SECTION VIII. PERCENTAGE. 509. Percentage is the process of computation in which the basis of comparison is a hundred. 510. The Term per cent.—from per, by, and centum, a hundred -means by or on the hundred ; thus, 6 per cent. of any quantity means 6 of every hundred of the quantity. 511. The Symbol of Percentage is %. The per cent. may also be indicated by a common fraction or a decimal; thus, 6%=16.=.06. 512. The Quantities considered in percentage are the Base, the Rate, the Percentage, and the Amount or Differ ence. 513. The Base is the number on which the percentage is computed. 514. The Rate is the number of hundredths of the base which are to be taken. 515. The Percentage is the result obtained by taking a certain per cent. of the base. 516. The Amount or Difference is the sum or difference of the base and percentage. They may both be embraced under the general term Proceeds. Note.-In computation the rate is usually expressed as a decimal. For the difference between Rate and rate per cent., see Brooks's Philosophy of drithmetic. OPERATION. EXPRESSION OF THE RATE. 1. Express 5% as a decimal and a common fraction. SOLUTION.–Since per cent. is so many on a hundred, 5% of a quantity is .05 of it; or, as a 5%=.05=1őv=z common fraction, Tó, or zo of it. Express 5. 10%. Ans. 10, or 1o. 3. 8% Ans. .08, or 25. 6. 12%. Ans. .12), or 5. 4. 4%. Ans. .04, or 25. 7. 163%. Ans. .163, or . 8. 11:%. Ans. .11), or . 12. 1%. Ans. .005, or to 9. 37%. Ans. .37, or . 13..004 Ans. 11% 10. 83%. Ans. .083, or so. 14. .028. Ans. 23%. 11. 429%. Ans..429, or . 15. .0025. Ans. %. 517. Cases.-There are Three Cases, as follows: 1. Given, the rate and the base, to find the percentage or the proceeds. 2. Given, the rate and the percentage or the proceeds, to find the base. 3. Given, the base and the percentage or the proceeds, to find the rate. Notes.-1. Authors usually present the subject in five or six cases, but it is thought that the method here adopted is to be preferred, on account of its logical accuracy and practical convenience. 2. A percentage deducted from the price of goods is called a Discount. Successive discounts, called Trade Discounts, are often taken off, as “ 10 and 5% off,” meaning 10% off and 5% off of the remainder. CASE I. 518. Given, the base and the rate, to find the percentage or the proceeds. 1. What is 25% of $480 ? 480 the number, 25% of $480 equals .25 times $480, which, .25 by multiplying, we find to be $120. $120.00 2. What is the amount of $480 increased by 25% of itself? OPERATION, SOLUTION.-A number increased by 25%, or . 25 $480 times itself, equals 1.25 times itself; 1.25 times $480 1.25 equals $600. $600.00 Rule 1.- Multiply the base by the rate, to find the percentage. Rule II.-Multiply the base by 1 plus the rate, to find the amount; or by 1 minus the rate, to find the difference. NOTES.-1. The method of solving by reducing the rate to a common fraction is simpler when the rate gives a small common fraction. 2. The amount equals the base plus the percentage; the difference equals the base minus the percentage. What is 3. 8% of $500 ? Ans. $40. 5. {% of $75? Ans. $.65 4. 25% of $960 ? Ans. $240. 6. 331% of 54 ? Ans. 18. OPERATION. 7. 124% of $900 ? Ans. $112.50. 8. 35% of $248 ? Ans. $86.80. 9. 663% of 596 lb.? Ans. 397{ lb. 10. % of 627 yd.? Ans. 5.016 yd. 11. 424 % of 343 acres ? Ans. 147 A. 12. 4541% of $165 ? Ans. $75. 13. I sold a lot of envelopes marked $7.20 ° M., at 10 and 10% off; what did I receive ? Ans. $5.83. 14. William bought a lot of base-balls at $12 a dozen and sold them for 20 and 5% on; what did he receive ? Ans. $15.12. 15. What is the difference between 20% off and 10 and 10% off; between 15% on and 10 and 5% on? Ans. 1% ; *%. 16. A clerk's salary is $2500 a year; if he pays 15% for board, 8% for clothing, 5% for books, and 12% for incidentals, how much will he save in a year? Ans. $1500. 17. A bought a house for $2500, and paid 62% of the price in cash, and gave a mortgage for the remainder; what was the amount of the mortgage ? Ans. $937.50. 18. A man contracted a debt of $570; he paid 331% of it the first quarter, 25% of the remainder the second quarter, and 163% of what was still due the third quarter; how much remained unpaid ? Ans. $237.50. 