Question

A fraction is such that if the numerator is multiplied by $3$ and the denominator is reduced by $3$, we get $18/11$, but if the numerator is increased by $8$ and the denominator is doubled, we get $2/5$. Find the fraction.

Answer 1

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

Then, according to the given conditions, we have

$\frac{3x}{y-3}=\frac{18}{11}$ and $\frac{x+8}{2y}=\frac{2}{5}$

$\Rightarrow 11x-6y-18$ and $5x+40=4y$

$\Rightarrow 11x-6y+18=0$ and $5x-4y+40=0$

By cross-multiplication, we have

$\frac{x}{(-6)\times 40-(-4)\times 18}=\frac{-y}{11\times 40-5\times 18}=\frac{1}{11\times (-4)-5\times (-6)}$

$\Rightarrow \frac{x}{-240+72}=\frac{-y}{440-90}=\frac{1}{-44+30}$

$\Rightarrow \frac{x}{-168}=\frac{y}{-350}=\frac{1}{-14}$

$\Rightarrow x=\frac{-168}{-14}$ and $y=\frac{-350}{-14}$

$\Rightarrow x=12$ and $y=25$

Hence, the fraction is $\frac{12}{25}$.