Question

The denominator of a fraction is $1$ more than twice its numerator. If $1$ is added to the numerator and the denominator, the ratio of the numerator to the denominator is $1:2$. Find the fraction.

Answer 1

Let the numerator be $x$ and denominator be $y$.

According to the first condition

$y=2x+1$

$2x-y=-1$ ...............$\left(i\right)$

According to the second condition

$\frac{(x+1)}{(y+1)}=\frac{1}{2}$

$\Rightarrow 2(x+1)=1(y+1)$

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

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

$\Rightarrow 2x-y=-1$...............$\left(ii\right)$

From $\left(i\right)$ and $\left(ii\right)$ we have

There can be infinite values that will satisfy.

One such possible value is $x=4$ and $y=9$

$\therefore$, the fraction is $\frac{4}{9}$