Question

If the arithmetic mean and the geometric mean of two numbers are in the ratio $5:4$ and the sum of two numbers is $30$, then find these numbers.

• A. $2,18$
  
• B. $6,24$
• C. $5,20$
• D. $7,28$

Answer 1

Answer: (B)

$6,24$

Let the two numbers be $A$ and $B.$

Given: $\frac{\left(\frac{A+B}{2}\right)}{\sqrt{\left(AB\right)}}=\frac{5}{4}$ ( AM and GM are in the ratio $5:4$)

$A+B=30$ (Sum of the numbers is 30) ..................... $\left(1\right)$

Therefore, $\frac{A+B}{2}=15$

$⟹\frac{15}{\sqrt{\left(AB\right)}}=\frac{5}{4}$

$⟹\sqrt{\left(AB\right)}=12$

$⟹AB=144$

$⟹B=\frac{144}{A}$

Putting this in $\left(1\right),$

$⟹A+\frac{144}{A}=30$

$⟹A^2-30A+144=0$

$⟹A^2-24A-6A+144=0$

$⟹(A-24)(A-6)=0$

$A=24\text{or}6$

and $B=6\text{or}24$

Thus, the two numbers are $6\text{and}24.$