Question

The denominator of a rational number is greater than its numerator by $7.$ If the numerator is increased by $17$ and the denominator is decreased by $6,$ the new number becomes $2.$ Find the original number.

Answer 1

Let numerator of a rational number be $x$

Then, its denominator $=x+7$

and, Original rational number $=\frac{x}{x+7}$

In second case, the numerator is increased by $17$ and the denominator is decreased by $6.$

Then, the new number $=\frac{x+17}{x+7-6}$

$=\frac{x+17}{x+1}$

New number is given as $2.$

According to the question, we have

$\frac{x+17}{x+1}=2$

$\Rightarrow x+17=2x+2$

$\Rightarrow 2x-x=17-2$

$\Rightarrow x=15$

$\therefore$ Original Rational number $=\frac{x}{x+7}=\frac{15}{15+7}=\frac{15}{22}$