Approximately how many years will it take to double an investment at 6% per annum compounded annually?

Introduction / Context:
This question asks for the time needed to double an investment at a fixed compound interest rate of 6% per annum. It is a classic problem in finance and aptitude exams and is often used to illustrate the power of compounding. The exact calculation involves solving an exponential equation, but there are also helpful rules of thumb, such as the rule of 72, that allow a quick estimate.

Given Data / Assumptions:

• Rate of interest r = 6% per annum.
• Interest is compounded annually.
• We want time n in years such that the amount is twice the principal.
• This means the growth factor over n years must be 2.

Concept / Approach:
The standard compound interest formula is A = P * (1 + r) ^ n. To double the investment, we set A = 2P. Substituting into the formula gives 2P = P * (1 + r) ^ n. Cancelling P from both sides yields 2 = (1 + r) ^ n. To solve this exactly, we would take logarithms and solve n = log(2) / log(1 + r). For exam purposes and whole number answer options, we calculate an approximate value and then match it to the nearest plausible integer number of years, which will give the correct option.

Step-by-Step Solution:
Step 1: Convert the rate to decimal form: r = 6% = 0.06. Step 2: Write the doubling condition: 2 = (1.06) ^ n. Step 3: Take logarithms to solve for n: n = log(2) / log(1.06). Step 4: Approximate log(2) as about 0.3010 and log(1.06) as about 0.0253. Step 5: Compute n = 0.3010 / 0.0253, which is approximately 11.9 years. Step 6: Since the options are given in whole years, the closest integer is 12 years. Step 7: Therefore, it will take approximately 12 years to double the investment at 6% compound interest.

Verification / Alternative check:
A quick rule of thumb is the rule of 72, which estimates the doubling time as 72 divided by the annual rate in percent. Here it gives 72 / 6 = 12 years, the same result as the more exact logarithmic calculation. We can also check powers of 1.06 directly: (1.06) ^ 10 is about 1.791, which is less than 2, and (1.06) ^ 12 is about 2.012, which is very close to exactly double. This confirms that about 12 years is the correct and reasonable answer.

Why Other Options Are Wrong:
An answer of 10 years would correspond to roughly (1.06) ^ 10, which is less than 2, so the investment would not quite double. Eleven years still gives a factor slightly below 2, while thirteen years would overshoot more than necessary. Fourteen years produces an even larger overshoot and is less efficient than required. Twelve years gives a factor almost exactly equal to 2, making it the only option that matches the mathematical requirement closely.

Common Pitfalls:
Some learners mistakenly use simple interest for doubling time, using 2 = 1 + r * n, which is not correct under compounding. Others may misapply the rule of 72 by using 6.0 in decimal form instead of percentage, which gives a nonsensical result. Another common error is to rely solely on rough mental estimates without checking powers of 1.06 or using logarithms, which can lead to choosing a nearby but incorrect option. Combining the rule of 72 with a quick check of powers is usually sufficient in exam situations.

Final Answer:
It will take approximately 12 years for an investment to double at 6% per annum with annual compounding, so the correct option is C.