At an annual interest rate of 3% compounded monthly, approximately how many years will it take for an investment to double in value?

Introduction / Context:
Time to double money is a classic compound interest question. It helps you understand how interest rate and compounding frequency affect growth. Here, the interest rate is 3% per annum, but it is compounded monthly, so we must work with a monthly rate and solve for time using the compound interest formula.

Given Data / Assumptions:

Annual nominal rate = 3% per annum.
Compounding frequency = monthly (12 times per year).
We want the time needed for the amount to double, that is, final amount = 2 * principal.
Compounding is at a constant rate with no deposits or withdrawals.

Concept / Approach:
For principal P, periodic rate i and n periods, the amount A is given by: A = P * (1 + i)^n Here, we set A = 2P, so: 2P = P * (1 + i)^n This simplifies to (1 + i)^n = 2. We solve for n, then convert the number of months into years by dividing by 12.

Step-by-Step Solution:
Step 1: Find the monthly rate. Annual nominal rate = 3%, so monthly rate i = 3% / 12 = 0.25% = 0.0025. Step 2: Set up the doubling equation. (1 + 0.0025)^n = 2. Step 3: Take logarithms to solve for n. n = log(2) / log(1.0025). Numerically, n is about 277.6 months. Step 4: Convert months to years. Years = 277.6 / 12 ≈ 23.1 years.

Verification / Alternative check:
We can quickly test reasonableness. At a small effective annual rate (slightly above 3% because of monthly compounding), the Rule of 72 estimates doubling time ≈ 72 / 3 = 24 years. Our precise calculation of about 23.1 years is close to this rule-of-thumb, so the answer is consistent and reasonable.

Why Other Options Are Wrong:
20.1, 21.1, and 22.1 years are all substantially less than both the Rule of 72 estimate and the exact logarithmic solution. They would imply a higher effective interest rate than 3% compounded monthly, so they do not match the given data.

Common Pitfalls:
A common mistake is to treat 3% as a monthly rate instead of annual, which dramatically understates the time to double. Another is to ignore the compounding and simply apply a simple interest calculation, which is not correct when interest is compounded.

Final Answer:
It takes approximately 23.1 years to double the money at 3% per annum compounded monthly.