In aptitude (Simple Interest), at what annual rate of simple interest (in % per annum) will a sum of money become exactly double of itself in 16 2/3 years? (Assume simple interest and that doubling means Amount = 2 * Principal.)

Introduction / Context:
This question explores how long term growth works under simple interest and asks you to connect the idea of doubling an amount with the simple interest rate. Many competitive exams include this type of problem to test whether you can interpret phrases like amount doubles and translate them into an algebraic condition involving principal, rate, and time.

Given Data / Assumptions:

• The amount becomes exactly double of the original principal.
• Time period t = 16 2/3 years.
• Interest is calculated using simple interest, not compound interest.
• We need to find the annual simple interest rate r in percent.
• Let the principal be P; then the final amount A is 2P.

Concept / Approach:
For simple interest, the amount A after time t is related to the principal P through: A = P + SI and SI = (P * r * t) / 100 If the amount doubles, then A = 2P, which means: 2P = P + (P * r * t) / 100 This simplifies to a direct relation between r and t. The goal is to insert the value of t in years and solve for r. The mixed fraction 16 2/3 years must be converted into an improper fraction before using it in the formula.

Step-by-Step Solution:
Step 1: Write the doubling condition: A = 2P. Step 2: Since A = P + SI, we have 2P = P + SI, so SI = P. Step 3: Use the simple interest formula SI = (P * r * t) / 100. Step 4: Substitute SI = P into the formula: P = (P * r * t) / 100. Step 5: Cancel P from both sides (P is nonzero), giving 1 = (r * t) / 100. Step 6: Rearrange to get r = 100 / t. Step 7: Convert 16 2/3 years to an improper fraction: 16 2/3 = (50 / 3) years. Step 8: Substitute t = 50 / 3 into r = 100 / t: r = 100 / (50 / 3) = 100 * 3 / 50. Step 9: Simplify: 100 * 3 / 50 = 300 / 50 = 6. Step 10: Therefore, r = 6% per annum.

Verification / Alternative check:
We can check the result numerically. Assume P = ₹100 for simplicity. With r = 6% and t = 16 2/3 years, simple interest is: SI = (100 * 6 * (50 / 3)) / 100 = (600 * 50 / 3) / 100 SI = (30000 / 3) / 100 = 10000 / 100 = 100 So the amount A = P + SI = 100 + 100 = 200, which is exactly double the principal. This confirms that 6% per annum is consistent with the doubling condition.

Why Other Options Are Wrong:
4%: At 4% per annum, the product r * t is less than 100, so the amount would not double in 16 2/3 years.
5%: With 5%, r * t becomes about 83.33, still not enough to double the principal.
6.5%: Here r * t is greater than 100, meaning the amount would increase to more than double.
3%: This rate is too low and would result in much less than doubling over the given time.

Common Pitfalls:
A frequent mistake is to treat the mixed fraction 16 2/3 as 16.2 or 16.67 without using the exact fractional form, which can introduce rounding errors. Some learners also forget that doubling implies SI equals the principal and instead try to write separate expressions for amount and interest. Always convert mixed fractions properly and simplify the condition A = 2P to SI = P before applying the simple interest formula.

Final Answer:
The required annual simple interest rate that doubles the sum in 16 2/3 years is 6% per annum.