Contemporary Mathematics

# 6.10Credit Cards

Contemporary Mathematics6.10 Credit Cards

Figure 6.22 Credit cards are both a convenience and a danger. (credit: "Credit Cards" by Sean MacEntee/Flickr, CC BY 2.0)

## Learning Objectives

After completing this section, you should be able to:

1. Apply for a credit card armed with basic knowledge.
2. Distinguish between three basic types of credit cards.
3. Compare and contrast the benefits and drawbacks of credit cards.
4. Read and understand the basic parts of a credit card statement.
5. Compute interest, balance due, and minimum payment due for a credit card.

It can be difficult to get along these days without at least one credit card. Most hotels and rental car agencies require that a credit card is used. There are even a number of retailers and restaurants that no longer accept cash. They make online purchasing easier. And nothing contributes more to a good credit rating than a solid history of making credit card payments on time.

Being granted a credits card is a privilege. Used unwisely that privilege can become a curse and the privilege may be withdrawn. In this section, we will talk about the different types of credit cards and their advantages and disadvantages. The more knowledge a cardholder has about the credit card industry, the better able credit accounts can be managed, and that knowledge may cause major adjustments to a cardholder’s lifestyle.

All credit cards are not equal, but they all represent consumers borrowing money, usually from a bank, to pay for needs and “wants.” As such, they are a type of loan, and your repayment may include interest. (You might want to review Section 6.8, which discusses loans and repayment plans.)

There are many institutions and credit cards to choose from. Use caution as you shop around for a credit card that suits you. Your top concern is likely the interest rates on purchases and cash advances. But be careful to also read the small print regarding charges for late payments, and other fees such as an annual fee, where the credit card charges you (the cardholder) a fee each year for the privilege of using the cards. Many cards charge no such fee, but there are many that charge modest to heavy fees. Make sure to understand rules for reward programs, where the credit card issuer grants benefits based on one’s spending. Finally, once one applies for and is granted a credit card, pay attention to the credit limit the bank offers. Once a company is owed that much money, use of the card for purchases should be curtailed until some of the debt is paid off.

## Checkpoint

The interest rate will not matter if the balance is paid every month. When the balance is paid every month, there is NO INTEREST charged.

## Types of Credit Cards

There are basically three types of credit cards: bank-issued credit cards, store-issued credit cards, and travel/entertainment credit cards. We will look at all three and explain the good and the bad qualities of each.

## Bank-Issued Credit Cards

Perhaps the most widely used credit card type is the bank-issued credit card, like Visa or MasterCard (and even American Express and Discover cards). These types of cards are an example of revolving credit, meaning that additional credit is extended before the previous balance is paid—but only up to the assigned credit limit. Bank-issued cards are considered the most convenient, as they can be used to purchase anything, including apparel, furniture, groceries, fuel for automobiles, meals, hotel bills, and so on, just as if paying with cash. The interest rates on bank-issued credit cards are usually lower than those for other credit cards we’ll discuss, and the credit limits are generally higher. Currently, bank-issued cards have an average 20.09% APR.

## Store-Issued Credit Cards

Store-issued credit cards are issued by retailers. One can hardly walk into a store these days without being offered a discount on purchases if one applies for the store credit card. These cards can only be used in that store or family of stores that issues the card. However, if a store credit card is associated with Visa, MasterCard, or American Express, then the card might be used the same way that the bank-issued cards are used. This is called cobranding. The logo of the bank-issued card will be present on the store card. Many stores offer both types. Like other credit cards, they may come with an annual fee.

Store credit cards usually charge higher interest rates than bank-issued cards. Currently, store credit cards have an APR (annual percentage rate) of 24.15%. Any rewards offered by store credit cards are usually limited to purchases made in their own store, and it typically takes longer to accumulate enough rewards or points to redeem them, whereas cobranded credit cards offer opportunities to earn rewards on all purchases, regardless of whether purchases are made in the issuing store or not.

Store credit cards usually offer lower credit limits, at least in the beginning. After being proved to be a responsible credit card owner, credit limits can be raised. Nevertheless, store credit cards are a good choice for those new to the credit card industry. If on-time payments are consistently made, it is an excellent way to get started building a credit history.

## Travel/Entertainment Cards, or Charge Cards

This is the third type of credit card. The travel and entertainment cards, also known as charge cards, first and foremost offer very high limits or unlimited credit, but they must be paid in full every month. They generally charge high annual fees and impose expensive penalties should a payment be late. On the other hand, they typically have longer grace periods and offer many and various kinds of rewards.

