Question

The arithmetic mean of two numbers is $18\frac{3}{4}$ and the positive square root of their product is $15.$ The larger of the two numbers is

• A. $24$
• B. $25$
• C. $20$
• D. $30$

Answer 1

Answer: (D)

Let $a$ and $b$ be the positive numbers.
Then $\frac{a+b}{2}=18\frac{3}{4}\Rightarrow a+b=\frac{75}{2}$
and $\sqrt{ab}=15\Rightarrow ab=225$
so, ${\left(a-b\right)}^2={\left(a+b\right)}^2-4ab={\left(\frac{75}{2}\right)}^2-4\times 225$
$=\left(\frac{75}{2}+30\right)\left(\frac{75}{2}-30\right)=\frac{135}{2}\times \frac{15}{2}$
$\Rightarrow a-b=\pm \left(\frac{15}{2}\times 3\right)=\pm \left(\frac{45}{2}\right)$
Case1 $\to a+b=\frac{75}{2},a-b=\frac{45}{2}$
$\Rightarrow a=30,b=\frac{45}{2}$
Case2 $\to a+b=\frac{75}{2},a-b=-\frac{45}{2}$
$\Rightarrow a=\frac{15}{2},b=30$
So, larger of the two numbers is $30$