An equal sum of money is invested in two different schemes for 2 years. One scheme offers simple interest and the other offers compound interest with annual compounding. The rate of interest for both schemes is 23% per annum. If the total interest received from both schemes together after 2 years is Rs. 2961, what is the first year interest (in rupees) from the simple interest scheme?

Introduction / Context: This question involves comparing simple interest and compound interest on equal principals at the same rate for the same time period. The total interest from both schemes together after 2 years is known, and we must find the first year interest from the simple interest scheme. This requires understanding how simple interest accumulates linearly while compound interest grows by adding interest on interest. Given Data / Assumptions:

• Principal P is invested in the simple interest scheme for 2 years at 23% per annum.
• The same principal P is invested in the compound interest scheme for 2 years at 23% per annum with annual compounding.
• Total interest from both schemes together after 2 years is Rs. 2961.
• Simple interest formula: SI = (P * R * T) / 100.
• Compound interest amount for 2 years: A = P * (1 + R / 100)^2, CI = A - P.

Concept / Approach: We write expressions for the 2-year simple interest and the 2-year compound interest on the same principal P. Adding these expressions and equating the result to Rs. 2961 allows us to solve for P. Once P is known, the first year interest from the simple interest scheme is just P * 23 / 100, since simple interest per year is constant. Step-by-Step Solution: Simple interest for 2 years: SI_2y = P * 23 * 2 / 100 = 0.46P. For compound interest, amount after 2 years: A = P * (1 + 23 / 100)^2 = P * (1.23)^2. Compute (1.23)^2 = 1.5129, so CI_2y = A - P = P * (1.5129 - 1) = 0.5129P. Total interest from both schemes = SI_2y + CI_2y = 0.46P + 0.5129P = 0.9729P. We are given 0.9729P = 2961, so P = 2961 / 0.9729. Using exact fractional reasoning, P = 70000 / 23 ≈ 3043.48. First year interest from simple interest scheme = P * 23 / 100 = (70000 / 23) * 23 / 100 = 70000 / 100 = Rs. 700. Verification / Alternative check: Check total interest with P = 70000 / 23. Simple interest for 2 years = 0.46P = 0.46 * 70000 / 23 = 32200 / 23. Compound interest for 2 years = 0.5129P = 0.5129 * 70000 / 23 = 35980 / 23. Sum = (32200 + 35980) / 23 = 68180 / 23 = 2961. This matches the given total interest, confirming that the first year simple interest of Rs. 700 is correct. Why Other Options Are Wrong: Rs. 500, Rs. 800, Rs. 900, and Rs. 1100 correspond to different assumed principals that do not produce a total interest of Rs. 2961 when both schemes are considered. Substituting these values back into the total interest expression would yield totals significantly different from 2961. Common Pitfalls: A common mistake is to treat compound interest as if it were simple interest and ignore the squaring of the growth factor (1.23)^2. Another error is to attempt separate trial and error for P instead of forming a single equation. Keeping the algebraic structure clear and using exact fractions or careful decimals avoids rounding errors and confusion. Final Answer: The first year interest from the simple interest scheme is Rs. 700.