{"id":3991,"date":"2023-06-05T16:40:27","date_gmt":"2023-06-05T16:40:27","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=3991"},"modified":"2025-10-31T14:03:36","modified_gmt":"2025-10-31T14:03:36","slug":"cars-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/cars-learn-it-2\/","title":{"raw":"Cars: Learn It 2","rendered":"Cars: Learn It 2"},"content":{"raw":"

Car Loans<\/h2>\r\n

Whether or not you buy a new car or a used car, if you finance the purchase, you are taking out a loan. A car loan is a type of installment loan<\/strong>. An installment loan has a fixed loan amount, fixed interest rate, and fixed repayment schedule. The borrower receives the entire loan amount upfront and agrees to make a series of scheduled payments, typically on a monthly basis, until the loan is fully repaid.<\/p>\r\n

\r\n
\r\n

installment loan<\/h3>\r\n

An installment loan<\/strong> is a type of loan that is repaid in regular, predetermined payments over a specific period of time.<\/p>\r\n<\/div>\r\n<\/section>\r\n

These loan payments work exactly the same way as other loans do as far as payments are concerned. Each payment made towards an installment loan consists of both principal (the original loan amount) and interest (the cost of borrowing).<\/p>\r\n

When purchasing a car, the total cost to obtain the car is not the only factor in your monthly price. You will also pay an interest rate for the loan you obtain. The interest rate you will get is dependent on your credit score. The interest rates available for used cars are frequently higher than those for new cars.<\/p>\r\n

\r\n

Different lenders will offer different interest rates. When purchasing a car at a dealership, the dealership will likely offer to finance your car loan. Frequently, dealerships offer special financing with very low rates. Even if the dealership offers financing, check with your bank or credit union to determine the interest rates they are offering. To reduce your payments, choose the lowest rate you can find.<\/p>\r\n<\/section>\r\n

\r\n
\r\n

loan payment formula<\/h3>\r\n

The payment, [latex]PMT[\/latex], per period to pay off a loan with with a beginning principal (loan amount) [latex]P[\/latex] is<\/p>\r\n

 <\/p>\r\n

[latex]PMT = \\dfrac{P(\\frac{r}{n})}{1-(1+\\frac{r}{n})^{-nt}}[\/latex]<\/center>\r\n

 <\/p>\r\n

where [latex]r[\/latex] is the annual interest rate in decimal form, [latex]t[\/latex] is the term in years of the loan, and [latex]n[\/latex] is the number of payments per year (typically, loans are paid monthly making [latex]n = 12[\/latex]).<\/p>\r\n

Note, payment to lenders is always rounded up to the next penny.<\/em><\/p>\r\n<\/div>\r\n<\/section>\r\n

Often, the formula takes the form [latex]PMT = \\frac{Pr(1+r)^n}{(1+r)^n -1}[\/latex], where [latex]r[\/latex] is the interest rate per period (annual rate divided by the number of periods per year), and [latex]n[\/latex] is the total number of payments to be made.It is important to note the difference between [latex]r[\/latex] in this formula and the [latex]r[\/latex] in the formula given previously.<\/section>\r\n
In the following, calculate the monthly payment using the given total to be financed, the interest rate, and the term of the car loan.\r\n\r\n
    \r\n\t
  1. Total to be financed is [latex]$31,885[\/latex], interest rate is [latex]2.9\\%[\/latex], for [latex]5[\/latex] years.<\/li>\r\n\t
  2. Total to be financed is [latex]$22,778[\/latex], interest rate is [latex]4.5\\%[\/latex], for [latex]6[\/latex] years.<\/li>\r\n<\/ol>\r\n

    [reveal-answer q=\"160930\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"160930\"]<\/p>\r\n

      \r\n\t
    1. The amount to be financed is the principal, [latex]P[\/latex], which is [latex]$31,885[\/latex]. The rate [latex]r[\/latex] is [latex]0.029[\/latex], and the term is [latex]t = 5[\/latex] years. These are monthly payments, so [latex]n = 12[\/latex]. Substituting and calculating, we find the monthly payment to be:
      [latex]\\begin{array}{ll}
      \r\nPMT && = \\frac{P(\\frac{r}{n})}{1-(1+\\frac{r}{n})^{-nt}} \\\\
      \r\n&& =\\frac{$31,885 \\times \\frac{0.029}{12}}{1-(1+\\frac{0.029}{12})^{-12 \\times 5}} \\\\
      \r\n&& =\\frac{$31,885 \\times .0024167}{1-(1+.0024167)^{-60}} \\\\
      \r\n&& =\\frac{$31,885 \\times .0024167}{1-.865172} \\\\
      \r\n&& = \\frac{77.06}{0.134828} \\\\
      \r\n&& = \\$571.54 \\\\
      \r\n\\end{array}[\/latex]<\/center><\/li>\r\n<\/ol>\r\n
        \r\n\t
      1. The amount to be financed is the principal, [latex]P[\/latex], which is [latex]$22,778[\/latex]. The rate [latex]r[\/latex] is [latex]0.045[\/latex], and the term is [latex]t = 6[\/latex] years. These are monthly payments, so [latex]n = 12[\/latex]. Substituting and calculating, we find the monthly payment to be:
        [latex]\\begin{array}{ll}
        \r\nPMT && = \\frac{P(\\frac{r}{n})}{1-(1+\\frac{r}{n})^{-nt}} \\\\
        \r\n&& =\\frac{$22,778 \\times \\frac{0.045}{12}}{1-(1+\\frac{0.045}{12})^{-12 \\times 6}} \\\\
        \r\n&& =\\frac{$22,778 \\times .00375}{1-(1+.00375)^{-72}} \\\\
        \r\n&& =\\frac{$22,778 \\times .00375}{1-.763765} \\\\
        \r\n&& = \\frac{85.4175}{0.236235} \\\\
        \r\n&& = \\$361.58 \\\\
        \r\n\\end{array}[\/latex]<\/center><\/li>\r\n<\/ol>\r\n

