Question

If $$a : b = c : d$$, prove that $$(9a + 13b) : (9a - 13b) = (9c + 13d) : (9c - 13d)$$.

Answer 1

Given: $$a : b = c : d$$

To prove: $$(9a + 13b) : (9a - 13b) = (9c + 13d) : (9c - 13d)$$

Proof:

1. $$\frac{a}{b} = \frac{c}{d}$$ [Given]
2. $$\frac{9a}{13b} = \frac{9c}{13d}$$ [Multiplying both sides by $$\frac{9}{13}$$]
3. $$\frac{9a + 13b}{9a - 13b} = \frac{9c + 13d}{9c - 13d}$$ [By componendo and dividendo]
4. $$(9a + 13b) : (9a - 13b) = (9c + 13d) : (9c - 13d)$$

Hence proved.