B. Number · Operations: Exponents, Factors, Fractions, Decimals, Integers, Percents and Proportions

Proportional and Non-Proportional Situations

B2.10 B2.1 A1About 45 minutes

Learning goals

I can tell whether a situation is proportional or non-proportional, and explain how I know.
I can solve proportional problems using unit rates, ratio tables and scale factors.
I can solve multi-step problems with ratios, rates, fractions, decimals and percents.

Words to know

ratio
A comparison of two quantities, like 3 cups of flour to 2 cups of milk (3:2).

rate
A comparison of two quantities with different units, like 270 km in 3 h.

unit rate
A rate for one unit, like 90 km per hour or $1.50 per notebook.

proportional
Two quantities are proportional when their ratio is always the same. Doubling one doubles the other, and 0 of one means 0 of the other.

proportion
A statement that two ratios are equal, like 3/4 = 9/12.

scale factor
The number you multiply both terms of a ratio by to get an equivalent ratio.

Remember from Grade 6

In Grade 6 you made equivalent ratios with ratio tables, found unit rates, and solved ratio, rate and percent problems. In Grade 7 you will learn to tell when a situation is proportional (when ratios stay the same) and when it is not, and you will use proportional reasoning to solve problems.

Let's start with two snow-day plans

Plan A: hot chocolate. Each mug takes 3 spoonfuls of mix. 1 mug → 3 spoons, 2 mugs → 6, 4 mugs → 12.

Plan B: bowling. Shoe rental is $4, plus $6 per game. 1 game → $10, 2 games → $16, 4 games → $28.

In Plan A, doubling the mugs doubles the spoons. Every pair has the same ratio: 3 spoons per mug. This is proportional.

In Plan B, doubling the games does not double the cost ($10 → $16). The $4 shoe fee is paid only once. This is non-proportional.

A double number line for the hot chocolate: every mug lines up with 3 spoonfuls, and 0 mugs lines up with 0. That is what proportional looks like.

Test	Proportional (hot chocolate)	Non-proportional (bowling)
Is the ratio the same for every pair?	3/1 = 6/2 = 12/4 = 3 ✓	10/1 = 10, 16/2 = 8 ✗
Does doubling one double the other?	2 mugs → 6, 4 mugs → 12 ✓	2 games → $16, 4 games → $28 ✗
Does 0 of one give 0 of the other?	0 mugs → 0 spoons ✓	0 games → still $4 for shoes ✗
Graph	Straight line through (0, 0)	Does not go through (0, 0)

Three quick tests for proportional situations.

Two graphs of cost. Apples at $2 per kg (solid line) start at 0 and go up $2 for each kg: proportional. A taxi that charges $4 to start plus $2 per km (dashed line) starts at $4: non-proportional, even though both go up by the same amount each step.

Solving proportional problems

When a situation is proportional, you can use any of these tools:

Unit rate: find the amount for 1, then multiply.
Ratio table: multiply or divide both terms by the same number.
Scale factor: find what one term was multiplied by, and multiply the other term by the same number.
Proportion: write two equal ratios as fractions and find the missing number, like 4/6 = ?/15.

When a situation is not proportional, you can't just scale it. Work with the parts separately (for example, a fixed fee plus a rate).

Worked example

Example 1: Unit rate

5 L of paint covers 40 m². How much paint do you need for 68 m²?

Is it proportional? Yes: no paint covers no wall, and twice the paint covers twice the wall.
Unit rate: 40 ÷ 5 = 8 m² per litre.
Paint needed: 68 ÷ 8 = 8.5 L.

8.5 L of paint

Worked example

Example 2: Is it proportional?

Ali is 12 and his sister is 4. Ali says, "I'm 3 times as old as my sister, so when I'm 24, she will be 8." Is he right?

Ages go up by the same amount, not the same multiple. Ali is always 8 years older.
When Ali is 24, his sister will be 24 − 8 = 16.
Ratios: 12:4 = 3:1, but 24:16 = 3:2. The ratio changes, so ages are not proportional.

No. His sister will be 16, not 8.

Worked example

Example 3: Proportion with a map scale

On a map, 1 cm represents 25 km. Two towns are 7.4 cm apart on the map. How far apart are they really?

Map distance and real distance are proportional.
Scale factor: 1 cm × 7.4 = 7.4 cm, so multiply 25 km by 7.4 too.
25 × 7.4 = 25 × 7 + 25 × 0.4 = 175 + 10 = 185.

