Question
If $a : b = c : d,$ prove that $: xa + yb : xc + yd = b : d.$

Answer

Given $\frac{ a }{ b }=\frac{ c }{ d }$
$\Rightarrow \frac{x a }{ yb }=\frac{x c }{ yd }\left(\right.$ Multiplying each side by $\left.\frac{x}{ y }\right.)$
$\Rightarrow \frac{x a + yb }{ yb }=\frac{x c + yd }{ yd }$ (By componendo)
$\Rightarrow \frac{x a + yb }{x c + yd }=\frac{ yb }{ yd } $
$$\Rightarrow \frac{x a + yb }{x c + yd }=\frac{ b }{ d }$

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