1. science
2. mathematics
3. arithmetic

2 Percentages, ratios and rates
number and algebra
2
Curriculum links
The proficiency strands Understanding,
Fluency, Problem solving and Reasoning
are fully integrated into the content of
the unit.
discover
The following content descriptions are
considered in this unit:
• Solve problems involving the use of
percentages, including percentage
increases and decreases, with
and without digital technologies
(ACMNA187).
• Solve a range of problems involving
rates and ratios, with and without
digital technologies (ACMNA188).
• Solve problems involving profit and loss,
with and without digital technologies
(ACMNA189).
Prerequisite knowledge
The following topics in this unit are related
to prerequisite knowledge and, while not
specified in the Australian Curriculum
document, it was felt that this needed to
be included to provide a comprehensive
overview of the unit.
• 2A Understanding percentages
• 2B Percentages, decimals and fractions
Discussion prompts
•
Direct students to examine the opening
photo for this unit on pages 44 and 45 of
their Student Book.
Percentages,
ratios and rates
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
Number and algebra
2A
Understanding
percentages
2B
Percentages, decimals
and fractions
2C
Percentage calculations
2D
Financial calculations
2E
Understanding ratios
2F
Working with ratios
2G
Dividing a quantity in
a given ratio
2H
Understanding rates
Before continuing this unit, complete the Preview
on pages 12–13 of your Student Progress Book.
• Brainstorm how percentages are used in
•
•
this photograph. (used to represent sales
and discounts)
Discuss the meaning of discount. (a
discount is a percentage of the price
that is subtracted from the original
price to find the new selling price, or the
discounted price)
Can you think of a different way to say:
– 50% discount? (half of the original
price)
– 75% discount? (one-quarter of
the price)
•
•
– 25% discount? (three-quarters of
the price)
If an item has an original price of $65,
can you calculate how much:
– 10% discount would be?
($6.50)
– 20% discount would be? ($13.00)
– 25% discount would be? ($16.25)
– 40% discount would be? ($26.00)
Why might it be a valuable skill to
be able to calculate these types of
percentage calculations without a
calculator? (So that sale prices can be
•
•
calculated on the spot. This would mean
that you do not need to rely on sales
staff to do the calculation for you, and
you can check that the correct amount
was deducted from the original price.)
Define the terms profit and loss. (Profit
is the money made by the retailer; that
is, the store sells the item for a price
greater than the purchase price. A loss is
the result when an item is sold at a price
lower than the purchase price.)
Discuss how shops can sell items at
a discount and still make a profit.
(Discuss wholesale price and mark-
•
•
➜
Percentages, ratios and rates are alternative
forms of presenting numbers that are both
practical and useful. Why are percentages
useful when making comparisons?
ups highlighting that, when selling
at a discount, a store is still generally
making a profit on the original price.)
Discuss any current/recent sales and
ask students to discuss any savings
made. It may be beneficial to bring in
some catalogues or local newspapers to
provide examples.
Discuss if students can think of any
more real-life scenarios in which
percentages are used. (Home loan
interest rates, percentage weight gain
or loss are some examples.)
Professional Support
52
Students need to be able to:
• solve problems involving the comparison,
addition and subtraction of whole
numbers, fractions and decimals
• solve problems involving multiplication
and division of whole numbers, fractions
and decimals
• multiply and divide whole numbers,
fractions and decimals by multiples of ten
• round decimals to a specified number of
decimal places
• convert between fractions, decimals and
percentages
• express one quantity as a fraction or
percentage of another
• use fractions, decimals and percentages,
and their equivalents
• compare the cost of items to make
financial decisions
• convert between units of measurement
• compare quantities.
Notes
2A Understanding percentages
53
obook
2 Percentages, ratios and rates
preview
2
resources
Deep Learning Kit
2A discover card 1
›
Writing fractions
Intervention task
2A discover card 2
›
Finding the highest common factor
Intervention task
Equivalent fractions
Intervention task
Multiplying and dividing by 10, 100 and
1000
Intervention task
Rounding
Intervention task
Converting fractions to decimals
Intervention task
1 What fraction of the carton contains Easter eggs?
Percentages, ratios and rates are
alternative forms of presenting
numbers that are both practical and
useful. Why are percentages useful
when making comparisons?
Converting units
Intervention task
2E discover card 2
›
Reproducible master
a
7
___
50
14
100
b
21
___
25
84
100
54
Each question relates to a prerequisite
skill for this unit. If students experience
difficulty with any of the questions, record
the matching discover card to be completed
(see below). You may like to advise students
individually of the discover cards they will
need to complete. Alternatively, students
can refer to the Preview reference card in
the Deep Learning Kit. Students can write
this information in the appropriate section
of My learning and then tick the discover
cards off as they are completed.
Those students that require intervention
can complete all the nominated discover
cards before commencing the first topic.
However, it may be more beneficial for
96
24
48
___
5
48
____
5000
c
3
___
15
20
100
16 Write each of these quantities in the required unit
a 28 cm in mm
b 3 kg in g
280 mm
3000 g
c 6.5 h in min
390 minutes
d 7.2 L in mL
7200 mL
8 Write the answer to each of these.
65 840
6.8
d 4870 ÷ 10
487
•
Questions 17 and 18 refer to the diagram provided with
questions 1 and 2.
1263
c 0.0068 × 1000
8
___
12
12
___
12
e 31.72 ÷ 100
0.3172
f 92684.8 ÷ 1000
92.6848
17 Describe the number of Easter eggs that have been
eaten to the number of remaining eggs.
5 to 7
•
12 to 7
7 to 5
9 What is 28.732 178 rounded to four decimal places?
2 What fraction of the Easter eggs are yellow?
18 Describe the number of remaining Easter eggs to
the total number of eggs.
12 to 7
7 to 12
28.7322
12 to 5
2
7
10 What is 518.7394 rounded to one decimal place?
4
10
7 to 5
19 Which unit of measurement relates to distance?
518.8
20
litres
dollars
11 What is $21.08 rounded to the nearest five cents?
4 Which of these fractions is not in simplest form?
6
___
17
1
__
5
12
7
___
100
kilometre
$21.10
20
___
50
9
20
them to complete each required discover
card just prior to commencing the
matching topic in the unit.
2A Understanding percentages
– preview Q1–7
➜ Focus: To write fractions to
represent a portion of a total amount,
to find the highest common factor
(HCF), to cancel fractions so that they
are written in their simplest form, to
convert given fractions into equivalent
fractions
hours
12 What is $1674.37 rounded to the nearest dollar?
45
5 Write ___
in simplest form.
100
•
•
Direct students to complete 2A discover
card 1: Writing fractions if they had
difficulty with Q1 or Q2, or require
more practice at this skill.
You may need to undertake some
explicit teaching so students understand
that the numerator of the fraction
represents the portion which is selected,
and the denominator represents the
total number of parts that the whole is
divided into. In reference to the diagram
of the egg carton in Preview Q1 and
Q2, there are 12 parts in the whole. It
may be beneficial to allow students to
20 What unit of measurement does not relate to time?
seconds
$1674
13
days
minutes
13 What is __43 written as a decimal?
0.34
Direct students to complete Preview in
the obook.
48 ___
15 What is ___
× 100
?
50
1
0.43
0.75
1.3
examine an actual egg carton so that
they can visualise this. Explain that 12
becomes the denominator.
Students need to recognise that the
numerator is the number of selected
parts, coloured parts, or shaded parts.
In the case of the egg carton, it is the
number of fi lled parts. This can be
modelled using the egg carton and
counters. Students may benefit from
writing fractions to describe different
scenarios that can be modelled using
the concrete example. Extra concrete
examples include packets of yoghurt
or packets of pudding which contain
grams
more than one part joined together, or a
multipack of mini breakfast cereals.
Guide students to the concept of:
number of shaded or selected parts
total number of parts
•
•
Direct students to complete 2A discover
card 2: Finding the highest common
factor if they had difficulty with Q3 or
require more practice at this skill.
You may need to undertake some
explicit teaching so students understand
how to find common factors. Students
may like to use counters to make a
model that can be used to find factors.
For example, give students 40 counters
7 × 2 = 14
50 × 2 100
b: To change 25 into 100, multiply by 4.
Then multiply the numerator by 4
also:
21 × 4 = 84
25 × 4 100
c: Students may be confused
because 15 does not divide evenly
into 100. Ask students to see if
they can simplify the fraction .
(Answer ) This can be converted
to an equivalent fraction with a
denominator of 100, by multiplying
the numerator and the denominator
by 20:
1 × 20 = 20
5 × 20 100
Professional Support
File 2.01: Conversion charts
7
___
12
4
___
12
2
Quantities and their units
Intervention task
Professional Support Online
0.625
7 What is the equivalent fraction with a denominator
of 100 for each of these?
3 What is the highest common factor of 40 and 100?
Comparing two quantities
Intervention task
2H discover card 1
›
100
2 Percentages, ratios and rates
Multiplying fractions by 100
Intervention task
2E discover card 1
›
25
14 Write __58 as a decimal.
4 to 8
2C discover card 1
›
5
b 12.63 × 100
2B discover card 3
›
4
a 6584 × 10
2B discover card 2
›
6 By what number do you need to multiply both the
7
to write it as an
numerator and denominator of ___
25
equivalent fraction with a denominator of 100?
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
Questions 1 and 2 refer to this carton of Easter eggs.
Some of the Easter eggs from the full carton have
been eaten.
2B discover card 1
›
preview
To answer each question, shade one bubble or write
your answer in the box provided.
2A discover card 3
›
Percentages,
ratios and rates
and ask them to divide the counters into
groups of equal size. A possible result
is four groups of 10 counters. Explain
that this represents a factor pair. Ask
students to find another factor pair by
rearranging the counters.
A CAS calculator can also be used to
find the HCF. Go to the Home screen:
c and highlight 1: Calculate. Press
the menu key: b, use the arrow keys
to scroll down to 2: Number, and click:
a. Scroll to 5: Greatest common
divisor and press ·.
Type in the numbers: (40, 100). Press
·. The HCF will appear to the right.
(20)
Direct students to complete 2A discover
card 3: Equivalent fractions if they had
difficulty with Q4–7, or require more
practice at this skill.
Students may need to be reminded
that they need to simplify fractions
by dividing the numerator and the
denominator by the HCF. They can find
the HCF using the methods described
above and can divide both parts of the
fraction to find the simplified equivalent
fraction. Alternatively, students can
type the fraction into their TI-30XB
Multiview calculator using the fraction
key: q. When they press <, the
simplified fraction will appear to the
right. When finding an equivalent
fraction, remind students that they need
to multiply both the numerator and
the denominator by the same factor.
Explicitly demonstrate the solution to
Preview Q7.
a: To change 50 into 100, multiply by 2.
Then multiply the numerator by 2
also.
55
obook
Fractions and ratios
preview
2A discover card 1
− count the number of zeros
− locate the decimal point
− if multiplying, move the decimal
point to the right, jumping to match
the number of 0s
− if dividing, move the decimal point
to the left, jumping to match the
number of 0s.
Writing fractions
Focus: To express the number of parts out of
a total as a fraction
Students review the terms numerator and
denominator and are guided to write a
fraction to describe the shaded parts as a
fraction of the total number of parts. This
concept is extended in extra questions in
which students write fractions to describe
different relationships.
Finding the highest common factor
Focus: To find the highest common factor of
two numbers
Students review the terms factor and factor
pair and are guided to identify factor
pairs for given numbers. They consider
the meaning of the term highest common
factor (HCF) and work through the steps
of a defined process to identify the HCF for
given numbers.
•
•
2A discover card 3
Equivalent fractions
Focus: To review that equivalent fractions
have the same value and to work to express
fractions in an equivalent form
2B Percentages, decimals and
fractions – preview Q8–14
•
•
•
➜ Focus: To multiply and divide by
multiples of 10, to round decimals and
to write fractions as decimals
56
•
•
Direct students to complete 2B discover
card 1: Multiplying and dividing by 10,
100 and 1000 if they had difficulty with
Q8, or require more practice at
this skill.
You may need to undertake some
explicit teaching so students understand
the shortcut that can be used when
multiplying or dividing by multiples of
10. Explain to students that they need to
complete the following steps:
•
•
2B discover card 1
Direct students to complete 2C discover
card 1: Multiplying fractions by 100 if
they had difficulty with this question
or require more practice at this skill.
You may need to undertake some
explicit teaching so students are
reminded that when changing a fraction
(or a decimal) to a percentage, they
need to multiply by 100. In the case
of changing a fraction, this can be
. (See
written as multiplying by 100
1
1D Operations with fractions.)
This calculation can be demonstrated:
•
Conversion charts can be printed
and laminated for students and can be
kept at hand for ready reference. This
reproducible master can also be used to
create a poster for display.
Students may need to be reminded how
to use the conversion charts.
Direct students to complete 2E discover
card 2: Comparing two quantities if
they had difficulty with Q17 or Q18, or
require more practice at this skill.
You may need to undertake some
explicit teaching so students
understand that a ratio is a way of
comparing quantities. Students need
to be reminded that the order is very
important when writing a ratio. Where,
in Q17, the order is given as ‘the number
of Easter eggs eaten to the number
remaining’, the numbers describing
each of these figures need to be in the
same order given in the description
(that is; 7 to 5).
The decimal point is after the 4.
Move one place to the right and add a
placeholder zero.
In Q8f:
92684.8 ÷ 1000 = 92.684.8 = 92.6848
Move the decimal point three places to
the left.
Direct students to complete 2B discover
card 2: Rounding if they had difficulty
with Q9–12, or require more practice at
this skill.
You may need to undertake some
explicit teaching so students understand
the conventions for rounding. Refer to
1A Estimating and rounding. It may be
appropriate for students to use their
scientific calculators to assist with
rounding. The key strokes required
to round on the TI-30XB Multiview
calculator are given in detail in the
notes for 1A Estimating and rounding.
Direct students to complete 2B discover
card 3: Converting fractions to
decimals if they had difficulty with
Q13 or Q14, or require more practice at
this skill.
You may need to undertake some
explicit teaching so students understand
how to convert fractions to decimals.
In the first instance, encourage students
to see if the denominator of a fraction
can be rewritten as 10, 100, 1000, etc.
For example, 34 can be rewritten with a
denominator of 100:
3 × 25 = 75 . Using the place-value
4 25 100
chart for decimals, this can be written
as 0.75. (See 1E Understanding
decimals.)
In cases when the denominator of a
fraction cannot be readily changed to
10, 100, 1000, etc. students will need
to be reminded that the vinculum of
a fraction is another way of saying
‘divided by’. So:
5=5÷8
8
Multiplying and dividing by 10, 100 and 1000
Focus: To explore the results when numbers
are multiplied or divided by 10, 100 and
1000
Resources: calculator
Students complete a number of
multiplication and division questions
using multiples of 10. They are guided to
recognise the relationship between the
number of zeros in the multiplication factor
or the divisor and the number of zeros in
the answer. Students are able to complete
extra questions independently to practise
this skill.
