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Counting coins

Ratio, rates and money · level 2, lesson 1 of 6

A question to start

Your piggy bank holds 4 coins: 3 quarters and 1 dime. Your friend's piggy bank holds 15 coins, all nickels. Who has more money?

15 coins sounds like a lot more than 4. But count the money, not the coins. Your 3 quarters are 75 cents, and the dime makes it 85 cents. Your friend's 15 nickels are 75 cents. You have more, with far fewer coins!

Here is another surprise. There are 29 different ways to make exactly 1 dollar out of nickels, dimes and quarters. In this lesson you will learn a fast, safe way to count any pile of coins, and you will find some of those 29 ways yourself.

The idea

Cents and dollars, a quick reminder

From Canadian coins:

Coin Worth
nickel 5 cents
dime 10 cents
quarter 25 cents
loonie 1 dollar
toonie 2 dollars

And the most important fact about money in Canada: 100 cents make 1 dollar. So a loonie is worth 100 cents, and a toonie is worth 200 cents.

Counting on from the biggest coin

Here is a pile of coins: 2 dimes, 3 quarters and 1 nickel, all mixed up. How much money is there?

Step 1. Sort. Put the coins in a line from the coin worth the most to the coin worth the least: quarters first, then dimes, then nickels.

Step 2. Count on. Say the first coin's worth. Then for each new coin, count on by that coin's worth.

252525 10105 255075 8595100 say the bold numbers as you touch each coin

"25, 50, 75" for the quarters. Then the dimes: "85, 95". Then the nickel: "100". There are 100 cents, which is 1 dollar.

Why start with the biggest?

You could count the coins in any order, and you would get the same total. Moving coins around does not add or remove any money. So why bother sorting?

Because it makes counting easier and safer. Try counting the same coins in a messy order: dime, quarter, nickel, quarter, dime, quarter. You would say "10, 35, 40, 65, 75, 100". Those jumps are hard to do in your head: 10 and 25 more? 40 and 25 more? It is easy to slip.

When the big coins come first, you count by 25s (25, 50, 75, which you can learn by heart), then by 10s, then by 5s. Each kind of coin is one simple counting pattern. That is the same "for every" idea from level 1: 25 cents for every quarter, 10 cents for every dime.

Counting with dollars and cents together

When the pile has loonies and toonies too, count the dollars first, then the cents.

Pile: 1 toonie, 1 loonie, 2 quarters, 1 dime.

  • Toonie: "2 dollars".
  • Loonie: "3 dollars".
  • Quarters: "3 dollars 25 cents, 3 dollars 50 cents".
  • Dime: "3 dollars 60 cents".

So there are 3 dollars and 60 cents. A price tag would show this as 3.60: the number before the dot counts dollars, and the two numbers after the dot count cents.

When the cents reach 100

Watch what happens with 5 quarters: "25, 50, 75, 100, 125". That is 125 cents. But 100 cents make 1 dollar, so 125 cents is 1 dollar and 25 cents. Whenever your cents count passes 100, take out 100 cents and call it 1 dollar.

Making 1 dollar in different ways

Here are some ways to make exactly 1 dollar (100 cents).

Quarters Dimes Nickels Check
4 0 0 25, 50, 75, 100
0 10 0 10, 20, ... , 100 (ten jumps of 10)
0 0 20 twenty jumps of 5
2 5 0 25, 50, then 60, 70, 80, 90, 100
3 2 1 25, 50, 75, then 85, 95, then 100
1 7 1 25, then 35, 45, 55, 65, 75, 85, 95, then 100

Why does it take 20 nickels but only 4 quarters? Because a nickel is a small amount. One quarter is worth 5 nickels (5, 10, 15, 20, 25), so 4 quarters are worth 4 groups of 5 nickels: 5, 10, 15, 20 nickels. The smaller the coin, the more of them you need to make the same amount.

In all there are 29 ways to make a dollar from nickels, dimes and quarters. (A loonie by itself is one more way!)

Worked examples

Example 1. Count the coins. 2 quarters, 4 dimes and 3 nickels.

