# If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes 6/5.

Question:

If the numerator of a fraction is multiplied by 2 and the denominator is reduced by 5 the fraction becomes 6/5. And, if the denominator is doubled and the numerator is increased by 8, the fraction becomes 2/5. find the fraction.

Solution:

Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$

If the numerator is multiplied by 2 and the denominator is reduced by 5, the fraction becomes $\frac{6}{5}$. Thus, we have

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

$\Rightarrow 10 x=6(y-5)$

$\Rightarrow 10 x=6 y-30$

$\Rightarrow 10 x-6 y+30=0$

$\Rightarrow 2(5 x-3 y+15)=0$

$\Rightarrow 5 x-3 y+15=0$

If the denominator is doubled and the numerator is increased by 8 , the fraction becomes $\frac{2}{5}$. Thus, we have

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

$\Rightarrow 5(x+8)=4 y$

$\Rightarrow 5 x+40=4 y$

$\Rightarrow 5 x-4 y+40=0$

So, we have two equations

$5 x-3 y+15=0$

$5 x-4 y+40=0$

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

$\frac{x}{(-3) \times 40-(-4) \times 15}=\frac{-y}{5 \times 40-5 \times 15}=\frac{1}{5 \times(-4)-5 \times(-3)}$

$\Rightarrow \frac{x}{-120+60}=\frac{-y}{200-75}=\frac{1}{-20+15}$

$\Rightarrow \frac{x}{-60}=\frac{-y}{125}=\frac{1}{-5}$

$\Rightarrow \frac{x}{60}=\frac{y}{125}=\frac{1}{5}$

$\Rightarrow x=\frac{60}{5}, y=\frac{125}{5}$

$\Rightarrow x=12, y=25$

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