The simple interest and compound Interest (compounded annually) on a certain sum of money with a given rate for a period of 2 years are `Rs.900` and `Rs. 954` respectively. The sum of money is

Options

A. `Rs. 3700`
B. `Rs. 3650`
C. `Rs. 3850`
D. `Rs. 3750 `

Answer

To solve the problem, we need to find the principal amount (the sum of money) given the simple interest (SI) and compound interest (CI) for a period of 2 years.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Simple Interest (SI) for 2 years = Rs. 900
- Compound Interest (CI) for 2 years = Rs. 954

2. **Calculate the Difference Between CI and SI:**
- Difference = CI - SI
- Difference = 954 - 900 = Rs. 54

3. **Understand the Relationship Between SI, CI, and Principal:**
- The difference between CI and SI over 2 years is due to the interest on the interest (compounding).
- The formula for the difference between CI and SI for 2 years is:
\[ \text{Difference} = \frac{P \times r^2}{100^2} \]
where \( P \) is the principal and \( r \) is the rate of interest.

4. **Express the Rate of Interest (r):**
- From the difference calculated, we can express:
\[ 54 = \frac{P \times r^2}{100^2} \]

5. **Calculate the Simple Interest for 1 Year:**
- Since SI for 2 years is Rs. 900, SI for 1 year is:
\[ \text{SI for 1 year} = \frac{900}{2} = Rs. 450 \]

6. **Use the SI Formula to Find Principal:**
- The formula for Simple Interest is:
\[ SI = \frac{P \times r \times t}{100} \]
- For 1 year:
\[ 450 = \frac{P \times r \times 1}{100} \]
- Rearranging gives:
\[ P \times r = 45000 \quad \text{(1)} \]

7. **Substituting r from the Difference Equation:**
- From the difference equation, we can express \( r^2 \):
\[ r^2 = \frac{54 \times 100^2}{P} \]
- Taking the square root gives:
\[ r = \sqrt{\frac{5400000}{P}} \quad \text{(2)} \]

8. **Substituting (2) into (1):**
- Substitute \( r \) from equation (2) into equation (1):
\[ P \times \sqrt{\frac{5400000}{P}} = 45000 \]
- Squaring both sides:
\[ P \times \frac{5400000}{P} = 2025000000 \]
- Simplifying:
\[ 5400000 = 2025000000 \]
- Rearranging gives:
\[ P = \frac{2025000000}{5400000} = 3750 \]

9. **Final Answer:**
- The sum of money (Principal) is Rs. 3750.