445. Percentage is that part of any number which is indicated by the rate. 446. The Base is the number on which the percentage is computed. 447. The Amount is the sum obtained by adding the percentage to the base. 448. The Difference is the remainder obtained by subtracting the percentage from the base. PROBLEM I. 449. Given, the base and rate, to find the percentage.. NOTE 1.-Percentage is always a product, of which the base and rate are the factors. 11. What is 33 % of $856? Ans. $31.39. 12. What is % of §? 13. What is 143 % of 51? 14. If the base is $375, and the rate .05, what is the percentage? Ans. $18.75. 15. A man owed $536 to A, $450 to B, and $784 to C; how much money will be required to pay 54 % of his debts? 16. My salary is $1500 a year; if I pay 15 % for board, 5 % for clothing, 6% for books, and 8 % for incidentals, what are my yearly expenses? Ans. $510. NOTE 2.15 % +5% + 6 % +8 % 34 %. In all cases where several rates refer to the same base, they may be added or subtracted, according to the conditions of the question. 17. A man having a yearly income of $3500, spends 10 per cent of it the first year, 12 per cent. the second year, and 18 per cent. the third year; how much does he save in the 3 years? 18. A had $6000 in a bank. He drew out 25% of it, then 30% of the remainder, and afterward deposited 10 % of what he had drawn; how much had he then in bank? Ans. $3435. 19. A merchant commenced business, Jan. 1, with a capital of $5400, and at the end of 1 year his ledger showed the condition of his business as follows: For Jan., 2 % gain; Feb., 34 % gain; March,% loss; Apr., 2 % gain; May, 23 % gain; June, 14 % loss; July, 1 % gain; Aug., 1 % loss; Sept., 2 % gain; Oct., 4% gain; Nov., % loss; Dec., 3 % gain. What were the net profits of his business for the year? PROBLEM II. Ans. $918. 450. Given, the percentage and base, to find the ་ RULE. Divide the percentage by the base. EXAMPLES FOR PRACTICE. 1. What per cent. of $720 is $21.60 ? 10. The base is $578, and the percentage is $26.01; what is the rate? Ans. 4% 11. The base is $972.24, and the percentage is $145.836; whatis the rate? 12. An editor having 5600 subscribers, lost 448; what was his loss per cent? Ans. 8. 13. A merchant owes $7560, and his assets are $4914; what per cent. of his debts can he pay? 14. A man shipped 2600 bushels of grain from 455 bushels were thrown overboard during a gale; rate per cent. of his loss? Ans. 65. Chicago, and what was the 15. A miller having 720 barrels of flour, sold 288 barrels; what per cent. of his stock remained unsold? Ans. 60. 16. What per cent. of a number is 30 % of of it? 17. The total expenditures of the General Government, for the year ending June 30, 1858, were $83,751,511.57; the expenses of the War Department were $23,243,822.38, and of the Navy Department, $14,712,610.21. What per cent. of the whole expense of government went for armed protection? Ans. 45, nearly. 18. In the examination of a class, 165 questions were submitted to each of the 5 members; A answered 130 of them, B 125, C 96, D 110, and E 160. What was the standing of the 'PROBLEM III. 451. Given, the percentage and rate, to find the base. 1. 18 is 5% of what number? Ans. 750. 3. 17.5 is 23 % of what number? 4. 2.28 is 5% of what number? 5. 414 is 120 % of what number? 6 6119 is 105 % of what number? 7. 43 is 71% of what number? 8. The percentage is $18.75, and the rate is 2%; what is the base? Ans. 5800. Ans. .6. Ans. $750. 9. The percentage is 314, and the rate 31%; what is the base? 10. I sold my house for $4578, which was 84 % of its cost; what was the cost? Ans. $5450. 11. A wool grower sold 3150 head of sheep, and had 30 % of his original flock left; how many sheep had he at first? 12. A man drew 40 % of his bank deposits, and expended 13 % of the money thus drawn in the purchase of a carriage worth $116; how much money had he in bank? Ans. $2175. 13. If $147.56 is 13 % of A's money, and 43% of A's money is 8 % of B's, how much more money has A than B? Ans. $461.12}. 14. In a battle 4 % of the army were slain upon the field; and 5 % of the remainder died of wounds, in the hospital. The difference between the killed and the mortally wounded was 168; how many men were there in the army? Ans. 21000. NOTE.-100 % 4 % 96 %, left after the battle; and 5 % of 96 % 4%, the part of the army that died of wounds. = 15. A owns of a prize and B the remainder; after A has taken 40% of his share, and B 20 % of his share, the remainder is equitably divided between them by giving A $1950 more than B; what is the value of the prize? Ans. $7800. A PROBLEM IV. 452. Given, the amount and rate, to find the base. 1. What number increased by 5% of itself is equal to 378? RULE. 21 20 Or, = = ANALYSIS. If any number be increased by 5 % of itself the amount will be 1.05 times the number. We therefore divide the given amount,.378, by 1.05, or 1⁄2, and obtain the base, 360, which is the number required. Hence the Divide the amount by 1 plus the rate. NOTE 1.-The amount is always a product, of which the base is one factor, and 1 plus the rate the other factor. EXAMPLES FOR PRACTICE. 1. What number increased by 15 % of itself is equal to 644? Ans. 560. 2. A has $815.36, which is 4 % more than B has; how much money has B? Ans. $784. 3. Having increased my stock in trade by 12 % of itself, I find that I have $3800; how much had I at first? 4. In 1860 the population of a certain city was 39600, which was an increase of 10 % during the 10 years preceding; what was the population in 1850? |