The compound interest for third and fourth year on a certain sum are ₹ 2880 and ₹ 3456 respectively. Find the simple interest on the same sum at the same rate for 8 years.

Options

A. ₹ 12000
B. ₹ 16000
C. ₹14000
D. ₹ 15000

Answer

To find the simple interest on a certain sum at the same rate for 8 years, given the compound interest for the third and fourth years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Compound Interest for the Years:**
- Given:
- Compound Interest (C.I.) for the 3rd year = ₹2880
- Compound Interest (C.I.) for the 4th year = ₹3456

2. **Calculate the Rate of Interest:**
- The difference in C.I. between the 4th year and the 3rd year gives us the interest earned on the principal amount during the 3rd year.
- Difference = C.I. for 4th year - C.I. for 3rd year
- Difference = ₹3456 - ₹2880 = ₹576
- This ₹576 is the interest earned on the principal amount during the 3rd year.

3. **Set Up the Relationship:**
- The C.I. for the 3rd year is calculated as:
\[ \text{C.I. for 3rd year} = \text{Principal} \times \frac{R}{100} \times 1 + \text{C.I. from previous years} \]
- The C.I. for the 4th year is calculated similarly:
\[ \text{C.I. for 4th year} = \text{Principal} \times \left(1 + \frac{R}{100}\right) \times \frac{R}{100} \]
- The difference in C.I. gives us:
\[ \text{C.I. for 4th year} - \text{C.I. for 3rd year} = \text{Principal} \times \frac{R}{100} \times \left(1 + \frac{R}{100}\right) - \text{Principal} \times \frac{R}{100} \]
- Simplifying this gives:
\[ \text{Principal} \times \frac{R}{100} \times \frac{R}{100} = ₹576 \]
- From the above equation, we can derive the rate \( R \).

4. **Calculate the Principal:**
- We know that the C.I. for the 3rd year is ₹2880. We can express this in terms of the principal and rate:
\[ \text{C.I. for 3rd year} = \text{Principal} \times \frac{R}{100} \]
- Rearranging gives:
\[ \text{Principal} = \frac{C.I. for 3rd year}{\frac{R}{100}} \]

5. **Calculate the Simple Interest for 8 Years:**
- The formula for Simple Interest (S.I.) is:
\[ S.I. = \frac{\text{Principal} \times R \times T}{100} \]
- Where \( T = 8 \) years.
- Substitute the values of Principal and R into the formula to find the S.I.

6. **Final Calculation:**
- Substitute the calculated values into the S.I. formula to find the final answer.