A certain sum of money amounts to Rs 2500 in 5 years and to Rs 3000 in 7 years at the same rate of simple interest. What is the value of the original principal (the sum initially invested) before interest is added?

Introduction / Context:
This problem uses two different future amounts of the same principal at the same simple interest rate but different time periods. From these two conditions, you can determine the annual simple interest and the original principal. Such questions reinforce the idea that under simple interest, the interest increases linearly with time, so the difference in amounts over different years reveals the yearly interest.

Given Data / Assumptions:

Principal P is unknown.
Amount after 5 years, A1 = Rs 2500.
Amount after 7 years, A2 = Rs 3000.
Same simple interest rate r% per annum for both periods.
Amount formula in simple interest: A = P + SI, and SI = (P * r * T) / 100.

Concept / Approach:
The difference between the two amounts over 2 extra years (from year 5 to year 7) equals the simple interest earned in those 2 years. From this, we can find the interest per year. Once we know yearly interest, we subtract 5 years of interest from the 5-year amount to recover the principal P. This approach avoids the need to explicitly find the rate r before finding P.

Step-by-Step Solution:
Step 1: Write amounts in terms of principal and interest. Step 2: A1 = P + SI_5, where SI_5 is interest for 5 years. Step 3: A2 = P + SI_7, where SI_7 is interest for 7 years. Step 4: Difference in amounts A2 - A1 = SI_7 - SI_5 = interest for additional 2 years. Step 5: Compute A2 - A1 = 3000 - 2500 = 500. Step 6: So, interest in 2 years is Rs 500, giving yearly interest = 500 / 2 = Rs 250. Step 7: Interest in 5 years is 5 * 250 = Rs 1250. Step 8: Since A1 = P + 5 years interest, we have 2500 = P + 1250. Step 9: Therefore, P = 2500 - 1250 = 1250.

Verification / Alternative check:
Check with the 7-year amount. If principal P is 1250 and yearly interest is 250, then interest for 7 years is 7 * 250 = 1750. Amount after 7 years = 1250 + 1750 = 3000, which matches A2. This confirms that the principal is correct.

Why Other Options Are Wrong:
If P were 1800, 2000, 1400, or 1500, the implied yearly interest would not be consistent with both given amounts at 5 and 7 years. These values either overshoot or undershoot one of the amounts, so they cannot satisfy both conditions simultaneously.

Common Pitfalls:
A common mistake is to try to form two separate equations with P and r and solve them simultaneously, which is possible but more tedious. Another frequent error is misinterpreting the difference in amounts as the total interest rather than only the additional interest for the extra years. Remember that in simple interest, interest per year is constant, so difference in amounts over extra years equals interest for those extra years only.

Final Answer:
The original principal invested is Rs 1,250.