Question

Consider the ratio $12:5$. If this ratio has to be reduced by $20\%$, which common number should be added to both the numerator and denominator?

Answer 1

Consider $\frac{12}{5}$. This has to be reduced by 20%. This means we have to consider 80% of this number. (Why?) Thus we must get
$\frac{12}{5}\times \frac{80}{100}=\frac{48}{25}$.
We have to find $a$ such that
$\frac{12+a}{5+a}=\frac{48}{25}$.
Cross multiplying, we obtain
$25(12+a)=48(5+a)$.
This simplifies to
$48a-25a=(25\times 12)-(48\times 5),$ or $23a=60$.
Hence,
$a=\frac{60}{23}$.
If we add $\frac{60}{23}$ to both terms of $12:5$ we get a ratio which is $20\%$ less than the original ratio.