You want to be able to withdraw $1,500 from your account quarterly for 20 years after you retire. You expect to retire in 35 years. If your account earns 6.1% interest, how much money will you need to deposit each quarter until retirement to achieve your retirement goals?

Do not round numbers while working on the problem. Only round the final answer.

Since we are working with money, round to the cent.

To solve this problem, we need to approach it in two steps:

1. **Determine the amount needed at retirement** to fulfill your withdrawal goals.
2. **Calculate the required quarterly deposit** from now until retirement to accumulate the necessary amount.

### Step 1: Determine the Amount Needed at Retirement

You want to withdraw $1,500 quarterly for 20 years. This is an annuity problem where we need to find the present value of these withdrawals at the time of retirement.

Given:
- Withdrawal amount per quarter ($ PMT $) = $1,500
- Number of quarters in 20 years ($ n $) = 20 years × 4 quarters/year = 80 quarters
- Quarterly interest rate ($ i $) = $\frac{6.1\%}{4}$ = 0.061 / 4 = 0.01525

The present value ($ PV $) of an annuity is given by the formula:

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

Substituting the values:

\[
PV = 1500 \times \left(\frac{1 - (1 + 0.01525)^{-80}}{0.01525}\right)
\]

Let's calculate the present value.

### Step 2: Determine the Quarterly Deposit

Now that we know the amount needed at retirement, we need to determine how much to deposit each quarter for the next 35 years to accumulate this amount.

Given:
- Future value needed ($ FV $) = $ PV $ calculated above
- Number of quarters in 35 years ($ n $) = 35 years × 4 quarters/year = 140 quarters
- Quarterly interest rate ($ i $) = 0.01525

The future value of a series of deposits is given by:

\[
FV = PMT \times \left(\frac{(1 + i)^n - 1}{i}\right)
\]

Solving for $ PMT $:

\[
PMT = \frac{FV}{\left(\frac{(1 + i)^n - 1}{i}\right)}
\]

Substituting the values:

\[
PMT = \frac{PV}{\left(\frac{(1 + 0.01525)^{140} - 1}{0.01525}\right)}
\]

Let's calculate the quarterly deposit amount.The amount needed at retirement to fulfill your withdrawal goals is approximately $69,052.97.

### Step 2: Calculate the Quarterly Deposit

Now, we'll calculate the quarterly deposit required to accumulate this amount by the time you retire in 35 years.

Let's proceed with this calculation.The quarterly deposit required to achieve your retirement goal is approximately $618,431.83.

Would you like any further details or have any questions?

Here are 5 related questions for further practice:

1. How would the required deposit change if the interest rate was 7% instead of 6.1%?
2. What would happen if you wanted to withdraw $2,000 instead of $1,500 per quarter after retirement?
3. How much would you need to deposit quarterly if you only had 30 years until retirement?
4. What if you decided to make monthly contributions instead of quarterly? How would this affect your calculations?
5. How does the present value change if the number of withdrawal years increases to 25 years?

**Tip:** When dealing with financial problems, always keep track of the time periods (monthly, quarterly, annually) to ensure your interest rates and time periods are consistent.

Learn how to calculate the quarterly deposit needed to achieve retirement savings goals using compound interest. This problem involves determining the amount needed at retirement and calculating the required quarterly deposit over a specified period with a given interest rate. Suitable for financial planning and college-level mathematics.

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