PERCENTAGE. 214. PERCENTAGE is an allowance made by the hundred, and is always a part of the number on which the allowance is made. THE BASE of percentage, is the number on which the percentage is reckoned. 215. PER CENT means by the hundred: thus, 1 per cent means 1 for every hundred; 2 per cent, 2 for every hundred ; 3 per cent, 3 for every hundred, &c. The numbers denoting the allowances, 1 per cent, 2 per cent, 3 per cent, &c., are called rates, and may be expressed decimally, as in the following 200 per cent is 2; for, 200 is equal to 2. per cent is .005; for, 2 is equal to .005. 3 per cent is .035; for, 31 = .03.005 = .035. 53 per cent is .0575; for, 53 .05+.0075 = .0575. 1. Write decimally, 91 per cent, and 83 per cent. 2. Write decimally, 12 per cent, and 97 per cent. 3. Write decimally, 208 per cent, 375 per cent, and 95 per cent? 4. Write decimally, 663 per cent. 214. What is percentage? What is the base? 215. What does per cent mean? What do you understand by 3 per cent? What is the rate, or rate per cent? 216. To find the percentage of any number. 1. What is the percentage of $450, the rate being 6 per cent? ANALYSIS.-The rate being 6 per cent, is expressed decimally by .06. We are then to take .06 of the base, $450; this we do by multiplying $450 by .06, giving $27. Hence, to find the percentage of a number, OPERATION. 450 .06 Multiply the number by the rate expressed decimally, and the product will be the percentage. EXAMPLES. 1. What is the percentage of $564, the rate being 5 per cent? NOTE.-When the rate cannot be reduced to an exact decimal, it is most convenient to multiply by the fraction, and then by that part of the rate which is expressed in exact decimals. OPERATION. 564 .051 188 per cent. 2840 5 per cent. $30285 per cent. Find the percentage of the following numbers: 12. 8 per cent of $3465,75. 13. 12 per cent of 126 cows. 9. 9 per cent of 186 miles. 10. 103 per cent of 460 sheep. 11. 5 per cent of 540 tons. 22. What is the difference and 7 per cent of $1500? 14. 50 per cent of 320 bales. 15. 37 per cent of 1275yds. 16. 95 per cent of $4573. 17. 105 per cent of 2500bar. 18. 1121 per cent of $4573. 19. 250 per cent of $5000. 20. 305 per cent of $1267,871. 21. 500 per cent of $3000. between 43 per cent of $1000 216. How do you find the percentage of any number? 23. If I buy 895 gallons of molasses, and lose 17 per cent by leakage, how much have I left? 24. A grocer purchased 250 boxes of oranges, and found that he had lost in bad ones 18 per cent: how many full boxes of good ones had he left? 25 A capitalist wishes to invest $25000; he invests 20 per cent in bank stock, 37 per cent in railroad stock, and the remainder in bonds and mortgages: what per cent, and what amount did he invest in the latter? 26. A man bought a house and lot for $3250; in three years time it increased in value 87 per cent: what was its value then? 27. A farmer having $1572,75, purchased cows with 25 per cent of it, sheep with 121 per cent of it, and lent 50 per cent of it to a friend; how much had he left? 217. To find the per cent which one number is of another. 1. What per cent of 64 is 16? ANALYSIS. In this example 16 is the percentage, 64 is the base, and we wish to find the rate. Since the percentage is equal to the base multiplied by the rate (Art. 216), the rate is equal to the percentage divided by the base; hence, OPERATION. 16 1 = .25 64 4 = .25; therefore, the rate is 25 per cent: hence, to find what per cent one number is of another, Divide the number denoting the percentage by the base, and the two first decimal places will express the rate per cent. NOTES.-1. The base is generally preceded by the word of. 2. There are three parts in percentage: 1st. The base; 2d. The rate; and 3d, their product, which is the percentage. 3. The percentage divided by the rate, gives the base; the percentage divided by the base, gives the rate. 217 How do you find the per cent which one number is of another? EXAMPLES. 1. What per cent of 10 dollars is 2 dollars? 2. What per cent of 32 dollars is 4 dollars? 3. What per cent of 40 pounds is 3 pounds? 4. Seventeen bushels is what per cent of 125 bushels? 5. Thirty-six tons is what per cent of 144 tons? 6. What per cent is $84 of $96? 7. What per cent is 275 of 440 ? 8. What per cent is 3 miles of 400 miles? 9. Eleven is what per cent of 800? 10. One hundred and four sheep is what per cent of a drove of 312 sheep? 11. A grocer has $325, and purchases sugars to the amount of $121,871: what per cent of his money does he expend? 12. Out of a bin containing 450 bushels of oats, 561 bushels were sold what per cent is this of the whole? 13. A merchant goes to New York with $2500; he first lays out 20 per cent for groceries, and then expends $1875 for dry goods what per cent of his money has he left? 14. Two persons invested in stocks $4500 each; one lost $562,50, and the other lost $405: what per cent more did one lose than the other? 15. A and B engage in different kinds of business with $5400 capital each; A gains $1350, and B loses $540 the first year: what per cent is B's money of A's? 218. To find the base when the percentage is added to or subtracted from the base. 1. Mr. Jones buys 8 hogsheads of sugar, sells them at an advance of 15 per cent, and receives $470: what did he pay for the sugar? 218. How do you find the base when the percentage is subtracted from the base? ANALYSIS. The amount received, $470, arises from adding the percentage to the base; that is, it arises from multiplying the base by 1 + plus the rate per cent; hence, to find the base, in such cases, OPERATION. 1.15) 470 ($400 470 Divide the given number by 1 plus the rate per cent, expressed decimally. 2. A cask of wine, out of which 37 per cent had leaked, was found to contain 33.39 gallons: how many gallons did the cask contain? lons as .63 is contained times in 33.39, viz., 53; hence; to find the base, in such cases, Divide the given number by the difference between 1 and the rate per cent, expressed decimally. EXAMPLES. 1. A farmer bought 40 sheep, and after keeping them for one year, sold them at an advance of 55 per cent, and received $248: what did he pay for the sheep per head? 2. A merchant bought a lot of goods and marked them at an advance of 26 per cent: when sold, he found that they brought him $6835,50: what did the goods cost him? 3. A son, who inherited a fortune, spent 37 per cent of it, when he found that he had only $31250 remaining: what was the amount of his fortune? 4. A grocer purchased a lot of teas and sugar, on which he lost 16 per cent. by selling them for $4200: what did he pay for the goods? |