Question

A rational number becomes $\frac{1}{3}$ on subtracting $1$ from its numerator and it becomes $\frac{1}{4}$when $8$ is added to its denominator, the rational number is:

• A. $\frac{5}{12}$
• B. $\frac{1}{12}$
• C. $\frac{7}{12}$
• D. $\frac{7}{17}$

Answer 1

The correct option is A

$\frac{5}{12}$

Step 1: Assumption and Apply first situation

Let the numerator of the rational number be $x$ and its denominator be $y$.

The rational number becomes $\frac{1}{3}$ on subtracting $1$ from its numerator.

So,

$\begin{array}{rcl}\frac{x-1}{y}& =& \frac{1}{3}\\ 3x-3& =& y\\ 3x-y& =& 3......\left(1\right)\end{array}$

Step 2: Apply second condition

The number becomes $\frac{1}{4}$when $8$ is added to its denominator.

$\begin{array}{rcl}\frac{x}{y+8}& =& \frac{1}{4}\\ 4x& =& y+8\\ 4x-y& =& 8...\left(2\right)\end{array}$

Step 3: Solve the system of equations.

Subtract $\left(1\right)$ from $\left(2\right)$ to get:

$\begin{array}{rcl}4x-y-(3x-y)& =& 8-3\\ 4x-3x-y+y& =& 5\\ x& =& 5\end{array}$

Putting this in $\left(1\right)$ to get:

$\begin{array}{rcl}3\left(5\right)-y& =& 3\\ y& =& 15-3\\ y& =& 12\end{array}$

The rational number is $\frac{5}{12}$.

Hence, option (a) is correct.