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

Solving Proportional Problems: Missing Values, Percents and Multi-Step Problems

B2.8 B2.1 A1About 45 minutes

Learning goals

I can find an unknown value in a proportional situation using a unit rate, a scale factor or a double number line.
I can solve a proportion like 3/5 = x/40.
I can solve percent problems, including discounts and 13% HST, as proportional problems.
I can solve multi-step problems that combine rates, ratios and percents.

Words to know

proportion
A statement that two ratios are equal, like 3/5 = 24/40.

scale factor
The number you multiply one ratio by to get an equivalent ratio. From 3/5 to 24/40 the scale factor is 8.

double number line
Two number lines lined up so that matching values of two quantities are directly above each other.

HST
Harmonized Sales Tax. In Ontario it is 13% of the price of most items.

percent change
How much a quantity grew or shrank, as a percent of the starting amount.

Remember from Grade 7

In Grade 7 you solved proportional problems with unit rates, ratio tables and proportions, and you increased and decreased amounts by a percent. In Grade 8 you will solve problems with less friendly numbers, with several steps, and with percents written as proportions.

Let's start with a canoe trip map

A map of a provincial park has the scale 1 cm : 2.5 km. Your canoe route measures 6.4 cm on the map. How far will you paddle?

The situation is proportional: every 1 cm on the map stands for 2.5 km. So 6.4 cm stands for 6.4 × 2.5 = 16 km.

A red canoe resting on pink granite rocks beside a calm lake with pine-covered islands. — Canoe routes in provincial parks are planned on maps with a scale. With a scale of 1 cm : 2.5 km, a 6.4 cm route means 6.4 × 2.5 = 16 km of paddling. AI-generated image

Strategies for finding a missing value

First ask: is the situation proportional? (If you doubled one quantity, would the other double? If one is 0, is the other 0?) If yes, choose a strategy:

Unit rate: find the value for 1, then multiply.
Scale factor: find what one quantity was multiplied by, and multiply the other by the same number.
Double number line or ratio table: line up matching values.
Write a proportion: two equal fractions with one unknown, like 3/5 = x/40. Use a scale factor, or use a unit rate (x = 40 × 3/5).

Percents are proportions too. "18 is 30% of what number?" means 18/x = 30/100.

A double number line for walking at a steady pace: 2.5 km takes 15 min. Matching values line up. 7.5 km lines up with 45 min.

Worked example

Example 1: Unit rate

A cookie recipe uses 3 cups of flour for 2 dozen cookies. How much flour is needed for 5 dozen?

Unit rate: 3 ÷ 2 = 1.5 cups per dozen.
5 × 1.5 = 7.5 cups.

7 1/2 cups

Worked example

Example 2: Scale factor

Solve 4/7 = x/56.

7 × 8 = 56, so the scale factor is 8.
x = 4 × 8 = 32.
Check: 32/56 simplifies (÷ 8) to 4/7. ✓

x = 32

Worked example

Example 3: A not-so-friendly scale factor

A photo 5 cm wide and 8 cm tall is enlarged so that it is 12 cm wide. How tall is it now?

Proportion: 5/8 = 12/h (width over height).
Unit rate: each 1 cm of width goes with 8 ÷ 5 = 1.6 cm of height.
h = 12 × 1.6 = 19.2.

19.2 cm tall

Worked example

Example 4: Discount, then tax

A hoodie costs $48. It is 25% off, and then 13% HST is added. What is the final price?

25% off means you pay 75%: 0.75 × 48 = $36.
Add 13% HST: 36 × 1.13 = $40.68. (Or 13% of 36 is 4.68, and 36 + 4.68 = 40.68.)

$40.68

Taking 25% off means multiplying by 0.75, and adding 13% HST means multiplying by 1.13. $48 × 0.75 = $36, and $36 × 1.13 = $40.68.

Worked example

Example 5: Find the whole

18 is 30% of what number?

Proportion: 18/x = 30/100.
Unit rate: if 30% is 18, then 10% is 6, so 100% is 60.

