Money is invested at 3% annual interest, compounded monthly. Approximately how many years will it take for the investment to double in value (that is, become twice the original principal)?

Introduction / Context:
This problem asks for the time required for an investment to double at a given nominal annual interest rate when the interest is compounded monthly. Rather than asking for the amount or the interest, the focus is on the time period. This is a classic application of the compound interest formula solved for time, and it helps you understand the long term effect of relatively small interest rates when compounding occurs frequently, such as every month.

Given Data / Assumptions:

• The nominal annual rate of interest R = 3% per annum.
• Interest is compounded monthly, so there are m = 12 compounding periods in a year.
• The initial principal is some amount P, and we want the amount to become 2P.
• We assume there are no withdrawals or extra deposits during the period.
• We seek the time T in years, to at least one decimal place, that yields doubling.

Concept / Approach:
For nominal annual rate R compounded m times per year, the compound amount after T years is:
A = P * (1 + (R / 100) / m)^(m * T)We want A = 2P, that is, the amount doubles. So:
2P = P * (1 + (R / 100) / m)^(m * T)After canceling P, we get:
2 = (1 + (R / 100) / m)^(m * T)We then solve for T using logarithms.

Step-by-Step Solution:
Step 1: Identify the periodic rate. Here R = 3% and m = 12, so monthly rate = 3 / 100 / 12 = 0.03 / 12 = 0.0025.Step 2: The equation for doubling is 2 = (1 + 0.0025)^(12 * T).Step 3: Take natural logarithms on both sides to isolate T: ln(2) = (12 * T) * ln(1.0025).Step 4: Solve for T: T = ln(2) / (12 * ln(1.0025)).Step 5: Numerically, ln(2) is about 0.6931 and ln(1.0025) is approximately 0.0024969, giving T approximately equal to 0.6931 / (12 * 0.0024969).Step 6: Compute the denominator: 12 * 0.0024969 is about 0.0299628.Step 7: Now T is roughly 0.6931 / 0.0299628, which is close to 23.1 years.

Verification / Alternative check:
As a rough rule of thumb, the time to double with annual compounding can often be approximated by the rule of 72, which says doubling time is about 72 / R years. With R = 3, this gives about 24 years. Because monthly compounding is slightly more beneficial than annual compounding, the actual doubling time should be a little less than 24 years. A value around 23 to 23.2 years therefore makes sense and confirms our detailed calculation of about 23.1 years.

Why Other Options Are Wrong:
20.1, 21.1, and 22.1 years are all too small, implying a faster doubling than what 3% annual interest compounded monthly can provide. They would require a higher effective interest rate. The choice 18.1 years is much too small and clearly unrealistic for such a low nominal rate. Only about 23.1 years aligns with the exact calculation and the rule of 72 approximation.

Common Pitfalls:
Many students mistakenly use simple interest calculations or forget to divide the nominal annual rate by the number of compounding periods. Some try to use the rule of 72 without considering that monthly compounding slightly reduces the doubling time compared to annual compounding. Others avoid using logarithms and guess randomly. A systematic use of the compound interest formula and basic logarithm properties leads to the correct answer.

Final Answer:
At 3% annual interest compounded monthly, the investment will double in approximately 23.1 years.