Buying on Credit

Center for Science, Mathematics, and Engineering Education

Buying on Credit

TASK. A credit card company, whose motto is "see the world on credit," charges 1.387% interest on the unpaid balance in an account each month, and requires a minimum payment of 2% of the outstanding balance each month. Suppose you charge $100 each month and make only the minimum payment each month. How much will you owe at the time of your 24th bill? Assuming you pay the whole bill at the end of that period, how much will be interest?

COMMENTARY. Consumer debt is a big issue in this country. Thanks to the widespread availability of calculators and computers, consumers can easily do the calculations themselves to better understand the cost of maintaining credit card balances.

Powerful tools for this job are spreadsheets and programmable and graphing calculators. Mathematical analysis is still necessary, of course, but technology provides an avenue for mathematical modeling that gives students straightforward access to some mathematics that once required much more background. Once the spreadsheet is set up, students can explore different payment options and see the consequences. Spreadsheets can help demystify mathematics and provide an exploratory medium for doing calculations that are both relevant and meaningful for students.

MATHEMATICAL ANALYSIS. The general mathematical structure of this task is the same as that of Drug Dosage (p. 80), and solutions can be obtained in similar ways. To analyze what happens when making only the minimum monthly payment, one might start with a table in order to obtain a recursion equation. That equation can be solved or used in a calculator or a spreadsheet.

In addition to charging interest on the previous month's unpaid balance, most credit card companies charge interest on new purchases, too, when the customer is carrying a balance. (The typical 25-day, interest-free grace period applies only if the entire balance is paid off each month.) The interest is usually computed based on an "average daily balance," so the actual amount of interest depends upon when the payment arrives and when the new charge is made. To simplify the calculations, assume that both the new charge and the payment come in at the end of the billing cycle and do not affect the interest that month.

To obtain a recursion equation, let x_n be the amount of the nth statement. Because the minimum payment is 2% of x_n, then, each month,

the amount of the payment is .02x_n,

the interest charge is .01387x_n, and

there is an additional $100 of purchases.

So to find x_n₊₁ (the amount of the (n + 1)st statement), the payment is subtracted from and the other amounts are added to x_n, the amount of the nth statement:

x_n₊₁ = x_n .02x_n + .01387x_n + 100

x_n₊₁ = .99387x_n + 100.

To find out how much will be owed at the time of the 24th bill, start with x₁ = 100, and repeat the calculation 23 times. Or, with a programmable calculator, define x₁ = 100 and x_n₊₁ by the formula given above and tell the program to calculate x₂₄.

To calculate using a spreadsheet, begin with the first row which probably has cells named as shown in Table 1. In a spreadsheet, these cells might be used to contain the information shown in Table 2.

Table 1:
Names of cells in a spreadsheet

A1 B1 C1 D1 E1

Table 2:
How data might be organized in a spreadsheet

The previous bill The minimum payment The interest The new charges The new balance

In the first month, there is no previous bill, no minimum payment, no unpaid balance, and no interest. Enter 100 in cell D1 representing the new charges and 100 in E1 to represent the balance at the end of month 1.

In the second row, enter the numbers and formulas shown in Table 3. Then formulas for the next 22 rows are the same, except that the row numbers will change. Most spreadsheet programs will change the row numbers automatically if these cells are copied and then pasted into the next 22 rows. The spreadsheet will create a table similar to Table 4.

