Question

The denominator of a fraction is 4 more than twice its numerator. Denominator becomes 12 times the numerator, if both the numerator and the denominator are reduced by 6. Find the fraction.

Answer 1

Let the fraction be $\frac{x}{y}$.
Denominator of a fraction is 4 more than twice its numerator.
So,
$$\begin{gathered} y=4+2x \\ \Rightarrow 2x-y=-4.....\left(\mathrm{I}\right) \end{gathered}$$
Also, denominator becomes 12 times the numerator, if both the numerator and the denominator are reduced by 6.
So,
$$\begin{gathered} y-6=12\left(x-6\right) \\ \Rightarrow y-6=12x-72 \\ \Rightarrow 12x-y=72-6=66 \\ \Rightarrow 12x-y=66.....\left(\mathrm{II}\right) \end{gathered}$$
Subtracting (I) from (II)
$$\begin{gathered} 12x-2x-y-\left(-y\right)=66-\left(-4\right) \\ \Rightarrow 10x=70 \\ \Rightarrow x=\frac{70}{10}=7 \\ \Rightarrow x=7 \end{gathered}$$
Putting the value of x = 7 in (I)
$$\begin{gathered} 2\left(7\right)-y=-4 \\ \Rightarrow 14-y=-4 \\ \Rightarrow y=14+4=18 \end{gathered}$$
Thus, the fraction obtained is $\frac{7}{18}$.