A sum of money at simple interest amounts to Rs. 815 in 3 years and Rs. 854 in 4 years. What is the original principal sum?

Introduction / Context:
This question uses the basic properties of simple interest to find the original principal when we know the amount at two different times. The key observation is that under simple interest, the interest added each year is constant. Therefore, the difference between the amounts at two times directly gives the interest for the additional period, which we can use to find the yearly interest and then the principal.

Given Data / Assumptions:

Amount after 3 years = Rs. 815.
Amount after 4 years = Rs. 854.
Interest is simple, not compound.
We need to find the principal (original sum) P.

Concept / Approach:
Under simple interest, interest per year is constant, so the difference in amounts between successive years equals the interest for one year. Once we know yearly interest I, we can write the amount after 3 years as A3 = P + 3I and solve for P. This approach avoids directly using the rate of interest, although it can be found if needed. The method is fast and standard for such questions.

Step-by-Step Solution:
Let the yearly simple interest be I rupees. Amount after 3 years A3 = P + 3I = 815. Amount after 4 years A4 = P + 4I = 854. Subtract the two equations: A4 - A3 = (P + 4I) - (P + 3I) = I. So I = 854 - 815 = 39 rupees per year. Now substitute I = 39 into P + 3I = 815. Then P + 3 * 39 = 815. Compute 3 * 39 = 117. So P + 117 = 815. Therefore P = 815 - 117 = 698.

Verification / Alternative check:
Check amounts using principal P = 698 and yearly interest I = 39. After 3 years, amount A3 = 698 + 3 * 39 = 698 + 117 = 815, which matches the given value. After 4 years, amount A4 = 698 + 4 * 39 = 698 + 156 = 854, also matching the problem statement. This confirms that the principal 698 rupees is correct.

Why Other Options Are Wrong:
Option 650 would require a different yearly interest to reach 815 and 854, which would not stay constant per year as required by simple interest.
Option 690 or 700 similarly would not produce the exact given amounts in 3 and 4 years when combined with a single fixed yearly interest value.
Only 698 satisfies both conditions A3 = 815 and A4 = 854 with constant annual interest.

Common Pitfalls:
Some learners try to find the interest rate directly using the formula, which is not necessary and can introduce fractions early in the solution.
Others mistakenly average the two amounts instead of taking their difference to find yearly interest.
A common error is using compound interest logic, where the increase from 3 to 4 years would not be constant, which is not appropriate here.

Final Answer:
The original principal sum is Rs. 698.