To determine the present value of an annuity using the given values, [tex]\(m = \$50\)[/tex], [tex]\(r = 3\% = 0.03\)[/tex], [tex]\(t = 5\)[/tex] years, and [tex]\(n = 12\)[/tex], we will calculate step-by-step according to the present value formula for an annuity:
[tex]\[
P = m \left[\frac{1 - (1 + \frac{r}{n})^{-nt}}{\frac{r}{n}}\right]
\][/tex]
Here is the step-by-step solution:
1. Identify the given values:
- Monthly payment ([tex]\(m\)[/tex]): \[tex]$50
- Annual interest rate (\(r\)): 3% or 0.03
- Time in years (\(t\)): 5 years
- Number of compounding periods per year (\(n\)): 12
2. Calculate the monthly interest rate:
\[
\frac{r}{n} = \frac{0.03}{12} = 0.0025
\]
3. Calculate the total number of payments:
\[
nt = 12 \cdot 5 = 60
\]
4. Substitute these values into the formula:
\[
P = 50 \left[\frac{1 - (1 + 0.0025)^{-60}}{0.0025}\right]
\]
5. Calculate the base of the exponent:
\[
1 + \frac{r}{n} = 1 + 0.0025 = 1.0025
\]
6. Calculate the exponent part:
\[
(1 + 0.0025)^{-60}
\]
- Evaluate \(1.0025^{-60}\).
7. Calculate \(1 - \text{result from step 6}\):
- Assume the result from the exponent calculation is approximately 0.8607.
\[
1 - 0.8607 = 0.1393
\]
8. Divide this result by the monthly interest rate:
\[
\frac{0.1393}{0.0025} = 55.72
\]
9. Multiply by the monthly payment to get \(P\):
\[
P = 50 \times 55.72 = 2782.617884340208
\]
10. Round the result to the nearest cent:
\[
P \approx 2782.62
\]
Therefore, the present value of the annuity is approximately \(\$[/tex]2782.62\).
