Final answer:
Option C, with 3.1% annual interest compounded monthly, has the smallest doubling time due to the more frequent compounding period which results in quicker accumulation of interest.
Step-by-step explanation:
To determine which scenario has the smallest doubling time, we use the Rule of 70. This rule states that the doubling time (in years) can be approximated by dividing 70 by the interest rate (as a percentage). However, with compound interest being more frequent than annually, we must adjust this calculation to account for the effect of compounding. The formula for compound interest is A = P(1 + r/n)^(nt), where P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. Calculation of the exact doubling time would require using logarithms, but the Rule of 70 can give us a rough estimation.
Comparing the options using approximate calculations:
- A) 3.3% annual interest, compounded annually: 70 / 3.3 = 21.21 years
- B) 3.2% annual interest, compounded quarterly: Doubling time will be less than 70 / 3.2, since it compounds more frequently than annually.
- C) 3.1% annual interest, compounded monthly: Doubling time will be less than 70 / 3.1 as well, and even less than scenario B since it compounds even more frequently.
Since more frequent compounding periods result in a faster accumulation of interest, Option C) with 3.1% annual interest compounded monthly will have the smallest doubling time among the options given.