ART. 24.—Percentage; Ratio. Rate of profit or loss is usually given in the form of a percentage, and there is frequently an ambiguity arising from the fact that it is not mentioned whether it is on cost or receipt. In such case it generally refers to the cost. A profit of 12 per cent. on the cost means or, 12 pence profit per 100 pence cost; 12 pence profit = 100 pence cost. Any other unit of value may be substituted for the penny, because it occurs on both sides of the equation. A rate which has the same unit on the two sides is called a ratio-rate; the two units differ only in quality. EXAMPLES. Ex. 1. If by selling at 7s. 6d. per yard you lose 10 per cent. on the outlay, what do you gain or lose per cent. when you sell at 8s. 6d. per yard? Ex. 2. A publisher sells books to a retail dealer at 5s. a copy, but allows 25 copies to count as 24. If the retailer sells each of the 25 copies for 6s. 9d., what profit per cent. does he make? Ex. 3. For a special sale a merchant gave his customers 40 per cent. off the marked price, but the goods had been marked at an advance of 60 per cent. on their cost. Did he gain or lose, and at what rate per cent. on the price received? 1+£ marked = £ cost, 6 Ex. 4. An article in passing from the producer to the consumer passed through the hands of three dealers, each of whom added for his own profit 10 per cent. on the price at which he bought. The final price was 365£; what was the original price? 1+ charged by 1st dealer = £ paid to producer, Τ 1 1+ charged by 2nd dealer = £ paid to 1st dealer, Ex. 5. In shipping ice 12 per cent. is destroyed, 45 per cent. of the shipped ice melts during the passage, 20 per cent of the remainder is lost in landing. At what increase per cent. on the original price per lb. must the residue be sold in order to yield a profit of 142 per cent. on the whole? 88 lbs. shipped = 100 lbs. bought, 55 lbs. arrived = 100 lbs. shipped, 88 x 55 x 80 lbs. sold = Let the buying price be a pence per lb., and the percentage of profit y pence 100' = pence paid, then the selling price is pence per lb. Hence y y 88 x 55 x 80 pence receipt = 1,000,000 pence cost, )88 × 55 × 80 pence receipt = 1,000,000 pence cost. i.l., (1 + 100 88 y 88 x 55 x 80 ...(1+100) 1,000,000 -1 pence profit = penny cost; 1. A grazier bought a sheep for £1 6s., and sold it for £3, and incurred a loss equal to half the cost, plus a quarter of the expense of feeding. What was the expense of feeding? 2. Formerly newspaper wrappers were sold at the rate of eight for fourpence halfpenny, and now they are sold at twelve for sevenpence. What is the percentage increase on the former price? 3. A corn dealer bought wheat at £2 1s. 3d. per quarter, which he subsequently sold at £2 9s. 7d. per quarter, and made a profit of £277 10s. upon the transaction. How many quarters did he buy and sell? 4. A house costs the landlord £130 a year for rent and taxes. For one third of the year it is let at £6 10s. a week, and for the remainder of the year at £5 a week. Find the landlord's profit, and how much it is per cent. 5. A man purchases 80 oxen for £1000, and sells 24 of them at a loss of 3 per cent. outlay. At what rate per head must he sell the remainder so as to lose nothing on the whole? 6. An article is sold for 12s. at a loss of 4 per cent. on cost. At what price must it be sold that a gain of 4 per cent. may be made? 7. A person by selling at 48. 14d. per lb. an article which cost £21 per cwt., cleared 2 per cent. more profit than if he had sold the whole for £162. How much was sold of the article? 8. If by selling at 20s. you lose 16 per cent. on your outlay, at what rate do you sell when you gain 16 per cent. on your outlay? 9. If by selling at 15s. 6d. you lose 7 per cent. on the outlay, what do you gain or lose per cent. when you sell at 16s. 6d. ? 10. A merchant buys cloth at 2s. 31d. per yard, and sells it at 3s. 44d. per yard. What is the percentage of profit on the outlay; and how many yards must he sell to gain a profit of £10? 11. A person bought a lot of land for $10,000. He sold one half of it at a gain of 50 per cent., two fifths of it at $40 an acre, and the remainder at a loss of 40 per cent. He gained 45 per cent. on the whole. Find the number of acres in the lot. 12. A merchant receives £300 for sales in one day; on £200 he gains 40 per cent. What does he gain or lose per cent. on the remaining £100 in order that his profit for the day may be £50? 13. A dealer is paid £23 19s. 2d. for an article. Assuming that it passed through the hands of three dealers and that each added 10 per cent. of the price at which he bought for his own profit, what did the first dealer pay? 14. A man has 1000 apples for sale; at first he sells so as to gain at the rate of 50 per cent. on the cost price; when he has done this for a time the sale falls off, so he sells the remainder for what he can get, and finds that by doing so he loses at the rate of 10 per cent. If his total gain is at the rate of 29 per cent., how many apples did he sell for what he could get? SECTION IV.-MIXTURE. ART. 25.-Composition. Suppose that a tea-dealer forms a mixture out of three kinds of tea, which we shall distinguish as A, B, C. Suppose that he takes a lbs. of A, b lbs. of B, c lbs. of C and mixes them thoroughly. The composition of the resulting mixture is fully given by the equivalence a lbs. of A + b lbs. of B + c lbs. of C = a + b + c lbs. of mixture. (1) From this complete equivalence certain partial equivalences may be derived; thus a lbs. of A = a+b+c lbs. of mixture, (2) that is, there are a and only a lbs. of A, in every a+b+c lbs. of mixture. Similarly, b lbs. of B = a+b+c lbs. of mixture, and c lbs. of Ca+b+c lbs. of mixture. (3) (4) From these partial equivalences three of a different kind can le ART. 26.-Price of Mixture. Suppose that the cost price of A tea is p shillings per lb.; of B, q shillings per lb. ; and of C, r shillings per lb. Then the cost of the a lbs. of A taken is pa shillings; of the b lbs. of B, qb shillings; of the c lbs. of C, rc shillings. Hence the resulting cost price of the mixture is or pa + qb + rc shillings = a+b+c lbs. of mixture, a+b+c shillings = lb. of mixture. If the cost price of the required mixture is known, say n shillings per lb., and the price of each of the components, but not the ratios of composition, we have for determining b/a and c/a the equation i. e., pa + qb + rc = n, If there are only two components, then c/a = 0, and the equation |