RATIO AND PROPORTION.

Art. 154. We arrive at a knowledge of particular quantities by comparing them with other quantities, which are either equal to, or greater or less than those which are the objects of inquiry. We may inquire, how much greater one quantity is than another; or how many times the one contains the other. The answer to either of these inquiries is termed a ratio of the two quantities. One is called arithmetical, and the other geometrical ratio.

Art. 155.-Arithmetical ratio is the difference between two quantities. Thus, the arithmetical ratio of 6 to 3 is 3. It is sometimes expressed by two points placed horizontally between the two quantities; thus, 633, which is the same as 6-3=3.

Art. 156.-Geometrical ratio is the quotient arising from dividing one quantity by another. Thus, the ratio of 6 to 3 is , or 2. Geometrical ratio is expressed by two points placed one over the other, between the two quantities compared; thus 6 32. If the ratio is not specified, it is always understood, to be geometrical.

The two quantities taken together, are called a couplet.

The number which is compared, being placed first, is called the antecedent, and that with which it is compared, the consequent.

Of these three, the antecedent, the consequent, and the ratio, any two being given, the other may be found.

EXAMPLES.

1. If the antecedent be 16, and the consequent 4, what is the ratio? Ans. 4. 2. If the antecedent be 18, and the ratio 3, what is the consequent ? Ans. 6.

Art. 157.-Inverse, or reciprocal ratio, is the ratio of the reciprocals of two quantities.

OBS.-The reciprocal of any quantity is a unit divided by that quantity. Thus, the reciprocal of 4 is 4, the reciprocal of 3 is 3.

The reciprocal ratio of 6 to isto; that is, 1÷}, which

QUESTIONS.-1. What is Ratio? 2. What is arithmetical ratio? 3. What is geometrical? 4. What is compound ratio?

is equal to . Hence, a reciprocal ratio is expressed by inverting the terms of the couplet. The reciprocal ratio of antecedent to consequent, is the direct ratio of consequent to antecedent. The direct ratio of 6 to 3 is 2. The recipro

cal ratio of 6 to 3 is

Art. 158.-Compound ratio is the ratio of the products of the corresponding terms of two or more simple ratios.

OBS. 1.-A compound ratio is not different in its nature from a simple ratio. The term compound is used merely to denote the origin of the ratio.

Art. 159. In a series of ratios, if the consequent of each preceding couplet is the antecedent of the following one, the ratio of the first antecedent to the last consequent is equal to that which is compounded of all the intervening ratios. Thus,

12: 6

6: 18

18: 3

3: 4

Art. 160.-If we multiply all the antecedents together, and all the consequents together, it will be found that the ratio of the products of the antecedents to the product of the consequents, is equal to the ratio of 12, the first antecedent, to 4, the last consequent, which is 12=3.

OBS. 2.-Rejecting all the antecedents but the first, and all the conse quents but the last, is cancelling equal factors from dividends and divisors. (See Art. 42.)

Art. 161.-If, in the several couplets, the ratios are equal, the sum of all the antecedents has the same ratio to the sum of all the consequents, which any one of the antecedents has to its consequent. Thus,

12 : 6-2
10: 5=2

8: 4=2

6: 3=2

36: 18=2

OBS. 3.-It will be observed, in this example, that the terms of the ratio are not used as factors. The ratio is, therefore, not a compound ratio.

It has already been shown (Art. 44) that to multiply the dividend with a given divisor, is the same as to multiply the

quotient, and to multiply the divisor with a given dividend, is the same as to divide the quotient. In Fractions the same principle was recognised, with this difference only in the mode of expression; we substituted numerator for dividend, and denominator for divisor. We shall now substitute antecedent for numerator or dividend, consequent for denominator or divisor, and ratio for value of the fraction.

Art. 162.-To multiply the antecedent, or to divide the consequent, is the same as to multiply the ratio.

Thus the ratio of

(See Art. 44.)

12 6 is 2 Multiply the antecedent by 2, the ratio of 24: 6 is 4 Divide the consequent by 2, the ratio of

Art. 163. To divide the antecedent, or to sequent, is the same as to divide the ratio.

Thus, the ratio of

Divide the antecedent by 2, the ratio of
Multiply the consequent by 2, the ratio of

12: 3 is 4 multiply the con

8 : 4 is 2

4 : 4 is 1

8: 8 is 1

Art. 164. To multiply both antecedent and consequent by the same quantity, does not affect the ratio.

