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Fair shares

Ratio, rates and money · level 1, lesson 1 of 5

A question to start

You have 12 crackers and 3 plates. Three friends are coming for a snack. You want every friend to get the same number of crackers, so nobody says "That's not fair!"

How many crackers go on each plate?

Here is a harder one. What if a fourth friend comes, and you need 4 plates? Will each friend get more crackers or fewer? And what if 5 friends come? Can you still share 12 crackers so that every plate is the same?

You do not need to know any big numbers to answer these questions. You only need your hands, the crackers and one clever trick. By the end of this lesson you will know the trick, and you will know why it always works.

The idea

What "fair" means

A share is your part of something. When we share fairly, every share is the same. Nobody gets more. Nobody gets less.

Look at these two ways to share 12 crackers on 3 plates.

6, 4 and 2: not fair 4, 4 and 4: fair

In the top row, all 12 crackers are on plates. But one plate has 6, one has 4 and one has only 2. That is not fair.

In the bottom row, every plate has 4 crackers. That is fair. Each friend gets 4.

The trick: one at a time

How do you make sure the plates come out the same? Here is the trick. It is called dealing, like dealing cards in a game.

  1. Put one cracker on the first plate.
  2. Put one cracker on the second plate.
  3. Put one cracker on the third plate.
  4. Go back to the first plate and start again.
  5. Keep going round and round until the crackers are all gone.

Each time you go round all the plates once, we call it a round. Here is what the plates look like after each round.

After round Plate 1 Plate 2 Plate 3 Crackers still in the box
1 1 1 1 9
2 2 2 2 6
3 3 3 3 3
4 4 4 4 0

After 4 rounds the box is empty, and every plate has 4 crackers.

Why the trick always works

This is the most important part of the lesson. Why does dealing one at a time make the plates the same?

Think about one round. In one round, plate 1 gets one cracker, plate 2 gets one cracker, and plate 3 gets one cracker. Every plate gets exactly the same: one each. No plate is skipped. No plate gets two.

So before a round, if the plates are the same, then after the round they are still the same. They all just went up by one.

And at the very start, before any round, every plate has 0 crackers. That is the same!

Same at the start. Every round keeps them the same. So at the end, after the last full round, they must still be the same. That is why dealing is fair. You do not have to hope. You can be sure.

Checking by counting

A good mathematician always checks. There are two checks.

Check 1: Are the plates the same? Count each plate. 4, 4, 4. Yes.

Check 2: Did we use every cracker? Count all the crackers on all the plates together. Count them in plates: "4 ... 8 ... 12". That makes 12, the same as the 12 we started with. So no cracker is lost and no cracker was added.

If both checks are good, the sharing is fair and complete.

Fewer friends, bigger shares

Let us share the same 12 crackers in different ways.

Number of plates Crackers on each plate Left over
2 6 0
3 4 0
4 3 0
6 2 0

Look down the table. When there are more plates, each plate gets fewer crackers. When there are fewer plates, each plate gets more. That makes sense: the same 12 crackers are spread over more friends, so each friend's share is smaller. It is like a pizza: if 2 people share it, the slices are big. If 8 people share it, the slices are small.

So the answer to the question at the start is: when a fourth friend comes, each friend gets fewer crackers, 3 instead of 4.

When there are some left over

Now share 12 crackers on 5 plates. Deal them one at a time. After 2 rounds, 10 crackers are on plates (2 on each) and 2 are left in the box. There are not enough for another full round, because a round needs 5.

2 left over

What can you do with the 2 left over? You have some choices:

  • Keep them in the box for later.
  • Break each one into pieces so everyone gets a little bit more. (That is how fractions start. You will meet them in Numbers.)
  • Give them to the dog. (Only if the dog is allowed crackers!)

What you must not do is give one extra to two of the friends. Then the shares are not the same any more.

Worked examples

Example 1. Strawberries for two. Mum has 10 strawberries for you and your brother. How many does each of you get?

  1. There are 2 bowls, so each round uses 2 strawberries: one for you, one for your brother.
  2. Deal: after round 1 you each have 1. Round 2: 2 each. Round 3: 3 each. Round 4: 4 each. Round 5: 5 each.
  3. The strawberries are all gone after 5 rounds.
  4. Check: each bowl has 5. And 5 and 5 make 10. Each of you gets 5.