48 × 100 = 48 × 2 = 96 = 96
50
1
1
1
1
•
Remind students that:
– cross-cancelling when multiplying
fractions will enable them to work
with smaller numbers
– fractions should always be written
in simplest form.
2C discover card 1
•
2B discover card 2
Multiplying fractions by 100
Rounding
Focus: To express 100 as a fraction and
multiply it by another fraction
2E discover card 1
Students review the multiplication of
fractions by a whole number and are guided
to complete the multiplication of a fraction
by 100. They are guided through the process
of cross-cancelling. If students fi nd the
process of cross-cancelling difficult, they
may find it beneficial to use their scientific
calculator.
Focus: To explore some different units for
various areas of measurement and convert
between the units
Focus: To round values to the nearest whole
number or a specified number of decimal
places and apply the concepts of rounding
to money
Resources: coloured pencils, highlighters
Students very briefly review the division
process required to convert fractions to
decimals. They then review the conventions
for rounding, and they complete extra
questions involving this skill independently.
Students consider the rounding conventions
for money and they complete questions in
which they round monetary values.
There is no preview question for
2D Financial calculations.
2E Understanding ratios –
preview Q16–18
2B discover card 3
➜ Focus: To convert between
units of measurement and to compare
quantities, writing the comparison of
the quantities as a ratio
Converting fractions to decimals
Focus: To write a fraction in its equivalent
decimal form
Students review the division process
required to convert fractions to decimals
in detail and are guided through two
examples, before practicing this skill by
completing questions independently.
2C Percentage calculations –
preview Q15
➜ Focus: To review the calculation
required to change a fraction into a
percentage; that is, to multiply a fraction
by 100
•
•
Direct students to complete 2E discover
card 1: Converting units if they had
difficulty with Q16, or require more
practice at this skill.
You may need to undertake some
explicit teaching so students are
reminded of how to convert between
units of length, mass, time and capacity.
It may be beneficial to provide students
with conversion charts. After the
next section is discussed, File 2.01:
Converting units
Resources: ruler, coloured pencils or
highlighters
Students consider types of measurement
and the units for each. They are guided
to convert between units of length,
discovering rules which can be applied
when converting from smaller to larger
units, and from larger to smaller units. File
2.01: Conversion charts could be provided.
Students complete extra questions
independently.
2E discover card 2
Comparing two quantities
There is no preview question for
2F Working with ratios or 2G Dividing
a ratio in a given quantity.
2H Understanding rates –
preview Q19 and Q20
➜ Focus: To identify which units
describe given quantities
•
•
Direct students to complete 2H discover
card 1: Quantities and their units if
they had difficulty with these questions
or require more practice at this skill.
You may need to undertake some
explicit teaching so students understand
which units describe different types
of measurement. Provide students
with File 2.01: Conversion charts and
ask them to write the names of each
unit of measurement onto each chart.
The charts can then be laminated and
retained by students for ready reference.
2H discover card 1
Quantities and their units
➜ Focus: To identify the units that
relate to a given quantity
Resources: ruler, coloured pencils or
highlighters
Students consider types of measurement
and units for each. They also consider
abbreviations for units. Students complete
extra questions in which the correct unit is
selected to represent a given measurement.
Professional Support
Students are guided to identify shaded
sections of diagrams as representing the
same, or equivalent, amounts. They are
guided to review the process for calculating
equivalent fractions, where the same
operation (multiplication or division) is
applied to both the numerator and the
denominator to create a fraction which is
equivalent to the original.
•
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
For example, in Q8a:
6584 × 10 = 65 840
2A discover card 2
Students can complete this division
to find the decimal equivalent of
the fraction, adding trailing zeros
as required (see 1F Operations with
decimals):
5 = 0.625
8
Focus: To write a comparison of two
quantities and express it in ratio form
Students consider ratios which describe
relationships between part to part and also
part to whole. Students may need to be
reminded that the order in which the ratio
is written is important. They also write
ratios to compare units of measurement,
and may need to be reminded to check that
the units are the same.
57
2 Percentages, ratios and rates
2A Understanding percentages
discover
•
resources
discover
2A discover card 1
Writing fractions
Intervention task
Finding the highest common factor
Intervention task
Equivalent fractions
Intervention task
2A discover card 4
›
Writing percentages
Additional skill practice
Professional Support Online
File 2.02: 100 Grid
Reproducible master
Assess
a 1 Write the proportion of the can that
has been filled as a fraction.
2 Find an equivalent fraction that has a
denominator of 100. (5 × 20 = 100.)
a 2 out of 5 L have been filled.
Fraction filled = _25
20
= _25 × __
20
40
= ___
3 Write the fraction as a percentage.
b 1 The percentage of a whole amount
is 100%. To find the percentage of
the can that is empty, subtract the
percentage that is filled from 100%.
2 Write your answer.
40% of the can has been filled.
b 100% – 40% = 60%
number of shaded parts
total number of parts
46
•
5L
4L
3L
2L
1L
a What percentage
of the can has
been filled?
b What percentage
is empty?
7 Erica’s class is raising money for charities. She
manages to collect $7 from her mother, $4 from
her aunt and $9 from her cousins.
a How much did she collect in total? $20
b Write the amount Erica received from her
7
mother as a fraction of the total. 20
c To convert to a percentage, we need a fraction
with a denominator of 100. Explain how
the fraction you obtained in part b can be
35
.
converted to ___
100
d What percentage does Erica’s mother contribute
to the total money? 35%
Friday
5 Repeat question 4 for the grid representing Friday’s
seating plan. a 31
b 31%
c 100%
60% of the can is empty.
•
key ideas
➜ A percentage represents how part of an amount is related to the total amount. The total of an amount
is equal to 100%.
➜ The term ‘per cent’ means ‘for every one hundred’ or ‘out of 100’.
27
➜ A percentage may be expressed as a fraction with a denominator of 100. For example, ___
= 27%.
100
➜ If a fraction does not have a denominator of 100, find an equivalent fraction that does.
•
7
7
28
= ___
× __4 = ___
.
For example, ___
25 25 4 100
8 a Follow the above steps to find the percentage of
money that is contributed by:
9
20
45
= 20% ii her cousins. 20 = 100 = 45%
i her aunt 204 = 100
b Add together these three percentages. What do
you find? 35% + 20% + 45% = 100%
now try these
1 Express these fractions as percentages.
a
9 Write a sentence explaining how converting
fractions so that they have a denominator of 100
helps in finding percentages.When a fraction has denominator of 100,
17
___
100 17%
b
42
___
100
c
42%
91
___
100
91%
2 What percentage of each square has been shaded?
b
a
60%
numerator is the percentage amount.
10 Erica’s class has collected $400 in total. The class
41
decides to give $200, $40 and $160 respectively to
as a percentage.
2 Explain how you would write ___
100
three different charities.
Which grid does this percentage appear to match?
41 means 41 out of 100 or 41%; matches grid for Thursday
a Write each amount to be given to the charities
100 3 a What percentage of the grid has been shaded on
200 40 160
as a fraction of the total raised. 400 , 400 , 400
Thursday’s seating plan? 41%
b Explain how these fractions can be converted so
b What percentage of the grid has been shaded on
that they have a denominator of 100.
Divide both numerator and denominator by 4 to obtain denominator of 100.
Friday’s seating plan? 69%
c Write each fraction with a denominator of 100. 50 , 10 , 40
100 100 100
4 Look at the grid representing Thursday’s seating plan.
d What percentage will be given to each of these
a How many squares have not been shaded? 59
charities? 50%, 10%, 40%
b What percentage of the grid has not been shaded? 59% e Find the sum of the percentages obtained in
c Add the percentage of the grid shaded with the
part d. What do you find? 100%
percentage of the grid not shaded. What do
11 How is finding the percentage of something with
these values add to? 100%
more than 100 parts different from, and how is it
100
d
36
___
100
36%
e
67
___
100
67%
c
40%
f
3
___
100
3%
60%
3 For each figure:
i write how many sections are shaded as a fraction of the total number of sections
ii write this fraction as an equivalent fraction with a denominator of 100
iii write the percentage of sections that are shaded.
a
16
i 20
ii
80
100
b
c
15
75
i 20
ii 100
iii 75%
iii 80%
e
4 For these cans, find the percentage:
a
5L
4L
3L
2L
1L
the same as, finding the percentage of something
with fewer than 100 parts?
i 80%
ii 20%
i that has been filled
b
10 L
8L
6L
4L
2L
•
7
i 10
70
47
ii 100
iii 70%
ii that is empty.
c
i 70%
ii 30%
10 L
8L
6L
4L
2L
i 30%
ii 70%
•
In both cases, aim to obtain a fraction with denominator of
100. With more than 100 parts, divide by a factor to obtain
denominator of 100. With fewer than 100 parts, multiply by a
factor to obtain denominator of 100.
denominator of the fraction by the same
factor. Once an equivalent fraction with
a denominator of 100 has been found,
the numerator can be written as a
percentage; that is, out of 100 parts.
whole class
➜ Focus: To consolidate student
understanding of a percentage as
representing how part of an amount is
related to the total of an amount, where
the total of the amount is 100%
•
After students complete the Discover
task, consolidate their understanding.
Ensure students understand that:
− per cent means ‘out of 100’
− the symbol for per cent is %
− a fraction with a denominator of 100
can be expressed as a fraction
− when they are working with a
fraction with a denominator that
is not 100, an equivalent fraction
can be found which does have a
denominator of 100
•
•
•
− when finding an equivalent fraction,
any operation which is performed
on the numerator of a fraction
must also be performed on the
denominator.
If students are experiencing difficulty
with these questions, or if they require
more support, refer to 2A discover card
1, 2A discover card 2 or 2A discover
card 3.
Demonstrate 2A eTutor, or direct
students to do this independently.
Direct students to the example. It shows
how to calculate what percentage of a
•
can has been fi lled, and also to calculate
what percentage remains unfilled, given
that the total percentage of the can is
equal to 100%.
Direct students to the key ideas. You
may like them to copy this summary.
•
now try these
➜ Focus: To use an understanding
of a percentage as representing a value
out of 100, and an understanding
of equivalent fractions to calculate
percentages
•
Students are then required to find
equivalent fractions with a denominator
of 100, and write a percentage to
represent the shaded amount.
In Q4 students calculate the percentage
of the can that has been fi lled and the
percentage that has not been filled,
given that the total percentage of the
can is 100%.
For those students experiencing
difficulty with these questions,
intervention through small group
teaching of the concept may be
necessary.
Some students may need to complete
further examples using a 100 grid, as
discussed in the Discover task. File
2.02: 100 Grid can be provided to
students. They can work through the
shading of specified percentages. For
example, ask students to shade the grid
to represent 28%. They would need
to shade 28 squares. It could then be
emphasised that there is a link between
28 and 28%. It can be said
the fraction 100
that the percentage sign is another way
of writing a denominator of 100.
Alternatively a practical activity can be
completed. Obtain a bottle or container
with no graduations. Ask the students
to mark a scale from 0% to 100% on the
side of the container, using masking
tape and a felt pen. Ask them to fi ll the
container to show a range of percentages
and record the result for the percentage
fi lled and the percentage empty each
time.
For extra practice, direct students to
2A discover card 4.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
…overmatter
Professional Support
•
A 5-L can has water
poured into it.
These grids both have a total of 100 parts, which makes
writing a fraction and a percentage easy. What can we
do to work in percentages if something does not have
100 parts? Let’s investigate.
Each quantity obtained in question 1 is called a
percentage. The term ‘per cent’ means ‘out of 100’ and
the symbol for percentage is %. Therefore 29% means
‘29 out of 100’. We also know that ‘29 out of 100’ can be
written as a fraction with a denominator of 100. This
29
.
means that 29% is equal to ___
100
discover task
58
➜ write
2A Understanding percentages
2A eTutor
2A Guided example
2A Test yourself
Students are guided to consider a 100
grid. They explore the grid in terms of
the real-life scenario of a seating plan
for a concert venue and represent the
number of allocated seats as percentages
of the total seats. Students are guided to
recognise that the percentages of both
the shaded and non-shaded squares
within the grid can be calculated and
that the total seating plan grid is equal
to 100%.
Students consider different scenarios
in which there are not 100 parts. They
are guided to discover that, to write
a percentage, the denominator of the
fraction needs to be 100, and that you
may need to find equivalent fractions.
Students may need to be explicitly
reminded that, when finding equivalent
fractions, they must multiply (or
divide) both the numerator and the
add to 100%.
1 Count the number of shaded squares in each grid
and then copy and complete these sentences.
a On Thursday night, 41 out of the 100
squares have been shaded.
b On Friday night, 69 out of the 100 squares
have been shaded.
File 2.03: Percentage mix and
match
•
➜ think
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
Thursday
Reproducible master
➜ Focus: To review the meaning
of ‘per cent’, and also to identify the
symbol used to represent percentage; to
review how to describe both: an amount
out of 100, and an amount not out of
100, as a percentage
6 Using the results from questions 4 and 5, write a
sentence explaining what the shaded and unshaded
sections of any shape will add to. Percentages always
A small theatre can seat 100 people. These grids
represent the theatre’s seating plan, where the shaded
squares illustrate the seats that have been reserved for
Thursday night and Friday night.
2A discover card 3
›
7c Numerator and denominator must be multiplied by same value to remain equivalent.
Multiplying denominator by 5 makes denominator 100. Multiplying numerator by 5 makes
7
35
numerator 35. So, 20 is equivalent to 100 .
Percentages surround us; for example, when we check
our favourite player’s stats, look at our computer or
iPod screen while downloading, or buy things in a sale.
But what is a percentage?
2A discover card 2
›
example
2A Understanding percentages
Deep Learning Kit
›
Q1 and Q2 involve students
representing as percentages:
– fractions with denominators of 100
– the number of squares shaded out
of 100.
In Q3 students count the number of
shaded sections and the total number of
sections to write a fraction representing
the shaded amount of each diagram.
This is written as a fraction:
59
2 Percentages, ratios and rates
2A Understanding percentages
explore
resources
Deep Learning Kit
•
explore
11 These containers each contain 50% of their total capacity.
2A explore card 1
Explain how each container holds a different volume of
The whole 100 story
Problem solving task
1 Find the percentage of balloons that are:
a yellow 20%
b not yellow 80%
2A explore card 2
›
6 Write each of these fractions as a percentage by first finding an equivalent fraction that has a denominator of 100.
a
g
60
•
•
b
h
11 = 55 = 55%
___
20 100
15 = 7.5 = 7.5%
___
200 100
c
i
13 = 52 = 52%
___
25 100
21 = 7 = 7%
___
300 100
d
j
1 = 25 = 25%
__
4 100
16 = 4 = 4%
___
400 100
e
k
8 = 80 = 80%
___
10 100
5 = 62.5 = 62.5%
__
8 100
f
l
4 = 80 = 80%
__
5 100
19 = 47.5 = 47.5%
___
40 100
ii 20%, 40%, 60%, 80%, 100%
b What percentage of battery life has been used for each
i 0%, 25%, 50%, 75%
ii 80%, 60%, 40%, 20%, 0%
icon?
iii 40%, 60%, 70%
A and B show c Explain how the icon in part iii requires a different
process compared to those in parts i and ii.
equal-sized
‘blocks’ in a panel
ii
iii
whereas C is one solid i
section of a panel. The
percentages for A and B
are found by first writing
the number of blocks as a
fraction of the total number
of blocks. In C, you need to measure the amount required as a fraction of the total panel length.
a 66% 13 Percentages can be added and subtracted like other numbers. Calculate each of these.
