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ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
513
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from ALG II-M3-TE-1.3.0-08.2015
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Lesson 31: Credit Cards
Student Outcomes
Students compare payment strategies for a decreasing credit card balance.
Students apply the sum of a finite geometric series formula to a decreasing balance on a credit card.
Lesson Notes
This lesson develops the necessary tools and terminology to analyze the mathematics behind credit cards and other
unsecured loans. Credit cards can provide flexibility to budgets, but they must be carefully managed to avoid the pitfalls
of bad credit. For young adults, credit card interest rates can be expected to be between 19.99% and 29.99% per year
(29.99% is currently the maximum allowable interest rate by federal law). Adults with established credit can be offered
interest rates around 8% to 14%. The credit limit for a first credit card is typically around $500, but these limits quickly
increase with a history of timely payments.
In this modeling lesson, students explore the mathematics behind calculating the monthly balance on a single credit card
purchase and recognize that the decreasing balance can be modeled by the sum of a finite geometric series (A-SSE.B.4).
We are intentionally keeping the use of rotating credit such as credit cards simple in this lesson. The students make one
charge of $1,500 on this hypothetical credit card and pay down the balance without making any additional charges.
With this simple example, we can realistically ignore the fact that the interest on a credit card is charged based on the
average daily balance of the account; in our example, the daily balance only changes once per month when the payment
is made.
The students need to recall the following definitions from Lesson 29:
SERIES: Let ๐1, ๐2, ๐3, ๐4, โฆ be a sequence of numbers. A sum of the form
๐1 + ๐2 + ๐3 + โฏ + ๐๐
for some positive integer ๐ is called a series (or finite series) and is denoted ๐๐. The ๐๐โs are called the terms of
the series. The number ๐๐ that the series adds to is called the sum of the series.
GEOMETRIC SERIES: A geometric series is a series whose terms form a geometric sequence.
The sum ๐๐ of the first ๐ terms of the finite geometric series ๐๐ = ๐ + ๐๐ + โฏ + ๐๐๐โ1 (when ๐ โ 1) is given
by
๐๐ = ๐ (1 โ ๐๐
1 โ ๐).
The sum formula of a geometric series can be written in summation notation as
โ ๐๐๐
๐โ1
๐=0
= ๐ (1 โ ๐๐
1 โ ๐).
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
514
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from ALG II-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Classwork
Opening (3 minutes)
Assign students to small groups, and keep them in the same groups throughout this lesson. In the first mathematical
modeling exercise, all groups work on the same problem, but in the second mathematical modeling exercise, the groups
are assigned one of three different payment schemes to investigate.
In the previous lesson, you investigated the mathematics needed for a car loan. What if you have decided to
buy a car, but you have not saved up enough money for the down payment? If you are buying through a
dealership, it is possible to put the down payment onto a credit card. For todayโs lesson, we investigate the
finances of charging $1โ,500 onto a credit card for the down payment on a car. We investigate different
payment plans and how much you end up paying in total using each plan.
The annual interest rates on a credit card for people who have not used credit in the past tend to be much
higher than for adults with established good credit, ranging between 14.99% and 29.99%, which is the
maximum interest rate allowed by law. Throughout this lesson, we use a 19.99% annual interest rate, and we
explore problems with other interest rates in the Problem Set.
One of the differences between a credit card and a loan is that you can pay as much as you want toward your
credit card balance, as long as it is at least the amount of the โminimum payment,โ which is determined by the
lender. In many cases, the minimum payment is the sum of the interest that has accrued over the month and
1% of the outstanding balance, or $25, whichever is greater.
Another difference between a credit card and a loan is that a loan has a fixed term of repaymentโyou pay it
off over an agreed-upon length of time such as five yearsโand that there is no fixed term of repayment for a
credit card. You can pay it off as quickly as you like by making large payments, or you can pay less and owe
money for a longer period of time. In the mathematical modeling exercise, we investigate the scenario of
paying a fixed monthly payment of various sizes toward a credit card balance of $1โ,500.
Mathematical Modeling Exercise (25 minutes)
In this exercise, students model the repayment of a single charge of $1,500 to a credit card that charges 19.99% annual
interest. Before beginning the Mathematical Modeling Exercise, assign students to small groups, and assign groups to be
either part of the 50-team, 100-team, or 150-team. The groups in each of the three teams investigate how long it takes
to pay down the $1,500 balance making fixed payments of either $50, $100, or $150 each month.
