Any time you enter data in a checkbook, a document, a spreadsheet, etc., you might make an error. The most common error, by far, is a single-digit error: entering an “8” instead of a “3” somewhere in the number, for example. Another common error is a transposition error, where you reverse the order of two adjacent digits: writing “83” instead of “38.”

Suppose you are balancing your checkbook. For simplicity, suppose all amounts in this checkbook are in dollars (no cents) from $1 to $9999.

- If you make one single-digit error, what is the greatest possible difference between the total numbers you have entered and the bankʼs total?
- What is the least possible difference?
- A common rule for spotting a transposition error is: “if the difference between your total and the bankʼs total is divisible by 9, look for a transposition error.” Using algebra, explain why this rule makes sense.

Hint: First consider the case where the transposition error is in the last two digits. If the correct amount of a check is 1000a + 100b + 10c + d, then the incorrect amount entered will be 1000a + 100b + 10d + c. What do you notice about the result of subtracting these two numbers? Then analyze the other two cases.