1
b 55 2 %
a 17% + 49%
b 23% + 32__12%
c 53__12% + 16__12% + 9__23%
2
c 79 3 %
d
7.9%
+
28.7%
+
46.5%
e
92%
−
68%
f
16__45% − 12__25%
d 83.1%
e 24%
g 37.4% + 52.9% − 15.6%
h 75.95% + 5.28% − 32.13% − 28.66%
2
f 4 5%
14
Seventy-seven
per
cent
of
the
reviewers
loved the new Harry Potter movie, 15% were undecided and the
g 74.7%
h 20.44%
remainder disliked the movie. What percentage of the reviewers disliked the movie? 8%
15 At a recent rock concert, 8% of the seats were classed as Platinum, 17% as Gold,
30% Silver and the remainder were General Admission seats. What percentage of
the seats were General Admission? 45%
a
48
9 Write a percentage to describe each of these situations by
Matthew
Thomas
first finding an equivalent fraction that has a denominator of 100.
3
60
9
45
a Andrew ate three out of five cupcakes. 5 = 100 = 60% b Nine people out of twenty went home. 20 = 100 = 45%
68
46
23
c Seventeen people out of 25 voted yes. 17
d
Mark
knew
46
people
out
of
200.
=
=
68%
=
=
23%
25 100
200 100
10 Remember that a whole amount can be expressed as 100%.
a What does 200% represent? What about 300%? twice an amount; three times an amount
b Write 100% as a fraction with a denominator of 100 and hence write 100% as a single number. 100
=1
100
c Use your answer to part b to write the numbers from 1 to 5 as percentages. 1 = 100%; 2 = 200%; 3 = 300%; 4 = 400%; 5 = 500%
d When might you use a percentage that is more than the whole amount? One possible answer is: when calculating daily intakes.
16 Draw this jar in your book and indicate the position described by each of the following.
a 15% full
200 squares in the grid to represent 80%.
8 Thomas and Matthew were asked to draw a diagram
representing 15%. Their answers are shown at right.
a Which diagram is correct? Clearly explain your
reasoning. Both are correct.
b Explain and illustrate another way of
possible answers are: shade 30 out of
representing 15%. Some
200, shade 60 out of 400.
➜ Focus: To find equivalent
fractions with a denominator of 100, in
order to write a percentage
iii 60%, 40%, 30%
b
b 80% full
c 50% full
d 20% empty
e 67% empty
17 Estimate and then calculate the percentage of each clock face that is shaded.
a 50%
11
12
b 25%
10
c
2
9
3
8
4
7
d
11
1
6
5
12
c 58 13 %
1
10
2
9
3
8
4
7
6
5
11
12
1
10
2
9
3
8
4
7
6
5
49
18 Explain the steps you could use to write __73 as a percentage. 18 Calculate 3 ÷ 7 then multiply by 100.
e
19 At a party of 20 people, 11 are female. Over the next hour, 8 females and 12 males arrive. By 10 pm, 13 females
and 11 males have gone home. What percentage of the party is male at 10 pm? 62.5%
20 Write your own percentage problem and swap with
a classmate.
reflect
Why is it important to be
able to visualise the fraction that a certain
percentage represents?
They are opposite processes: percentage is expressed as a fraction with denominator of 100, compared to expressing a fraction with
denominator of 100 as a percentage. Both rely on the fraction having denominator of 100.
•
•
•
to find equivalent fractions with a
denominator of 100, and then write a
percentage to represent each fraction.
Q7 explores the 100 grid, a smaller grid
and a grid consisting of 200 squares.
Students may need to be directed that
they could need to find equivalent
fractions.
Q8 explores different representations of
the same percentage on a 100 grid.
In Q9 students are asked to write
a fraction to represent each of four
different scenarios. These then need
to be expressed as percentages. For
•
•
•
students experiencing difficulty with
this concept, strategies are discussed in
the small group section that follows.
Q10 explores percentages larger than
100%.
In Q11 and Q12 students are asked to
write percentages to represent different
quantities. Students may be familiar
with the battery icons on mobile phones
and may be able to relate personal
experience to Q12.
In Q13 students explore the addition of
percentages and discover that they can
be added and subtracted. If students are
•
•
experiencing difficult with the addition
and subtraction, encourage them to use
their scientific calculator.
Students apply understanding of
percentages, in relation to a total of
100%, to solve Q14 and Q15. They
need to be reminded to read worded
questions carefully and they may need
to be advised to use a highlighter or a
pen to highlight the mathematics which
needs to be extracted from the words
and used for calculations.
In Q16 students represent different
percentage quantities using the visual
•
•
•
representation of the quantity of
material in a jar.
In Q17 students explore a clock face
and are required to state the percentage
shaded. They may need to be reminded
that the total number of degrees in a
circle is equal to 360°, and that each
five-minute segment on the clock
represents 30°.
Q18 requires students to list the
steps required to write a fraction as a
percentage.
In Q19 students write fractions to
describe scenarios and then convert
Students who experience difficulty in
progressing independently to finding
equivalent fractions with a denominator of
100 may benefit from a concrete task using
counters. For example, give the students
4 red counters and 20 blue counters. Th is
4.
represents 20
Ask the students, how many of these groups
of blue counters they would need to make
100. (5) Tell them they will need to count
out the same number of groups of red
counters to keep the fraction equivalent.
20 .
The equivalent fraction will be 100
Students should then write the relationship:
4 = 20 = 20%. This can be repeated. Use
20 100
only those denominators that will convert
to 100 easily.
Deep Learning Kit
2A explore card 1
The whole 100 story
Focus: To consider percentages as telling a
story for amounts out of 100
Resources: Internet access (optional)
Students interpret current statistical
information given as percentages and
convert the percentage information to
amounts out of 100. Students also reconvert
information from amounts out of 100
and then state the equivalent percentage
quantities.
Professional Support
•
3 = 6 = 6%
___
50 100
70 = 14 = 14%
___
500 100
7 a Explain how you could represent these percentages using a grid consisting of 100 squares.
i 80% shade
ii 92% shade
iii 25% shade
iv 17% shade
v 43% shade
vi 66% shade
80 squares
92 squares
25 squares
17 squares
43 squares
66 squares
b Explain how you could represent the percentages in part a using a rectangle of length 10 cm and width of
10 cm = 100 mm, so a length of 1 mm would represent 1%. For 80%, shade 80 mm (or 8 cm); for 92%,
2 cm. Show an example of this. shade 92 mm (or 9.2 cm); for 25%, shade 25 mm (or 2.5 cm); for 17%, shade 17 mm (or 1.7 cm).
c Explain how you could represent the percentages in part a using a grid consisting of 200 squares. Show an
example of this. Double the percentage amount. For example, 80% means 80 out of 100 or 160 out of 200. Shade 160 of the
explore questions
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
Q1 and Q2 relate to a photograph of
10 balloons. Students are asked to
write percentages representing the
different colours. Students may need
to be instructed to find the fraction
representing each colour first, and then
to find an equivalent fraction with a
denominator of 100.
In Q3 and Q4 students are required to
subtract the given percentage
from 100%.
In Q5 students write percentages as
fractions. In Q6 students are asked
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
4 What percentage of its original value is a car worth if it depreciates (loses value) by these amounts?
a 20% 80%
b 19% 81%
c 8% 92%
d 12% 88%
e 28.5% 71.5% f 33__13% 66 23 %
5 In the Discover task, we saw that a fraction with a denominator of 100 can be written as a percentage;
29
= 29%.
for example, ___
100
a Write these percentages as fractions with a denominator of 100.
11
47
27
73
51
89
i 11% 100
ii 47% 100
iii 27% 100
iv 73% 100
v 51% 100
vi 89% 100
b Explain how this process is both similar to, and different from, question 1 of Now try these on page 47.
a What percentage of battery life is left for each of the
battery icons shown in parts i–iii? i 100%, 75%, 50%, 25%
2A Understanding percentages
Remind students of what was learnt in
the Discover section. For this whole class
activity, students will need the three
remaining cards they cut out yesterday.
As a class, brainstorm a list of the factors
of 100. On the first of their cards the
students need to write a fraction that does
not have a denominator of 100, but does
have a denominator which is a factor of
100. The numerator can be anything, but
must be smaller than the denominator, so
that the fraction is a proper fraction. On
their second card students need to write
their fraction as an equivalent fraction
with a denominator of 100 and on the third
card students write the percentage which
represents their fraction. A game of ‘Mix
and match’ can be played. (See the previous
section for instructions on how to play.)
small group
12 Most mobile phones have an icon that tells you how much
battery life is left.
3 Answer each of these.
a Huynh has downloaded 76% of a song onto her iPod. What percentage remains to be downloaded? 24%
b If a product has been discounted by 23%, what percentage of the original price is paid? 77%
c Richmond wins 17% of their games. What percentage do they lose? 83%
d Forty-two per cent of homes were affected by floods. What percentage were not affected? 58%
e Morgan received 93% on her Japanese test. What percentage of marks was lost? 7%
whole class
•
d not green. 90%
2 From question 1, what do you notice about the percentages found when looking at a
group of 10 objects? Percentage is ten times the number of objects.
Considerable totals
Investigative task
➜ Focus: To apply understanding
of a percentage as representing a value
out of 100, and an understanding
of equivalent fractions to calculate
percentages
c blue 70%
liquid, but they are the same percentage full. As capacity of each container is
different, 50% (or 50 mL out of every 100 mL) would be a different volume of liquid for each.
➜
›
these to percentages, they may like to
refer to the steps listed in the previous
question.
Q20 is an open-ended question in which
students write their own percentage
problem and swap it with a classmate.
2A explore card 2
Considerable totals
Focus: To investigate percentages when
presented with comparable data
Resources: calculator
Students compare three different sets
of school data: student numbers, male
and female totals and student canteen
purchases. They examine the quantities in
terms of percentages, to meet a directive
from the school board to maintain the
viability of running each school’s canteen.
…overmatter
61
2 Percentages, ratios and rates
2B Percentages, decimals and fractions
discover
•
2B Percentages, decimals and fractions
actions
resources
Deep Learning Kit
discover
•
2B discover card 1
›
example 2
Multiplying and dividing by 10, 100 and
1000
Intervention task
Percentages, decimals and fractions are often written together and used
interchangeably. But how do we compare them? Consider the three numbers
9
and 0.45.
45%, ___
20
9
2 Write the equivalent fraction for ___
with a denominator of 100.
20
›
3 Write 0.45 as a fraction with a denominator of 100.
1 Multiply 0.285 by 100%. An easy way to multiply by 100 is to ‘move’
the decimal point two places to the right.
•
45
100
45
100
0.285 = 0.285 × 100%
= 0.285
2 Write the answer.
All represent same value.
5 What relationship can you see between percentages, A percentage is a quantity out of 100.
decimals and fractions? (Hint: how is the This same amount can also be expressed as a fraction
with denominator of 100 and a decimal in hundredths.
number 100 important?)
Converting percentages, fractions and
decimals
Additional skill practice
Decimal
6 Copy and complete this table.
0.375
37.5%
0.75
75%
b To convert a fraction to a percentage, obtain an
equivalent fraction with a denominator of 100 .
0.8
80%
Reproducible master
File 2.04: Section 2B Discover task
0.7
70%
c A short cut for converting fractions and
decimals to percentages is to multiply by 100.
0.68
68%
Reproducible master
Assess
0.95
95%
8 Copy and complete the following sentences.
key ideas
➜ Percentages, decimals and fractions are all closely related.
➜ To convert a percentage to a decimal, a short cut is to divide by 100. (Move the decimal point two places
Fraction with a
denominator
d
i t off 100
3
5
3
8
3
__
4
4
__
5
7
10
17
25
19
20
➜ To convert a percentage to a fraction, the number becomes the numerator and the denominator is 100.
60
100
37.5
100
75
100
80
100
Simplify the fraction if required by dividing the numerator and the denominator by the highest common
factor (HCF).
➜ To convert a decimal to a percentage, a short cut is to multiply by 100. (Move the decimal point two places
to the right.)
➜ To convert a fraction to a percentage, express the fraction with a denominator of 100 if possible.
70
___
100
Otherwise, convert the fraction to a decimal and multiply it by 100.
68
100
95
100
now try these
1 Write each of these percentages as a decimal.
a To convert a percentage to a fraction, first write the percentage as a fraction with a denominator of 100
and simplify if necessary.
2B eTutor
2B Guided example
2B Test yourself
a 17% 0.17
b To convert a percentage to a decimal, simply divide by 100.
As different organisations use fractions, decimals and/or percentages to communicate, it is useful when comparing the different
representations.
discover task
b 21% 0.21
c 35% 0.35
d 8% 0.08
e 15% 0.15
2 Write each of the percentages in question 1 as a fraction in simplest form.
9 Why might it be important to be able to convert between percentages, decimals and fractions?
a
e1
17
100
b
21
100
c
7
20
d
3 Write each of these percentages as a decimal.
a 6.78% 0.0678 b 3.1% 0.031
c 162% 1.62
g 10.57% 0.1057 h 3% 0.03
i 472.38% 4.7238 j
2
25
e
3
20
f 98% 0.98
f
49
50
d 72.1% 0.721 e 347% 3.47
215.7% 2.157 k 0.1155%
l
d 0.08 8%
e 0.9 90%
f 0.31 31%
e 7.057 705.7%
f 1.859 12 185.912%
0.001 155
f 81.09% 0.8109
0.04% 0.004
4 Write each of these decimals as a percentage.
•
62
•
•
Students are guided to consider the
way in which fractions, decimals and
percentages are used interchangeably
to represent the same amount in
everyday life.
Students are guided to discover that the
number 100 is very important when
working with fractions, decimals and
percentages.
Students can be provided with a copy
of File 2.04: Section 2B Discover task
(one between two). This reproducible
master contains the table from the
Discover task. Students can complete
the table and then paste it into their
work book.
Students are guided to summarise in
words the different steps required to
convert between percentages, decimals
and fractions.
a 0.71 71%
50
➜ think
➜ write
1 Write 3.75% as a fraction to show what it means.
2 To express this fraction as a decimal, we divide the numerator by
the denominator. An easy way to divide by 100 is to ‘move’ the
decimal point two places to the left.
