Question

If $$a : b = c : d$$, prove that $$(3a + 2b) : (3a - 2b) = (3c + 2d) : (3c - 2d)$$.

Answer 1

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

To prove: $$(3a + 2b) : (3a - 2b) = (3c + 2d) : (3c - 2d)$$

Proof:

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

Hence proved.