Question

If geometric mean and harmonic mean of two numbers $a$ and $b$ are 16 and $\frac{64}{5}$ respectively, then the value of $a:b$ is

• A. $4:1$
• B. $3:2$
• C. $2:3$
• D. $1:4$

Answer 1

Answer: (A)

Geometric mean of $a$ and $b=\sqrt{ab}$
$\Rightarrow \sqrt{ab}=16$ (given)
$\Rightarrow ab=256$ ... (i)
And harmonic mean of $a$ and $b=\frac{2ab}{a+b}$
$\therefore \frac{2ab}{a+b}=\frac{64}{5}$ (given)
$\Rightarrow \frac{2\times 256}{a+b}=\frac{64}{5}[$ from Eq. (i) $]$
$\Rightarrow a+b=40$ ... (ii)
Now, $(a-b)=\sqrt{(a+b{)}^2-4ab}$
$=\sqrt{(40{)}^2-4\times 256}$
$=\sqrt{1600-1024}$
$=\sqrt{576}$
$\Rightarrow a-b=24$ ...(ii)
On solving Eqs. (ii) and (iii), we get
$a=32$ and $b=8$
$a:b=32:8$
$=4:1$