Question

The denominator of a rational number is greater than its numerator by $8$. If the numerator is increased by $17$ and the denominator is decreased by $1$, the number obtained is $\frac{3}{2}$. Find the rational number.

• A. $\frac{13}{21}$
• B. $\frac{10}{9}$
• C. $\frac{2}{3}$
• D. $\frac{15}{13}$

Answer 1

The correct option is A $\frac{13}{21}$

Let the numerator of the rational number be $x$.
Then, the denominator of the rational number $=x+8.$
$\therefore$ the rational number $=$ $\frac{x}{x+8}$.
If the numerator is increased by $17$ and the denominator is decreased by $1$, the number becomes $\frac{3}{2}$.
$\Rightarrow \frac{x+17}{x+7}=\frac{3}{2}$
$\Rightarrow 2(x+17)=3(x+7)$ ....[On cross multiplying]
$\Rightarrow 2x+34=3x+21$
$\Rightarrow 2x-3x=21-34$ ....[Transposing $3x$ to LHS and $34$ to RHS]
$\Rightarrow -x=-13\Rightarrow x=13$
$\Rightarrow \frac{x}{x+8}=\frac{13}{13+8}=\frac{13}{21}$
Hence, the required, rational number $=$ $\frac{13}{21}$.