The simple interest and compound interest of a certain sum compounded annually for 2 years are ₹ 900 and ₹ 954 respectively. What is the principal amount?

Correct option is D

1. Given:

- Simple Interest (SI) for 2 years = ₹900
- Compound Interest (CI) for 2 years = ₹954
- We need to find the principal amount ( P )

2. Concept/Formula Used:

- Simple Interest Formula:
$$SI = \frac{P \times R \times T}{100}$$​
Where P = Principal, R = Rate of interest, T = Time in years.
- Compound Interest Formula:
$$CI = P \left(1 + \frac{R}{100}\right)^T - P$$​
Where P = Principal, R = Rate of interest, T = Time in years.
- From the given SI , calculate the rate R first and then solve for P .

3. Solution:

- From SI :
$$900 = \frac{P \times R \times 2}{100}$$​
Simplify:
$$P \times R = 45000 \quad \text{(1)}$$​
- From CI - SI = ₹54 :
$$\left(P \left(1 + \frac{R}{100}\right)^2 - P\right) - \frac{P \times R \times 2}{100} = 54$$​
Simplify:
$$P \left(1 + \frac{R}{100}\right)^2 - \frac{2P \times R}{100} - P = 54$$​
- Approximate:
Substitute P = ₹3750 and solve:
- For SI :
$$SI = \frac{3750 \times R \times 2}{100} = ₹900 \quad \Rightarrow R = 12\%$$​
- For CI :
$$CI = 3750 \left(1 + \frac{12}{100}\right)^2 - 3750 = ₹954$$​
- Verification matches.

4. Answer:

Option D: ₹3750