Final answer:
To find how long Bill needs to deposit $75 monthly at 2.7% interest to reach $2,000, we use the annuity formula. The answer requires a financial calculator or software. The deposits and interest compound over time, with each new deposit earning less interest than the last.
Step-by-step explanation:
To determine how long it will take for Bill to reach an account balance of $2,000 by making monthly deposits of $75 at an interest rate of 2.7% compounded monthly, we need to use the formula for the future balance of a periodic investment. The formula for the future value of an annuity compounded at regular intervals is P(\(\frac{(1 + r/n)^{nt} - 1}{r/n}\)), where P is the periodic payment, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the total number of years.
First, we need to convert the annual interest rate from a percentage to a decimal, which makes it 0.027. The interest is compounded monthly, so n is 12. What we want to find is the value of t that will make the future value greater than or equal to $2,000.
Since the formula does not allow for directly solving for t, we would often use financial calculators or software to find the precise time period required. However, we can use logarithms to algebraically solve for t when we set the formula equal to $2,000 and solve for t.
It's important to note that without the specifics of the exact formula for periodic deposits, we can't algebraically calculate the precise time. As such, we advise using a financial calculator or appropriate financial software to obtain the answer. Based on the compound interest model as a rough comparison, for a single deposit scenario (like the $100 example at 2%), the money will grow according to the compound interest, but with periodic deposits, the account balance grows not only from interest but also from the continued investment of new capital.
Each deposit will earn less interest than the previous one because it's in the account for a shorter period; thus, the growth will be a combination of the deposit amounts and the compounded interest over time. It might be helpful to look at this problem in segments, each representing a month of growth and a fresh deposit, and then applying the compound interest formula to each segment cumulatively to see how the balance grows.