Question

If $HM:GM=4:5$ for two positive numbers then the ratio of the numbers is

• A. 4:1
• B. 3:2
• C. 3:4
• D. 2:3

Answer 1

The correct option is A 4:1

Let the two numbers be $a$ and $b$
Given
$\frac{2ab}{(a+b)\sqrt{ab}}=\frac{4}{5}$
Squaring, $\frac{4ab}{(a+b{)}^2}=\frac{16}{25}$
$\Rightarrow 100ab=16a^2+32ab+16b^2$
$\Rightarrow 16a^2-68ab+16b^2=0$
$\Rightarrow 4a^2-17ab+4b^2=0$
$\Rightarrow (4a-b)(4b-a)=0$
Hence, $a=4b$ or $b=4a$
So the ratio is $4:1$