519. The Base is either an abstract or denominate number; the rate per cent. is always abstract, and the percentage, amount, and difference are always like the base.

REMARKS. 1. In all operations where a decimal rate is used, too great care cannot be taken to express all decimal terms with exactness.

2. As the greater part of commercial calculations are based upon percentage, the importance of a thorough mastery of its principles will be readily perceived.

520. Since per cent. is any number of hundredths, it may be expressed either as a decimal or as a common fraction, and the table of aliquot parts can be used with little variation and to great advantage in many operations in percentage. Hence, the rules given under SPECIAL APPLICATIONS may be applied in this subject.

Table.

521. The relation between the elements of Percentage is such, that by the application of the General Principles of Multiplication and Division, if any two of the elements, except amount per cent. and difference per cent., are given, the other three may be found.

522. To find the Percentage, the Base and Rate being given. EXAMPLE.-What is 25% of $468 ?

OPERATION.

$468 base.

.25 rate per cent.

$117.00 = percentage.

FIRST EXPLANATION.-25 per cent. equals .25; therefore, 25 per cent. of $468 equals $468 multiplied by .25, equals $117.

SECOND EXPLANATION.-$468 is 100 per cent. of itself; and since 25 per cent, equals of 100 per cent., 25 per cent.

of $468 will be of that sum, or $117.

Rules.-1. Multiply the base by the rate expressed decimally. Or, 2. Take such a part of the base as the number expressing the rate is part of 1.

REMARK.-When the rate is an aliquot part of 100, the percentage may be found by taking a like part of the base; thus, for 10% take, for 25% take, for 331% take †, etc.

524. 1. A man owning 250 acres of land, sold 20% at one time, and 25% of the remainder at another time. How many acres did he have left ?

2. If a ranchman having 5450 sheep, lost 20% by a storm and afterwards sold 20% of those remaining, how many sheep did he sell?

3. A collector deposited $13500 in coin, and 121% more than that amount in bank bills. What was the total of his deposit?

4. Find 114% of 1680 lb. of wool.

5. Find 16% of 12 lb. 3 oz. of silver.

6. From a charge of $675, made for a bill of goods, 8% was deducted. What was the net amount of the bill?

7. If 526 barrels of salt were bought for $1.10 per bar., and sold at an advance of 15%, what was gained?

8. Two men, each having $12500, made investments, from which one gained 15%, and the other lost 35%. How much did each then have?

9. How much greater is 12% of $1550, than 7% of $2150?

10. Having raised 1240 bushels of wheat, a farmer used 5% of it for seed and 5% for bread; he then sold to one man 10% and to another 25% of what remained. How many bushels had he left?

11.

Having $75000 to invest, a gentleman bought United States bonds with 334% of his money, a home with 20%, and invested the remainder equally in farm lands and manufacturing stock. How much did he pay for the farm lands?

12. I owed John Smith $1750, and paid at one time 20% of the debt, at another time 35% of the remainder, and at another time 25% of what then remained unpaid. How much of the debt did I still owe?

13. A capitalist owning of a coal mine, sold 324% of his share for $65000. At that rate, what was the entire mine worth?

14. A jobber having bought 2160 bags of coffee, sold at one time 83%, at another 25% of what remained, and at a third sale 15% of what still remained. Find the value of what was left, at $18 per bag.

15. Of a farm containing a half section of land, 15% was in wheat, 32% in oats, 5% in potatoes, and the remainder devoted equally to orchard, corn, beans, and pasture. How many acres were in pasture?

16. A farmer having 156 sheep to shear, agreed to pay for their shearing 4% of the sum received for their wool. If the fleeces averaged 7 lb. and sold for 304 per pound, how much was paid for shearing?

17. A speculator having $41820, invested 50% of it in oil, on which he lost 16%; the remainder he invested in cotton, which he sold at 9% below cost. How much was received from both sales?

18. A trader bought 12 mustangs for $400, and after selling 25% of the number at a gain of 50%, and 33% of those remaining at a gain of 121%, sold those still on hand at $30 per head. Did he gain or lose, and how much?

525. To find the Base, the Percentage and Rate being given.

REMARK.-Since the base multiplied by the rate produces the percentage, percentage must be a product; if, therefore, it is divided by either factor, the quotient will be the other factor. EXAMPLE.-By selling 4% of a stock of goods, a merchant realized $644. What was the value of the entire stock?

OPERATION.

Rate. Percentage.

.04) 644.00

16100 base.

EXPLANATION.-If the value of 4 per cent. is $644, the value of 1 per cent. will be $161; and if the value of 1 per cent. is $161, the value of 100 per cent. will be $16100.

