Question

The numerator of a rational number is less than its denominator by $3$. If the numerator becomes three times and the denominator is increased by $20$, the new number becomes $\frac{1}{8}$. Find the original number.

• A. Original rational number is $\frac{1}{4}$
• B. Original rational number is $\frac{7}{4}$
• C. Original rational number is $\frac{8}{3}$
• D. Original rational number is $\frac{4}{9}$

Answer 1

The correct option is A Original rational number is $\frac{1}{4}$

Let the numerator of the rational number be $x$.
Then, the denominator of the rational number $=x+3$
According to the given condition,
New numerator $=3x$ and new denominator $=(x+3)+20=x+23$
Given, $\frac{3x}{x+23}=\frac{1}{8}$

By cross multiplying, we get

$8\left(3x\right)=x+23$

$\Rightarrow 24x=x+23$

$\Rightarrow 23x=23$ ....[On Transposing]

or $x=1$

Therefore, numerator of the rational number $=1$ and denominator $=1+3=4$.
Hence, the original rational number is $\frac{1}{4}$.