235. To find the percentage, amount, and difference of any number. 1. What is 4% of $125, and what are the amount and difference? = $5.00, Percentage. Or, of $125 $5, Percentage. $5-$120, Diff. $125 +$5-$130, Amt. $125 Or, 100% +4% 104% = 1.04 of $125 $ 130, Amt. = SOLUTION.4% of $125 = .04 of $125, $125 $5, percentage. = Since the amount is the base plus the percentage, it is $125 + $5 $130; and since the difference is the rate minus the percentage, it is $125 $5 $120. Or, the base 100% + the rate 4% = 104%, the amount expressed in per cent. 104% of $125 = 1.04 of $125 $130, Amt.; and the base 100% - the rate 4% = 96%, the difference expressed in per cent. 96% of $125 = .96 of $125 : $120, Diff. = RULE.I. Multiply the given number or quantity by the rate per cent, expressed decimally, and point off as in decimals; or, reduce the decimal to a common fraction, aud take that fractional part of the given number, and the result will be the percentage. II. Add the percentage and base to find the amount, and subtract the percentage from the base to find the difference. Or, To find the amount or difference directly from the base; add the rate per cent to 100% and multiply the base by this per cent, expressed decimally, to find the amount, and subtract the rate from 100% and multiply the base by the result to find the difference. 2. What is 6% of $320 ? Ans. $19.20. Ans. $4.11. 4. What are the amount and difference of $320 at 6%? Ans. Amt. $339.20; Diff. $300.80. 5. What is 2450 pounds +12% ? 6. What is 846 gallons less 33%? Ans. 564 gallons. 7. What is 105% of $5760? Ans. $6048. Ans. 25% equals 0%, and × 1 = 32. 8. What is 25% of 7? 9. What is 15% of ? 10. What is 21% of 68? Ans. Ans. 11. A man has a capital of $24500; he invests 18% of it in bank stock, 30% of it in railroad stocks, and the remainder in bonds and mortgages. How much does he invest in bonds and mortgages? Ans. $12740. 12. A speculator bought 1576 barrels of apples, and upon opening them he found 121% of them spoiled. How many barrels did he lose? 13. Two men engaged in trade, each with $2760. One of them gained 331% of his capital, and the other gained 75%. How much more did the one gain than the other? 236. To find what per cent one number is of another. 1. A man, having $125, lost $5. What per cent of his money did he lose? OPERATION. 5+125.04 = 4%. Or, SOLUTION. He lost his money. of Reducing this to a decimal, we have .04 4% loss. We divide the percentage by the base to obtain the rate per cent. Or, since $125 is 100% of his money, $5 is 15, or 2 of 100%" which is 4%. 125 2 of 100% =4%. RULE. Divide the percentage by the base, and the quotient will be the rate per cent expressed decimally. Or, Take such a part of 100% as the percentage is part of the base. 2. What per cent of $450 is $90? Ans. 20%. Ans. 121%. Ans. 51%. 7. What per cent of 6 bushels 1 peck is 4 bushels 2 pecks 6 quarts? Ans. 75%. 8. What per cent of 15 pounds is 5 pounds 10 ounces, avoirdupois weight? Ans. 371%. 9. What per cent of 250 head of cattle is 40 head? 10. From a hogshead of sugar containing 760 pounds, 100 pounds were sold at one time, and 90 pounds at another. What per cent of the whole was sold? 11. A man, having 600 acres of land, sold of it at one time, and of the remainder at another time. What per cent remained unsold? Ans. 50%. 237. To find a number when a certain per cent of it is given. 1. A man lost $5, which was 4% of all the money he had. How much had he at first? OPERATION. $5.04 $125. Or, = SOLUTION. - We are here required to find the base of which $5 is the percentage. Since the percentage equals the base multiplied by the rate per cent (235), the base must equal the percentage divided by the rate per cent. Dividing $5 by .04, we have $125, the base. Or, since $5 is 4% of all he had, 1% of all he had will be of $5, or $, and 100% (or all he had) will be 100 times $ = $125. $x 100 = $125. RULE. Divide the percentage by the rate per cent, expressed decimally, and the quotient will be the base, or number required. Or, Take as many times 100 as the number expressing the percentage is times that expressing the rate per cent. 2. 16 is 8% of what number? Ans. 200. Ans. 600. Ans. 1200. 6. $281.25 is 371% of what sum of money? Ans. $750. 7. A farmer sold 50 sheep, which was 20% of his whole flock. How many sheep had he at first? 8. I loaned a man a certain sum of money; at one time he paid me $59.75, which was 121% of the whole sum loaned to him. How much did I loan him? 9. A merchant invested $975 in dry goods, which was 15% of his entire capital. What was the amount of his capital? Ans. $6500. 238. To find a number when the number, increased by a certain per cent of itself, is given. 1. A man's income this year is $525, which is 5% more than it was last year. What was it last year? OPERATION. $525 ÷ 105 $5 x 100 = $500. come, Or, $525 1.05 = $500. SOLUTION. Since his income this year is .05 more than it was last year, this year's income must be 1.05 times the income of last year. Since $525 is 185 of last year's in 100 100% will be of $525 = $5, and 18%, or his last year's income, will be 100 times $5, or $500. Or, we divide this year's income by 1.05 and it gives the income of last year. RULE. Divide the amount by 1 plus the rate expressed decimally, and the quotient will be the base or number required. 2. What number increased by 18% of itself is equal to 1475 ? Ans. 1250. 3. A merchant sells broadcloth for $4 per yard, and thereby gains 25%. What did the cloth cost him? 4. A expended a certain sum for a house, and 15% of the purchase price on repairs, and then found that the whole cost was $6900. What was the purchase price? 5. A certain manufacturing company has sold this year $432250 worth of goods, which is 8% more than it sold last year. How much did it sell last year? Ans. $400000. 239. To find a number when the number, diminished by a certain per cent of itself, is given. 1. A man lost 8% of his sheep and had 368 left. How many had he at first? SOLUTION. left; hence 368 is OPERATION. 368÷ 92 = 4 × 100 = 400. 100 Since the man lost 8% of his sheep, he has 92% 92% of his original flock. I will be of 368 his original flock, will be 100 times 4 = 400. Or, we divide the remainder, 368, by .92 and obtain the required number of sheep. = 4 and 100, or RULE. - Divide the given number by 1 minus the rate expressed decimally, and the quotient will be the base or number required. 2. What number diminished by 15% of itself is equal to 340? Ans. 400. 3. A having a certain sum on deposit drew out 20%, and then found he had $1000 left. How much had he on deposit at first? Ans. $1250. 4. My income this year is $4028, which is 24% less than it was last year. How much was it last year? 5. What number diminished by 1% of itself is equal to 298? Ans. 300. |