RATIO AND PROPORTION.

189. Two numbers having the same unit, may be compared 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 contained in the other, which is shown by their quotient.

In comparing two numbers, one with the other, by means of their difference, the less is always taken from the greater.

In comparing two numbers, one with the other, by means of their quotient, one of them must be regarded as a standard which measures the other, and the quotient which arises by dividing by the standard, is called the ratio.

190. Every ratio is derived from two terms: the first is called the antecedent, and the second the consequent; and the two, taken together, are called a couplet. The antecedent will be regarded as the standard.

If the numbers 3 and 12 be compared by their difference, the result of the comparison will be 9; for, 12 exceeds 3 by 9. If they are compared by means of their quotient, the result will be 4; for, 3 is contained in 12, 4 times; that is, 3 measuring 12, gives 4.

191. The ratio of one number to another is expressed in two ways:

1st. By a colon; thus, 3:12; and is read, 3 is to 12; or, 3 measuring 12.

189. In how many ways may two numbers, having the same unit, be compared with each other? If you compare by their difference, what do you do? If you compare by the quotient, how do you regard one of the numbers ? What is the ratio?

190. From how many terms is a ratio derived? What is the first term called? What is the second called? Which is the standard?

191. How may the ratio of two numbers be expressed? How read?

2d. In a fractional form, as 12; or, 3 measuring 12.

192. If two couplets have the same ratio, their terms are said to be proportional: the couplets,

4 : 20 and 1 : 5,

have the same ratio 5; hence, the terms are proportional, and are written,

4 : 20 :: 1 : 5,

by simply placing a double colon between the couplets. The terms are read,

4 is to 20 as 1 is to 5,

and taken together, they are called a proportion: hence,

A proportion is a comparison of the terms of two equal ratios What are the ratios of the proportions?

193. The 1st and 4th terms of a proportion are called the extremes; the 2d and 3d terms, the means. Thus, in any proportion, 6 : 24 :: 8 : 32,

6 and 32 are the extremes, and 24 and 8 the means :

we shall have, by reducing to a common denominator,

But since the fractions are equal, and have the same denomi

nators, their numerators must be equal, viz. :

24 × 8 = 32 × 6; that is,

192. If two couplets have the same ratio, what is said of the terms? How are they written? How read? What is a proportion?

193. Which are the extremes of a proportion? What is the product of the extremes equal to ?

Which the means?

In any proportion, the product of the extremes is equal to the

194. Since, in any proportion, the product of the extremes is equal to the product of the means, it follows that,

1st. If the product of the means be divided by one of the extremes, the quotient will be the other extreme.

Thus, in the proportion,

4 : 16 :: 6 : 24, and 4 × 24 16 × 6 = 96; then, if 96, the product of the means, be divided by one of the extremes, 4, the quotient will be the other extreme, 24; or, if the product be divided by 24, the quotient will be 4.

2d. If the product of the extremes be divided by either of the means, the quotient will be the other mean.

Thus, if 4 × 24 = 16 × 696 be divided by 16, the quotient will be 6; or if it be divided by 6, the quotient will be 16.

EXAMPLES.

1. The first three terms of a proportion are 5, 10, and 19? what is the fourth term?

2. The first three terms of a proportion are 6, 24, and 14: what is the fourth term?

3. The first, second and fourth terms of a proportion are 9, 12 and 16: what is the third term?

4. The first, third and fourth terms of a proportion are 16, 8, and 20: what is the second term?

5. The second, third and fourth terms of a proportion are 48, 90, and 45: what is the first term?

194. If the product of the means be divided by one of the extremes, what will the quotient be? If the product of the means be divided by either extreme, what will the quotient be?

SIMPLE AND COMPOUND RATIO.

195. The ratio of two single numbers is called a Simple Ratio, and the proportion which arises from the equality of two such ratios, a Simple Proportion.

If the terms of one ratio be multiplied by the terms of another, antecedent by antecedent, and consequent by consequent, the ratio of the products is called a Compound Ratio. Thus, if the two ratios

3 : 6 and 4 : 12

be multiplied together, we shall have the compound ratio

3 x 4 : 6 x 12, or 12 : 72;

in which the ratio is equal to the product of the simple ratios.

A proportion formed from the equality of two compound ratios, or from the equality of a compound ratio and a simple ratio, is called a Compound Proportion.

196. What part one number is of another.

When the standard, or antecedent, is greater than the number which it measures, the ratio is a proper fraction, and is such a part of 1, as the number measured is of the standard.

NOTE. The standard is generally preceded by the word of, and in comparing numbers, may be named second, as in examples 7, 8, 9, 10, and 11, but it must always be used as a divisor, and should be placed first in the statement.

1. What part of 25 is 5? that is, what part of the standard 25, is 5?

that is, the standard is to the number measured as 1 to }; or, the number measured is one-fifth of the standard.

195. What is a simple ratio? What is the proportion called which comes from the equality of two simple ratios? What is a compound ratio? What is a compound proportion?

196. When the standard is greater than the consequent, what kind of a number is the ratio? What part is 3 of 4 6 of 12? What part of 4 is 16 12 of 36 ?

2. What part of 6 is 4?
3. What part of 10 is 5?
4. What part of 34 is 17?
5. What part of 450 is 300?
6. What part of 96 is 16?

7. 8 is what part of 12? 8. 16 is what part of 48? 9. 18 is what part of 90? 10. 15 is what part of 165? 11. 9 is what part of 11?

DIRECT AND INVERSE PROPORTION.

197. It often happens that two numbers which are compared with each other, undergo, or may undergo, certain changes of value, in which case they represent variable and not fixed quantities. Thus, when we say, that the amount of work done, in a single day, will be proportional to the number of men employed, we mean, that if we increase the number of men the amount of work done will also be increased; or, if we diminish the number of men employed, the work done will also be diminished. This is called Direct Proportion.

If we say that a barrel of flour will serve 12 men a certain time, and ask how long it will serve 24 men, there is a certain relation between the number of men and time; but that relation is such that the time will decrease if the number of men is increased, and will increase if the number of men is decreased. This is called Inverse Proportion; hence,

1. Two numbers are directly proportional when they increase or decrease together; in which case their ratio is always the same. 2. Two numbers are inversely or reciprocally proportional when one increases as the other decreases; in which case their product is always the same.

NOTE. This is sometimes called Reciprocal Proportion.

198. If we refer to the numeration table of integral and decimal numbers (Art. 146), we see that the unit of the first

197. When are two numbers directly proportional? When are two numbers inversely proportional? Does their product then vary?

198. What relation exists between the units of place in the integral and decimal numeration table? Give an example?