Car Loan Payment D30718

1. **State the problem:** We need to find the monthly payment for a car loan of $54,276.47 over 5 years with an annual interest rate of 4.5% compounded monthly. 2. **Formula used:** The monthly payment $M$ for a loan amount $P$ with monthly interest rate $r$ over $n$ months is given by the amortization formula: $$ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} $$ 3. **Identify variables:** - Principal $P = 54276.47$ - Annual interest rate = 4.5% = 0.045 - Monthly interest rate $r = \frac{0.045}{12} = 0.00375$ - Loan term = 5 years = $5 \times 12 = 60$ months 4. **Calculate $(1+r)^n$:** $$ (1 + 0.00375)^{60} = 1.00375^{60} $$ Using a calculator: $$ 1.00375^{60} \approx 1.2434 $$ 5. **Calculate numerator:** $$ r(1+r)^n = 0.00375 \times 1.2434 = 0.00466275 $$ 6. **Calculate denominator:** $$ (1+r)^n - 1 = 1.2434 - 1 = 0.2434 $$ 7. **Calculate monthly payment $M$:** $$ M = 54276.47 \times \frac{0.00466275}{0.2434} $$ Simplify the fraction: $$ \frac{0.00466275}{0.2434} \approx 0.01916 $$ So, $$ M = 54276.47 \times 0.01916 \approx 1039.88 $$ **Final answer:** The monthly car payment is approximately **$1039.88**.