maths

1.1.1 Introduction

Everyone needs to have an understanding of consumer mathematics

and its use in real-life applications. Accountants, financial planners,

bookkeepers and managers all use consumer arithmetic in their jobs. We

all need to know how to check wages and salaries, and how to calculate

overtime payments based on time-and-a-half or double time.

You will see percentages used for discounts at shops, interest rates for

bank accounts and loans, rates of property growth or loss, statistics for

sports matches, data used in the media, and company statements about

profit and loss. Many employees, such as salespeople, factory workers

and fruit pickers, are paid a commission or by piecework for doing their

jobs. Piecework means that workers are paid for the amount of work that

they have completed rather than an hourly or weekly rate. Government

allowances and pensions are also an important form of income for many

people.

1.1.2 Syllabus links

Lesson Lesson title

1.2

1.3 Wages

1.4

Earning wages

1.5 Working overtime

1.6

1.7

1.8

Earnings — commission

and piecework

Payments — government

allowances and pensions

Personal budgets

Answer questions

and check results

Syllabus links

Rates and percentages ● Understand the meanings of rates and percentages.

● Calculate weekly, fortnightly or monthly wages from an annual

salary.

● Calculate wages from an hourly rate.

● Calculate wages from situations involving overtime and other

allowances.

● Calculate earnings based on commission or piecework.

● Calculate income support payments based on government

allowances and pensions.

● Prepare a personal budget for a given income, taking into account

fixed and discretionary spending.

● Use a spreadsheet to display examples of the above computations

when multiple or repeated computations are required, e.g. preparing

a wage sheet displaying the weekly earnings of workers in an

organisation, preparing a budget, investigating the potential cost

of owning and operating a car over a year.1.2 Rates and percentages

SYLLABUS LINKS

• Understand the meaning of rates and percentages.

Source: General Mathematics Senior Syllabus 2024 © State of Queensland (QCAA) 2024; licensed under CC BY 4.0.

1.2.1 Definition of rates

Rates are used to compare two related quantities. A rate is a measure

of change between two different units. Examples of rates are a car’s

speed in kilometres per hour (km/h), the number of loaves of bread a

baker makes in a day (loaves/day) or the cost of a concert ticket for

each person (cost/person). A rate is calculated per unit or per item.

Running heart rate: 145 bpm

If a rider travels at a constant speed of 20km/h, then they will travel

20km in 1 hour, 40km in 2 hours, 60km in 3 hours and so on.

The units used for the rate depend on the units used to measure each

quantity.

WORKED EXAMPLE 1 Identifying rates

After a school assembly, 689 students leave the assembly hall in 13 minutes. Determine the rate in

students per minute.

THINK

WRITE

1. Identify the two quantities: number of students and

time. The time is measured in minutes.

2. Write the rate as a fraction in terms of number of

students over number of minutes.

3. Simplify.

4. Write the answer, including the units.

WORKED EXAMPLE 2 Calculating a rate in km/h

eles-3036

Calculate the rate (in km/h) at which you are moving if you are on a bus that travels 11.5km in

12 minutes.

THINK

1. Identify the two quantities: distance and time.

As the question asks for the answer in km/h, convert

the time quantity units from minutes to hours.

2. Write the rate as a fraction in terms of number of

kilometres over number of hours.

WRITE

The quantities are 11.5km and 12 minutes.

12

60

=

1

5

or 0.2 hours

11.5 km

0.2 h

The quantities are 689 students and 13 minutes.

689 students

13 minutes

689 students

13 minutes

=53 students per minute

Students leave the assembly hall at a rate of

53 students per minute.3. Simplify.

11.5 km

0.2 h

4. Write the final answer, including the units.

=57.5km/h

You are travelling at 57.5km/h.

WORKED EXAMPLE 3 Calculating a rate of pay per hour

James works as a barista at the local café and is paid $99 for

6 hours. Calculate his rate of pay per hour.

THINK

WRITE

1. Identify the two quantities: money and time.

2. Write the rate as a fraction in terms of money

over the number of hours.

3. Simplify.

4. Write the answer.

1.2.2 Percentages

The term per cent means ‘per hundred’ or ‘out of a hundred’ and

can be written as a fraction or a decimal.

For example, 50%=

50

100

=0.5.

The quantities

are $99 and 6 hours.

99

6

99

6

= $16.5 per hour

James is paid $16.50 per hour.

Digital technology

Scientific calculators have a % button that can be used to compute

calculations involving percentages.

Percentages can be converted into decimals and fractions.