19. Mr. Martin deposited $1250 in bank; he drew out 15% of it the first month, 20% of the remainder the next month, and having realized 184% on what he had drawn, deposited it; what was his bank deposit then ? Ans. $925. 20. If 48% of whisky is alcohol, how much alcohol does a man swallow in 35 years who drinks half a gill of whisky 6 times a day? Ans. 574.875 gal. 21. A mechanic contracts to supply dressed stone for a church for $87,560, if the rough stone cost him 18 cents a cubic foot; but if he can get it for 16 cents a cubic foot, he will deduct 5% from his bill; required the number of cubic feet, and the charge for dressing the stone. Ans. 218900 cubic feet; charge 22 cents per cu. ft. CASE II. OPERATION. OPERATION. 519. Given, the rate and the percentage or the proceeds, to find the base. 1. 75 is 5% of what number? SOLUTION.—If 75 is 5% of some number, then .05 times some number equals 75; if .05 75;.05=1500 times some number equals 75, the number equals 75:-.05, which is 1500. 2. What number, increased by 25% of itself, equals 450, or diminished by 25% of itself, equals 270 ? SOLUTION.-A number, increased by 25% or .25 of itself, equals 1.25 times the number; 450 = 1.25= 360 and if some number multiplied by 1.25 equals 450, the number equals 450:-1.25, or 360. A number diminished by 25% of itself, equals .75 times the number; and if some number multiplied by .75 equals 270, the number equals 270.75, or 360. Rule I.-- Divide the percentage by the rate, to find the base. Rule II.—Divide the amount by 1 plus the rate, or the difference by 1 minus the rate, to find the base. 3. 96 is 564% of what number? Ans. 1703 4. 101 is 683% of what number? Ans. 14611 5. 3752 is 811 % of what number? Ans. 462 6. 784 is 831% of what number? Ans. 940.8. 7. Bought 2 doz. cast steel riveting hammers @ $32.061 at 25 and 5% off; what was the marked price? Ans. $45. 8. Sold 200 carriage bolts @ $2.91 p C. for 10 and 5% on; what was the cost ? Ans. $2.52. 9. The fraction il is 5% more or 4% less than what fractions ? Ans. 44, or 35 10. I draw 45% of my bank deposit to pay a note of $5670; what did I have at first ? Ans. $12600. 11. A man collecting $75 of the money due him, increases his funds 163%; how much had be at first? Ans. $450. 12. In 99gal. of alcohol the water is 81% of the spirits; how many gallons are there of each ? Ans. 92; 74 13. A farmer's crop of oats this year is 71% greater than his crop of last year; what was this year's crop if in the two years he raised 725 bushels ? Ans. 34933 bushels. 14. A lady's cloak cost $40; the making cost 333% less than the cloth, and the trimmings 25% more than the cloth; what did each cost ? Ans. Cloth, $13%; making, $91; trimmings, $177. 15. A and B together have 1320 acres of land, 314% of A's equaling 373% of B's, and 561% of B's equaling 66% of C's; how much land has C? Ans. 5064 acres. 16. Mr. Howard drew 75% of his money from bank, and paid 871% of it for a house worth $5600; how much money had he remaining in bank? Ans. $2133.33 17. In an engagement 5% of an army were killed, 121% of the remainder were wounded, and 163% of the wounded died; there were 290 more killed than mortally wounded; how many men were in the army? Ans. 9600. 18. A, wishing to sell a cow and horse to B, asked 150% more for the horse than the cow; he then reduced the price of the cow 25%, and the horse 331%, at which price B took them, paying $290; what was the price of each? Ans. Cow, $90; horse, $200. 19. In building a church, the trustees paid three times as much for material as for labor; had they paid 43% more for material and 7% more for labor, the church would have cost $14700; what was its cost ? Ans. $14,000. CASE III. OPERATION. 520. Given, the base and the percentage or the proceeds, to find the rate. 1. 24 is what per cent. of 96 ? SOLUTION.-If 24 is some per cent. of 96, then 96 multiplied by some rate per cent. equals 24; if 96 24:-965.25 multiplied by some rate equals 24, the rate equals 24 divided by 96, which is .25 or 25%. Rule I.—Divide the percentage by the base, to find the rate. Rule II.-Divide the difference between the proceeds and base by the base, to find the rate. NOTE.—The rate may also be found by dividing the proceeds by the base and taking the difference between 1 and the quotient. |