3 Insert a place-holding zero in the empty space (tenths column).
3.75
3.75% = ___
100
= 3.75 ÷ 100
= . 375
4 Write your answer. Remember to show a digit before the decimal
point. In this case as there are zero ones, we write 0.
e2
= .0375
c 0.05 5%
a 0.369 36.9%
b 0.248 24.8%
c 10.06 1006%
g 12.1 1210%
h 2.7 270%
i 0.9523 95.23% j
d 0.81 81%
0.005 0.5%
k 0.244 61 24.461% l
0.000 504 0.0504%
51
6 Write each of these fractions as a percentage by first converting to a fraction with a denominator of 100.
a
= 0.0375
b 0.28 28%
5 Write each of these decimals as a percentage.
17 = 0.17
17% = 100
Students who are experiencing difficulty
with the arithmetic can use their
scientific calculators to convert between
decimals, fractions and percentages.
For example, using the TI-30XB
Multiview for Q1c:
3
__
5 60%
b
7
___
10 70%
c
19
___
20
95%
d
23
___
25 92%
e
47
___
100 47%
f
3
__
2 150%
f
3
__
37.5%
8
35
35% = 100
This can be simplified on the calculator.
Type the fraction using the q key. Press
<and the simplified fraction will
7 Write each of these fractions as a percentage by first converting to a decimal number.
a
5
__
62.5%
8
b
27
___
67.5%
40
c
7
___
80
8.75%
d
3
___
18.75%
16
e
11
___
8.8%
125
8 Check your answers to questions 6 and 7 with a calculator.
whole class
➜ Focus: To consolidate student
understanding of the importance of,
and the steps involved in converting
between, percentages, decimals and
fractions
• After students complete the Discover
task, consolidate their understanding.
Ensure students understand that:
− percentages, decimals and fractions
are closely related
•
− to convert from a percentage to a
decimal, a shortcut is to divide
by 100
− to convert from a percentage to a
fraction, the percentage is written as
the numerator and the denominator
of the fraction is 100. The fraction
may be simplified.
− to convert a decimal or a fraction to
a percentage, multiply by 100.
If students are experiencing difficulty
with these questions, or if they require
more support, refer to 2B discover card
1, 2B discover card 2 or 2B discover
card 3.
•
•
•
Demonstrate 2B eTutor, or direct
students to do this independently.
Direct students to the examples.
Example 1 shows how to write a
percentage as a decimal. Example 2
shows how to convert a decimal to a
percentage.
Direct students to the key ideas. You
may like them to copy this summary.
now try these
➜ Focus: To convert between
fractions, decimals and percentages
•
•
•
•
Q1 involves writing percentages as
decimals. The shortcut is to divide by
100 (move the decimal point two places
to the left). Some students may find
it easier to write each percentage as a
fraction with a denominator of 100 first.
Q2 requires students to simplify
the fractions that correspond to the
percentages in Q1.
In Q3 students write percentages as
decimals.
In Q4 and Q5 students write decimals
as percentages.
•
•
7
appear to the right: 20
For Q3a, type in 6.78, press %_.
Press < and the decimal will appear
to the right.
For Q5a, type in 0.369, press %R.
Press < and the percentage will
appear to the right.
For Q7a, type in the fraction using the
q key, Press %R. Press < and the
percentage will appear to the right.
For further practice, direct students to
2B discover card 4.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
…overmatter
Professional Support
•
example 1
Write 3.75% as a decimal.
•
to the left.)
2B Percentages, decimals and fractions
60%
a To convert a decimal to a percentage, first
express the decimal as a fraction with a
denominator of 100 .
File 2.02: 100 Grid
Percentage
age Fraction
0.6
7 Copy and complete these sentences.
Professional Support Online
➜ Focus: To review the link between
fractions, decimals and percentages, and
to review the methods used to convert
between these different representations
of the same amount
•
= 28.5%
4 What do you notice about the answers to questions 1, 2 and 3?
Converting fractions to decimals
Intervention task
2B discover card 4
›
➜ write
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
1 Express 45% as a fraction with a denominator of 100.
›
2B discover card 3
➜ think
45
100
2B discover card 2
Rounding
Intervention task
Write 0.285 as a percentage.
Q6 requires students to write each
fraction as a percentage, by first
finding an equivalent fraction with a
denominator of 100.
In Q7 students convert fractions to
decimals and then write them as
percentages.
In Q8 students are encouraged to check
their answers using a calculator.
For those students experiencing
difficulty with these questions,
intervention through small group
teaching of the concept may be
necessary.
It may be necessary to take some
students back to the 100 grid so that
they can visualise the link between
fractions, decimals and percentages.
For example, in Q1a, ask students to
shade 17 squares. Explain that, to write
this as a fraction, the number of shaded
squares will be represented as the
numerator and the denominator will
be the total number of squares.
Remind students of the link between
a denominator of 100 and the
decimal place-value chart. (See 1E
Understanding decimals.)
63
2 Percentages, ratios and rates
2B Percentages, decimals and fractions
explore
The calculator can be used to convert some
of the more difficult options. For example,
in Q9a:
resources
Deep Learning Kit
e Express the value obtained in part c as:
2B explore card 1
i a percentage 25%
Best one – Percentages, decimals or
fractions
Problem solving task
1 A packet of gumballs is emptied onto a table.
a Write the number of blue gumballs as a fraction of the totall number of gumballs.
f Calculate the girl’s energy expenditure as a percentage of the boy’s energy expenditure.
3
13
g Calculate the boy’s energy expenditure as a percentage of the girl’s energy expenditure.
b Convert this to a decimal number and round to four decimal places. 0.2308
d What percentage of the gumballs (correct to two decimal places) are:
i orange? 30.77%
ii green? 7.69%
iii red or yellow? 38.46% iv not red? 84.62%
b
c
d
15
20
2
5
12
15
3
8
23
e 268 ___
% 2.6846
50
i 400% 4
64
•
c
4
__
9
d
44.44%
6
__
7
85.71%
e
8
___
15
53.55%
f
ii 500% 5
iii 600% 6
1
5
__
% 160
8
e
a
192 4
=
240 5
7
3
4
__
__
__
5 , 0.89, 4 , 200%, 5 , 1.3
13
3 ___
b 4.9, ___
, 560%, 7.2, 4 ___
, 22
10 4
2
15 A group of 580 P-platers were surveyed on their
views about organ donation. The results are given
below.
31
50
2
9
100%
Start
10
11 %
0
The remainder was undecided.
4138 %
60%
3
4 10
7
11
4
5
13
2
560% 6
7
16 Write three fractions between 0.15 and 16%.
1
=4
31
157
ii a decimal. 0.2
•
8
53
159
Some possible answers are: 200, 1000, 1000.
18
Some possible answers are: 0.251, 0.2655, 0.27.
76
Write three percentages between ___
and 0.847.
95
4
Some possible answers are: 81%, 82%, 84%.
reflect Why do you think it is
important to be able to convert between
fractions, decimals and percentages?
16
5
3
4
In Q10 students are required to write
a range of whole number and fraction
percentages as fractions in simplest
form. For students experiencing
difficulty with these conversions,
strategies are discussed in the small
group section that follows.
Q11 and Q12 are real-life applications in
which students are required to subtract
the given percentage from 100% to find
the missing amount.
Q13 involves organising a collection
of fractions, decimals and percentages
along a number line; that is, in
•
•
•
0.89 1
1.3
7
5
200%
0 0.28 65% 1
ascending order. Remind students that,
before making comparisons, all the
amounts need to be converted to the
same type.
Q14 is an interesting game. While some
students may find this challenging,
the more able students will enjoy the
context.
In Q15 students are required to write
fractions and then convert fractions
to percentages in order to compare
quantities.
Q16–18 are open-ended questions
in which students are asked to write
3
2
145 2
Press n and the simplified fraction will
3
)
appear to the right (700
When students are using their calculator
for a number of different conversions, they
should be encouraged to make a summary
of the different types and the steps involved,
for reference.
Deep Learning Kit
2B explore card 1
Best one – percentages, decimals or
fractions
7.2
One possible answer is: just over half feel that organ donation should be compulsory and just under half feel it should not be compulsory.
0
•
93
105
c Write a short paragraph about the values obtained and comment on the differences between each group.
1
=5
13
17 Write three decimals between 25% and ___
.
48
•
88
115
0.76
22
4
4.9
101
100
0.52 33% 0.49 100% Finish
5
60
5
•
Type in the fraction using q%_.
This should look exactly the same as the
question. Press < and the decimal will
appear to the right. (0.004 285 714 2857)
compulsory: 52.59%, non-compulsory: 43.28%, undecided: 4.14%
2500
12 500
2500
10 500
d Express the value obtained in part b as:
•
12
28%
b Express each of the views as a percentage, correct to two decimal places.
a Find the difference between these two values. 2500 kilojoules
100%.
Q5–8 involve within real-life
applications of percentages, decimals
and fractions. Students may
require some guidance as to how to
extract from the written question
the information required for the
calculation. Encourage students to use
a highlighter or a pen to highlight the
mathematical information.
In Q9 students are required to convert
mixed number percentages to decimals.
For Q10d:
a How many P-platers were undecided? 24
8 On average, a boy will expend (use) 12 500 kilojoules of energy per day while a girl will expend 10 000 kilojoules.
• Q3 explores percentages larger than
11
11
___
% 300
3
8
9
72
90
0.75 72% 132%
be compulsory.
1
3
8
5
0.5
3
11
61
___
of the P-platers believed that organ donation should
116
a What percentage of the medals he won were bronze? 50%
b What fraction of the medals he won were silver? 16
c Write the difference as a fraction of the girl’s energy expenditure in simplest form.
h
16 __
3
4
___
__
5 , 2 , 340%, 0.28, 15, 65%
0.25
115
251 P-platers believed that organ donation should not
be compulsory.
7 Whilst competing in the Olympics, the Russian gymnast Alexei Nemov won
a total of four gold, two silver and six bronze medals.
b Write the difference as a fraction of the boy’s energy expenditure in simplest form.
1
4
__
7% 175
g
3
25
c
14 Starting at zero in this diagram, find the shortest path 0, 10%, 0.52,
11
required to get to 100%, moving in ascending order. 7 , 0.76, 88 ,
11
b As a decimal, what part of the tub remains? 0.38
9
9
__
% 400
4
f
13 For each of these lists, draw a number line and place each number in its
correct position.
a What fraction of the tub is required? Write your answer in simplest form.
i a percentage 20%
3
12
___
% 700
28
b What percentage will make it through to the finals? 12.5%
6 A recipe requires 62% of a 2-litre tub of ice-cream.
c What fraction of the medals he won were gold?
d
a What fraction of players (in simplest form) will not make it through to
the finals? 78
c Write the answers to parts a and b as decimals. 0.8, 0.2
52
47
100
b What percentage of marks were lost? 12%
a What fraction of the tickets had they sold? Write your answer in simplest form.
(37.5%)
9
b 245% 220 c 47%
12 Of the 128 tennis players competing in the Australian Open, only 16 will make
the finals.
5 Katharine and Genevieve are able to sell 192 of the 240 raffle tickets for their dancing school fundraiser.
(80%)
17
25
a What fraction of marks were lost? Write the answer in simplest form.
v 375% 3.75
b Another cereal has 4.75 times the amount of the daily recommended intake of sugar. Express this number
as a percentage. 475%
b What percentage of the tickets did they still have to sell?
Type in the mixed number using
%N%_. This should look exactly the
same as the question. Press < and the
decimal will appear to the right. (2.6846)
11
h 14 ___
% 0.1455
20
11 In a piano exam, a student begins with a score of 100% and has marks taken away for mistakes.
Georgio received a score of 88% on his piano exam.
a One particular cereal contained 350% of the daily recommended intake in sugar. Express this percentage
as a decimal number. 3.5
20%
16
g 38 ___
% 0.3864
25
10 Write these percentages as fractions in simplest form. (Hint: to divide fractions, convert the division sign to a
1
1
= ___
.
multiplication sign and turn the fraction that follows upside down.) For example, __41% = __41 ÷ 100 = __14 × ___
100 400
11
___
12 91.67%
a 68%
iv 250% 2.5
47
f 92 ___
% 0.9294
50
23
%
268 50
19
d 62 ___
% 0.6295
20
13
c 21___
% 0.2152
25
Focus: To determine the best choice to
describe a quantity from either a percentage,
a decimal or a fraction
Resources: newspapers (optional)
3 340%
fractions, decimals and percentages
respectively, between set boundaries.
small group
➜ Focus: To develop a strategy that
can be used to convert more complex
percentages into decimals and fractions
Students consider everyday scenarios
and how they would be reported in either
percentage, decimal or fraction terms.
They are guided to consider which form is
the more appropriate for each situation, to
mostly clearly convey information.
2B explore card 2
65
Switch!
Focus: To use percentages, fractions
and decimals to determine and compare
solutions to two-step problems
Resources: calculator (optional)
Some students will be finding the
arithmetic required extremely challenging
and can be encouraged to use their
calculators to convert between fractions,
decimals and percentages.
Professional Support
•
63.64%
4 A recent study found that many of the popular breakfast cereals contain more than the recommended daily
intake of sugar.
(40%)
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
Q1 involves students writing fractions
to represent different coloured gumballs
as a fraction of the total number of
gumballs. Students then convert
the fractions to decimals and round
the decimals to four decimal places.
Remind students that they can use the
mode setting on their calculator to give
four decimal places.
In Q2 students convert fractions to
percentages correct to two decimal
places. If students have set their
calculator for the first question, remind
them that they will need to change
the setting.
7
___
11
100
b To express 100% as a number, simply divide it by 100. For example, 100% = ___
= 1. Use this method
100
to express each of the following percentages as a number.
explore questions
•
b
a If 200% is twice the original amount, what does 400% mean? four times the original amount
(75%)
➜ Focus: To apply understanding
of the relationship between fractions,
decimals and percentages
33.33%
For Q9e:
2B Percentages, decimals and fractions
a
1
__
3
9
b 5___
% 0.059
10
a 17 __25% 0.174
3 Percentages greater than 100% occur when we have more than the whole amount.
Remind students of what was learnt in the
Discover section. Some prompts are:
• Convert the following decimals to
percentages:
a 0.8795 (87.95%)
b 0.76
(76%)
c 1.429 (142.9%)
d 0.056 (5.6%)
• Convert the following fractions to
percentages:
5
= 4 = 125%
9 Write each of these percentages as a decimal. (Hint: in some cases you may need to change the fraction part to
a decimal first.)
2 Write each of these fractions as a percentage correct to two decimal places. (Hint: convert each fraction
to a decimal number first.)
a
4
= 5 = 80%
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
c Convert this decimal to a percentage. What percentage of these gumballs are blue? 23.08%
›
whole class
10 000
12 500
12 500
10 000
h Comment on the values obtained in parts f and g. Girl’s energy expenditure is 80% of boy’s; boy’s is 125% of girl’s.
2B explore card 2
Switch!