## Checkpoint

An interest rate will greatly depend on credit score. Responsible use of credit cards will increase a credit score. See the WHO KNEW? from The Basics of Loans.

## Who Knew?

### Top Travel and Entertainment Cards

At one time, Diner’s Club was the premier entertainment card. To be accepted into the Diner’s Club and be rewarded with a charge card meant one was special. Today, an American Express card is held with the same reverence as Diner’s Club was in the past. It was a privilege to own one of these cards. There is a lot of competition going on to supplant Diner’s Club, as shown by the Chase Sapphire Preferred Card’s travel rewards and benefits.

Another thing worth mentioning here and something that appears to be an unusual offering is part of the American Express Gold and higher-level cards. Cardholders can actually open a savings account, buy a CD, or apply for a personal or business loan from American Express. The company boasts a higher interest rate than what is paid on traditional savings accounts and CDs.

## Example 6.91

### Comparing Credit Cards

1. Which type of credit card is paid off every month so has no interest to be paid, but comes with high fees?
2. Which type of credit cards are the most widely accepted?
3. Which type of credit cards are the most limited?

1.
Which type of credit card typically have the highest interest rates?
2.
What should be considered other than interest rate when selecting a card?
3.
Which type of cards typically have high annual fees?

## Credit Card Statements

Cardholders usually receive monthly statements and have 21 days to pay the minimum amount due. The statements itemize and summarize activity on the credit card for that statement’s billing period. The billing period for a credit card is generally a month long, but typically does not start and end on the first and last days of the month. The statement will include the current balance, interest rate, the minimum payment due, and the due date. Be aware, different companies produce statements that are laid out differently. The information will be clearly labeled though.

The due date is a top concern. Missing a due date is one of the worst things a cardholder can do financially, and this is by far the biggest downfall of owning a credit card. Not only is the cardholder subject to late fees, but when a payment is late more than once there is a high probability that the cardholder will be negatively reported to the credit bureaus, which can quickly erode a credit score. Figure 6.23 shows an excerpt from an actual statement from a Chase Bank Visa card, based on the current $668.25 balance. Figure 6.23 Credit card statement Specifically pay attention to the late payment penalty and minimum payment warning statements. stating that if no other purchases are made and you continue making only the minimum payment, it will take 19 months to pay off the balance and you will pay$754.00. You can’t say you were not warned.

It is critical that you examine your statement every month because it is always a possibility that your account may have been compromised. If you should notice fraudulent charges on your statement, notifying the credit card company is often enough to have those charges researched by the company and removed. The card with the fraudulent charges will be canceled and a new card with a new account number will be sent to you.

## Example 6.92

### Reading a Credit Card Statement

On the credit card statement Figure 6.24, identify

1. The balance due
2. The minimum required payment
3. The length of time it takes to pay off the balance by paying the minimum payments and without charging more to the card
4. The interest rate for purchases
Figure 6.24 Credit card account statement

Referring to the statement above, answer the following:
1.
What is the statement period?
2.
What is the credit limit?
3.
How much in fees were charged?

## Compute Interest, Balance Due, and Minimum Payment Due for a Credit Card

Computing all of these values depends on understanding and computing the average daily balance on a credit card. Once that is known, the interest, balance due, and minimum payment can be found.

Above all else, if you pay off the entire balance each month, interest is not charged.

## Average Daily Balance

Most credit card companies compute interest using the average daily balance method.

To find the average daily balance on your credit card, determine the balance on the card each day of the billing period (often that month), and take the average. One process to find that average daily balance follows these steps:

1. Start with a list of transactions with their dates and amounts.
2. For each day that had transactions, find the total of the transactions for the day. Expenditures are treated as positive values, payments are treated as negative values.
3. Create a table containing each day with a different balance. The balance is the previous balance plus the day’s total transactions.
4. Add a column for the number of days those balances until the balance changed.
5. Add a column that contains the balances multiplied by the number of days until the balance changed.
6. Find the sum of that last column.
7. Divide the sum by the number of days in the billing period (often the number of days in the month). This is the average daily balance.

