Question

If x:y =a:b, prove that a:x=(a+b):(x:y)

Answer 1

We start by writing the ratio $x:y$ and $a:b$ as fraction

$\frac{x}{x+y}: \frac{y}{x+y} = \frac{a}{a+b}: \frac{b}{a+b}$

This shows two pairs of equivalent fractions

$\frac{x}{x+y}= \frac{a}{a+b}$ and $\frac{y}{x+y}= \frac{b}{a+b}$

We will use the first pair of fraction
$\frac{x}{x+y}= \frac{a}{a+b}$ ⇒ cross multiply gives
$x(a+b)=a(x+y)$ ⇒ then rearrange
$\frac{a+b}{x+y} = \frac{a}{x}$ ⇒ which we can also write in form of ratio
$(a+b):(x+y)=a:x$

Hence it is proved that $a:x=(a+b):(x+y)$ as required