Question

If the numerator of a fraction is multiplied by $2$ and the denominator is reduced by $5$, then fraction becomes $\frac{6}{5}$. And, if the denominator is doubled and the numerator increased by $8$, the fraction becomes $\frac{2}{5}$. Find the fraction.

Answer 1

Let the fraction be $\frac{x}{y}$

Given that,

$\frac{2x}{y-5}=\frac{6}{5}$

$10x=6y-30$

$-10x+6y=30$

$-5x+3y=15$-----(1)

and

$\frac{x+8}{2y}=\frac{2}{5}$

$5x+40=4y$

$-5x+4y=40$------(2)

Solving eq 1 & 2

$-5x+4y=40$

$-5x+3y=15$

+ - -

_______________

$0+y=25$

$\therefore y=25$

$\Rightarrow -5x+75=15$

$x=\frac{60}{5}=12$

the fraction is $\frac{12}{25}$