Making Change

The most direct way to find change is to subtract the price of the item, \(P\), from the amount of money paid, \(A\). This gives the change owed, \(C\):

\( C = A - P \)

For example, suppose an item costs \(3.65\) and the customer pays with a \(5\) dollar bill. The change owed is:

\( C = 5.00 - 3.65 = 1.35 \)

The subtraction method works well when you already have paper and pencil handy, but in real life (at a cash register, for instance) most people find it faster to count up instead.

Counting up starts at the price of the item and adds small, easy amounts until you reach the amount paid. Each amount you add along the way becomes part of the change. This method mirrors how change is actually handed back in coins and bills, one denomination at a time, so it is often the quickest way to check that a customer receives the correct change.

Using the same example, start at \(3.65\) and work up to \(5.00\):

• Add \(0.05\) to reach \(3.70\)
• Add \(0.30\) to reach \(4.00\)
• Add \(1.00\) to reach \(5.00\)

Adding up everything given back, \(0.05 + 0.30 + 1.00 = 1.35\), matches the answer found with subtraction. Both methods should always agree, which makes counting up a great way to double check a subtraction answer, or the other way around.

Once you know the total change owed, the next step is deciding which coins and bills to actually hand over. The usual strategy is to start with the largest denomination that fits into the amount, then work down to smaller coins until the full amount is covered. For \(1.35\) in change, a natural combination is one dollar bill, one quarter, and one dime, since \(1.00 + 0.25 + 0.10 = 1.35\).

There is often more than one correct combination for the same amount of change. For instance, \(1.35\) could also be given as one dollar bill, two dimes, and three nickels, since \(1.00 + 0.20 + 0.15 = 1.35\). Exploring these different combinations connects directly to equivalent money combinations, where the same total value is built from different sets of coins and bills.

An item costs \(7.42\) and the customer pays with a \(10\) dollar bill. Find the change.

Using subtraction: \( C = 10.00 - 7.42 = 2.58 \).

Checking with counting up: from \(7.42\), add \(0.08\) to reach \(7.50\), add \(0.50\) to reach \(8.00\), then add \(2.00\) to reach \(10.00\). Adding the steps together, \(0.08 + 0.50 + 2.00 = 2.58\), which matches the subtraction answer, so \(2.58\) is the change owed. That change could be handed back as two dollar bills, two quarters, and eight pennies.

Always double check your work by trying both methods when possible. Line up the decimal points carefully when subtracting money amounts, since a misplaced decimal is one of the most common mistakes. Practicing with a range of prices and payment amounts (including cases where the change involves several different coins) builds speed and confidence with making change.

Related lessons

• Math 2