As you circulate the room while students are working, take note of groups that are working well together on this set of
problems. Select at least one group on each team to present their work at the end of the exercise period.
Mathematical Modeling Exercise
You have charged $๐, ๐๐๐ for the down payment on your car to a credit card that charges
๐๐. ๐๐% annual interest, and you plan to pay a fixed amount toward this debt each month until it
is paid off. We denote the balance owed after the ๐th payment has been made as ๐๐.
a. What is the monthly interest rate, ๐? Approximate ๐ to ๐ decimal places.
๐ =๐. ๐๐๐๐
๐๐โ ๐. ๐๐๐๐๐
Scaffolding:
For struggling students, use an interest rate of 24.00% so that ๐ = 0.02 and ๐ = 1.02.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
515
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b. You have been assigned to either the ๐๐-team, the ๐๐๐-team, or the ๐๐๐-team, where the number indicates
the size of the monthly payment ๐น you make toward your debt. What is your value of ๐น?
Students will answer ๐๐, ๐๐๐, or ๐๐๐ as appropriate.
c. Remember that you can make any size payment toward a credit card debt, as long as it is at least as large as
the minimum payment specified by the lender. Your lender calculates the minimum payment as the sum of
๐% of the outstanding balance and the total interest that has accrued over the month, or $๐๐, whichever is
greater. Under these stipulations, what is the minimum payment? Is your monthly payment ๐น at least as
large as the minimum payment?
The minimum payment is ๐. ๐๐($๐๐๐๐) + ๐. ๐๐๐๐๐($๐๐๐๐) = $๐๐. ๐๐. All given values of ๐น are greater
than the minimum payment.
d. Complete the following table to show ๐ months of payments.
Month, ๐ Interest Due
(in dollars)
Payment, ๐น
(in dollars)
Paid to Principal
(in dollars)
Balance, ๐๐
(in dollars)
๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐
Month, ๐ Interest Due
(in dollars)
Payment, ๐น
(in dollars)
Paid to Principal
(in dollars)
Balance, ๐๐
(in dollars)
๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐. ๐๐ ๐, ๐๐๐. ๐๐
Month, ๐ Interest Due
(in dollars)
Payment, ๐น
(in dollars)
Paid to Principal
(in dollars)
Balance, ๐๐
(in dollars)
๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐, ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐๐๐. ๐๐ ๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐๐๐. ๐๐
e. Write a recursive formula for the balance ๐๐ in month ๐ in terms of the balance ๐๐โ๐.
To calculate the new balance, ๐๐, we compound interest for one month on the previous balance ๐๐โ๐ and
then subtract the payment ๐น:
๐๐ = ๐๐โ๐(๐ + ๐) โ ๐น, with ๐๐ = ๐๐๐๐.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
516
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f. Write an explicit formula for the balance ๐๐ in month ๐, leaving the expression ๐ + ๐ in symbolic form.
We have the following formulas:
๐๐ = ๐๐(๐ + ๐) โ ๐น
๐๐ = ๐๐(๐ + ๐) โ ๐น
= [๐๐(๐ + ๐) โ ๐น](๐ + ๐) โ ๐น
= ๐๐(๐ + ๐)๐ โ ๐น(๐ + ๐) โ ๐น
๐๐ = ๐๐(๐ + ๐) โ ๐น
= [๐๐(๐ + ๐)๐ โ ๐น(๐ + ๐) โ ๐น](๐ + ๐) โ ๐น
= ๐๐(๐ + ๐)๐ โ ๐น(๐ + ๐)๐ โ ๐น(๐ + ๐) โ ๐น
โฎ
๐๐ = ๐๐(๐ + ๐)๐ โ ๐น(๐ + ๐)๐โ๐ โ ๐น(๐ + ๐)๐โ๐ โ โฏ โ ๐น(๐ + ๐) โ ๐น
g. Rewrite your formula in part (f) using ๐ to represent the quantity (๐ + ๐).
๐๐ = ๐๐๐๐ โ ๐น๐๐โ๐ โ ๐น๐๐โ๐ โ โฏ โ ๐น๐ โ ๐น
= ๐๐๐๐ โ ๐น(๐ + ๐ + ๐๐ + โฏ + ๐๐โ๐)
h. What can you say about your formula in part (g)? What term do we use to describe ๐ in this formula?
