Grade by Grade
B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Solving Problems with Ratios and Rates

B2.12 B2.1 A1About 45 minutes
Learning goals
  • I can write part-to-part and part-to-whole ratios.
  • I can use ratio tables and unit rates to solve problems.
  • I can compare ratios and rates to decide which is greater or which is the better buy.

Words to know

ratio
A comparison of two quantities of the same kind, like 3 cups of cereal to 1 cup of raisins (3:1).
rate
A comparison of two different kinds of quantities, like 80 km in 1 hour.
unit rate
A rate for one unit, like $3 per notebook or 75 km per hour.
equivalent ratios
Ratios that make the same comparison, like 2:3 and 8:12.
ratio table
A table that shows equivalent ratios, made by multiplying or dividing both terms by the same number.
scale
A ratio that compares a distance on a map or model to the real distance.

Remember from Grade 5

In Grade 5 you made equivalent ratios and rates by multiplying or dividing both terms by the same number (2:5 = 6:15), found unit rates, and used ratios to find a better buy. In Grade 6 you'll use these tools to solve harder problems, including problems where you need to go through the unit rate to get the answer.

Let's start with trail mix

A trail mix recipe uses 3 cups of cereal for every 2 cups of raisins. You have 12 cups of cereal. How many cups of raisins do you need?

12 is 3 × 4, so you need 2 × 4 = 8 cups of raisins. The ratios 3:2 and 12:8 are equivalent: the trail mix tastes the same.

A glass bowl of trail mix made of cereal and raisins, with one measuring cup of cereal and one of raisins beside it.
A recipe ratio such as 3 cups of cereal for every 2 cups of raisins keeps every batch tasting the same, whether you make a little or a lot. AI-generated image
Trail mix: 3 cups cereal for every 2 cups raisinsbatch 1batch 2batch 3batch 412 cups cereal8 cups raisins1 cup of cereal1 cup of raisins1 batch: 3 : 2× 44 batches: 12 : 8
Four batches of 3 : 2 give 12 : 8. Every batch has the same mix, so the ratios are equivalent and the trail mix tastes the same.
Cereal (cups)3691215
Raisins (cups)246810
A ratio table for the trail mix. Each column is the first column multiplied by the same number.

Tools for ratio problems

  • Ratio table: multiply or divide both terms by the same number.
  • Unit rate: find the amount for 1, then multiply. This is handy when one number isn't a multiple of the other.
  • Part-to-part or part-to-whole: in a class with 12 girls and 15 boys, girls to boys is 12:15 = 4:5. Girls to all students is 12:27 = 4:9.
  • Simplest form: divide both terms by a common factor until they share no factor except 1.
Worked example

Example 1: Go through a simpler ratio

6:8 = 15:?

  1. 15 is not a whole-number multiple of 6, so you can't multiply 6:8 straight to 15:?. Go through a smaller ratio first.
  2. Divide both terms of 6:8 by 2: 3:4.
  3. 15 = 3 × 5, so multiply both terms by 5: 3:4 = 15:20.

6:8 = 15:20

Worked example

Example 2: Unit rate

A car travels 240 km in 3 hours on the highway. At the same speed, how far does it go in 5 hours?

  1. Find the unit rate: 240 ÷ 3 = 80 km per hour.
  2. Multiply: 80 × 5 = 400 km.

400 km in 5 hours.

A double number line for 240 km in 3 hourskmhours008011602240332044005givenunit rate: 80 km/hanswer
Kilometres on top, hours below, lined up. Divide 240 by 3 to get the unit rate (80 km in 1 hour), then count up to 400 km in 5 hours.
Worked example

Example 3: Which is the better buy?

A 750 mL bottle of maple syrup costs $12.00. A 1 L bottle costs $15.00. Which is the better buy?

  1. Compare the price for the same amount. Use 250 mL (750 ÷ 3 = 250 and 1 000 ÷ 4 = 250).
  2. 750 mL bottle: $12.00 ÷ 3 = $4.00 per 250 mL.
  3. 1 L bottle: $15.00 ÷ 4 = $3.75 per 250 mL.

The 1 L bottle is the better buy (if you will use all of it).

Worked example

Example 4: Compare two ratios

Juice A is 2 cups of concentrate to 3 cups of water. Juice B is 3 cups of concentrate to 5 cups of water. Which tastes stronger?

