If compound interest for third year is ₹1000 @10% p.a compounded annually, find the compound interest for seventh year.

Options

A. ₹ 1464.10
B. ₹ 1331
C. ₹ 1440
D. ₹ 1610.5

Answer

To solve the problem of finding the compound interest for the seventh year given that the compound interest for the third year is ₹1000 at a rate of 10% per annum compounded annually, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The compound interest for the third year is ₹1000.
- The rate of interest is 10% per annum.

2. **Determine the Principal Amount (P)**:
- The compound interest for the third year can be expressed as:
\[ \text{CI for 3rd year} = A_3 - A_2 \]
- Where \( A_3 \) is the amount at the end of the third year and \( A_2 \) is the amount at the end of the second year.
- Given that \( \text{CI for 3rd year} = ₹1000 \), we can express \( A_3 \) and \( A_2 \) in terms of the principal \( P \):
\[ A_3 = P(1 + r)^3 \quad \text{and} \quad A_2 = P(1 + r)^2 \]
- Substituting these into the equation gives:
\[ 1000 = P(1 + 0.10)^3 - P(1 + 0.10)^2 \]

3. **Calculate the Amounts**:
- First, calculate \( (1 + 0.10)^2 \) and \( (1 + 0.10)^3 \):
\[ (1 + 0.10)^2 = 1.1^2 = 1.21 \]
\[ (1 + 0.10)^3 = 1.1^3 = 1.331 \]
- Thus, we have:
\[ 1000 = P(1.331 - 1.21) = P(0.121) \]
- Solving for \( P \):
\[ P = \frac{1000}{0.121} \approx 8264.46 \]

4. **Calculate the Compound Interest for the Seventh Year**:
- The compound interest for the seventh year can be expressed as:
\[ \text{CI for 7th year} = A_7 - A_6 \]
- Where:
\[ A_7 = P(1 + r)^7 \quad \text{and} \quad A_6 = P(1 + r)^6 \]
- Thus:
\[ \text{CI for 7th year} = P(1 + r)^7 - P(1 + r)^6 = P(1.1^7 - 1.1^6) \]
- Calculate \( (1 + 0.10)^6 \) and \( (1 + 0.10)^7 \):
\[ (1 + 0.10)^6 = 1.1^6 \approx 1.771561 \]
\[ (1 + 0.10)^7 = 1.1^7 \approx 1.948717 \]
- Now substituting:
\[ \text{CI for 7th year} = P(1.948717 - 1.771561) = P(0.177156) \]
- Substituting \( P \):
\[ \text{CI for 7th year} = 8264.46 \times 0.177156 \approx 1464.10 \]

5. **Final Result**:
- Therefore, the compound interest for the seventh year is approximately ₹1464.10.