Final answer:
To find the amount that should be invested to total $50,000 in 7 years at an annual interest rate of 12% compounded quarterly, the principal amount should be approximately $28,055.79.
Step-by-step explanation:
To find the amount that should be invested to total $50,000 in 7 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where A is the final amount, P is the principal amount (the initial investment), r is the annual interest rate (in decimal form), n is the number of times the interest is compounded per year, and t is the number of years.
In this case, the principal amount P is what we want to find, the final amount A is $50,000, the annual interest rate r is 0.12 (12%), the number of times the interest is compounded per year n is 4 (quarterly compounding), and the number of years t is 7.
Plugging in these values into the formula:
$50,000 = P(1 + 0.12/4)^(4*7)
Simplifying:
$50,000 = P(1.03)^28
Dividing both sides by (1.03)^28:
$50,000 / (1.03)^28 = P
Using a calculator, the principal amount P is approximately $28,055.79.