: a a chaser receives £6 a-year (interest) on £120 (principal). On £100 (principal), therefore, he receives £ 5 a-year (interest): (Prin.) (Prin.) (Int.) (Int.) a=(100 X 6-1205) £5. So that the actual per-centage is greater in the second case than in the first by nearly £}. A comparison could also be instituted in this way: The yearly interest of £230 being £10, what is the yearly interest of £120-at the same rate? The proportion is (Prin.) (Prin.) (Int.) (Int.) a=(120x10+230=) £52 On £120, therefore, the yearly interest would be (£6–£5*=) £is more in the 6-per-cent stock than in the 10-per-cent. stock; so that £100 would produce nearly £z a-year more in the one stock than in the other: £120 : £100 : : £33 : a : 18 15 0231 = EXAMPLE V.- What quantity of New Threes, at 90), could be obtained for £1,500 of 33 per-cent. stock, at 964 ? The lower the price of stock, the larger the quantity required to realize a certain sum of money, and vice versa. As small, therefore, as 90} is, compared to 96, so large must be the quantity of New Threes, compared to that of the 32-per-cent. stock-in other words, so small must be the quantity of the latter stock compared to that of the former. "We thus have the (inverse) proportion 90} : 96} : : 1,500 : a a=(1,500 X 961:90=) £1,595 6s. Id. nearly. Here is a more round-about solution : £1,500 of stock, if sold at 96, would realize, in cash, £1,4438: (Stock.) (Stock.) (Cash.) (Cash) £100 : £1,500 . : £961 : a=(1,500 x 964--100=) £1,4431. And £1,443å, if invested in New Threes at 90}, would purchase £1,595 6s. id. (nearly) of that stock: iCash. (Cąsh.) (Stock.) (Stock.) £ a = а a PROFIT AND LOSS. 201. Under this head, we apply our knowledge of Proportion to questions relating to the gains and losses of people in trade. Such gains and losses are usually expressed as so much PER CENT.—that is, (at the rate of) so much on £100, cost-price. When an article which cost £ 5 is sold for £6, there is a gain of £1; and when an article which cost £20 is sold for £22, there is a gain of £2. The first of these gains, although absolutely less, is relatively greater, than the second; a gain of £1 on an outlay of £ 5 being at the same rate as a gain of (not £2, but) £4 on an outlay of £20: £5: £20::£1:a; a=(20--5=) £4. Again : when an article which cost £i is sold for 193. 8d., there is a loss of 4d.; and when an article which cost 35. 4d. is sold for 38., there is a loss of 4d. also. The first loss, however, although absolutely the same as the second, is relatively smaller; a loss of 4d. on an outlay of £1 being at the same rate as a loss of two-thirds of id. on an outlay of 38. 4d.: 240d. : 40d. :: 4d. : 0; a=(40x4--240=)}d. . . So that, to form a just estimate of gains and losses, we must, in every case, take the cost-price into account; and the adoption of £100, as a standard cost-price, enables us at once to contrast one gain with another, or one loss with another. The following examples explain themselves : EXAMPLE I.-A draper bought a piece of cloth, 50 yards long, for £27 1os., and realized by the sale of it a gain of £315. ; what was the selling-price per yard? Adding the gain to the cost-price, we find that the sellingprice of the piece was (£27 ios. +£3 158.=) £31 58.; so that the selling-price per yard was £31 55.-50=128. 6d. EXAMPLE II.-A quantity of merchandise which cost £640 was sold for £725; what was the gain per cent. ? On £640 the gain was (£725 - £640=) £85; therefore, to find the gain on £100, we say (Cost-price.) (Cost-prico.) (Gain.) (Gain.) £640 : £100 :: £85 : a a : a EXAMPLE III.-A person bought a quantity of hay for £250, and sold it for £235; how much per cent. did he lose? On £250 he lost (£250-£235=) £15; therefore, to find his loss on £100, we say (Cost-price.) (Cost-price.) (Logs. (Loss.) £250 : £100 :: £15 : a=(15 x 100_250=) £6, the loss per cent. EXAMPLE IV.-A quantity of tea, which cost £365, was sold at a profit of 20 per cent. ; what was the selling price? Every £100, cost-price, realized (£100+£20=) £120, sellingprice; therefore, to find the selling-price realized by £365, costprice, we say- (Cost-price.) (Cost-price.) (Selling price.) (Selling-price.) £100 : £365 :: £120 =(365 x 120+100=) £438, the required selling-price. EXAMPLE V.