Question

Let $\alpha ,\beta$ be two number where $\alpha <\beta$. The geometric mean of these numbers exceeds the smaller number $\alpha$ by $12$ and the arithmetic mean of the same number is smaller by $24$ than the larger number $\beta$, then the value of $|\beta -\alpha |$ is

• A. $48$
• B. $45$
• C. $44$
• D. $27$

Answer 1

The correct option is A $48$

Given $\alpha$ and $\beta$ be two numbers where $\alpha <\beta$
$\because$ The geometric mean of these numbers exceeds the smaller number $\alpha$ by $12$
i.e., $\sqrt{\alpha \beta }=\alpha +\mathrm{12...}\left(i\right)$
and the arithmetic mean of the same number is smaller by $24$ than the number $\beta$
i.e., $\frac{\alpha +\beta }{2}=\beta -\mathrm{24...}\left(ii\right)$
By solving Eqs (i) and (ii) we get
$\alpha =6,\beta =54$
$\therefore |\beta -\alpha |=|54-6|=48$