A sum of money is invested at simple interest and amounts to Rs. 415 in 2 years and to Rs. 514 in 4 years. Using the change in amount over time, what is the value of the original principal sum deposited?

Introduction / Context:
This simple interest problem uses the fact that the amount increases linearly with time when the interest is simple. We are given the total amounts at the end of 2 years and 4 years and are asked to determine the original principal. The difference between these two amounts directly reveals the interest earned in an additional 2-year period, from which we can find the annual interest and then the principal.

Given Data / Assumptions:

• Amount after 2 years, A2 = Rs. 415.
• Amount after 4 years, A4 = Rs. 514.
• The same principal P and rate R are used throughout.
• Simple interest is applied, so interest per year is constant.
• Relation: Amount A = P + SI, where SI = (P * R * T) / 100.

Concept / Approach:
Under simple interest, the extra amount between year 2 and year 4 is just the interest earned for 2 more years. That difference allows us to calculate the yearly interest, and then we subtract the total 2-year interest from the 2-year amount to recover the principal P.

Step-by-Step Solution:
Difference between amounts: A4 - A2 = 514 - 415 = Rs. 99. This extra Rs. 99 is the simple interest for 2 additional years. So yearly interest = 99 / 2 = Rs. 49.50 per year. Interest for 2 years = 2 * 49.50 = Rs. 99. Amount after 2 years A2 = P + interest for 2 years. So P = A2 - 2-year interest = 415 - 99. P = Rs. 316. Therefore, the original principal sum is Rs. 316.
Verification / Alternative check:
If P = 316 and interest per year = 49.50, then after 2 years, amount = 316 + 2 * 49.50 = 316 + 99 = 415, matching A2. After 4 years, amount = 316 + 4 * 49.50 = 316 + 198 = 514, matching A4. This confirms that our principal calculation is correct.

Why Other Options Are Wrong:
Rs. 251, Rs. 294, Rs. 350, and Rs. 154 do not produce consistent amounts of Rs. 415 and Rs. 514 when a constant yearly interest is applied. Substituting these values for P leads to either different amounts or non-integer yearly interest values that do not match the given data for both 2 years and 4 years simultaneously.

Common Pitfalls:
Learners sometimes try to find the rate first, which introduces extra steps and possible errors. Another frequent mistake is to divide the difference 99 by 4, thinking it spans 4 years, instead of recognizing it is the interest for only the extra 2 years. Forgetting that simple interest grows linearly with time also leads to confusion.

Final Answer:
The original principal sum is Rs. 316.