OF THE RATIO AND PROPORTION OF NUMBERS

183. Two numbers having the same unit may be compared together in two ways.

1st. By considering how much one is greater or less than the other, which is shown by their difference; and

2d. By considering how many times one is greater or less than the other, which is shown by their quotient.

Thus, in comparing the numbers 3 and 12 together with respect to their difference, we find that 12 exceeds 3 by 9; and in comparing them together with respect to their quotient, we find that 12 contains 3 four times, or that 12 is four times as great as 3.

The quotient which arises from dividing the second number by the first, is called the ratio of the numbers, and shows how many times the second number is greater than the first, or how many times it is less.

Thus, the ratio of 3 to 9 is 3, since 9÷3 3. The ratio of 2 to 4 is 2, since 42 = 2.

1

We may also compare a larger number with a less. For example, the ratio of 4 to 2 is ; for, 2 ÷ 4 = 1. The ratio of 9 to 3 is, since 39

2

18

EXAMPLES.

1. What is the ratio of 9 to 18? 2. What is the ratio of 6 to 24?

Ans.
Ans.

QUEST.-183. In how many ways may two numbers having the same unit be compared? How do you determine how much one number is greater than another? How do you determine how many times it is greater or less? How much does 12 exceed 3? How many times is 12 greater than 3? What is the quotient called which arises from dividing the second number by the first? What does it show? When the second number is the least, what does it show?

3. What is the ratio of 12 to 48?

4. What is the ratio of 11 to 13?

Ans.

Ans.

5. What part of 20 is 4? Or what is the ratio of 20 to 4? 6. What part of 100 is 30? Or what is the ratio of 100 to 30?

NOTE.-In determining what part one number is of another, it is plain that the number to be measured, must be written in the numerator; while the standard, or number with which it is compared, and of which it forms a part, is written in the denominator. fraction, reduced to its lowest terms, will expross the part.

This

184. If one yard of cloth cost $2, how many dollars will 6 yards of cloth cost at the same rate?

It is plain that 6 yards of cloth will cost 6 times as much as one yard; that is, the cost will contain $2 as many times as 6 contains 1. Hence the cost will be $12.

In this example there are four numbers considered, viz., 1 yard of cloth, 6 yards of cloth, $2, and $12: these numbers are called terms.

Now the ratio of the first term to the second is the same as the ratio of the third to the fourth.

This relation between four numbers is called a proportion; and generally

Four numbers are said to be in proportion when the ratio of

QUEST.-How do you determine what part one number is of another? 184. If one yard of cloth cost $2, what will 6 yards cost? How many numbers are here considered? What are they called? What is the ratio of the first to the second equal to? What is this relation between numbers called? When are four numbers said to be in proportion?

the first to the second is the same as that of the third to the fourth. Hence,

A PROPORTION is an equality of ratios between numbers compared together two and two.

185. We express that four numbers are in proportion thus: 1 : 6 :: 2 : 12.

That is, we write the numbers in the same line and place two dots between the 1st and 2d terms, four between the 2d and 3d, and two between the 3d and 4th terms. We read the proportion thus,

as 1 is to 6, so is 2 to 12.

The 1st and 2d terms of a proportion always express quantities of the same kind, and so likewise do the 3d and 4th terms. As in the example,

yd. yd.

1 : 6 :: 2 : 12.

This is implied by the definition of a ratio; for, it is only quantities of the same kind which can be divided the one by the other. The ratio of the first term to the second, or of the third to the fourth, is called the ratio of the proportion. 1. What are the ratios of the proportions

186. When two numbers are compared together, the first is called the antecedent, and the second the consequent; and when four numbers are compared, the first antecedent and consequent are called the first couplet, and the second antecedent and consequent the second couplet. Thus, in the last

QUEST.-How do you define proportion? 185. How do you indicate that four numbers are in proportion? How is the proportion read? What do you remark of the first and second terms? Also of the third and fourth ? 186. When two numbers are compared together, what is the first called? What the second? When four numbers are compared, what are the two first called? What the two second?

proportion, 16 and 48 are the antecedents, and 15 and 45 the consequents; also, 16 and 15 make the first couplet, and 48 and 45 the second.

187. We have said that proportion is an equality of ratios. (Art. 184). Besides the method above, we may express that equality thus:

188. If 4lb. of tea cost $8, what will 127b. cost at the same rate?

It is evident that the 4th term, or cost of 1276. of tea, must be as many times greater than $8, as 1276. is greater than 4lb. But the ratio of 4lb. to 12lb. is 3; hence, 3 is the number of times which the cost exceeds $8: that is, the cost is

QUEST.-187. What has proportion been called? By what second method ray this equality be expressed? 188. Explain this example mentally.

equal to $8 × 3 = $24. But instead of writing the numbers

and as the same may be shown for every proportion, we conclude,

That the 4th term of every proportion may be found by multiplying the 2d and 3d terms together, and dividing their product by the 1st term.

EXAMPLES.

1. The first three terms of a proportion are 1, 2, and 3: what is the fourth?

Ans.

2. The first three terms are 6, 2, and 1: what is the 4th? Ans.

3. The first three terms are 10, 3, and 1: what is the 4th ?

Ans.

189. The 1st and 4th terms of a proportion are called the two extremes, and the 2d and 3d terms are called the two

means.

Now, since the 4th term is obtained by dividing the product of the 2d and 3d terms by the 1st term, and since the product of the divisor by the quotient is equal to the dividend, it follows,

That in every proportion the product of the two extremes is equal to the product of the two means.

Thus, in the example, Art. 184 we have

1 : 6 : 2:12; and 1 x 12 = 2 × 6;

also, 4 : 12 :: : 8: 24;

QUEST.-How may the fourth term of every proportion be found? 189. What are the first and fourth terms of a proportion called? What are the second and third terms called? In every proportion, what is the product of the extremes equal to?