gives the value of b/a. Otherwise the equation is indeterminate; it is then a question of finding the simplest solution. For the meaning of the slant bar see Art. 11. ART. 27.-Sharing. Denote a bankrupt's creditors by A, B, C, and their claims by a£, b£, c£, respectively. Then a£ of A+ b£ of B + c£ of C = a + b + c£ liability. From this equivalence partial equivalences may be derived as in Art. 25. It is a convention in the commercial world, that the assets are to be distributed according to the ratios supplied by the above equivalence. EXAMPLES. Ex. 1. A grocer mixes four kinds of tea, in the proportions of 1, 1.5, 2, 2.5 parts by weight of the several kinds respectively. Required the number of pounds of each kind in one hundredweight of the mixture. 2 lbs. 1st + 3 lbs. 2nd + 4 lbs. 3rd + 5 lbs. 4th = 14 lbs. mixture, 112 lbs. of mixture, Similarly 24 lbs. of 2nd, 32 lbs. of 3rd, 40 lbs. of fourth. Ex. 2. The four kinds of tea in the preceding question, having cost the grocer at the rates of 5, 4, 3, 2 shillings per lb. respectively; required the price per lb. at which he must sell the mixture in order to realize 25 per cent. profit on his outlay? 2 lbs. 1st + 3 lbs. 2nd + 4 lbs. 3rd + 5 lbs. 4th = 14 lbs. mixture, 5s. per lb., 4s. per lb., 3s. per lb., 2s. per lb. 10+12+12+10 shillings cost = 14 lbs. mixture, Now i.e., or 1+ shilling receipt = 1 shilling cost; (1+1) 22 shilling receipt = 1 lb. mixture; Ex. 3. Oranges are bought for half-a-crown a hundred; some are sold at 3s. 6d. a hundred, and the rest at 2s. 101⁄2d. a hundred : the same profit is made as if they had all been sold at 3s. 11⁄2d. a hundred. Of a thousand oranges sold, how many fetch 3s. 6d. a hundred ? xoranges 1st kind + 1000 - x oranges 2d kind = 1000 oranges mixed. 42 pence per 100 of 1st, 34.5 per 100 of 2nd, 42x, 34.5 (1000-x) 100 + 100 pence receipt = 1000 oranges mixed. But this is equal to 375 pence receipt = 100 oranges mixed; therefore we get the equation 42 34.5 (1000) + Hence 400 oranges out of 1000 sold fetch 3s. 6d. a hundred. Observe-It is not necessary to give the original price. Ex. 4. Out of a cask containing 360 quarts of pure alcohol a quantity is drawn off and replaced by water. Of the mixture, a second quantity, 84 quarts more than the first, is drawn off and replaced by water. The cask now contains as much water as alcohol. Find how many quarts were taken out the first time. Show that the problem has only one solution. 2 quarts water+ (360-x) quarts alcohol 360 quarts 1st mixture, (360-x-84) quarts 1st mixture left; therefore (360-x-84) quarts water + (360 − x − 84) 360 360-x 360 The second mixture is formed by filling up with water, quarts alc. = 360 qrts. 2nd mixture. -qrts. alcohol = 1 quart alcohol = 2 qrts. 2nd mixture; which reduces to the quadratic equation x2 – 636x + 96 × 360 = 0, the roots of which are x = 576 and x = 60. Only the latter is possible, for 576 is greater than 360. Ex. 5. A creditor received 16s. 3d. in the pound, and thereby lost 1357. 10s.; how much was due to him? 1. A vessel is filled with a mixture of spirit and water in which there is 70 per cent. of spirit; 19 gallons are taken out, and the vessel is filled up again with water; the proportion of spirit is now found to be 56.7 per cent.; find how much the vessel contains. 2. If apples are bought for 4 a penny, and mixed with an equal number bought for 3 a penny, and then sold at the rate of 5 for 2 pence; what is the gain per cent. on the outlay? 3. If gunpowder is composed of nitre, charcoal, and sulphur, in the proportions of 16, 3, 21; how much of each is required for one cwt. of gunpowder? 4. An estate is divided into three portions of 250 acres 62 acres 2 roods, and 19 acres 1 rood 20 poles; these portions are let at 17. 5s. 4d., 17. 1s. 8d., and 31. per acre respectively. At what uniform rent per acre might the whole estate be let so as to bring in the same rental? 5. A milk-dealer buys pure milk at 11d. per gallon. How much water must he add that he may sell at 5d. a quart, and obtain a gross profit of 100 per cent.? 6. A tobacconist pays 4s., 3s. 6d., and 2s. 6d. per lb. for three kinds of tobacco. He mixes them, and, by selling at 4s. per lb., obtains a gross profit of 25 per cent. on his receipts. If he omitted the cheapest tobacco from his mixture, keeping the others in the same proportion, his profit would be only 24d. per lb. What was his mixture? 