185 km

Worked example

Example 4: A multi-step problem

A muffin recipe uses 2 1/2 cups of flour for 24 muffins. The class wants 60 muffins. How much flour is needed?

Proportional? Yes: twice the muffins needs twice the flour.
Scale factor: 60 ÷ 24 = 2.5 (= 2 1/2).
Flour: 2 1/2 × 2 1/2 = 5/2 × 5/2 = 25/4 = 6 1/4 cups.

6 1/4 cups of flour

Think like a mathematician

Before you scale, ask: "If I had zero of one thing, would I have zero of the other?" and "If I doubled one, would the other double?" If either answer is no, the situation is not proportional, and you need to deal with the fixed part separately.

Hands-on activity

Pacing proportions

You need: a measuring tape or metre stick, some open space away from traffic (a hallway, backyard or park), paper and a pencil.

Measure 10 m. Walk it at your normal pace and count your steps. Do it twice and use the average.
Write your rate: ___ steps per 10 m. Then find your unit rate (steps per metre).
Measure a longer distance, like 25 m or the length of a hallway. Predict the number of steps using proportional reasoning.
Walk it and check. How close were you?
Now think: is the time it takes to walk proportional to the distance? What if you stop to tie your shoe?

What to look for: for example, 14 steps per 10 m → 1.4 steps per metre → 25 m is about 35 steps. Walking at a steady pace is about proportional; stopping adds time that doesn't depend on distance, so it is no longer proportional.

Math talk: "Why wasn't your prediction exactly right?"

Just for fun

Good thing ages aren't proportional. If they were, a 2-year-old and a 4-year-old would one day be 50 and 100. Birthdays add one year each, they don't multiply!

Your turn

Apples cost $2 per kg. Is the cost proportional to the mass? Explain.

A taxi charges $3.50 to start plus $2 per km. Is the cost proportional to the distance? Explain.

Is this table proportional? x: 2, 4, 6 and y: 5, 10, 15. If yes, give the unit rate.

Is this table proportional? x: 1, 2, 3 and y: 4, 7, 10.

3 notebooks cost $4.50. How much do 8 notebooks cost?

A train travels 270 km in 3 h at a steady speed. How far does it go in 5 h?

Find the missing number: 4/6 = ?/15.

More practice: warm-up

2 tickets cost $16. How much is 1 ticket?

Using the same price, how much do 5 tickets cost?

Is a person's height proportional to their age? Explain.

Complete the ratio: 3:5 = 6:?

Eggs come 12 per carton. Is the number of eggs proportional to the number of cartons? How many eggs in 4 cartons?

More practice: on your own

A map scale is 1 cm : 20 km. Two lakes are 6.5 cm apart on the map. How far apart are they?

A cookie recipe uses 2 1/2 cups of flour for 24 cookies. How much flour for 36 cookies?

Which is the better buy: 750 g of cereal for $6.00 or 1 kg for $7.50?

A phone plan costs $20 a month plus $0.10 per text. Is the monthly cost proportional to the number of texts? What is the cost for 50 texts?

Snow falls at a steady 2.5 cm per hour. How much falls in 4.5 hours?

Challenge

A class trip costs $350 for the bus plus $12 per student for admission. 28 students go and share the total cost equally. How much does each student pay?

On a map, 8 cm represents 150 km. What real distance does 20 cm represent?

3 cans of paint cover 45 m². How many cans do you need to buy for 110 m²? (You can only buy whole cans.)

Check your understanding

Quiz

Which situation is proportional?

Which table is proportional?

4 L of milk cost $6. How much do 10 L cost?

What is the missing number? 3/8 = 12/?

A graph of a proportional situation is a straight line that...

Ella is 5 and her brother is 10. When Ella is 20, how old will her brother be?

For parents and teachers

Big idea: The Ontario curriculum (B2.10) asks Grade 7 students to identify proportional and non-proportional situations and apply proportional reasoning to solve problems. This lesson also practises B2.1: solving multi-step problems with ratios, rates, fractions, decimals and percents.

Common difficulties:

Scaling everything, even when there is a fixed fee or starting amount.
Additive thinking: thinking 3:5 = 6:8 because 3 was added to both terms. Equivalent ratios come from multiplying, not adding.
Rounding "can" problems the wrong way: when buying whole items, 7.33 cans means you must buy 8.

How to help: Try the pacing activity, compare unit prices while shopping, and scale recipes together. Ask, "Would zero of this give zero of that?" This lesson builds A1 skills: reasoning and proving, making connections and reflecting on whether an answer makes sense.