6. 25% of of an article is how many % of 3 of it?

Ans. 13%.

7. The base is 28.35 gal., the percentage 5.31561 gal.; what is the rate?

8. A cask containing 25 gal. 2 qt. leaked escaped 9 gal. 2 qt.; what % leaked out?

Ans. 183%.

so that there Ans. 37%.

9. Standard gold or silver of the United States is 9 parts

pure and 1 part alloy; what % is pure? 10. A pound of English standard silver of alloy; what % is pure?

Ans. 90%. contains 18 pwt. Ans. 921%.

11. English standard gold is 22 carats fine; what % of alloy is there in a sovereign?

Ans. 81%. inches; how

12. The dry gallon contains 268.8 cubic many % larger is it than the wine gallon, or smaller than the old beer gallon? Ans. 16,4% larger; 43% smaller. 13. I bought a quantity of valentines, wholesale, at a discount of 50, 50, and 25%; what is the rate of discount? Ans. 811%.

14. A manufacturer sold a quantity of slates at 60, 10, 10, 10, 10, and 5% discount; what was the rate of discount? Ans. 75.0682%.

15. What is the difference between a discount of 40% and 10% taken 4 times? between 40% and 20% taken twice? Ans. 5.61%; 4%.

16. A person deposited $6000 in bank, checked out 331% of it, deposited $6000 more and checked out 15% of what was then in; what per cent. of his deposit remains in bank? Ans. 70%.

17. Mr. Johnson drew 331% of his money from the bank, and paid 621% of it for a horse worth $125, and then deposited the remainder; what per cent. of his entire deposit was the sum then remaining in bank? Ans. 791%.

GENERAL FORMULAS.

521. Formulas.-These methods and rules may all be presented in general formulas. Let b represent the base, r the rate, p the percentage, A the amount, D the difference, and we have the following:

CASE I.

CASE II.

CASE III.

1. p÷r=b.

1. p÷b=r.

1. bxr=p. 2. b× (1+r)=A. 2. A÷(1+r)=b. 2. A÷b=1+r. 3. bx(1-r)=D. 3. D÷(1—r)=b. 3. D÷b=l—r.

522. The second and third formulas of each case may be united in one; thus, using P for proceeds, P=b×(1±r); b=P÷(1±r); r=P÷b−1, or r=1−P÷b.

NOTE. These formulas apply to all the cases in practical applications, and may be used instead of the rules, or with them, as the teacher prefers.

APPLICATIONS OF PERCENTAGE.

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

524. The Method of Treating the cases of the Applications of Percentage is the same as in Percentage itself. 525. 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.

1. Profit and Loss.

2. Commission.

3. Stocks, Dividends, etc.

4. Premium and Discount.

5. Brokerage.

6. Stock Investments.

7. Taxes.

8. Duties or Customs.

2D CLASS.

1. Simple Interest.

2. Partial Payments.

3. True Discount.

4. Discounting and Banking.

5. Exchange.

6. Compound Interest.

7. Annuities.

8. Insurance.

NOTES.-1. In the different cases of the application of percentage, care should be taken to see clearly the base upon which the percentage is reckoned.

2. The subject of percentage has been greatly extended by the fact of our money system reckoning a hundred cents to a dollar. Pupils should remember, however, that per cent. and cents are two distinct things.

PROFIT AND LOSS.

526. Profit and Loss are terms which denote the gain or loss in business transactions.

527. The Quantities considered are: 1. The Cost; 2. The Rate of Profit or Loss; 3. The Profit or Loss; 4. The Proceeds or Selling Price.

NOTES.-1. Profit and Loss are not always estimated upon things bought and sold.

2. 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.

528. Given, the cost and the rate of profit or loss, to find the profit or loss, or the selling price.

1. A house was bought for $5780, and sold at a gain of 12%; what was the gain?

SOLUTION.-If a house was bought for $5780 and sold at a gain of 12%, the gain was .12 times $5780, which is $693.60.

OPERATION.

$5780 .12

$693.60

Rule I-Multiply the cost by the rate, to find the profit or loss.

Rule II. Multiply the cost by 1 plus the rate of profit, or by 1 minus the rate of loss, to find the selling price.

