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OPERATION,

OPERATION.

12. The population of a certain village is 2700, which is 124 % 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 ?

11. A gentleman gave his daughter $50 as a Christmas present, which is 624% 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 papilg, which was 11}% less than were registered; how many were registered ?

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

SOLUTION.-If 60 is 5% of some nuniber, then .05 times some number equals 60; if .05 times some

60-.05=1200 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

360 - 1.20=300 1.20 times a number equals 360, the number equals 360 - 1.20, or 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.

Of what number is 2. 45 20%? Ans. 225. 6. 333% ? 3. 75 25 %? Ans. 300. 7. 74 75% ? 4. 112 lb. 40% ? Ans. 280 lb. 8. $645 621%? Ans. $1032. 5. 456 A. 30% ? Ans. 1520A. 9. $450 12%? Ans. $3606. 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 18?

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

Ans. * Ans. 105

Ans..

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

Ans. 256 A. 32 P.

16. 14 lb. 10 oz. 16 pwt. is 331% more than what nun ber?

Ans. 11 lb. 2 oz. 2 pwt. 16. A bookkeeper spends $600 per year, which is 24% of nis salary; required his salary.

Ans. $2500 17. A young farmer owns 320 acres of land, which is 15% of what his father owns; how much bas the father?

Ans. 2133} A. 18. A newsboy earned $15, wbich was 30% of what he then had in bank; how much had he in bank? Ans. $50.

19. A teacher spends 24% of his salary, and can thus save $760 a year ; what was his salary? Ans. $1000.

20. Mr. Hays drew 35% of his bank deposit to pay a debt of $4788.56; what was his deposit? Ans. $13681.60.

21. A man bought some flour and sold 25% of it to A, and 33% of the remainder to C; how much did he buy if he sold C 640 barrels ?

Ans. 2560 bar. 22. Mr. Herr drew 62% of his money from the bank, and paid 33}% of it for a house worth $4500 ; how much money had he remaining in bank?

Ans. $8100. 23. A lady invested 90% of her money in bank stock, and some time after sold 33% of the stock, and still had $4500 invested ; required the whole amount of her money.

Ans. $7500.

CASE III.

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

MENTAL EXERCISES.

1. 12 is what per cent. of 48?

SOLOTION.-48 is 100 per cent. of 48, and 12, which is 1 of 48, is of c00 per cent., or 25 per cent. of 48. What per cent. 2. Of 75 is 15?

6. Of is ß ? 3. Of $12 are $3?

7. Of is ş? 4. Of 81 is 27?

8. Of 75% is 25% ? 5. Of 16qt. are 5 qt.?

9. Of 2.5% is 1.5% ? 10. From a hogshead of wine containing 90 galions, 12 gallons leaked out; what per cent. was lost?

11. II a man's income is $2000 a year and he saves $300; what per cent. does he spend ?

12. The standard for gold and silver coin in the United States is 9 parts pure to 1 of alloy; what % of pure metal is there?

13. A merchant having put $5000 into a certain speculation, finds on settling up the business that he has received $5250; what % dio he gain?

WRITTEN EXERCISES.

OPERATION.

1. 20 is what per cent. of 80 ? SOLUTION.-If 20 is some per cent. of 80, then 80 OPERATION. multiplied by some rate equals 20; if 80 multiplied

20:-805.25 by some rate equals 20, the rate equals 20 divided by 80, which is .25, or 25%.

2. 240 yd. being increased by a certain per cent. of itself equals 300 yd.; required the rate.

SOLUTION.—300 yd. minus 240 yd. equals 60 yd., which is the percentage. If 240 yd. multiplied by

300—240—60 some rate equals 60 yd., the rate equals 60 divided

60-7-2405.25 by 240, 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 buse 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. What per cent. of 3. .360 is 90 ? Ans. 25%., 6. $880 is $528 ? Ans. 60%. 4. 675 is 135? Ans. 20%. 7. is f?

Ans. 90% 5. 900 is 360? Ans. 40%. 8. is * ? Ans. 45%. 9. 32% is 5%?