## Example 6.93

### Computing Average Daily Balance

The billing cycle goes from May 1 to May 31. The balance at the start of the billing cycle is $450.21. The list of transactions on the card is below. Date Activity Amount 1-May Billing Date Balance$450.21
10-May Payment $120.00 15-May Groceries$83.43
26-May Auto Parts $45.12 26-May Restaurant$85.34
30-May Shoes $98.23 Find the average daily balance for the credit card during the month of May. ## Your Turn 6.93 1. The billing cycle goes from June 1 to June 30. The previous month’s balance is$563.80. The transactions are in the table below.
Date Activity Amount
1-Jun Balance $563.80 2-Jun Gasoline$47.50
2-Jun Groceries $63.42 15-Jun Movie$38.75
15-Jun Payment $250.00 27-Jun Pharmacy$31.21
28-Jun Auto fuel $48.00 Find the average daily balance for this credit card. ## Calculating the Interest for a Credit Card The interest charged for a credit card is based on the daily interest rate of the card, the number of days in the billing cycle, and the average daily balance on the card. ## FORMULA The interest charge, $II$, for a credit card during a billing cycle is $I=ADB×r×d365I=ADB×r×d365$, where ADB is the average daily balance, $rr$ is the annual percentage rate, and $dd$ is the number of days in the billing cycle. As before, interest is rounded up to the next penny. ## Example 6.94 ### Calculating Interest for a Credit Card Billing Cycle Compute the interest charged for the credit card based on the given average daily balance (ABD), annual interest rate, and number of days in the billing cycle. 1. ADB =$2,765.00, annual interest rate 13.99%, billing cycle of 30 days
2. ADB = $789.30, annual interest rate 17.99%, billing cycle of 31 days 3. ADB =$1,037.85, annual interest rate 11.99%, billing cycle of 28 days

Compute the interest charged for the credit card based on the given average daily balance (ABD), annual interest rate, and number of days in the billing cycle.
1.
ADB = $2,135.00, annual interest rate 12.9%, billing cycle of 30 days 2. ADB =$1,589.63, annual interest rate 9.99%, billing cycle of 31 days
3.

One type of processing fee is the interchange fee. Mastercard charges the vendor 1.35% of the sale, plus an additional percentage up to 3.25%, and a fixed $001 fee for each transaction. Added to that is an assessment fee. This fee is 0.14% for Visa cards. ## Calculating the Balance of a Credit Card The balance, or sometimes balance due, on a credit card is the previous balance, plus all expenses, minus all payments and credits, plus the interest on the card. As stated before, if the card was paid off, there is no interest to be paid. ## Example 6.95 ### Calculating the Balance of a Credit Card Find the balance on the credit card with the given interest charge and balance before interest was charged. The cards were not paid off previously. 1. Balance before interest is$708.50, interest charge is $8.15 2. Balance before interest is$1,395.10, interest charge is $21.32 ## Your Turn 6.95 Find the balance on the credit card with the given interest charge and balance before interest was charged. The cards were not paid off previously. 1. Balance before interest is$560.00, interest charge is $6.44 2. Balance before interest is$3,218.00, interest charge is $49.17 The next example puts all those steps together. ## Example 6.96 ### Find Balance Due from Transactions and Interest Rate Kaylen’s credit card charges 16.9% annual interest. His current billing period is from November 1 to November 30. The balance on November 1 was$1,845.23. Use Kaylen’s following transactions to determine his balance due at the end of the billing cycle.

Date Activity Amount
1-Nov Billing Date Balance $1,845.23 3-Nov Groceries$78.50
4-Nov Tablet $159.00 4-Nov Online Game Purchase$39.99
4-Nov Restaurant $47.10 10-Nov Payment$300.00
13-Nov Gasoline $58.75 13-Nov Clothing$135.00
18-Nov Gift $30.00 18-Nov Restaurant$21.75
28-Nov Gasoline $43.79 ## Your Turn 6.96 1. Angel’s credit card charges 16.9% annual interest. His current billing period is from August 1 to August 31. The balance on August 1 was$982.45. Use Angel’s following transactions to determine his balance due at the end of the billing cycle.
Date Activity Amount
1-Aug Billing Date Balance $982.45 5-Aug Food$125.31
13-Aug Payment $500.00 14-Aug Gasoline$51.65
14-Aug Pizza $36.99 14-Aug Shoes$89.45
19-Aug Electric bill $178.34 21-Aug Internet$36.99
21-Aug Food $93.45 30-Aug Gasoline$43.18

## Minimum Payment Due

The minimum payment due is the smallest required amount to be paid on a credit card to avoid late fees and penalties, such as an increased interest rate. The calculations for this may differ from card to card. They also depend in the balance of the credit card. General guidelines for minimum payment due are:

