Quppose you want to have $$ \$ 300,000 $$ for retirement in 20 years. Your account earns $$ 6 \% $$ interest. Round your answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?

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We are given a future value goal of $$ \$300,\!000 $$ in 20 years with an account earning $$ 6\% $$ annual interest compounded monthly. We want to find

1. The monthly deposit $$ P $$
2. The total interest earned

**Step 1. Determine the monthly interest rate and the number of deposits.**

- The monthly interest rate is
  $$
  r = \frac{6\%}{12} = 0.5\% = 0.005.
  $$
- The total number of months is
  $$
  n = 20 \times 12 = 240.
  $$

**Step 2. Write the future value formula for an annuity.**

The future value $$ FV $$ of an annuity with monthly deposits $$ P $$ is given by
$$
FV = P \left( \frac{(1+r)^n - 1}{r} \right).
$$
Here, $$ FV = 300,\!000 $$.

**Step 3. Solve for the monthly deposit $$ P $$.**

Rearrange the formula to solve for $$ P $$:
$$
P = \frac{FV \cdot r}{(1+r)^n - 1}.
$$
Substitute the known values:
$$
P = \frac{300,\!000 \times 0.005}{(1.005)^{240} - 1}.
$$

**Step 4. Compute $$(1.005)^{240}$$.**

Using logarithms or a calculator,
$$
(1.005)^{240} \approx 3.310.
$$
Thus,
$$
(1.005)^{240} - 1 \approx 3.310 - 1 = 2.310.
$$

**Step 5. Calculate $$P$$.**

Substitute back:
$$
P \approx \frac{300,\!000 \times 0.005}{2.310} = \frac{1500}{2.310} \approx 649.35.
$$
So, the monthly deposit is approximately $$ \$649.35 $$.

**Step 6. Determine the total amount deposited over 20 years.**

Total deposits made over 240 months are:
$$
\text{Total deposits} = P \times n \approx 649.35 \times 240 \approx 155,\!844.00.
$$

**Step 7. Compute the total interest earned.**

The interest earned is the difference between the future value and the total deposits:
$$
\text{Interest earned} = \$300,\!000 - \$155,\!844.00 \approx \$144,\!156.00.
$$

**Final Answers:**

a) You need to deposit approximately $$ \$649.35 $$ each month.

b) The interest earned over the 20 years is approximately $$ \$144,\!156.00 $$.
a) You need to deposit approximately \$649.35 each month. b) The interest earned over 20 years is approximately \$144,156.00.

Quppose you want to have for retirement in 20 years. Your account earns
interest. Round your answers to the nearest cent.
a) How much would you need to deposit in the account each month?
b) How much interest will you earn?

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Quppose you want to have for retirement in 20 years. Your account earns interest. Round your answers to the nearest cent. a) How much would you need to deposit in the account each month? b) How much interest will you earn?