Question

The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2 : 3. Determine the fraction. [CBSE 2010]

Answer 1

Let the fraction be $\frac{x}{y}$.
As per the question
$$\begin{gathered} x+y=4+2x \\ \Rightarrow y-x=4.....\left(i\right) \end{gathered}$$
After changing the numerator and denominator
New numerator = x + 3
New denominator = y + 3
Therefore
$$\begin{gathered} \frac{x+3}{y+3}=\frac{2}{3} \\ \Rightarrow 3\left(x+3\right)=2\left(y+3\right) \\ \Rightarrow 3x+9=2y+6 \\ \Rightarrow 2y-3x=3.....\left(ii\right) \end{gathered}$$
Multiplying (i) by 3 and subtracting (ii), we get
$$\begin{gathered} 3y-2y=12-3 \\ \Rightarrow y=9 \end{gathered}$$
Now, putting y = 9 in (i), we get
$9-x=4\Rightarrow x=9-4=5$
Hence, the fraction is $\frac{5}{9}$.