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XXV. RATIO AND PROPORTION.

376. The Ratio of one number to another is the quotient obtained by dividing the first number by the second.

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Thus, the ratio of a to b is; and it is also expressed a: b.

377. A Proportion is an equality of ratios.

Thus, if the ratio of a to b is equal to the ratio of c to d, they form a proportion, which may be written in either of the forms:

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378. The first term of a ratio is called the antecedent, and the second term the consequent.

Thus, in the ratio a: b, a is the antecedent, and b is the consequent.

The first and fourth terms of a proportion are called the extremes, and the second and third terms the means.

Thus, in the proportion a:bc:d, a and d are the extremes, and b and c the means.

379. In a proportion in which the means are equal, either mean is called a Mean Proportional between the first and last terms, and the last term is called a Third Proportional to the first and second terms.

A Fourth Proportional to three quantities is the fourth term of a proportion whose first three terms are the three quantities taken in their order.

Thus, in the proportion a: b=b: c, b is a mean propor tional between a and c, and c is a third proportional to a and b.

In the proportion a: bc:d, d is a fourth proportional to a, b, and c.

380. A Continued Proportion is a series of equal ratios, in which each consequent is the same as the following antecedent; as,

a:b=b:cc:d = d: e.

PROPERTIES OF PROPORTIONS.

381. In any proportion the product of the extremes is equal to the product of the means.

Let the proportion be a:bc: d.

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382. A mean proportional between two quantities is equal to the square root of their product.

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That is, in any proportion either extreme is equal to the product of the means divided by the other extreme; and either mean is equal to the product of the extremes divided by the other mean.

384. (Converse of Art. 381.) If the product of two quantities is equal to the product of two others, one pair may be made the extremes, and the other pair the means, of a proportion.

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385. In any proportion the terms are in proportion by Alternation; that is, the first term is to the third as the second term is to the fourth.

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386. In any proportion the terms are in proportion by Inversion; that is, the second term is to the first as the fourth term is to the third.

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387. In any proportion the terms are in proportion by Composition; that is, the sum of the first two terms is to the first term as the sum of the last two terms is to the third term.

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388. In any proportion the terms are in proportion by Division; that is, the difference of the first two terms is to the first term as the difference of the last two terms is to the third

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389. In any proportion the terms are in proportion by Composition and Division; that is, the sum of the first two terms is to their difference as the sum of the last two terms is to their difference.

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390. In a series of equal ratios, any antecedent is to its consequent as the sum of all the antecedents is to the sum of

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In like manner the theorem may be proved for any num ber of equal ratios.

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392. In any number of proportions, the products of the corresponding terms are in proportion.

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