Decimals and fractions can be converted into percentages.Applying discount

A discount is a reduction in price, commonly used by businesses aiming to clear out old stock or attract new

customers.

Calculating discount

In general, if an r% discount is applied:

discount=

r

100

×original price

Calculating the selling price of a discounted item

• Method 1

Use the percentage remaining after the percentage discounted has been subtracted from 100%; that is, if an

item for sale has a 10% discount, then the price must be 90% of the marked price.

• Method 2

The new sale price of the item can be solved by calculating the amount of the discount, then subtracting the

discount from the marked price.

For example, see the two different methods used to calculate the sale price on a pair of shoes marked $95 if

a 10% discount is given.

Method 1

Sale price = 90% of $95

= $85.50

Method 2

Discount =

=

100

= $9.50

Sale price = marked price−discount

= $95.00−$9.50

= $85.50

In other words, reducing the price by 10% is the same as multiplying by 90% or (100−10)%.

Increasing or decreasing a quantity by x%

To decrease a quantity by x%, multiply by (100−x) %.

To increase a quantity by x%, multiply by (100+x) %.

Note: Convert the percentage to a decimal or fraction before multiplying.

WORKED EXAMPLE 4 Calculating a percentage increase

Calculate an increase to $76 by 15%.

THINK

WRITE

1. The original percentage is always 100%, so an

increase of 15% means a total of 115%.

2. Write 115% as a fraction and then express it

as a decimal.

3. Multiply the amount $76 by 1.15.

4. Write the answer.

(100+15) %=115%

115% =

115

100

= 1.15

76×1.15=87.40

$87.40

CHAPTER 1 Consumer arithmetic — earning and budgeting 7

r

100

10

× original price

×95.00WORKED EXAMPLE 5 Calculating the sales price

eles-6361

Sarah bought a car for $7500 and sold it 4 years later for 30% less than she

paid for it. Calculate the price she sold the car for.

THINK

WRITE

1. The original percentage is always 100%, so a

discount of 30% means a total of 70%.

2. Write 70% as a fraction and then express it

as a decimal.

3. Multiply the amount $7500 by 0.7.

4. Write the answer.

(100−30) %=70%

70% =

70

100

= 0.70 or 0.7

7500×0.7=5250

Sarah sold it for $5250.

1.2.3 Percentage increase and decrease

Percentage increase and decrease can be used to calculate sale prices, discounts, profits and many other

quantities. It is calculated as a percentage of the original amount.

Percentage change

The formula for calculating the percentage increase/decrease is:

Percentage increase/decrease=

amount of increase/decrease

original amount

WORKED EXAMPLE 6 Calculating the percentage discount

Ramon bought a laptop in a sale for $774.40. If the original price was $968, calculate the percentage

discount.

THINK

WRITE

1. Calculate the amount of the discount by

subtracting $774.40 from $968.

2. Calculate the percentage discount by using

the formula: percentage discount

=

amount of discount

original amount

×100.

3. Write the answer as a percentage.

The percentage discount is 20%.

Discount = $968−$774.40

= $193.60

Percentage discount =

193.60

968

= 20

×100

×100When a large number of values are being considered in a problem involving percentages, spreadsheets or other

technologies can be useful to help carry out most of the associated calculations.

For example, a spreadsheet can be set up so that entering the original price of an item will automatically

calculate several different percentage increases for comparison.

Exercise 1.2 Rates and percentages

1.2 Exercise

Simple familiar

1, 2, 3, 4, 5, 6, 7,

8, 9

Simple familiar

1. WE1 Using the units stated, calculate the rates for:

a. 4-kg bag of apples that cost $23.60 expressed in $/kg

b. a tank that loses 1320mL of water in 2 hours expressed in mL/h

c. a 3.6-metre-long carpet that costs $67.14 expressed in $/m

d. a basketball player who has scored a total of 833 points in 68 games expressed in points/game.

2. Calculate the following rates when the units are changed as indicated. Where necessary, give answers correct

to 2 decimal places.

a. 1.5m/s to km/h

b. 60 km/h to m/s

c. 65 cents per gram to $/kg

d. $5.65 per kilogram to cents per gram

3. WE2 Calculate the rate (in km/h) that you are moving if you are in a passenger aircraft that travels 1770km

in 100 minutes.

Complex familiar

10, 11, 12

1.2 Exam questions

Complex unfamiliar

13, 14

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4. WE3 Michael works as an apprentice chef and is paid $424.80 for 36 hours. Determine his rate of pay

per hour.