Game
Type in the mixed number using
%N%_. This should look exactly the
same as the question. Press < and the
decimal will appear to the right. (0.174)
ii a decimal. 0.25
➜
›
17 25 %
explore
Students are given a series of questions that
involve basic operations with percentages,
fractions and decimals. Students convert
all forms in turn, to either percentages,
fractions or decimals and compare the ease
…overmatter
2 Percentages, ratios and rates
2C Percentage calculations
discover
resources
2C Percentage calculations
Deep Learning Kit
•
discover
2C discover card 1
›
Multiplying fractions by 100
Intervention task
Zahra’s younger sister Amina has been made captain of the school
basketball team and she asks Zahra to help her put together a
team list based on the results of a training session.
2C discover card 2
›
1
Professional Support Online
File 2.01: Conversion charts
2
Reproducible master
3
File 2.05: Section 2C Discover task
Reproducible master
4
Assess
5
number of successful shots.
number of attempts
•
•
66
•
Students convert each fraction to a
percentage for ease of comparison.
Students also consider an extra factor
(throws from the free-throw line) and
then calculate the percentage of success
including this new consideration.
Students are required to select players
for a new team, giving detailed reasons
for their decisions.
Calculate 7% of
220 m.
1 Convert the percentage to a fraction, replace ‘of ’ with a
multiplication sign and write the amount as a fraction with
a denominator of 1.
2 Look for any common factors in the numerators and
denominators. The numbers 100 and 220 have a HCF of 20.
3 Cancel 100 and 220 by dividing both numbers by 20
(shown in pink).
7% of 220 m
7
220
× ___
= ___
100
1
5
8 8
3
1010
11
2020
7
1010
2
3 3
5
1212
4
or 236
6
9
3
15 or15
5
8
11
11
3
5 5
62.5%
30%
55%
70%
66.67%
41.67%
66.67%
60%
72.73%
60%
5
3
11
7
2
5
4
9
8
3
6
3
6
7
4 Write the result obtained after cancelling.
7
5 Multiply the numerators together and then multiply the
denominators together.
6 Convert the fraction to a decimal and write the answer
with the appropriate unit.
4
7
6
7
7
×
5 100
11
= _75 × __
1
77
__
= 5
22011
1
= 15.4 m
key ideas
➜ To express one value as a percentage of another:
the seven girls with the highest percentages (Penelope, Evelyn, Fatima, Huan, Bianca, Joey, Tamika)
Number of
attempts
Number of
successful shots
11
6
8
12
7
9
20
10
5
4
7
4
3
11
6
3
16
8
5
2
1 write the values as a fraction
2 convert the fraction to a percentage.
Remember to include any units
required with your answers to
percentage calculations.
➜ To calculate the percentage of an amount:
1
2
3
4
Bianca 63.64%,
Brittney 66.67%,
Claire 37.5%,
convert the percentage to a fraction
replace ‘of ’ with a multiplication sign
write the amount as a fraction with a denominator of 1
perform the calculation and write the final answer in the correct form.
Evelyn 91.67%,
now try these
Fatima 85.71%,
Georgina 33.33%,
e1
1 Express these values as percentages.
Huan 80%,
a 20 as a percentage of 80 25%
b 45 as a percentage of 50 90%
c 19 as a percentage of 25 76%
Joey 80%,
d 90 as a percentage of 120 75%
e 48 as a percentage of 160 30%
f 18 as a percentage of 40 45%
Penelope 100%,
g 63 as a percentage of 315 20%
h 300 as a percentage of 60 500%
i 240 as a percentage of 40 600%
Tamika 50%
2 Express these values as percentages.
Fatima, Huan, Bianca, Joey. Remaining position to Tamika (60%, 50%) or Brittney (30%, 66.67%).
a 140 as a percentage of 800 17.5% b 72 as a percentage of 36 000 0.2% c 46 as a percentage of 125 36.8%
54
example 1
➜ think
➜ write
Express 99 as a
percentage of 500.
1 Write 99 as a fraction out of 500.
2 To convert to a percentage, multiply the fraction by 100%.
3 Simplify the fraction by dividing the numerator and
denominator by 100 (the HCF).
99
___
500
99
= ___
× 100%
4 Divide 99 by 5 and write your answer.
=
99
5 500
99
= ___
5%
×
1001
%
1
d 57 as a percentage of 200 28.5%
e 133 as a percentage of 500 26.6% f 39 as a percentage of 40 97.5%
g 58 as a percentage of 80 72.5%
h 282 as a percentage of 48 587.5% i 6450 as a percentage of 80 8062.5%
3 Calculate each of these.
500
= 19.8%
e2
➜ Focus: To consolidate student
understanding of percentage
calculations, including conversions
between fractions, decimals and
percentages
• After students complete the Discover
task, consolidate their understanding.
Ensure students understand:
− how to convert written text into
a fraction
•
•
− how to convert a fraction to a
percentage by multiplying by 100
− how to calculate a percentage of
an amount
− that, where units are included in the
question, they must also be included
in the answer.
If students are experiencing difficulty
with these questions, or if they require
more support, refer to 2C discover
card 1.
Demonstrate 2C eTutor, or direct
students to do this independently.
55
a 10% of 840 84
b 65% of 360 234
c 42% of 1800 756
d 6% of 150 9
e 18% of 6400 1152
f 25% of 18 4.5
g 125% of 580 725
h 9% of 1250 112.5
•
•
4 Calculate each of these.
a 63% of 720 m
b 150% of 90 seconds
c 35% of 660 cm
d 12% of 840 L
e 70% of 250 people
f 40% of 190 buttons
g 45% of 450 m
h 10.5% of 112 mL
453.6 m
175 people
whole class
•
6
It is also important to be able to throw from the free-throw line. Zahra tries to predict the number of successful
shots each girl would get from the free-throw line based upon
Table 2
their 3-point results.
Name
7 How would you use these percentages to predict how many
successful shots each girl would get from 10 attempts from
the free-throw line? Divide the percentage result by 10 (see answer to Q5). Bianca
Brittney
8 Complete the last column of table 1 by dividing each
Claire
percentage by 10 (for the number of attempts) and rounding
Evelyn
to the nearest whole number.
Fatima
Amina then gives Zahra the actual results of the girls’ attempts
Georgina
from the free-throw line, shown in Table 2.
Huan
9 Copy the table into your book and use it to find the
Joey
percentage of successful shots from the free-throw line.
Penelope
10 List who you would advise Amina to pick for the team.
Tamika
Explain your decision. Top six in both lists are: Penelope, Evelyn,
=
•
•
135 seconds
76 buttons
Direct students to the examples.
Example 1 shows how to express one
quantity as a fraction compared to a
total, and then demonstrates how to
convert this fraction to a percentage.
Example 2 shows how to calculate a
percentage of an amount.
Direct students to the key ideas. You
may like them to copy this summary.
231 cm
202.5 m
•
•
•
now try these
➜ Focus: To apply the appropriate
percentage calculation to a range of
questions
•
100.8 L
11.76 mL
Q1 and Q2 require that students express
values as percentages. Students may
need to be reminded to write a fraction
first, and then convert the fraction to
a percentage.
In Q3 students calculate the percentage
of an amount.
In Q4 students calculate the percentage
of an amount, but they need to be
reminded to include units as part of
their answer because there are units
given in the question.
For those students experiencing
difficulty with these questions,
POTENTIAL DIFFICULT Y
Students need to be able to clearly
identify the type of calculation required.
A summary of the different appearance
of each type may be useful for some
students.
Deep Learning Kit
2C discover card 1
Multiplying fractions by 100
Focus: To express 100 as a fraction and
multiply it by another fraction
Students review the multiplication of
fractions by a whole number and are guided
…overmatter
Professional Support
•
➜ write
2C Percentage calculations
•
➜ think
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
Bianca
Brittney
Why might the girl who had the most successful shots not be
the best shooter? depends on total number of attempts made
Claire
Evelyn
Copy the table and add three columns to the right.
Fatima
Express each girl’s number of successful shots from the
Georgina
3-point line as a fraction of her number of attempts. Write
Huan
your answers in the first of the three new columns.
Joey
Convert each fraction to a percentage to complete the second
Penelope
of the new columns.
proportion of attempts
What do these percentages represent? that produced successful shots; Tamika
6 Which girls would you pick to fill the team (seven girls) based on these results?
➜ Focus: To discover how
percentage calculations can be applied
to real-life situations
Students are guided to consider the reallife application of selecting team-mates
based upon a training session in which
the number of attempts at goal and the
number of successful shots at goal were
recorded for each player.
File 2.05: Section 2C Discover task can
be provided to students for this task. It
contains the tables required, complete
with extra columns.
Students are guided to write a fraction
for each player, comparing the
number of successful shots at goal
compared to the total number of shots:
Number of
NumberSuccessful
of
Fraction
Percentage
shots
10 attempts
attempts
successfulfrom
shots
that is, percentage shows how many successful shots would expect to shoot in 100 attempts
discover task
•
Name
Table 1 shows the list of girls trying out for the team and their
efforts at shooting from the 3-point line.
Calculating percentages
Additional skill practice
2C eTutor
2C Guided example
2C Test yourself
Table 1
example 2
intervention through small group
teaching of the concept may be
necessary.
Encourage students to read the
questions aloud to identify the type
of calculation which is required.
Where a question reads ‘out of’ or ‘as a
percentage of’, the key strokes required
are as shown for Q1a in the point below.
Where the question reads ‘per cent of’,
the key strokes required are as shown
for Q3e.
Some students may be finding the
arithmetic increasingly difficult and
can be encouraged to use their scientific
calculator to assist them with these
calculations. It is important that
the students can identify the type of
calculation required and then use their
calculator appropriately.
For example, to express 20 as a
percentage of 80 for Q1a, the key strokes
described below are necessary.
20
.
Students must first write a fraction: 80
Then they use the q key to enter
the percentage, and press %R to
calculate the percentage. (25%)
To calculate a percentage of an amount
as in Q3e, students must complete the
following key strokes:
Type 18 %_. Remind students that
the word ‘of’ can be replaced with a
multiplication, so press V and type
in the amount, 6400. Press < to
complete the calculation:
18% of 6400 = 1152
For extra practice, direct students to
2C discover card 2.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
67
2 Percentages, ratios and rates
2C Percentage calculations
explore
scaffolding which can be followed if
required.
resources
Percentage increase:
Deep Learning Kit
explore
›
Percentages within
Problem solving task
1 The tallest living horse is a Belgian draft horse named Big Jake, who is
210 cm tall. The smallest living horse is a dwarf miniature horse named
Thumbelina who is 44.5 cm tall. As a comparison, an average horse is about
165 cm tall and an average Labrador dog is about 60 cm tall.
89
a Write Thumbelina’s height as a fraction of Big Jake’s in simplest form. 420
b Convert the answer from part a to a percentage rounded to two decimal places.
21.19%
c Find the percentage of:
i Thumbelina’s height compared to an average horse’s height 26.97%
ii Thumbelina’s height compared to an average Labrador’s height 74.17%
iii Big Jake’s height compared to an average horse’s height. 127.27%
The height of a horse (and dogs) is measured at its withers (shoulders) because this height does not
change like the height of the head can with movement.
d Estimate how tall these two horses would be if they were measured to the top of their heads. One possible answer is: Big
e How would this change the percentage you calculated in part b? Jake may be 252 cm and Thumbelina may be 53 cm. At these
2C explore card 2
›
Of and As what’s the diff?
Problem solving task
whole class
➜ Focus: To apply understanding
of different types of percentage
calculations to real-life scenarios
•
68
•
•
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
Q1 involves students comparing the
height of the tallest Belgian draft
horse to other horses, and also to a
Labrador. Students will need to write
the comparisons as fractions and then
convert them to percentages.
Q2 requires students to express
comparisons as percentages. Students
may need to be reminded that, where
heights, the percentage from part b changes to 21.03% (not
3 Express each of these as a percentage. Remember to convert them first so that they are in the same units.
a 50 g out of 4 kg 1.25%
b 48 minutes out of 8 hours 10% c 8 months out of 3 years 22.2%
d 18 hours out of one day 75%
e 125 mm out of 60 cm 20.83%
f 90 min out of 24 hours 6.25%
Since the Nintendo Wii console went on sale, medical staff have
recorded an increase in the number of video gamers arriving at hospital
for treatment. This table shows the percentage (to one decimal place) of
Wii gamers treated for each category of injury.
If 380 gamers have been treated at a hospital to date, calculate how
many people suffered injuries in each category.
5 A breakfast cereal contains 24% rice, 15% wheat bran and 6% whole
wheat. How many grams of each of these ingredients are there in a
600 g cereal box? rice 144 g, wheat bran 90 g, whole wheat 36 g
56
Nature of injury
Percentage
Sprain or strain
Open wound
Superficial injury
Fracture
Dislocation
Other/unspecified
34.2%
26.3%
10.5%
10.5%
7.9%
10.6%
6 Ella downloads the game Angry Birds onto her phone. On her first
day playing it (Monday), her highest score is 50 000. On Tuesday she
increases this by 2250.
a What is her new highest score? 52 250
b What is her new highest score as a percentage of her old highest score? 104.5%
c What is the increase as a percentage of her old highest score? 4.5%
d We would say that Ella has increased her highest score by 4.5%. How does this
relate to your answers to parts b and c? Her new score as a percentage of the old score (104.5%) is equivalent to a 4.5% increase.
(100% + 4.5% = 104.5%)
On Wednesday, Ella increases her highest score to 54 500.
e By how much did Ella increase her score? (Hint: subtract your answer to part a from 54 500.) 2250
f Calculate this increase as a percentage of her last highest score (from Tuesday). Round to one decimal place. 104.3%
g Explain why, even though she increased her highest score by the same amount on Tuesday and Wednesday,
her percentage increase is better on Tuesday. Same amount (2250) is being divided by higher previous score (54 500
compared to 52 250) so the percentage is lower.
there are units in the question, there
must also be units in the answer. Q3
continues with similar questions;
however students need to inspect the
units given. When expressing one
amount as a percentage of another, the
units must be the same. Students may
need to complete some conversions
before expressing the amounts as
fractions. File 2.01: Conversion charts
can be provided to students if they
require assistance with different types
of conversions.
•
•
•
Q4 and Q5 involve real-life scenarios in
which students are required to calculate
percentages of amounts.
Q6 requires students to correctly
identify and apply both types of
calculations (‘out of’ and ‘a per cent of’)
at different stages.
Q7 introduces students to the
calculations percentage increase and
percentage decrease. A percentage
increase is when the percentage amount
is added to the original amount.
A percentage decrease is when the
18
100 × 300 = 54
Step 2: Because this is an increase, the
amount is added to the original amount.
300 + 54 = 354
The new amount is 354.
Percentage decrease:
Calculate the new amount if 650 was
decreased by 25%
8 For each of these situations:
ii state whether it is an increase or decrease
i find the amount of change that has occurred
iii calculate the percentage increase or decrease.
a A room that was holding 25 people now has 31 people. i six people ii increase iii 24%
b A shelf that had 40 chocolate bars now has only 5 chocolate bars. i 354 choc bars ii decrease iii 87.5%
c Stacey moves house and now instead of walking 1.5 km to school, she walks 2 km. i 0.5 km ii increase
d Ahmed was 150 cm tall in Year 7 and now he is 165 cm tall. i 15 cm ii increase iii 10%
e A jogger improves her lap time from 20 minutes to 18 minutes. i 2 minutes ii decrease iii 10%
Step 1: Calculate 25% of 650.
