Question

you would like to have $3,000 in 4 years for a special vacation following graduation by making deposits at the end of every six months in an annuity that pays 6% compounded semiannually.
a. determine how much you should deposit at the end of every six months.
b. how much of the $3,000 comes from deposits and how much comes from interest?
a. in order to have $3,000 in 4 years, you should deposit $\square$ at the end of every six months.
(do not round until the final answer. then round up to the nearest dollar.)
b. $\square$ of the $3,000 comes from your deposits and $\square$ comes from interest.
(use the answer from part a to find this answer. round to the nearest dollar as needed.)

Explanation:

Step1: Define given values

Future value $FV = 3000$, annual rate $r=0.06$, compounding periods/year $n=2$, time $t=4$ years.

Step2: Calculate periodic rate & total periods

Periodic rate $i=\frac{r}{n}=\frac{0.06}{2}=0.03$
Total periods $N=n\times t=2\times4=8$

Step3: Use ordinary annuity formula

Ordinary annuity payment formula: $PMT = \frac{FV \times i}{(1+i)^N - 1}$
Substitute values:
$PMT = \frac{3000 \times 0.03}{(1+0.03)^8 - 1}$
Calculate $(1.03)^8 \approx 1.26677$
$(1.03)^8 -1 \approx 0.26677$
$PMT = \frac{90}{0.26677} \approx 337.36$, round up to $\$338$

Step4: Calculate total deposits

Total deposits $= PMT \times N = 338 \times 8 = 2704$

Step5: Calculate total interest

Interest $= FV - \text{Total deposits} = 3000 - 2704 = 296$

Answer:

a. $\$338$
b. $\$2704$ of the $\$3,000$ comes from your deposits and $\$296$ comes from interest.