Juliana has $54,500 in her "World Tour Savings Plan" right now. She will not be able to make any more contributions to the Plan but she is planning to let the money accumulate until she has enough so that he can take out $2,500 per month for 3 years while she travels around the world. She will take out the first $2,500 one month after she leaves on her trip. If her investments earn 9.9% compounded monthly, how many months will it be before she leaves on her trip?

Question

Quizplus · Accepted Answer

To solve this problem, we need to calculate how long it will take for Juliana's savings to grow to an amount that can sustain $2,500 monthly withdrawals for 3 years (36 months) at an interest rate of 9.9% compounded monthly. The future value of an annuity formula can be used to calculate the required savings amount to support these withdrawals:$$FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)$$Where:- $FV$ is the future value of the annuity (the amount needed to support the withdrawals),- $P$ is the payment amount ($2,500),- $r$ is the monthly interest rate (9.9% annual rate / 12 months = 0.825% or 0.00825 as a decimal),- $n$ is the total number of payments (36 months).First, calculate the amount needed to support 36 months of withdrawals:$$FV = 2500 \times \left( \frac{(1 + 0.00825)^{36} - 1}{0.00825} \right)$$$$FV ≈ 2500 \times 45.762 ≈ 114,405$$This means Juliana needs approximately $114,405 in her savings to support her withdrawals for 36 months.Next, we need to calculate how long it will take for her current savings of $54,500 to grow to $114,405 at an interest rate of 9.9% compounded monthly. We use the future value of a single sum formula:$$FV = PV \times (1 + r)^n$$Rearranging the formula to solve for $n$:$$n = \frac{\log(FV / PV)}{\log(1 + r)}$$Where:- $FV$ is the future value needed ($114,405),- $PV$ is the present value of her savings ($54,500),- $r$ is the monthly interest rate (0.00825),- $n$ is the number of periods (months) until the savings reach the required amount.$$n = \frac{\log(114,405 / 54,500)}{\log(1 + 0.00825)}$$$$n ≈ \frac{\log(2.099)}{\log(1.00825)} ≈ \frac{0.322}{0.0036} ≈ 89.44$$Since Juliana will start withdrawing one month after she leaves, we subtract one month from the total:$$89.44 - 1 ≈ 88.44$$Since we cannot have a fraction of a month, we round up to the nearest whole month, which gives us 89 months. However, the question asks for how many months before she leaves on her trip, not the total time until the savings are depleted. Given the options provided do not match this calculation, it indicates a need to reevaluate the approach or the interpretation of the options. The correct approach involves understanding the time needed for the savings to grow to support the withdrawals, and the discrepancy suggests a misunderstanding in the calculation or the application of the formula. The correct answer should reflect the time needed for the savings to accumulate to the required amount before withdrawals begin, considering the compounding interest. Given the options and the typical approach to such problems, there might have been an error in the calculation steps or assumptions made. The correct procedure involves calculating the time until the savings reach the amount needed to sustain the withdrawals, but without the exact calculations matching the options, the explanation provided aims to outline the correct method to approach the problem.

Juliana Has $54,500 in Her "World Tour Savings Plan" Right