You have $$ \$ 300,000 $$ saved for retirement. Your account earns $$ 9 \% $$ interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 25 years? Round your ansmer to the nearest cent.

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UpStudy AI · Accepted Answer

We can use the annuity payment formula

$$
P = PV \times \frac{r}{1 - (1 + r)^{-n}},
$$

where:
- $$ PV $$ is the present value (initial amount saved),
- $$ r $$ is the periodic interest rate,
- $$ n $$ is the total number of withdrawals (periods).

**Step 1: Identify the values**

The given values are:
- $$ PV = 300,\!000 $$,
- Annual interest rate $$ = 9\% $$. The monthly rate is

$$
r = \frac{0.09}{12} = 0.0075.
$$

- The retirement period is 25 years, so the number of months is

$$
n = 25 \times 12 = 300.
$$

**Step 2: Substitute into the formula**

Substitute the known values into the annuity formula:

$$
P = 300,\!000 \times \frac{0.0075}{1 - (1 + 0.0075)^{-300}}.
$$

**Step 3: Evaluate the denominator**

First, calculate $$ (1 + 0.0075)^{-300} $$:
- $$ 1 + 0.0075 = 1.0075 $$.
- Raise to the power $$-300$$:

$$
(1.0075)^{-300} = \frac{1}{(1.0075)^{300}}.
$$

Using logarithms or a calculator, we find approximately:

$$
(1.0075)^{300} \approx 9.406,
$$

so

$$
(1.0075)^{-300} \approx \frac{1}{9.406} \approx 0.1063.
$$

Now compute:

$$
1 - (1.0075)^{-300} \approx 1 - 0.1063 = 0.8937.
$$

**Step 4: Compute the monthly withdrawal**

Now substitute back:

$$
P = 300,\!000 \times \frac{0.0075}{0.8937}.
$$

Calculate the numerator:

$$
300,\!000 \times 0.0075 = 2250.
$$

Now,

$$
P \approx \frac{2250}{0.8937} \approx 2517.90.
$$

**Final Answer:**

Each month you can withdraw approximately \$$$2517.90$$ rounded to the nearest cent.
Each month, you can withdraw approximately \$2,517.90.

You have \( \$ 300,000 \) saved for retirement. Your account earns \( 9 \% \) interest. How much will you be able to pull out each month, if you want to be able to take withdrawals for 25 years? Round your ansmer to the nearest cent.

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