Skip to content

Play: Recipes, sharing and "for every"

Ratio and rates · step 1 of 5

A question to start

Your favourite lemonade is made with 1 cup of lemon juice for every 3 cups of water. Today, lots of friends are coming, so you use 4 cups of lemon juice. How much water do you need so that the lemonade tastes exactly the same?

If you said 12 cups, you are right. If you said 6 cups, you are in good company, because that is the most common guess, but the lemonade would come out far too sour. On this page you will see why, and you will learn to think in "for every", one of the most useful ideas in everyday maths.

The idea

"For every" groups

"1 cup of juice for every 3 cups of water" describes a group: 1 juice and 3 water. To make more lemonade that tastes the same, you make more of the same group. Never change what is inside a group.

Groups Juice (cups) Water (cups) Total (cups)
1 1 3 4
2 2 6 8
3 3 9 12
4 4 12 16

With 4 cups of juice you have made 4 groups, so you need 4 lots of 3 cups of water: 12 cups. The guess of 6 comes from adding: the juice went from 1 cup to 4 cups, which is 3 more, so people add 3 more cups of water too, going from 3 to 6. But adding the same amount to both changes the taste. Making more groups keeps the taste.

This "for every" comparison is called a ratio. Two mixtures with the same ratio taste the same, look the same colour, or behave the same way.

Seeing it with beads

A necklace has 2 orange beads for every 3 blue beads. One group is 2 orange and 3 blue.

Count the groups and everything else follows:

  • 3 groups have 6 orange and 9 blue beads, 15 beads in all.
  • If a necklace has 10 orange beads, it has 5 groups (5 lots of 2 make 10), so it has 5 lots of 3 blue: 15 blue.
  • If a necklace has 25 beads in all, it has 5 groups (each group is 5 beads), so 10 orange and 15 blue.

The trick every time: find how many groups there are.

Sharing fairly by groups

Grandma gives 20 sweets to two children, and says the older one gets 3 for every 2 the younger one gets. How many does each child get?

Hand them out in rounds. In each round, the older child gets 3 and the younger gets 2, so 5 sweets go out each round. 20 sweets make 4 rounds (4 lots of 5 make 20). After 4 rounds the older child has 4 lots of 3, which is 12, and the younger has 4 lots of 2, which is 8. Check: 12 and 8 make 20.

Maps: small for big

A map is a small picture of a big place, drawn so that every distance is shrunk by the same amount. On a park map where 1 square stands for 10 steps, a path 4 squares long is 40 steps long, and a pond 2 squares wide is 20 steps wide. If the map shrank some paths more than others, it would be useless for finding your way.

Speed: "for every" with time

"The bike goes 3 metres for every second" is a "for every" too, comparing distance with time. In 10 seconds it goes 10 lots of 3 metres: 30 metres. When a "for every" compares two different kinds of thing, such as metres and seconds, it is called a rate.

Worked example

The recipe: pancakes for 4 people need 2 eggs, 1 cup of milk and 2 cups of flour. How much of each do you need for 8 people? And for 6 people?

  1. For 8 people. 8 people is two lots of 4 people, so make the recipe twice: 4 eggs, 2 cups of milk and 4 cups of flour.
  2. For 6 people, step one: make a smaller group. Half the recipe feeds half as many people: 2 people need 1 egg, half a cup of milk and 1 cup of flour.
  3. Step two: build 6 out of pieces you know. 6 people is 4 people and 2 people. Add the two recipes:
    • eggs: 2 and 1 make 3 eggs;
    • milk: 1 cup and half a cup make one and a half cups;
    • flour: 2 cups and 1 cup make 3 cups.
  4. Check the taste. In the original recipe there is twice as much flour as milk, and the same number of eggs as cups of flour. For 6 people: 3 cups of flour is twice one and a half cups of milk, and there are 3 eggs and 3 cups of flour. The recipe still matches, so the pancakes will taste the same.

Try it

  1. Orange squash is made with 1 cup of squash for every 5 cups of water. How much water goes with 3 cups of squash?
  2. A necklace has 1 red bead for every 4 white beads. It has 8 red beads. How many white beads does it have? How many beads altogether?
  3. Share 30 stickers between Ali and Bea, so that Ali gets 2 for every 1 that Bea gets.
  4. On a map, 1 square stands for 5 metres. The path from the gate to the swings is 7 squares long. How long is it really?
  5. A snail crawls 2 centimetres every minute. How far does it go in a quarter of an hour (15 minutes)?
  6. Paint is mixed with 2 tins of blue for every 1 tin of yellow, to make green. Sam mixes 4 tins of blue with 4 tins of yellow. Will Sam's green be the same colour? If not, will it be more blue or more yellow?
Answers
  1. 15 cups. 3 cups of squash is 3 groups, and each group needs 5 cups of water: 5, 10, 15.
  2. 32 white beads, 40 beads in all. 8 red beads is 8 groups; each group has 4 white, so 8 lots of 4 is 32. Each group has 5 beads, and 8 lots of 5 is 40.
  3. Ali 20, Bea 10. Each round hands out 3 stickers (2 for Ali, 1 for Bea), and 30 stickers make 10 rounds. Ali gets 10 lots of 2 and Bea 10 lots of 1. Check: 20 and 10 make 30.
  4. 35 metres. 7 lots of 5 metres.
  5. 30 centimetres. 15 lots of 2 centimetres.
  6. No. The right mix has twice as much blue as yellow, so 4 tins of blue need only 2 tins of yellow. Sam used 4 tins of yellow, 2 too many, so the green will be more yellow.

Common mistake

Adding instead of making more groups. To go from 1 cup of juice to 4 cups, you did not "add 3 cups"; you made "4 times as much". The water must also be 4 times as much, not "3 cups more". Adding the same amount to both parts of a mixture changes its taste, colour or strength. Making more of the same group never does.

Where it leads