If the compound interest of second year on some amount at 12% rate is 5600 then find the CI for third year at same rate?

Options

A. ₹ 6272
B. ₹6373
C. ₹ 6171
D. ₹ 6263

Answer

To solve the problem step by step, we need to find the compound interest (CI) for the third year given that the CI for the second year is 5600 at a rate of 12%.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The compound interest for the second year is given as 5600.
- The rate of interest is 12%.

2. **Calculate the Fractional Value of the Rate:**
- The fractional value of 12% is \( \frac{12}{100} = \frac{3}{25} \).

3. **Identify the Compound Interest for Each Year:**
- Let \( P \) be the principal amount.
- The interest for the first year (CI1) is \( \frac{3}{25} P \).
- The interest for the second year (CI2) is also calculated on the amount after the first year, which is \( P + CI1 \).
- The second year's interest is given as 5600, so we can express it in terms of \( P \).

4. **Set Up the Equation for the Second Year:**
- The amount at the end of the first year is \( P + \frac{3}{25} P = P \left(1 + \frac{3}{25}\right) = P \left(\frac{28}{25}\right) \).
- The interest for the second year is calculated as:
\[ CI2 = \text{Amount after first year} \times \text{Rate} = P \left(\frac{28}{25}\right) \times \frac{3}{25} = \frac{84}{625} P \]
- We know that \( CI2 = 5600 \), so:
\[ \frac{84}{625} P = 5600 \]

5. **Solve for Principal Amount \( P \):**
- Rearranging the equation gives:
\[ P = 5600 \times \frac{625}{84} \]
- Calculate \( P \):
\[ P = 5600 \times \frac{625}{84} = 5600 \times 7.4405 \approx 41600 \]

6. **Calculate the Interest for the Third Year (CI3):**
- The amount at the end of the second year is:
\[ A = P + CI1 + CI2 = P + \frac{3}{25} P + 5600 = P \left(1 + \frac{3}{25} + \frac{84}{625}\right) \]
- The interest for the third year is calculated as:
\[ CI3 = \text{Amount after second year} \times \text{Rate} = A \times \frac{3}{25} \]
- Substitute the value of \( A \):
\[ CI3 = P \left(1 + \frac{3}{25} + \frac{84}{625}\right) \times \frac{3}{25} \]

7. **Final Calculation:**
- We can find \( CI3 \) using the relationship:
\[ CI3 = CI2 \times \left(1 + \frac{3}{25}\right) = 5600 \times \frac{28}{25} = 6272 \]

### Conclusion:
The compound interest for the third year at the same rate is **6272**.