The formula in part (g) contains the sum of a finite geometric series with common ratio ๐.
i. Write your formula from part (g) in summation notation using ๐บ.
๐๐ = ๐๐๐๐ โ ๐น(๐ + ๐ + ๐๐ + โฏ + ๐๐โ๐)
= ๐๐๐๐ โ ๐น โ ๐๐๐โ๐
๐=๐
j. Apply the appropriate formula from Lesson 29 to rewrite your formula from part (g).
Using the sum of a finite geometric series formula,
๐๐ = ๐๐๐๐ โ ๐น(๐ + ๐ + ๐๐ + โฏ + ๐๐โ๐)
= ๐๐๐๐ โ ๐น (๐ โ ๐๐
๐ โ ๐)
Scaffolding:
Ask advanced learners to develop a generic formula for the balance ๐๐ in terms of the payment amount R and the growth factor ๐.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
517
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
k. Find the month when your balance is paid off.
The balance is paid off when ๐๐ = ๐. (The final payment is less than a full payment so that the debt is not
overpaid.)
Students will likely do this calculation with the values of ๐, ๐๐, and ๐น substituted in.
๐๐๐๐ โ ๐น (๐ โ ๐๐
๐ โ ๐) = ๐
๐๐๐๐ = ๐น (๐ โ ๐๐
๐ โ ๐)
(๐ โ ๐)(๐๐๐๐) = ๐น(๐ โ ๐๐)
(๐ โ ๐)(๐๐๐๐) + ๐น๐๐ = ๐น
๐๐(๐๐(๐ โ ๐) + ๐น) = ๐น
๐๐ =๐น
(๐๐(๐ โ ๐) + ๐น)
๐ ๐ฅ๐จ๐ (๐) = ๐ฅ๐จ๐ (๐น
(๐๐(๐ โ ๐) + ๐น))
๐ = ๐ฅ๐จ๐ (
๐น(๐๐(๐ โ ๐) + ๐น)
)
๐ฅ๐จ๐ (๐)
If ๐น = ๐๐, then ๐ โ ๐๐. ๐๐๐. The debt is paid off in ๐๐ months.
If ๐น = ๐๐๐, then ๐ โ ๐๐. ๐๐. The debt is paid off in ๐๐ months.
If ๐น = ๐๐๐, then ๐ โ ๐๐. ๐๐๐๐. The debt is paid off in ๐๐ months.
l. Calculate the total amount paid over the life of the debt. How much was paid solely to interest?
For ๐น = ๐๐: The debt is paid in ๐๐ payments of $๐๐, and the last payment is the amount ๐๐๐ with interest:
๐๐(๐๐) + (๐ + ๐)๐๐๐ = ๐๐๐๐ + ๐ (๐๐๐๐ โ ๐น (๐ โ ๐๐
๐ โ ๐))
โ ๐๐๐๐ + ๐(๐๐. ๐๐)
โ ๐๐๐๐. ๐๐.
The total amount paid using monthly payments of $๐๐ is $๐, ๐๐๐. ๐๐. Of this amount, $๐๐๐. ๐๐ is interest.
For ๐น = ๐๐๐: The debt is paid in ๐๐ payments of $๐๐๐, and the last payment is the amount ๐๐๐ with interest.
๐๐๐(๐๐) + (๐ + ๐)๐๐๐ = ๐๐๐๐ + ๐ (๐๐๐๐๐ โ ๐น (๐ โ ๐๐๐
๐ โ ๐))
โ ๐๐๐๐ + ๐(๐๐. ๐๐)
โ ๐๐๐๐. ๐๐
The total amount paid using monthly payments of $๐๐๐ is $๐, ๐๐๐. ๐๐. Of this amount, $๐๐๐. ๐๐ is interest.
For ๐น = ๐๐๐: The debt is paid in ๐๐ payments of $๐๐๐, and the last payment is the amount ๐๐๐ with interest.
๐๐๐(๐๐) + (๐ + ๐)๐๐๐ = ๐๐๐๐ + ๐ (๐๐๐๐ โ ๐น (๐ โ ๐๐
๐ โ ๐))
โ ๐๐๐๐ + ๐(๐. ๐๐)
โ ๐๐๐๐. ๐๐
The total amount paid using monthly payments of $๐๐๐ is $๐, ๐๐๐. ๐๐. Of this amount, $๐๐๐. ๐๐ is interest.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
518
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Discussion (9 minutes)
Have students from each team present their solutions to parts (k) and (l) to the class. After the three teams have made
their presentations, lead students through the following discussion, which should help them to make sense of the
different results that arise from the different payment values ๐ .
