Question

A fraction becomes $\frac{9}{11}$, if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes $\frac{5}{6}$. Find the fraction.

Answer 1

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

If 2 is added to both the numerator and the denominator,

$\frac{(x+2)}{(y+2)}=\frac{9}{11}$

$\Rightarrow 11x+4=9y\dots \dots \left(i\right)$

3 is added to both the numerator and the denominator

$\frac{(x+3)}{(y+3)}=\frac{5}{6}$

$\Rightarrow 6x+3=5y\dots \dots \left(ii\right)$

On solving (i) and (ii) we get,

$11x=9y-4\Rightarrow x=\frac{9y-4}{11}$

Substituting value of x in (ii)

$6x+3=5y$

$6\left(\frac{9y-4}{11}\right)+3=5y$

$\Rightarrow y=9$

Now, $x=\frac{9y-4}{11}$

$x=\frac{9\left(9\right)-4}{11}$

$\Rightarrow x=11$

$\therefore x=7,y=9$

$\therefore$ Required fraction is $\frac{7}{9}$