Math RWLO Template Title Placeholder



Logarithms and Car Payments

Project Overview

Is there a new car you are interested in? Do you know its approximate price? Do you know how much of a car payment you could afford each month? Based on the payment you can afford, is this car within your reach? In this project, students will use a formula to calculate the number of months it will take them to pay off a car loan based on the amount of the loan, the amount of the monthly car payment, and an interest rate which they will get from an internet resource.

This problem-solving based RWLO is intended for use in an Intermediate Algebra, College Algebra, or Business Math course after students have been exposed to exponential equations and logarithms.

Student Learning Objectives

For this RWLO, students will be able to:

• Utilize properties of logarithms

• Solve exponential equations using logarithms

Procedure

Time: Approximately 30 minutes.

Materials: Pencil, paper, scientific calculator, and computers with internet access.

Implementation: This RWLO is recommended as a project that is done outside of class as homework, with a suggested in-class introduction to the project.  However, this project could be done entirely as an in-class project.  Students may discuss their procedures with other students, but it is recommended that each student goes through each step and does his/her own calculations. 

Steps:

1. Review properties of logarithms with students and methods for using logarithms for solving exponential equations.

2. Print off a copy of the Car Loan Worksheet (in the content materials section).  Students will need to print off a copy of this worksheet for themselves or you may provide copies for them.

3. On the Car Loan Worksheet, students should fill in the data, show their calculations, and answer the questions.

4. One week after assigning the project, collect the Car Loan Worksheet.

Content Material

Is there a new car you are interested in? Do you know its approximate price? Do you know how much of a car payment you could afford each month? Based on the payment you can afford, is this car within your reach? Print off the Car Loan Worksheet (in the Content Materials folder) and fill it in using the steps below.

OPTIONAL: Go to or and build your car. Then use the price from this site.

1. Write down the price of the car you are interested in. (Note: We are assuming this is the final price after tax and down payment or trade in.)

2. Write down the monthly payment you can afford.

3. Look up NEW CAR interest rates at brm/rate/brm_autosearch.asp

• Choose auto loan (you may already be at the auto loan page)

• Choose your state

• Choose the city near where you live

You will have the choice between interest rates for 36, 48, 60, and 72 months.

4. Now you will calculate the number of months it would actually take you to pay off your car loan based on the price of the car, the monthly payment you can afford, and the interest rate you have selected.

Use the following formula: [pic]

PV = present value of the loan or the amount of the loan (cost of car)

PMT = monthly payment

n = number of payments (this is what you are finding)

[pic]

(Since you will be making monthly payments, the number of times compounded per year =12.)

5. How many months would it take you to pay off your loan? Is the number of months within the restrictions of the interest rate (if you chose a 60 month rate can you actually pay off the car in 60 months?).

6. If it would take you longer than the interest rate would allow, calculate how many months it would take you to pay off your loan if you went to the next interest rate (60 months to 72 months).

If you could pay off the loan within the number of months specified by the rate, see if you could pay off your loan if you went down to the next interest rate (60 months to 48 months).

7. Check your calculations. You will be able to check your calculations by using a calculator at . To get to the calculator, repeat the process in step 3 on the Content Materials page. The page with the interest rates has a calculator on the right side of the screen (you are looking for the calculator similar to what you see below). Enter the loan amount, the number of months it should take you to pay off your loan according to your calculation (from step 4), and the interest rate you used. Click on “Calculate” and see if the monthly payment is what you estimated you could afford (from step 2). Now check your calculation from step 6 by repeating this process.

|Calculate your payment |

|Loan amount: |

|$ |

|[pic] |

| |

|Loan term: |

|[pic]months |

| |

|Interest rate: |

|[pic]% |

| |

| |

|[pic] |

| |

| |

| |

SEE EXAMPLE BELOW:

1. PRICE of car: $13,000

2. Estimated MONTHLY PAYMENT you can afford: $250

3. NEW CAR interest rate: 3.45% (for a 48 month loan)

4. Use the following formula: [pic]

PV = present value of the loan or the amount of the loan (cost of car) = $13,000

PMT = $250

n = number of payments (this is what you are finding)

[pic] = [pic]

[pic]

[pic]

5. How many months would it take you to pay off your loan? Is the number of months within the restrictions of the interest rate (if you chose a 48 month rate can you actually pay off the car in 48 months?).

It would take me approximately 56.41 months to pay off my loan. Since it would take me approximately 56.41 months to pay off my loan of $13,000 with monthly payments of $250 and an interest rate of 3.75%, I will not be able to pay off the loan within the required 48 months.

