Sitaram, a money lender lent apart of Rs 200000 to Shyam at simple interest 6% p.a. and the remaining to Sushil at 10% p.a. at simple interest. Sitaram earned an annual interest income of Rs. 18000. Based on the given information answer the following questions :

What is the mean rate of interest ?

9% p.a.

The correct answer is Option (3) → 9% p.a.

Let,

x = Amount lent to Shyam

$(2,00,000-x)$ = Amount lent to Sushil (at 10% pa)

Total annual interest earned = Rs 18,000

Simple interest = $\frac{P×R×T}{100}$

Interest from Shyam (6% p.a.) = $\frac{x×6×1}{100}=0.06x$

Interest from Sushil (at 10% p.a.) = $\frac{(2,00,000-x×10×1)}{100}$

$=0.10(2,00,000-x)$

and,

$0.06x+0.10(2,00,000-x)=18,000$

$⇒0.04x=2000$

$⇒x=\frac{2000}{0.04}=5000$

$P_1$ Amount lent to Shyam = Rs 50,000 → 6% p.a.

$P_2$ Amount lent to Sushil = Rs 1,50,000 → 10% p.a.

The mean rate of interest $R_{mean}$ is,

$R_{mean}=\frac{(P_1×R_1)+(P_2×R_2)}{P_1+P_2}$

$=\frac{(50,000×6)+(1,50,000×10)}{50,000+1,50,000}$

$=\frac{18,00,000}{2,00,000}=9\%$