Verify Proportions

Using means and extremes and more

Proportion says two ratios (or fractions) are equal in value.

Example:

$Two circle diagrams representing fractions 1 third and 2 sixths next to each other$

Both 13 and 26 are ratios

We see that 1-out-of-3 is equal to 2-out-of-6

The ratios are the same, so they are in proportion.

Learn more at Proportion. Here we focus on checking or verifying a proportion.

Proportions can be written in different ways. These are the same:

35 = 610
3 / 5 = 6 / 10
3 : 5 = 6 : 10

What Are Means and Extremes?

When a proportion is written like this:

a : b = c : d

a and d are the extremes (the outside numbers)
b and c are the means (the inside numbers)

The words means and extremes describe the positions of the numbers, not their sizes.

Verifying a Proportion

To verify whether two ratios are proportional, we use this rule:

The product of the means equals the product of the extremes

This is cross multiplication. Here we use it to check the proportion.

Example: Verify this

3 / 5 = 6 / 10

Step 1: Identify means and extremes

Means: 5 and 6
Extremes: 3 and 10

Step 2: Multiply

Means: 5 × 6 = 30

Extremes: 3 × 10 = 30

They are equal! So the ratios are proportional.

Example 2: Verify this

4 : 7 = 6 : 10

Means: 7 × 6 = 42

Extremes: 4 × 10 = 40

The products are not equal, so the ratios are not proportional!

Checking Proportions Using Equivalent Fractions

Another way to check a proportion is to simplify each fraction.

Example

Check whether:

68 = 1520

Simplify both fractions:

68 ⇒ 34

1520 ⇒ 34

Since both fractions simplify to the same value, the ratios are proportional.

This method uses number sense and is often quicker when the numbers are small.

Solving vs Verifying a Proportion

It is important to know the difference:

Verifying a proportion means checking whether two ratios are equal
Solving a proportion means finding an unknown value

Example: Solve this

35 = x10

We need to solve for x: see Cross Multiply to learn how to do this.

Ratio and Rate

A ratio compares two quantities.

A rate is a special kind of ratio that compares quantities with different units, such as distance and time.

Ratio: 3 boys to 5 girls
Rate: 60 kilometers per hour

Summary

Equivalent ratios can be written as a proportion
The outside numbers are the extremes
The inside numbers are the means
If the product of the means equals the product of the extremes, the ratios are proportional
Proportions can also be checked by simplifying fractions