Answer :
To determine the equal yearly payments required to repay a loan of [tex]$8,000 borrowed at an interest rate of 5% over five years, we can use the formula for an annuity payment:
\[ A = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( A \) is the annuity payment (the amount of each yearly payment),
- \( P \) is the principal amount (loan amount),
- \( r \) is the interest rate per period,
- \( n \) is the number of periods.
Here are the steps to calculate it:
1. Identify and assign the variables based on the problem:
- \( P = 8000 \) dollars (principal/loan amount)
- \( r = 0.05 \) per year (annual interest rate)
- \( n = 5 \) years (period in years)
2. Plug these values into the annuity payment formula:
\[ A = 8000 \times \frac{0.05(1 + 0.05)^5}{(1 + 0.05)^5 - 1} \]
3. Calculate the individual terms step-by-step:
- \( (1 + r)^n \):
\[ (1 + 0.05)^5 = 1.05^5 \]
\[ 1.05^5 \approx 1.276 \]
- Calculate \( r(1 + r)^n \):
\[ 0.05 \times 1.276 \approx 0.0638 \]
- Calculate \( (1 + r)^n - 1 \):
\[ 1.276 - 1 = 0.276 \]
4. Substitute these back into the formula:
\[ A = 8000 \times \frac{0.0638}{0.276} \]
\[ A = 8000 \times 0.2312 \]
\[ A \approx 1849.60 \]
5. Round the result to the nearest whole number:
\[ A \approx 1850 \]
Therefore, the amount of each yearly payment is $[/tex]1850.
