Question

7.
Find the present value.
Amount Needed: $42,000
Time (Years): 8
Rate: 3%
Compounded: semiannually


$37,283.82

$36,960.00

$33,097.26

$33,155.22

Answer 1

To solve this we are going to use the formula for a future value: $FV=PV(1+ \frac{r}{n} )^{nt}$
where
$FV$ is the future value 
$PV$ is the present value 
$r$ is the interest rate in decimal form
$n$ is the number of times the interest is compounded per year
$t$ is the time in years

Since the amount needed is $42,000, $FV=42000$. We also know for our problem that $t=8$ and $r= \frac{3}{100} =0.03$. Since the interest is compounded semiannually, it is compounded two times per year; therefore, $n=2$. Lets replace those values in our formula and solve for $PV$:
$FV=PV(1+ \frac{r}{n} )^{nt}$
$42000=PV(1+ \frac{0.03}{2})^{(2)(8)}$
$PV= \frac{42000}{(1+ \frac{0.03}{2})^{(2)(8)}}$
$PV=33097.30$

We can conclude that the present value needed to make $42000 after 8 years according to your given choices is $33,097.26