Compound Interest Sum Calculation

The problem asks us to find the principal sum (P) when the compound interest (CI), rate (R), and time (T) are given. We are given:

• Compound Interest (CI) = ₹6,800
• Rate (R) = $12\frac{1}{2}\%$ per annum = 12.5% per annum
• Time (T) = 2 years
• Compounding is done annually.

Compound Interest Formula

The formula for compound interest is:

CI = P $\left[ \left(1 + \frac{R}{100}\right)^T - 1 \right]$

Where:

• P = Principal sum
• R = Annual interest rate
• T = Time in years

Step-by-Step Calculation

1. Convert Rate to Fraction: The rate R = 12.5% can be written as a fraction: $R = 12.5\% = \frac{12.5}{100} = \frac{125}{1000} = \frac{1}{8}$
2. Calculate the Growth Factor: Calculate the term $(1 + \frac{R}{100})$: $1 + \frac{R}{100} = 1 + \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8}$
3. Calculate the Compounding Factor: Calculate $(1 + \frac{R}{100})^T$ for T=2 years: $\left(\frac{9}{8}\right)^2 = \frac{9^2}{8^2} = \frac{81}{64}$
4. Calculate the Interest Factor: Find the value of $\left[ \left(1 + \frac{R}{100}\right)^T - 1 \right]$: $\frac{81}{64} - 1 = \frac{81 - 64}{64} = \frac{17}{64}$
5. Solve for Principal (P): Substitute the known values into the CI formula: $6,800 = P \times \frac{17}{64}$
6. Isolate P: Rearrange the equation to solve for P: $P = 6,800 \times \frac{64}{17}$
7. Final Calculation: Perform the multiplication: $P = (6,800 \div 17) \times 64$ $P = 400 \times 64$ $P = 25,600$

Conclusion

The principal sum on which the compound interest will be ₹6,800 is ₹25,600. This corresponds to Option B.