What happens to the number of payments as you increase the amount ๐ of the recurring monthly payment?
As the amount ๐ of the payment increases, the number of payments decreases.
What happens to the total amount of interest paid as you increase the amount ๐ of the recurring monthly
payment?
As the amount ๐ of the payment increases, the number of payments decreases.
What is the largest possible amount of the payment ๐ ? In that case, how many payments are made?
The largest possible payment would be to pay the entire balance in one payment: (1 + ๐)$1500 = $1524.99.
Ask students about the formulas that they developed in the Mathematical Modeling Exercise to calculate the balance of
the debt in month ๐. Students may use different notations, but they should have come up with a formula similar to
๐๐ = ๐0๐๐ โ ๐ (1โ๐๐
1โ๐). Depending on what notation the students used, you may need to draw the parallel from this
formula to the present value of an annuity formula developed in Lesson 30. If we substitute ๐๐ = 0 as the future value
of the annuity when it is paid off in ๐ payments, and ๐ด๐ = ๐0 as the present value/initial value of the annuity, then we
have
๐๐ = ๐0๐๐ โ ๐ (1 โ ๐๐
1 โ ๐)
0 = ๐ด๐๐๐ โ ๐ (1 โ ๐๐
1 โ ๐)
๐ด๐๐๐ = ๐ (1 โ ๐๐
1 โ ๐)
๐ด๐(1 + ๐)๐ = ๐ (1 โ (1 + ๐)๐
1 โ (1 + ๐))
๐ด๐(1 + ๐)๐ = ๐ (1 โ (1 + ๐)๐
โ๐)
๐ด๐ = ๐ ((1 + ๐)๐ โ 1
๐) โ (1 + ๐)โ๐
๐ด๐ = ๐ (1 โ (1 + ๐)โ๐
๐).
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
519
This work is derived from Eureka Math โข and licensed by Great Minds. ยฉ2015 Great Minds. eureka-math.org This file derived from ALG II-M3-TE-1.3.0-08.2015
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Closing (3 minutes)
Ask students to summarize the main points of the lesson either in writing or with a partner. Some highlights that should
be included are listed below.
Calculating the balance from a single purchase on a credit card requires that we sum a finite geometric series.
We have a formula from Lesson 29 that calculates the sum of a finite geometric series:
โ ๐๐๐
๐โ1
๐=0
= ๐ (1 โ ๐๐
1 โ ๐).
When you have incurred a credit card debt, you need to decide how to pay it off.
If you choose to make a lower payment each month, then both the time required to pay off the debt
and the total interest paid over the life of the debt increases.
If you choose to make a higher payment each month, then both the time required to pay off the debt
and the total interest paid over the life of the debt decreases.
Exit Ticket (5 minutes)
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
520
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Name Date
Lesson 31: Credit Cards
Exit Ticket
Suppose that you currently have one credit card with a balance of $10,000 at an annual rate of 24.00% interest. You
have stopped adding any additional charges to this card and are determined to pay off the balance. You have worked
out the formula ๐๐ = ๐0๐๐ โ ๐ (1 + ๐ + ๐2 + โฏ + ๐๐โ1), where ๐0 is the initial balance, ๐๐ is the balance after you have
made ๐ payments, ๐ = 1 + ๐, where ๐ is the monthly interest rate, and ๐ is the amount you are planning to pay each
month.
a. What is the monthly interest rate ๐? What is the growth rate, ๐?
b. Explain why we can rewrite the given formula as ๐๐ = ๐0๐๐ โ ๐ (1โ๐
๐
1โ๐).
c. How long does it take to pay off this debt if you can afford to pay a constant $250 per month? Give the
answer in years and months.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
521
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Exit Ticket Sample Solutions
Suppose that you currently have one credit card with a balance of $๐๐, ๐๐๐ at an annual rate of ๐๐. ๐๐% interest. You
have stopped adding any additional charges to this card and are determined to pay off the balance. You have worked out
the formula ๐๐ = ๐๐๐๐ โ ๐น(๐ + ๐ + ๐๐ + โฏ + ๐๐โ๐), where ๐๐ is the initial balance, ๐๐ is the balance after you have
made ๐ payments, ๐ = ๐ + ๐, where ๐ is the monthly interest rate, and ๐น is the amount you are planning to pay each
month.
a. What is the monthly interest rate ๐? What is the growth rate, ๐?