7. The estate of a bankrupt pays 4s. 44d. in the £; what loss will a creditor sustain whose claim is for 3371. 6s. 8d. ? 8. A bankrupt owes 7,3577. 128., and his assets for distribution among his creditors amount to 3.0657. 13s. 4d. How much in the pound will they receive? 9. A bankrupt whose estate is worth 1,8237 18s. 9d. owes to three persons sums of 1,031. 5s., 8147., and 5867. 13s. 4d. respectively; how much can he pay in the pound? 10. A statement of affairs showed liabilities 14,6637. and assets 8,4617. A composition of 10s. in the £ was accepted. What was the theoretical value of the composition? 11. A merchant mixes a lbs. of one kind of tea, b lbs. of a second, and c lbs. of a third, the cost prices of the three kinds being respectively p, q, r shillings per lb.; find the percentage of profit in his receipts when he sells at m shillings per lb. 12. A grocer mixes a lbs. of tea at p shillings per lb., 6 lbs. at q shillings per lb., c lbs. at r shillings per lb., and d lbs. at s shillings per lb. Required the retail price of the mixture in order that he may realize k per cent. profit on his outlay. 13. An apple-woman, having purchased / dozen of apples at p pence per dozen, m dozen at q pence per dozen, and n dozen at r pence per dozen, has to dispose of the three lots afterwards at p+q+r pence per three dozen. Required, the con-dition that she should just realize her original outlay. 14. A fruit-dealer found that he had left on hand a quantity of peaches, of which he had bought one half at the rate of 5 for a shilling and the other half at the rate of 6 for a shilling, and, thinking to escape without loss, sold them all at the rate of 11 for 2 shillings, but found that by so doing he had incurred a loss of eighteenpence. At what rate ought he to have sold them in order to have made a profit of a guinea upon the transaction? 15. A grocer can sell coffee at 30 cents per lb., and realize a profit of 25 per cent. He, however, mixes the coffee with chicory, which cost him 6 cents per lb., and selling the mixture at 25 cents per lb. realizes a profit of 40 per cent. How much per cent. of coffee does the adulterated mixture contain? 16. How must a grocer mix tea at 2s. 6d. per lb. and 3s. per lb. in order to pro duce a mixture worth 2s. 8d. per lb. ? 17. A sum of 237. 14s. is to be divided between A, B, and C; if B gets 20 per cent. more than A, and 25 per cent. more than C, how much does each get? SECTION V.-SIMPLE INTEREST. ART. 28.-Rate of Interest. Interest is money paid for the loan of a sum of money. When the interest is made proportional to the sum lent and to the time during which it is lent, it is called simple interest. The rate of interest has then the form r£ interest per £ principal per annum; but as the interest is a quantity of a lower order than the principal, the rate is usually expressed in the percentage form, as for example 5£ interest per 100£ principal per annum, which is equivalent to 1/20£ interest per £ principal per annum. Expressed in the form of an equivalence, it is— 5£ interest = 100£ principal by year, 1£ interest: 20£ principal by year, or or or or = 1£ interest per 20£ principal = year, 1£ interest per year 20£ principal. = When the dependent quantity in a rate and one of the independent quantities are expressed in terms of the same unit, the value of the rate is independent of the size of that unit. Thus, if "shilling" be substituted for "pound" in the rate of interest, the value of the rate remains unaltered. In such cases the size of the unit is indifferent, but its kind is determinate; in the case of interest it must be some unit of value. ART. 29.—Amount and Present Value. Interest is an increment; by adding it to the principal we get the amount, that is, the new value of the principal. If, for one year, then r£ interest = £ principal, 1 + r£ at end of year = £ at beginning. This is called the rate of improvement of money. Let t denote any integral or fractional number, then tr£ interest£ principal, and 1+tr£ at end of t years = £ at beginning. The reciprocal of (1) is 1 1+r £ at beginning of year = £ at end of year, and this is the rate of present value for one year. The reciprocal of (3) is (1) (2) (3) (4) |