At what annual rate of simple interest (in percent per annum) will a sum of money double itself in 16 2/3 years?

Introduction / Context:
This question asks for the simple interest rate at which a sum of money will double in a given period of 16 2/3 years. In simple interest, the interest grows linearly with time, so doubling the sum means the interest earned equals the original principal. This kind of problem is common in aptitude tests and helps you quickly link doubling time and rate using a direct proportional relationship between rate, time, and interest.

Given Data / Assumptions:

• Time T = 16 2/3 years, which is 16 + 2/3 = 50/3 years.
• The amount becomes double of the principal, so A = 2P.
• Interest is calculated using simple interest.
• The interest earned equals the principal when the sum doubles.

Concept / Approach:
For simple interest, the amount A after time T is A = P + SI, where SI = P * R * T / 100. If the sum doubles, then A = 2P, so SI = A - P = P. This gives P * R * T / 100 = P. We can cancel P on both sides to get R * T / 100 = 1, then solve for R: R = 100 / T. Here, T is a mixed fraction, so we must convert it to an improper fraction before dividing 100 by it. This simple relationship makes doubling-time questions very quick to solve once understood.

Step-by-Step Solution:
Step 1: Let the principal be P rupees. Step 2: If the sum doubles, then A = 2P, so the interest SI = A - P = P. Step 3: Simple interest formula: SI = P * R * T / 100. Step 4: Substitute SI = P: P = P * R * T / 100. Step 5: Cancel P from both sides (P is non-zero): 1 = R * T / 100. Step 6: Rearranging gives R = 100 / T. Step 7: Time T = 16 2/3 years = 16 + 2/3 = 50/3 years. Step 8: So R = 100 / (50/3) = 100 * 3 / 50. Step 9: Simplify: 100 * 3 / 50 = 300 / 50 = 6. Step 10: Therefore, the required annual simple interest rate is 6% per annum.

Verification / Alternative check:
Assume P = Rs. 100 for easy checking. At R = 6% and T = 50/3 years, the simple interest SI = 100 * 6 * (50/3) / 100. Simplify: 100 cancels with 100, leaving SI = 6 * 50 / 3 = 300 / 3 = 100. So SI = 100, and the amount A = P + SI = 100 + 100 = 200, which is exactly double the principal. This confirms that 6% per annum is correct.

Why Other Options Are Wrong:
If R = 4%, then SI over 50/3 years would be 100 * 4 * (50/3) / 100 = 200 / 3, which is less than the principal. If R = 5%, SI would be 250 / 3, still less than 100. If R = 6 2/3%, SI would be more than the principal, causing the sum to more than double. Only R = 6% yields interest exactly equal to the principal over 16 2/3 years.

Common Pitfalls:
Many students forget to convert the mixed fraction into an improper fraction correctly or misinterpret the condition for doubling the sum. Others may accidentally use compound interest formulas instead of the simple interest relationship. Always remember that, for simple interest, doubling means SI = P, and the quick formula R = 100 / T is a powerful shortcut once T is correctly expressed in years.

Final Answer:
The rate of simple interest is 6% per annum.