Question

The denominator of a fraction is greater than its numerator by 12. If the numerator is decreased by 2 and the denominator is increased by 7, the new fraction is equivalent with $\frac{1}{2}$. Find the fraction.

Answer 1

Let the numerator of the fraction be x.

∴ Denominator = x + 12

∴ The fraction = $\frac{x}{x+12}$

If the numerator is decreased by 2 and the denominator is increased by 7, the new fraction obtained is $\frac{1}{2}$.
From the given information,
$$\begin{gathered} \frac{x-2}{x+12+7}=\frac{1}{2} \\ \therefore 2\left(x-2\right)=1\left(x+12+7\right) \\ \therefore 2x-4=x+19 \\ \therefore 2x-x=19+4 \\ \therefore x=23 \end{gathered}$$

∴ Numerator of the fraction is 23.
∴ Denominator of the fraction is 23 + 12 = 35.

Hence, the fraction is $\frac{23}{35}$.