If the compound interest on a certain sum for 2 years is Rs 308 and simple interest at same rate of interest and for same time is 280. Find the sum?

Options

A. A)Rs 700
B. B)Rs 1200
C. C)Rs.1100
D. D)Rs 500

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Compound Interest for 2 years (CI) = Rs. 308
- Simple Interest for 2 years (SI) = Rs. 280

2. **Calculate Simple Interest for 1 Year:**
- Since the total simple interest for 2 years is Rs. 280, the simple interest for 1 year (SI for 1 year) is:
\[ \text{SI for 1 year} = \frac{280}{2} = Rs. 140 \]

3. **Calculate Total Interest for 2 Years:**
- The total interest for 2 years using simple interest is Rs. 280.
- The total interest for 2 years using compound interest is Rs. 308.

4. **Determine the Interest on Interest:**
- The difference between the compound interest and simple interest for 2 years gives us the interest on interest:
\[ \text{Interest on Interest} = \text{CI} - \text{SI} = 308 - 280 = Rs. 28 \]

5. **Relate Interest on Interest to Principal:**
- The interest on interest for the second year is calculated based on the principal. Since the simple interest for 1 year is Rs. 140, we can say that:
\[ \text{Interest on Interest} = \frac{P \times r}{100} \quad \text{(where r is the rate of interest)} \]
- From the first year, we know that Rs. 140 is the interest for the principal amount P.

6. **Set Up the Equation:**
- The interest on interest (Rs. 28) can be expressed as:
\[ \frac{P \times r}{100} = 28 \]
- We also know that:
\[ \frac{P \times r}{100} = 140 \quad \text{(for the first year)} \]

7. **Find the Rate of Interest:**
- From the first year, we can express the rate of interest (r) in terms of P:
\[ r = \frac{140 \times 100}{P} \]

8. **Substitute r into the Interest on Interest Equation:**
- Substitute r into the equation for interest on interest:
\[ \frac{P \times \frac{140 \times 100}{P}}{100} = 28 \]
- Simplifying this gives:
\[ 140 = 28 \]
- This means that the interest on interest is 1/5 of the first year's interest (140).

9. **Calculate the Principal:**
- Since the interest on interest is 1/5 of the first year’s interest, we can set up the equation:
\[ \frac{P \times r^2}{100^2} = 28 \]
- From the earlier calculations, we can find:
\[ P = 700 \]

### Final Answer:
The principal sum (P) is Rs. 700.