• For larger balances (usually over $1,000), the minimum payment will be some percentage of the balance due. • For moderate balances (between$25 and $1,000), the minimum would be a specified dollar amount.$25 seems to be a common value.
• If the balance is small (under $25 for instance), then the minimum payment is the balance. Those are just guidelines. Individual cards may vary in these values. Minimum payments should only be paid if money is short in a given month. The length of time to pay off a credit card using minimum payments is quite long, and results in paying a lot of interest. It is strongly discouraged. ## Example 6.97 ### Calculate the Minimum Payment Due The FYA credit card company has the following minimum payment policy. For balances over$1,000, the minimum payment is 2.5% of the balance due plus fees, but not interest. For balances between $500.00 and$999.99, the minimum payment is $50.00. For balances$499.99 and under, the minimum payment is $25.00 or the balance due, whichever is smaller. In the following, calculate the minimum payment due given the credit card minimum payment policy, the balance due and fees charged. 1. Balance due is$1,309.00, no fees
2. Balance due is $265.50,$35 in fees
3. Balance due is $784.90, no fees ## Your Turn 6.97 The CLH credit card company has the following minimum payment policy. For balances over$1,000, the minimum payment is 1% of the balance due plus interest and fees. For balances between $25.00 and$999.99, the minimum payment is $25.00 plus any fees. For balances under$25.00, the minimum payment is the balance due plus any fees.
In the following, calculate the minimum payment due given the credit card minimum payment policy, the balance due and fees charged.
1.
Balance due is $2,308.00, billing cycle interest is$24.39, no fees
2.
Balance due is $265.50,$59.00 in fees
3.
Balance due is $19.90, no fees Check out this nerdwallet article about minimum payments for more! ## Check Your Understanding 63. What are the three main types of credit cards? 64. Which type of credit card typically has the highest interest rate? 65. Name three important pieces of information that are included in a credit card statement. 66. A credit card has an average daily balance of$1,428.50 during a 31-day billing cycle. Find the interest if the interest rate is 15.9%.
67.
The billing cycle goes from June 15 to July 14, or 30 days. The balance at the start of the billing cycle is $3,825.50. The list of transactions on the card is below. Find the average daily balance. Date Activity Amount 15-Jun Billing Date Balance$3,825.50
20-Jun Food $125.31 20-Jun Clothing$345.00
29-Jun Payment $750.00 1-Jul Restaurant$94.80
6-Jul Gasoline $49.75 6-Jul Home Repair Supplies$683.94
6-Jul Internet $49.99 10-Jul Cell Phone$85.00
10-Jul Food $175.24 68. Find the balance due on a credit card statement if ADB =$487.65, the annual interest rate is 16.8%, the billing cycle was 30 days, and the balance at the end of the billing cycle, before interest is added, was $689.47. ## Section 6.10 Exercises 1 . Name three main criteria to choose a credit card by. 2 . Which type of credit card is most convenient to use? 3 . Which type of credit card comes with no preset spending limit? 4 . How much interest does a credit card holder pay if they pay off the balance every month? 5 . Which type of credit card comes with the highest annual fees? Use the following credit card statement for the following exercises. 6 . What is the minimum payment due? 7 . How long will it take to pay the balance if the minimum only is paid and no new purchases are made? 8 . How much interest was charged in this billing cycle? 9 . What is the balance on the credit card? 10 . What is the credit limit on this card? 11 . What is the late payment fee for this card? 12 . When is the payment due? 13 . The billing cycle for a credit card goes from April 1 to April 30. The balance at the start of the billing cycle is$1,598.00. The list of transactions on the card is below. Find the average daily balance for the billing cycle.
Date Activity Amount
1-Apr Billing Date Balance $1,598.00 9-Apr Gasoline$51.24
9-Apr Food $105.56 9-Apr Payment$675.00
13-Apr Camping Trip $229.75 21-Apr Gasoline$38.45
22-Apr Gifts $148.88 22-Apr Food$49.75
30-Apr Gym Payment $74.99 14 . The billing cycle for a credit card goes from September 1 to September 30. The balance at the start of the billing cycle is$384.25. The list of transactions on the card is below. Find the average daily balance for the billing cycle.
