INTRODUCTION TO PERCENTAGE. MENTAL EXERCISES. 1. A gain of $2 on $5 is a gain of how many dollars on the hun dred? SOLUTION.-If the gain on $5 is $2, on $100, which is 20 times $5, tho gain is 20 times $2, or $40. 2. A gain of $3 on $5 is a gain of how many dollars on the hundred? 3. What is the gain on a hundred when the gain is 4 on 20? 5 on 20? 4 on 25 ? 4. If the gain on $100 is $25, what is the gain on $4? on $12? on $20? 5. If the gain on $100 is $20, what is the gain on $5? on $15? on $25 ? 6. If the gain on $100 is $40, what is the gain on $1? on $12? on $36? 7. If the gain on $100 is $25, what part of the $100 equals the gain ? 8. If the gain on $100 is $40, what part of the $100 equals the gain? 9. If the gain on $24 is at the rate of 25 on the 100, what is the gain? 10. If the gain on $25 is at the rate of 20 on the 100, what is the gain? 11. What is the gain on $50 at the rate of 10 on the hundred ? 16. Per cent. means the same as per hundred; what then can we call the rate per hundred ? Ans. Rate per cerit. 17. A gain of $20 on $80 is a gain of what per cent.? 18. A loss of $15 on $75 is a loss of what per cent. 19. What per cent. is a gain of 20 on 40? 5 on 25 ? 4 on 801 3 on 80? 8 on 2007 20. What is 5 per cent. of 80? 4 per cent. of 24? 20 per cent of 40? 25 per cent. of 48? SOLUTION.—5 per cent. is at the rate of 5 on the 100, and since 5 is st of 100, 5 per cent. of 80 is zo of 80, which 4. 21. What is 50 per cent. of 24? 30 per cent. of 60 ? 40 per cent. of 85? 60 per cent. of 45 ? 22. What per cent. is a gain of 15 on 60? 18 on 722 12 on 48? 18 op 80? 20 on 60? 15 on 90 ? SECTION VIII. PERCENTAGE. 376. Percentage is the process of computation in which the basis of comparison is a hundred. 377. 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. 378. The Symbol of Percentage is %. The per cent. may also be indicated by a common fraction or a decimal; thus 6%=160=.06. 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 differ. ence of the base and percentage. They may both be em. braced 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 Arithnietic. EXPRESSION OF THE RATE, OPERATION. 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 4%=.04–17. a common fraction, tér or as of it. Express 2. 5%. Ans. .05 or zo. 4. 7%. Ans. .07 or to 3. 6%. Ans. .06 or 6 5. 8%. Ans. .08 or at . 6. 10%. Ans. .10 or to. 10. 331%. Ans. .33} or . 7. 11390. Ans. .113 or 5. 11. 37%. Ans. .374 or 8. 121%. Ans. .121 or $. 12. 1%. Ans. .005. 9. 167%. Ans. .16f or 5. 13. 1%. Ans. .0025. 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 too or 4 of it; and 4 of 120 yards is 30 yards. Therefore, etc. What is 2. 20% of 757 6. 15% of 60? 3. 25% of 80? 7. 35% of 120? 4. 50% of 132? 8. 12}% of 1441 5. 75% of 96? 9. 163 % of 108? 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, 331% 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 113%; what did he then receive a month? 14. Mr. Smith paid a tax of 1% 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? What is the amount of $275, increased by 6% of itself? OPERATION. $275 SOLUTION.—6% of $275 equals .06 times $275, .06 which, by multiplying, we find to be $16.50. $16.50 OPERATION, SOLUTION.-A number increased by 6%, or.06 times $275 itself, equals 1.06 times itself; 1.06 times $275 equals 1.06 $291.50. $291.50 Rule 1.-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. What is Ans. 57. 3. 8% of 1875? Ans. 150. 4. 25% of 948 miles ? Ans. 237. 5. 124% of 1256 rd. ? Ans. 157. 6. 35% of 1840 yd. ? Ans. 644. 7. 663% of $124.65 ? Ans. $83.10. 8. 331% of $234.54 ? Ans. $78.18. 9. 45% of 184? Ans. 8.431. 10. *% of $348 ? Ans. $2.61. 11. $% of 1 lb. ? Ans. .12 oz. 12. Find 25% of 46 lb. 124 oz., Av.? Ans. 11 lb. 11 % oz. 13. How much is 42% of 6 lb. 8 oz. 12 pwt., Troy? Ans. 2 lb. 10 oz. 57 pwt. 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 33 per cent. more than the flour, what is the weight of the bread ? Ans. 2613 lb. 17. Mr. Hamlin bad 360 acres of land, and sold 339% 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 123% to Mr. Chase and 372% of the remainder to Mr. Dunn; how much remained ? Ans. 5553 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. 1.313 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 wbich he spends 30% for board, 125% 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, sold 169% of his share to his brother; what part of the whole shop did he still retain, and what was its value ? Ans. ; value, $6250. CASE II. 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 to, or } of that number; if 24 is f of some number, ß, or the number, equals 5"times ap 120. Of what number is 2. 16 25% ? 6. 4 lb. 6% ? 3. 19 163 %? 7. 2 bu. 121 % ? 4, 27 334 % ? 8. 8 A. 663% ? 5. 1.5 50% ? 9. £7.5 75% ? 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? 60% less than what dumber ? 100% less than what number? |