Hunan bought a car priced at $\$ 15,600$ for $15 \%$ down and equal monthly payments for five years. If interest is $8 \%$ compounded monthly, what is the size of the monthly payment?
The monthly payment is $\$ \square$.
(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)
Answer
Rounding to the nearest cent, the monthly payment is \(\boxed{\$268.86}\).
Step 1 ：Hunan bought a car priced at \$15,600. The down payment is 15\% of the total price, which is \$15,600 \times 0.15 = \$2,340.
Step 2 ：The amount that needs to be financed is the total price minus the down payment, which is \$15,600 - \$2,340 = \$13,260. This is the principal for our loan.
Step 3 ：The annual interest rate is 8\%, but it is compounded monthly, so the monthly interest rate is 0.08 ÷ 12 = 0.006666666666666667.
Step 4 ：The loan is for 5 years, so the total number of payments is 5 \times 12 = 60.
Step 5 ：We can then use the formula for the monthly payment on a loan, which is: P = [r*PV] / [1 - (1 + r)^-n], where P is the monthly payment, r is the monthly interest rate, PV is the present value or principal of the loan, and n is the number of payments.
Step 6 ：Substituting the values into the formula, we get: P = [0.006666666666666667 * 13260] / [1 - (1 + 0.006666666666666667)^-60] = 268.8649882643677.
Step 7 ：Rounding to the nearest cent, the monthly payment is \(\boxed{\$268.86}\).
