Create and Use Equivalent Ratio Tables

People have been comparing quantities for thousands of years. Ancient bakers needed to know how much flour goes with how much water. Builders measured lengths of wood against stone. Traders exchanged goods at agreed-upon rates. All of these situations involve ratios — a way of comparing two numbers.

Over time, mathematicians discovered that organizing ratios into tables made it much easier to spot patterns, find missing values, and compare different rates. Let's look at some key moments in that story.

Today, you use the same core idea: organize ratios in a table, find patterns, and even graph them. The big question this lesson answers is: How do we create a table of equivalent ratios, find missing values, and plot those pairs on a coordinate plane?

Before we build ratio tables, let's make sure we understand the key vocabulary. These four ideas are the building blocks of everything in this lesson.

The key rule is: to get from one row to another in a ratio table, you multiply or divide both quantities by the same number. If you only change one quantity, the ratio breaks.

Let's look at a ratio table for the ratio 2 : 3. The diagram below shows how multiplying both quantities by the same number creates equivalent ratios.

Look at the table above. The first row is the original ratio, 2 : 3. The second row multiplies both numbers by 2, giving 4 : 6. The third row multiplies both by 3, giving 6 : 9. Every row keeps the same relationship between the two quantities.

There are two main strategies for building and using ratio tables. Both rely on one golden rule: whatever you do to one quantity, you must do the same to the other.

Strategy 1: Multiply Both Quantities

For example, start with 3 : 4. If n = 5, then the new pair is 3 × 5 = 15 and 4 × 5 = 20. So 15 : 20 is equivalent to 3 : 4.

Strategy 2: Add the Ratio to Itself

With 3 : 4, the pattern would be 3, 6, 9, 12, … for quantity A and 4, 8, 12, 16, … for quantity B. You are adding 3 each time to A and 4 each time to B.

Finding a Missing Value

Suppose the ratio is 2 : 7 and you know that quantity A is now 10. The multiplier is 10 ÷ 2 = 5. So quantity B is 7 × 5 = 35. The missing value is 35.

One of the coolest things about ratio tables is that you can graph them. Each pair of values becomes an ordered pair (x, y) on the coordinate plane. When you plot all the pairs from a ratio table, they line up in a straight line that passes through the origin (0, 0).

Notice how every plotted point sits on the same straight line. This is the signature of equivalent ratios. If a point does not land on the line, the pair is not equivalent to the others.

A lemonade recipe calls for 3 cups of water for every 1 cup of lemon juice. You want to make enough for a party. Let's build a ratio table, find a missing value, and plot the pairs.

Ratio tables aren't just for extending one ratio. You can also use them to compare two different ratios. The trick is to find a common value in one of the columns. Then you can see which ratio gives a bigger or smaller value in the other column.

For example, Store A sells 3 erasers for $2. Store B sells 5 erasers for $4. Which store gives you more erasers for your money? Let's compare.

Look at the rows where the cost is the same. For $4, Store A gives you 6 erasers, but Store B gives you only 5. For $8, Store A gives 12 erasers, while Store B gives only 10. Store A is the better deal because you get more erasers for the same price.

Ratio tables are a stepping stone to bigger ideas you'll see later in math. Here is how this concept connects to what comes next.

Mastering ratio tables now makes all of these future topics much easier. You're building a strong foundation that will carry you through 7th and 8th grade math and beyond.