A sum of money is invested at a simple interest rate of 7.5% per annum for 4 years, and if the same sum were invested for 5 years the interest would increase by Rs 375; what was the initial principal amount invested in this simple interest situation?

Introduction / Context:
This question uses the idea that in simple interest, the interest amount is directly proportional to time. The same principal and rate are applied, but the time is increased from 4 years to 5 years. The extra year of investment adds a known additional interest of Rs 375. By using this extra year of interest, we can determine the principal. This is a common style of shortcut problem in simple interest chapters.

Given Data / Assumptions:
- Simple interest rate R = 7.5 percent per annum. - Original time T1 = 4 years. - New time T2 = 5 years. - Increase in interest when time increases by one year = Rs 375. - Principal P is unknown and constant across both cases.

Concept / Approach:
For simple interest, SI = (P * R * T) / 100. Since principal and rate are unchanged, the interest is directly proportional to time. The difference in interest between 5 years and 4 years is simply the interest for 1 extra year. Therefore extra interest = (P * R * 1) / 100. Setting this equal to Rs 375 allows us to solve for P directly without computing full interest for multiple years.

Step-by-Step Solution:
Step 1: Extra time between the two cases is 1 year (5 − 4). Step 2: Extra interest for that 1 year is given as Rs 375. Step 3: For 1 year, simple interest SI1 = (P * R * 1) / 100. Step 4: Substitute R = 7.5 percent and SI1 = 375. Step 5: Write the equation: 375 = (P * 7.5 * 1) / 100. Step 6: Rearrange for P: P = (375 * 100) / 7.5. Step 7: Compute numerator: 375 * 100 = 37,500. Step 8: Divide: 37,500 / 7.5 = 5,000. Step 9: Therefore, the principal invested is Rs 5,000.

Verification / Alternative check:
Using P = Rs 5,000, compute interest for 4 and 5 years. For 4 years, SI4 = (5,000 * 7.5 * 4) / 100. First 5,000 * 7.5 = 37,500, then multiply by 4 to get 1,50,000, and divide by 100 to get Rs 1,500. For 5 years, SI5 = (5,000 * 7.5 * 5) / 100 = (37,500 * 5) / 100 = 1,87,500 / 100 = Rs 1,875. The difference SI5 − SI4 = 1,875 − 1,500 = Rs 375, which matches the given increase in interest.

Why Other Options Are Wrong:
If P were Rs 4,750, interest for one extra year would be (4,750 * 7.5 * 1) / 100 = Rs 356.25, not Rs 375. For P = Rs 4,500, extra interest would be Rs 337.50. For P = Rs 3,750, extra interest would be Rs 281.25. None of these equal the stated increase of Rs 375. Only Rs 5,000 yields the correct additional interest for the extra year.

Common Pitfalls:
A common mistake is to compute full interest for both 4 and 5 years separately without noticing that a one year shortcut exists. Another error is to treat 7.5 percent as 0.75 instead of 7.5 divided by 100. Remember that for simple interest, the extra interest due to an extra year is directly proportional to the principal and rate only.

Final Answer:
The initial principal amount invested was Rs 5,000.