Question

If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$ prove that $$\frac{a^3 + c^3 + e^3}{b^3 + d^3 + f^3} = \frac{ace}{bdf}$$.

Answer 1

Given: $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$

To prove: $$\frac{a^3 + c^3 + e^3}{b^3 + d^3 + f^3} = \frac{ace}{bdf}$$

Proof:

1. Let $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k$$, which gives $$a = bk$$, $$c = dk$$, $$e = fk$$.
2. $$\text{L.H.S.} = \frac{a^3 + c^3 + e^3}{b^3 + d^3 + f^3} = \frac{b^3 k^3 + d^3 k^3 + f^3 k^3}{b^3 + d^3 + f^3} = \frac{k^3(b^3 + d^3 + f^3)}{b^3 + d^3 + f^3} = k^3$$
3. $$\text{R.H.S.} = \frac{ace}{bdf} = \frac{(bk)(dk)(fk)}{bdf} = \frac{bdf k^3}{bdf} = k^3$$
4. $$\text{L.H.S.} = \text{R.H.S.}$$

Hence proved.