Rule.-Divide the percentage by the rate, expressed decimally.

6.

A man sold 25% of his farm for $2120. How much was the farm worth at that rate ?

7. What is the value of a house renting for $360 per year, if the rent equals 9% of its value ?

8. How many acres in a farm of which 12.5 acres is but 5%.

9.

Of what sum is $36 but 33% ?

EXAMPLES FOR WRITTEN PRACTICE.

527. 1. A planter sold 76 bales of cotton, which was 19% of his crop. How many bales did he raise ?

2. I paid $123.48, which was 163% of a debt. What amount did I owe? 3. A lady paid for millinery, $17.50; for shoes, $11.40; for jewelry, 113.80; for furs, $78.55; and had expended but 15% of her money. How many dollars

had she at first ?

4. A clerk's present salary of $520 per year is only 75% of what he formerly received. How much was formerly paid him ?

5. A grocer, after increasing his stock to the amount of $6448, found that the new purchase was but 16% of the old stock on hand. What was the value of his old stock?

6. The owner of 68% of a mine, received $91510 from the sale of 25% of his share. Find the value of the entire mine at that rate ?

7. A, B, C, and D are partners; A furnished 15% of the capital, B 25%, C 42%, and D $16200. What was the capital of the firm?

8. A Wyoming ranchman lost 1684 cattle during a blizzard. How many had he at first, if his loss was only 13% of his herd?

9. The population of a county increased 22% in ten years. If the births exceeded the deaths by 2166, and the county received 13234 immigrants during the time, what must have been its population before the increase?

10. A speculator owned a quarter interest in a mill, and sold one-quarter of his part for $11250. What was the mill worth, on that basis of value ?

528. To find the Rate, the Percentage and Base being given. REMARK.-The percentage is a product, the base being one of its factors. EXAMPLE.-What per cent. of 480 is 120 ?

FIRST OPERATION.

4.80) 120.00 (25 times.

960 2400

1%=.01 2400

FIRST EXPLANATION.-Since 480 is 100 per cent. of itself, 1 per cent. of 480 would be part of it, or 4.80; and since 4.80 is 1 per cent. of 480, 120 would be as many times 1 per cent. as 4.80 is contained times in 120, which is 25 times; and 25 times 1 per cent. = 25 per cent.

SECOND EXPLANATION.-Since the percentage is a product of the base and rate, the quotient obtained by dividing the percentage by the base will be the rate. Or, 120 is 18, or of 480; and since 480 is 100 per cent. of itself, 120, which is of 480, must be of 100 per cent., or 25 per cent.

Rule.-Divide the percentage by the base, carrying the quotient to two

decimal places.

530. 1. From a herd of 1184 cattle, 296 were sold. What

per cent. was sold?

2. R. G. Dun & Co. charged $21 for collecting an account of $600. What rate was charged?

3. Sold of a stock of goods for what the entire stock cost. rate of gain?

4. What per cent. of 12 lb. 8 oz. is 2 lb. 8 oz., Avoirdupois ?

What was my

5. From a half section, 120 acres were sold, and afterwards 80 acres more. What per cent. was sold ?

6. Of a stock of 800 bushels of potatoes, 240 bushels were sold at one time, and 135 bushels at another. What per cent. was still unsold?

7. A merchant failed, owing $27984, his assets amounting to $16090.80. What per cent. of his debts can he pay?

8. At a normal school there were enrolled 855 male pupils and only 185 female pupils. What per cent. more were the male than the female pupils ?

9. A girl having $5.40, expended $1.35 for gloves, 45¢ for flowers, and onehalf of the remainder for a pair of slippers. What per cent. of her money had she left?

10. From a cask of lard of 314 lb., 78.5 lb. were sold at one time, and 25% of the remainder at another. What per cent. of the whole remained unsold?

11. Of a regiment of men entering battle, 1040 strong, only 260 came out unhurt, of the remainder having been killed. What per cent. of the whole were killed?

531. To find the Amount Per Cent., the Rate being given. EXAMPLE.-If the rate be 7%, what is the amount per cent.?

100%

OPERATION.

= 1, = a unit

77%= .07 rate

107% = 1.0% amount per cent.

EXPLANATION.-Since the amount per cent. (definition, page 160), is always 100 per cent. increased by the rate, we may find it by adding 100 per ent., or 1, to the per cent. given. Hence, if the rate is 7 per cent., the amount per cent. will be 107 per cent.

Rule. Add the rate to the unit 1.

Formula.-Amount Per Cent. = 1+ Rate.