GM Lesson 001 Understanding Rates and Percentages

Learning Intentions

By the end of this lesson, students will be able to:

• Understand the meaning of rates and percentages.
• Recognise that a rate compares quantities with different units.
• Represent common financial quantities using percentages.

Prerequisites

Students should already be able to:

• Multiply and divide whole numbers and decimals.
• Interpret fractions as parts of a whole.
• Recognise that .
• Use basic units such as dollars, hours, kilograms, litres and kilometres.

Key Idea Summary

A rate compares two quantities with different units.

Examples include:

• $18 per hour
• km per hour
• $4.50 per kilogram
• litres per km

A percentage compares a quantity to .

For example:

Percentages are common in financial contexts, including:

• discounts
• interest rates
• tax rates
• commission rates
• profit and loss
• price increases

A useful way to interpret percentages is:

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

A rate compares two quantities that are measured in different units.

For example, if someone earns $60 in hours, then their hourly rate compares dollars with hours.

The word per means “for each one”.

So:

• $20 per hour means $20 for each hour.
• $3 per kilogram means $3 for each kilogram.
• km per hour means kilometres for each hour.

A percentage is a special comparison where the whole is divided into equal parts.

For example:

This means parts out of every parts.

In consumer arithmetic, percentages are often used to describe parts of money amounts.

For example:

• A discount means out of every dollars is removed.
• A interest rate means out of every dollars is added over a given time period.
• A commission means out of every dollars of sales is earned.

Worked Example 1: Identifying a Rate

A worker earns $72 for hours of work.

Find the rate of pay per hour.

The quantities being compared are dollars and hours.

Therefore, the rate of pay is $18 per hour.

Worked Example 2: Interpreting a Unit Price

A kg bag of rice costs $7.80.

Find the cost per kilogram.

Therefore, the rice costs $3.90 per kilogram.

Worked Example 3: Representing a Financial Quantity as a Percentage

A shop charges a service fee on a bill.

A bill is $80 before the service fee.

Find the service fee.

Therefore, the service fee is $8.

Worked Example 4: Understanding a Percentage in Context

A jacket is advertised as off.

This means out of every dollars of the original price is removed.

If the original price is $100, then the discount is $25.

So the sale price would be:

Therefore, a discount on $100 gives a sale price of $75.

Understanding Checks

Check 1

Which of the following are rates?

a. $15 per hour

b.

c. km per hour

d. $42

e. $5 per kilogram

Check 2

Complete the sentence:

A rate compares two quantities with __________ units.

Check 3

What does mean?

Write it as a fraction out of .

Check 4

A phone plan costs $45 per month.

What two units are being compared?

Check 5

A store advertises a discount.

Explain what this means in words.

Exercises

Simple Familiar Exercises

Exercise 1

Identify whether each quantity is a rate, a percentage, or neither.

a. $25 per hour

b.

c. $80

d. km per hour

e. $6.50 per kilogram

f.

Exercise 2

Write each percentage as a fraction out of .

a.

b.

c.

d.

e.

Exercise 3

Write the two units being compared in each rate.

a. $18 per hour

b. km per hour

c. $4.20 per litre

d. pages per minute

e. $12 per ticket

Exercise 4

Calculate each unit rate.

a. $40 for hours

b. $90 for hours

c. km in hours

d. $18 for kg

e. pages in minutes

Complex Familiar Exercises

Exercise 5

A worker earns $156 for hours of work.

a. Identify the two quantities being compared.

b. Calculate the worker’s hourly rate.

c. Write the answer using correct units.

Exercise 6

A car travels km using litres of petrol.

a. Find the distance travelled per litre.

b. Explain what your answer means in context.

Exercise 7

A cafe sells muffins for $18.

a. Find the cost per muffin.

b. Another cafe sells muffins for $4.80 each. Which cafe has the cheaper muffin price?

Exercise 8

A shop charges a card surcharge on a purchase.

The purchase price is $120.

a. Write as a fraction out of .

b. Calculate the surcharge.

c. Explain the meaning of the surcharge in context.

Homework Problems

Homework 1

Identify whether each is a rate, a percentage, or neither.

a. $22 per hour

b.

c. $65

d. km per hour

e. $9 per person

Homework 2

Write each percentage as a fraction out of .

a.

b.

c.

d.

Homework 3

A cleaner earns $95 for hours of work.

a. Find the hourly rate.

b. Write your answer using correct units.

Homework 4

A kg bag of apples costs $10.50.

a. Find the cost per kilogram.

b. Explain what the answer means.

Homework 5

A shop offers a discount on a $150 item.

a. Write as a fraction out of .

b. Calculate the discount amount.

c. Calculate the sale price.

Homework 6

A delivery driver travels km in hours.

a. Calculate the average speed in kilometres per hour.

b. Explain why this is a rate.

Homework 7

A worker earns $24 per hour.

a. How much would the worker earn in hours?

b. How much would the worker earn in hours?

c. Explain how the rate helps you calculate earnings for any number of hours.

Homework 8

A product has a GST included in its price.

Explain, in words, what the means in this financial context.

Next: GM Lesson 002 Comparing Quantities Using Rates