2. I bought fish at $4.50 a quintal, and sold the same at a gain of 8%; what was my gain? Ans. $0.36.

3. A furrier sold a set of furs which cost $87.50, at a gain of 121%; what did he receive for them? Ans. $98.43.

4. A train of cars was running 24 miles an hour, when the conductor, to make up lost time, increased the speed 25%; how fast did he then run? Ans. 30 miles.

5. The price of a certain lot of drugs is $96; if I buy at 10% off and sell at 25% on, what do I gain? Ans. $33.60.

6. My key for marking my goods is "Charleston;" if I buy cassimere @ $3.75, what will be the mark for the selling price if I intend to gain 15%? Ans. r.ac.

с

r

7. Bought 50 yards of paper muslin @ 8 and marked it

at a profit of 25%; what will be my profit if I sell at 121% less than the selling mark? Ans. 371.

8. A gentleman bought a yacht for $3500, sold it at a loss of 20%, and the buyer sold it at a gain of 25%; what Idid the latter receive for it? Ans. $3500.

9. A drover bought 75 cows at $241 a head; if 9 of them were killed by an accident, how must he sell the remainder to gain 20%, the expenses being $75? Ans. $34.4321

10. A cistern containing 230 barrels of water, receives by one pipe 73% of its contents in an hour, and loses by another 16%; how much water is in the cistern at the end of an hour? Ans. 209.49 bar.

11. A merchant marks down some old-fashioned goods 121%; how should he mark to the nearest half-cent those selling @ 12, 183, 621, 75, $1.621, $1.871, 2.371?

Ans. 11, 161, 541, 651, $1.42, $1.64, $2.08. 12. I buy 7 lots of English prints averaging 75 yd. in a lot, marked 10g, at a discount of 10, 121, 15, 10 and 5, 20, 25, and 20 and 20%, and sell them all at 7% below marked price; what is my clear profit? Ans. $6.30.

13. A began business with $25,000; he cleared 25% the first year, and added it to his capital; the 2d year he cleared 25% and added it to his capital; the 3d year he did the same; what was his entire gain? Ans. $23,828.12.

CASE II.

529. Given, the rate and the profit or loss, or the selling price, to find the cost.

1. A man sold a house for $870 above cost, and gained

25%; required the cost.

SOLUTION.-At a gain of 25%, .25 times the cost equals the gain, which is $870; if the cost multiplied by .25 equals $870, the cost equals $870 divided by .25, or $3480.

OPERATION.

$870.25 $3480

Rule I. Divide the profit or loss by the rate, to find the

cost.

Rule II.-Divide the selling price by 1 plus the rate of profit, or by 1 minus the rate of loss, to find the cost.

2. I sell chintzes @ 25 and gain 25%, and also @ 20 and lose 20%; what was their cost? Ans. 20g; 259.

3. A drover lost 2% of his cattle by disease, and 31% by accident, losing altogether 46; how many were in the drove at first? Ans. 800.

4. I sold my horse at a gain of 163%, and with the proceeds bought another which I sold for $180.32, at a loss of 8%; what did each horse cost? Ans. 1st, $168; 2d, $196.

5. Two newsboys invested, during the year, equal sums of money in papers; the one gained 63% and the other 81%; what amount did each invest, if the gain of the second was $14.50 more than that of the first? Ans. $750.

6. A merchant bought some water-proof cloth @ $1, and marked it so that he could fall 5% on his asking price, and gain 25% on cost; how did he mark it?

Ans. $1.32.

7. A merchant bought a lot of alpaca @30; what must the goods be marked that he may throw off 25% from the marking price, and still make 25% profit? Ans. 50¢.

8. Brown sold Jones some goods for $585 and lost 21%, and Jones sold them to Robinson and made 21%; did Robinson pay more or less than Brown? Ans. $0.37 less.

9. I purchased a lot of ingrain carpets from the manufacturer, and marked them $1.25 retail, which is 11% above the rate at which I actually sold them; if I gained 28%, what was the cost? Ans. 871.

10. My gain this year was $1413, which was 781% of my gain last year, and that was 1121% of my gain the year before; required my gain last year and the year before.

Ans. $1800 last; $1600 before.

11. Mr. Brenner offered his house for sale at an advance of 20%, but afterwards sold it for $5250, which was 12% less than his original offer; what was the first cost of the house? Ans. $5000. each of 4 years, year, and at the

12. A man's capital increased 25% for on what he had at the beginning of each end of the time he was worth $12207.03; capital?

what was his Ans. $5000.