Question

If the numerator of a fraction is increased by 2 and its denominator is decreased by 1, it becomes $\frac{2}{3}$. If the numerator is increased by 1 and the denominator is increased by 2, it becomes $\frac{1}{3}$. Find the fraction.
[3 Marks]

Answer 1

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

$\therefore \frac{x+2}{y-1}=\frac{2}{3}$ and $\frac{x+1}{y+2}=\frac{1}{3}$

$\Rightarrow 3x-2y=-8\dots \left(i\right)$
and $3x-y=-1\dots \left(ii\right)$ [1 Mark]

Subtract equation (ii) from equation (i), to eliminate $x$,

$\begin{array}{cc}3x-2y& =-8\\ -(3x-y& =-1)\\ -y& =-7\\ \Rightarrow y& =7\end{array}$
[1 Mark]

Substitute $y=7$ in equation (i), to get the value of $x$,

$\begin{array}{cc}3x-\left(14\right)& =-8\\ 3x& =-8+14\\ 3x& =6\\ \therefore x& =2\end{array}$

Now, we have $x=2$ and $y=7$.

Therefore, the given fraction is $\frac{2}{7}$. [1 Mark]