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the reddens want to start saving for sarahs college education. they estimate college tuition will be $20,000 per year for four years, starting in 19 years. the investment account they will use earns 9.6% annual interest. round answers to the nearest cent.
a. how much money will they need to have saved at the beginning of sarahs college?
$ 67788.80
b. if they invest a fixed amount now, how much will they need to invest to build up to the amount needed at the start of college?
$ 11977.82
c. if they invest monthly in an annuity to save for her education, how much should they invest each month?
$ 120.85

Explanation:

Step1: Calculate present value of annuity (part a)

This is a present value of an ordinary annuity problem, since tuition is paid at the start/end of each year. The formula for present value of an ordinary annuity is:
$$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$
Where $PMT = \$20,000$, $r = 0.096$, $n = 4$
$$PV = 20000 \times \frac{1 - (1 + 0.096)^{-4}}{0.096}$$
First calculate $(1.096)^{-4} \approx 0.6938$
$$1 - 0.6938 = 0.3062$$
$$\frac{0.3062}{0.096} \approx 3.1896$$
$$PV = 20000 \times 3.1896 = 63792.00$$

Step2: Calculate present value today (part b)

We need to find the present value of the amount from part a, 19 years before. Use the present value formula:
$$PV_0 = \frac{FV}{(1 + r)^t}$$
Where $FV = 63792.00$, $r = 0.096$, $t = 19$
$$(1.096)^{19} \approx 5.598$$
$$PV_0 = \frac{63792.00}{5.598} \approx 11395.50$$

Step3: Calculate monthly annuity payment (part c)

Use the future value of an ordinary annuity formula, solve for $PMT$:
$$FV = PMT \times \frac{(1 + \frac{r}{12})^{12t} - 1}{\frac{r}{12}}$$
Where $FV = 63792.00$, $r = 0.096$, $t = 19$, so $\frac{r}{12} = 0.008$, $12t = 228$
First calculate $(1 + 0.008)^{228} \approx e^{228 \times \ln(1.008)} \approx 5.598$
$$(1.008)^{228} - 1 \approx 4.598$$
$$\frac{4.598}{0.008} \approx 574.75$$
$$PMT = \frac{63792.00}{574.75} \approx 111.00$$

Answer:

a. $\$63792.00$
b. $\$11395.50$
c. $\$111.00$