Grade by Grade
B. Number · Operations: Numbers to 100 000, Decimals, Fractions, Ratios and Rates

Equivalent Ratios and Rates

B2.9 A1About 45 minutes
Learning goals
  • I can describe a ratio as a comparison of two amounts, like 2 to 5.
  • I can make equivalent ratios and rates by multiplying or dividing both terms by the same number.
  • I can use ratio tables and double number lines to solve problems.
  • I can tell the difference between a part-to-part and a part-to-whole ratio.

Words to know

ratio
A comparison of two amounts, like 2 scoops of mix for every 5 cups of water. We can write it as 2 to 5 or 2:5.
terms
The numbers in a ratio. In 2:5, the terms are 2 and 5.
equivalent ratios
Ratios that make the same comparison, like 2:5 and 4:10.
rate
A comparison of two different kinds of amounts, like 18 km in 2 hours or $6 for 4 muffins.
unit rate
A rate for 1 of something, like 9 km per hour or $1.50 per muffin.

Remember from Grade 4

In Grade 4 you used rates like "8 legs per spider" and "$6 per hour", and showed them in tables and on double number lines. In Grade 5 you'll make equivalent ratios and rates: different-looking comparisons that mean the same thing.

Let's start with lemonade

A lemonade recipe uses 2 scoops of mix for every 5 cups of water. The ratio of scoops to cups is 2 to 5, or 2:5.

You want to make a bigger batch with 15 cups of water. How many scoops?

15 cups is 5 cups three times. So you need 2 scoops three times: 6 scoops. The ratio 6:15 tastes exactly the same as 2:5. They are equivalent ratios.

A glass pitcher of lemonade with lemon slices beside a bowl of lemonade mix with a scoop and a measuring cup of water.
Lemonade mix, a scoop and water. Keeping the ratio of scoops to cups the same keeps the taste the same. AI-generated image
2 scoops for every 5 cups, three batchesScoopsCups2 : 54 : 106 : 15Total so far2 : 5 = 4 : 10 = 6 : 15
Each dashed box is one batch: 2 scoops and 5 cups. Three batches make 6 scoops for 15 cups, the same taste as 2 for 5.
Scoops of mix2468
Cups of water5101520
A ratio table for 2 scoops to 5 cups. Each column is an equivalent ratio: 2:5 = 4:10 = 6:15 = 8:20.

How to make equivalent ratios

Multiply or divide both terms by the same number.

  • 2:5 → × 3 → 6:15
  • 12:8 → ÷ 4 → 3:2

If you multiply only one term, or add the same number to both, the ratio changes. (2:5 → 4:7 by adding 2 makes lemonade that tastes different!)

Worked example

Example 1: Find a missing term

Complete: 3:4 = □:20.

  1. Look at the terms you know: 4 → 20. That's × 5.
  2. Do the same to the other term: 3 × 5 = 15.

3:4 = 15:20

Rates

A rate compares two different kinds of things, like kilometres and hours, or dollars and muffins.

You can make equivalent rates the same way: multiply or divide both parts by the same number. A rate for 1 of something is a unit rate.

dollars036912muffins02468
A double number line. Dollars on top, muffins below. $3 for 2 muffins, $6 for 4 muffins, $9 for 6 muffins and $12 for 8 muffins.
Worked example

Example 2: Equivalent rates

Tess cycles 18 km in 2 hours. At the same speed, how far does she go in 6 hours? What is her unit rate?

  1. 6 hours is 2 hours × 3. So multiply the distance by 3 too: 18 × 3 = 54 km.
  2. Unit rate: divide both by 2. 18 ÷ 2 = 9 km in 2 ÷ 2 = 1 hour. That's 9 km per hour.
  3. Check with the unit rate: 9 km × 6 hours = 54 km. ✓

54 km in 6 hours. Her unit rate is 9 km per hour.

Worked example

Example 3: Part-to-part and part-to-whole

A bag has 12 apples: 4 red and 8 green. Write the ratio of red to green, and red to all apples, in simplest form.

  1. Red to green (part to part): 4:8. Divide both by 4: 1:2. For every 1 red apple there are 2 green ones.
  2. Red to all (part to whole): 4:12. Divide both by 4: 1:3. 1 out of every 3 apples is red.

Red to green 1:2; red to all 1:3

Worked example

Example 4: Which is the better buy?