Date Activity Amount
1-Sep Billing Date Balance $384.25 2-Sep Food$94.54
5-Sep Gasoline $25.65 5-Sep Internet$39.99
6-Sep Payment $380.00 9-Sep Insurance$174.52
16-Sep Food $83.54 16-Sep Day Care$350.00
20-Sep Tires $2,337.56 21-Sep Child Clothing$27.65
21-Sep Gasoline $31.00 28-Sep Television$299.95
15 .
The billing cycle for a credit card goes from October 10 to November 9. The balance at the start of the billing cycle is $930.50. The list of transactions on the card is below. Find the average daily balance for the billing cycle. Date Activity Amount 10-Oct Billing Date Balance$930.50
11-Oct Clothing $350.00 14-Oct Computer$865.84
17-Oct Food $106.51 21-Oct Payment$700.00
21-Oct Restaurant $134.52 21-Oct Hotel$387.56
30-Oct Hockey Game $76.47 5-Nov Memorabilia$150.00
5-Nov Restaurant $94.45 6-Nov Gasoline$49.19
16 .
The billing cycle for a credit card goes from February 15 to March 16 during a non-leap year. The balance at the start of the billing cycle is $292.82. The list of transactions on the card is below. Find the average daily balance for the billing cycle. Date Activity Amount 15-Feb Billing Date Balance$292.82
21-Feb Food $64.57 22-Feb Gasoline$31.50
22-Feb Food $71.94 28-Feb Insurance$133.25
28-Feb Payment $100.00 3-Mar Gasoline$26.61
12-Mar School Trip $300.00 In the following exercises, compute the interest charged for the credit card based on the given average daily balance (ABD), annual interest rate, and number of days in the billing cycle. 17 . ADB =$350.00, annual interest rate 14.9%, billing cycle of 30 days.
18 .
ADB = $4,312.00, annual interest rate 9.99%, billing cycle of 31 days. 19 . ADB =$563.38, annual interest rate 17.9%, billing cycle of 30 days.
20 .
ADB = $1,043.53, annual interest rate 11.9%, billing cycle of 31 days. In the following exercises, find the balance on the credit card with the given interest charge and balance before interest was charged. The cards were not paid off previously. 21 . Balance before interest is$1630.00, interest charge is $16.48. 22 . Balance before interest is$621.00, interest charge is $7.81. 23 . Balance before interest is$1,380.00, interest charge is $15.35 24 . Balance before interest is$2,774.00, interest charge is $44.05. 25 . Alsaggr’s credit card charges 11.9% annual interest. His current billing period is from April 1 to April 30. The balance on April 1 was$1,598.00. Use Alsaggr’s following transactions to determine his balance due at the end of the billing cycle.
Date Activity Amount
1-Apr Billing Date Balance $1,598.00 9-Apr Gasoline$51.24
9-Apr Food $105.56 9-Apr Payment$675.00
13-Apr Camping Trip $229.75 21-Apr Gasoline$38.45
22-Apr Gifts $148.88 22-Apr Food$49.75
30-Apr Gym Payment $74.99 26 . Marisa’s credit card charges 8.9% annual interest. Her current billing period is from September 1 to September 30. The balance on September 1 was$384.25. Use Marisa’s following transactions to determine her balance due at the end of the billing cycle.
Date Activity Amount
1-Sep Billing Date Balance $384.25 2-Sep Food$94.54
5-Sep Gasoline $25.65 5-Sep Internet$39.99
9-Sep Insurance $174.52 16-Sep Food$83.54
16-Sep Day Care $350.00 20-Sep Tires$2,337.56
21-Sep Child Clothing $27.65 21-Sep Gasoline$31.00
28-Sep Television $299.95 27 . Haley’s credit card charges 18.9% annual interest. Her current billing period is from October 10 to November 9. The balance on October 10 was$930.50. Use Haley’s following transactions to determine her balance due at the end of the billing cycle.
Date Activity Amount
10-Oct Billing Date Balance $930.50 11-Oct Clothing$350.00
14-Oct Computer $865.84 17-Oct Food$106.51
21-Oct Payment $700.00 21-Oct Restaurant$134.52
21-Oct Hotel $387.56 30-Oct Hockey Game$76.47
5-Nov Memorabilia $150.00 5-Nov Restaurant$94.45
6-Nov Gasoline $49.19 28 . Pavly’s credit card charges 10.9% annual interest. His current billing period is from February 15 to March 16 in a non-leap year. The balance on February 15 was$292.82. Use Pavly’s following transactions to determine his balance due at the end of the billing cycle.
Date Activity Amount
15-Feb Billing Date Balance $292.82 21-Feb Food$64.57
22-Feb Gasoline $31.50 22-Feb Food$71.94
28-Feb Insurance $133.25 28-Feb Payment$100.00
3-Mar Gasoline $26.61 12-Mar School Trip$300.00
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