The simple interest on a certain sum of money for (5)/(4) yr at 12% per annum is X 20 less than the simple interest on the same sum for (7)/(2) yr at 10% per annum. Find the sum

Options

A. X800
B. X750
C. X625
D. X400

Answer

To solve the problem step by step, we will use the formula for simple interest and set up the equations based on the information given in the question.

### Step 1: Understand the Problem
We need to find the principal amount (sum of money) based on the simple interest for two different time periods and interest rates.

### Step 2: Define Variables
Let:
- \( P \) = Principal amount (the sum we need to find)
- \( R_1 = 12\% \) (rate for the first case)
- \( T_1 = \frac{5}{4} \) years (time for the first case)
- \( R_2 = 10\% \) (rate for the second case)
- \( T_2 = \frac{7}{2} \) years (time for the second case)

### Step 3: Write the Simple Interest Formula
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]

### Step 4: Calculate Simple Interest for Both Cases
1. **For the first case**:
\[ SI_1 = \frac{P \times 12 \times \frac{5}{4}}{100} = \frac{15P}{100} = \frac{3P}{20} \]

2. **For the second case**:
\[ SI_2 = \frac{P \times 10 \times \frac{7}{2}}{100} = \frac{70P}{200} = \frac{7P}{20} \]

### Step 5: Set Up the Equation Based on the Given Condition
According to the problem, the simple interest of the first case is 20 less than that of the second case:
\[ SI_2 - SI_1 = 20 \]
Substituting the values we calculated:
\[ \frac{7P}{20} - \frac{3P}{20} = 20 \]

### Step 6: Simplify the Equation
Combine the terms on the left side:
\[ \frac{7P - 3P}{20} = 20 \]
This simplifies to:
\[ \frac{4P}{20} = 20 \]

### Step 7: Solve for \( P \)
Multiply both sides by 20:
\[ 4P = 400 \]
Now, divide both sides by 4:
\[ P = 100 \]

### Step 8: Conclusion
The principal amount (sum of money) is \( P = 100 \).