CHAPTER XV.

RATIO. PROPORTION.

RATIO.

Art. 297. Ratio is the relation, in respect to value, which one quantity has to another of the same kind.

A ratio is a number which expresses ratio.

Ratio implies the comparison of one number with another, in respect to their value. Unlike numbers can have no ratio.

Like numbers may be compared, in respect to their value, in two ways, namely:

FIRST.-By subtraction, to find how much one number exceeds the other. The difference is sometimes called the arithmetical ratio, but the name difference is generally preferred.

SECOND. By division, to find how many times one number contains the other. The quotient is sometimes called the geometrical ratio, but the name ratio is generally preferred.

Art. 298. In estimating ratio, many mathematicians find the value of the first number as compared with the second. Thus, they consider the question "What is the ratio of 2 to 10?" as meaning "What is 2 as compared with 10?" and answer "Two is of 10, and the ratio of 2 to 10 is ." A few mathematicians find the value of the second number as compared with the first. To them the question "What is the ratio of 2 to 10?" means "What is 10 as compared with 2?" and they answer "Ten is 5 times 2, and the ratio of 2 to 10 is 5." In this work ratio will be considered as a comparison of the first number with the second, in accordance with the custom of nearly all French, German, and English writers.

ILLUSTRATIONS OF RATIO.

1. What relation has 3 to 6? 2. What relation has 6 to 3?

Ans. It is , or as much.

Ans. It is g, or twice as much.

3. What is 4, compared with 12?

Ans. It is, or as much.

4. What is 12, compared with 4?

Ans. 12, or 3 times as much.

Art. 299. The terms of a ratio are the two numbers which are compared.

The antecedent is the first term of a ratio.

The consequent is the second term of a ratio.

When both terms are mentioned together, they are sometimes called a couplet.

:

Art. 300. Ratio is indicated in two ways, namely:FIRST.-By a fraction whose numerator is the antecedent, and whose denominator is the consequent.

SECOND. By writing the consequent after the antecedent, with a colon between them.

Thus, the ratio of 3 to 7 is written either, or 3: 7, and is read "three is to seven."

The value of a ratio is the quotient of the antecedent divided by the consequent.

NOTE. Those who take the second view of ratio mentioned in Art. 298, write the consequent for the numerator, and the antecedent for the denominator, when writing ratio as a fraction; but, when using the colon, they write the antecedent before and the consequent after the colon.

Art. 301. The terms of a ratio must express not only the same kind of quantity, but the same denomination, if either is denominate. Thus, the ratio of 4 inches to 5 feet is not ; but, if we reduce 5 ft. to 60 in., the ratio is o

60

1

15'

Art. 302. Since the antecedent of a ratio is the numerator of a fraction, and the consequent is the denominator,

FIRST.-The value of a ratio varies directly as the antecedent. Therefore, multiplying the antecedent by a number multiplies the value of the ratio by that number; and dividing the antecedent by a number divides the value of a ratio by that number.

SECONDLY.-The value of a ratio varies inversely as a consequent. Therefore, multiplying the consequent by a number divides the value of the ratio by that number; and dividing the consequent by a number multiplies the value of the ratio by that number.

THIRDLY.-The value of a ratio is constant when the terms vary proportionally. Therefore, multiplying both terms by the same number, or dividing both terms by the same number, does not affect the value of the ratio.

EXERCISES.

8 9 16 24

7 4 5

1. Illustrate the first principle with 8, 32, 4, 12, 14. 2. Illustrate the second principle with 10, 28, 18, 30, 41. 3. Illustrate the third principle with, 18, 3, 4, 81

10 12 16 54

Art. 303. In reference to the number of its terms, a ratio is either simple or compound.

A simple ratio is a ratio which has only one antecedent and one consequent. Thus 3: 4, or å, is a simple ratio.