The monthly interest rate ๐ is given by ๐ =๐.๐๐๐๐
= ๐. ๐๐, and ๐ = ๐ + ๐ = ๐. ๐๐.
b. Explain why we can rewrite the given formula as ๐๐ = ๐๐๐๐ โ ๐น (๐โ๐
๐
๐โ๐).
Using summation notation and the sum formula for a finite geometric series, we have
๐ + ๐ + ๐๐ + โฏ + ๐๐โ๐ = โ ๐๐๐โ๐
๐=๐
=๐ โ ๐๐
๐ โ ๐.
Then the formula becomes
๐๐ = ๐๐๐๐ โ ๐น(๐ + ๐ + ๐๐ + โฏ + ๐๐โ๐)
= ๐๐๐๐ โ ๐น (๐ โ ๐๐
๐ โ ๐).
c. How long does it take to pay off this debt if you can afford to pay a constant $๐๐๐ per month? Give the
answer in years and months.
When the debt is paid off, ๐๐ โค ๐. Then ๐๐๐๐ โ ๐น (๐โ๐๐
๐โ๐) = ๐, and ๐๐๐๐ = ๐น (
๐โ๐๐
๐โ๐). Since ๐๐ = ๐๐๐๐๐,
๐น = ๐๐๐, and ๐ = ๐. ๐๐, we have
๐๐๐๐๐(๐. ๐๐)๐ โค ๐๐๐ (๐ โ ๐. ๐๐๐
๐ โ ๐. ๐๐)
๐๐๐๐๐(๐. ๐๐)๐ โค โ๐๐๐๐๐(๐ โ ๐. ๐๐๐)
๐๐๐๐๐(๐. ๐๐)๐ โค ๐๐๐๐๐(๐. ๐๐๐ โ ๐)
(๐. ๐๐)๐ โค ๐. ๐๐(๐. ๐๐)๐ โ ๐. ๐๐
๐. ๐๐ โค ๐. ๐๐(๐. ๐๐)๐
๐ โค ๐. ๐๐๐
๐ฅ๐จ๐ (๐) โค ๐ ๐ฅ๐จ๐ (๐. ๐๐)
๐ โฅ๐ฅ๐จ๐ (๐)
๐ฅ๐จ๐ (๐. ๐๐)
๐ โฅ ๐๐. ๐๐
It takes ๐๐ months to pay off this debt, which means it takes ๐ years and ๐๐ months.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
522
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Problem Set Sample Solutions
Problems 1โ4 ask students to compare credit card scenarios with the same initial debt and the same monthly payments
but different interest rates. Problems 5, 6, and 7 require students to compare properties of functions given by different
representations, which aligns with F-IF.C.9 and F-LE.B.5.
The final two problems in this Problem Set require students to do some online research in preparation for Lesson 32, in
which they select a career and model the purchase of a house. Have some printouts of real-estate listings ready to hand
to students who have not brought their own to class. Feel free to add some additional constraints to the criteria for
selecting a house to purchase. The career data in Problem 9 can be found at http://themint.org/teens/starting-
salaries.html. For additional jobs and more information, please visit the U.S. Bureau of Labor Statistics at
http://www.bls.gov/ooh and http://www.bls.gov/ooh/about/teachers-guide.htm. The salary for the โentry-level full-
timeโ position is based on the projected federal minimum wage in 2016 of $10.10 per hour and a 2,000-hour work year.
1. Suppose that you have a $๐, ๐๐๐ balance on a credit card with a ๐๐. ๐๐% annual interest rate, compounded
monthly, and you can afford to pay $๐๐๐ per month toward this debt.
a. Find the amount of time it takes to pay off this debt. Give your answer in months and years.
