{"id":4029,"date":"2023-06-05T17:57:26","date_gmt":"2023-06-05T17:57:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=4029"},"modified":"2026-03-23T17:49:54","modified_gmt":"2026-03-23T17:49:54","slug":"cars-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/cars-learn-it-3\/","title":{"raw":"Cars: Learn It 3","rendered":"Cars: Learn It 3"},"content":{"raw":"

The Basics of Leasing a Car<\/h2>\r\n

Leasing a car is an alternative to purchasing a car. Leasing offers the convenience of driving a new car without the long-term commitment and financial responsibility of owning it. It is similar to renting an apartment, where you pay for the time you use it. It is still a loan, and acts like one in many respects.\u00a0<\/p>\r\n

When the lease is over, the car is returned to the dealer. At that time there may be fees that have to be paid, such as for damage to the car or for extra miles driven over the limit. You may also have the choice to purchase the vehicle at a predetermined price, known as the residual value.\u00a0<\/p>\r\n

\r\n
\r\n

leasing a car<\/h3>\r\n

Leasing a car is a process where you essentially rent a vehicle for a specified period of time, typically a few years. Instead of buying the car outright, you make monthly lease payments to the leasing company or dealership. During the lease term, you have the right to use the car, but you do not own it.<\/p>\r\n<\/div>\r\n<\/section>\r\n

There are two components to lease costs. One is the monthly payment for the lease. The other is the fees for leasing, These often are paid before the lease is complete. These include:<\/p>\r\n

    \r\n\t
  • a down payment<\/strong>, which is your initial payment that is applied to the price of the car. It reduces the amount you finance, much the same as when you purchase a car. It is recommended that this be negotiated away.<\/li>\r\n\t
  • the acquisition fee<\/strong>, sometimes called the bank fee. This is the money charge for the company to set up the lease. It is essentially a paperwork fee. It is not likely that this can be negotiated.<\/li>\r\n\t
  • a security deposit<\/strong>, which might be required. It is about the same as [latex]1[\/latex] month\u2019s payment for the lease. The deposit is returned to you if the car is in good shape at the end. This can be negotiated away.<\/li>\r\n\t
  • disposition fees<\/strong>, which cover the cost the company will incur when they take your car back and are typically between [latex]$200[\/latex] and [latex]$450[\/latex].<\/li>\r\n\t
  • the title, registration, and license fees<\/strong>, just as with the purchase of a car.<\/li>\r\n\t
  • sales tax,<\/strong> which will likely be applied. The sales tax only covers the depreciated portion of the car (more on depreciation later) in many states. Since this depends on the state in which the car is leased, you should determine the sales tax rules for where you lease the car.<\/li>\r\n<\/ul>\r\n

    As you can imagine, this can come to a fairly high dollar amount.<\/p>\r\n

    Donna wants to lease a Subaru Outback in Eden, New York. Find the total cost of obtaining her lease if there is no down payment, [latex]$175.00[\/latex] in acquisition fees, a security deposit of [latex]$300.00[\/latex], [latex]$350.00[\/latex] in disposition fees, [latex]$102.50[\/latex] in title and registration fees, and sales tax of [latex]$3,536.05[\/latex].
    \r\n[reveal-answer q=\"160930\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"160930\"]
    \r\nAdding these values together we find the total cost is [latex]$4,463.55[\/latex].
    \r\n[\/hidden-answer]<\/section>\r\n

    You have some obligations when you lease a car. You must keep the car in good condition, cleaned, maintained, and free of anything more than minor damage. If the car is in poor condition when the car is returned, you will be responsible for the cost to bring the car to an acceptable condition. Leases often come with mileage restrictions, which limit the number of miles you can drive each year without incurring additional fees. You are also expected to keep the mileage under its limit. If you go over, you will pay [latex]10[\/latex] to [latex]25[\/latex] cents per mile over.<\/p>\r\n

    Lease Payments<\/h3>\r\n

    Lease payments are similar to regular loan payments, but have some other details. Calculating a lease payment involves knowing the following values:<\/p>\r\n

      \r\n\t
    • The price of the car<\/strong>. This is the cost you would pay for the car after applying all discounts, incentives, and negotiations.<\/li>\r\n\t
    • Residual Value<\/strong>. This is the manufacturer's estimate of the car's value after a set period of time. The residual value is expressed as a percentage of the manufacturer\u2019s suggested retail price (MSRP).<\/li>\r\n\t
    • Months<\/strong>. This is the length of the lease. Most leases are either [latex]24[\/latex]- or [latex]36[\/latex]-month leases, but other terms are available.<\/li>\r\n\t
    • Monthly Depreciation<\/strong>. The monthly depreciation is the difference between the price of the car and the residual value, divided by the number of months of the lease, and represents the monthly loss of value of the car while it\u2019s being used.<\/li>\r\n\t
    • Money Factor (MF)<\/strong>. This is the interest rate, but expressed in a different way for a lease. Converting from the money factor to the annual percentage rate (APR) is done by multiplying the MF by [latex]2400[\/latex]. Naturally, converting an APR to a MF is done by dividing the APR by [latex]2400[\/latex].<\/li>\r\n<\/ul>\r\n
      \r\n
      \r\n