Multiply both terms by 3, 18: 9: 2 the same ratio.
Divide both terms by 3,

2:1-2

The ratio of two fractions, which have a common denominator, is the ratio of their numerators. (See Art. 76.) Thus, 2.

The direct ratio of two fractions, which have a common numerator, is the reciprocal ratio of their denominators. (See Art. 77.) Thus, 3:3=4.

Art. 165.-A factor may be transferred from antecedent to consequent, and from consequent to antecedent, without altering the ratio; observing, that when a factor is transferred, it becomes a divisor, and when a divisor is transferred, it becomes a factor. (See Art. 150.)

Thus, the ratio of
16: 2×4=2
Transferring the factor 2, 16: 4

2

}

the same ratio.

Art. 166.-It may be observed, in regard to ratio, that it exists only between quantities of the same nature, or, the things compared must be so far alike that one may be said to be larger or smaller than the other. For example, a rod can.

not be said to be longer than an hour, nor can there be a comparison between them in any respect, for there is no common property. But a rod can be said to be longer than a foot, for it is made up of feet. There may be, however, a relation between the numbers which stand for quantities of a dissimilar nature. Thus, the ratio of 16 to 8 is 2. Now, 16 may stand for rods, and 8 for hours, which things bear no relation to each other.

The subject of ratio is of incalculable importance, since it lies at the foundation of all arithmetical investigation. The practical nature of ratio will be seen by the following example.

1. If 6 yards of cloth cost 30 dollars, what will 12 yards cost? The ratio of 12: 6 is 2, which shows that 12 is twice as large as 6. It is, therefore, plain that the cost of 12 yards will be as much greater than the cost of 6 yards, as 12 is greater than 6. Therefore, 30×2=60, the cost of 12 yards. Again, if we know the price of 1 yard, we can repeat this price 12 times, and thus obtain the price of 12 yards. If 6 yards cost 30 dollars, it is evident that one-sixth of 30 will be the cost of 1 yard. Although, strictly speaking, there is no relation between the cost and the number of yards, yet the ratio of 30 to 6, considered as numbers merely, is a number which will represent the cost of 1 yard. Therefore, 30: 6=5, the cost of 1 yard, and 5×12=60, the cost of 12 yards, as before.

If we now compare the cost of the second with the cost of the first piece, we shall find that the ratio is equal to the ratio of the length of the second piece, to the length of the first piece. Thus, 12: 6=2, and 60:30=2.

When two or more couplets of numbers have equal ratios, these numbers are said to be proportionals. Hence, (Art. 167,) Proportion is an equality of ratios.

Arithmetical Proportion is an equality of arithmetical ratios, and Geometrical Proportion is an equality of geometrical ratios. Proportion may be expressed, either by the common sign of equality, or by four points placed between the couplets. Thus

8.6

4

..

2, or 8 6:4

2, arithmetical proportion. 12: 68: 4, or 12: 6::8: 4, geometrical proportion.

The latter is read, the ratio of 12 to 6 equals the ratio of 8 to 4, or 12 is to 6 as 8 is to 4.

The first and last terms are called the extremes, and the others the means.

Art. 167.-The number of terms must be at least four, for the equality is between the ratios of the couplets; and each couplet must have an antecedent and consequent. There may be, however, a proportion between three quantities; for one of the quantities may be repeated, so as to form the two terms. Thus, 6:12::12:24.

Art. 168.-If four numbers are in geometrical proportion, the product of the extremes is equal to the product of the means. Thus, 128: 15: 10, for 12 × 10-8 × 15.

Art. 169. By multiplying the extremes and means together, a proportion is reduced to an equation. When the product of any two numbers is equal to the product of any other two, the numbers may be formed into a proportion by taking the factors on one side of the equation for the extremes, and those on the other for the means. Thus, 4×3=6×2. Making 4 and 3 constitute the extremes, and 6 and 2 the means, we have the following proportion; 4: 2:: 6:3. Form proportions of the following equations:

Art. 170.-In compounding proportion, equal factors may be rejected from antecedents and consequents. Thus :

12: 4::9: $

4: 8:3: 6
$:20:6:15

12:20:9: 15

Art. 171.-If the corresponding terms of two or more ranks of proportiona' quantities be multiplied together, the products will be proportional. Thus:

12 4: 6:2

10: 5: 8:4

120:20:48:8

Art. 172.-If the terms in one rank of proportionals be divided by the corresponding terms in another rank, the quotients will be proportional.