To find the principal amount (P) that grows to [tex]$40 over 5 years with an annual interest rate of \(4.5\%\) compounded annually, we can use the compound interest formula. Here's the detailed step-by-step solution:
1. Identify the given variables:
- \(A\): The future value or amount after interest, which is $[/tex]40.
- [tex]\(t\)[/tex]: The time in years, which is 5 years.
- [tex]\(r\)[/tex]: The annual interest rate in decimal form.
The interest rate given is [tex]\(4.5\%\)[/tex]. We need to convert this percentage to a decimal by dividing by 100:
[tex]\[
r = \frac{4.5}{100} = 0.045
\][/tex]
2. Recall the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
where:
- [tex]\(A\)[/tex] is the amount of money accumulated after [tex]\(n\)[/tex] years, including interest.
- [tex]\(P\)[/tex] is the principal amount (the initial amount of money).
- [tex]\(r\)[/tex] is the annual interest rate (decimal).
- [tex]\(n\)[/tex] is the number of times the interest is compounded per year.
- [tex]\(t\)[/tex] is the time the money is invested for in years.
Since the interest is compounded annually, [tex]\(n\)[/tex] is 1.
3. Simplify the formula for [tex]\(n = 1\)[/tex]:
[tex]\[
A = P (1 + r)^t
\][/tex]
4. Rearrange the formula to solve for the principal [tex]\(P\)[/tex]:
[tex]\[
P = \frac{A}{(1 + r)^t}
\][/tex]
5. Plug in the known values:
- [tex]\(A = 40\)[/tex]
- [tex]\(r = 0.045\)[/tex]
- [tex]\(t = 5\)[/tex]
So the principal [tex]\(P\)[/tex] can be calculated as:
[tex]\[
P = \frac{40}{(1 + 0.045)^5}
\][/tex]
6. Calculate [tex]\( (1 + 0.045)^5 \)[/tex]:
[tex]\[
(1 + 0.045)^5 \approx 1.246182
\][/tex]
7. Divide the future value [tex]\(A\)[/tex] by this calculated value:
[tex]\[
P \approx \frac{40}{1.246182} \approx 32.10
\][/tex]
Therefore, the principal amount [tex]\(P\)[/tex] that amounts to [tex]$40 after 5 years at an annual interest rate of \(4.5\%\) compounded annually is approximately \( \$[/tex]32.10 \).
