Part 26

(f) Division.

(g) Composition and division.

(h) In a series of equal ratios, the sum of the antecedents is to the sum of the consequents etc.

(i) Like powers or like roots of the terms of a proportion etc.

6. If x : m :: 13 : 7, write all the possible proportions that can be derived from it. [See (5) above.]

7. Given rs = 161m; write the eight proportions that may be derived from it, and quote your authority.

8. (a) What theorem allows you to change any proportion into an equation?

(b) What theorem allows you to change any equation into a proportion?

9. If xy = rg, what is the ratio of x to g? of y to r? of y to g?

10. Find two numbers such that their sum, difference, and the sum of their squares are in the ratio 5 : 3 : 51. (Yale.)

~Reference:~ The chapter on Ratio and Proportion in any algebra.

An easy and powerful method of proving four expressions in proportion is illustrated by the following example:

Given a : b = c : d;

prove that 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.

Let a/b = r. Therefore a = br.

Also c/d = r. Therefore c = dr.

Substitute the value of a in the first ratio, and c in the second:

Then

(3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3b^3r^3 + 5b^3r)/(3b^3r^3 - 5b^3r) = [b^3r(3r^2 + 5)]/[b^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).

Also

(3c^3 + 5cd^2)/(3c^3 - 5cd^2) = (3d^3r^3 + 5d^3r)/(3d^3r^3 - 5d^3r) = [d^3r(3r^2 + 5)]/[d^3r(3r^2 - 5)] = (3r^2 + 5)/(3r^2 - 5).

Therefore (3a^3 + 5ab^2)/(3a^3 - 5ab^2) = (3c^3 + 5cd^2)/(3c^3 - 5cd^2).

Axiom 1.

Or, 3a^3 + 5ab^2 : 3a^3 - 5ab^2 = 3c^3 + 5cd^2 : 3c^3 - 5cd^2.

If a : b = c : d, prove:

1. a^2 + b^2 : a^2 = c^2 + d^2 : c^2.

2. a^2 + 3b^2 : a^2 - 3b^2 = c^2 + 3d^2 : c^2 - 3d^2.

3. a^2 + 2b^2 : 2b^2 = ac + 2bd : 2bd.