  1. Make the water amounts the same. Multiply A by 5: 10:15. Multiply B by 3: 9:15.
  2. With 15 cups of water, A has 10 cups of concentrate and B has only 9.

Juice A is stronger.

Order matters

A ratio can be written three ways: 3 to 2, 3:2 or 3/2. Always keep the terms in the same order as the words. "Cereal to raisins" is 3:2, but "raisins to cereal" is 2:3.

Just for fun

Funny but true: it takes about 40 L of maple sap to make just 1 L of maple syrup. That's a 40 : 1 ratio, which is why you should never waste a drop on the kitchen floor.

Hands-on activity

Heartbeat rates

You need: a clock or timer that shows seconds, paper and a pencil.

  1. Sit quietly. Find your pulse on your wrist or neck (a grown-up can help).
  2. Count the beats for 15 seconds. Multiply by 4 to find your rate in beats per minute. Why multiply by 4?
  3. Make a ratio table: beats in 15 s, 30 s, 45 s and 60 s.
  4. Do 20 jumping jacks in a safe, open space. Measure your rate again. How did it change?
  5. Find a grown-up's resting rate too and compare.

Math talk: "If your heart beats 18 times in 15 seconds, how many times in an hour?" (72 per minute × 60 = 4 320)

Your turn

1

2:3 = 8:?

2

5:4 = ?:20

3

Write 12:16 in simplest form.

4

6 notebooks cost $18. What is the unit price?

5

A train goes 150 km in 2 hours. What is its speed in km per hour?

6

A class has 12 girls and 15 boys. What is the ratio of girls to all students, in simplest form?

7

9:12 = 6:?

More practice: warm-up

1

There are 4 cats and 7 dogs. Write the ratio of cats to dogs.

2

3:5 = 6:?

3

Write 10:15 in simplest form.

4

You earn $5 per hour. How much in 4 hours?

5

Is 2:3 the same as 3:2?

More practice: on your own

1

A recipe uses 3 cups of flour for 2 eggs. How many eggs for 9 cups of flour?

2

Lee swims 4 laps in 6 minutes. At that rate, how long for 10 laps?

3

500 g of cheese costs $4.50. 750 g costs $6.00. Which is the better buy?

4

On a map, 1 cm stands for 5 km. Two towns are 7.5 cm apart on the map. How far apart are they really?

5

Sam scores on 12 of 20 shots. Ali scores on 14 of 25. Who has the better ratio of goals to shots?

Challenge

1

A sugar shack uses 40 L of sap to make 1 L of maple syrup. How much sap for 2.5 L of syrup? How much syrup from 600 L of sap?

2

A bead pattern uses 2 red beads for every 5 white beads. A finished bracelet has 28 beads. How many of each colour?

3

Paint mix A is 3 cans of yellow to 4 cans of blue. Mix B is 5 cans of yellow to 7 cans of blue. Which mix has a greater ratio of yellow to blue?

Check your understanding

Quiz

1.

Which ratio is equivalent to 3:4?

2.

A car goes 210 km in 3 hours. What is the unit rate?

3.

Which is the better buy: 4 juice boxes for $6.00 or 6 juice boxes for $8.40?

4.

What is 15:25 in simplest form?

5.

2:7 = ?:35

6.

A bag has 5 red and 3 blue marbles. What is the ratio of red marbles to all marbles?

For parents and teachers

Big idea: B2.12 asks Grade 6 students to solve problems involving ratios, including percents and rates, using appropriate tools and strategies. This lesson covers ratios and rates; the next covers percents. Grade 5 introduced equivalent ratios and unit rates, so Grade 6 focuses on solving problems: going through a unit rate, comparing ratios and choosing the better buy.

Common difficulties:

  • Adding instead of multiplying (2:5 → 4:7). Equivalent ratios come from multiplying or dividing both terms.
  • Switching the order of the terms.
  • Mixing part-to-part and part-to-whole. Ask "Compared with what?"

How to help: Compare unit prices at the grocery store, scale recipes up or down, and read map scales. The 40 L of sap to 1 L of syrup in the challenge is one sugar shack's ratio given for the problem; the real amount varies with the sap. Ask your child to explain their strategy and why it works (A1).