Question

Precalculus Compound Interest/Annuities Name: 1. Abby is saving for a vacation to Europe. She has budgeted $40 a month to put towards the trip. She researched travel costs and found that the trip will cost her $9000 if she wants to stay for two weeks. Abby found an annuity that would give her 7.5% on her investment compounded daily to help her achieve her goal. If Abby starts right now, how many years until she can afford her vacation?​

Answer 1

11.7 years

Explanation:

You want the number of years it takes for an annuity of $40 per month earning 7.5% interest compounded daily to achieve a value of $9000.

Annuity

The formula for the future value of an ordinary annuity is ...

FV = P((1 +r)^n -1)/(r -1)

where P is the periodic payment, r is the periodic interest rate, and n is the number of periods.

This problem is a bit tricky in that the payment period is 1 month, but the compounding of interest is done daily. That means the effective monthly interest rate (r) is ...

1 +r = (1 +0.075/365)^(365/12) ≈ 1.00626892594

This is the 12th root of the effective annual rate.

Solution

Solving for n, we get ...

9000 = 40(1.0063^n -1)/0.0063

0.0063·225 = 1.0063^n -1

log(0.0063·225 +1)/log(1.0063) = n ≈ 140.7885 . . . . . . months

We want the time in years, so we need to divide by 12:

years = 140.7885/12 ≈ 11.7324 ≈ 11.7

It will be about 11.7 years until Abby can afford her vacation.