Question

If $2$ is added to the numerator of a fraction, it reduces to $\frac{1}{2}$ and if $1$is subtracted from the denominator, it reduces to$\frac{1}{3}$.

Find the fraction.

Answer 1

Step $1$: Framing the equations.

Assume $x$ is the numerator and$y$is the denominator.

Given: $2$ is added to the numerator of a fraction, it reduces to $\frac{1}{2}$.

Thus, $\frac{x+2}{y}=\frac{1}{2}$

$$\begin{gathered} \Rightarrow 2(x+2)=y \\ \Rightarrow 2x+4=y \end{gathered}$$

$\Rightarrow 2x-y=-4$……………$\left(1\right)$

Also given that $1$ is subtracted from the denominator of a fraction, it reduces to $\frac{1}{3}$.

Thus, $\frac{x}{y-1}=\frac{1}{3}$

$\Rightarrow 3x=y-1$

$\Rightarrow 3x-y=-1$…………… $\left(2\right)$

Step $2$ : Solving the simultaneous equations.

Subtract equation $\left(2\right)$ from equation $\left(1\right)$.

$$\begin{gathered} 2x-y-3x+y=-4+1 \\ \Rightarrow -x=-3 \\ \Rightarrow x=3 \end{gathered}$$

Now, put the value of $x$ in equation $\left(2\right)$.

$$\begin{gathered} 2\left(3\right)-y=-4 \\ \Rightarrow 6-y=-4 \\ \Rightarrow -y=-10 \\ \Rightarrow y=10 \end{gathered}$$

Hence, the fraction is $\frac{3}{10}$.