RATIO, in mathematics, is the relation which one quantity bears to another in respect of magnitude, the comparison be ing made by considering how often one contains, or is contained by the other. Thus, in comparing 6 with 3, we observe that it has a certain magnitude with res pect to 3, which it contains twice ; again, In comparing it with 2, we see that it has a different relative magnitude, for it con tains 2 three times, or it is greater when compared with 2 than it is when compar ed with 3. The ratio of a lob is usually expressed by two points placed between them, thus, a : b ; and the former is called the antecedent of the ratio, the latter the consequent. When one antecedent is the tame multiple part, or parts, of its conse quent, that another antecedent is of its consequent, the ratios are equal. Thus, the ratio of 4: 6 is equal to the ratio of 2: 3, i. e. 4 has the same magnitude when compared with 6, that 2 has when com 4 pared with 3, since = the ratio of a : b is equal to the ratio of c : d, if because and represent the multiple, part, or parts, that a is of b, and c If the terms of a ratio be multiplied or divided by the same quantity, the ratio is a m a not altered. For =-- to b That ratio is greater than mother, whose antecedent is the greater multiple, part, or parts of its consequent. Thus, the ratio of 7 : 4 is greater than the ratio 35 of 8 : 5; because - 7 or — is greater than 4 20 8 32 — or —. These conclusiens follow im 5 20 mediately from our idea of ratio.
"A ratio is called a ratio of greater in equality, of less inequality, or of equality, according as the antecedent is greater, less than, or equal to the consequent." "A ratio of greater inequality is dimi nished, and of less inequality increased, by adding any quantity to both its terms. If to the terms of the ratio 7 : 4, 1 he add ed, it becomes the ratio of 8 : 5, which is less than the former. And in general, let x be added to the terms of the ratio a : 6, and it becomes a + x b x, which is greater or less than the for a b + x mer, according as — is greater or less than a ; or by reducing them, to a common denominator, as -- is b. b x
ab+ax greater or less than --; that is, as b .b x b is greater or less than a. Hence, a tio of greater inequality is increased, and of ess inequality diminished, by taking from the terms a quantity less than either of them.
If the antecedents of any ratios be together, and also the co.ise quents, a new ratio results, which is said to be compounded of the former. Thus, a c : b d is said to be compounded of the two a : b and c : d. It is also sometimes called the sum of the ratios ; and when the ratio a : b is compounded with itself, the resulting ratio, a' : is called the double of the ratio of a: b; and if three of these ratios be compounded together, the result as : 63, is called the triple of the first, &c. Also, the ratio of a : b is said to be one third of the ratio of at : 1.3 ; 1 1 and a; : 67n is said to be an mth part of the ratio of a : b.
Let the first ratio be a : 1; then a' : 1, a3 : 1, ....an : 1, are twice three times, ....n times the first ratio ; where n the index of a, shows what multiple, or part, of the ratio an : 1, the first ratio, a : 1, is. On this account, the indices, 1, 2, 3, ...n, are called measures of the ratios : 1, : 1, a3 : 1, an : 1.
"If the consequent of the preceding ratio be the antecedent of the succeeding one, and any number of such ratios be ta ken, the ratio which arises from their composition is that of emit' antece dent to the last consequent." Let a : b, b : c, c : d, &c. be the ratios, the com pound ratio is a X x c: 6 x c X d; or dividing by b X c, a : d.
"A ratio of greater inequality, com pounded with another, increases it ; and a ratio of less inequality diminishes it." Let the ratio of x: y oe compounded with the ratio of a : b, and the resulting ratio ax : by is greater or less than the ratio a : b, according as is greater or less b y than e. according as x is greater or less than y.