10. From a hogshead of sugar containing 760 pounds, 100 pounds were sold at one time, and 90 pounds at another; what per cent. of the whole was sold?

11. A man, having 600 acres of land, sold of it at one time, and of the remainder at another time; what per cent. "emained unsold? Ans. 50 per cent.

CASE III.

237. To find a number when a certain per cent. of it is given.

1. A man lost $5, which was 4 per cent. of all the money he had; how much had he at first?

OPERATION.

$5 ÷ .04 = $125, Or,

× 100 = $125.

ANALYSIS. We are here required to find the base, of which $5 is the percentage. Now, percentage equals base multiplied by the rate per cent.; conversely, base equals percentage divided

by rate per cent. Or, $5 is 4 per cent. of all he had; † of $5, or 1⁄2, equals 1 per cent. of all he had, and 100 times equals 100 per cent., or all he had. Hence the

RULE. Divide the percentage by the rate per cent., expressed decimally, and the quotient will be the base, or number required. Or,

Take as many times 100 as the percentage is times the rate per cent.

EXAMPLES FOR PRACTICE.

2. 16 is 8 per cent. of what number? 3. 42 is 7 per cent. of what number?

4. 75. is 12 per cent. of what number?

5. 33 is 22 per cent. of what number?

Ans. 200.

Ans. 600.

Ans. 1200.

Ans. $750.

6. $281.25 is 37 per cent. of what sum of money?

7. A farmer sold 50 sheep, which was 20 per cent. of his

whole flock; how many sheep had he at first?

Case III is what? Give explanation. Rule.

8. I loaned a man a certain sum of money; at one time he paid me $59.75, which was 12 per cent. of the whole sum loaned to him; how much did I loan him?

9. A merchant invested $975 in dry goods, which was 15 per cent. of his entire capital; what was the amount of his capital? Ans. $6500. 10. If a man, owning 40 per cent. of an iron foundery, sell 25 per cent. of his share for $1246.50, what is the value of the whole foundery? Ans. $12465.

11. A produce buyer, having a quantity of corn, bought 2000 bushels more, and he found that this purchase was 40 per cent. of his whole stock; how much had he before he bought this last lot? Ans. 3000 bushels.

238. To find a number when the number, increased by a certain per cent. of itself, is given.

1. A man's income this year is $525, which is 5 per cent. more than it was last year; what was it last year?

OPERATION.

ANALYSIS. Since his income

$2551.05 = $5.00. this year is .05 more than it was last year, this year's income must be 1.05 times the income of last year; therefore divide this year's income by 1.05 and it gives the income of last year. Hence the

RULE. Divide the amount by 1 plus the rate expressed decimally, and the quotient will be the base or number required. Or,

Take as many times 100 as the amount is times 1 plus the rate per cent.

*The changes that have been wrought in financial and commercial operations the past few years, require some new methods and applications in the subject of percentage. Several pages have therefore been inserted in this work, including U. S. Securities, Stock, and Gold Investments, and their comparative values in commercial transactions shown by practical examples, &c.

All the changes that have been made, and the new matter inserted, are contained in the following sixteen pages, except the repaging, and renumbering the Articles, which is continued through the remainder of the book.

What is Case IV.? Explanation? Rule?

EXAMPLES FOR PRACTICE.

2. What number increased by 18 per cent. of itself is equal to 1475? Ans. 1250. 3. A merchant sells broadcloth for $4 per yard, and thereby makes 25 per cent.; what did the cloth cost him? Ans. $3.20.

4. A. expended a certain sum for a house, and 15 per cent. of the purchase price on repairs, and then found that the whole cost was $6900. What was the purchase price?

5. A certain manufacturing company have sold $432,250 worth of goods, which is 8 per cent. more than they sold last year. How much did they sell last year? Ans. $400,000.

6. A merchant bought a stock of goods, and paid 41 per cent. of the purchase price for freight, when he found that the goods cost him $8757. What was the purchase price ?

7. A merchant increased his capital by 20 per cent. each year for two years, when he found he had $9360 invested. How much had he at first? Ans. $6500.

CASE V.

239. To find a number when the number, diminished by a certain per cent. of itself, is given.

1. A man lost 8 per cent. of his sheep and had 368 left;

how many

had he at first?

OPERATION.

368.92 = 400.

ANALYSIS. Since the man lost 8 per cent. of his sheep, he has

92 per cent. left; hence 368 is .92 times his original flock. We therefore divide 368 by .92 and obtain the required number of sheep. Hence the

RULE. Divide the given number by 1 minus the rate ex pressed decimally, and the quotient will be the base or number required. Or,

Take as many times 100 as the given number is times 1 minus the rate.

What is Case V.? Explanation? Rule?

EXAMPLES FOR PRACTICE.

2. What number diminished by 15 per cent, of itself is equal to 340? Ans. 400.

3. A. having a certain sum on deposit drew out 20 per cent., when he found he had $1000 ieft. How much had he on deposit at first? Ans. $1250.

14. My income this year is $4028, which is 24 per cent. less than it was last year. How much was it last year?

5. What number diminished by per cent. of itself is equal to 298 ? Ans. 300.

6. A sells his horse for $198, which is 10 per cent. less than his asking price, and his asking price was 10 per cent. more than he cost him; what did the horse cost him?

COMMISSION AND BROKERAGE.

Ans. $200.

240. An Agent, Factor, or Broker is a person who transacts business for another, or buys and sells money, stocks, notes, &c.

241. Commission is the percentage, or compensation allowed an agent, factor, or commission merchant, for buying and selling goods or produce, collecting money, and transacting other business.

242. Brokerage is the fee, or allowance paid to a broker or dealer in money, stocks, or bills of exchange, for making exchanges of money, buying and selling stocks, negotiating bills of exchange, or transacting other like business.

NOTE. The rates of commission and brokerage are not regulated by law, but are usually reckoned at a certain per cent. upon the money employed in the transaction.

Define an agent, factor, or broker. What is meant by commission? Brokerage?

CASE I.

243. To find the commission or brokerage on any sum of money.

1. A commission merchant sells butter and cheese to the amount of $1540; what is his commission at 5 per cent.?

a dollar, on $1540 it is $1540 X .05 is 150

ANALYSIS. Since the com. mission on $1 is

5 cents or .05 of

$77. Or, since 5 per cent of the sum received, the commission is of $1540 =$77. Hence the

RULE. Multiply the given sum by the rate per cent. expressed decimally, and the result will be the commission or brokerage. Or,

the

Take such a part of the given sum as the number expressing per cent. is part of 100.

EXAMPLES FOR PRACTICE.

2. A commission merchant sells goods to the amount of $6756; what is his commission at 2 per cent.? Ans. $135.12. 3. What commission must be paid for collecting $17880, at 3 per cent.? Ans. $608.30.

4. An agent in Chicago purchased 4700 bushels of wheat, at 75 cents a bushel; what was his commission at 11 per cent. on the purchase money?

5. A broker in New York exchanged $25875 on the Suffolk Bank, Boston, at/per cent.; how much brokerage did he receive? Ans. $64.6875.

6. An auctioneer sold at auction a house for $3284, and the furniture for $2176.50; what did his fees amount to at 21 per cent.?

7. A broker negotiates a bill of exchange of $2890 for/ per cent. commission; how much is his brokerage?

Ans. $23.12.

Case I is what? Give explanation. Rule.