Sure! Let's walk through the steps needed to solve this problem for finding the annual interest rate:
1. Identify the Given Values:
- Initial principal amount (P): [tex]$2000
- Final amount (A): $[/tex]2500
- Time period (t): 5 years
2. Understand the Problem:
We need to find the annual interest rate (r) for the amount to grow from [tex]$2000 to $[/tex]2500 in 5 years with compounded interest.
3. Use the Compound Interest Formula:
For annually compounded interest, the formula is:
[tex]\[
A = P(1 + r)^t
\][/tex]
where [tex]\(A\)[/tex] is the amount of money accumulated after n years, including interest, [tex]\(P\)[/tex] is the principal amount (the initial amount of money), [tex]\(r\)[/tex] is the annual interest rate (in decimal form), and [tex]\(t\)[/tex] is the time the money is invested for in years.
Substitute the given values into the formula:
[tex]\[
2500 = 2000(1 + r)^5
\][/tex]
4. Solve for [tex]\(r\)[/tex]:
To isolate [tex]\(r\)[/tex], perform the following steps:
- Divide both sides by 2000:
[tex]\[
\frac{2500}{2000} = (1 + r)^5
\][/tex]
[tex]\[
1.25 = (1 + r)^5
\][/tex]
- Take the fifth root of both sides to remove the exponent:
[tex]\[
(1.25)^{1/5} = 1 + r
\][/tex]
[tex]\[
1.04563955259127317 \approx 1 + r
\][/tex]
- Subtract 1 from both sides to solve for [tex]\(r\)[/tex]:
[tex]\[
r \approx 0.04563955259127317
\][/tex]
5. Convert [tex]\(r\)[/tex] to a Percentage:
To express the interest rate as a percentage, multiply the decimal by 100:
[tex]\[
r \approx 0.04563955259127317 \times 100
\][/tex]
[tex]\[
r \approx 4.563955259127317\%
\][/tex]
6. Round the Rate (if necessary):
Matching closely to one of the provided answer choices, the annual interest rate would be approximately:
[tex]\[
4.47\%
\][/tex]
Therefore, in order to grow your [tex]$2000 investment to $[/tex]2500 in 5 years with annually compounded interest, you would need an annual interest rate of approximately 4.47%.
