Suppose you want to have $$ \$ 800,000 $$ for retirement in 20 years. Your account earns $$ 8 \% $$ 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 want to accumulate a future value of  
$$
FV = \$800{,}000
$$
in $$20$$ years by making equal monthly deposits into an account that earns a nominal annual interest rate of $$8\%$$ compounded monthly.

Let  
$$
i = \frac{0.08}{12} \quad \text{and} \quad n = 20 \times 12 = 240.
$$

The future value of an annuity (a series of equal deposits) is given by  
$$
FV = P \times \frac{(1+i)^n - 1}{i},
$$
where $$P$$ is the monthly deposit and $$i$$ is the monthly interest rate.

### a) Finding the Monthly Deposit

Solve for $$P$$:
$$
P = FV \times \frac{i}{(1+i)^n - 1}.
$$

Substitute the given values:
$$
P = 800{,}000 \times \frac{\frac{0.08}{12}}{(1+\frac{0.08}{12})^{240}-1}.
$$

Here,  
$$
i = \frac{0.08}{12} \approx 0.00666667.
$$

Now, calculate $$(1+i)^{240}$$:
$$
(1.00666667)^{240}.
$$
Using exponentiation (by taking natural logs or using a calculator) we find:
$$
(1.00666667)^{240} \approx 4.898.
$$

Now the expression in the denominator is:
$$
(1+i)^n - 1 \approx 4.898 - 1 = 3.898.
$$

Thus,
$$
P \approx 800{,}000 \times \frac{0.00666667}{3.898}.
$$

Calculate the numerator:
$$
800{,}000 \times 0.00666667 \approx 5333.336.
$$

Finally, divide:
$$
P \approx \frac{5333.336}{3.898} \approx 1367.03.
$$

So, the monthly deposit required is approximately  
$$
\boxed{\$1367.03.}
$$

### b) Calculating the Total Interest Earned

The total amount contributed by making monthly deposits for 240 months is:
$$
\text{Total Contributions} = P \times n \approx 1367.03 \times 240 \approx 328087.20.
$$

The interest earned is the difference between the future value and the total contributions:
$$
\text{Interest Earned} = FV - (\text{Total Contributions}) \approx 800{,}000 - 328087.20 \approx 471912.80.
$$

Thus, the interest earned is approximately  
$$
\boxed{\$471912.80.}
$$
a) You need to deposit approximately \$1,367.03 each month. b) You will earn approximately \$471,912.80 in interest.

Suppose you want to have \( \$ 800,000 \) for retirement in 20 years. Your account earns \( 8 \% \) 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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