Percentages and Ratios

Lessons

Primary 6 · Mathematics

Percentages and Ratios

6 lessons

Lesson 5 of 6

Ratios and Equivalent Ratios

15 min

Let's begin

To make passion-fruit juice for a family visit, Umutoni's mother mixes 1 cup of juice concentrate with 4 cups of water. Today more visitors are coming, so she uses 3 cups of concentrate. How much water should she add so the juice tastes exactly the same?

What a ratio is

A ratio compares two or more quantities. For the juice, we say "the ratio of concentrate to water is 1 to 4", and we write 1 : 4. It means that for every 1 cup of concentrate there are 4 cups of water.

Order matters. The ratio of water to concentrate is 4 : 1, not 1 : 4.

A ratio also tells you fractions of the whole. The mixture has 1 + 4 = 5 cups, so concentrate is 15 of the juice and water is 45.

Equivalent ratios

If you multiply (or divide) both numbers of a ratio by the same number, you get an equivalent ratio. Equivalent ratios describe the same mixture, just bigger or smaller.

Cups of concentrate	1	2	3	5
Cups of water	4	8	12	20

So with 3 cups of concentrate, Umutoni's mother needs 3 × 4 = 12 cups of water. 1 : 4 = 3 : 12.

Simplifying a ratio

Divide both numbers by their greatest common factor (GCF).

Worked example

Simplify 18 : 24. The GCF of 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 18 : 24 = 3 : 4.

Same units first. Simplify 50 cm : 2 m. Change 2 m into 200 cm. Then 50 : 200. Divide both by 50 to get 1 : 4.

Finding a missing number

Worked example

3 : 5 = 12 : ?
3 was multiplied by 4 to make 12, so multiply 5 by 4 too: 5 × 4 = 20. The missing number is 20.

Is it equivalent? Is 18 : 15 equivalent to 6 : 5? 6 × 3 = 18 and 5 × 3 = 15, so yes. Is 8 : 12 equivalent to 3 : 4? 8 : 12 simplifies to 2 : 3, not 3 : 4, so no.

Explore

Open the Ratio and Proportion simulation and move the two hands so they keep a 1 : 2 ratio. Watch the screen turn green when the ratio is right, even as the hands move up and down together.

Tip

A ratio is a team: whatever you do to one number (× 3, ÷ 2 …), you must do to the other.

✦

Try it yourself

Simplify 24 : 36.
Find the missing number: 2 : 7 = 10 : ?
A class has 14 girls and 16 boys. Write the ratio of girls to boys in its simplest form.
Simplify 400 g : 2 kg.
Is 4 : 6 equivalent to 6 : 9?

Answers

The GCF is 12, so 24 : 36 = 2 : 3.
2 × 5 = 10, so 7 × 5 = 35.
14 : 16. Divide both by 2 to get 7 : 8.
2 kg = 2 000 g, so 400 : 2 000. Divide both by 400 to get 1 : 5.
Yes. Both simplify to 2 : 3.

Quick check

Simplify 15 : 25.

What you learned today

A ratio a : b means "a for every b", and the order matters.
Multiply or divide both parts by the same number to get equivalent ratios.
Simplify by dividing by the GCF, after changing to the same units.

Sources and credits

Grade 6, Unit 2, Lesson 1: Introducing Ratios and Ratio Language, by Illustrative Mathematics and Open Up Resources (IM 6–8 Math), provided by Illustrative Mathematics / Kendall Hunt, licensed under CC BY 4.0. Adapted by Ganzaa.
Grade 6, Unit 2, Lesson 5: Defining Equivalent Ratios, by Illustrative Mathematics and Open Up Resources (IM 6–8 Math), provided by Illustrative Mathematics / Kendall Hunt, licensed under CC BY 4.0. Adapted by Ganzaa.
Ratio and Proportion, version 1.2.0. Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC BY 4.0 (https://phet.colorado.edu)

1What a ratio is

2Equivalent ratios

3Simplifying a ratio

4Finding a missing number

5Explore