A financial institution offers a "double-your-money" savings account in which you will have $2 in 9 years for every dollar you invest today. What annual interest rate does this account offer?
Please specify your answer in decimal terms and round your answer to the nearest thousandth (e.g., enter 12.3 percent as 0.123).
In order to calculate the annual interest rate offered by the account, we can use the formula for compound interest:
Future Value = Present Value * (1 + Interest Rate)^Number of Years
Given that the account doubles the money invested in 9 years, we can set up the equation as follows:
2 = 1 * (1 + Interest Rate)^9
To find the interest rate, we need to isolate it on one side of the equation. We can do this by taking the 9th root of both sides:
(1 + Interest Rate) = 2^(1/9)
Next, we subtract 1 from both sides:
Interest Rate = 2^(1/9) - 1
Using a calculator, we find that 2^(1/9) is approximately 1.082576175, and subtracting 1 gives us approximately 0.082576175.
Therefore, the annual interest rate offered by the account is approximately 0.083.
To determine the annual interest rate offered by the savings account, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = Total amount after t years
P = Principal amount (initial investment)
r = Annual interest rate (decimal)
n = Number of times that interest is compounded per year
t = Number of years
In this case, we want to find the interest rate (r) that will double the initial investment after 9 years. So we can rewrite the formula as:
2 = 1(1 + r/n)^(n*9)
Let's solve for r by trial and error. Start with an initial guess, and we'll refine it until we find an interest rate that results in a doubling of the initial investment.
Let's start with an initial guess of 0.05 (5%):
2 = 1(1 + 0.05/1)^(1*9)
Using a calculator to perform the calculations:
2 = 1(1.05)^9
2 = 1.538624078
As the value is less than 2, we need to increase the interest rate.
Let's try with an interest rate of 0.06 (6%):
2 = 1(1 + 0.06/1)^(1*9)
Calculating:
2 = 1(1.06)^9
2 = 1.601030461
As the value is greater than 2, we can conclude that the annual interest rate offered by the savings account is approximately 0.06 or 6% (rounded to the nearest thousandth).