๐๐๐๐ (๐ +๐. ๐๐๐๐
๐๐)
๐
โ ๐๐๐ (๐ โ (๐ +
๐. ๐๐๐๐๐๐
)๐
โ๐. ๐๐๐๐
๐๐
) = ๐
๐๐๐๐ (๐ +๐. ๐๐๐๐
๐๐)
๐
= ๐๐๐ ((๐ +
๐. ๐๐๐๐๐๐
)๐
โ ๐
๐. ๐๐๐๐๐๐
)
๐๐๐๐
๐๐๐๐(๐ +
๐. ๐๐๐๐
๐๐)
๐
= (๐ +๐. ๐๐๐๐
๐๐)
๐
โ ๐
(๐ +๐. ๐๐๐๐
๐๐)
๐
(๐๐๐๐
๐๐๐๐โ ๐) = โ๐
(๐ +๐. ๐๐๐๐
๐๐)
๐
(๐ โ๐๐๐๐
๐๐๐๐) = ๐
๐ โ ๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐) + ๐ฅ๐จ๐ (
๐๐๐๐
๐๐๐๐) = ๐ฅ๐จ๐ (๐)
๐ โ ๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐) = โ๐ฅ๐จ๐ (
๐๐๐๐
๐๐๐๐)
๐ = โ๐ฅ๐จ๐ (
๐๐๐๐๐๐๐๐
)
๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐)
๐ โ ๐๐. ๐๐๐
So it takes ๐ year and ๐ months to pay off the debt.
b. Calculate the total amount paid over the life of the debt.
๐๐. ๐๐๐ โ $๐๐๐ = $๐๐๐๐. ๐๐
c. How much money was paid entirely to the interest on this debt?
$๐๐๐. ๐๐
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
523
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
2. Suppose that you have a $๐, ๐๐๐ balance on a credit card with a ๐๐. ๐๐% annual interest rate, and you can afford to
pay $๐๐๐ per month toward this debt.
a. Find the amount of time it takes to pay off this debt. Give your answer in months and years.
๐๐๐๐ (๐ +๐. ๐๐๐๐
๐๐)
๐
โ ๐๐๐ (๐ โ (๐ +
๐. ๐๐๐๐๐๐
)๐
โ๐. ๐๐๐๐
๐๐
) = ๐
๐๐๐๐ (๐ +๐. ๐๐๐๐
๐๐)
๐
= ๐๐๐ ((๐ +
๐. ๐๐๐๐๐๐
)๐
โ ๐
๐. ๐๐๐๐๐๐
)
๐๐๐๐
๐๐๐๐(๐ +
๐. ๐๐๐๐
๐๐)
๐
= (๐ +๐. ๐๐๐๐
๐๐)
๐
โ ๐
(๐ +๐. ๐๐๐๐
๐๐)
๐
(๐๐๐๐
๐๐๐๐โ ๐) = โ๐
(๐ +๐. ๐๐๐๐
๐๐)
๐
(๐ โ๐๐๐๐
๐๐๐๐) = ๐
๐ โ ๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐) + ๐ฅ๐จ๐ (
๐๐๐๐
๐๐๐๐) = ๐ฅ๐จ๐ (๐)
๐ โ ๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐) = โ๐ฅ๐จ๐ (
๐๐๐๐
๐๐๐๐)
๐ = โ๐ฅ๐จ๐ (
๐๐๐๐๐๐๐๐
)
๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐)
๐ โ ๐๐. ๐๐๐
The loan is paid off in ๐ year and ๐ months.
b. Calculate the total amount paid over the life of the debt.
๐๐. ๐๐๐ โ $๐๐๐ = $๐, ๐๐๐. ๐๐
c. How much money was paid entirely to the interest on this debt?
$๐๐๐. ๐๐
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
524
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3. Suppose that you have a $๐, ๐๐๐ balance on a credit card with a ๐. ๐๐% annual interest rate, and you can afford to
pay $๐๐๐ per month toward this debt.
a. Find the amount of time it takes to pay off this debt. Give your answer in months and years.
