A church congregation has raised ​$37 comma 625 for future outreach work. If the money is invested in a fund paying​ 7% compounded​ quarterly, what annual payment can be made for ten years from the fund if the first payment is to be made four years from the date of investment in the​ fund?

Question

Part 1
The size of the payment is ​$

​(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as​ needed.)

Answer 1

To calculate the size of the annual payment that can be made for ten years from the fund, we can use the formula for the future value of an annuity:

FV = Pmt * ((1 - (1 + r)^-n) / r)

Where:
FV = Future value of the annuity ($37,625)
Pmt = Annual payment
r = Quarterly interest rate = 0.07/4 = 0.0175
n = Number of quarters from the first payment to the end of ten years = 10 * 4 = 40

Substitute the values into the formula:

37625 = Pmt * ((1 - (1 + 0.0175)^-40) / 0.0175)

Solve for Pmt:

37625 = Pmt * ((1 - (1.0175)^-40) / 0.0175)
37625 = Pmt * ((1 - 0.572597) / 0.0175)
37625 = Pmt * (0.427403 / 0.0175)
37625 = Pmt * 24.457
Pmt = 37625 / 24.457
Pmt ≈ $1538.47

Therefore, the annual payment that can be made for ten years from the fund is approximately $1538.47.