25
100 × 650 = 162.5
iii 33%
Step 2: Because this is a decrease, the
amount is subtracted from the original
amount.
9 Calculate the new amount if:
a a high score of 400 was increased by 15% 460
b a water tank holding 2500 L decreased by 30% 1750 L
c a playlist of 60 songs was increased by 75% 105 songs
d the waiting time of 15 minutes was decreased by 40% 9 minutes
e the crowd at a rugby match was 11 000 one week and increased by 5% the next week. 11 550
10 In question 6 part b we found that Ella’s new high score was 104.5% of her old score. This 4.5% increase can be
written as (100% + 4.5%).
a What do you think a 4.5% decrease might be written as? 100% − 4.5% = 95.5%
b Copy and complete this table. The first row has been done for you.
Percentage
Increase or decrease
Expanded form
Full percentage
Number form
4.5%
7%
25%
increase
decrease
decrease
100% + 4.5%
100% − 7%
104.5%
1.045
40%
increase
15%
decrease
100% + 40%
100% − 15%
20%
increase
100% + 20%
100% – 25%
93%
0.93
75%
0.75
140%
85%
1.40
0.85
120%
57
of situations that involve percentage
calculations?
For a percentage increase, add the percentage increase to 100% and convert the result to a decimal. Multiply the quantity
by this decimal to obtain the increased amount. For a percentage decrease, subtract the percentage decrease amount from
100% and convert the result to a decimal. Multiply the quantity by this decimal to obtain the decreased amount.
•
percentage amount is subtracted from
the original amount. For students
experiencing difficulty with these
calculations, strategies are discussed in
the small group section that follows.
Q8 and Q9 involve applications of
percentage increases and decreases,
and in Q10 students are introduced to a
method which can be used as a shortcut
for these calculations.
In Q11 students are encouraged to
discuss the shortcut with a classmate.
Discussing strategies can be very
beneficial for students because they
If students are finding this concept tricky,
they should stick with this method of
calculation rather than taking shortcuts,
and should follow the steps as shown. As
previously, students can be encouraged to
use their calculators if required.
Deep Learning Kit
2C explore card 1
Focus: To determine percentages of
percentages
Resources: calculator
70 × 1.4 = 98
11 Explain to a classmate a quick way to calculate percentage
increase or decrease of a quantity.
•
The new amount is 487.5.
Percentages within
1.2
c How are the fourth and fifth columns related? Final column is decimal form of the percentage (fourth column).
d The fifth column can be used to easily calculate percentage increases and decreases. For example,
a 4.5% increase on 50 000 is 50 000 × 1.045 = 52 250. Use the fifth column to quickly find:
i a 25% decrease on 200
ii a 20% increase on 150
200 × 0.75 = 150
150 × 1.2 = 180
iii a 7% decrease on 2500
iv a 40% increase on 70.
reflect What are some examples
2500 × 0.93 = 2325
650 – 162.5 = 487.5
need to have their own understanding
clear, before they can explain it to
others.
small group
➜ Focus: To develop understanding
of percentage increases and percentage
decreases
Link the concept of an increase with
addition, and the concept of a decrease with
a subtraction. Explicitly demonstrate both
types of calculations for students, providing
Students consider the numbers of different
types of people at a concert as a percentage
of percentages. They break down what
quantity a percentage of a percentage
actually represents.
Professional Support
explore questions
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
much difference!).
2 Express each of these as a percentage.
a 72 marks out of 90 marks 80%
b 37 marks out of 40 marks 92.5% c 560 grams out of 640 grams 87.5%
d 21 girls out of a class of 70 30% e 130 m out of 5200 m 2.5%
f 60 grams out of 2000 grams 3%
sprain or strain4
130, open
wound 100,
superficial
injury 40,
fracture 40,
dislocation
30, other/
unspecified 40
Step 1: Calculate 18% of 300.
57 225
7 The type of percentage change in question 6 is called a percentage increase because
the percentage amount is added to the original amount to give a bigger number.
A change in percentage can also be a percentage decrease. This is when the
percentage amount is subtracted from the original amount to give a smaller amount.
a What is 20% of 50? 10
b If a platter originally had 50 cupcakes and 20% were eaten, how many cupcakes would be left? 40
c By the end of the night, 20% of the remaining cupcakes were taken from the platter. How many cupcakes
is this? (Hint: this is different from your answer to part a.) 8
d Find how many cupcakes are still on the platter at the end of the night. 32
2C Percentage calculations
Remind students of what was learnt in
the Discover section. Some possible
prompts are:
• A student achieved a score of 19 out of
25 for a mathematics test. Convert this
to a percentage. (76%)
• Your friend won $750 at a fair. Answer
the questions below.
a Your friend gave you 30% of the
winnings. How much did you
receive? ($225)
b If another friend was given 15% of
the winnings, how much was that
share? ($112.50)
c What percentage did the winner
keep? (55%)
d How much money did the winner
keep? ($412.50)
e Is there a way to check that your
calculation is correct?
(Check:
$225 + $112.50 + $412.50 = $750.
This is the correct total that was won
Calculations are correct.)
Calculate the new amount if 300 was
increased by 18%.
Ella decides that she wants to improve by at least 5% every day.
h What is 5% of 54 500? 2725
i What highest score would Ella need to get on Thursday in order to improve by 5%?
➜
2C explore card 1
2C explore card 2
Of and As what’s the diff?
Focus: To consider the difference between
calculating the ‘percentage of ’ as compared
to calculating ‘as a percentage’
Resources: calculator
Students consider a variety of questions
exploring the number of 14-year-old
school students who have a bedroom to
themselves or share with a sibling. Students
differentiate between the questions asking
to write ‘as a percentage’ and those that
require finding ‘a percentage of’.
…overmatter
69
2 Percentages, ratios and rates
2D Financial calculations
discover
2D Financial calculations
resources
Deep Learning Kit
•
discover
2D discover card 1
›
example 2
Financial calculations involving
percentages
Additional skill practice
Rachel makes earrings as a hobby and decides that she will sell them on eBay.
Before she can start, she needs to make some financial decisions.
A music system is sold for $1200 when it originally cost $1500.
b Calculate the profit or loss.
a State if a profit or loss has been made.
c Find the percentage profit or loss.
A standard pair of earrings costs $8.00 to make. Rachel obviously wants to make
money so she needs to decide how much she is going to sell them for. She decides
to add on 75% of the original cost to make the selling price.
Assess
2D eTutor
2D Guided example
2D Test yourself
2 Add your answer from question 1 to $8.00 to find the selling price of a standard pair of earrings. $14.00
Adding an amount onto an original price is called a mark-up. We would say that Rachel has added a 75% mark-up
to her earrings.
➜ write
a The selling price is less than the wholesale price, so it is a loss.
b Find the difference between the two prices.
a A loss has been made.
b loss = $1500 − $1200
= $300
300
c percentage loss = _____
× 100%
1500
c 1 Express the loss amount as a percentage of the original amount.
= __15 × 100%
= 20%
The system sold for a 20% loss.
3 A customer buys 10 pairs of Rachel’s earrings. How much money will this sale make for Rachel in total? $140.00
4 How much will it cost Rachel to make 10 pairs of earrings? $80.00
discover task
5 Does Rachel make a profit, and if so, how much? Use your answers from questions 3 and 4 to help you. profit of $60.00
Rachel decides to offer a percentage discount to people who buy five or more pairs of earrings. She decides to offer
a 20% discount off the selling price.
7 What will each pair of earrings be worth after this discount? $11.20
8 How much will the person who bought 10 pairs of earrings now pay? $112.00
9 What profit will Rachel make from this transaction now? $32.00
now try these
a 30% of $1200 $360
e 8% of $200 $16
e1
10
80
– 18 = 12.5%
d 15% of $10 500 $1575
h 19% of $775 $147.25
g 63% of $999 $629.37
2 Calculate each of these.
d a 20% mark-up on $99 $118.80
loss
g a 47% mark-up on $645 $948.15 h a 38% discount on $750 $465
percentage loss = original cost × 100%
17 Can you use the process in questions 10 and 11 to write a formula to calculate percentage profit or loss?
c 40% of $250 $100
profit
percentage profit = original cost × 100%,
16 Find the percentage loss Rachel made on this batch of earrings.
b 25% of $5000 $1250
f 21% of $2745 $576.45
a a 30%
mark-up on $1200 $1560 b a 25% discount on $449 $336.75 c a 40% discount on $2250 $1350
å
14 Did Rachel make a profit or a loss on these earrings? Explain how you know. loss; selling price is less than original price
15 Find how much Rachel lost in this transaction. $10
e2
e a 15% discount on $959 $815.15 f a 75% mark-up on $149 $260.75
i a 54% discount on $1200 $648
3 For each of these situations:
i state if a profit or loss has been made
58
ii calculate the profit or loss
example 1
➜ think
➜ write
Calculate a 25%
mark-up on a
$350 camera.
1 Find 25% of $350.
25% of $350 = 25% × 350
25
350
× ___
= ___
100
1
350
1 ___
_
= ×
iii find the percentage profit or loss correct to two decimal places.
4
2 Add this to the original price to find the
new marked-up price.
1
= $87.50
new price = $350 + $87.50
= $437.50
a Scarves bought at $5.98 each were sold for $13.50 each. i profit
b A car purchased for $28 899 was sold for $17 580. i loss
ii $7.52
ii $11 319
iii 39.17%
c Fabric bought at $5.99 per metre (wholesale) was sold for $12.99 per metre. i profit
d A toy bought at $8.75 was sold for $2.60. i loss
ii $6.15
profit or loss × 100%
being: original
cost
•
now try these
•
•
added to the students’ glossary.
Demonstrate 2D eTutor, or direct
students to do this independently.
Direct students to the examples.
Example 1 shows how to calculate
the dollar value of a mark-up and the
new price of a camera. Example 2
demonstrates how to consider if a profit
or loss has been made and shows how to
calculate the percentage loss.
➜ Focus: To consolidate
understanding of financial calculations
involving percentages
•
Q1 reviews calculations which require
students to calculate percentages of
dollar values. Students may need to
be reminded that they need to include
a unit with their answer. They may
also need to be reminded to use the
•
•
pfunction on their calculator to set
the number of decimal places to two;
which is appropriate when working
with money.
In Q2 students calculate the mark-up on
dollar values. Remind students that they
not only need to calculate the dollar
value of the mark-up, they will need to
add this value to the original price to
find the new price.
In Q3 students determine whether
a profit or loss has been made in
a number of scenarios. Once the
difference has been identified as a profit
ii $7
iii 116.86%
•
POTENTIAL DIFFICULT Y
Students may need to be reminded to
calculate the new price when considering
mark-ups and discounts. They can often
calculate the dollar value of the markups or discount, and forget to add or
subtract as appropriate in reference to
the original price.
iii 70.29%
e A sandwich maker that was originally priced at $59 was sold for $45. i loss
Direct students to the key ideas. You
may like them to copy this summary.
59
iii 125.75%
ii $14
f A supermarket buys milk at $0.80 per litre and sells it for $2.24 per litre. i profit
− the method used to calculate the
percentage profit or loss, the formula
•
1 Calculate each of these.
12 How much would a customer pay for 10 of these discounted pairs of earrings? $70.00
13 How much money did it cost Rachel to make these earrings? $80.00
• The definitions for all new terms can be
After students complete the Discover
task, consolidate their understanding.
Ensure students understand:
− how to calculate the mark-up on an
item, and the new price of the item
− how to calculate the value of a
discount and the new price of
the item
− the meaning of the terms profit
and loss
original cost
= 25
Rachel discovers that some of the materials she used on a batch of earrings were faulty. Because of this, she offers
a single discount of 50% on this batch of earrings.
➜ Focus: To consolidate student
understanding of financial calculations
involving percentages
•
32
80
11 Convert this to a percentage. This is the percentage profit that Rachel made on this transaction. Is it more
or less than you expected? 40%; it is less than the 75% mark up
whole class
70
➜ The amount added to the wholesale (original) price is called a mark-up.
➜ The difference between the regular price and a lower price is called a discount.
➜ A profit occurs when the selling price is greater than the wholesale price.
➜ A loss occurs when the selling price is less than the wholesale price.
profit or loss
➜ The percentage profit or loss of a transaction can be calculated using the formula ___________ × 100%.
6 What is 20% of the selling price of $14.00? $2.80
10 Express this profit as a fraction of the cost of the earrings to Rachel.
•
key ideas
2D Financial calculations
➜ Focus: To consider percentages in
terms of financial calculations
• Students are guided to consider a
real-life scenario in which financial
decisions need to be made to ensure the
viability of a business.
• Students discover the terms mark-up,
profit, discount and loss.
• Students discover that a mark-up on
an item is calculated in the same way
as calculating a percentage increase; in
that the price of the item increases when
the mark-up is added to the original
amount. Students are then guided to
consider the profit made if the markedup items are sold.
• Students are guided to consider the
scenario in which a discount is given for
bulk purchases, and discover that the
calculation of a discount is similar to
the calculation of a percentage decrease;
in that the price of the discount is
subtracted from the original amount.
• Students consider the scenario in
which goods must be sold below the
manufacturing cost, resulting in a loss.
2 Write your final answer.
A profit occurs when the selling price is greater than the original price.
•
•
•
iii 23.73%
ii $1.44
iii 180%
or loss, students calculate both the
dollar value of the profit or loss and also
the percentage profit or loss.
For those students experiencing
difficulty with these questions,
intervention through small group
teaching of the concept may be
necessary.
Students may benefit from a whole class
activity where they form manufacturing
businesses in small groups. Suggestions
are a bakery for a cake stall, or
manufacturing simple bracelets. Allow
students to consider the actual cost of
Deep Learning Kit
2D discover card 1
Financial calculations involving percentages
Focus: To perform percentage calculations
involving mark-ups, discounts, profits
and losses
Resources: calculator
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
1 What is 75% of $8.00? $6.00
➜ think
manufacturing items for sale. They also
need to factor in the costs of packaging
and marketing.
Students can then calculate the markup required on each sale item, so that
their business makes a profit. They need
to ensure that the price of each item is
reasonable.
If time permits, once students have
prepared a business plan, there may
be a day on which the students could
organise a lunchtime sale, so students
from other year levels could purchase
items. Students may need to consider
discounts if their items don’t sell, or if
closing time is approaching.
By completing an activity such
as this, students develop a strong
understanding of the types of financial
calculations which have been considered
in this section. Any money raised could
be donated to the school council, or to a
charity of the class’s choosing.
For extra practice, direct students to
2D discover card 1.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
71
Students are taken through the steps of the
process in which calculations involving
‘percentages of amounts’ are applied to
real-life scenarios. Extra practice questions
similar to now try these Q1–3 are provided.
whole class
Some possible questions are suggested
below.
