If the difference between the CI and SI on some amount is 170 at the rate of `9 (1)/(11)%` interest for 3 years then find the amount?

Options

A. ₹ 6655
B. ₹6556
C. ₹ 5665
D. ₹ 5566

Answer

To solve the problem step by step, we will follow the mathematical principles of compound interest (CI) and simple interest (SI).

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Difference between CI and SI: \( CI - SI = 170 \)
- Rate of interest: \( R = 9 \frac{1}{11}\% = \frac{100}{11}\% \)
- Time period: \( T = 3 \) years

2. **Convert the Rate of Interest**:
- Convert \( R \) into a decimal for calculations:
\[ R = \frac{100}{11} = 9.0909\% \]

3. **Calculate Simple Interest (SI)**:
- The formula for SI is:
\[ SI = \frac{P \times R \times T}{100} \]
- Substituting the values:
\[ SI = \frac{P \times \frac{100}{11} \times 3}{100} = \frac{3P}{11} \]

4. **Calculate Compound Interest (CI)**:
- The formula for CI is:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
- Substituting the values:
\[ CI = P \left(1 + \frac{100}{11 \times 100}\right)^3 - P = P \left(1 + \frac{1}{11}\right)^3 - P \]
- Simplifying:
\[ CI = P \left(\frac{12}{11}\right)^3 - P \]
- Expanding \( \left(\frac{12}{11}\right)^3 \):
\[ \left(\frac{12}{11}\right)^3 = \frac{1728}{1331} \]
- Therefore:
\[ CI = P \left(\frac{1728}{1331} - 1\right) = P \left(\frac{1728 - 1331}{1331}\right) = P \left(\frac{397}{1331}\right) \]

5. **Set Up the Equation**:
- From the information given:
\[ CI - SI = 170 \]
- Substitute the expressions for CI and SI:
\[ P \left(\frac{397}{1331}\right) - \frac{3P}{11} = 170 \]

6. **Find a Common Denominator**:
- The common denominator for \( 1331 \) and \( 11 \) is \( 1331 \):
\[ \frac{3P}{11} = \frac{3P \times 121}{1331} = \frac{363P}{1331} \]
- Now, the equation becomes:
\[ P \left(\frac{397}{1331} - \frac{363}{1331}\right) = 170 \]
- Simplifying:
\[ P \left(\frac{34}{1331}\right) = 170 \]

7. **Solve for P**:
- Multiply both sides by \( 1331 \):
\[ 34P = 170 \times 1331 \]
- Calculate \( 170 \times 1331 = 226270 \):
\[ P = \frac{226270}{34} = 6655 \]

8. **Final Answer**:
- The principal amount \( P \) is \( 6655 \).