Question

A gift of ​$9000 to a city grew to ​$​4,000,000 in 200 years. At what interest rate compounded annually would this growth​ occur?

Answer 1

Bank One offered a 19​-year certificate of deposit​ (CD) at ​4.59% interest compounded quarterly. On the same day on the​ Internet, First Bank offered a 19​-year CD at ​4.58% compounded monthly. Find the APY for each CD. Which bank paid a higher​ APY?

Answer 2

We can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where:
A = final amount
P = initial amount
r = interest rate (as a decimal)
n = number of times compounded per year
t = time in years

We know that P = $9000, A = $4,000,000, n = 1 (compounded annually), and t = 200 years. So we can solve for r:

$4,000,000 = $9000(1 + r/1)^(1*200)
(1 + r)^200 = $4,000,000/$9000
(1 + r)^200 = 444.44
log(1 + r)^200 = log(444.44)
200 log(1 + r) = log(444.44)
log(1 + r) = log(444.44)/200
1 + r = 1.028
r = 0.028, or 2.8%

Therefore, the interest rate compounded annually would be 2.8%.

Answer 3

The APY (annual percentage yield) takes into account the effect of compounding on the interest rate. To find the APY for each CD, we can use the formula:

APY = (1 + r/n)^n - 1

where:
r = annual interest rate (as a decimal)
n = number of times compounded per year

For Bank One's CD:
r = 0.0459
n = 4 (compounded quarterly)

APY = (1 + 0.0459/4)^4 - 1 = 0.0471, or 4.71%

For First Bank's CD:
r = 0.0458
n = 12 (compounded monthly)

APY = (1 + 0.0458/12)^12 - 1 = 0.0471, or 4.71%

Both CDs have the same APY of 4.71%, so the interest rates are equivalent in terms of their effect on the investment.