A certain sum 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 percent per annum simple interest. The interest earned from scheme M is twice the interest earned from scheme N. What is the value of k?

Introduction / Context:
This problem compares compound interest and simple interest on the same principal over the same time period. The interest from the compound interest scheme is given to be twice the interest from the simple interest scheme, and we are asked to find the unknown simple interest rate k. Such questions test understanding of basic interest formulas and the difference between simple and compound interest growth.

Given Data / Assumptions:

• The same principal amount is invested in both schemes M and N.
• Scheme M: rate = 20 percent per annum compound interest, compounded annually, for 2 years.
• Scheme N: rate = k percent per annum simple interest, for 2 years.
• Interest from scheme M is twice the interest from scheme N.
• We assume rates remain fixed and there are no additional deposits or withdrawals.

Concept / Approach:
For compound interest with annual compounding, the interest for 2 years on principal P at rate r percent is: CI = P * [(1 + r/100)^2 - 1] For simple interest on the same principal and time with rate k percent, the interest is: SI = P * k * 2 / 100 We are told that CI from scheme M is twice SI from scheme N. This gives an equation in terms of k, which we solve to find the required rate.

Step-by-Step Solution:
Let the principal be P. Scheme M: r = 20 percent, time = 2 years. CI from M = P * [(1 + 20/100)^2 - 1] (1 + 20/100) = 1.20, so (1.20)^2 = 1.44 Therefore CI from M = P * (1.44 - 1) = P * 0.44 Scheme N: rate = k percent, time = 2 years, simple interest. SI from N = P * k * 2 / 100 = 0.02 * k * P Given: CI from M = 2 * SI from N So P * 0.44 = 2 * (0.02 * k * P) 0.44 = 0.04 * k k = 0.44 / 0.04 = 11

Verification / Alternative check:
To verify, pick a convenient value, for example P = Rs. 100. For scheme M, CI for 2 years at 20 percent is 44 rupees, since 100 grows to 144. For scheme N with k = 11 percent per annum simple interest, the interest in 2 years is: SI = 100 * 11 * 2 / 100 = 22 Twice of 22 is 44, which exactly matches the CI from scheme M. This confirms that k = 11 percent is correct.

Why Other Options Are Wrong:
A rate of 7 percent or 9 percent would produce significantly lower simple interest and would not give a compound interest that is exactly double. Thirteen percent or fifteen percent would lead to higher simple interest, and twice that value would exceed the compound interest earned at 20 percent for 2 years. Only 11 percent fits the condition that compound interest at 20 percent for 2 years is exactly twice the simple interest at k percent for 2 years on the same principal.

Common Pitfalls:
One common error is to confuse the formulas for simple and compound interest and treat both as if they grow linearly. Another mistake is to forget that the compounding period is annual and misuse a more complex formula. Students also sometimes cancel the principal incorrectly or make algebraic mistakes when solving for k. Carefully writing down the formulas and simplifying step by step helps avoid these issues.

Final Answer:
The required simple interest rate k for scheme N is 11 percent per annum.