The Mathematics of Money

422 Chapter 10 Consumer Mathematics

that the account carried each balance; each balance amount would count once for each

day that it was the card’s balance. The $500 balance with which we started the month will

be included 4 times, since that was the balance for 4 days, while the $2,200 at which the

month ended will only count once, reflecting the fact that it was only the balance for a

single day. Thus, we can calculate the ADB by adding up all the daily balances and divid-

ing the result by 30 (since there are 30 days in the month). Thus:

ADB 

$500  $500  $500  $500  $800  ...  $2,200

_____ 30

$21,000

____ 30
$700

which seems a much better representation of the “average” that the $1025 figure was.

The dots, indicating that “you get the idea so please don’t make me write it all out,”

reflect the fact that, while this is a logical approach, it is also a tedious one. We can simplify

this calculation simply by noting that our total included $500 four times, since it was the

balance for 4 days, the $800 nine times, since it was the balance for 9 days, and so on. So

we could more easily calculate the ADB as:

ADB 

4($500)  9($800)  16($600)  1($2,200)

____ 30

$21,000

____ 30
$700

This is an example of something known in mathematics as a weighted average. In a

weighted average, things being “averaged” are given different “weights,” which gives

some amounts more importance in the average than others. In this case, the weights are the

days that the account carries each of the different balances.

Calculating Average Daily Balances Efficiently

Expanding on the table we originally created for the balance history can provide an efficient

way to calculate the ADB. We repeat that table, this time adding columns for the number

of days at each balance and for the result of multiplying each balance by the number of its

days. We also total both of these new columns:

Effective Date Activity Balance Days at Balance (Balance) (Days)

April 1 Start of month $500 4 $2,000

April 5 Charged $300 $800 9 $7,200

April 14 Paid $200 $600 16 $9,600

April 30 Charged $1,600 $2,200 1 $2,200

Totals: 30 $21,000

Notice that the total of the (Balance)(Days) column is just the top of the ADB fraction,

while the Days at Balance total is just the bottom. So we can just divide these column totals

to get:

ADB

Total of (Days)(Balance)column

___________________________Total of Days column 

$21,000

____ 30
$700

In this calculation, the credit card’s billing month coincided with a calendar month.

This generally won’t be the case, and the following example will illustrate how to deal

with this:

Example 10.1.1 Joanna has a credit card whose billing period begins on the 17th

day of each month. On July 17 her balance was $815.49. She made a $250 payment

on July 28. She also made new charges of $27.55 on July 21, $129.99 on August 5, and

$74.45 on August 8. Find her average daily balance.

The fi rst step is to organize the information in a table like the one we used before. We will

leave the last two columns blank for the moment while we put the information in order. Also,

to save space, we will indicate charges simply with a “” and payments with a “”.