14. A lady's cloak cost $40; the making cost 331% less than the cloth, and the trimmings 25% more than the cloth; what did each cost?

Ans. Cloth, $135; making, $94; trimmings, $174. 15. A and B together have 1320 acres of land, 31% of A's equaling 37% of B's, and 561% of B's equaling 663% of C's; how much land has C? Ans. 506 acres.

16. Mr. Howard drew 75% of his money from bank, and paid 87% of it for a house worth $5600; how much money had he remaining in bank? Ans. $2133.331.

17. In an engagement 5% of an army were killed, 121% of the remainder were wounded, and 163% of the wounded died; there were 290 more killed than mortally wounded; how many men were in the army? Ans. 9600.

18. A, wishing to sell a cow and horse to B, asked 150% more for the horse than the cow; he then reduced the price of the cow 25%, and the horse 33%, at which price B took them, paying $290; what was the price of each?

Ans. Cow, $90; horse, $200.

19. In building a church, the trustees paid three times as much for material as for labor; had they paid 41% more for material and 7% more for labor, the church would have cost $14700; what was its cost? Ans $14,000.

CASE III.

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

1. 24 is what per cent. of 96?

SOLUTION.-If 24 is some per cent. of 96, then 96 multiplied by some rate per cent. equals 24; if 96 multiplied by some rate equals 24, the rate equals 24 divided by 96, which is .25 or 25%.

OPERATION. 24-96.25

Rule I. Divide the percentage by the base, to find the

rate.

Rule II.—Divide the difference between the proceeds and base 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.

11. English standard gold is 22 carats fine; what % of alloy is there in a sovereign? Ans. 81%.

12. The dry gallon contains 268.8 cubic inches; how 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 33% 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 33% of his money from the bank, and paid 62% 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. 79%.

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=1-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, Cast should be taken to see clearly the base upon which the percentage in 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. $98.433. 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

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. 37.

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 did 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. $34.43

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 @ 121, 1837, 6217, 759, $1.621, $1.871, 2.371⁄2?

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 109, 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

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.