# The difference between the compound interest and the simple interest on a certain sum for 2 years

Question:

The difference between the compound interest and the simple interest on a certain sum for 2 years at 6% per annum is Rs 90. Find the sum.

Solution:

Let $\operatorname{Rs} P$ be the sum.

Then SI $=\left(\frac{P \times 2 \times 6}{100}\right)=$ Rs. $\frac{12 P}{100}=$ Rs. $\frac{3 P}{25}$

Also, CI $=\left\{P \times\left(1+\frac{6}{100}\right)^{2}-P\right\}$

$=$ Rs. $\left\{P \times\left(\frac{100+6}{100}\right)^{2}-P\right\}$

$=$ Rs. $\left\{P \times\left(\frac{53}{50}\right)^{2}-P\right\}$

$=$ Rs. $\left\{\left(\frac{2809 P}{2500}\right)-P\right\}$

$=$ Rs. $\left\{\frac{2809 P-2500 P}{2500}\right\}=$ Rs. $\frac{309 P}{2500}$

Now, $(\mathrm{CI}-\mathrm{SI})=\mathrm{Rs} .\left(\frac{309 P}{2500}-\frac{3 P}{25}\right)$

$=$ Rs. $\left(\frac{309 P-300 P}{2500}\right)$

$=$ Rs. $\frac{9 P}{2500}$

Now, Rs. $90=\frac{9 P}{2500}$

$\Rightarrow P=\left(\frac{90 \times 2500}{9}\right)=$ Rs. 25000

Hence, the required sum is Rs. 25000 .