3. Two numbers are in the ratio of 2 to 3, and if 9 be added to each they are in the ratio of 3 to 4. Find the numbers.

4. Shew that the ratio a : b is the duplicate of the ratio a+c: b+c if c2 = ab.

5. There are two roads from A to B, one of them 14 miles longer than the other, and two roads from B to C, one of them 8 miles longer than the other. The distances from A to B and from B to C along the shorter roads are in the ratio of 1 to 2, and the distances along the longer roads are in the ratio of 2 to 3. Determine the distances.

and equal to each of the former; and that each fraction

be equal, prove that

ca'-c'a

are equal,

T. A.

15

385. Four quantities are said to be proportionals when the first is the same multiple, part, or parts, of the second, as the the four quantities

third is of the fourth; that is, when

a с

=

b d'

́a, b, c, d, are called proportionals. This is usually expressed by saying, a is to b as c is to d, and is represented thus, a : b::c: d, or thus, a :

b

= c : d

The terms a and d are called the extremes, and b and c the

means.

386. When four quantities are proportionals, the product of the extremes is equal to the product of the means.

Let a, b, c, d be the four quantities; then since they are pro

portionals

α C

b

d

(Art. 385); and by multiplying both sides of

the equation by bd, we have ad=bc.

387. Hence if the first be to the second as the second is to the third, the product of the extremes is equal to the square of the mean.

388. If any three terms in a proportion are given, the fourth may be determined from the equation ad = bc.

389. If the product of two quantities be equal to the product of two others, the four are proportionals; the terms of either product being taken for the means, and the terms of the other product for the extremes.

390. If a b :: cd, and c d e f, then

:

a: be: f.

391. If four quantities be proportionals, they are proportionals when taken inversely.

thus

If a b c : d, then ba :: d: c.

=

d

; divide unity by each of these equal quantities

; or b: a :: d : c.

392. If four quantities be proportionals, they are proportionals when taken alternately.

Unless the four quantities are of the same kind the alternation cannot take place; because this operation supposes the first to be some multiple, part, or parts, of the third. One line may have to another line the same ratio as one weight has to another weight, but there is no relation, with respect to magnitude, between a line and a weight. In such cases, however, if the four quantities be represented by numbers, or by other quantities which are all of the same kind, the alternation may take place.

393. When four quantities are proportionals, the first together with the second is to the second as the third together with the fourth is to the fourth.

If a b c d, :

then a+b: b :: c + d : d.

For=; add unity to both sides; thus

or

a+b b c +d: d.

This operation is called componendo.

394. Also the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth.

395. Also the first is to the excess of the first above the second as the third is to the excess of the third above the fourth.

396. When four quantities are proportionals, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference.

397. When any number of quantities are proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.

also

ab = ba;

that is,

hence ab+ad + af = ba+bc + be ; a (b+d+f) = b (a+c+e).

Hence, by Art. 389, a: b :: a+c+e : b+d+f.

Similarly the proposition may be established when more quantities are taken.

398. When four quantities are proportionals, if the first and second be multiplied, or divided, by any quantity, as also the third and fourth, the resulting quantities will be proportionals.

399. If the first and third be multiplied, or divided, by any quantity, and also the second and fourth, the resulting quantities will be proportionals.