If the simpld interset on a certain sum of money for 15 months at `7(1)/(2) %` per annum exceeds the simple interset on the same sum for 8 months at `12(1)/(2)` % per annum by Rs 32.50, then the sum of money (in Rs) is :

Options

A. 312
B. 312.5
C. 3120
D. 3120.5

Answer

To solve the problem, we will use the formula for Simple Interest (SI):

\[ \text{SI} = \frac{P \times R \times T}{100} \]

where:
- \( P \) = Principal amount (the sum of money)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 1: Calculate the Simple Interest for the first scenario
The first scenario involves a time period of 15 months at a rate of \( 7.5\% \) per annum.

Convert 15 months into years:
\[ T_1 = \frac{15}{12} = 1.25 \text{ years} \]

Now, using the formula for SI:
\[ \text{SI}_1 = \frac{P \times 7.5 \times 1.25}{100} \]

### Step 2: Calculate the Simple Interest for the second scenario
The second scenario involves a time period of 8 months at a rate of \( 12.5\% \) per annum.

Convert 8 months into years:
\[ T_2 = \frac{8}{12} = \frac{2}{3} \text{ years} \]

Now, using the formula for SI:
\[ \text{SI}_2 = \frac{P \times 12.5 \times \frac{2}{3}}{100} \]

### Step 3: Set up the equation based on the given information
According to the problem, the difference between the two simple interests is Rs 32.50:
\[ \text{SI}_1 - \text{SI}_2 = 32.50 \]

Substituting the expressions for \(\text{SI}_1\) and \(\text{SI}_2\):
\[ \frac{P \times 7.5 \times 1.25}{100} - \frac{P \times 12.5 \times \frac{2}{3}}{100} = 32.50 \]

### Step 4: Simplify the equation
Multiply through by 100 to eliminate the denominator:
\[ P \times 7.5 \times 1.25 - P \times 12.5 \times \frac{2}{3} = 3250 \]

### Step 5: Calculate the coefficients
Calculating \(7.5 \times 1.25\):
\[ 7.5 \times 1.25 = 9.375 \]

Calculating \(12.5 \times \frac{2}{3}\):
\[ 12.5 \times \frac{2}{3} = \frac{25}{3} \approx 8.3333 \]

Now substitute these values back into the equation:
\[ P \times 9.375 - P \times 8.3333 = 3250 \]

### Step 6: Factor out \(P\)
\[ P(9.375 - 8.3333) = 3250 \]

Calculating the difference:
\[ 9.375 - 8.3333 = 1.0417 \]

### Step 7: Solve for \(P\)
\[ P \times 1.0417 = 3250 \]
\[ P = \frac{3250}{1.0417} \approx 3125 \]

Thus, the sum of money is approximately Rs 3125.

### Final Answer:
The sum of money is Rs 3125.

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