Question

If the ratio of $H.M.$and $G.M$. between two numbers $a$ and $b(a>b)$is$4:5$, then the ratio of the two numbers will be

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

Answer 1

The correct option is C

$4:1$

Step 1. Find the ratio of the two numbers:

HM of $a$and $b=\frac{2ab}{(a+b)}$

GM of $a$ and$b$ $=\sqrt{\left(ab\right)}$

Ratio of $HMandGM=4:5$

$\Rightarrow$ $2\sqrt{\frac{\left(ab\right)}{(a+b)}}=\frac{4}{5}$

$\Rightarrow$ $2a+2b=5\sqrt{\left(ab\right)}$

$\Rightarrow$ $\frac{(a+b)}{\sqrt{\left(ab\right)}}=\frac{5}{2}$

$\Rightarrow$ $\frac{a}{\sqrt{\left(ab\right)}}+\frac{b}{\sqrt{\left(ab\right)}}=\frac{5}{2}$

$\Rightarrow$ $\sqrt{\left(\frac{a}{b}\right)}+\sqrt{\left(\frac{b}{a}\right)}=\frac{5}{2}$

Put $\sqrt{\frac{a}{b}}=x$

$\Rightarrow$ $x+\frac{1}{x}=\frac{5}{2}$

$\Rightarrow$ $2x^2-5x+2=0$

Solving we get $x=\frac{1}{2}or2$

Since $a>b,\sqrt{\left(\frac{a}{b}\right)}=2$

$\Rightarrow$ $\frac{a}{b}=4$

$\Rightarrow$So the ratio is $4:1.$

So, the option (C) is correct.