…overmatter
2 Percentages, ratios and rates
2D Financial calculations
explore
small group
resources
explore
1 a i $6.40
d i $21.70
2D explore card 1
›
The power of BIG!
Investigative task
c i $12.50
f i $87.40
Just how much GST?
Investigative task
Property 2: valued at $1 200 000, sold for $1 600 000. property 1 increased by 38.56%,
a Monopoly board game $64
b watch $54.95
c concert ticket $124.90
d CAS calculator $217
e facial $225
f iPad $874
a $3680 $257.60
c $490 $34.30
d $108 $7.56
e $7390 $517.30
ii an increase of 17%1.17 iii a 75% increase 1.75
iv a reduction of 32% 0.68
vii a reduction of 7% 0.93 viii a discount of 28%. 0.72
10 Switched On Electrics is having a ‘Scratch and Dent’ sale. Any damaged stock will be discounted by 35%.
A refrigerator has a scratch on its door and is discounted to $1820.
b Granny Smith apples $3.90 (per kilogram)
d broccoli $5.98 (per kilogram)
e carrots $2.99 (per kilogram)
f capsicum $4.50 (per kilogram)
i
i
i
i
i
i
$3.11
$1.37
$4..38
$2.09
$1.05
$1.58
ii
ii
ii
ii
ii
ii
$12.00
$5.25
$16.90
$8.05
$4.05
$6.10
b Fifteen per cent of the frames are damaged and so Daphne reduces the selling price of these to 50% of the
planned selling price. How many frames are damaged and what is the new selling price for these frames? 6 frames; $6.25
e Find the percentage profit she has made. 36.03%
f Compare the value obtained in part e with the percentage profit Daphne would have made if none of the
frames had been damaged. undamaged 47.06%; damaged 36.04%
When you increase price by 40%, you find 40% of original price and add it to original price. When you increase price by 30% then 10%,
you first find 30% of original price and add it to original price, then you find 10% of this second price and add this amount to the second
price to find final price. Increasing by 30% then 10% will always give a larger amount than just increasing by 40%.
students to experiment and explore
the differences.
Q3 explores the concept of commission
as a method of payment for work.
In Q4 students consider the impact of
an insect plague on the price of fresh
fruit and vegetables.
Q5 considers a real-life scenario
involving a store owner, who purchases
frames in bulk and sells them at a
profit. Students are guided to consider
the impact on the profit if some of the
frames are discounted due to damage.
•
•
•
•
Step 1: Add 18% to 100% (adding as it is an
increase) = 118%
Step 2: 118% of 300 = 1.18 × 300 = 354
e A dishwasher has a dent and is discounted to $700. This price is 70% of its original price.
c Does Daphne make a profit or loss on the damaged frames? Calculate this amount. loss of $2.25
d If all 40 frames are sold, calculate the total profit she has made. $122.50
•
i a 25% discount 0.75
v a mark-up of 62% 1.62 vi a 200% mark-up 3.0
d How much money is saved because of the scratch on the door? $980
The method you have used in parts b and c is called the unitary method. In the unitary method, we find 1%
(one unit) of the original amount and then multiply by 100 to obtain 100% (the original amount or 100 units).
c cherries $12.50 (per kilogram)
a What is the expected profit on each frame? $4
•
b State what you would multiply the original prices by to obtain the new prices if there was to be:
c Use your answer to part b to find 100% of the original price. $2800
a grapes $8.88 (per kilogram)
5 Daphne purchased 40 picture frames for her store at a cost of $8.50 each.
She plans to sell them for $12.50 each.
Where, previously, students calculated 18%
of 300 and then added this to the original
amount to find the new amount, in this
alternate method, the following steps are
taken.
(15%). This is 115%, which as a decimal is 1.15.
b This means that $1820 is 65% of the original price. What would 1% of the original price be? $28
4 a
b
c
d
e
f
Calculate the new amount if 300 is
increased by 18%.
a Explain why this works. A 15% increase is 15% on top of original amount. That is, whole amount (100%) plus the increased amount
g If Matthew wishes to earn a minimum of $1440, what must his sales total? $16 000
ii the new price (rounded to the nearest 5 cents).
•
30%: $489.30; 20% then 10%: $503.28; customer is correct
9 In the previous Explore section on pages 56–7, we found a short cut to expressing percentage increase and
decrease. For example, in order to find a 15% increase, rather than finding 15% of an amount and then adding
it to the original amount, we can instead multiply by 1.15.
a If it is discounted by 35%, what percentage of the original price remains? (Hint: subtract 35% from 100%.) 65%
i the amount each item has been increased by
60
(Multiplying by a percentage larger
than 100)
f If Matthew makes sales totalling $8500, how much will he be paid? $765
4 A plague of locusts reduces the supply of fresh fruit and vegetables, and a high
demand for this produce causes the prices to increase by 35%. Calculate:
the percentage profit or loss, if a stereo
with a wholesale value of $1049 is sold
for $899. (14.3% loss)
Percentage increase:
8 During a sale, Julian accidentally discounted a $699 food mixer by 20% instead of the
advertised 30%. Once he realised his mistake he took a further 10% off the discounted
price. The customer claimed that this would not be the same as reducing the original
price of the mixer by 30%. Who is correct? Provide calculations to support your answer.
In Q6 students explore percentage profit
made on properties being sold.
In Q7 students calculate the wholesale
price and then calculate percentage
profit.
In Q8 students explore the difference
in price when a one-step discount of
30% is compared to the case of a 20%
discount followed by a further 10%
being taken off the price.
In Q9 students consider a shorter
method of calculating percentage
increases and decreases, and apply this
i Find 1% of the original price. (Hint: divide $700 by 70.) $10
ii Find 100% of the original price. (Hint: multiply your answer to part i by 100.) $1000
iii What is the original price of the dishwasher? $1000
f
Dividing by 65 and multiplying by 100 is equivalent to dividing by 0.65.
2D Financial calculations
•
b $6925 $484.75
Matthew also works as a telemarketer and receives 9% commission on the sales he makes.
➜ Focus: To apply understanding
of financial calculations involving
percentages to real-life scenarios
•
ii 42.70%
3 Vesna has a part-time job as a telemarketer and is paid a percentage of the sales she makes. This type of
payment is called a commission. Calculate the amount (rounded to the nearest 5 cents) Vesna receives if she
earns a 7% commission on these sales.
explore questions
72
ii 43.14%
g Use your answers to explain why increasing a price by 30% then 10% is different from increasing it by 40%.
the new price on a book which was
$29.99 and is marked-up by 15%.
($34.49)
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
Q1 explores the concept of an example
relating to a goods and services tax
(GST).
In Q2 students calculate a mark-up of
30% and then an additional 10% of GST.
They will need to calculate the new
prices for items in two stages. They are
then directed to consider the difference
between the calculation of the new price
in two stages and the calculation of a
one-step increase of 40%. Encourage
i $25.75
e rice crackers 90 g originally $2.49 i $3.55 ii 42.57% f raisin toast 520 g originally $4.59 i $6.55
b the new price on a doona cover which
was $89.99 and is discounted by 30%.
($62.99)
•
b Calculate the percentage profit Tony made, by first writing the profit as a
fraction of the price he paid. 16.67%
ii Compare the original price with the final price of each item and calculate the actual percentage increase.
a toothpaste 160 g originally $3.88 i $5.55 ii 43.04% b tuna 95 g originally $1.24 i $1.75 ii 41.13%
Remind students of what was learnt in the
Discover section. They can calculate the
following:
d the percentage profit or loss, if a plasma
television with a wholesale value of $699
is sold for $749. (7.15% profit)
a If Tony sells a power drill for $420, what price did he pay for it? $360
i Calculate the final price of the following items once the mark-up and GST have been added. Where
appropriate, round to the nearest five cents.
Inflating rates
Investigative task
Explicitly demonstrate both types of
calculations for students, providing
scaffolding which can be followed if
required.
property 2 increased by 33.33%
11 Use the unitary method to find the total amount if:
a 20% of an amount is $160 $800
b 15% of an amount is $89.25 $595
61
c 39% of an amount is $6435 $16 500 d 6% of an amount is $121.50. $2025
12 During its stock-take sale a department store offers a 22 __12% discount on all items
in the store. If a pair of designer shoes was discounted by $282.60, what was the
original price? $1256
13 Consider the two statements: ‘All stock discounted by 45%’
and ‘Stock discounted by up to 45%’. Explain to a friend
whether these statements mean the same thing.
‘All stock discounted by 45%’ means all stock will receive a discount of 45%. ‘Stock
discounted by up to 45%’ means the discount can be any amount from 0% to 45%; for
example, 10%, 20%, 35%, 45%.
•
•
method to mark-ups and discounts. For
students experiencing difficulty with
this shortcut, strategies are discussed in
the small group section that follows.
Q10 involves exploration of the unitary
method of calculating the original
amount. In this method, 1% (or one
unit) of the original amount is found,
and then multiplied by 100, to fi nd the
original price.
Q11 requires students to use the unitary
method introduced in Q10, to calculate
the total amount when a percentage
•
•
(Multiplying by a percentage smaller
than 100)
Calculate the new amount if 650 is
decreased by 25%.
Where, previously, students calculated 25%
of 650 and then subtracted this from the
original amount to find the new amount, in
this alternate method, the following steps
are taken.
Step 1: Subtract 25% from 100% = 75%
reflect
What are the benefits of
using percentages when comparing profit
and loss?
Percentage decrease:
Step 2: 75% of 650 = 0.75 × 650 = 487.5
of the original amount is provided.
Some students may find this concept
challenging.
Q12 requires students to calculate the
original price of a product. They need to
extract the required information from
the worded question.
In Q13 students consider two
commonly used expressions which are
often used in advertising sales. Students
will need to carefully consider each
word in both phrases to recognise the
difference between them.
This strategy maintains the link between
adding for percentage increases (mark-ups)
and subtracting for percentage decreases
(discounts). If students are fi nding this
concept tricky, they should stick with the
usual method of calculation, rather than
using this shortcut, and should follow
the steps originally shown. As previously,
students can be encouraged to use their
calculators if required.
Deep Learning Kit
2D explore card 1
The power of BIG!
Focus: To investigate and compare prices,
discounts and profits of bulk orders
…overmatter
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
7 Tony makes a $60 profit on each power drill sold.
2 A local grocery store applies a 30% mark-up on all its goods. A GST of 10% is then added to this marked-up price.
whole class
c
6 Thomas and Ivo were asked to compare the increase in value of two properties and state which had made the
greatest percentage profit. Thomas said that the first property had increased the most but Ivo thought that the
second one had. Use your understanding of percentages to show who was correct.
Property 1: valued at $695 000, sold for $963 000.
c cauliflower (half) originally $1.99 i $2.85 ii 43.22% d oil 4 L originally $17.99
a
ii $137.40
ii $961.40
ii the price it will be sold for (round to the nearest five cents).
2D explore card 4
›
ii $60.45
ii $247.50
i the GST to be added
Taking a further percentage discount
Exploration
2D explore card 3
›
b i $5.50
e i $22.50
1 A goods and service tax (GST) is added to many items we
buy. This means 10% is added to the cost of most of the goods
bought and services provided. For each item, calculate:
2D explore card 2
›
ii $70.40
ii $238.70
➜
Deep Learning Kit
➜ Focus: To develop an
understanding of a shortcut which can
be used when calculating percentage
increases (or mark-ups) and percentage
decreases (discounts)
73
2 Percentages, ratios and rates
2E Understanding ratios
discover
•
resources
2E Understanding ratios
Deep Learning Kit
discover
2E discover card 1
›
example
Converting units
Intervention task
Ratios deal with the comparison of two or
more quantities of the same kind. What
can we compare in this photo of buttons?
2E discover card 2
›
Comparing two quantities
Intervention task
›
➜ think
b There are 3
purple buttons compared to
4
green buttons.
Professional Support Online
c There are
File 2.01: Conversion charts
7
jumbo buttons compared to
4
tiny buttons.
➜ A ratio is a comparison of two or more quantities of the same kind.
➜ Before writing a ratio, the numbers must be in the same unit of measurement.
➜ Ratios do not require units. They are written as whole numbers with no units shown.
➜ A ratio must be written in the order of the worded description given. For example, two parts red paint
In the photo, there are two orange buttons compared to three red buttons,
so the ratio can be written in shorthand as 2:3. The numbers in a ratio are
separated by a colon.
Assess
2E eTutor
2E Guided example
2E Test yourself
b 3:4
These ratios compare one part to another part (for example, blue to orange). Ratios can also be written to
represent part to whole (for example, blue to total number).
a number of orange buttons compared to the total number of buttons
b number of small buttons compared to the total number of buttons
•
•
•
74
whole class
➜ Focus: To consolidate student
understanding of ratios as a way of
comparing two or more quantities of the
same kind
2
2:16, 16
4
4:16, 16
c number of buttons with two holes compared to the total number of buttons 9:16,
i the number of shaded sections to the number of non-shaded sections as a ratio
ii the number of non-shaded sections to total number of sections as a fraction.
a
i 17:8
8
ii 25
9
16
b
c
i 7:11
ii 11
18
i 5:4
ii 49
6 Why might it be easiest to always write a part-to-part comparison as a ratio and a part-to-whole comparison
as a fraction? Denominator of fraction generally represents total number of parts; best to use for ‘part to whole’ comparisons.
Let’s see what other important information we can find about ratios.
7 Cordial is made up by mixing one part of cordial to five parts of water. Write this information as a ratio. 1:5
8 What would a cordial mixture of 2:4 represent? 2 parts cordial to 4 parts water
2 In question 1, why is part i expressed as a ratio and part ii as a fraction?
9 What would a cordial mixture of 4:2 represent? 4 parts cordial to 2 parts water
Part-to-part comparisons are best expressed as ratios and part-to-whole comparisons are best expressed as fractions.
3 Write each of these comparisons as a ratio in the given order.
10 If you were to drink the cordial mixture in question 7, how would it compare, in colour and taste, to one with
a ratio of 4:2? lighter in colour, weaker flavour
62
11 Explain why the order of each part listed in the ratio is important. Can you think of a different example that
requires the ratio to be correctly followed? Explain. Ratio describes order in the quantities compared. For example, when comparing
amount of white paint to blue paint, ratio of 1:2 will produce different paint colour from 2:1.
12 Greg makes a cordial mix using 400 mL of cordial and 2 L of water. He writes this as a ratio of 400:2. Explain
what a ratio of 400:2 means and why this does not represent the cordial mixture that Greg has made.
400:2 means 400 mL to 2 mL or 400 L to 2 L. Units in a ratio must be same before they can be compared.
