Question

The arithmetic mean of two numbers is $17$ and their geometric mean is $15$. Find the numbers.

Answer 1

We know that the arithmetic mean between the two numbers $a$ and $b$ is $AM=\frac{a+b}{2}$ and the geometric mean is $GM=\sqrt{ab}$.

Here, it is given that the arithmetic mean of $a$ and $b$ is $17$, therefore,

$AM=\frac{a+b}{2}\Rightarrow 17=\frac{a+b}{2}\Rightarrow 17\times 2=a+b\Rightarrow a+b=34...\left(1\right)$

Also, it is given that the geometric mean of $a$ and $b$ is $15$, therefore,

$GM=\sqrt{ab}\Rightarrow 15=\sqrt{ab}\Rightarrow ab={15}^2\Rightarrow ab=\mathrm{225.....}\left(2\right)$

Now consider $(a-b{)}^2$ and useequations 1 and 2 as follows:

$(a-b{)}^2=(a+b{)}^2-4ab=(34{)}^2-(4\times 225)=1156-900=256$ implies that

$a-b=\mathrm{16.....}\left(3\right)$

Adding equations 1 and 3, we have

$(a+a)+(b-b)=34+16\Rightarrow 2a=50\Rightarrow a=\frac{50}{2}=25$

Substitute the value of $a$ in equation 1 as follows:

$25+b=34$

$b=34-25=9$

Hence, the numbers are $25$ and $9$.