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342. Percentage is an allowance for every hundred, consisting of a part of that on which it is reckoned.
Per cent., a contraction of per centum, signifies by the hundred. By 1 per cent. is to be understood 1 out of a hundred, or 1 hundredth; by 2 per cent. is to be understood 2 out of a hundred; or 2 hundredths, &c.
343. The rate per cent. is the number denoting the part allowed for every hundred, as 5 per cent., 6 per cent., &c.
344. The basis of percentage is the number on which percentage is reckoned.
345. The amount is the basis of percentage increased by the percentage; and the remainder is the basis decreased by the percentage.
346. The rates per cent, may be written decimally as in the following
\ per cent. may be written .005, or .00^, and be read T6T of 1 per cent., or \ of 1 per cent.
\ per cent. may be written .0025, or .00^, and be read of 1 per cent., or \ of 1 per cent.
\ per cent. may be written .00125, or .001, and be read -jj2^ of 1 per cent., or \ of 1 per cent.
1\ per cent, may be written .075, or .07^, and be read 7^ per cent., or 1\ per cent.
Note. — If a fraction of 1 per cent. cannot be exactly expressed in decimal figures, it may be written as a part of a mixed decimal. Thus, 41 per cent. may be written .04s; 67 per cent. may be written .06^; and 11| per cent may be written .11|.
1. Express decimally 19 per cent. Ans. .19.
2. Express decimally 27 per cent.
3. Express decimally 13£ per cent.
4. Express decimally 1J per cent.
5. Express decimally 7§ per cent.
6. Express decimally 77£ per cent.
7. Express decimally 106 per cent.
8. Express decimally 107 per cent.
9. Express decimally 305 per cent . 10. Express decimally 999|§ per cent.
347. To find the percentage any given rate per cent, is of any number or quantity.
Ex. 1. Bought J of a ship for $ 15650, and sold the same at a gain of 12 per cent. How much did I make by the transaction? Ans. $ 1878.
»».,„«,. Since 12 per cent . equals
Basis of percentage, $, 1 5 6 o 0 .12 of the original cost, we
Rate per cent. .12 multiply $15650 by the
Percentage, $18 7 8.00 Ans. decimal expression .12.
Rule. —Multiply the given number by the rate per cent. expressed decimally, and the product will be the percentage. Or,
As 100 per cent. is to the given rate per cent., so is the given basis of percentage to the percentage required.
2. What is 15 per cent, of 500 bushels? Ans. 75bu.
3. What is 20 per cent, of 75cwt.? Ans. 15cwt.
4. What is 30 per cent, of 150 tons? Ans. 45 tons.
5. What is 75 per cent. of $ 500?
6. What is 95 per cent, of 700 chaldrons?
7. What is 2 per cent, of 40 miles? Ans. .8 mile.
8. What is 99 per cent, of $ 1000? Ans. $ 990.
9. What is 33£ per cent. of 144 barrels? Ans. 48bbl.
10. What is 66| per cent. of 90 hogsheads?
11. What is i per cent. of $ 100? Ans. $ 0.25.
12. What is £ per cent. of 17281b.? Ans. 15.121b.
13. A certain colonel, whose regiment consisted of 900 men, lost 8 per cent. of them in battle, and 50 per cent, of the remainder by sickness. How many had he remaining?
Ans. 414 men.
14. A merchant, having $ 1728 in the Union Bank, wishes to withdraw 15 per cent.; how much will remain?
15. A gentleman, who had an estate of $ 25,000, in his will gave to his wife 40 per cent, of his property, and to his son Samuel 30 per cent. of the remainder. The residue he divided equally among his daughters, Marcia, Isabella, and Clara, after having deducted $ 6O as a present to his clergyman. What did each receive?
Ans. Wife, $ 10,000; son, $ 4,500; daughters, $ 3,480 each.
348i To find what rate per cent, one given number is of another.
Ex. 1. What per cent, of 50 is 12? Ans. 24 per cent.
Operation. Since the percentage equals
= 2P5 = .24, Ans. the product of the basis of
2 percentage by the number
12 X X (i (S denoting the rate per cent.
0r — = 24 per cent. (Art. 343), the quotient
0 arising from dividing the
percentage by the number denoting the basis must equal the rate per cent. We therefore divide 12 by the 50, and obtain W = A, which, expressed decimally, equals .24, or 24 per cent. Since the question, evidently, is the same as to find \\ of 100 per cent., we multiply the 12 by 100, or annex two ciphers and divide by 50, and obtain the same result as before.
