26. x (bc-xy) y (xy-ac), xy (ay + bx-xy) = abc (x+y-c).

369. Ratio is the relation which one quantity bears to another with respect to magnitude, the comparison being made by considering what multiple, part, or parts, the first is of the second.

Thus in comparing 6 with 3, we observe that 6 has a certain magnitude with respect to 3, which it contains twice; again, in comparing 6 with 2, we see that 6 has now a different relative magnitude, for it contains 2 three times; or 6 is greater when compared with 2 than it is when compared with 3.

370. The ratio of a to b is usually expressed by two points placed between them, thus, a : b; and a is called the antecedent of the ratio, and b the consequent of the ratio.

371. A ratio is measured by the fraction which has for its numerator the antecedent of the ratio, and for its denominator the consequent of the ratio. Thus the ratio of a to b is measured

then for shortness we may say that the ratio of a to b is

372. Hence we may say that the ratio of a to b is equal to

373. If the terms of a ratio be multiplied or divided by the same quantity the ratio is not altered.

374. We may compare two or more ratios by reducing the fractions which measure these ratios to a common denominator. Thus suppose one ratio to be that of a to b, and another ratio to

be that of c to d; then the first ratio

ratio

с

d

=

bc

bd'

a

=

ad

b bd

and the second

,

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.

375. 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.

376. A ratio of greater inequality is diminished, and a ratio of less inequality is increased, by adding any quantity to both terms of the ratio.

α

Let the ratio be

b' x to both terms of the original ratio; then

and let a new ratio be formed by adding

α

b (a + x)
b (b + x)

a + x
b + x
is greater or less than

is greater or less

than according as is, according as b (a + x) is greater or less than a (b + x); that is, according as xb is greater or less than xa; that is, according as b is greater or less than a.

377. A ratio of greater inequality is increased, and a ratio of less inequality is diminished, by taking from both terms of the ratio any quantity which is less than each of those terms.

a- х

x from both terms of the original ratio; then

is greater or

that is, 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.

378. 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.

379. The ratio compounded of two ratios is sometimes called the sum of those two ratios. When the ratio a: b is compounded with itself, the resulting ratio a3: b2 is sometimes called the double of the ratio a: b. Also the ratio a3 : b3 is called the triple of the ratio ab. Similarly, the ratio ab is sometimes said

1 1

to be half of the ratio a2: b3, and the ratio a' :b" is sometimes

This language, however, is now not used; the following terms are in conformity with it, and some of them are still retained. The ratio a b2 is said to be the duplicate ratio of a: b, and the ratio a3 3 the triplicate ratio of a b. Similarly, the ratio Jab is called the subduplicate ratio of a : b, and the ratio

:

:

ab the subtriplicate ratio of a b. is called the sesquiplicate ratio of a : b.

And the ratio a: b

380. If the consequent of the preceding ratio be the antecedent of the succeeding ratio, and any number of such ratios be taken, the ratio which arises from their composition is that of the first antecedent to the last consequent.

Let there be three ratios, namely a : b, b: c, cd; then the compound ratio is a xbxc: bxcxd (Art. 378), that is, a : d.

Similarly, the proposition may be established whatever be the number of ratios.

381. A ratio of greater inequality compounded with another increases it, and a ratio of less inequality compounded with another diminishes it.

Let the ratio x : y be compounded with the ratio a : b; the compound ratio is xa : yb, and this is greater or less than the

ха

α

ratio a b, according as is greater or less than that is,

yb

according as x is greater or less than y.

382. If the difference between the antecedent and the consequent of a ratio be small compared with either of them, the ratio of their squares is nearly obtained by doubling this difference.

2

Let the proposed ratio be a +x : a, where x is small compared with a; then a2 + 2ax+x3 : a3 is the ratio of the squares of the antecedent and consequent. But x is small compared with a, and therefore x2 or xxx is small compared with 2ɑ × x, and much smaller than a×a. Hence a2+2ax: a3, that is, a + 2x : a, will nearly express the ratio (a+x): a3.

Thus the ratio of the square of 1001 to the square of 1000 is nearly 1002: 1000. The real ratio is 1002.001 : 1000, in which the antecedent differs from its approximate value 1002 only by one-thousandth part of unity.

383. Hence we may infer that the ratio of the square root of a+2x to the square root of a is the ratio a +x: a nearly, when x is small compared with a. That is; if the difference of two quantities be small compared with either of them, the ratio of their square roots is nearly obtained by halving this difference.

In the same manner as in Art. 382 it may be shewn when x is small compared with a, that a + 3x a is nearly equal to the ratio (a + x)3 : a3, and a + 4x: a is nearly equal to the ratio (a + x)* : a*.

These results may be generalised by the student when he is. acquainted with the Binomial Theorem.

384. We will place here a theorem respecting ratios which

is often of use.

therefore p (kb)" + q (kd)" + r (kf)" = pa” + qc′′ + re" ;

The same mode of demonstration may be applied, and a similar

a с e

result obtained, when there are more than three ratios b'd' f

given equal. It may be observed that p, q, r, n are not necessarily positive quantities.

As a particular example we may suppose n = 1, then we see

and then as a special case we may suppose p=q=r, so that each

of the given equal ratios is equal to

a+c+e

b+d+Ĩ

EXAMPLES OF RATIO.

1. Write down the duplicate ratio of 2: 3, and the subduplicate ratio of 100 144.

:

2. Write down the ratio which is compounded of the ratios 35 and 79.