If a certain sum becomes Rs. 1237.50 in four years with some annual rate of simple interest and it becomes Rs. 1443.75 in six years, then the principal amount is:

Question

Accepted Answer

To determine the principal amount, we need to use the information given about the simple interest accrued over different periods.

1. **Understanding the Problem:**
   - The sum becomes Rs. 1237.50 in four years.
   - The sum becomes Rs. 1443.75 in six years.
   - We need to find the principal amount.

2. **Using the Simple Interest Formula:**
   - Simple Interest (SI) = Principal (P) × Rate (R) × Time (T) / 100
   - From the information given, we can set up two equations based on the simple interest formula.

3. **Setting Up the Equations:**
   - Let the principal amount be P and the rate of interest be R.
   - After 4 years, the amount is Rs. 1237.50:
     $$
     P + \frac{P \times R \times 4}{100} = 1237.50
     $$
   - After 6 years, the amount is Rs. 1443.75:
     $$
     P + \frac{P \times R \times 6}{100} = 1443.75
     $$

4. **Finding the Simple Interest for 2 Years:**
   - The difference in the amounts after 6 years and 4 years is due to the interest for 2 years:
     $$
     1443.75 - 1237.50 = 206.25
     $$
   - This Rs. 206.25 is the interest for 2 years.

5. **Calculating the Rate of Interest:**
   - Using the simple interest formula for 2 years:
     $$
     \frac{P \times R \times 2}{100} = 206.25
     $$
   - Solving for $ P \times R $:
     $$
     P \times R = \frac{206.25 \times 100}{2} = 10312.5
     $$

6. **Substituting Back to Find Principal:**
   - Substitute $ P \times R = 10312.5 $ into one of the original equations:
     $$
     P + \frac{P \times R \times 4}{100} = 1237.50
     $$
     $$
     P + \frac{10312.5 \times 4}{100} = 1237.50
     $$
     $$
     P + 412.5 = 1237.50
     $$
     $$
     P = 1237.50 - 412.5 = 825
     $$

7. **Conclusion:**
   - The principal amount is Rs. 825, which corresponds to Option 3.

**Explanation for Incorrect Options:**
- **Option 1 (Rs. 860.50):** This amount does not satisfy the equations derived from the given conditions.
- **Option 2 (Rs. 870):** This amount also does not satisfy the equations derived from the given conditions.
- **Option 4 (Data is insufficient):** The data provided is sufficient to calculate the principal amount using the simple interest formula.

Therefore, the correct answer is Option 3: Rs. 825.