Find equivalent ratios and simplify ratios

What you will learn

• Find equivalent ratios and simplify ratios

Models used: number sentences, stories, diagrams.

Before you start

• What a ratio compares

Resize every ratio unit together

Four red counters and six blue counters have ratio 4 : 6. Can we describe the same comparison with smaller numbers?

1. All the small boxes represent equally sized units, even when they belong to different rows.

   Start with four red counters and six blue counters. At first, each box stands for one counter.

   4 : 6

   The counts of equal units describe the comparison. Their actual size comes from a given quantity or total.

   Red11114 altogether

   Blue1111116 altogether

   Red : Blue = 4 : 6 · 10 units = 10 · 1 unit = 1

2. Each box now represents two counters. The total amounts stay four and six.

   Bundle counters in twos in both groups. Red has two bundles and blue has three; no counters have been lost.

   4 : 6 = 2 : 3

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red224 altogether

   Blue2226 altogether

   Red : Blue = 2 : 3 · 5 units = 10 · 1 unit = 2

3. All the small boxes represent equally sized units, even when they belong to different rows.

   Both ratio terms were divided by the same number. Multiplying every term by the same number reverses the change.

   4 ÷ 2 : 6 ÷ 2 = 2 : 3

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red224 altogether

   Blue2226 altogether

   Red : Blue = 2 : 3 · 5 units = 10 · 1 unit = 2

Worked example: A new example, together

Simplify 24:30.

1. All the small boxes represent equally sized units, even when they belong to different rows.

   Keep the quantity labels in order. Distinguish one part from the sum of all ratio parts.

   Dividing every term by the same common factor preserves the comparison.

   Red

   Blue

   Total amount: 54

2. All the small boxes represent equally sized units, even when they belong to different rows.

   Find the greatest factor common to every term: 6.

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red

   Blue

   Total amount: 54

   Simplify 24:30.

   4:5

3. All the small boxes represent equally sized units, even when they belong to different rows.

   Find the greatest factor common to every term: 6.

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red

   Blue

   Red : Blue = 4 : 5 · 9 units = 54

4. All the small boxes represent equally sized units, even when they belong to different rows.

   Divide every term by 6.

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red

   Blue

   Red : Blue = 4 : 5 · 9 units = 54

5. All the small boxes represent equally sized units, even when they belong to different rows.

   The simplest ratio is 4:5.

   One unit must have the same value in every row. That lets us compare amounts by counting their units.

   Red

   Blue

   Red : Blue = 4 : 5 · 9 units = 54

6. All the small boxes represent equally sized units, even when they belong to different rows.

   Check the result against the question we started with.

   Multiply all simplified terms by 6 to recover 24:30.

   We found 4:5. Dividing every term by the same common factor preserves the comparison.

   Red

   Blue

   Red : Blue = 4 : 5 · 9 units = 54

Common mistakes to discuss

• Adding the same amount to both. Equivalent ratios multiply or divide both parts by the same number. 2 : 3 equals 4 : 6, not 3 : 4.

Check your understanding

• Find equivalent ratios and simplify ratios
• explain a ratio through labelled equal units
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.