Finding Compound Interest Principal Sum

The problem provides the amount a sum grows to after 1 year and after 2 years with compound interest. We need to find the original principal sum.

Step 1: Calculate the Annual Interest Rate

Let the principal sum be $P$, and the annual interest rate be $r\%$. The amount after 1 year ($A_1$) is given by the formula: $A_1 = P \left(1 + \frac{r}{100}\right)$ We are given $A_1 = \text{₹ } 650$. So, $P \left(1 + \frac{r}{100}\right) = 650$. (Equation 1)

The amount after 2 years ($A_2$) is: $A_2 = P \left(1 + \frac{r}{100}\right)^2$ We are given $A_2 = \text{₹ } 676$. So, $P \left(1 + \frac{r}{100}\right)^2 = 676$. (Equation 2)

To find the rate, divide Equation 2 by Equation 1:

$ \frac{P \left(1 + \frac{r}{100}\right)^2}{P \left(1 + \frac{r}{100}\right)} = \frac{676}{650} $

$ 1 + \frac{r}{100} = \frac{676}{650} $

Simplify the fraction:

$ \frac{676}{650} = \frac{26 \times 26}{25 \times 26} = \frac{26}{25} $

Now, substitute this back:

$ 1 + \frac{r}{100} = \frac{26}{25} $

Solve for $\frac{r}{100}$:

$ \frac{r}{100} = \frac{26}{25} - 1 = \frac{26 - 25}{25} = \frac{1}{25} $

The annual interest rate is $r = \frac{1}{25} \times 100 = 4\%$.

Step 2: Calculate the Principal Sum

Now, use Equation 1 with the calculated rate factor ($1 + \frac{r}{100} = \frac{26}{25}$):

$ P \left(\frac{26}{25}\right) = 650 $

Solve for the principal $P$:

$ P = 650 \times \frac{25}{26} $

Calculate the value:

$ P = \frac{650}{26} \times 25 $

$ P = 25 \times 25 $

$ P = 625 $

The certain sum (principal) is ₹ 625.