      monthly depreciation and money factor formulas<\/h3>\r\n

      The monthly depreciation for a car, [latex]MD[\/latex], is,<\/p>\r\n

      [latex]MD=\\frac{P\u2212R}{n}[\/latex],<\/center>\r\n

       <\/p>\r\n

      where [latex]P[\/latex] is the price paid for the car, [latex]R[\/latex] is the residual value of the car, and [latex]n[\/latex] is the number of months of the lease.
      \r\n\u00a0
      \r\nThe annual percentage rate for a lease is [latex]APR=2400\u00d7MF[\/latex], where [latex]MF[\/latex] is the money factor of the lease. The [latex]MF[\/latex] for a lease is [latex]MF=\\frac{APR}{2,400}[\/latex].
      \r\n
      \r\nNote:\u00a0 When converting between APR and money factor, the APR should remain as a percentage (not converted to decimal form) in these calculations.<\/em><\/p>\r\n<\/div>\r\n<\/section>\r\n

      \r\n
        \r\n\t
      1. The purchase price of a car is [latex]$25,000[\/latex]. Its residual price is [latex]$14,500[\/latex]. What is its monthly depreciation for a [latex]36[\/latex]-month lease?<\/li>\r\n\t
      2. The purchase price of a car is [latex]$30,000[\/latex]. Its residual price is [latex]$18,600[\/latex]. What is its monthly depreciation for a [latex]24[\/latex]-month lease?<\/li>\r\n<\/ol>\r\n

        [reveal-answer q=\"160931\"]Show Solution[\/reveal-answer]
        \r\n[hidden-answer a=\"160931\"]<\/p>\r\n

          \r\n\t
        1. The monthly depreciation formula is [latex]MD=\\frac{P\u2212R}{n}[\/latex], Substituting [latex]$25,000[\/latex] for [latex]P[\/latex], [latex]$14,500[\/latex] for [latex]R[\/latex], and [latex]36[\/latex] for [latex]n[\/latex], we find [latex]MD[\/latex] to be
          [latex]MD=\\frac{P\u2212R}{n}=\\frac{$25,000\u2212$14,500}{36}=\\frac{$10,500}{36}=$291.67[\/latex]
          \r\n
          \r\n<\/center><\/li>\r\n\t
        2. The monthly depreciation formula is [latex]MD=\\frac{P\u2212R}{n}[\/latex], Substituting [latex]$30,000[\/latex] for [latex]P[\/latex], [latex]$18,600[\/latex] for [latex]R[\/latex], and [latex]24[\/latex] for [latex]n[\/latex], we find [latex]MD[\/latex] to be
          [latex]MD=\\frac{P\u2212R}{n}=\\frac{$30,000\u2212$18,600}{24}=\\frac{$11,400}{24}=$475.00[\/latex]<\/center><\/li>\r\n<\/ol>\r\n

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

          \r\n
            \r\n\t
          1. Find the annual percentage rate if the money factor is [latex]0.00001875[\/latex].<\/li>\r\n\t
          2. Find the money factor if the APR is [latex]6.25\\%[\/latex].<\/li>\r\n<\/ol>\r\n

            [reveal-answer q=\"160932\"]Show Solution[\/reveal-answer]
            \r\n[hidden-answer a=\"160932\"]<\/p>\r\n

              \r\n\t
            1. The APR is the money factor times [latex]2400[\/latex], so [latex]APR=2400\u00d7MF=2400\u00d70.00001875=0.045[\/latex]. The APR is [latex]4.5\\%[\/latex].<\/li>\r\n\t
            2. The [latex]MF[\/latex] is the APR divided by [latex]2400[\/latex], so [latex]MF=\\frac{APR}{2400}=\\frac{0.0625}{2400}=0.000026041\\overline{6}[\/latex].<\/li>\r\n<\/ol>\r\n

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

              Once the monthly depreciation and money factor values are found, the payment for the lease can be calculated.<\/p>\r\n

              \r\n
              \r\n

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

              The payment, [latex]PMT[\/latex], for a lease is,<\/p>\r\n

              [latex]PMT=\\frac{(P\u2212R)}{n}+(P+R)\u00d7MF[\/latex],<\/center>\r\n

               <\/p>\r\n

              where [latex]P[\/latex] is the price paid for the car, [latex]R[\/latex] is the residual value of the car, [latex]n[\/latex] is the number of months of the lease, and [latex]MF[\/latex] is the money factor for the lease.<\/p>\r\n<\/div>\r\n<\/section>\r\n

              Calculate the lease payments for car with the following price, residual price, length of lease, and money factor or APR.\r\n\r\n
                \r\n\t
              1. Price is [latex]$28,344[\/latex], residual price is [latex]$18,140.16[\/latex], [latex]24[\/latex]-month lease, money factor is [latex]0.000025[\/latex].<\/li>\r\n\t
              2. Price is [latex]$22,500[\/latex], residual price is [latex]$13,050[\/latex], [latex]36[\/latex]-month lease, APR is [latex]7.5\\%[\/latex].<\/li>\r\n<\/ol>\r\n

