Skip to content

Ratio tables

Ratio, rates and money · level 2, lesson 5 of 6

A question to start

Your lemonade stand uses a family recipe: 2 lemons for every 5 cups of water. It is a hot day, and you want to fill a big jug that holds 20 cups of water. How many lemons do you need?

Many people guess "17 lemons", because 20 is 15 more than 5, so they add 15 lemons to the 2. That lemonade would be so sour nobody could drink it! Others guess "8". They are right.

In this lesson you will meet a simple tool that makes questions like this easy: the ratio table. It is just a "for every" list written in rows, and it never lets you down.

The idea

One batch

A batch is the amount you make with the recipe once. One batch of our lemonade is

2 lemons and 5 cups of water.

2 lemons5 cups of water

The recipe says for every 2 lemons, use 5 cups of water. To make more lemonade that tastes the same, make more batches. Never change what goes into one batch.

Building a ratio table

A ratio table has one column for each thing in the recipe, and one row for each number of batches.

Batches Lemons Cups of water
1 2 5
2 4 10
3 6 15
4 8 20
5 10 25

Each new row is one more batch, so it adds 2 lemons and 5 cups. The lemons column counts by twos: 2, 4, 6, 8, 10. The water column counts by fives: 5, 10, 15, 20, 25.

Now answer the question from the start. Look down the water column for 20. It is in the row for 4 batches. Look across: 8 lemons. That is the answer.

Why every row tastes the same

Every row is made of whole batches. And every batch is exactly the family recipe. If you poured each batch into its own glass, every glass would taste the same. Pour them all into one jug, and the jug tastes the same too, because nothing else was added.

So every row of a ratio table describes lemonade that tastes the same. Only the amount changes.

Shortcuts: you do not have to go one row at a time

Sometimes you need a row far down the table. Counting one row at a time is slow. Here are three shortcuts. Each one works because each one is still "just more batches".

Double a row. 2 batches are 4 lemons and 10 cups. Double them: 4 batches are 8 lemons and 20 cups. Doubling both numbers in a row means making that many batches twice.

Add two rows. 2 batches (4 lemons, 10 cups) and 3 batches (6 lemons, 15 cups) together make 5 batches: 4 and 6 lemons make 10, and 10 and 15 cups make 25. Adding two rows means pouring two jugs together.

Times a row by any number. 10 batches: 10 lots of 2 lemons is 20, and 10 lots of 5 cups is 50.

Batches Lemons Cups of water How we got it
1 2 5 the recipe
2 4 10 double row 1
4 8 20 double row 2
6 12 30 add rows 2 and 4
10 20 50 10 lots of row 1

There is one thing you must never do: add the same number to both columns. Adding 1 lemon and 1 cup of water to a batch is not a batch any more.

What goes wrong when you add

The wrong guess at the start was 17 lemons for 20 cups. Let us see why it is so bad. The right amount is 8 lemons. 17 lemons is more than twice that! In the right lemonade, the 20 cups of water share 8 lemons. In the wrong lemonade, they share 17. Each cup would get more than twice as much lemon, and the lemonade would be far too sour.

The mistake came from comparing by how many more instead of how many times as many (see Twice as many, half as much). The water went from 5 cups to 20 cups. That is 4 times as much water, so the lemons must be 4 times as much too: 4 lots of 2 is 8.

Going either way

A ratio table works from either column.

  • Know the water, want the lemons: find the water in its column, then read across.
  • Know the lemons, want the water: find the lemons in their column, then read across.

If the number you know is not in the table yet, keep adding rows (or use a shortcut) until it appears.

Worked examples

Example 1. Lemons to water. You have 12 lemons. How many cups of water do you need?

  1. Look for 12 in the lemons column. From the shortcut table: 6 batches is 12 lemons.
  2. Read across: 30 cups of water.
  3. Check: 12 lemons is 6 groups of 2 lemons, so 6 batches. 6 batches need 6 groups of 5 cups: 5, 10, 15, 20, 25, 30.

Example 2. Water to lemons. You have a giant bucket that holds 35 cups of water. How many lemons?

