The SI on certain sum of money for 15 months t the rate of `7.5%` per annum exceeds the SI on the same sum at `12.5%` per annum for 8 months by Rs. 3250. Then find the sum

Options

A. Rs. 316000
B. Rs. 320000
C. Rs. 312000
D. Rs. 324000

Answer

To find the sum of money based on the given conditions, we can follow these steps:

### Step 1: Understand the Simple Interest Formula
The formula for Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal (the sum of money)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Convert Time into Years
The time periods given in the problem are in months. We need to convert them into years:
- 15 months = \( \frac{15}{12} = 1.25 \) years
- 8 months = \( \frac{8}{12} = \frac{2}{3} \) years

### Step 3: Write the SI for Both Cases
1. For the first case (15 months at 7.5%):
\[ \text{SI}_1 = \frac{P \times 7.5 \times 1.25}{100} \]
\[ \text{SI}_1 = \frac{P \times 9.375}{100} \]

2. For the second case (8 months at 12.5%):
\[ \text{SI}_2 = \frac{P \times 12.5 \times \frac{2}{3}}{100} \]
\[ \text{SI}_2 = \frac{P \times 8.3333}{100} \]

### Step 4: Set Up the Equation
According to the problem, the SI for the first case exceeds the SI for the second case by Rs. 3250:
\[ \text{SI}_1 - \text{SI}_2 = 3250 \]
Substituting the expressions for SI:
\[ \frac{P \times 9.375}{100} - \frac{P \times 8.3333}{100} = 3250 \]

### Step 5: Simplify the Equation
Combine the terms:
\[ \frac{P \times (9.375 - 8.3333)}{100} = 3250 \]
Calculating the difference:
\[ 9.375 - 8.3333 = 1.0417 \]
So the equation becomes:
\[ \frac{P \times 1.0417}{100} = 3250 \]

### Step 6: Solve for Principal (P)
Multiply both sides by 100:
\[ P \times 1.0417 = 325000 \]
Now, divide by 1.0417:
\[ P = \frac{325000}{1.0417} \]
Calculating this gives:
\[ P \approx 312000 \]

### Conclusion
The sum of money (Principal) is Rs. 3,12,000.