Ans. 163% 10. 4.5% is 3.371% ?

Ans. 75% 11. 936 yd. is 312 yd.?

Ans. 331% 12. 18 lb. is 5 lb. 8 oz., Av.?

Ans. 30%. 13. The base is $14.10, the percentage $2.35; what is the rate ?

Ans. 163% 14. If a miller takes 10 quarts of every bushel he grinds for toll, what per cent. does be take for toll? Ans. 314%.

15. My income last year was $1800 and my expenses $1356 ; what % of my income did I expend ? Ans. 751%.

16. A regiment went into battle with 960 men, and came out with 600 men; wbat per cent.was lost? Ans. 371%.

17. A merchant's liabilities are $15760, and his assets $7289; what % of his debts can be pay? Ans. 461%

18. A merchant bought 275 barrels of flour, and after losing 20% of it, he sold 25% of the remainder; what per cent. of the whole remained ?

Ans. 60%. 19. d put $780 in a savings bank, wbich was 15% of all his money, and afterward deposited 25% of the rest of his money; what per cent. of all his money had he then in bank?

Ans. 361%. 20. A gold eagle of the United States weighs 258 gr. and the alloy in it weighs 25.8 gr. ; what per cent of the coin is alloy ?

Ans. 10%. 21. 35 per cent. of a regiment being sick, only 637 men were able to enter battle, of whom { were killed; bow many did the regiment number, and what per cent. of the whole number were killed ? Ans. 980 men ; 94%.

GENERAL FORMULAS. 388. These methods and rules may all be represented in general formulas as follows:

CASE I. 1. Basex rate=Percentage. 1. Percentage---rate=base. 2. Basex (1+rate)=Amount. 2. Amount:(1+rate)=base. 3. Basex (1-rate)=Difference. 3. Difference:-(1-rate)=-base.

CASE II.

CASE III.

Percentage:-base=rate.
Amount; base=1+rate.

Difference: base=1-rate. NOTE.—These formulas apply to all the cases in the practical applica tions, and may be used instead of the rules, or with them, as the teache prefers.

APPLICATIONS OF PERCENTAGE.

389. The Applications of Percentage are extensive, owing to the great convenience of reckoning by the hundred . in business transactions.

390. These Applications of Percentage are of two

classes; those not involving time and those involving time The following are the most important of these applications 1ST CLASS.

2D CLASS. 1. Profit and Loss.

1. Simple Interest. 2. Commission.

2. Partial Payments. 3. Stocks, Dividends, etc.

3. True Discount. 4 Premium and Discount. 4. Discounting and Banking. 5. Brokerage.

5. Exchange. 6. Stock Investments.

6. Compound Interest. 7. Taxes.

7. Annuities. 8. Duties or Customs.

8. Insurance. NOTES.-1. In the different cases of the application of percentage, care ehould be taken to see clearly the base upon which the percentage is rock oned.

2. A percentage deducted from the price of goods is called a Discount, and is treated under Profit and Loss. Successive Discounts called “Trade Discounts” are often taken off, as “10 and 5 per cent. off,” meaning 10 per cent. off and 5 per cent. off of the remainder.

PROFIT AND LOSS. 391. Profit and Loss are terms which denote the gain or loss in business transactions.

392. The Quantities considered are as follows: 1. The Cost, which is the base. 2. The Rate of profit or loss. 3. The Profit or Loss, which is the percentage. 4. The Selling Price, which is the amount or difference. NOTE.-In marking goods it is customary to take one or more words or a phrase or sentence, consisting of ten different letters, and let each letter in succession represent one of the Arabic figures. The prices marked thus can only be read by those who have the key.

CASE I. 393, Given, the cost and rate of profit or loss, to find the profit or, loss, or the selling price.

MENTAL EXERCISES. 1. A lady paid $10 for a shawl, and sold it at a loss of 20% · ro quired the loss.

SOLUTION.–At a loss of 20 per cent., to, or of the cost equals the loss of $10 is $2. Therefore, etc.

3. A grocer bought tea at 80 cents a pound, and sold it so as to gain 25% ; what was the gain?

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