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?
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%.
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?
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.
To find the interest rate at which the gift grew, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
where:
A is the final amount ($4,000,000),
P is the principal amount ($9,000),
r is the interest rate (in decimal form),
n is the number of times interest is compounded per year, and
t is the time in years (200).
Since the interest is compounded annually, we can assume n = 1.
Substituting the given values into the formula, we get:
4,000,000 = 9,000(1 + r/1)^(1 * 200)
Simplifying the equation:
(1 + r)^(200) = 4,000,000 / 9,000
(1 + r)^(200) = 444.44...
To solve for r, we need to take the 200th root of both sides of the equation:
(1 + r) = (444.44...)^1/200
Now we can find (444.44...)^1/200 using a calculator. The value of (444.44...)^1/200 is approximately 1.0326.
Subtracting 1 from both sides of the equation:
r = 1.0326 - 1
r = 0.0326
Finally, multiplying the result by 100 to convert it to a percentage:
r = 0.0326 * 100
r ≈ 3.26%
Therefore, the growth in the gift would occur at an interest rate of approximately 3.26% compounded annually.