased by 10% of itself is equal to 110% of itself. If 110% of a number is equal to 385, 1% of it is equal to of 385, or 3.50, and 100%, or the number, is equal to 100 times 3.50, or 350. Or, since 110%, or 1.10, of some number is equal to 385, that number 3851.10, which is 350. RULE. Divide the given number, according as it may be more or less than the required number, by 1, increased or diminished by the rate per cent. expressed decimally. Explain the operation. Repeat the Rule. |
