B. Number · Number: Number Sets, Powers and Rational Numbers in Action

Rates and Proportions: Unit Rates, Scales, Similar Figures and Direct Variation

B3.5 A1 A2About 70 minutes

Learning goals

I can find and compare unit rates to make decisions.
I can solve proportions using unit rates, scale factors and equivalent ratios, and explain my method.
I can use proportional reasoning with scale drawings, maps, similar triangles, unit conversions and data.
I can connect a constant rate to a linear relation like d = 18t, and pose my own proportional problems.

Words to know

ratio
A comparison of quantities with the same kind of unit, like 3 : 5 or 3/5.

rate
A comparison of quantities with different units, like 90 km/h or $6.49 per 4 L.

unit rate
A rate for 1 unit of the second quantity, like $1.62/L.

proportion
An equation stating that two ratios or rates are equal, like 3/5 = x/40.

scale factor
The number you multiply by to go from one figure (or quantity) to a proportional one.

similar figures
Figures with the same shape: matching angles are equal and matching sides are proportional.

direct variation
A linear relation through (0, 0), like d = 18t. The constant rate (18) is called the constant of variation.

Remember from Grade 8

In Grade 8 you compared rates and ratios to find the better buy, and you solved proportional problems with maps, recipes and discounts. Grade 9 asks you to use proportions across all of math: data, measurement, geometry, linear relations and money, and to pose your own problems.

Sugar shack math

In the spring, maple producers in Ontario and Québec collect sap and boil it down. It takes about 40 L of sap to make 1 L of maple syrup (the exact amount depends on how sweet the sap is). If a sugar bush collects 140 L of sap, how much syrup can it make?

The amounts grow together at the same rate: double the sap, double the syrup. That is a proportional relationship.

A double number line for about 40 L of sap per 1 L of syrup. 140 L of sap lines up with 3.5 L of syrup, because 140 ÷ 40 = 3.5.

Three ways to solve a proportion

Solve 3/5 = x/40:

Scale factor: 5 × 8 = 40, so x = 3 × 8 = 24.
Unit rate: 3 ÷ 5 = 0.6 per unit, so x = 0.6 × 40 = 24.
Algebra: multiply both sides by 40: x = 40 × 3/5 = 24.

Choose the method that makes the numbers easiest. Always check that your answer is reasonable.

Worked example

Example 1: Better buy

In Ontario, milk is often sold in bags. A 4 L bag costs $6.49 and a 2 L carton costs $4.29 (example prices). Which is the better buy?

Unit rate for the bag: 6.49 ÷ 4 ≈ $1.62 per L.
Unit rate for the carton: 4.29 ÷ 2 ≈ $2.15 per L.
$1.62 < $2.15.

The 4 L bag is the better buy (if you can use it before it expires).

Worked example

Example 2: Map scale

A trail map has a scale of 1 : 50 000. Two lookouts are 6.4 cm apart on the map. What is the real distance in km?

1 cm on the map is 50 000 cm in real life.
6.4 × 50 000 = 320 000 cm.
320 000 cm ÷ 100 = 3200 m = 3.2 km.

3.2 km

Worked example

Example 3: Similar triangles and shadows

At the same time of day, a 1 m stick casts a 1.6 m shadow and a maple tree casts a 12 m shadow. How tall is the tree?

The sun's rays make the same angle, so the triangles are similar and their sides are proportional.
height/shadow: 1/1.6 = h/12.
h = 12 × 1/1.6 = 12 ÷ 1.6 = 7.5.

The tree is about 7.5 m tall.

The stick and the tree, with their shadows, form similar right triangles (drawn to scale). Each height is 1/1.6 = 0.625 of its shadow, so the tree is 0.625 × 12 = 7.5 m tall.

Worked example

Example 4: Unit conversion as a proportion

The speed limit on many Ontario highways is 100 km/h. What is that in m/s?

100 km = 100 000 m and 1 h = 3600 s.
100 000 m / 3600 s ≈ 27.8 m/s.

About 27.8 m/s: a car at 100 km/h covers the length of a hockey rink (about 60 m) in roughly 2 seconds.

Worked example

Example 5: Direct variation

A cyclist rides at a constant 18 km/h. Write an equation for distance d after t hours. How far does she go in 2 1/2 h? How long does 63 km take?

Distance = rate × time, so d = 18t. Its graph is a line through (0, 0) with rate of change 18.
d = 18 × 2.5 = 45 km.
63 = 18t, so t = 63 ÷ 18 = 3.5 h.

d = 18t; 45 km; 3.5 h (3 h 30 min)

Worked example

Example 6: Proportions with data

In a survey of 250 students at one school, 85 walk to school. If the same rate holds for a school of 1100 students, about how many walk?

