An antecedent and consequent, when spoken of as one, are called a couplet; when spoken of as two, the terms of the ratio.

Thus, when the ratio of 2 to 6 is spoken of, 2 and 6 together, form a couplet, of which 2 is the first term, and 6 the second.

234. Ratio is expressed in two ways:

1st. In the form of a fraction, of which the antecedent is the denominator, and the consequent the numerator.

Thus, the ratio of 2 to 6, is expressed by ; the ratio of 3 to 12, by 12, etc.

2d. By placing two points () between the terms.

Thus, the ratio of 2 to 6, is written 26; the ratio of 3 to 8, 3: 8, etc.

235. The ratio of two quantities may be either a whole number, a common fraction, or an interminate decimal.

From the last illustration, it is obvious that the ratio of two quantities can not always be expressed exactly, except by symbols; but, by employing decimals, we may find the approximate ratio to any required degree of exactness.

236. Since the ratio of two numbers is expressed by a fraction, it follows that whatever is true of a fraction, is true of the terms of a ratio. Hence,

REVIEW.-234. When are the antecedent and consequent of a ratio called a couplet? When called terms? By what two methods is ratio expressed? Example. 235. What forms may the ratio of two quantities have?

1st. To multiply the consequent, or divide the antecedent by any number, multiplies the ratio by that number. Arts. 122, 125.

Thus, the ratio of 4 to 12, is 3.

The ratio of 4 to 12×5, is 3×5.

The ratio of 4-2 to 12, is 6, which is equal to 3×2.

2d. To divide the consequent, or multiply the antecedent by any number, divides the ratio by that number. Arts. 123, 124.

Thus, the ratio of 3 to 24, is 8.

The ratio of 3 to 24-2, is 4, which is equal to 8÷÷÷2.
The ratio of 3X2 to 24, is 4, which is equal to 8÷2.

3d. To multiply or divide both the antecedent and consequent by any number, does not alter the ratio. Arts. 126, 127.

Thus, the ratio of 6 to 18, is 3.
The ratio of 6×2 to 18×2, is 3.
The ratio of 6÷÷2 to 18÷÷2, is 3.

237. When the two numbers are equal, the ratio is called a ratio of equality; when the second is greater than the first, a ratio of greater inequality; when less, a ratio of less inequality.

Thus, the ratio of 4 to 4, is a ratio of equality.

The ratio of 4 to 8, is a ratio of greater inequality.

The ratio of 4 to 2, is a ratio of less inequality.

We see, from this, that a ratio of equality may be expressed by 1; a ratio of greater inequality, by a number

REVIEW.-236. How is a ratio affected by multiplying the consequent, or dividing the antecedent? Why? By dividing the consequent, or multiplying the antecedent? Why? By multiplying or dividing both antecedent and consequent by the same number? Why?

237. What is a ratio of equality? Of greater inequality? Of less inequality? Examples.

greater than 1; and a ratio of less inequality, by a number less than 1.

238. A Compound Ratio is the product of two or more ratios.

Thus, the ratio 10, compounded with the ratio g, is 10X0=60=4. In this case, 3 multiplied by 5, is said to have to 10 multiplied by 6, the ratio compounded of the ratios of 3 to 10 and 5 to 6.

239. Ratios may be compared by reducing the fractions which represent them to a common denominator.

Thus, the ratio of 2 to 5 is less than the ratio of 3 to 8, for 5 or 15 is less than 8 or 16.

PROPORTION.

240. Proportion is an equality of ratios.

b

Thus, if a, b, c, d are four quantities, such that is d

a

equal to then a, b, c, d form a proportion, and we say č

that a is to b, as c is to d; or, that a has the same ratio to b, that c has to d.

Proportion is written in two ways, by using,

1st. The colon and double colon; thus, ab::c: d. 2d. The sign of equality; thus,

a: bc: d.

The first is read, a is to b as c is to d; the second is read, the ratio of a to b equals the ratio of c to d.

From the preceding definition, it follows, that when four quantities are in proportion, the second divided by the first gives the same quotient as the fourth divided by the third.

REVIEW.-238. When are two or more ratios said to be compounded? Examples.

239. How may ratios be compared to each other? 240. What is proportion? Example. How are four quantities in proportion written? How read? Examples.

This is the test of the proportionality of four quantities. Thus, if a, b, c, d are the four terms of a true proportion, b d

so that a b c d, we must have a с

If these fractions are equal to each other, the proportion is true; if they are not equal, it is false.

Let it be required to find whether 3: 82: 5.

Since is not a true equation, the proportion is false.

REMARK.—The words ratio and proportion are often misapplied. Thus, two quantitics are said to be in the proportion of 3 to 4, instead of, in the ratio of 3 to 4.

A ratio subsists between two quantities, a proportion only between four. It requires two equal ratios to form a proportion.

241. In the proportion a: b:: c: d, each of the quantities a, b, c, d is called a term. The first and last terms are called the extremes; the second and third, the means.

242. Of four proportional quantities, the first and third are called antecedents; and the second and fourth, consequents, Art. 233. The last is said to be a fourth proportional to the other three, taken in their order.

243. Three quantities are in proportion, when the first has the same ratio to the second that the second has to the third. In this case, the middle term is called a mean proportional between the other two. Thus, if we have the proportion

abb c

then b is called a mean proportional between a and C1 and c is called a third proportional to a and b.

REVIEW.-240. Give examples of a true and false proportion. What is a test of the proportionality of four quantities? 241. What are the first and last terms of a proportion called? The second and third?

242. What are the first and third terms of a proportion called? The second and fourth? 243. When are three quantities in proportion? Example. What is the second term called? The third ?

244. Proposition I.-In every proportion, the product of the means is equal to the product of the extremes.

Then, since this is a true proportion, we must have

Clearing of fractions, we have ad=bc.

Illustration by numbers, 3:6:: 5 : 10, and 6×5=3×10.

a=

bc

d'

from which we see, that if any three terms of a proportion

are given, the fourth may be readily found.

The first three terms of a proportion, are ac, bd, and acxy; what is the fourth? Ans. bdxy.

REMARK. This proposition furnishes a more convenient test of the proportionality of four quantities, than the method given in Article 240. Thus, 3:8 :: 2 : 5 is a false proportion, since 3×5 is not equal to 8×2.

245. Proposition II.-Conversely, If the product of two quantities is equal to the product of two others, two of them may be made the means, and the other two the extremes of a proportion.

Dividing each of these equals by ac, we have

Illustration, 5×8=4×10, and 4:5::8: 10.

In applying this PROP., take either factor on either side of the equation for the first term of the proportion, pass to the other side of the equation