  1. Count the water by fives until you reach 35, counting the batches: 5, 10, 15, 20, 25, 30, 35. That is 7 batches.
  2. 7 batches need 7 groups of 2 lemons: 2, 4, 6, 8, 10, 12, 14.
  3. You need 14 lemons.

Example 3. A new recipe. Trail mix uses 3 handfuls of nuts for every 1 handful of raisins. Make a ratio table up to 4 batches. How many handfuls of nuts go with 4 handfuls of raisins?

Batches Nuts Raisins
1 3 1
2 6 2
3 9 3
4 12 4

4 handfuls of raisins are in the row for 4 batches: 12 handfuls of nuts.

Example 4. A money table. Juice boxes cost 2 dollars for every 3 boxes. How much do 12 boxes cost?

Boxes Dollars
3 2
6 4
12 8
  1. Start with the deal: 3 boxes for 2 dollars.
  2. Double: 6 boxes for 4 dollars.
  3. Double again: 12 boxes for 8 dollars.

A price is a "for every" too: 2 dollars for every 3 boxes. Ratio tables work for money just as well as for lemonade.

Example 5. Does it taste the same? Ali makes lemonade with 6 lemons and 12 cups of water. Is it the family recipe?

  1. With the family recipe, 6 lemons is 3 batches, which need 15 cups of water.
  2. Ali used only 12 cups. That is less water for the same lemons.
  3. So Ali's lemonade is more sour than the family recipe. Not the same.

Try it

  1. Make a ratio table for "3 apples for every 2 pears", up to 5 batches.
  2. A paint mix uses 1 cup of blue for every 4 cups of white. How much white goes with 3 cups of blue?
  3. Using the lemonade recipe (2 lemons for every 5 cups of water), how many lemons for 15 cups of water?
  4. Using the lemonade recipe, how much water goes with 16 lemons? (Try doubling.)
  5. Stickers cost 1 dollar for every 4 stickers. How many stickers can you buy with 6 dollars?
  6. A bead pattern has 2 red beads for every 3 white beads. A necklace has 12 white beads. How many red beads?
  7. Sara's lemonade has 4 lemons and 8 cups of water. Is it more sour or less sour than the family recipe?
  8. Leo says: "2 lemons for 5 cups, so 3 lemons for 6 cups, so 4 lemons for 7 cups." What is wrong?
Answers
  1. Apples count by threes and pears by twos, one row for each batch: 1 batch is 3 apples and 2 pears; 2 batches are 6 and 4; 3 batches are 9 and 6; 4 batches are 12 and 8; 5 batches are 15 and 10.
  2. 12 cups of white. 3 cups of blue is 3 batches; each batch has 4 cups of white: 4, 8, 12.
  3. 6 lemons. 15 cups is 3 batches (5, 10, 15), and 3 batches need 2, 4, 6 lemons.
  4. 40 cups. 8 lemons go with 20 cups (4 batches). Double that row: 16 lemons go with 40 cups (8 batches).
  5. 24 stickers. 6 dollars is 6 batches of "1 dollar, 4 stickers": 4, 8, 12, 16, 20, 24.
  6. 8 red beads. 12 white beads is 4 groups of 3 (3, 6, 9, 12), so 4 batches. 4 batches need 4 groups of 2 red: 2, 4, 6, 8.
  7. More sour. With the family recipe, 4 lemons (2 batches) need 10 cups of water. Sara used only 8 cups, so there is more lemon in each cup.
  8. Leo added 1 to both each time. That is not making batches. 3 lemons for 6 cups is more sour than the recipe, and 4 lemons for 7 cups is even more sour. The real table goes 2 and 5, then 4 and 10, then 6 and 15.

Common mistake

Adding the same number to both columns. It feels natural, because it keeps the gap the same. But a ratio table is built from whole batches, so each column counts by its own number: the lemons by 2s, the water by 5s.

Mixing up the columns. Before you read across, say what each column means: "lemons here, water there". Reading "5 lemons and 2 cups" instead of "2 lemons and 5 cups" makes a completely different drink.

Where it leads