262. Ratios may be compared with each other by reducing the fractions which represent them to a common denominator.

Thus, the ratio of 2 to 7 is greater than the ratio of 3 to 10, for the fractions and 10, reduced to a common denominator, are 21 and 20, and the first is greater than the second.

PROPORTION.

263. Proportion is an equality of ratios; that is, when two ratios are equal, their terms are said to be proportional. Thus, if the ratio of a to b is equal to the ratio of c to d; that b d is, if then, a, b, c, d, form a proportion.

α

Proportion is written in two ways:

1st. By placing a double colon between the ratios; Thus, ab::c: d.

Read, a is to b as c is to d.

2d. By placing the sign of equality between the ratios; Thus, a b c : d.

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, must give the same quotient as the fourth divided by the third. This is the primary test of the proportionality of four quantities.

Thus, if 3, 5, 6, 10, are the four terms of a proportion, so that 3:5:: 6:10, we must have §=10.

If these fractions are not equal to each other, the proportion is false.

Thus, the proportion 3: 8:: 2 : 5 is false, since >.

REMARK. The words ratio and proportion should not be confounded. Thus, two quantities are not in the proportion of 2 to 3, but in the ratio of 2 to 3. A ratio subsists between two quantities, a proportion between four.

264. Each of the four quantities in a proportion is called a term. The first and last terms are called the extremes; the second and third terms, the means.

265. Of four quantities in proportion, the first and third are called the antecedents, and the second and fourth, the consequents (Art. 257); and the last is said to be a fourth proportional to the other three taken in their order.

266. Three quantities are in proportion when the first has the same ratio to the second, that the second has to the third. The middle term is a mean proportional between the other two.

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

When several quantities have the same ratio between each two that are consecutive, they are said to form a continued proportion.

267. Proposition I.—In every proportion, the product of the means is equal to the product of the extremes.

Since this is a true proportion, we must have (Art. 263)

Illustration by numbers. 2: 6:5:15; and 6×5=2×15.

If any three terms of a proportion be given, the remaining term may be found.

1. The first three terms of a proportion are x+y, x3—y3, and x-y; what is the fourth? Ans. x2-2xy+y3.

2. The 1st, 3d, and 4th terms of a proportion are (m—n)3, m2—n2, and m+n; required the 2d.

Ans. m-n.

This proposition furnishes a more convenient test of proportionality than the method given in Art. 263.

Thus, 2:35:8, is not a true proportion, since 3×5 is not equal to 2×8.

268. 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 ać, we have

By dividing each of the equals by ab, cd, bd, etc., we may have the proportion in other forms.

Or, since one member of the equation must form the extremes and the other the means, we have the following

Rule.-Take either factor on either side of the equation for the first term of the proportion, the two on the other side for the second and third, and the remaining factor for the fourth.

Thus, from each of the equations bc-ad, and 3×12=4×9, we may have the eight following forms:

269. Proposition III.-If three quantities are in proportion, the product of the extremes is equal to the square of the mean.

It follows from Art. 268, that the converse of this proposition is also true.

If the product of the first and third of three quantities is equal to the square of the second, the second is a mean proportional between the first and third.

270. Proposition IV.-If four quantities are in proportion, they will be in proportion by ALTERNATION; that is, the first will be to the third as the second to the fourth.

NOTE. This proposition is true, only when the four quantities are of the same kind.

271. Proposition V.-If four quantities are in proportion, they will be in proportion by INVERSION; that is, the second will be to the first as the fourth to the third.

converted into a proportion in either of two ways, thus: 1

abcd, or b: a:: d: c.

272. Proposition VI.-If two sets of proportions have an antecedent and consequent in the one, equal to an antecedent and consequent in the other, the remaining terms will be proportional.

If 4 8 10 20 and 4:8:: 6:12; then, 10:20 :: 6:12.

:

273. Proposition VII.-If four quantities are in proportion, they will be in proportion by COMPOSITION; that is, the sum of the first and second will be to the first or second, as the sum of the third and fourth is to the third or fourth.