Question

The sum of two numbers is equal to $15$ and their arithmetic mean is $25$ percent greater than their geometric mean. Find the numbers.

Answer 1

Suppose two numbers are $m$ and $n$.

Then, $m+n=15$
Arithmetic mean (AM) $=\frac{(m+n)}{2}=\frac{15}{2}=7.5$
GM $=\sqrt{mn}$
According to the given condition AM is $25\%$ greater than GM.
AM $=$ GM $\times \frac{125}{100}$

$7.5\times 100=GM\times 125$
$GM=\frac{750}{125}=6$
GM $=6$

Then, $mn$ will be $36$.
$m+n$ is given as $15$

$\Rightarrow mn+n^2=15n$

$\Rightarrow n^2-15n+36=0$

$\Rightarrow (n-12)(n-3)=0$
Thus, the numbers are $12$ and $3$.