B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional Reasoning

Comparing Rates and Ratios: Which Is Better?

B2.8 B2.1 A1About 45 minutes

Learning goals

I can compare two proportional situations using unit rates.
I can compare ratios by making a common term, or by using fractions, decimals or percents.
I can find the better buy using a unit price, and explain when the better buy might not be the best choice.
I can explain my comparison clearly.

Words to know

ratio
A comparison of two quantities with the same kind of unit, like 2 cups of concentrate to 5 cups of water (2:5).

rate
A comparison of two quantities with different units, like 360 km per 30 L.

unit rate
A rate for 1 unit of the second quantity, like 12 km per litre or $1.55 per litre.

unit price
The cost for 1 unit (or 100 units) of an item, like $0.32 per 100 mL.

proportional situation
A situation in which two quantities change at a constant rate, so their ratio always stays the same.

Remember from Grade 7

In Grade 7 you decided whether a situation was proportional and solved problems with unit rates, ratio tables and proportions. In Grade 8 you will compare two proportional situations to decide which is faster, cheaper, stronger or better, and explain how you know.

Let's start in the grocery aisle

Maple syrup comes in two sizes:

250 mL for $8.75
540 mL for $17.28

The bigger bottle costs more, but is it a better deal? Compare the cost for the same amount, like 100 mL.

Small: 8.75 ÷ 250 = $0.035 per mL, so $3.50 per 100 mL.
Large: 17.28 ÷ 540 = $0.032 per mL, so $3.20 per 100 mL.

The large bottle is the better buy, by 30 cents per 100 mL.

Comparing the cost for the same amount (100 mL) shows the large bottle is cheaper: $3.20 instead of $3.50.

Ways to compare

Unit rate. Find the amount for 1 (or 100) of the same thing in both situations, then compare. Example: km per litre, dollars per kilogram, words per minute.
Common term. Scale both ratios so one term matches. 2:5 and 3:8 become 6:15 and 6:16. With the same concentrate (6), the first has less water, so it is stronger.
Fraction, decimal or percent. Write each as part of a whole. 18 out of 24 = 75% and 21 out of 30 = 70%.

The key rule: compare the same thing per the same amount.

Situation	Unit rate	Which is better?
Ava runs 5 km in 30 min	30 ÷ 5 = 6 min per km
Ben runs 8 km in 44 min	44 ÷ 8 = 5.5 min per km	Ben: fewer minutes per kilometre means faster
Job A pays $66 for 4 h	66 ÷ 4 = $16.50 per hour	Job A: more dollars per hour
Job B pays $117 for 7.5 h	117 ÷ 7.5 = $15.60 per hour

Unit rates make different-sized situations easy to compare. Watch whether a bigger or a smaller unit rate is better.

Worked example

Example 1: Faster runner

Ava runs 5 km in 30 min. Ben runs 8 km in 44 min. Who runs faster on average?

Ava: 30 ÷ 5 = 6 min per km.
Ben: 44 ÷ 8 = 5.5 min per km.
Fewer minutes per kilometre means faster.

Ben is faster on average.

Worked example

Example 2: Stronger juice

Mix A uses 2 cups of concentrate to 5 cups of water. Mix B uses 3 cups of concentrate to 8 cups of water. Which tastes stronger?

Make a common term for concentrate: 2:5 = 6:15 and 3:8 = 6:16.
With 6 cups of concentrate, Mix A has 15 cups of water and Mix B has 16.
Less water for the same concentrate means stronger.

Mix A is stronger.

With the same 6 cups of concentrate, Mix A uses 15 cups of water and Mix B uses 16. Less water means Mix A is stronger.

Worked example

Example 3: Better shooter

Leila made 18 of 24 free throws. Sam made 21 of 30. Who has the better success rate?

Leila: 18/24 = 3/4 = 75%.
Sam: 21/30 = 7/10 = 70%.

Leila has the better rate.

Is the better buy always best?

A bigger package is often cheaper per unit, but not always. Always check. And a better unit price only helps if you will actually use it all before it spoils. Explaining this kind of thinking is part of making good decisions (and links to your Financial Literacy unit).

