Question

A fraction becomes $\frac{4}{5}$ if 1 is added to both numerator and denominator of how ever s is subtracted from both numerator and denominator , the fraction becomes $\frac{1}{2}$. what is the fraction?

Answer 1

Let the dfraction is $\frac{x}{y}$

Then, according to the given question,

$\frac{x+1}{y+1}=\frac{4}{5}$

$\Rightarrow 5x+5=4y+4$

$\Rightarrow 5x-4y+1=0........\left(1\right)$

Again, according to the given question,

$\frac{x-1}{y-1}=\frac{1}{2}$

$\Rightarrow 2x-2=y-1$

$\Rightarrow 2x=y+1$

$\Rightarrow x=\frac{y+1}{2}.......\left(2\right)$

Put the value of x by equation (2) in to (1), we get,

$5x-4y+1=0$

$\Rightarrow 5\left(\frac{y+1}{2}\right)-4y+1=0$

$\Rightarrow 5y+5-8y+2=0$

$\Rightarrow -3y+3=0$

$\Rightarrow y=1$

Put $y=1$ in equation (2) and we get,

$x=\frac{1+1}{2}=1$

Hence, fraction is $\frac{x}{y}=\frac{1}{1}$

This is the answer