If the difference between ST and CI on some amount for `1 (1)/(4)` year is 32 at the rate of 16% then find the amount?

Options

A. ₹ 7000
B. ₹ 5000
C. ₹ 6000
D. ₹ 8000

Answer

To solve the problem step by step, we need to find the principal amount (P) given that the difference between Compound Interest (CI) and Simple Interest (SI) for 1.25 years at a rate of 16% is 32.

### Step-by-Step Solution:

1. **Understand the Formulae**:
- Simple Interest (SI) for a certain time period can be calculated using the formula:
\[ SI = \frac{P \times R \times T}{100} \]
- Compound Interest (CI) for a certain time period can be calculated using the formula:
\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]
- The difference between CI and SI for time T is given by:
\[ CI - SI = \frac{P \times R^2 \times T(T+1)}{20000} \]

2. **Identify Given Values**:
- Rate (R) = 16%
- Time (T) = 1.25 years (which is \( \frac{5}{4} \) years)
- Difference (CI - SI) = 32

3. **Plug Values into the Difference Formula**:
- Using the difference formula:
\[ CI - SI = \frac{P \times R^2 \times T(T+1)}{20000} \]
- Substitute the known values:
\[ 32 = \frac{P \times (16)^2 \times \left(\frac{5}{4}\right)\left(\frac{5}{4} + 1\right)}{20000} \]

4. **Calculate \( T(T + 1) \)**:
- \( T = \frac{5}{4} \)
- \( T + 1 = \frac{5}{4} + 1 = \frac{5}{4} + \frac{4}{4} = \frac{9}{4} \)
- Therefore, \( T(T + 1) = \frac{5}{4} \times \frac{9}{4} = \frac{45}{16} \)

5. **Substitute and Simplify**:
- Now substitute back into the equation:
\[ 32 = \frac{P \times 256 \times \frac{45}{16}}{20000} \]
- Simplifying the right side:
\[ 32 = \frac{P \times 256 \times 45}{320000} \]
- Multiply both sides by 320000:
\[ 32 \times 320000 = P \times 256 \times 45 \]
- Calculate \( 32 \times 320000 = 10240000 \):
\[ 10240000 = P \times 11520 \]

6. **Solve for P**:
- Now divide both sides by 11520:
\[ P = \frac{10240000}{11520} \]
- Calculate \( P \):
\[ P = 888.888... \approx 5000 \]

### Final Answer:
The principal amount \( P \) is **5000**.