A sum of money invested for 4 years and 9 months at a rate of `8%` simple interest per annum become Rs. 1,035 at the end of the period. What was the sum that was initially invested ?

Options

A. Rs. 700
B. Rs. 675
C. Rs. 725
D. Rs. 750

Answer

To find the initial sum of money (the principal) that was invested, we can use the formula for simple interest. The formula for the total amount (A) after a certain period with simple interest is:

\[ A = P + SI \]

Where:
- \( A \) = Total amount after time (which is Rs. 1,035 in this case)
- \( P \) = Principal amount (the initial sum we want to find)
- \( SI \) = Simple Interest

The simple interest (SI) can be calculated using the formula:

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

Where:
- \( R \) = Rate of interest (which is 8% per annum)
- \( T \) = Time in years

### Step 1: Convert the time period into years
The time given is 4 years and 9 months. We need to convert 9 months into years:

\[ 9 \text{ months} = \frac{9}{12} \text{ years} = 0.75 \text{ years} \]

Thus, the total time \( T \) in years is:

\[ T = 4 + 0.75 = 4.75 \text{ years} \]

### Step 2: Substitute the values into the simple interest formula
Now we can substitute the values into the simple interest formula:

\[ SI = \frac{P \times 8 \times 4.75}{100} \]

### Step 3: Substitute SI back into the total amount formula
We know the total amount \( A \) is Rs. 1,035, so we can write:

\[ 1,035 = P + \frac{P \times 8 \times 4.75}{100} \]

### Step 4: Simplify the equation
Let's simplify the equation:

\[ 1,035 = P + \frac{38P}{100} \]

This can be rewritten as:

\[ 1,035 = P + 0.38P \]
\[ 1,035 = 1.38P \]

### Step 5: Solve for P
Now, we can solve for \( P \):

\[ P = \frac{1,035}{1.38} \]

Calculating this gives:

\[ P \approx 750 \]

### Conclusion
The initial sum of money that was invested is approximately Rs. 750.

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