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Making change

Ratio, rates and money · level 2, lesson 2 of 6

A question to start

A notebook costs 3 dollars and 60 cents. You pay with a 5-dollar bill. The cashier gives you some coins back. How much should you get?

Most people think "5 take away 3 dollars 60 cents" and get stuck, because taking away with dollars and cents feels hard. But shopkeepers have used a trick for hundreds of years that needs no taking away at all. They count up. Watch: "3 dollars 60 ... 70, 75, 4 dollars, 5 dollars." By the end of this lesson you will know exactly what that means, and you will be able to check any cashier.

The idea

What change is

When you pay more than the price, the shop gives you the extra money back. The money you get back is called change.

The price and the change together make what you paid.

You paid 5 dollars. The notebook kept 3 dollars and 60 cents of it. The rest comes back to you as change.

Counting up

Here is the trick. Start at the price. Add coins, one at a time, until you reach the amount paid. The coins you added are the change.

The best way is to add small coins first to reach a "round" number, then bigger coins and bills.

Start at 3 dollars 60.

  1. Add a dime: 3 dollars 70.
  2. Add a nickel: 3 dollars 75.
  3. Add a quarter: 4 dollars. Now we are at a round number of dollars.
  4. Add a loonie: 5 dollars. We reached what you paid. Stop.
3.603.703.75 4.005.00 dimequarterloonie nickel

The coins you added are 1 dime, 1 nickel, 1 quarter and 1 loonie. Count them (biggest first, as in Counting coins): "1 dollar, 1 dollar 25, 1 dollar 35, 1 dollar 40".

So your change is 1 dollar and 40 cents.

Why counting up works

Think of the number line in the picture. The price, 3 dollars 60, is one place on the line. What you paid, 5 dollars, is another place. The change is the gap between them: how far you must go to get from the price to what you paid.

You can measure a gap two ways:

  • Start at the big end and take away (5 dollars take away 3 dollars 60).
  • Start at the small end and count up to the big end.

Both measure the same gap, so both give the same answer. Counting up is just easier, because adding a nickel or a dime is easy, and reaching a round number like 4 dollars is easy too. The hard part of taking away (borrowing across the dot) never happens.

That is also why a good cashier counts the change into your hand, saying the numbers out loud: "3.60 ... 3.70, 3.75, 4, and 5." You hear the count reach the amount you paid, so you know the change is right.

Checking your change

Always check with the rule from the start:

The price and the change together make what you paid.

3 dollars 60 and 1 dollar 40: the dollars make 4, and 60 cents and 40 cents make 100 cents, which is 1 more dollar. So 5 dollars. It checks!

Rounding to 5 cents when you pay cash

Canada has had no pennies since 2013 (see Canadian coins). But some prices still end in 1, 2, 3 and so on, like 3 dollars 62. When you pay with coins and bills, the total is rounded to the nearest 5 cents:

The total ends in It becomes Example
1 or 2 down to 0 3.62 becomes 3.60
3 or 4 up to 5 3.63 becomes 3.65
6 or 7 down to 5 3.67 becomes 3.65
8 or 9 up to 10 3.68 becomes 3.70

The rounding always goes to whichever of the nearby 0 or 5 is closest. If you pay by card, nothing is rounded: the shop takes the exact amount.

Worked examples

Example 1. A 10-dollar bill. A book costs 6 dollars and 25 cents. You pay with a 10-dollar bill. What is your change?

  1. Start at 6.25.
  2. Quarter: 6.50. Quarter: 6.75. Quarter: 7.00. (Three quarters to reach a round number.)
  3. Toonie: 9.00. Loonie: 10.00. We reached what you paid.
  4. The change is 3 quarters, 1 toonie and 1 loonie. Count biggest first: 2 dollars, 3 dollars, 3.25, 3.50, 3.75.
  5. Change: 3 dollars and 75 cents.
  6. Check: 6.25 and 3.75. Cents: 25 and 75 make 100, which is 1 dollar. Dollars: 6 and 3 and that 1 make 10. Yes.

