Final answer:
To determine the time needed for a $1200 investment to earn $200 interest at 9% interest compounded quarterly, the compound interest formula is used. By solving this formula, we find that it takes approximately 3 years for the investment to grow to $1400, which is the principal plus interest. Hence, the correct answer is C. 3 years.
Step-by-step explanation:
To find how long it takes a $1200 investment to earn $200 interest when invested at 9% interest compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount (the initial amount of money).
r = the annual interest rate (decimal).
n = the number of times that interest is compounded per year.
t = the time the money is invested for, in years.
In this case, our goal is to have the investment grow to $1200 + $200 = $1400. So, we can set this up as:
1400 = 1200(1 + 0.09/4)^(4*t)
Now we can solve for t, which is the time in years:
1400/1200 = (1 + 0.09/4)^(4*t)
1.1667 = (1 + 0.0225)^(4*t)
After solving for t, we find that t is approximately 3 years. Therefore, the correct answer is C. 3 years.
It's important to note that compound interest can significantly affect the growth of an investment compared to simple interest, particularly over longer periods and with larger sums of money.