At 6% simple interest per annum a certain sum yields a total amount of Rs 3,228 at the end of `5 "" (3)/(4)` years . What was the sum invested ?

Options

A. Rs 2, 550
B. Rs 2,660
C. Rs 2,350
D. Rs 2,400

Answer

To find the sum invested (the principal), we can use the formula for the amount in simple interest:

\[ A = P \left(1 + \frac{RT}{100}\right) \]

Where:
- \( A \) is the total amount after time \( T \)
- \( P \) is the principal amount (the sum invested)
- \( R \) is the rate of interest per annum
- \( T \) is the time in years

### Step-by-step Solution:

1. **Identify the given values:**
- Total amount \( A = 3228 \)
- Rate of interest \( R = 6\% \)
- Time \( T = 5 \frac{3}{4} \) years

2. **Convert the time into an improper fraction:**
- \( T = 5 \frac{3}{4} = \frac{23}{4} \) years

3. **Substitute the values into the formula:**
\[ 3228 = P \left(1 + \frac{6 \times \frac{23}{4}}{100}\right) \]

4. **Calculate the interest factor:**
- First, calculate \( \frac{6 \times \frac{23}{4}}{100} \):
\[ \frac{6 \times 23}{4 \times 100} = \frac{138}{400} = \frac{69}{200} \]
- Now substitute back into the equation:
\[ 3228 = P \left(1 + \frac{69}{200}\right) \]
- Convert \( 1 \) into a fraction:
\[ 1 = \frac{200}{200} \]
- Thus:
\[ 3228 = P \left(\frac{200 + 69}{200}\right) = P \left(\frac{269}{200}\right) \]

5. **Rearranging to find \( P \):**
\[ P = \frac{3228 \times 200}{269} \]

6. **Calculate \( P \):**
- First, calculate \( 3228 \times 200 \):
\[ 3228 \times 200 = 645600 \]
- Now divide by \( 269 \):
\[ P = \frac{645600}{269} \approx 2400 \]

7. **Conclusion:**
- The sum invested (the principal) is \( Rs 2400 \).