                [reveal-answer q=\"160933\"]Show Solution[\/reveal-answer]
                \r\n[hidden-answer a=\"160933\"]<\/p>\r\n

                  \r\n\t
                1. Substituting the values [latex]P = $28,344[\/latex], [latex]R = $18,140.16[\/latex], [latex]n = 24[\/latex] and [latex]MF = 0.000025[\/latex] into the formula and calculating, the monthly lease payment is
                  [latex]\\begin{array}{ll}
                  \r\nPMT &&=\\frac{(P\u2212R)}{n}+(P+R)\u00d7MF \\\\
                  \r\n&& =\\frac{($28,344\u2212$18,140.16)}{24}+($28,344+$18,140.16)\u00d70.000025 \\\\
                  \r\n&& =\\frac{($10,203.84)}{24}+($46,484.16)\u00d70.000025 \\\\
                  \r\n&& = \\$426.33 \\\\
                  \r\n\\end{array}[\/latex]
                  \r\n
                  \r\n<\/center><\/li>\r\n\t
                2. Given the APR, we find the [latex]MF[\/latex] which is [latex]MF=\\frac{APR}{2400}=\\frac{7.5}{2400}=0.003125[\/latex]. Substituting the values [latex]P = $22,500[\/latex], [latex]R = $13,050[\/latex], [latex]n = 36[\/latex] and the [latex]MF[\/latex] into the formula and calculating, the monthly lease payment is
                  [latex]\\begin{array}{ll}
                  \r\nPMT &&=\\frac{(P\u2212R)}{n}+(P+R)\u00d7MF \\\\
                  \r\n&& =\\frac{($22,500\u2212$13,050)}{36}+($22,500+$13,050)\u00d70.003125 \\\\
                  \r\n&& =\\frac{($9,450)}{36}+($35,550)\u00d70.003125 \\\\
                  \r\n&& = \\$373.59 \\\\
                  \r\n\\end{array}[\/latex]<\/center><\/li>\r\n<\/ol>\r\n

                  [\/hidden-answer]<\/p>\r\n<\/section>","rendered":"

                  The Basics of Leasing a Car<\/h2>\n

                  Leasing a car is an alternative to purchasing a car. Leasing offers the convenience of driving a new car without the long-term commitment and financial responsibility of owning it. It is similar to renting an apartment, where you pay for the time you use it. It is still a loan, and acts like one in many respects.\u00a0<\/p>\n

                  When the lease is over, the car is returned to the dealer. At that time there may be fees that have to be paid, such as for damage to the car or for extra miles driven over the limit. You may also have the choice to purchase the vehicle at a predetermined price, known as the residual value.\u00a0<\/p>\n

                  \n
                  \n

                  leasing a car<\/h3>\n

                  Leasing a car is a process where you essentially rent a vehicle for a specified period of time, typically a few years. Instead of buying the car outright, you make monthly lease payments to the leasing company or dealership. During the lease term, you have the right to use the car, but you do not own it.<\/p>\n<\/div>\n<\/section>\n

                  There are two components to lease costs. One is the monthly payment for the lease. The other is the fees for leasing, These often are paid before the lease is complete. These include:<\/p>\n

                    \n
                  • a down payment<\/strong>, which is your initial payment that is applied to the price of the car. It reduces the amount you finance, much the same as when you purchase a car. It is recommended that this be negotiated away.<\/li>\n
                  • the acquisition fee<\/strong>, sometimes called the bank fee. This is the money charge for the company to set up the lease. It is essentially a paperwork fee. It is not likely that this can be negotiated.<\/li>\n
                  • a security deposit<\/strong>, which might be required. It is about the same as [latex]1[\/latex] month\u2019s payment for the lease. The deposit is returned to you if the car is in good shape at the end. This can be negotiated away.<\/li>\n
                  • disposition fees<\/strong>, which cover the cost the company will incur when they take your car back and are typically between [latex]$200[\/latex] and [latex]$450[\/latex].<\/li>\n
                  • the title, registration, and license fees<\/strong>, just as with the purchase of a car.<\/li>\n
                  • sales tax,<\/strong> which will likely be applied. The sales tax only covers the depreciated portion of the car (more on depreciation later) in many states. Since this depends on the state in which the car is leased, you should determine the sales tax rules for where you lease the car.<\/li>\n<\/ul>\n

                    As you can imagine, this can come to a fairly high dollar amount.<\/p>\n

                    Donna wants to lease a Subaru Outback in Eden, New York. Find the total cost of obtaining her lease if there is no down payment, [latex]$175.00[\/latex] in acquisition fees, a security deposit of [latex]$300.00[\/latex], [latex]$350.00[\/latex] in disposition fees, [latex]$102.50[\/latex] in title and registration fees, and sales tax of [latex]$3,536.05[\/latex].<\/p>\n