21. Ogden wants to save for a trip to Australia. She will need $12,000 at the end of four years. She can invest a certain amount at the beginning of each of the next four years in a bank account that will pay her 6.8 percent annually. How much will she have to invest annually to reach her target?
To determine how much Ogden needs to invest annually to reach her target of $12,000 in four years at 6.8% interest, we'll need to use the concept of present value and annuities.
The formula to calculate the present value of an annuity is as follows:
PV = P * [(1 - (1+r)^(-n)) / r]
Where:
PV = Present value (target goal)
P = Payment made annually
r = Interest rate per period
n = Number of periods
In this case, Ogden wants to calculate how much she needs to invest annually, so we'll rearrange the formula to solve for P:
P = PV * (r / (1 - (1+r)^(-n)))
Now, let's plug in the values we have:
PV = $12,000
r = 6.8% or 0.068 (as a decimal)
n = 4
Using the formula, we can calculate the payment Ogden needs to make annually:
P = $12,000 * (0.068 / (1 - (1+0.068)^(-4)))
Calculating this using a calculator or a spreadsheet, Ogden will need to invest approximately $2,677.69 annually to reach her target of $12,000 in four years at a 6.8% annual interest rate.