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. Lowest Terms. .06 = 18o reducible to 3. per cent. = .08 18 reducible to 9 per cent. = .09 = To reducible to do 10% reducible to to. 16 reducible to the 25 reducible to 1. 30 per cent. = .30 10% reducible to Pio 50 per cent. = .50 reducible to 1. reducible to . 14 per cent. .0125 125. reducible to go 50%, reducible to be 100 reducible to go 64 per cent. .0625 6 2 , reducible to 16 8} per cent. .0833 25.0.0 reducible to iz. 124 per cent. .125 125 reducible to š. 163 per cent. = .1663 5.0.0 reducible to H. 33} per cent. = .3333 1000 reducible to }. 62} per cent. = 6 25 reducible to . 663 per cent. .663 reducible to s. 874 per cent. = .875 8.76 reducible to . .25 100 100 30 000 3000 10000 30000 1000 3000 3000 .625 1000 = 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., given, the other three may be found. are 522. To find the Percentage, the Base and Rate being given, EXAMPLE.— What is 25% of $468 ? FIRST EXPLANATION.—25 per cent. equals .25; therefore, OPERATION 25 per cent. of $468 equals $468 multiplied by .25, equals $468 = base. $117. .25 = rate per cent. SECOND EXPLANATION.—$468 is 100 per cent. of itself; and since 25 per cent. equals 1 of 100 per cent., 25 per cent. $117.00 = percentage. 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 io, for 25% take 1, for 33}% take ţ, etc. Formula.—Percentage = Base X Rate. EXAMPLES FOR MENTAL PRACTICE. 523. What is 1. per cent. of 100 ? 2. 12 per cent. of 600 ? 3. 15 per cent. of 800 ? 46 20 per cent. of 500 ? 7. 25 per cent. of 1440 ? 8. 8 per cent. of 450 ? 9. 50 per cent. of 680 ? EXAMPLES FOR WRITTEN PRACTICE. 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? A collector deposited $13500 in coin, and 127% more than that amount in bank bills. What was the total of his deposit? 4. Find 114% of 1680 lb. of wool. 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 ? i 11. Having $75000 to invest, a gentleman bought United States bonds with 33% 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 32% 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 81%, 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 73 lb. and sold for 30¢ per pound, how much was paid for shearing ? 17. A speculator having $41820, invested 50% of it in oil, on which he lost 163%; 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 12%, 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. EXPLANATION.-If the value of 4 per cent. is $644, the value of .04 ) 644.00 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. 16100 base. Rule.- Divide the percentage by the rate, expressed decimally. Formula.—Base = Percentage - Rate. EXAMPLES FOR MENTAL PRACTICE. 526. 1. 846 = 6% of what number? *% of what number? 6. A man sold 25% of his farm for $2120. How much was the farm worth at that rate ? 150 = 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%, 0 42%, and D $16200. What was the capital of the firm ? 8. A Wyoming ranchman lost 1684 cattle during a blizzard. 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 ? How many 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. FIRST EXPLANATION.—Since 480 is 100 per cent. of itself, 1 960 per cent. of 480 would be to part of it, or 4.80; and since 4.80 2400 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 1% =.01 2400 times 1 per cent. = 25 per cent. 25 .25 = 25%. SECOND OPERATION. 480) 120.00 ( .25 = 25%. 960 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 16, or ; of 480; and since 480 is 100 per cent. of itself, 120, which is 1 of 480, must be 1 of 100 per cent., or 25 per cent. 2400 Rule.-Divide the percentage by the base, carrying the quotient to two decimal places. Formula. -Rate = Percentage • Base. EXAMPLES FOR MENTAL PRACTICE. 529. What per cent. is 1. 25 of 125 ? 2. 40 of 160 ? 3. 18 of 36 ? 4. 124 of 100 ? 7. 374 of 150 ? EXAMPLES FOR WRITTEN PRACTICE. 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 i of a stock of goods for what the entire stock cost. What was my rate of gain ? 4. What per cent. of 12 lb. 8 oz. is 2 lb. 8 oz., Avoirdupois ? 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. Ata 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. ? OPERATION. EXPLANATION.-Since the amount per cent. 100% = 1, (definition, page 160), is always 100 per cent. in= a unit creased by the rate, we may find it by adding 7% = .07 = rate 100 per "ent., or 1, to the per cent. given. Hence, if the rate is 7 per cent., the amount per cent. will 207% = 1.07 = amount per cent. be 107 per cent. Rule.- Add the rate to the unit 1. Formula.—Amount Per Cent. = 1+ Rate. |