Question

A person lent out a certain sum on simple interest and the same sum on compound interest at certain rate of interest per annum. He noticed that the ratio between the difference of compound interest and simple interest of 3 years and that of $2$ years is $25:8$. The rate of interest per annum is........

• A. $10\%$
• B. $11\%$
• C. $12\%$
• D. $12\frac{1}{2}\%$

Answer 1

The correct option is D $12\frac{1}{2}\%$

$SI=Prt$

$CI=P{(1+\frac{r}{n})}^{nt}-P$

$n=1\left\{Parannum\right\}$

$CI=P(1+r{)}^t-P$

$CI-SI=P(1+r{)}^t-P-Prt$

at $t=3$

$CI-SI=P(1+r{)}^3-P-3Pr...(1)$

at $t=2$

$CI-SI=P(1+r^2)-P-2Pr...\left(2\right)$

$1÷2$

$\frac{P(1+r{)}^3-P-3Pr}{P(1+r{)}^2-P-2Pr}=\frac{25}{8}$

$\frac{(1-r{)}^3-1-3r}{(1+r{)}^2-1-2r}=\frac{25}{8}$

$\frac{1+r^3+3r+3r^2-1-3r}{1+r^2+2r-1-2r}=\frac{25}{8}$

$\frac{r^3+3r^2}{r^2}=\frac{25}{8}$

$r+3=\frac{25}{8}$

$r=.125$

$\%r=12.5\%$

$=12\frac{1}{2}\%$