๐๐๐๐ (๐ +๐. ๐๐๐๐
๐๐)
๐
โ ๐๐๐ (๐ โ (๐ +
๐. ๐๐๐๐๐๐
)๐
โ๐. ๐๐๐๐
๐๐
) = ๐
๐๐๐๐ (๐ +๐. ๐๐๐๐
๐๐)
๐
= ๐๐๐ ((๐ +
๐. ๐๐๐๐๐๐
)๐
โ ๐
๐. ๐๐๐๐๐๐
)
๐๐๐
๐๐๐๐(๐ +
๐. ๐๐๐๐
๐๐)
๐
= (๐ +๐. ๐๐๐๐
๐๐)
๐
โ ๐
(๐ +๐. ๐๐๐๐
๐๐)
๐
(๐๐๐
๐๐๐๐โ ๐) = โ๐
(๐ +๐. ๐๐๐๐
๐๐)
๐
(๐ โ๐๐๐
๐๐๐๐) = ๐
๐ โ ๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐) + ๐ฅ๐จ๐ (
๐๐๐๐
๐๐๐๐) = ๐ฅ๐จ๐ (๐)
๐ โ ๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐) = โ๐ฅ๐จ๐ (
๐๐๐๐
๐๐๐๐)
๐ = โ๐ฅ๐จ๐ (
๐๐๐๐๐๐๐๐
)
๐ฅ๐จ๐ (๐ +๐. ๐๐๐๐
๐๐)
๐ โ ๐๐. ๐๐๐
The loan is paid off in ๐ year and ๐ months.
b. Calculate the total amount paid over the life of the debt.
๐๐. ๐๐๐ โ $๐๐๐ = $๐๐๐๐. ๐๐
c. How much money was paid entirely to the interest on this debt?
$๐๐๐. ๐๐
4. Summarize the results of Problems 1, 2, and 3.
Answers will vary but should include the fact that the total interest paid in each case dropped by about half with
every problem. Lower interest rates meant that the loan was paid off more quickly and that less was paid in total.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
525
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
5. Brendan owes $๐, ๐๐๐ on a credit card with an interest rate of ๐๐%. He is making payments of $๐๐๐ every month
to pay this debt off. Maggie is also making regular payments to a debt owed on a credit card, and she created the
following graph of her projected balance over the next ๐๐ months.
a. Who has the higher initial balance? Explain how you know.
Reading from the graph, Maggieโs initial balance is between $๐, ๐๐๐ and $๐, ๐๐๐, and we are given that
Brendanโs initial balance is $๐, ๐๐๐, so Maggie has the larger initial balance.
b. Who will pay their debt off first? Explain how you know.
From the graph, it appears that Maggie will pay off her debt between months ๐๐ and ๐๐. Brendanโs balance
in month ๐ can be modeled by the function ๐๐ = ๐๐๐๐(๐. ๐๐)๐ โ ๐๐๐ (๐.๐๐
๐โ๐
๐.๐๐), which is equal to zero
when ๐ โ ๐๐. ๐. Thus, Brendanโs debt will be paid in month ๐๐, so Maggieโs debt will be paid off first.
6. Alan and Emma are both making $๐๐๐ monthly payments toward balances on credit cards. Alan has prepared a
table to represent his projected balances, and Emma has prepared a graph.
Alanโs Credit Card Balance
Month, ๐ Interest Payment Balance, ๐๐
๐ ๐, ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐, ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐, ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐, ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐, ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐, ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐
๐ ๐๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐
๐๐ ๐. ๐๐ ๐๐๐ ๐๐๐. ๐๐
๐๐ ๐. ๐๐ ๐๐๐ ๐๐. ๐๐
a. What is the annual interest rate on Alanโs debt? Explain how you know.
One monthโs interest on the balance of $๐, ๐๐๐ was $๐๐. ๐๐, so ๐๐. ๐๐ = ๐(๐๐๐๐). Then the monthly
interest rate is ๐ = ๐. ๐๐๐๐๐๐, and the annual rate is ๐๐๐ = ๐. ๐๐๐๐, so the annual rate on Alanโs debt is
๐๐. ๐๐%.
Month
Cre
dit
Car
d B
alan
ce
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
526
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b. Who has the higher initial balance? Explain how you know.
From the table, we can see that Alanโs initial balance is $๐, ๐๐๐, while Emmaโs initial balance is the
๐-intercept of the graph, which is above $๐, ๐๐๐. Thus, Emmaโs initial balance is higher.
c. Who will pay their debt off first? Explain how you know.
Both Alan and Emma will pay their debts off in month ๐๐ because both of their balances in month ๐๐ are
under $๐๐๐.
d. What do your answers to parts (a), (b), and (c) tell you about the interest rate for Emmaโs debt?
Because Emma had the higher initial balance, and they made the same number of payments, Emma must
have a lower interest rate on her credit card than Alan does. In fact, since the graph decreases apparently
linearly, this implies that Emma has an interest rate of ๐%.
