Comparing Present Value of $8000 in 10 Years with $6433 Now
The question asks you to find the present value of $8000 in 10 years given an interest rate of 2.2% compounded quarterly. To decide which is larger, $6433 now or $8000 in 10 years, we need to calculate the present value of the $8000 using the present value formula for compound interest.
The present value formula is given by:
\[ PV = \frac{FV}{(1 + \frac{r}{n})^{n \times t}} \]
where:
PV = present value
FV = future value ($8000)
r = annual interest rate (2.2% or 0.022)
n = number of times the interest is compounded per year (quarterly, so 4 times a year)
t = number of years (10)
Now let's calculate PV:
\[ PV = \frac{8000}{(1 + \frac{0.022}{4})^{4 \times 10}} \]
\[ PV = \frac{8000}{(1 + 0.0055)^{40}} \]
\[ PV = \frac{8000}{(1.0055)^{40}} \]
Using a calculator:
\[ PV \approx \frac{8000}{(1.0055)^{40}} \approx \frac{8000}{2.48832} \approx 3215.77 \]
Therefore, the present value of $8000 in 10 years at an interest rate of 2.2% compounded quarterly is approximately $3215.77.
Now, comparing the present value of $8000 in 10 years ($3215.77) with $6433 now, it is clear that $6433 now is larger.
Please note: The final answer has been rounded to the nearest cent as requested.