Example 2. A toonie. A juice costs 1 dollar 35. You pay with a toonie. What is your change?

  1. Start at 1.35.
  2. Nickel: 1.40. Dime: 1.50. Quarter: 1.75. Quarter: 2.00.
  3. Change: 1 nickel, 1 dime and 2 quarters. Count biggest first: 25, 50, 60, 65.
  4. Change: 65 cents.

Example 3. Almost exact. A snack costs 4 dollars 85. You pay with a 5-dollar bill.

  1. Start at 4.85.
  2. Nickel: 4.90. Dime: 5.00.
  3. Change: 15 cents. Only two small coins. When the price is close to what you paid, the change is small.

Example 4. Rounding first. Your total is 2 dollars 63. You pay cash with a toonie and a loonie (3 dollars).

  1. Round first: 63 ends in 3, so it rounds up to 2.65.
  2. Count up from 2.65: dime 2.75, quarter 3.00.
  3. Change: a dime and a quarter, 35 cents.

Try it

  1. You pay with a loonie for a pencil that costs 80 cents. What is your change?
  2. You pay with a toonie for a muffin that costs 1 dollar 50. What is your change?
  3. You pay with a 5-dollar bill for a toy that costs 2 dollars 75. What is your change? Which coins could it be?
  4. You pay with a 10-dollar bill for a lunch that costs 7 dollars 40. What is your change?
  5. A cashier gives you 1 dollar 50 change when you pay 5 dollars for something costing 3 dollars 60. Is that right? Who gets the mistake, you or the shop?
  6. Paying cash, what does each total become: 2.98, 4.12, 6.06?
  7. Your total is 1 dollar 33. You pay cash with a toonie. What is your change?
  8. You bought a comic and paid with a 5-dollar bill. Your change was 1 dollar 25. How much did the comic cost?
Answers
  1. 20 cents. Count up from 80: dime, 90; dime, 100. That is 2 dimes. (Check: 80 and 20 make 100 cents, a dollar.)
  2. 50 cents. Count up from 1.50: quarter, 1.75; quarter, 2.00. Two quarters make 50 cents.
  3. 2 dollars and 25 cents. Count up from 2.75: quarter, 3.00; toonie, 5.00. So the change is a quarter and a toonie. (Or a quarter and 2 loonies. Any coins worth 2 dollars 25 are right.)
  4. 2 dollars and 60 cents. Count up from 7.40: dime 7.50, quarter 7.75, quarter 8.00, toonie 10.00. Change: 2 dollars, then 25, 50, 60 cents. Check: 7.40 and 2.60 make 10.00.
  5. No. The right change is 1 dollar 40. The cashier gave 1 dollar 50, which is 10 cents too much. The shop loses 10 cents. (An honest customer gives the extra dime back.)
  6. 2.98 ends in 8, so it rounds up to 3.00. 4.12 ends in 2, so it rounds down to 4.10. 6.06 ends in 6, so it rounds down to 6.05.
  7. 65 cents. First round 1.33 up to 1.35 (it ends in 3). Count up from 1.35: nickel 1.40, dime 1.50, quarter 1.75, quarter 2.00. Change: 25, 50, 60, 65.
  8. 3 dollars and 75 cents. This time, start at what you paid and go back by the change. 5 dollars take away 1 dollar is 4 dollars. 4 dollars take away 25 cents is 3 dollars 75. Check by counting up from 3.75: quarter 4.00, loonie 5.00. The change is a loonie and a quarter, 1 dollar 25. Yes.

Common mistake

Stopping at the round number. Counting up from 3.60 to 4.00 and saying "40 cents change" forgets the last jump from 4 dollars up to the 5 dollars you paid. Keep going until you reach what you paid.

Counting the coins, not their worth. "I got 4 coins back" says nothing. Count the change itself, biggest coin first, and check that price and change add up to what you paid.

Where it leads