--An estate, which cost £3,000, was sold at a loss of 15 per cent. ; what was the selling-price? Every £100, cost-price, realized only (£100-£15=) £85, selling-price; therefore, to find the selling-price realized by £3,000, cost-price, we say (Cost-price.) (Cost-price.) (Selling-price.) (Selling-price.) £100 : £3,000 :: £85 a=(3,000 X 85-100=) £2,550, the required selling price. EXAMPLE VI.-A merchant sold a quantity of rice for £585, and gained 12} per cent. ; what did the rice cost him? Every £100, cost-price, produced (£100+£12)=) £112, selling price; therefore, to find the cost-price which produced £585, selling-price, we say (Selling-price.) (Selling-price.) (Cost-price.) Cost-price.) £112] : £585 :: £100 : a=(585 1001124=) { 520, the required cost-price. EXAMPLE VII.-A grazier sold a flock of 200 sheep for £270, and lost 10 per cent. ; how much--on an average- did each sheep cost him ? Every £100, cost-price, produced only (£100-£10=) £90. selling-price; therefore, to find what cost-price produced £270, selling-price, we say (Selling price.) (Selling-price.) (Cost-price.) (Cost-prico.) £90: £270 :: £100 a=(270 x 100-90=) £300, the cost-price of the flock. On an average, therefore, each sheep cost the grazier £300-200=£1 108. : a = a a DIVISION INTO PROPORTIONAL PARTS. 202. Under this head, we employ our knowledge of Proportion in dividing a number into two or more parts proportional to other numbers. EXAMPLE I.—Divide £40 between A and B, so that A's share shall bear to B's the ratio of 3 to 2. Out of every (£3+£2=) £5 in the given amount, £3 must be given to A, and £2 to B. Consequently, as £40 contains 8 sums of £5 each, A must get 8 sums of £3 each, and B 8 sums of £2 each; in other words, £3 must bear to A's share, and £2 to B's share, the ratio which £5 bears to £40. We thus have the proportions -- £,5 : ,40 :: £3 : A's share. £5 : £40 :: £2 : B's Or* 5 : 3 :: £,40 : A's share (£24.) 5 : 2 :: £40 : B's (£16.) EXAMPLE II.-Divide an estate of 2,160 acres between A, B, and C, so that A's share shall bear to B's the ratio of 7 to 8, and B's share to C's the ratio of 8 to 9. Out of every (77.8+9=) 24 acres, 7 must be given to A, 8 to B, and 9 to C. As large, therefore, as the estate is, compared to 24 acres, so large must be A's share compared to 7 acres, B’s share compared to 8 acres, and C's share compared to 9 acres. We thus have- 9 : C's Or* Acres. Acres. 24 : 7 :: 2160 : A's share (630.) 8 :: 2160 : B's (720.) (810.) EXAMPLE III.-Divide 1,000 into four parts proportional to the fractions }, à, }, and } : that is, into four such parts that the first shall bear to the second the ratio of } to ; the second to the third, the ratio of į to }; and the third to the fourth, the ratio of } to . 20 19 20 19 20 19 20 19 20 : 5 fourth , 6 DIVISION INTO PROPORTIONAL PARTS. 217 The preceding examples suggest the following solution : }+*+*+d=18; } :: 1,000 : c; c=21019, , third : 1 :: 1,000: d; d=17535 , This solution is easily explained. It is obvious that, if the fractions 1, 4, 5, and } were all multiplied by the same number, the resulting products would be proportional to the fractions (p. 165); and it is equally obvious that, if their sum were exactly 1,000, those products would be the proportional parts required. So that, m being put for the multiplier which would give products amounting in the aggregate to 1,000, we have {xm+ixm+xm+&xm=1,000; (3+*+}+5) xm=1,000; 20xm=1,000 ; m=. The proportional parts being represented by a, b, c, and d, respectively, we also have a=ixm=1 .; ax19=}x 1,000 1,000 19 19 b= } xm=ix-20; bx8=} x 1,000 ni 1,000 19 20 1,000 19 20 1,000 19 20 1,000 19 20 C=} xm=}x d=}xm=jx ; dx:28=X. 1,000 Hence the above proportions ( 167). a 203. To divide a number into Proportional Parts : Set down, as the second terms of so many proportions, the numbers to which the parts are to be proportional; make the sum of these numbers the first term, in every case; and write, as the third term of each proportion, the number to be divided. The fourth terms will be the proportional parts required. NOTE.—We verify the work by adding the proportional parts together, and finding their sum equal to the number proposed for division. Thus, £24+£16= £40 (Ex. I.); 630 acres +720 acres +810 acres=2,160 acres (Ex. II.); 35059 + 26315 +2101% +17535=1,000 (Ex. III.) |