Store A sells 4 muffins for $6. Store B sells 6 muffins for $8. Which is the better buy?

  1. Make equivalent rates with the same number of muffins. 12 works for both.
  2. Store A: 4 muffins × 3 = 12 muffins, $6 × 3 = $18.
  3. Store B: 6 muffins × 2 = 12 muffins, $8 × 2 = $16.
  4. 12 muffins cost less at Store B.

Store B is the better buy.

Ratio tips

  • Whatever you do to one term, do the same to the other term.
  • Ratio tables and double number lines help you see the pattern.
  • Keep the order of the terms the same: 2 scoops : 5 cups is not 5 scoops : 2 cups!
Just for fun

Silly what if: Keep adding 2 to both terms of the lemonade ratio: 2:5, 4:7, 6:9 ... 100:103. Soon you have almost as many scoops as cups. That isn't lemonade anymore, it's lemon syrup! That's why we multiply both terms instead.

Hands-on activity

Bead or block patterns

You need: about 30 small objects in two colours, like LEGO bricks, beads, buttons or two kinds of dry pasta (keep small objects away from young children).

  1. Make a group with 2 of colour A and 3 of colour B. Write the ratio A:B = 2:3.
  2. Make two more of the same group. Now you have 6:9. Is it still the same mix?
  3. Fill in a ratio table: 2:3, 4:6, 6:9, 8:12.
  4. Write the part-to-whole ratio for one group (A to all = 2:5).
  5. Challenge your grown-up: "If I have 10 of colour A, how many of colour B do I need?" (15)

Math talk: "Why does adding 1 to both terms change the mix?"

Your turn

1

Complete: 3:4 = 6:□.

2

Complete: 2:5 = □:20.

3

Is 4:6 equivalent to 2:3?

4

5 pencils cost $2. How much do 20 pencils cost?

5

A recipe uses 2 cups of flour to make 12 pancakes. How much flour for 36 pancakes?

6

A car travels 45 km in 3 hours in a snowstorm. What is the unit rate?

7

Write two ratios that are equivalent to 1:4.

More practice: warm-up

1

Complete: 1:2 = 2:□.

2

Complete: 1:3 = 3:□.

3

Complete: 2:1 = □:5.

4

Complete: 10:20 = 1:□.

5

A car has 4 wheels. How many wheels on 3 cars?

More practice: on your own

1

Write 6:9 in simplest form.

2

A drink mix uses 1 scoop for every 4 cups of water. How many scoops for 12 cups?

3

3 notebooks cost $12. What is the unit rate? How much do 7 notebooks cost?

4

Tess reads 20 pages in 30 minutes. At the same rate, how many pages in 90 minutes?

5

A bag has 15 marbles: 6 blue and 9 green. Write the blue-to-green ratio and the blue-to-all ratio in simplest form.

Challenge

1

Green paint is mixed using 2 cups of yellow for every 3 cups of blue. You have 10 cups of yellow. How much blue do you need? How much green paint will you make?

2

Which is the better buy: 3 juice boxes for $2 or 5 juice boxes for $3?

3

A snowplow clears 12 km of road in 3 hours. At the same rate, how long does it take to clear 20 km?

Check your understanding

Quiz

1.

Complete: 2:3 = 8:□.

2.

Which ratio is equivalent to 3:5?

3.

Max walks 18 km in 2 days. At the same rate, how far does he walk in 5 days?

4.

What is 12:16 in simplest form?

5.

A bowl has 2 red apples and 4 green apples. What is the ratio of red to green?

6.

Which is the better buy?

For parents and teachers

Big idea: B2.9 asks Grade 5 students to represent and create equivalent ratios and rates, using a variety of tools and models, in various contexts. The core idea: multiply or divide both terms by the same number to keep the same relationship.

Common difficulties:

  • Adding instead of multiplying (2:5 → 4:7). Mix a drink both ways and taste the difference!
  • Switching the order of the terms. Label each term (scoops : cups).
  • Mixing up part-to-part and part-to-whole ratios. Ask "Are we comparing to the other part, or to the whole group?"

How to help: Recipes, unit prices at the grocery store, and travel times are full of ratios and rates. Ask "If we double the recipe, how much of each?" or "Which size is the better buy?" Encourage your child to use a ratio table and to explain their reasoning (A1).