Question

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

Answer 1

Let the numerator of the fraction =x

Then denominator=12+x

So, fraction will be

$\frac{x}{12+x}$

New fraction:

numerator =x-2

denominator=12+x+7

=19+x

So, fraction will be

$\frac{x-2}{19+x}$

$\begin{array}{c}\frac{x-2}{19+x}=\frac{1}{2}\\ 2\left(x-2\right)=19+x\\ 2x-x=19+4\\ x=23\end{array}$

So, old fraction is

$\frac{23}{12+23}=\frac{23}{35}$

and new fraction is,

$\begin{array}{c}\frac{23-2}{19+23}=\frac{21}{42}\\ =\frac{1}{2}\end{array}$