A certain sum of money is invested for 2 years in Scheme M at 20% per annum compound interest, compounded annually. The same sum is also invested for the same period in Scheme N at k% per annum simple interest. The interest earned from Scheme M is twice the interest earned from Scheme N. What is the value of k (the simple interest rate per annum)?

Introduction / Context:
This is an important conceptual question comparing compound interest (C.I.) with simple interest (S.I.) on the same principal and time period. You are asked to find an unknown simple interest rate k such that the C.I. at 20% is exactly twice the S.I. at k%. This type of problem is common in banking and finance aptitude tests.

Given Data / Assumptions:
- Principal in both schemes is the same, say P rupees.
- Scheme M: 20% per annum compound interest for 2 years, compounded annually.
- Scheme N: k% per annum simple interest for 2 years.
- Interest from Scheme M is twice the interest from Scheme N.
- We must find the value of k (in percent per annum).

Concept / Approach:
We separately write expressions for the interest in each scheme. For compound interest (annually) for 2 years at 20%, the amount is P * (1.2)^2. The compound interest is this amount minus P. For simple interest, the interest is P * k * 2 / 100. Equating the C.I. from Scheme M to two times the S.I. from Scheme N gives an equation in k that we can solve easily.

Step-by-Step Solution:
Step 1: Scheme M amount after 2 years: A_M = P * (1 + 20/100)^2 = P * (1.2)^2 = 1.44P.Step 2: Compound interest in Scheme M: I_M = A_M - P = 1.44P - P = 0.44P.Step 3: Scheme N simple interest for 2 years at k%: I_N = P * k * 2 / 100 = 0.02kP.Step 4: Given condition: I_M = 2 * I_N.Step 5: So 0.44P = 2 * 0.02kP = 0.04kP.Step 6: Cancel P (P ≠ 0), giving 0.44 = 0.04k.Step 7: Solve for k: k = 0.44 / 0.04 = 11.Step 8: Therefore, k = 11% per annum.

Verification / Alternative check:
Assume P = Rs. 100 for quick checking. C.I. at 20% for 2 years is Rs. 44. If k = 11%, S.I. for 2 years is 100 * 11 * 2 / 100 = Rs. 22. Twice of 22 is 44, exactly matching the C.I. from Scheme M. This confirms that k = 11% is correct.

Why Other Options Are Wrong:
Values 7%, 9%, 13% or 15% produce simple interest amounts that do not satisfy the condition that C.I. is exactly twice the S.I. For example at 10% (near the middle), S.I. for 2 years would be 20% of P, and twice S.I. would be 40% of P, but the C.I. is 44% of P, not 40%. Only 11% makes the equality hold.

Common Pitfalls:
Students sometimes mistakenly equate the amounts instead of the interests, or they ignore the “twice” condition and set C.I. equal to S.I. Others incorrectly use the simple interest formula for Scheme M as well. Always read carefully whether the comparison is between interest figures or total amounts and ensure you use the correct formula for each scheme.

Final Answer:
The required simple interest rate k is 11% per annum.