What a ratio compares

What you will learn

• Read a ratio as a comparison in equal units
• Find equivalent ratios and the simplest form
• Connect a ratio to the fraction of the total

Models used: ratio bar models, bar models, fraction strips, stories.

Before you start

• Simplest form
• What a ratio compares

A ratio is a comparison in equal units

There are 2 red counters, 3 blue counters and 1 green counter. How do we write their ratio?

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

   Count the red and blue groups first. In the order red to blue, their ratio is 2 to 3.

   Red : blue = 2 : 3

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

   Red112 altogether

   Blue1113 altogether

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

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

   Include the green group as the third quantity. Keep red first, blue second and green third.

   Red : blue : green = 2 : 3 : 1

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

   Red112 altogether

   Blue1113 altogether

   Green11 altogether

   Red : Blue : Green = 2 : 3 : 1 · 6 units = 6 · 1 unit = 1

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

   The order matters. If we name blue before red, the first two numbers must change places too.

   Blue : red : green = 3 : 2 : 1

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

   Blue1113 altogether

   Red112 altogether

   Green11 altogether

   Blue : Red : Green = 3 : 2 : 1 · 6 units = 6 · 1 unit = 1

Worked example

Simplify 12 : 18

1. Find the largest number that divides both.

   What is the largest common factor of 12 and 18?

   6

2. Divide both parts by 6.

   12 ÷ 6 = 2 and 18 ÷ 6 = 3

   What is 12 : 18 in its simplest form?

   2:3

3. So 12 : 18 = 2 : 3.

   First

   Second

   First : Second = 2 : 3

4. Check: multiplying 2 : 3 by 6 gives back 12 : 18.

   2 : 3 = 12 : 18 ✓

Common mistakes to discuss

• Reading a ratio as a difference. 3 : 2 does not mean “one more”. It means three units to two units, whatever one unit is worth.
• Treating ratio numbers as the actual amounts. 3 : 2 could be 3 and 2, or 30 and 20. The units keep the comparison the same.

Check your understanding

• read 3 : 2 as three units to two units
• simplify 12 : 18 to 2 : 3
• find the missing term in 3 : 2 = 12 : □
• say what fraction of the whole each part is

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.