A compound ratio is a ratio which expresses the product of two or more ratios. Thus, if the ratio 3:4 be compounded with the ratio 5: 6, the compound ratio is 3 × 5:4 × 6, = 3 × = }}. X

A ratio which is compounded of two equal ratios is said to be duplicate of either of them. Thus, the ratio compounded of 2:4 and 3:6, each being equal to, is equal to the ratio 12:22, or, and 1:4 is the duplicate ratio of each constituent of the compound ratio. A ratio compounded of three equal ratios is said to be triplicate of the constituent ratios, and is the ratio of the cubes of its terms; a ratio

compounded of four equal ratios is said to be quadruplicate of the constituent ratios, and is the ratio of the fourth powers of its terms; &c. (See Art. 44.)

Art. 304. The ratio of the reciprocals of two numbers is called the reciprocal ratio, or inverse ratio, of those numbers. Thus, the reciprocal ratio of 3: 4, or 3, is, or

The reciprocal ratio of two numbers is equal to the reciprocal of their ratio. Thus, the reciprocal of is 3.

The ratio of two numbers is sometimes called direct, to distinguish it from their inverse, or reciprocal ratio. Thus, the direct ratio of 3 to 4 is 3:4, or ; and their reciprocal ratio is 4:3, or 3.

Art. 305. A ratio of equality is a ratio whose value is 1. Thus, 3:3, 2 × 9: 3 × 6 are ratios of equality.

A ratio of greater inequality is a ratio whose value is greater than 1. Thus, 6:4, 4×7:3 × 5 are ratios of greater inequality.

A ratio of less inequality is a ratio whose value is less than 1. Thus, 5:7, 2 × 3:5×8 are ratios of less inequality.

Art. 306. If the same number be added to both terms of a ratio of equality, the value of the ratio is not changed. Thus, 2:2 2+1:2+1 3:3, a ratio of equality.

Art. 307. If the same number be added to both terms of a ratio of greater inequality, the value of the ratio is diminished; and, if the same number be subtracted from both terms of a ratio of greater inequality, the value of the ratio is increased. Thus, 2; 164:8+ 4 = 20:12 13; and 16 - 4:8 12:4

16:8

4

=

3.

=

Art. 308. If the same number be added to both terms of a ratio of less inequality, the value of the ratio is increased; and, if the same number be subtracted from both terms of a ratio of less inequality, the value of the ratio is diminished. Thus, 8:16 =84:16 + 4 = 12:20; and 8 — 4: 16 44:12 1.

=

૩.

NOTE. The fact that an equal increase or decrease of the less of two numbers is more in proportion to the value of that number than the other, explains the facts of Articles 306 and 307.

Art. 309. The ratio of the sum of the antecedents of two or more equal ratios to the sum of the consequents is the same as the ratio of any one of the antecedents to its consequent. Thus, in each of the couplets 15:3, 25:5, and 30:6, the ratio is 5; and in 15+ 25+ 30: 3 + 5+ 6, or 70:14, the ratio is 5. Again, in 1, §, and , each ratio is 2; also, in 18, (formed as described,) the ratio is 2.

Thus,

Art. 310. The ratio of the difference of the antecedents of two equal ratios to the difference of the consequents is the same as the ratio of either antecedent to its consequent. in each couplet 45: 5, and 18: 2, the ratio is 9; 18:52, or 27: 3, the ratio is 9. Again, in and 10 each ratio is 2; also, in §, (formed as described,) the ratio is 2.

6

PROPORTION.

and in 45

Art. 311. Proportion, in general, is similarity of the relations of quantities. Proportion in two magnitudes, two structures, two systems, or two mixtures, exists when the parts of one are related to each other and the whole just as the parts of the other are related to each other and the whole. Proportion, in Arithmetic, is equality of ratios.

A proportion is a statement of the equality of ratios. Thus, the equation 10-12 is a proportion. A statement of the equality of the value of ratios is not always a proportion. Thus, in the above proportion, the equation, 2 = 2, expressing the value of the ratios, is not a proportion. Every term of the ratios must be a term in a proportion formed by them. Art. 312. The equality of one ratio to another is indicated in three ways, namely:

FIRST. As the equality of one fraction to another, as =§. SECOND. By writing the ratios with colons, and placing the sign of equality between them; as 8:46:3.

THIRD. By writing the ratios with colons, and placing a double colon between them; as 8:4:: :6:3.