and another ratio to be that of c to d; then the first ratio a ad and the second ratio с bc d bd = Hence the first ratio is greater than, equal to, or less than the second ratio, according as ad is greater than, equal to, or less than bc. 352. A ratio is called a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less,than, or equal to the consequent. 353. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding any number to both terms of the ratio. a Let the ratio be and let a new ratio be formed by adding a to both terms of the original ratio; then a + x b+x is greater or less than, according as b(a+a) is greater or less than a (b+x); that is, according as ba is greater or less than ax, that is, according as b is greater or less than a. 354. A ratio of greater inequality is increased, and a ratio of less inequality is diminished, by taking from both terms of the ratio any number which is less than each of those terms. Let the ratio be a b and let a new ratio be formed by taking a from both terms of the original ratio; then a-x a b -X is greater or less than, according as b(a−x) is greater or less than a (b-x); that is, according as be is less or greater than ax, that is, according as b is less or greater than a. 355. If the antecedents of any ratios be multiplied together, and also the consequents, a new ratio is obtained which is said to be compounded of the former ratios. Thus the ratio ac: bd is said to be compounded of the two ratios a b and c : d. : When the ratio ab is compounded with itself the resulting ratio is a2: b2; this ratio is sometimes called the duplicate ratio of a : b. And the ratio a3: b3 is sometimes called the triplicate ratio of a : :b. 356. The following is a very important theorem concerning equal ratios. where p, q, r, n are any numbers whatever. The same mode of demonstration may be applied, and a similar result obtained when there are more than three ratios given equal. As a particular example we may suppose n=1, then we see that if pa+qc+re pb+qd+rf ; and then as a special case we may suppose p=q=r, so that each of the given cqual ratios is equal to a+c+e b+d+f' EXAMPLES. XXXV. 1. What is the ratio of fourteen shillings to three guineas? 2. Arrange the following ratios in the order of magnitude; 3:4, 7:12, 8:9, 2:3, 5:8. 3. Find the ratio compounded of 4 : 15 and 25:36. 4. Two numbers are in the ratio of 2 to 3, and if 7 be added to each the ratio is that of 3 to 4: find the numbers. 5. Two numbers are in the ratio of 4 to 5, and if each be diminished by 6 the ratio is that of 3 to 4: find the numbers. 6. Two numbers are in the ratio of 5 to 8; if 8 be added to the less number, and 5 taken from the greater number, the ratio is that of 28 to 27: find the numbers. 7. Find the number which added to each term of the ratio 5 3 makes it three-fourths of what it would have become if the same number had been subtracted from each term. 8. Find two numbers in the ratio of 2 to 3, such that their difference has to the difference of their squares the ratio of 1 to 25. 9. Find two numbers in the ratio of 3 to 4, such that their sum has to the sum of their squares the ratio of 7 to 50. 10. Find two numbers in the ratio of 5 to 6, such that their sum has to the difference of their squares the ratio of 1 to 7. 11. Find a so that the ratio : 1 may be the duplicate of the ratio 8: x. 12. Find x so that the ratio a-x: b-x may be the duplicate of the ratio a: b. 13. A person has 200 coins consisting of guineas, halfsovereigns, and half-crowns; the sums of money in guineas, half-sovereigns, and half-crowns are as 14 8:3; find the numbers of the different coins. 14. If b-ab+a=4a-b: 6a-b, find a b. XXXVI. Proportion. 357. Four numbers are said to be proportional when the first is the same multiple, part, or parts of the second as the third is of the fourth; that is when a с b d' the four numbers a, b, c, d are called proportionals. This is usually expressed by saying that a is to b as c is to d; and it is represented thus, a : b :: c : d, or thus a : b=c : d. The terms a and d are called the extremes, and b and c the means. 358. Thus when two ratios are equal, the four numbers which form the ratios are called proportionals; and the present Chapter is devoted to the subject of two equal ratios. 359. When four numbers are proportionals the product of the extremes is equal to the product of the means. Let a, b, c, d be proportionals; If any three terms in a proportion are given, the fourth may be determined from the relation ad=bc. = If b c we have ad=b2; that is, if the first be to the second as the second is to the third, the product of the extremes is equal to the square of the mean. In this case a, b, c are said to be in continued proportion. 360. If the product of two numbers be equal to the product of two others, the four are proportionals, the terms of either product being taken for the means, and the terms of the other product for the extremes. : 361. If a b c d, and c: def, then a bef. 362. If four numbers be proportionals, they are proportionals when taken inversely; that is, if a b c d, then bad For thus α с b c. =; divide unity by each of these equals; = d d α с or bad c. 363. If four numbers be proportionals, they are proportionals when taken alternately; that is, if a b c : d, then a cb: d. b For; multiply by thus or a cb d. α b : 364. If four numbers are proportionals, the first together with the second is to the second as the third together with the fourth is to the fourth; that is if a b c d, then a+bb :: c+d: d. 365. Also the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth. |