Question

If $$(3a + 5b) : (3a - 5b) = (3c + 5d) : (3c - 5d)$$, prove that $$a : b = c : d$$.

Answer 1

Given: $$(3a + 5b) : (3a - 5b) = (3c + 5d) : (3c - 5d)$$

To prove: $$a : b = c : d$$

Proof:

1. $$\frac{3a + 5b}{3a - 5b} = \frac{3c + 5d}{3c - 5d}$$ [Given]
2. $$\frac{(3a + 5b) + (3a - 5b)}{(3a + 5b) - (3a - 5b)} = \frac{(3c + 5d) + (3c - 5d)}{(3c + 5d) - (3c - 5d)}$$ [By componendo and dividendo]
3. $$\frac{6a}{10b} = \frac{6c}{10d}$$
4. $$\frac{a}{b} = \frac{c}{d}$$ [Multiplying both sides by $$\frac{10}{6}$$]
5. $$a : b = c : d$$

Hence proved.