Math Core

You are making lemonade for a party. The recipe uses 2 lemons for every 5 cups of water, but you need a much bigger batch. A ratio table keeps track of the amounts as the recipe grows, so you never lose count.

Building a ratio table

A ratio table lists pairs of amounts that all follow the same ratio. Each column is one possible batch.

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

Each new column adds another group of 2 lemons and 5 cups. You can also think of each column as the first column multiplied by a number: the column with 8 lemons is $2 \times 4$ lemons and $5 \times 4$ cups of water.

Ratio tables

To make a new column in a ratio table, do the same thing to both rows:

Multiply both numbers by the same number, or
Divide both numbers by the same number, or
Add two columns you already have.

Every column then shows the same ratio.

Filling in missing values

When a number is missing, look for what the known row was multiplied or divided by. Then do the same to the other row.

Worked example: Multiplying to fill a gap

A car wash uses 3 gallons of soap for every 40 cars. Complete the table.

Gallons of soap	3	6	?	30
Cars	40	?	200	?

$3 \times 2 = 6$ , so the cars also double: $40 \times 2 = 80$ .
$40 \times 5 = 200$ , so the soap is also multiplied by 5: $3 \times 5 = 15$ gallons.
$3 \times 10 = 30$ , so the cars are $40 \times 10 = 400$ .

Gallons of soap	3	6	15	30
Cars	40	80	200	400

Going down, then up

Sometimes it helps to divide first to reach a small, friendly column, then multiply to reach the amount you want.

Worked example: Dividing first

Twelve bottles of juice cost $18. How much do 10 bottles cost?

Bottles	12	2	10
Cost (dollars)	18	3	15

Divide both rows by 6: $12 \div 6 = 2$ bottles and $18 \div 6 = 3$ dollars.
Multiply both rows by 5: $2 \times 5 = 10$ bottles and $3 \times 5 = 15$ dollars.

Ten bottles cost $15.

Common mistake

Don't add the same number to both rows. Starting from 2 lemons and 5 cups, adding 1 to each gives 3 lemons and 6 cups. That is a different ratio, so the lemonade tastes different! Adding a whole column (2 more lemons and 5 more cups) is fine, because that is one more batch.

Graphing a ratio table

You can turn each column of a ratio table into a point $(x, y)$ on a coordinate plane. Use the top row for $x$ and the bottom row for $y$ . Here are the lemonade batches:

Lemons (x) and cups of water (y) for five batches of lemonadeOpen in grapher →

Notice two things:

The points lie on a straight line.
If you extended that line, it would pass through $(0, 0)$ . Zero lemons go with zero cups of water.

This is true for every ratio table. If a point is not on the line, it does not belong to the same ratio.

Worked example: Reading a graph

The graph shows the number of pages Leo reads and the minutes it takes him.

Pages read (x) and minutes (y)Open in grapher →

How many minutes does it take Leo to read 9 pages? How long for 12 pages?

The point $(9, 36)$ means 9 pages take 36 minutes. Each step to the right of 3 pages goes up 12 minutes, so 12 pages take $36 + 12 = 48$ minutes. You can check with a table: $3 : 12$ times 4 is $12 : 48$ . ✓

Tip

To check a column, divide the bottom number by the top number. In the lemonade table, $5 \div 2$ , $10 \div 4$ and $25 \div 10$ all equal $2.5$ . If one column gives a different answer, it has a mistake.