Question

A certain amount becomes Rs. 17,400 in
2 years under simple interest and
becomes Rs. 17,496 in 2 years under
compound interest, at the same rate of
interest. What is the rate of interest?
(1) 8%
(2) 9%
(3) 10%
(4) 11%​

Answer 1

Let the  principal is P.

$\boxed{\displaystyle\sf{Simple\:Interest=\frac{P\times Rate\times Time}{100}}}$

Simple Interest = Amount - Principal

$\implies\displaystyle\sf{17400-P=\frac{P\times r\times 2}{100}}$

 $\implies\displaystyle\sf{2Pr=100(17400-P)---(1)}$

$\implies\displaystyle\sf{r=\frac{100(17400-P)}{2P}---(2)}$

$\boxed{\displaystyle\sf{Compounded\:Amount=P(1+\frac{r}{100})^t}}$

$\implies\displaystyle\sf{17496=P(1+\frac{r}{100})^2}$

$\implies\displaystyle\sf{17496=P(1+\frac{r^2}{10000}+\frac{2r}{100})}$

$\implies\displaystyle\sf{17496=P+\frac{Pr^2}{10000}+\frac{2Pr}{100}}$

$\displaystyle\sf{From\:(1)\:,}$

$\implies\displaystyle\sf{17496=P+\frac{Pr^2}{10000}+\frac{100(17400-P)}{100}}$

$\implies\displaystyle\sf{17496=P+17400-P+\frac{Pr^2}{10000}}$

$\implies\displaystyle\sf{96=\frac{Pr^2}{10000}}$

$\displaystyle\sf{From\:(2)\:,}$

$\implies\displaystyle\sf{96=\frac{Pr(\frac{100(17400-P)}{2P})}{10000}}$

$\implies\displaystyle\sf{960000=\frac {P(1740000r-100Pr)}{2P}}$

$\implies\displaystyle\sf{1920000=1740000r-100Pr}$

$\displaystyle\sf{Divide\:by\:100\:,}$

$\implies\displaystyle\sf{19200=17400r-Pr$

$\displaystyle\sf{From\:(1)\:,}$

$\implies\displaystyle\sf{19200=17400r-50(17400-P)}$

$\displaystyle\sf{From\:(2)\:,}$

$\implies\displaystyle\sf{19200=17400(\frac{100(17400-P)}{2P})-50(17400-P)}$

$\displaystyle\sf{Equating\:,}$

$\implies\displaystyle\sf{(17400-P)^2=384P}$

$\implies\displaystyle\sf{302760000-34800P+P^2=384P}$

$\implies\displaystyle\sf{P^2-34416P+302760000=0}$

$\implies\displaystyle\sf{P=15000}$

$\displaystyle\sf{Substituting\:in\:(2)\:,}$

$\implies\displaystyle\sf{r=\frac{100(17400-15000)}{2\times 15000}}$

$\implies\displaystyle\sf{r=\frac{240000}{30000}}$

$\implies\displaystyle\sf{r=8\%}$

$\boxed{\displaystyle\sf{Rate\:of\:Interest=8\%}}$