Table 3:
Formulas for one row of the spreadsheet¹

E1 .02*A2 .01387*A2 $100 A2B2+C2+D2

Table 4:
The completed spreadsheet²

Month Previous Balance Payment Interest Purchases New Balance

A B C D E

1 $100.00 $100.00

2 $100.00 $2.00 $1.39 $100.00 $199.39

3 $199.39 $3.99 $2.77 $100.00 $298.16

4 $298.16 $5.96 $4.14 $100.00 $396.34

5 $396.34 $7.93 $5.50 $100.00 $493.91

6 $493.91 $9.88 $6.85 $100.00 $590.88

7 $590.88 $11.82 $8.20 $100.00 $687.26

8 $687.26 $13.75 $9.53 $100.00 $783.04

9 $783.04 $15.66 $10.86 $100.00 $878.24

10 $878.24 $17.56 $12.18 $100.00 $972.86

11 $972.86 $19.46 $13.49 $100.00 $1,066.90

12 $1,066.90 $21.34 $14.80 $100.00 $1,160.36

13 $1,160.36 $23.21 $16.09 $100.00 $1,253.24

14 $1,253.24 $25.06 $17.38 $100.00 $1,345.56

15 $1,345.56 $26.91 $18.66 $100.00 $1,437.31

16 $1,437.31 $28.75 $19.94 $100.00 $1,528.50

17 $1,528.50 $30.57 $21.20 $100.00 $1,619.13

18 $1,619.13 $32.38 $22.46 $100.00 $1,709.21

19 $1,709.21 $34.18 $23.71 $100.00 $1,798.73

20 $1,798.73 $35.97 $24.95 $100.00 $1,887.70

21 $1,887.70 $37.75 $26.18 $100.00 $1,976.13

22 $1,976.13 $39.52 $27.41 $100.00 $2,064.02

23 $2,064.02 $41.28 $28.63 $100.00 $2,151.37

24 $2,151.37 $43.03 $29.84 $100.00 $2,238.18

Total $527.97 $366.14 $2,400.00

The spreadsheet can total the columns, too, as illustrated. Table 4 shows that if you pay in the way suggested, you will have made $527.97 in payments on your $2,400 in purchases and still owe $2,238.18 because of $366.14 in interest.

If the calculations in Table 4 are correct, the purchases plus the interest minus the payments should give the outstanding balance.

Purchases + Interest Payments =
$2,400 + $366.14 $527.97 = $2,238.17

so the calculations are off by a penny somewhere. By asking the spreadsheet to display its results more accurately, it becomes clear that there is not a mistake, just "round-off" error.

Purchases + Interest Payments =
$2,400 + $366.1438 $527.9651 = $2,231.1787.

See Rounding Off (p. 119) for further discussion of this issue.

EXTENSIONS. The solution above made several assumptions to simplify the calculations. Some extensions can bring the solution closer to the way the credit card companies actually do their computations:

For which of the above calculations is rounding necessary? Fix the table in the spreadsheet to properly account for rounding.

For many credit card companies, the minimum payment is always a whole-dollar amount. Incorporate this idea into the solution.

Expand on the solution above so that the dates of payments and new charges could be varied and so that the interest would be calculated according to an average daily balance. Investigate how the dates of payments and new charges affect the interest charges.

Other extensions could investigate alternative scenarios. Suppose, for example, you spent $2,400 up front and then made the minimum payment for each of 24 months. What would the accrued interest be?

The commentary mentioned that the mathematical structure of this task is the same as that of Drug Dosage (p. 80). Lottery Winnings (p. 111) also shares the same basic structure. Once students have sufficient experience with tasks like these, they might explore the difference and similarities among the procedures, formulas, and solutions of these tasks.

NOTES

¹ Here, as in many spreadsheet programs, an asterisk serves as a multiplication sign. In some spreadsheet programs, formulas must be preceded with an equals sign, so that, for example, the contents of the second cell would instead be: =.02*A1.

² The column headings are provided for clarity, but are not part of the spreadsheet. Conveniently, the row numbers can function as month numbers.

Table 1: Names of cells in a spreadsheet

Table 2: How data might be organized in a spreadsheet

Table 3: Formulas for one row of the spreadsheet1

Table 4: The completed spreadsheet2

NOTES

Table 1:
Names of cells in a spreadsheet

Table 2:
How data might be organized in a spreadsheet

Table 3:
Formulas for one row of the spreadsheet¹

Table 4:
The completed spreadsheet²