If the difference between ST and CI on some amount for `1 (1)/(2)` years is 40 at the rate of `12(1)/(2)%` then find the amount?

Options

A. ₹ 5220
B. ₹ 5020
C. ₹ 5210
D. ₹ 5120

Answer

To solve the problem of finding the amount when the difference between Simple Interest (SI) and Compound Interest (CI) for 1.5 years at a rate of 12.5% is 40, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formulae**:
- Simple Interest (SI) for 1 year is given by:
\[ SI = \frac{P \times R \times T}{100} \]
- Compound Interest (CI) for 1.5 years can be calculated using the formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

2. **Calculate SI for 1 Year and 0.5 Year**:
- For the first year (T = 1):
\[ SI_1 = \frac{P \times 12.5 \times 1}{100} = \frac{12.5P}{100} = \frac{P}{8} \]
- For the next half year (T = 0.5):
\[ SI_2 = \frac{P \times 12.5 \times 0.5}{100} = \frac{12.5P}{200} = \frac{P}{16} \]

3. **Total Simple Interest for 1.5 Years**:
- Total SI for 1.5 years:
\[ SI_{total} = SI_1 + SI_2 = \frac{P}{8} + \frac{P}{16} \]
- To add these fractions, find a common denominator (16):
\[ SI_{total} = \frac{2P}{16} + \frac{P}{16} = \frac{3P}{16} \]

4. **Calculate CI for 1.5 Years**:
- Using the formula for CI:
\[ CI = P \left(1 + \frac{12.5}{100}\right)^{1.5} - P \]
- First, calculate \(1 + \frac{12.5}{100} = 1.125\).
- Now calculate \(1.125^{1.5}\):
\[ 1.125^{1.5} \approx 1.125 \times 1.125^{0.5} \approx 1.125 \times 1.0607 \approx 1.194 \]
- Therefore:
\[ CI \approx P \times 1.194 - P = 0.194P \]

5. **Difference Between CI and SI**:
- Given that the difference between CI and SI is 40:
\[ CI - SI = 0.194P - \frac{3P}{16} = 40 \]
- Convert \(\frac{3P}{16}\) to a decimal:
\[ \frac{3P}{16} = 0.1875P \]
- So:
\[ 0.194P - 0.1875P = 40 \implies 0.0065P = 40 \]

6. **Solve for P**:
- Rearranging gives:
\[ P = \frac{40}{0.0065} \approx 6153.85 \]

### Final Amount:
The amount \(P\) is approximately **6153.85**.