Hands-on activity

Best-buy detective

You need: two sizes of the same product at home or in a store flyer (cereal, juice, rice, pasta or yogurt), a calculator, paper.

Write the size and price of each.
Find the unit price per 100 g or per 100 mL for each. (Divide the price by the size, then multiply by 100.)
Decide which is the better buy and by how much.
Discuss: would your family use the bigger size before it goes stale or expires?

Model answer: cereal 425 g for $5.49 → 5.49 ÷ 425 × 100 ≈ $1.29 per 100 g. Cereal 750 g for $8.99 → 8.99 ÷ 750 × 100 ≈ $1.20 per 100 g. The larger box is about 9 cents cheaper per 100 g.

Just for fun

Riddle: Which is the better deal, 1 joke for free or 10 jokes for free?

Unit price check: both cost $0 per joke. The second deal just takes ten times longer to groan through.

Your turn

Milk: 4 L for $6.20 or 2 L for $3.50. Which is the better buy?

A car goes 360 km on 30 L of gas. A truck goes 480 km on 48 L. Which uses gas more efficiently?

Ana types 270 words in 6 min. Ravi types 352 words in 8 min. Who types faster?

In Class A, 18 of 30 students walk to school. In Class B, 15 of 24 walk. Which class has a greater fraction of walkers?

Lemonade 1 uses 2 scoops for every 3 cups of water. Lemonade 2 uses 3 scoops for every 5 cups. Which is stronger?

Yogurt: 650 g for $3.90 or 175 g for $1.25. Which is the better buy?

Town A gets 12 cm of snow in 8 h. Town B gets 9 cm in 6 h. Which snowfall rate is greater?

More practice: warm-up

5 pens cost $4.00. What is the unit price?

A train travels 120 km in 2 h. What is its speed?

Which is greater: 3/4 or 7/10?

12 apples for $6 or 8 apples for $4. Which is cheaper per apple?

You earn $45 in 3 hours. What is your hourly rate?

More practice: on your own

Nuts: 400 g for $6.00 or 1 kg for $13.50. Which is the better buy?

Swimmer A swims 100 m in 80 s. Swimmer B swims 150 m in 114 s. Who is faster on average?

Recipe A uses 3 cups of flour for 24 cookies. Recipe B uses 2 1/2 cups for 18 cookies. Which uses more flour per cookie?

Team Maple scored on 9 of 40 shots. Team Pine scored on 7 of 28. Which team had the better shooting percentage?

Phone data: $36 for 6 GB or $50 for 10 GB. Which costs less per GB? Is it always the better choice?

Challenge

Rice comes in 2 kg for $6.98, 4 kg for $12.48 and 8 kg for $25.92. Which is the best buy?

Trail A climbs 150 m over 2 km. Trail B climbs 240 m over 3.2 km. Which trail is steeper on average?

Juice mix A is 3 cans of concentrate to 7 cans of water. Mix B is 5 cans to 12 cans. Which is stronger? Show two ways.

Check your understanding

Quiz

Which is the better buy: 500 g for $4.00 or 750 g for $5.25?

Car A travels 240 km in 3 h. Car B travels 320 km in 4 h. Which is faster on average?

6 tickets cost $45. What is the unit price?

Which success rate is higher: 14 out of 20 or 17 out of 25?

Which drink is stronger: 1 scoop to 4 cups of water, or 2 scoops to 6 cups?

Cheese costs $1.45 per 100 g. How much does 1 kg cost?

For parents and teachers

Big idea: Ontario's Grade 8 expectation B2.8 includes comparing proportional situations. The main tool is the unit rate: compare the same thing per the same amount. This also uses B2.1 (problems with ratios, rates and percents).

Common difficulties:

Dividing the wrong way round (km per minute versus minutes per km). Both work, but the student must know whether a bigger or smaller number is better.
Assuming the biggest package is always cheapest.
Rounding too early. Keep extra decimal places until the end.

How to help: At the store, look at the unit price on shelf labels together, then check one yourselves. Talk about whether the better buy is the right choice for your family.