The difference between simple interest and compound interest on a sum for 3 years at `5%` , when the interest is compound annually is `Rs. 122`. Find the principal value.

Options

A. Rs. 10000
B. Rs.16000
C. Rs.18000
D. Rs.24000

Answer

To find the principal value given the difference between simple interest (SI) and compound interest (CI) over 3 years at a rate of 5% is Rs. 122, we can follow these steps:

### Step 1: Understand the formulas
- **Simple Interest (SI)** is calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal, \( R \) is the rate of interest, and \( T \) is the time in years.

- **Compound Interest (CI)** is calculated using the formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
This can be simplified to:
\[ CI = P \left(\left(1 + \frac{R}{100}\right)^T - 1\right) \]

### Step 2: Calculate Simple Interest for 3 years
Given:
- \( R = 5\% \)
- \( T = 3 \) years

Using the SI formula:
\[ SI = \frac{P \times 5 \times 3}{100} = \frac{15P}{100} = \frac{3P}{20} \]

### Step 3: Calculate Compound Interest for 3 years
Using the CI formula:
\[ CI = P \left(\left(1 + \frac{5}{100}\right)^3 - 1\right) \]
Calculating \( \left(1 + \frac{5}{100}\right)^3 \):
\[ \left(1 + 0.05\right)^3 = (1.05)^3 = 1.157625 \]
Thus,
\[ CI = P \left(1.157625 - 1\right) = P \times 0.157625 \]

### Step 4: Express CI in terms of P
\[ CI = 0.157625P \]

### Step 5: Find the difference between CI and SI
Given that the difference between CI and SI is Rs. 122:
\[ CI - SI = 122 \]
Substituting the values:
\[ 0.157625P - \frac{3P}{20} = 122 \]

### Step 6: Convert SI to a decimal
Convert \(\frac{3P}{20}\) to a decimal:
\[ \frac{3P}{20} = 0.15P \]

### Step 7: Set up the equation
Now we have:
\[ 0.157625P - 0.15P = 122 \]
This simplifies to:
\[ 0.007625P = 122 \]

### Step 8: Solve for P
To find \( P \):
\[ P = \frac{122}{0.007625} \]
Calculating this gives:
\[ P \approx 16000 \]

### Conclusion
The principal value is Rs. 16000.