85/250 = x/1100.
Unit rate: 85 ÷ 250 = 0.34 (34% of students).
x = 0.34 × 1100 = 374.

About 374 students. This assumes the two schools are similar (for example, similar distances from home).

When is a situation NOT proportional?

Proportional reasoning only works when one quantity is a constant multiple of the other. A taxi that charges $4 to start plus $2/km is not proportional: 10 km costs $24, but 20 km costs $44, not $48. Its graph is a line that does not pass through (0, 0). You will study these partial-variation relations in the Linear and Non-Linear Relations unit.

Hands-on activity

Pose your own proportion problem

You need: a grocery flyer or receipt, a map (paper or app), a tape measure, a sunny day (optional).

Better buy: Find two sizes of the same product. Compare unit prices.
Map: Use the scale on a map to find the real distance between two places you know.
Shadow (sunny day): Measure your own height and your shadow, then the shadow of a tall object (a lamp post, a tree). Find the object's height with a proportion. Never look directly at the Sun.
Pose a problem: Write one proportion problem about your own life (a recipe, a car trip, a sports statistic), solve it, and give it to someone at home to solve too.

Model answer for 3: You are 1.6 m tall with a 2.4 m shadow; a lamp post's shadow is 9 m. h/9 = 1.6/2.4, so h = 9 × 1.6 ÷ 2.4 = 6 m.

Just for fun

A funny-but-true fact: maple syrup is serious business in Canada. In Québec, producers keep a strategic reserve of maple syrup in barrels, so there is syrup to sell even after a bad sugaring season.

Your turn

Solve 4/7 = x/63.

Which is the better buy: 12 granola bars for $5.49 or 30 for $12.99?

A map scale is 1 cm : 2.5 km. Two towns are 7.2 cm apart on the map. How far apart are they?

A 1.5 m student casts a 2 m shadow. At the same time, a flagpole casts a 10 m shadow. How tall is the flagpole?

Convert 72 km/h to m/s.

A printer prints 45 pages in 3 min. Write an equation for pages p after t minutes and find p when t = 7.

Is a phone plan costing $15 plus $0.10 per text proportional? Explain.

More practice: warm-up

Find the unit rate: $9 for 3 kg.

Solve 2/3 = x/12.

A car travels 240 km in 3 h. What is its average speed?

A recipe uses 2 cups of flour for 12 muffins. How much for 30 muffins?

Write 15 : 25 in lowest terms.

More practice: on your own

Two similar rectangles: the first is 4 cm by 6 cm. The second has a long side of 15 cm. Find its short side.

In a sample of 400 light bulbs, 6 were faulty. Predict the number of faulty bulbs in a shipment of 15 000.

A car uses 7.2 L of gas per 100 km. How much gas for a 350 km trip? At $1.55/L, what does it cost?

Water flows at 18 L/min. How long to fill a 450 L hot tub? Write the relation and solve.

A model of a CN Tower is built at a scale of 1 : 400. The real tower is about 553 m tall. How tall is the model, in cm?

Challenge

Two runners: Aya runs 5 km in 24 min; Ben runs 8 km in 40 min. Who is faster? By how much in km/h?

A recipe for 4 people uses 3/4 cup of rice. Write a direct-variation equation for cups of rice r for p people, and find r for 10 people.

Pose and solve a proportion problem using this fact: about 40 L of sap makes 1 L of syrup, and a typical tap gives about 40 L of sap per season (example value). How many taps are needed for 25 L of syrup?

Check your understanding

Quiz

Which is a unit rate?

Solve 5/8 = x/56.

A map scale is 1 : 25 000. What real distance is 4 cm on the map?

A 2 m pole casts a 3 m shadow. At the same time a building casts a 27 m shadow. How tall is it?

Which relation is proportional?

What is 36 km/h in m/s?

For parents and teachers

Big idea: Expectation B3.5 asks students to pose and solve problems involving rates, percentages and proportions in contexts connected to data, measurement, geometry, linear relations and financial literacy. This lesson covers rates and proportions; the next covers percentages.

Common difficulties:

Setting up a proportion with mismatched parts (height/shadow = shadow/height). Label each ratio with units.
Mixing units (cm and km on a map). Convert before comparing.
Using proportions where they do not apply, like a taxi fare with a starting fee.

How to help: At the store, ask "Which size is the better deal per unit?" On a road trip, ask "At this speed, how long until we arrive?" Ask your teen to invent a proportion problem for you to solve; posing problems is part of this expectation.