Three years from now, a couple plans to spend 4 months traveling in China, Japan, and Southeast Asia. When they take their trip, they would like to withdraw $15,000 at the beginning of each month to cover their expenses for that month. Starting now, how much must they deposit at the beginning of each month for the next 3 years so that the account will provide the money they want while they are traveling (in $)? Assume that such an account pays 5.4% compounded monthly. (Round your answer to the nearest cent.)
$ _____

To solve this problem, we need to calculate how much the couple must deposit at the beginning of each month for the next 3 years in order to accumulate enough to withdraw $15,000 at the beginning of each month during their 4-month trip, assuming an account paying 5.4% compounded monthly.

### Step 1: Understanding the timeline
- The couple will withdraw $15,000 each month for 4 months during their trip, totaling 4 withdrawals.
- They have 3 years to make deposits (which is 36 months).
- Interest is compounded monthly at 5.4% per year.

### Step 2: Monthly interest rate
The nominal annual interest rate is 5.4%, so the monthly interest rate is:

\[
r = \frac{5.4\%}{12} = \frac{0.054}{12} = 0.0045
\]

### Step 3: Present value of the withdrawals during the trip
We need to find the present value of the withdrawals ($15,000 per month for 4 months), discounted back to the point when the couple starts their trip.

The formula for the present value of an annuity is:

\[
PV = W \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $ PV $ is the present value of the withdrawals
- $ W = 15,000 $ (the amount withdrawn each month)
- $ r = 0.0045 $ (monthly interest rate)
- $ n = 4 $ (number of months of withdrawals)

Let’s calculate this.The present value of the withdrawals during the trip is approximately \$59,331.03. This is the amount the couple needs to have in their account right when they start their trip to cover the 4 months of withdrawals.

### Step 4: Calculate the required monthly deposits

Now, we need to calculate how much the couple must deposit each month for 36 months in order to accumulate this amount. The formula for the future value of an ordinary annuity is:

\[
FV = D \times \left( \frac{(1 + r)^t - 1}{r} \right)
\]

Where:
- $ FV = 59331.03 $ (the amount they need by the time they start their trip)
- $ D $ is the monthly deposit amount we want to find
- $ r = 0.0045 $ (monthly interest rate)
- $ t = 36 $ (number of months of deposits)

Rearranging the formula to solve for $ D $:

\[
D = \frac{FV \times r}{(1 + r)^t - 1}
\]

Let’s calculate the required monthly deposit.The couple must deposit approximately \$1,521.89 at the beginning of each month for the next 3 years to accumulate enough for their trip.

Would you like further details or clarifications on this solution?

Here are 5 related questions for deeper exploration:
1. How would the deposit amount change if the interest rate were higher or lower?
2. What if the couple wanted to withdraw different amounts during each month of their trip?
3. How would the required monthly deposit change if they had more or less time to save?
4. What impact does compound interest have on long-term savings?
5. How would the plan differ if the couple wanted to continue withdrawing money after the trip?

**Tip:** When saving for future expenses, starting earlier and benefiting from compound interest can significantly reduce the required monthly savings!

Three years from now, a couple plans to spend 4 months traveling in China, Japan, and Southeast Asia. When they take their trip, they would like to withdraw $15,000 at the beginning of each month to cover their expenses for that month. Starting now, how much must they deposit at the beginning of each month for the next 3 years so that the account will provide the money they want while they are traveling (in $)? Assume that such an account pays 5.4% compounded monthly. (Round your answer to the nearest cent.)

Learn how to calculate the required monthly deposit for a couple planning to travel in 3 years. The problem involves compound interest, annuities, and calculating the present and future value of withdrawals. A step-by-step solution is provided, suitable for advanced high school and undergraduate students.

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