A sum of Rs. 400 amounts to Rs. 448 at simple interest in 2 years. At the same simple interest rate, in how many years will a sum of Rs. 550 amount to Rs. 682 in this aptitude question on simple interest and time?

Introduction / Context:
This question examines your ability to work with simple interest by first determining the rate of interest from one scenario and then using that rate to find the time required in a second scenario. The principal and amount change between the two scenarios, but the rate of interest remains the same. This type of problem is common in banking and aptitude exams where quick manipulation of the simple interest formula is required.

Given Data / Assumptions:

• First situation: Principal P₁ = Rs. 400, Amount A₁ = Rs. 448, Time t₁ = 2 years.
• Second situation: Principal P₂ = Rs. 550, Amount A₂ = Rs. 682, Time t₂ = ? years.
• The rate of simple interest r is the same in both situations.
• Interest is simple interest only, with no compounding.

Concept / Approach:
Simple interest (SI) is given by:
SI = (Principal * Rate * Time) / 100.
Amount (A) is related to principal (P) and simple interest (SI) as A = P + SI. We first use the information from the first situation to calculate the rate r. Once we know r, we use it in the second situation formula where principal and final amount are known, and solve for time t₂. This two-stage approach is standard for chained simple interest questions like this.

Step-by-Step Solution:
Step 1: In the first situation, simple interest SI₁ = A₁ − P₁ = 448 − 400 = Rs. 48. Step 2: Use SI₁ = (P₁ * r * t₁) / 100 to find r: 48 = (400 * r * 2) / 100. Step 3: Simplify: 48 = (800r) / 100 = 8r, so r = 48 / 8 = 6 percent per annum. Step 4: Now, in the second situation, simple interest SI₂ = A₂ − P₂ = 682 − 550 = Rs. 132. Step 5: Use SI₂ = (P₂ * r * t₂) / 100 with r = 6: 132 = (550 * 6 * t₂) / 100. Step 6: Simplify the right-hand side: (550 * 6 / 100) * t₂ = (33) * t₂. Step 7: Therefore, 132 = 33 * t₂, so t₂ = 132 / 33 = 4 years. Step 8: Hence, it will take 4 years for Rs. 550 to amount to Rs. 682 at the same simple interest rate.

Verification / Alternative check:
To verify, we recompute the amount for P₂ = Rs. 550, r = 6 percent, t₂ = 4 years. Simple interest = (550 * 6 * 4) / 100 = (550 * 24) / 100 = 132. The amount becomes 550 + 132 = 682, which matches the required amount. This confirms that our calculated time of 4 years is correct and consistent with the given data.

Why Other Options Are Wrong:
If t₂ were 2 or 3 years, the interest would be too small to reach Rs. 682. For example, at 3 years, interest would be (550 * 6 * 3) / 100 = 99, giving an amount of Rs. 649. At 3.5 years, the interest would be 115.5, still short of 132. For 5 years, the interest would be 165, giving an amount of 715, which is too high. Only 4 years produces exactly the required amount of Rs. 682.

Common Pitfalls:
Students sometimes miscalculate the rate from the first scenario by confusing amount and interest, or by forgetting to subtract principal from amount. Another common mistake is to plug values into the formula without simplifying step by step, which can lead to arithmetic errors. Some also mistakenly assume that the same time is used in both scenarios, instead of properly solving for the new time. Being methodical about computing interest and then using the same rate avoids these issues.

Final Answer:
At the same simple interest rate, Rs. 550 will amount to Rs. 682 in 4 years.