Rule. — Annex two ciphers to the number denoting the percentage, and divide by the number on which the percentage is reckoned; and the quotient will be the rate per cent. Or,
As the gircn basis of percentage is to the given percentage, so is 100 per cent. to the rate per cent. required.
2. What per cent. of 16 is 2? Ans. 12 J per cent.
3. What per cent, of 110 is 11? Ans. 10 per cent.
4. What per cent. of 2£ is £?
5. ^ of 18 per cent, is what per cent. of 24 per cent.?
Ans. 25 per cent.
6. What per cent, of $ 150 is 25 per cent. of $ 36?
N 22 Ans. 6 per cent.
7. 36 bushels is what per cent, of 48 bushels?
Ans. 75 per cent.
8. What per cent, of 4 years is 1 year 6 months?
9. 31 gallons 2 quarts is what per cent. of 1 hogshead?
Ans. 50 per cent .
10. Of 160 yards of cloth there have been sold 128 yards; what per cent. of the whole remains unsold?
Ans. 20 per cent.
11. What per cent, of £ of j of § is -J? Ans. 20 per cent.
12. In a certain school the number of pupils studying geography is 40 per cent, more than the number studying grammar. What per cent. less is the number studying grammar than the number studying geography? Ans. 28f per cent.
13. If a miller takes out 4 quarts toll from every bushel he grinds, what per cent. does he take for toll?
14. If a certain coin is made of 22. parts copper and 3 parts nickel, what per cent, of it is copper, and what per cent . nickel? Ans. 88 per cent. copper; 12 per cent, nickel.
15. In a certain orchard there are 250 trees, of which 40 per cent, are apple-trees, 12 per cent, cherry-trees, 8 per cent, plum-trees, and the remainder, with the exception of 25 peartrees, consist of peach-trees. What per cent, of the whole are the peach-trees? Ans. 30 per cent.
349. To find a number when a given number is known to be a certain per cent. of it.
Ex. 1. I have bought two house-lots; for the one I paid $ 300, which was 60 per cent, of what I paid for the other. What did I pay for the latter? Ans. $ 500.
Operation. Since $ 300
$300 -r- 60 = $5; $5 X 100 = $500, Ans. is 60 per cent.
g of the unknown
. „ sum, 1 per
Or §^0X1O0-$5OO. cent. of it
60 fflust equal «V
vv of S300, or
$ 5; and 100 per cent., or the whole of it, must equal 100 times $ 5, or $ 500, Ans. And since the question is evidently the same as to find the value of >M of $ 300, we multiply the $ 300 by 100, and divide by (i0, and obtain the same result as before.
Rule. — Annex two ciphers to the number denoting the percentage, and divide by that denoting the rate per cent. Or,
As the given rate per cent. is to 100 per cent., so is the given percentage to the basis of percentage required.
2. 25 is 10 per cent. of what number? Ans. 250.
3. 16£ is 8 per cent, of what number? Ans. 203|-.
4. 72 is 12 per cent, of what number?
5. J is 40 per cent. of what number? Ans. 2^.
6. $1.624 is 12£ per cent, of how many dollars? Ans. $ 13.
7. $ l^r is \ per cent, of what sum? Ans. $ 140.
8. A flock of sheep has lost 154, per cent. of its number, 17 sheep having been killed by the dogs, and 6 more having been drowned. What was the original number? Ans. 150.
9. If a man owning 45 per cent. of a mill should sell 334, per cent. of his share for $ 450, what would be the value of the whole mill?
10. 124. per cent, of the length of a certain railroad is equal to 3m. lfur. lrd. What is its entire length?
Ans. 25m. Ofur. 8rd.
11. Gave to a benevolent society 19 bushels of corn, which was 174 per cent, of all I raised. How many bushels had I left? Ans. 91$ bushels.
12. Dalton says to Turner, $36.89 is 13§ per cent. of the sum you borrowed of me; and Turner replies, It is just 16§ per cent, of the amount I have repaid you. How much of the money that was borrowed remains unpaid? Ans. $ 57.66.
350. To find the number on which the percentage is reckoned, when the amount, or the remainder, and the rate per cent. are given.
Ex. 1. Sold a horse for $ 200, which was 25 per cent, more than he cost. What did he cost? Ans. $ 160.
Operation. Since the
1 + .25 = 1.25; $ 200 -i- 1.25 = $ 160, Ans. $ 200 is evidently the
cost and 25 per cent. of the cost, it must equal the cost taken 1 25 times. Therefore, the given amount, $ 200, divided by 1 increased by .25, or the given per cent. expressed decimally, equals $ 160, the basis of percentage required.