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