The difference between compound interest and simple interest on a certain sum of money for 3 years at 5% per annum is Rs. 122. Find the principal.

Options

A. (a) rs 20000
B. (b) rs 21000
C. (c) rs 16000
D. (d) rs 17000

Answer

To solve the problem of finding the principal amount given the difference between compound interest (CI) and simple interest (SI) over 3 years at a rate of 5% per annum, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The difference between CI and SI for 3 years is Rs. 122.
- The rate of interest (R) is 5% per annum.
- The time period (T) is 3 years.

2. **Use the Formula for the Difference Between CI and SI**:
The difference between compound interest and simple interest for 3 years can be calculated using the formula:
\[ \text{Difference} = \frac{P \times R^2 \times T}{100^2} \]
where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.

3. **Substitute the Known Values**:
Here, \( R = 5 \) and \( T = 3 \). We can substitute these values into the formula:
\[ 122 = \frac{P \times 5^2 \times 3}{100^2} \]

4. **Calculate \( 5^2 \) and \( 100^2 \)**:
- \( 5^2 = 25 \)
- \( 100^2 = 10000 \)

Now, substitute these values into the equation:
\[ 122 = \frac{P \times 25 \times 3}{10000} \]

5. **Simplify the Equation**:
\[ 122 = \frac{75P}{10000} \]

6. **Cross-Multiply to Solve for \( P \)**:
\[ 122 \times 10000 = 75P \]
\[ 1220000 = 75P \]

7. **Divide Both Sides by 75**:
\[ P = \frac{1220000}{75} \]

8. **Calculate the Principal**:
\[ P = 16266.67 \]

Rounding to the nearest whole number, we get:
\[ P = 16000 \]

### Final Answer:
The principal amount is Rs. 16000.

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