  1. Biggest first: quarters. "25, 50."
  2. Dimes: "60, 70, 80, 90."
  3. Nickels: "95, 100, 105."
  4. 105 cents is more than 100, so take out 1 dollar: 1 dollar and 5 cents.

Example 2. Dollars and cents. 2 toonies, 3 quarters and 1 nickel.

  1. Toonies: "2 dollars, 4 dollars."
  2. Quarters: "4 dollars 25, 4 dollars 50, 4 dollars 75."
  3. Nickel: "4 dollars 80."
  4. Total: 4 dollars and 80 cents, written 4.80 on a price tag.

Example 3. Make 50 cents three ways.

  1. Quarters: 2 quarters make 50 (25, 50).
  2. Dimes: 5 dimes make 50 (10, 20, 30, 40, 50).
  3. A mix: 1 quarter, 2 dimes and 1 nickel make 25, 35, 45, 50.
  4. Check each one by counting on. All three make 50 cents. (There are more ways too.)

Example 4. Fewest coins. What is the smallest number of coins that make 40 cents?

  1. Start with the biggest coin that fits. A quarter (25) fits in 40. Now 15 cents are still needed.
  2. The biggest coin that fits in 15 is a dime (10). Now 5 cents are needed.
  3. A nickel (5) finishes it.
  4. 3 coins: a quarter, a dime and a nickel. (Four dimes also make 40, but that is 4 coins.)

Try it

  1. Count: 3 dimes and 2 nickels.
  2. Count: 1 quarter, 1 dime and 1 nickel.
  3. Count: 4 quarters and 1 dime. Write the answer in dollars and cents.
  4. Count: 1 toonie, 2 loonies and 3 dimes.
  5. How many nickels make 1 quarter? How many nickels make 1 dollar?
  6. Find two different ways to make 75 cents.
  7. What is the smallest number of coins that make 90 cents?
  8. Kai has 7 coins worth 1 dollar altogether. They are all quarters and dimes. How many of each?
Answers
  1. 40 cents. Dimes: 10, 20, 30. Nickels: 35, 40.
  2. 40 cents. 25, then 35, then 40.
  3. 1 dollar and 10 cents (1.10). Quarters: 25, 50, 75, 100. Dime: 110. 110 cents is 100 cents (1 dollar) and 10 cents.
  4. 4 dollars and 30 cents (4.30). Toonie: 2 dollars. Loonies: 3, 4 dollars. Dimes: 4 dollars 10, 20, 30.
  5. 5 nickels make a quarter (5, 10, 15, 20, 25). 20 nickels make a dollar, because a dollar is 4 quarters and each quarter is 5 nickels: 5, 10, 15, 20.
  6. Some ways: 3 quarters (25, 50, 75). Or 7 dimes and 1 nickel (10, 20, ..., 70, then 75). Or 2 quarters and 2 dimes and 1 nickel (25, 50, 60, 70, 75). Or 15 nickels. Any way that counts on to 75 is right.
  7. 5 coins. Biggest first: 3 quarters make 75, leaving 15. A dime leaves 5. A nickel finishes it. That is 3 + 1 + 1 = 5 coins.
  8. 2 quarters and 5 dimes. Try each number of quarters and fill the rest with dimes. 1 quarter and 6 dimes: 25, then 35, 45, 55, 65, 75, 85. Only 85 cents, too little. 2 quarters and 5 dimes: 25, 50, then 60, 70, 80, 90, 100. Exactly 1 dollar! 3 quarters and 4 dimes: 75, then 85, 95, 105, 115. Too much. So the answer is 2 quarters and 5 dimes. Notice: each time you swap a dime for a quarter, the total goes up by 15 cents, so only one answer can land exactly on 100.

Common mistake

Counting coins instead of cents. "I have 6 coins, so 6 something" is not counting money. Every coin must be counted by its worth: 25 for a quarter, 10 for a dime.

Mixing up the counting jumps. When switching from quarters to dimes, some children keep going by 25: "25, 50, 75, 100" for 3 quarters and a dime. Say the coin's name before you count it: "quarter, 25; quarter, 50; quarter, 75; dime, 85".

Where it leads