Approximately how many years will it take for an investment to double in value if it grows at a compound interest rate of 6% per annum, compounded annually?

Introduction:
Here we are asked to determine how long it takes for an investment to double in value under compound interest at 6% per annum, compounded annually. This is a classic time value of money question and is closely related to the idea of the doubling time of an investment.

Given Data / Assumptions:

• Initial principal = P (any positive amount; result does not depend on the exact value).
• Annual interest rate r = 6% per annum.
• Compounding frequency = once per year (annually).
• We want the time t (in years) for the final amount to be 2P, that is, twice the principal.

Concept / Approach:
For annual compounding, the amount after t years is:
A = P * (1 + r/100)^tWe need A = 2P. Substituting gives:
2P = P * (1 + r/100)^tDividing both sides by P yields:
2 = (1 + r/100)^tThen we solve for t using logarithms. For r = 6%, 1 + r/100 = 1.06, so we must solve 2 = (1.06)^t.

Step-by-Step Solution:
Step 1: Set up the equation for doubling: 2 = (1.06)^t.Step 2: Take logarithms on both sides: log(2) = t * log(1.06).Step 3: Solve for t: t = log(2) / log(1.06).Step 4: Using approximate values, log(2) ≈ 0.3010 and log(1.06) ≈ 0.0253.Step 5: Compute t ≈ 0.3010 / 0.0253 ≈ 11.9 years.This is approximately 12 years, so we select 12 years as the closest option.

Verification / Alternative check:
A useful mental rule is the Rule of 72, which estimates the doubling time by dividing 72 by the interest rate. For 6% we get 72 / 6 = 12 years. This matches our logarithmic calculation and further confirms that 12 years is a good approximation for the doubling time at 6% compound interest.

Why Other Options Are Wrong:
10 and 11 years are too short. At 6%, (1.06)^10 is about 1.79 and (1.06)^11 is about 1.90, both still below 2. Thirteen years is slightly longer than required, since (1.06)^13 is already above 2. The 9 years choice is much smaller and clearly underestimates the required time.

Common Pitfalls:
Many students guess the answer without using the formula or Rule of 72, leading to underestimates like 8 or 9 years. Others mistakenly divide by 6 directly without using the rule of 72 or logs. Always remember: for precise work use t = log(2) / log(1 + r/100), and for quick estimates use the Rule of 72 as a shortcut.

Final Answer:
It will take approximately 12 years for the investment to double at 6% per annum compounded annually.