7. Both Gary and Helena are paying regular monthly payments to a credit card balance. The balance on Garyโs credit
card debt can be modeled by the recursive formula ๐๐ = ๐๐โ๐(๐. ๐๐๐๐๐) โ ๐๐๐ with ๐๐ = ๐๐๐๐, and the balance
on Helenaโs credit card debt can be modeled by the explicit formula ๐๐ = ๐๐๐๐(๐. ๐๐๐๐๐)๐ โ ๐๐๐ (๐.๐๐๐๐๐
๐โ๐
๐.๐๐๐๐๐)
for ๐ โฅ ๐.
a. Who has the higher initial balance? Explain how you know.
Gary has the higher initial balance. Helenaโs initial balance is $๐, ๐๐๐, and Garyโs is $๐, ๐๐๐.
b. Who has the higher monthly payment? Explain how you know.
Helena has the higher monthly payment. She is paying $๐๐๐ every month while Gary is paying $๐๐๐.
c. Who will pay their debt off first? Explain how you know.
Helena will pay her debt off first since she starts at a lower balance and is paying more per month.
Additionally, they appear to have the same interest rates.
8. In the next lesson, we will apply the mathematics we have learned to the purchase of a house. In preparation for
that task, you need to come to class prepared with an idea of the type of house you would like to buy.
a. Research the median housing price in the county where you live or where you wish to relocate.
Answers will vary.
b. Find the range of prices that are within ๐๐% of the median price from part (a). That is, if the price from part
(a) was ๐ท, then your range is ๐. ๐๐๐ท to ๐. ๐๐๐ท.
Answers will vary.
c. Look at online real estate websites, and find a house located in your selected county that falls into the price
range specified in part (b). You will be modeling the purchase of this house in Lesson 32, so bring a printout
of the real estate listing to class with you.
Answers will vary.
ALGEBRA II
NYS COMMON CORE MATHEMATICS CURRICULUM M3 Lesson 31
Lesson 31: Credit Cards
527
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
9. Select a career that interests you from the following list of careers. If the career you are interested in is not on this
list, check with your teacher to obtain permission to perform some independent research. Once it has been
selected, use the career to answer questions in Lesson 32 and Lesson 33.
Occupation Median Starting Salary Education Required
Entry-level full-time
(wait staff, office clerk,
lawn care worker, etc.)
$๐๐, ๐๐๐ High school diploma or GED
Accountant $๐๐, ๐๐๐ ๐-year college degree
Athletic Trainer $๐๐, ๐๐๐ ๐-year college degree
Chemical Engineer $๐๐, ๐๐๐ ๐-year college degree
Computer Scientist $๐๐, ๐๐๐ ๐-year college degree or more
Database Administrator $๐๐, ๐๐๐ ๐-year college degree
Dentist $๐๐๐, ๐๐๐ Graduate degree
Desktop Publisher $๐๐, ๐๐๐ ๐-year college degree
Electrical Engineer $๐๐, ๐๐๐ ๐-year college degree
Graphic Designer $๐๐, ๐๐๐ ๐- or ๐-year college degree
HR Employment Specialist $๐๐, ๐๐๐ ๐-year college degree
HR Compensation Manager $๐๐, ๐๐๐ ๐-year college degree
Industrial Designer $๐๐, ๐๐๐ ๐-year college degree or more
Industrial Engineer $๐๐, ๐๐๐ ๐-year college degree
Landscape Architect $๐๐, ๐๐๐ ๐-year college degree
Lawyer $๐๐๐, ๐๐๐ Law degree
Occupational Therapist $๐๐, ๐๐๐ Masterโs degree
Optometrist $๐๐, ๐๐๐ Masterโs degree
Physical Therapist $๐๐, ๐๐๐ Masterโs degree
PhysicianโAnesthesiology $๐๐๐, ๐๐๐ Medical degree
PhysicianโFamily Practice $๐๐๐, ๐๐๐ Medical degree
Physicianโs Assistant $๐๐, ๐๐๐ ๐ years college plus ๐-year program
Radiology Technician $๐๐, ๐๐๐ ๐-year degree
Registered Nurse $๐๐, ๐๐๐ ๐- or ๐-year college degree plus
Social WorkerโHospital $๐๐, ๐๐๐ Masterโs degree
TeacherโSpecial Education $๐๐, ๐๐๐ Masterโs degree
Veterinarian $๐๐, ๐๐๐ Veterinary degree