A company will need $65,000 in 6 years for a new addition. To meet this goal the company deposits money in an account today that pays 5% annual intrest compound quarterly. Find the amount that should be invested to total $65,000 in 6 years.
the company should invest?
treated in the related questions below.
To find the amount that should be invested to total $65,000 in 6 years, we can use the formula for compound interest:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
A = the future value of the investment
P = the principal amount (the initial investment)
r = the annual interest rate (expressed as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
In this case, the principal amount (P) is what we need to find. The future value (A) is given as $65,000. The annual interest rate (r) is 5% or 0.05, and the interest is compounded quarterly, so n is 4. The investment period (t) is 6 years.
Let's plug in these values into the formula and solve for P:
\[\$65,000 = P \left(1 + \frac{0.05}{4}\right)^{(4 \times 6)}\]
Now, we can simplify the equation and solve for P:
\[\$65,000 = P(1.0125)^{24}\]
To isolate P, divide both sides of the equation by \((1.0125)^{24}\):
\[\frac{\$65,000}{(1.0125)^{24}} = P\]
Using a calculator, we get:
\[\frac{\$65,000}{(1.0125)^{24}} \approx \$45,736.676\]
Therefore, the company should invest approximately $45,736.68 to have $65,000 in 6 years.