6. If it would take you longer than the interest rate would allow calculate how many months it would take you to pay off your loan if you went to the next interest rate (in this case 48 months to 60 months).

Since I could not pay off my loan in 48 months, I must go back to to find out what the rate is for a 60 month new car loan. If the interest rate is the same, then it will take me 56.41 months to pay off my loan. However, if the interest rate is different, I must calculate how many months it will take me to pay off my loan at the new interest rate.

Upon checking the interest rate for a 60 month new car loan at , I find that the rate is now 5.5%. Using the formula from above, I now calculate that it will take me approximately 59.54 months to pay off my loan of $13,000 with monthly payments of $250 and an interest rate of 5.5%. Therefore, I will be able to pay off the loan within the required 60 months. (NOTE: It will take me 3 months longer to pay off the loan at this interest rate.)

If you could pay off the loan within the number of months specified by the rate, see if you could pay off your loan if you went down to the next interest rate (60 months to 48 months).

7. Check your calculations. You will be able to check your calculations by using a calculator at . To get to the calculator, repeat the process in step 3 on the Content Materials page. The page with the interest rates has a calculator on the right side of the screen (you are looking for the calculator similar to what you see below). Enter the loan amount, the number of months it should take you to pay off your loan according to your calculation (from step 4), and the interest rate you used. Click on “Calculate” and see if the monthly payment is what you estimated you could afford (from step 2). Now check your calculation from step 6 by repeating this process.

|Calculate your payment |

|Loan amount: |

|$ |

|[pic] |

| |

|Loan term: |

|[pic]months |

| |

|Interest rate: |

|[pic]% |

| |

| |

|[pic] |

| |

| |

| |

Assessment

The following is a recommended form of assessment:

Allow students one week to complete this project then collect the Car Loan Worksheet. Require students to show their calculations. They should have used logarithms to solve the exponential equation. Suggested point values are listed in red (a total of 15 points).

Car Loan Worksheet

Name_________________________________________

8. PRICE of car: _____________

9. Estimated MONTHLY PAYMENT you can afford: ____________

10. NEW CAR interest rate: __________ Number of months specified for this rate:_______

11. Use the following formula: [pic]

PV = ___________

PMT = ___________

i=___________ (2 points)

SHOW WORK BELOW or attach a sheet of paper with your work.

n = ___________ (5 points)

12. How many months would it take you to pay off your loan? Is the number of months within the restrictions of the interest rate (if you chose a 48 month rate can you actually pay off the car in 48 months?).

(1 point)

13. If it would take you longer than the interest rate would allow, calculate how many months it would take you to pay off your loan if you went to the next interest rate (in this case 48 months to 60 months).

If you could pay off the loan within the number of months specified by the rate, see if you could pay off your loan if you went down to the next interest rate (60 months to 48 months).

(5 points)

14. Check your calculations. You will be able to check your calculations by using a calculator at . To get to the calculator, repeat the process in step 3 on the Content Materials page. The page with the interest rates has a calculator on the right side of the screen (you are looking for the calculator similar to what you see below). Enter the loan amount, the number of months it should take you to pay off your loan according to your calculation (from step 4), and the interest rate you used. Click on “Calculate” and see if the monthly payment is what you estimated you could afford (from step 2). Now check your calculation from step 6 by repeating this process.

(2 points)

|Calculate your payment |

|Loan amount: |

|$ |

|[pic] |

| |

|Loan term: |

|[pic]months |

| |

|Interest rate: |

|[pic]% |

| |

| |

|[pic] |

| |

| |

| |

Links to Course Competencies

This project is appropriate for an Intermediate Algebra course or a College Algebra course.

Students should be able to:

• Demonstrate an understanding of and simplify exponential expressions

• Work on the internet

• Work collaboratively (if the instructor allows group work)

• Evaluate functions

• Use a scientific calculator

• Assess the reasonableness of their calculations

Supplementary Resources

• - an online resource for car loan interest rates

• Ask Dr. Math

• Present Value Formula

•

•

Recommendations

Recommendations for Integration:

This problem-solving based RWLO can be used in an Intermediate Algebra, College Algebra, or Business Math course. This RWLO is intended for use as a supplement to a unit with logarithms and/or exponential equations. Students will use a formula to calculate the number of months it will take them to pay off a car loan based on the amount of the loan, the amount of the monthly car payment, and an interest rate which they will get from an internet resource.

Back-up:

Print off a copy of the Car Loan Worksheet so that you may distribute copies to your students if this website is inaccessible. Also, use an alternative website (a bank website), a newspaper, or call a bank to get approximate interest rates for car loans for students to use in their calculations. Be sure to get rates for 36, 48, 60, and 72 months.

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