To determine Katelyn’s monthly payment for a \[tex]$10,000 loan at an interest rate of 5.6% over 5 years, we need to use the monthly payment formula for an installment loan.
Given:
- Principal amount (\(P\)) = \$[/tex]10,000
- Annual interest rate ([tex]\(r\)[/tex]) = 5.6%
- Loan term ([tex]\(t\)[/tex]) = 5 years
First, convert the annual interest rate to a monthly interest rate by dividing it by 12:
[tex]\[ r_{\text{monthly}} = \frac{5.6\%}{12} = \frac{0.056}{12} = 0.0046667 \][/tex]
Next, calculate the total number of monthly payments by multiplying the number of years by 12:
[tex]\[ n = t \times 12 = 5 \times 12 = 60 \][/tex]
Using the formula for the monthly payment ([tex]\(M\)[/tex]):
[tex]\[ M = \frac{P \cdot r_{\text{monthly}} \cdot (1 + r_{\text{monthly}})^n}{(1 + r_{\text{monthly}})^n - 1} \][/tex]
Plugging in the values:
[tex]\[ M = \frac{10000 \cdot 0.0046667 \cdot (1 + 0.0046667)^{60}}{(1 + 0.0046667)^{60} - 1} \][/tex]
This gives us the monthly payment:
[tex]\[ M \approx 191.47353285954674 \][/tex]
Rounded to the nearest cent, the monthly payment is:
[tex]\[ M \approx \$191.47 \][/tex]
Therefore, Katelyn's monthly payment for the loan is \$191.47.
