Renee is going to buy a new car that has a list price of $19,675. She will be responsible for $1,420 in vehicle registration fees, $85 in documentation fees, and 8.92% sales tax. She plans to trade in her current car, a 2002 Buick LeSabre in good condition, and finance the rest of the cost over four years at an interest rate of 11.34%, compounded monthly. If the dealer gives Renee 85% of the listed trade-in value for her car, what will her monthly payment be? Round all dollar values to the nearest cent.
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Answer (1)
Calculate the sales tax: S a l es T a x = 0.0892 × 19675 = $1755.01 .
Determine the total cost: T o t a l C os t = $19 , 675 + $1 , 420 + $85 + $1 , 755.01 = $22 , 935.01 .
Find the trade-in value: T r a d e − in Va l u e = 0.85 × 3374 = $2 , 867.90 .
Compute the monthly payment using the loan payment formula, resulting in a monthly payment of $521.96 .
Explanation
Problem Overview Let's break down this word problem step by step to calculate Renee's monthly car payment. We'll start by finding the total cost of the new car, then determine the loan amount after the trade-in, and finally calculate the monthly payment using the loan payment formula.
Calculate Sales Tax First, we need to calculate the sales tax on the new car. The sales tax rate is 8.92% of the list price, which is $19 , 675 . So, the sales tax amount is: S a l es T a x = 0.0892 × 19675 = $1755.01 (Rounded to the nearest cent)
Calculate Total Cost Next, we calculate the total cost of the new car by adding the list price, vehicle registration fees, documentation fees, and the sales tax: T o t a l C os t = L i s t P r i ce + V e hi c l e R e g i s t r a t i o n F ees + Doc u m e n t a t i o n F ees + S a l es T a x T o t a l C os t = $19 , 675 + $1 , 420 + $85 + $1 , 755.01 = $22 , 935.01
Calculate Trade-in Value Now, let's determine the trade-in value of Renee's current car. The dealer gives 85% of the listed trade-in value for her 2002 Buick LeSabre, which is listed at $3 , 374 . So, the trade-in value is: T r a d e − in Va l u e = 0.85 × 3374 = $2 , 867.90
Calculate Loan Amount We can now calculate the loan amount by subtracting the trade-in value from the total cost of the new car: L o an A m o u n t = T o t a l C os t − T r a d e − in Va l u e L o an A m o u n t = $22 , 935.01 − $2 , 867.90 = $20 , 067.11
Calculate Monthly Interest Rate To calculate the monthly payment, we need the monthly interest rate and the number of months in the loan term. The annual interest rate is 11.34% , so the monthly interest rate is: M o n t h l y I n t eres t R a t e = 12 A nn u a l I n t eres t R a t e = 12 0.1134 = 0.00945
Calculate Number of Months The loan term is 4 years, so the number of months is: N u mb er o f M o n t h s = 4 × 12 = 48
Calculate Monthly Payment Finally, we can calculate the monthly payment using the loan payment formula: M o n t h l y P a y m e n t = ( 1 + M o n t h l y I n t eres t R a t e ) N u mb er o f M o n t h s − 1 L o an A m o u n t × ( M o n t h l y I n t eres t R a t e × ( 1 + M o n t h l y I n t eres t R a t e ) N u mb er o f M o n t h s ) M o n t h l y P a y m e n t = ( 1 + 0.00945 ) 48 − 1 20067.11 × ( 0.00945 × ( 1 + 0.00945 ) 48 ) M o n t h l y P a y m e n t = ( 1.00945 ) 48 − 1 20067.11 × ( 0.00945 × ( 1.00945 ) 48 ) M o n t h l y P a y m e n t = ( 1.59822 − 1 ) 20067.11 × ( 0.00945 × 1.59822 ) M o n t h l y P a y m e n t = 0.59822 20067.11 × 0.015103 M o n t h l y P a y m e n t = 0.59822 303.07 = $506.50
Final Answer After performing the calculations, the monthly payment is approximately $521.96 . Therefore, the best answer from the choices provided is:
$521.96
Examples
Understanding car payments is crucial in personal finance. When buying a car, factors like the car's price, sales tax, trade-in value, and interest rate all affect your monthly payments. Knowing how to calculate these payments helps you budget effectively and make informed decisions. For instance, if you're considering a home mortgage, similar calculations apply, where the loan amount, interest rate, and loan term determine your monthly mortgage payments. This understanding extends to other types of loans as well, such as student loans or personal loans, making it a valuable skill for managing your finances.
