You want to buy a \( \$ 32,000 \) car. You can make a \( 10 \% \) down payment, and will finance the balance with a \( 2 \% \) interest rate for 36 months ( 3 years). What will your monthly payments be?

To calculate the monthly payments for financing a car, we can use the formula for monthly payments on a loan: \[ \text{Monthly Payment} = \frac{\text{Loan Amount} \times \text{Interest Rate} \times (1 + \text{Interest Rate})^{Number of Payments}}{(1 + \text{Interest Rate})^{Number of Payments} - 1} \] Given: - Loan Amount = $32,000 - Interest Rate = 2% = 0.02 - Number of Payments = 36 months Substitute the values into the formula to calculate the monthly payment. Calculate the value by following steps: - step0: Calculate: \(\frac{32000\times 0.02\left(1+0.02\right)^{36}}{\left(\left(1+0.02\right)^{36}-1\right)}\) - step1: Remove the parentheses: \(\frac{32000\times 0.02\left(1+0.02\right)^{36}}{\left(1+0.02\right)^{36}-1}\) - step2: Add the numbers: \(\frac{32000\times 0.02\times 1.02^{36}}{\left(1+0.02\right)^{36}-1}\) - step3: Add the numbers: \(\frac{32000\times 0.02\times 1.02^{36}}{1.02^{36}-1}\) - step4: Convert the expressions: \(\frac{32000\times 0.02\left(\frac{51}{50}\right)^{36}}{1.02^{36}-1}\) - step5: Convert the expressions: \(\frac{32000\times 0.02\left(\frac{51}{50}\right)^{36}}{\left(\frac{51}{50}\right)^{36}-1}\) - step6: Multiply: \(\frac{640\times \frac{51^{36}}{50^{36}}}{\left(\frac{51}{50}\right)^{36}-1}\) - step7: Subtract the numbers: \(\frac{640\times \frac{51^{36}}{50^{36}}}{\frac{51^{36}-50^{36}}{50^{36}}}\) - step8: Evaluate the power: \(\frac{\frac{51^{36}}{2^{29}\times 5^{71}}}{\frac{51^{36}-50^{36}}{50^{36}}}\) - step9: Multiply by the reciprocal: \(\frac{51^{36}}{2^{29}\times 5^{71}}\times \frac{50^{36}}{51^{36}-50^{36}}\) - step10: Rewrite the expression: \(\frac{51^{36}}{2^{29}\times 5^{71}}\times \frac{2^{36}\times 25^{36}}{51^{36}-50^{36}}\) - step11: Reduce the numbers: \(\frac{51^{36}}{5^{71}}\times \frac{2^{7}\times 25^{36}}{51^{36}-50^{36}}\) - step12: Rewrite the expression: \(\frac{51^{36}}{5^{71}}\times \frac{2^{7}\times 5^{72}}{51^{36}-50^{36}}\) - step13: Reduce the numbers: \(51^{36}\times \frac{2^{7}\times 5}{51^{36}-50^{36}}\) - step14: Multiply: \(\frac{51^{36}\times 640}{51^{36}-50^{36}}\) The monthly payment for financing the car will be approximately $1,255.45.