6. What number diminished by 20% of itself is 40?

ANALYSIS. Since 40 is the number diminished by 20% of itself, it is, or of the number; and if of the number is 40, the number itself is 5 times of 40, or 50.

7. What number diminished by 5% of itself is 38?
8. What sum diminished by 50% of itself is $20.50?
9. 68 yd. are 15% less than what number?

10. A tailor, after using 75% of a piece of cloth, had 9₫ yards left. How many yards in the whole piece?

11. A sells tea at $.90 a pound, which is 10% less than he paid for it. What did he pay for it?

520. 1. What sum increased by 37% of itself is $2055 ?

OPERATION.

1+.37=1.37

$20551.37=$1500

Or,

ANALYSIS.-Since the number is increased 37%, or by .37 of itself, $2055 is 137%, or 1.37 the

189 of $2055=$2055÷137 × 100=$1500 number. Hence

1.37, is the base or required number. Or,

$2055 divided by

Since $2055, the amount, is 187 of the base, 100 times of $2055, or $1500, is the base.

2. What number increased by 18% of itself equals 2950? 3. What sum increased by 15% of itself is $6900 ?

4. What number diminished by 12% of itself is 2640 ?

OPERATION. 1.12.88

2640.88=3000

Or, 264022 x 25=3000

ANALYSIS.-Since une number is diminished 12%, or by .12 of itself, 2640 is 88%, or .88 of the number. Hence 2640 divided by .88 is the base or required number. Or,

Since 2640, the difference, is or of the base, 25 times

of 2640, or 3000, is the base.

88 700

5. If the difference is $1000 and the rate 20%, what is the base?

6. What sum diminished by 35% of itself equals $4810?

RULE.-Divide the amount by 1 plus the rate; or, divide the difference by 1 minus the rate.

15. A man sold 160 acres of land for $4563.20, which was 8% less than it cost. What did it cost an acre?

16. A speculator bought 48 bales of cottor, and afterward sold the whole for $2008.80, losing 7%. What was the cost of each bale?

17. A dealer bought a quantity of grain by measure and sold it by weight, thereby gaining 14% in the number of bushels. He sold at 10% above the purchase price, and received $4910.976 for the grain. Required the cost.

18. A merchant, after paying 60% of his debts, found that $3500 would discharge the remainder. What was his whole indebtedness?

19. The net profits of a mill in two years were $6970, and the profits the second year were 5% greater than the profits the first year. What were the profits each year?

20. A man sold two houses at $2500 each; for one he received 20% more than its value and for the other 20% less. Required his loss.

APPLICATIONS OF PERCENTAGE.

521. The applications of percentage are those which are independent of time, as, Profit and Loss, Commission, Stocks, etc.; and those in which time is considered, as, Interest, Discount, Exchange, etc.

Since some one of the four formulas of percentage already considered will apply to any of these applications, the following will serve as a general

RULE.-Note what elements of Percentage are given in the problem, and what element is required, and then apply the special rule or formula for the corresponding case.

PROFIT AND LOSS.

522. Profit and Loss are terms used to express the gain or loss in business transactions.

523. Gains and losses are usually estimated at a rate per cent. on the cost, or the money or capital invested.

524. The operations involve the same principles as those of Percentage.

525. The corresponding terms are the following:

1. The Base is the Cost, or capital invested. 2. The Rate is the per cent. of profit or loss.

3. The Percentage is profit or loss.

4. The Amount is the cost plus the profit, or the Selling Price.

5. The Difference, is the cost minus the loss, or the Selling Price.

526. 1. A horse that cost $200 was sold at a gain of 12%. What was the gain, and the selling price?

ANALYSIS.—Since the gain was 12%, it was % of $200, which is $24; and the selling price was $200+$24-$224. Hence, etc. (510.) 2. A saddle that cost $25 sold at a loss of 10%. What was the loss, and the selling price?

3. A tailor bought cloth at $6 a yard, and wished to sell it at a gain of 25%. At what price must he sell it? 4. For how much must a grocer sell tea that cost $.60 a pound, to gain 30%?

5. A merchant buys gloves at $.75 a pair, and sells them at a profit of 33%. For how much does he sell them?

6. Bought a carriage for $160, and, after paying 10% for repairs, sold it at 121% profit. What was the gain, and the selling price?

7. If butter bought at 36 cents a pound is sold at a loss of 16%, what is the selling price?

8. What must be the selling price of coffee that cost 25 cents a pound, in order to gain 20%?

9. At what price must an article that cost $5 be sold, to gain 100%? 120%? 150%? 200%?

527. 1. A merchant bought cloth at $5 a yard, and sold it at $6 a yard. What was the gain per cent.?

ANALYSIS.-The whole gain is the difference between $6 and $5, which is $1. Since $5 gain $1, or of itself, the gain per cent. is of 100% or 20%. Hence, etc. (513.)

2. What is gained per cent. by selling coal at $7 a ton, that cost $6 a ton ?

3. Sold a piano for $300, which was 4 of what it cost. What was the loss per cent.?

4. Sold melons for $.75 that cost $.50. What was the gain per cent.?

5. What is gained per cent. by selling pine-apples at 30 cents each, that cost $15 a hundred?

6. Sold a sewing machine at a loss of of what it cost. What was the loss per cent.?

7. What % is gained on goods sold at double the cost? 8. What % is lost on goods sold at one-half the cost? 9. What per cent. profit does a grocer make who buys sugar at 10 cents and sells it at 12 cents?

10. What per cent. is gained on an article bought at $3 and sold at $5 ?

528. 1. A dealer sold flour at a profit of $2 a barrel, and gained 25%. What was the cost?

ANALYSIS. Since the gain was 25%, or 4, $2 is of the cost; $2 is of 4 times $2, or $8. Hence, etc. (516.)

2. Sold hats for $1 less than cost, and lost 16%. What did they cost?

3. A merchant sells silk at a profit of $1 a yard, which is 40% gain. What did it cost, and what is the selling price?

4. If corn selling for 21 cents a bushel more than cost gives a profit of 30%, what did it cost?

5. Sold sheep at $24 more than cost, which was a profit of 50%. What did they cost, and what is the selling price? 6. Shoes sold at $.50 above cost give a profit of 121%. What did they cost?

7. A farmer, by selling a cow for $12 less than she cost, lost 331%. What did she cost?

8. A grocer sells a certain kind of tea for 6 cents a pound more than cost and gains 5%. What did it cost?