Question

The denominator of a rational number is greater than its numerator by $6$. If the numerator is increased by $5$ and the denominator is decreased by $3$, the number obtained is $\frac{5}{4}$. Find the rational number.

• A. $\frac{2}{9}$
• B. $\frac{5}{11}$
• C. $\frac{7}{6}$
• D. $\frac{10}{11}$

Answer 1

The correct option is B $\frac{5}{11}$

Let the numerator of the rational number be $x$.
Then, the denominator of the rational number $=x+6$.
It is given that the numerator is increased by $5$ and the denominator is decreased by $3$.
$\therefore$ the numerator of the new rational number $=x+5$.
Denominator of the new rational number $=(x+6)-3=x+3$
$\therefore$ new rational number = $\frac{x+5}{x+3}$
But the new rational number is given as $\frac{5}{4}$.
$\therefore \frac{x+5}{x+3}=\frac{5}{4}$
By cross multiplying, we get,
$4(x+5)=5(x+3)$
$\Rightarrow 4x+20=5x+15$
$\Rightarrow 4x-5x=15-20$ ....[Transposing $5x$ to LHS. and $20$ to RHS]
$\Rightarrow -x=-5$ or $x=5$
$\therefore$ numerator of the rational number is $5$ and denominator $=$ $5+6$ i.e. $11$.
$\therefore$ the required rational number is $\frac{5}{11}$.