To determine the monthly payment for a loan of $9,900 over five years with an annual interest rate of 0.10 (10%) compounded monthly, we can use the installment loan formula. Here is the step-by-step solution:
1. Identify the given values:
- Loan amount ([tex]$P$[/tex]): $9,900
- Annual interest rate ([tex]$r$[/tex]): 10%, or 0.10 as a decimal
- Loan term ([tex]$n$[/tex]): 5 years
2. Calculate the monthly interest rate:
- The annual interest rate is converted to a monthly interest rate because the interest is compounded monthly.
- Monthly interest rate ([tex]$r_{monthly}$[/tex]) = Annual interest rate / 12
[tex]\[
r_{monthly} = \frac{0.10}{12} = 0.0083333\ldots \text{ (rounded to } 0.008333)
\][/tex]
3. Calculate the total number of monthly payments:
- Number of monthly payments ([tex]$N$[/tex]): Number of years * 12
[tex]\[
N = 5 \times 12 = 60
\][/tex]
4. Use the installment loan monthly payment formula:
The formula for calculating the monthly payment ([tex]$M$[/tex]) is given by:
[tex]\[
M = P \times \frac{r_{monthly} \times (1 + r_{monthly})^N}{(1 + r_{monthly})^N - 1}
\][/tex]
Substituting the known values:
[tex]\[
M = 9900 \times \frac{0.008333 \times (1 + 0.008333)^{60}}{(1 + 0.008333)^{60} - 1}
\][/tex]
5. Calculate the monthly payment:
- Applying the formula, the exact calculation would yield the monthly payment.
- The final result is rounded to two decimal places for practical use.
The calculated monthly payment is approximately $210.35.
So, the monthly payment for a loan of [tex]$9,900 over five years at an interest rate of 0.10 compounded monthly is $[/tex]210.35.
