A sum of Rs. 8000 is divided into two parts. The simple interest on the first part at the rate of 21% per annum for one year is equal to the simple interest on the second part at the rate of 35% per annum for one year. What is the simple interest, in rupees, earned on each part in that year?

Introduction / Context:
This problem illustrates how to divide a total sum into two parts such that the simple interest earned on each part, despite different rates, is the same. Such questions are common in aptitude exams because they involve both proportional reasoning and simple interest formulas. The question states that the interest from the first part at 21 percent per annum equals the interest from the second part at 35 percent per annum for one year, and asks for the interest value on each part.

Given Data / Assumptions:
Total sum is Rs. 8000.
First part is invested at 21 percent simple interest per annum for one year.
Second part is invested at 35 percent simple interest per annum for one year.
Simple interest on both parts for one year is equal.
Interest rates and principal are fixed over the year.

Concept / Approach:
For simple interest for one year, I = P * r / 100. If the interest on both parts is the same, then P1 * 21 / 100 = P2 * 35 / 100, where P1 and P2 are the two parts. The common division factor 1 / 100 cancels, leading to a ratio between P1 and P2. Once the ratio is known and the total of P1 and P2 is 8000, we can find each part. Since the question asks for the interest on each part, we finally compute simple interest for one year on either P1 or P2 using the corresponding rate.

Step-by-Step Solution:
Let the first part be P1 rupees and the second part be P2 rupees.Given that P1 + P2 = 8000.Equality of simple interest for one year gives P1 * 21 / 100 = P2 * 35 / 100.Cancel 1 / 100 to get 21P1 = 35P2.This simplifies to P1 / P2 = 35 / 21 = 5 / 3.Therefore, let P1 = 5k and P2 = 3k for some k.Then P1 + P2 = 5k + 3k = 8k = 8000, so k = 1000.Hence P1 = 5000 rupees and P2 = 3000 rupees.Interest on the first part for one year is I1 = 5000 * 21 / 100 = 1050 rupees.Interest on the second part for one year is I2 = 3000 * 35 / 100 = 1050 rupees.Thus the simple interest on each part is Rs. 1050.

Verification / Alternative check:
We can verify the equality directly. For P1 = 5000 at 21 percent, simple interest for one year is 1050 rupees. For P2 = 3000 at 35 percent, interest for one year is also 1050 rupees. Both interest amounts match, satisfying the condition that the simple interest on each part is equal. The total sum remains 5000 + 3000 = 8000 rupees, consistent with the problem statement.

Why Other Options Are Wrong:
Rs. 840 would correspond to different principal values and rates that do not sum to Rs. 8000 with equal interests at 21 percent and 35 percent.
Rs. 1400 and Rs. 1220 are larger interest amounts that would require higher principal values or different rates than those specified.
Rs. 700 is too small to arise as the simple interest on both parts under the given conditions. None of these options satisfy the equation P1 * 21 / 100 = P2 * 35 / 100 with P1 + P2 equal to 8000.

Common Pitfalls:
Some learners mistakenly assume that each part is half of 8000, which is not correct because the rates differ. Others equate the principals instead of equating the interests. A further source of error is mishandling the ratio 21 to 35 while simplifying. Working slowly through the ratio and total sum helps avoid these errors.

Final Answer:
The simple interest earned on each part of the divided sum in one year is Rs. 1050.