        [\/hidden-answer]<\/p>\r\n<\/section>\r\n

        [ohm2_question hide_question_numbers=1]7888[\/ohm2_question]<\/section>\r\n
        [ohm2_question hide_question_numbers=1]7889[\/ohm2_question]<\/section>\r\n

        To lower car payments, offering a larger down payment or trading in a vehicle can be effective, as this reduces the financed amount.<\/p>\r\n

        \r\n
        \r\n

        down payment<\/h3>\r\n

        A down payment<\/strong> is a sum of money paid upfront by a buyer as a partial payment towards the total cost of a purchase. The down payment is usually a percentage of the total purchase price and is paid at the time of the transaction or contract signing.<\/p>\r\n<\/div>\r\n<\/section>\r\n

        The purpose of a down payment is to reduce the amount of financing needed and provide assurance to the seller or lender. The larger your down payment, the smaller your subsequent monthly payments will be, which is an important factor during price negotiations.<\/p>\r\n

        Gabriela negotiates a [latex]$19,800[\/latex] price for her new car. The sales tax is [latex]9.5\\%[\/latex] in her area, and the dealership charges her [latex]$300[\/latex] in documentation fees. Her title, plates, and registration come to [latex]$321.50[\/latex]. The dealership adds to this a destination fee of [latex]$1,100[\/latex]. If she places a down payment of [latex]$5,000[\/latex] on the car, what is the total she will finance for the car?
        \r\n[reveal-answer q=\"160931\"]Show Solution[\/reveal-answer]
        \r\n[hidden-answer a=\"160931\"]
        \r\n

        The price was [latex]$19,800[\/latex]. The sales tax of [latex]9.5\\%[\/latex] is based on this number. The sales tax comes to [latex]$19,800\u00d70.095=$1,881[\/latex]. Adding all the fees to the price and the sales tax brings the total cost of the car to:<\/p>\r\n

        [latex]$19,800+$1,881+$300+$321.50+$1,100=$23,402.50[\/latex].<\/center>\r\n

         <\/p>\r\n

        Her down payment is applied to this number, so the [latex]$5,000[\/latex] is subtracted from [latex]$23,402.50[\/latex]. The subtraction yields the amount to be financed, which is [latex]$18,402.50[\/latex].[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"

        Car Loans<\/h2>\n

        Whether or not you buy a new car or a used car, if you finance the purchase, you are taking out a loan. A car loan is a type of installment loan<\/strong>. An installment loan has a fixed loan amount, fixed interest rate, and fixed repayment schedule. The borrower receives the entire loan amount upfront and agrees to make a series of scheduled payments, typically on a monthly basis, until the loan is fully repaid.<\/p>\n

        \n
        \n

        installment loan<\/h3>\n

        An installment loan<\/strong> is a type of loan that is repaid in regular, predetermined payments over a specific period of time.<\/p>\n<\/div>\n<\/section>\n

        These loan payments work exactly the same way as other loans do as far as payments are concerned. Each payment made towards an installment loan consists of both principal (the original loan amount) and interest (the cost of borrowing).<\/p>\n

        When purchasing a car, the total cost to obtain the car is not the only factor in your monthly price. You will also pay an interest rate for the loan you obtain. The interest rate you will get is dependent on your credit score. The interest rates available for used cars are frequently higher than those for new cars.<\/p>\n

        \n

        Different lenders will offer different interest rates. When purchasing a car at a dealership, the dealership will likely offer to finance your car loan. Frequently, dealerships offer special financing with very low rates. Even if the dealership offers financing, check with your bank or credit union to determine the interest rates they are offering. To reduce your payments, choose the lowest rate you can find.<\/p>\n<\/section>\n

        \n
        \n

        loan payment formula<\/h3>\n

        The payment, [latex]PMT[\/latex], per period to pay off a loan with with a beginning principal (loan amount) [latex]P[\/latex] is<\/p>\n

         <\/p>\n

        [latex]PMT = \\dfrac{P(\\frac{r}{n})}{1-(1+\\frac{r}{n})^{-nt}}[\/latex]<\/div>\n

         <\/p>\n

        where [latex]r[\/latex] is the annual interest rate in decimal form, [latex]t[\/latex] is the term in years of the loan, and [latex]n[\/latex] is the number of payments per year (typically, loans are paid monthly making [latex]n = 12[\/latex]).<\/p>\n

        Note, payment to lenders is always rounded up to the next penny.<\/em><\/p>\n<\/div>\n<\/section>\n

        Often, the formula takes the form [latex]PMT = \\frac{Pr(1+r)^n}{(1+r)^n -1}[\/latex], where [latex]r[\/latex] is the interest rate per period (annual rate divided by the number of periods per year), and [latex]n[\/latex] is the total number of payments to be made.It is important to note the difference between [latex]r[\/latex] in this formula and the [latex]r[\/latex] in the formula given previously.<\/section>\n
        In the following, calculate the monthly payment using the given total to be financed, the interest rate, and the term of the car loan.<\/p>\n
          \n
        1. Total to be financed is [latex]$31,885[\/latex], interest rate is [latex]2.9\\%[\/latex], for [latex]5[\/latex] years.<\/li>\n
        2. Total to be financed is [latex]$22,778[\/latex], interest rate is [latex]4.5\\%[\/latex], for [latex]6[\/latex] years.<\/li>\n<\/ol>\n