SECTION VIII.

PERCENTAGE.

376. Percentage is the process of computation in whic} the basis of comparison is a hundred.

377. The Term per cent.-from per, by, and centum, ú hundred-means by or on the hundred; thus, 6 per cent. of any quantity means 6 of every hundred of the quantity.

378. The Symbol of Percentage is %. The per cent. may also be indicated by a common fraction or a decimal; thus 6% 18.06. = 00

379. The Quantities considered in percentage are the Base, the Rate, the Percentage, and the Amount or Differ.

ence.

380. The Base is the number on which the percentage is computed.

381. The Rate is the number of hundredths of the base which are taken.

382. The Percentage is the result obtained by taking a certain per cent. of the base.

383. 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 Arithmetic.

EXPRESSION OF THE RATE.

1. Express 4% as a decimal and common fraction.

SOLUTION. Since per cent. is so many on a hundred, 4% of a quantity is .04 of it; or, as a common fraction, or of it.

OPERATION.

4%=.0413=15.

384. Cases. The subject of percentage is conveniently treated under three distinct cases:

1. Given the rate and base, to find the percentage or pro ceeds.

2. Given the rate and percentage or proceeds, to find the

base.

3. Given the base and percentage or proceeds, to find the

rate.

NOTE.-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.

CASE I.

385. Given, the base and the rate, to find the percentage or the proceeds.

MENTAL EXERCISES.

1. What is 25% of 120 yards?

SOLUTION.-25% of anything is or of it; and of 120 yards is 30 yards. Therefore, etc.

10. Out of a purchase of 120 dozen eggs, 20% turned out to be bad; how many were good?

11. From a hogshead of kerosene containing 108 gallons, 33% leaked out; how many gallons remained?

12. A train of cars running 20 miles an hour, increases its speed 15%; what is the rate of running after the increase?

13. A clerk's salary is $45 a month, but at the beginning of the year it was raised 11%; what did he then receive a month?

14. Mr. Smith paid a tax of 4% on $3000; what was the amount of his tax?

15. In the 10th problem, which is the base, which the rate. and which the percentage?

WRITTEN EXERCISES.

1. What is 6% of $275? increased by 6% of itself?

What is the amount of $275,

SOLUTION.-6% of $275 equals .06 times $275, which, by multiplying, we find to be $16.50.

SOLUTION.-A number increased by 6%, or .06 times itself, equals 1.06 times itself; 1.06 times $275 equals $291.50.

OPERATION.

$275

.06

$16.50

OPERATION.

$275

1.06

$291.50

Rule I.-Multiply the base by the rate, to find the per centage.

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. When the rate gives a small common fraction, take such a part of the base as is indicated by this fraction.

2. The amount equals the base plus the percentage; the difference equals the base minus the percentage.

14. A grain dealer bought 600 bar. of Western flour, and sold 16% of it; how many barrels remained? Ans. 500. 15. A man's income is $1800 a year, of which he pays 12% for house rent; what rent does he pay? Ans. $216. 16. If the bread made from a barrel of flour weighs 331 per cent. more than the flour, what is the weight of the bread? Ans. 261 lb.

17. Mr. Hamlin had 360 acres of land, and sold 33% of it, how many acres remained? Ans. 240 acres.

18. The silver coin of the United States contains 10% of alloy; how much pure silver is there in 163 oz. of silver coin? Ans. 15 oz.

19. A land agent bought 1016 acres of land, and sold 12% to Mr. Chase and 37% of the remainder to Mr. Dunn; how much remained? Ans. 555 acres.

20. How much linseed oil can be extracted from 1 cwt. 27 lb. of flaxseed, if flaxseed contains 11% of oil, and a pint of oil weighs of a pound? Ans. 2 gal. 1.31 qt.

21. A clerk's salary is $2000 a year; he spends 10% of it the first quarter, 15% the second, 6% the third, and 4% the fourth; how much did he save ? Ans. $1300.

22. Mr. Walton's income is $2500 a year, of which he spends 30% for board, 12% for clothes and books, and 10% for incidentals; what does he save in a year?

Ans. $1187.50. 23. A man owning of a machine shop worth $10,000, Bold 16% of his share to his brother; what part of the whole shop did he still retain, and what was its value?

CASE II.

Ans. ; value, $6250.

386. Given, the rate and the percentage or proceeds, to find the base.

MENTAL EXERCISES.

1. Twenty-four is 20% of what number?

SOLUTION.-If 24 is 20 per cent. of some number, it is %, or of that aumber; if 24 is of some number, §, or the number, equals 5 times 4 ar 120.

10. Thirty is 25% more than what number? 50% more than what number? 100% more than what number?

11. Sixty is 25% less than what number? 50% less than wha number? 100% less than what number?

12. The population of a certain village is 2700, which is 12% more than it was 5 years ago; what was its yearly gain?

13. Mr. A. paid a tax of $60, which was % of the value of his property; what was the value of his property?

14. A gentleman gave his daughter $50 as a Christmas present, which is 62% of what he gave to his wife; what did he give his wife?

15. On a certain day there were present at a graded school 160 pupils, which was 113% less than were registered; how many were registered?

WRITTEN EXERCISES.

1. 60 is 5% of what number? What number, increased by 20% of itself, equals 360?

SOLUTION.-If 60 is 5% of some number, then .05 times some number equals 60; if .05 times some number equals 60, the number equals 60.05, which is 1200.

SOLUTION.-A number increased by 20%, or .20 of itself, equals 1.20 times the number; and if 1.20 times a number equals 360, the number equals 3601.20, or 300.

OPERATION.

60.05 1200

OPERATION. 3601.20=300

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.

10. What number increased by 40% of itself equals 1694? Ans. 1210,

11. What number diminished by 20% of itself equals 468!

Ans. 585.

12. What fraction increased by 16% of itself equals ?

Ans.

13. What fraction diminished by 36% of itself equals ?

14. 42 A. 112 P. is 163% of how much land?

Ans. §.

Ans. 256 A. 32 P,