Final answer:
The present value needed to grow to $3,000 in 10 years with an annual interest rate of 4.5% compounded quarterly is approximately $2,084.03.
Step-by-step explanation:
The student is asking to find the present value that will grow to $3,000 given an annual interest rate of 4.5% compounded quarterly over a period of 10 years. To solve this problem, we will use the formula for the present value (PV) of an investment using compound interest:
PV = FV / (1 + r/n)nt
where:
- FV = future value ($3,000)
- r = annual interest rate (0.045)
- n = number of times interest is compounded per year (4, for quarterly)
- t = number of years (10)
We can now plug the values into the formula:
PV = 3,000 / (1 + 0.045/4)4*10
PV = 3,000 / (1 + 0.01125)40
PV = 3,000 / (1.01125)40
Calculating the right-hand side with a calculator:
PV = $2,084.03 approximately
Therefore, the present value needed to grow to $3,000 in 10 years at an annual interest rate of 4.5% compounded quarterly is approximately $2,084.03.