A certain sum amounts to Rs. 9243.20 in `1^(1/2)` years at `4%` p.a. simple interest. What will be the simple interest on the same sum for `7^(1/2)` years at `8%` per annum?

Options

A. Rs. 5580
B. Rs. 5223
C. Rs. 5232
D. Rs. 5508

Answer

To solve the problem step-by-step, we will first determine the principal amount using the information provided about the first case, and then calculate the simple interest for the second case.

### Step 1: Understand the first case
We know that the amount (A) after 1.5 years at 4% per annum is Rs. 9243.20. The formula for the amount in simple interest is:
\[ A = P + I \]
Where \( I \) (Interest) can be calculated using the formula:
\[ I = \frac{P \times R \times T}{100} \]
Here, \( R \) is the rate of interest and \( T \) is the time in years.

### Step 2: Set up the equation
From the problem, we have:
- \( A = 9243.20 \)
- \( R = 4\% \)
- \( T = 1.5 \) years (which can be written as \( \frac{3}{2} \))

Substituting these values into the equation, we get:
\[ 9243.20 = P + \frac{P \times 4 \times \frac{3}{2}}{100} \]

### Step 3: Simplify the equation
Let's simplify the interest part:
\[ I = \frac{P \times 4 \times \frac{3}{2}}{100} = \frac{12P}{200} = \frac{3P}{50} \]

So, we can rewrite the equation as:
\[ 9243.20 = P + \frac{3P}{50} \]

### Step 4: Combine like terms
To combine the terms involving \( P \):
\[ 9243.20 = P \left(1 + \frac{3}{50}\right) \]
\[ 9243.20 = P \left(\frac{50 + 3}{50}\right) \]
\[ 9243.20 = P \left(\frac{53}{50}\right) \]

### Step 5: Solve for \( P \)
Now, we can solve for \( P \):
\[ P = 9243.20 \times \frac{50}{53} \]
Calculating this gives:
\[ P = 8720 \]

### Step 6: Calculate the simple interest for the second case
Now that we have the principal amount \( P = 8720 \), we will calculate the simple interest for 7.5 years at 8% per annum.

### Step 7: Set up the new interest calculation
For the second case:
- \( R = 8\% \)
- \( T = 7.5 \) years (which can be written as \( \frac{15}{2} \))

Using the simple interest formula:
\[ I = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ I = \frac{8720 \times 8 \times \frac{15}{2}}{100} \]

### Step 8: Simplify the interest calculation
Calculating the interest:
\[ I = \frac{8720 \times 8 \times 15}{200} \]
\[ I = \frac{8720 \times 120}{200} \]
\[ I = \frac{1046400}{200} \]
\[ I = 5232 \]

### Final Answer
The simple interest on the same sum for 7.5 years at 8% per annum is Rs. 5232.