Question

In a rational number, twice the numerator is $2$ more than denominator. If $3$ is added to each, the numerator and the denominator, the new fraction is $\frac{2}{3}$. Find the original number.

Answer 1

Let numerator$=x$

Then denominator $=2x-2$

Fraction $=\frac{x}{2x-2}$

According to the condition :

$\frac{x+3}{2x-2+3}=\frac{2}{3}$

$\Rightarrow \frac{x+3}{2x+1}=\frac{2}{3}$

By cross multiplication

$3(x+3)=2(2x+1)$

$\Rightarrow 3x+9=4x+2$

$\Rightarrow 9-2=4x-3x$
$\therefore x=7$

Thus, the rational number $=\frac{x}{2x-2}=\frac{7}{(2\times 7)-2}=\frac{7}{14-2}=\frac{7}{12}$.