Sharing between 2 is called halving. When you halve 10, you get 5. You might notice a shortcut: 5 and 5 make 10, so half of 10 is 5. Doubles facts you already know help you halve fast.

Example 2. Grapes for three. 15 grapes are shared by 3 children. How many does each child get?

  1. Each round uses 3 grapes, one for each child.
  2. Count the grapes used, round by round: 3, 6, 9, 12, 15. That is 5 rounds.
  3. Each round gave every child one grape, and there were 5 rounds. So each child has 5 grapes.
  4. Check: 5 and 5 and 5 make 15. Yes.

Notice the clever part in step 2. You did not have to move every grape one by one to know the answer. You only had to count the rounds. The number of rounds is the same as the number each child gets, because each round gives each child one.

Example 3. Cookies for four. 9 cookies are shared by 4 friends. Is it fair if each friend gets 2?

  1. Deal one at a time. Round 1 uses 4 cookies (5 left). Round 2 uses 4 more (1 left).
  2. A third round would need 4 cookies, but only 1 is left. So we stop.
  3. Each friend has 2 cookies, and 1 cookie is left over.
  4. Check: 2, 4, 6, 8 cookies on plates, and 1 left. 8 and 1 make 9. Good.
  5. To use the last cookie fairly, cut it into 4 pieces the same size, one piece for each friend.

Try it

  1. Share 6 apples between 2 children. How many does each child get?
  2. Share 8 balls into 4 baskets, the same number in each. How many in each basket?
  3. Share 12 stickers between 2 friends. Then share 12 stickers among 6 friends. Which way gives each friend more? Why?
  4. Share 7 cars between 2 children. How many does each get? Is any car left over?
  5. Max shares 12 crackers on 3 plates. He puts 5 on his plate, 4 on Lia's and 3 on Tom's. "All 12 are used, so it's fair," says Max. Is he right? How could he fix it?
  6. Share 20 blocks among 4 children by counting rounds. How many rounds did you deal?
  7. Three friends shared some crayons fairly. Each friend got 4 crayons, and none were left over. How many crayons were there at the start?
  8. You have 10 grapes. Can you share them fairly among 3 people with none left over? Among 5 people? Among 4 people?
Answers
  1. 3 each. Deal one to each child: after round 1 each has 1, after round 2 each has 2, after round 3 each has 3, and the 6 apples are used. Check: 3 and 3 make 6.
  2. 2 in each basket. One round uses 4 balls, so 2 rounds use 8 balls. Two rounds means 2 in each basket. Check: 2, 4, 6, 8.
  3. Between 2 friends: 6 each. Among 6 friends: 2 each. Sharing between 2 gives more, because the same 12 stickers are spread over fewer people. Fewer friends means bigger shares.
  4. 3 each, and 1 car is left over. Each round uses 2 cars. After 3 rounds, 6 cars are given out and 1 is left. One car is not enough for another round. (The children could take turns with it!)
  5. No. Using all 12 is good, but the plates are not the same: 5, 4 and 3. To fix it, Max should move 1 cracker from his plate to Tom's plate. Then the plates are 4, 4 and 4. That is fair.
  6. 5 rounds, and each child gets 5 blocks. Each round uses 4 blocks: 4, 8, 12, 16, 20. That is 5 rounds. Check: 5 and 5 and 5 and 5 make 20.
  7. 12 crayons. Put the 3 shares back together: 4 and 4 and 4 make 12. Going backwards from the shares to the whole is the same as counting all the plates together.
  8. Among 3: no, each gets 3 and 1 is left over (3, 6, 9, then only 1 left). Among 5: yes, each gets 2 (2 rounds use 10). Among 4: no, each gets 2 and 2 are left over (4, 8, then only 2 left).

Common mistake

Thinking "all used up" means "fair". Putting every cracker on a plate is only half the job. A share is fair only when every plate has the same number. Always do both checks: count each plate, then count all of them together.

Giving out handfuls. If you grab a handful for each plate, the handfuls will almost never be the same size. Dealing one at a time (or two at a time, as long as every plate gets the same in every round) is what makes it fair.

Where it leads