Split the whole into ten parts of 10%. If 3 parts (30%) make 18, each part is 6, so the whole (100%) is 60.

Multiply instead of two steps

This uses the relationship between operations: adding 13% is the same as multiplying by 1.13 (100% + 13% = 113%). Taking 25% off is the same as multiplying by 0.75. One multiplication is faster and gives fewer chances to make a mistake.

Hands-on activity

Step and pace predictions

You need: a measuring tape (or a 1 m stick), a hallway, driveway or park path, paper, a pencil. If you go outside, stay on the sidewalk or path with a grown-up.

Measure 10 m. Walk it at a normal pace and count your steps. Do it twice and use the average.
Write your rate: ___ steps per 10 m.
Use proportional reasoning to predict your steps for 25 m, 100 m and 1 km.
Time yourself walking 10 m. Predict how long 1 km would take.
Test one prediction (for example 25 m). How close were you?

Model answer: if you take 14 steps per 10 m, then 25 m → 14 × 2.5 = 35 steps; 100 m → 140 steps; 1 km = 1 000 m → 1 400 steps. If 10 m takes 8 s, then 1 km takes 8 × 100 = 800 s, which is about 13 minutes.

Just for fun

A recipe for pancakes serves 4. You're feeding 12, so you triple everything: 3 times the flour, 3 times the milk, 3 times the eggs... and, sadly, 3 times the dishes. Proportions are fair like that.

Your turn

5 notebooks cost $8.75. How much do 8 notebooks cost?

Solve 3/5 = x/40.

A map scale is 1 cm : 4 km. Two towns are 7.5 cm apart on the map. How far apart are they?

14 is 20% of what number?

A jacket costs $80 and is 15% off. What is the sale price (before tax)?

Find the total for a $68 item with 13% HST.

A recipe for 6 servings uses 450 g of pasta. How much pasta for 4 servings?

More practice: warm-up

2 kg of apples cost $6. How much do 5 kg cost?

Solve x/9 = 2/3.

What is 10% of 250?

4 tickets cost $30. What does 1 ticket cost?

A car travels 90 km in 1 hour. How far does it go in 3 hours at the same speed?

More practice: on your own

Solve 7/4 = 21/x.

A 1.5 L bottle fills 6 glasses. How many of the same glasses can you fill from 4 L?

What is the total for a $24.00 item with 13% HST?

In a survey, 45 of 120 students bike to school. What percent is that?

A model canoe is built at a scale of 1 : 20. The model is 26 cm long. How long is the real canoe?

Challenge

A price goes from $40 to $46. What is the percent increase?

Zara walks 2.4 km in 30 minutes. At the same pace, how long will 6 km take? If she leaves at 3:20 p.m., when will she arrive?

Store A sells a game for $60 with 30% off. Store B sells it for $50 with 10% off. Both add 13% HST. Which store is cheaper, and by how much after tax?

Check your understanding

Quiz

Solve 2/5 = x/35.

A recipe uses 6 eggs for 2 batches. How many eggs for 5 batches?

What is the total for a $50 item with 13% HST?

21 is 30% of what number?

A map scale is 1 cm : 5 km. A road is 3.2 cm long on the map. How long is the real road?

Which situation is not proportional?

For parents and teachers

Big idea: Ontario's Grade 8 expectation B2.8 asks students to compare proportional situations and determine unknown values in proportional situations, and apply proportional reasoning to solve problems in various contexts. B2.1 adds multi-step problems with ratios, rates and percents. Ontario's HST is 13%, which makes a realistic multi-step context.

Common difficulties:

Scaling a non-proportional situation (like a taxi with a starting fee).
Adding percents in the wrong order or adding them together (25% off then 13% tax is not the same as 12% off).
Percent change compared with the wrong amount. Always divide by the starting value.

How to help: When shopping, estimate the price after a discount and after HST before you reach the till. Use the step-and-pace activity on a family walk.