13 Convert 2 L into millilitres and hence write the correct ratio for Greg’s cordial mixture.
2 L = 2000 mL so ratio is 400:2000
14 Why is it important that comparisons be in the same units for ratios to be written?
so quantities can be compared correctly
15 Discuss with a classmate three important things you learned about ratios.
•
1 For each of these, write:
e
a 20 seconds to 33 seconds 20:33
b 95 kg to 57 kg 95:57
c 113 m to 167 m 113:167
d 256 mL to 175 mL 256:175
e $499 to $575 499:575
f 23 days to 2 days 23:2
g 4 months to 13 months 4:13
h 19 cans to 21 cans 19:21
i 42 km to 117 km 42:117
2E Understanding ratios
➜ Focus: To build on student
knowledge of ratios and to discover how
to write ratios
now try these
Four button sizes:
jumbo, medium,
small and tiny.
5 Write these comparisons as both a ratio and a fraction.
discover task
Students are guided to complete
sentences in which the words ‘compared
to’ are used instead of the ratio symbol.
Students are then introduced to
the ratio symbol and rewrite their
comparisons from the first part of the
task in mathematical notation.
Students discover that the ratios
they have written to this point have
compared part to part, and that ratios
can also be written to compare part
to whole. Students consider that
fractions are another way of writing a
comparison of part to whole.
Students are guided to discover that the
units used when comparing amounts
using ratios must be the same. When
they are not, one of the units may
need to be converted. Students may
refer to File 2.01: Conversion charts
if experiencing difficulty with their
conversions.
➜ Ratios do not contain fractions or decimals. They only contain whole numbers.
c 7:4
4 How do we usually write a comparison of a part out of a whole? as a fraction
•
to three parts white paint is written as 2:3.
3 Rewrite the comparisons from questions 1 and 2 as ratios using numbers and a colon.
a 3:1
13:180
key ideas
2 Write four more comparisons using the photo provided.
Reproducible master
13 seconds to 3 minutes
13 seconds to 180 seconds
task, consolidate their understanding.
Ensure students understand:
− that a ratio is a way to compare two
or more quantities
− the quantities being compared must
be of the same kind
− when writing a ratio which involves
units, the units must be in the same
unit of measure
− a ratio must match the order given
in the worded description
•
•
•
− ratios only contain whole numbers,
they do not include decimals or
fractions
If students are experiencing difficulty
with these questions, or if they require
more support, refer to 2E discover card
1 or 2E discover card 2.
Demonstrate 2E eTutor, or direct
students to do this independently.
Direct students to the example. It shows
how to write a ratio comparing seconds
to minutes. The working demonstrates
the conversion of minutes to seconds so
•
•
63
4 Write each of these comparisons as a ratio in the given order.
a 12 days to 3 weeks 12:21
b 2 hours to 17 min 120:17
c 27 cents to $5 27:500
POTENTIAL DIFFICULT Y
d 4000 mL to 3 L 4:3
e 1.7 km to 793 m 1700:793
f 7 months to 3 years 7:36
g 509 g to 1.3 kg 509:1300
h 472 kg to 2.371 tonnes 472:2371
i 61 mm to 7.9 cm 61:79
Students must read each question
carefully and make sure that the order
of their ratio matches the comparison
written in words. Reassure students that,
in some instances, the larger number
may appear first, in others the smaller
number may appear first. The important
thing is that the numbers in the ratio
must correspond with the order in
the question.
j
38 min to 2 hours 11 min 38:131 k $4.20 to 97 cents 420:97
l
0.6 L to 53 mL 600:53
Some possible answers are: order, units the same before comparing, no units written in ratio, whole
numbers only, part-to-whole comparison better expressed as a fraction.
• After students complete the Discover
•
•
that the units of measurement are the
same.
Direct students to the key ideas. You
may like them to copy this summary.
•
now try these
➜ Focus: To use ratios as a method
of comparing two quantities
•
Q1 involves students examining
diagrams; and writing ratios and
fractions to compare different aspects
of the diagrams. For example, the
number of shaded parts to the number
•
•
of non-shaded parts; and part to whole
relationships.
In Q2 students explain the difference
between ratios and fractions, in that the
ratios are used to compare part to part
and the fractions are used to compare
part to whole.
In Q3 students write ratios to compare
measurements in which the units are
the same.
Q4 requires students to convert one of
the units of measuement to match the
other, before expressing the comparison
as a ratio.
Deep Learning Kit
2E discover card 1
Converting units
Focus: To explore some different units for
various types of measurement and to convert
between the units
…overmatter
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
a There are 3
red buttons compared to
1
yellow button.
Writing ratios
Additional skill practice
➜ write
1 Ratios must have quantities in the same unit of measurement.
Converting the larger unit to the smaller unit usually avoids dealing
with decimals and fractions. Convert minutes into seconds.
2 Write the ratio in the order of the worded description given.
Do not include the units.
1 Copy and complete the following sentences
about the buttons.
2E discover card 3
•
Write the comparison 13 seconds to 3 minutes as a ratio.
For those students experiencing
difficulty with these questions,
intervention through small group
teaching of the concept may be
necessary.
Some students may need to use a
concrete model in the early stages of
the topic, so that they can develop an
understanding of the concept. For
example, give the students a range of
coloured counters, or buttons which
match those in the Discover task, and
ask specific questions requiring ratios as
answers, such as:
– How many blue counters are there
compared to green counters?
– How many yellow counters are there
compared to green counters?
– How would you compare red
counters to blue counters?
You may like to discuss the difference
between comparing part to part and
part to whole, and the concept that part
to whole comparisons should be written
as a fraction.
This strategy can be extended to ask
the students if they can use ratios
to compare counters and blocks,
reinforcing that the objects have to
be similar.
For extra practice, direct students to
2E discover card 3.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
75
2 Percentages, ratios and rates
2E Understanding ratios
explore
good example of part to part comparisons.
Refer them to the types of comparisons
made in the Discover task, when they
compared coloured buttons and buttons of
different size.
resources
Deep Learning Kit
explore
8 The produce manager finds that in a box of 50 nectarines, 18% have blemishes
es
on their skin. Write the:
2E explore card 1
Aspects of framing
Investigative task
1 Write these comparisons as ratios or fractions as appropriate.
a number of blue push pins compared to number of yellow push pins
ns 11:9
2E explore card 2
b number of white push pins compared to number of green push pins
ns 2:7
›
c number of yellow push pins compared to total number of push pins
ns
The ratios of bike design
Exploration
d number of red and green push pins compared to number of blue push pins 18:11
a Rafael won 11 tournaments in a year compared to Andy, who won
n 7.
7 11:7
b Fedora kicked 89 goals last season while Sophie kicked 101 goals. 89:101
c Mark trains five times a week while Christian trains three times a week. 5:3
d Two-stroke petrol fuel is made by mixing 25 parts petrol to 1 part oil. 25:1
e A pancake mixture requires two cups of flour to three cups of milk. 2:3
f Concrete is made by mixing one part cement, two parts sand and three parts screenings. 1:2:3
g A hair dye solution is made up in the ratio of three parts dye to seven parts water. 3:7
h Sarah works 8-hour shifts on the weekend and 3-hour shifts on a weekday. 8:3
3 For the winning Lotto numbers shown, write the following
comparisons as ratios.
a odd numbers to even numbers 3:5
b prime numbers to composite numbers 2:5
c numbers that are multiples of 3 to numbers that are
multiples of 2 1:5
d numbers less than 20 to numbers greater than 20 4:3
4 Give an example that could illustrate each of these ratios.
➜ Focus: To apply understanding of
ratios to application questions
•
76
•
•
•
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
Q1 involves identification of
comparisons which compare part to
part, and those which compare part
to whole. Students then need to write
comparisons in the appropriate format;
that is, part to part comparisons as a
ratio and part to whole comparisons as
a fraction.
Q2 requires that students write ratios.
They may need to be reminded that the
order of their ratio needs to match the
order in the written description.
In Q3 students compare odd, even,
prime and composite numbers, and also
explore multiples.
In Q4 students write written
descriptions to suit given ratios. As this
c 3:1
d 4:9
e 17:25
nine games she plays, she wins four and loses five.
6 Consider this group of animals.
b What is the ratio of mammals to birds? 5:1
b What needs to be done to one of the measurements before a comparison can be
b made?
d ?
convert it to same unit as other measurement
c Write the ratio of the Rialto Tower’s height to Taipei 101’s height to the nearest metre.
(Hint: 1 foot = 0.3048 m.) 1670 feet = (1670 × 0.3048) m = 509 m; 270:509
is open-ended there are many possible
answers.
In Q5 students are asked to explain a
win–loss ratio, a term commonly used
in sporting commentary.
In Q6 students explore a photograph
which includes a variety of animals.
Q7 involves the real-life context of
an iceberg and requires students to
calculate the percentage of the iceberg
visible above the water. Once the
students have the two percentages (the
visible part of the iceberg and the part
of the iceberg below the surface of the
•
Aspects of framing
c Sharon thinks the solution of bleach and water is quite strong. Suggest an alternative ratio which would
result in a weaker solution. Clearly explain your logic. More water will result in weaker solution so increase second value
Focus: To investigate how the aspect ratio
of pictures can be related to the size of
the frame
in ratio. Or less bleach will also result in weaker solution so decrease first value in ratio. Both options could also apply.
13 One hundred people attended an auction on the weekend. Seven people expressed an interest in bidding but
on the day only three actually bid.
a Write a fraction for the number of bidders to the total number of people.
3
100
b Write a fraction for the number of non-bidders to the total number of people.
97
100
c Write a ratio for the number of bidders to the number of non-bidders. 3:97
d Write a sentence explaining when a ratio can be expressed as a fraction.
when comparing part to whole
Don’t
communicate
15%
email/facebook/Skype: 120; text/call: 72; don’t communicate: 36; postcard/letter: 12
water), the comparison of the portion
of the iceberg above the water and the
portion of the iceberg below the water
can be written as a ratio.
Q8–10 require students to calculate
values which are then used in ratios.
Students may need to be reminded
that, when given one percentage, the
other complementary percentage can
be found by subtracting the given
percentage from 100. For example,
in Q8, 18% of nectarines have
blemishes, therefore there will be
100% − 18% = 82% of the nectarines
blemish free.
2E explore card 1
a Can the two quantities be expressed as a ratio as they are? No, units are not the same.
b If each capful measures 19 mL, what is the ratio of bleach to water? 57:2500
b Write the number of people communicating via
email/Facebook/Skype compared to those communicating
via postcards/letters as a ratio. 120:12
b Write as a ratio the percentage of the iceberg submerged
to the percentage visible. 87:13
Deep Learning Kit
12 Whiter Than White bleach recommends using three capfuls of bleach for each 2 __12 L of water.
a Determine the number of people in each category.
a What percentage of the iceberg is visible? 13%
•
a Explain why the ratio for the heights of Rialto Tower and Taipei 101
cannot be written as 270:1670. units are different
d What is the fraction of lizards to the total number
of animals? 19
e Dane said that the ratio of dogs to cats was 1:2.
Explain where Dane went wrong and what he should
have written. incorrect order; 2:1
When students are considering Q13, parts
a and b are part to whole comparisons and
need to be written as a fraction. But part c
is asking students to compare two groups
within the population, and this can be
written as a ratio.
11 Consider the height of Melbourne’s Rialto Tower (270 m) and Taiwan’s
Taipei 101 (1670 feet).
c What is the ratio of rabbits to fish? 1:2
7 Eighty-seven per cent of an iceberg is submerged in water.
•
a Determine the lengths of the other two fish. 81 cm, 72 cm
b Write the lengths of the fish in the order they were caught as a ratio. 60:81:72
14 Two hundred and forty travellers returning from holidays were asked
how they communicated whilst away. Their responses are shown on
the pie chart.
a What is the ratio of dogs to cats? 2:1
•
10 Pete caught three fish on the weekend. The first was 60 cm long, the second was
35% longer than the first and the third was __98 the length of the second fish.
Text/
Call
30%
Email
Facebook
Skype
50%
65
c Write the proportion of people in each category as a fraction in
simplest form.email/facebook/Skype: 12 ; text/call: 103 ; don’t communicate: 203 ; postcard/letter: 201
email/facebook/Skype: 0.5; text/call: 0.3; don’t communicate: 0.15; postcard/letter: 0.05
One possible answer is: half the people surveyed communicate via
email/facebook/Skype while away. Technological communication
makes up 80% of the communication.
•
•
Q11 and Q12 involve scenarios in which
one of the units of measurement needs
to be converted before the comparison
can be made as a ratio.
Q13 and Q14 require students to
recognise the difference between a
comparison of part to part and part
to whole. For students experiencing
difficulty distinguishing the difference
between these comparisons, strategies
are discussed in the small group section
that follows.
reflect What similarities exist
between ratios and fractions?
Students measure and determine the aspect
ratio of different pictures and determine
the most appropriate frame size. Students
then calculate the reduction or enlargement
percentage factor required to fit certain
pictures into the desired frame sizes.
2E explore card 2
The ratios of bike design
Focus: To determine the ratios of bike
designs and relate these ratios to the purpose
of the bike
d Write the proportion of people in each category as a decimal.
e Write a brief paragraph explaining what the
information suggests.
Resources: calculator
small group
➜ Focus: To develop an
understanding of the difference
between part to part and part to whole
comparisons
Students may be finding it difficult to
distinguish between part to part and part to
whole comparisons. It may be beneficial to
explain some of the key words that provide
an indication of the type of comparison.
When students are required to count
subsets within a larger group, this is a
Resources: ruler, protractor, calculator,
books or magazines with pictures of
different bikes (optional)
Students explore the ratios and angle
measurements in the designs of different
bikes to determine bike purpose and actual
lengths of bike parts.
whole class: reflect
Possible answer: Ratios and fractions are
both ways of comparing quantities.
Direct students to complete the appropriate
section of My learning in the obook.
Professional Support
•
b 2:7
28 11
20
44 18
1
13 38
5 Samantha’s win–loss ratio against Venus is 4:5.
What does this mean? For every four games she wins, she loses five; or out of every
64
9 The mark-up on a pair of jeans originally purchased for $80 was 315%.
What is the ratio of the original price to the selling price? 80:332
2E Understanding ratios
a 5:2
explore questions
9
50
Postcard/Letter
5%
Remind students of what was learnt in the
Discover section. Some prompts are:
• Write the following comparisons as a
ratio:
a 27 seconds compared to 45 seconds
(27:45)
b 11 grams compared to 27 grams
(11:27)
c 3 hours compared to 31 minutes
(180:31)
d 57 mm compared to 12 cm
(57:120)
• It may be appropriate to discuss as a
class why, in part d, the conversion was
not performed on the 57 mm. (This
would have resulted in a decimal and
there are no decimals in ratios.)
b fraction of blemished nectarines out of the total number of nectarines.
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
9
40
2 Write each of these as a ratio in the given order.
whole class
When students are asked to compare
an amount with the whole, the question
will include words such as total, whole,
all of, population and entire. When the
comparison includes these terms, it is an
indication that the comparison should be
represented as a fraction.
a ratio of unblemished nectarines to blemished nectarines 41:9
➜
›
77
2 Percentages, ratios and rates
2F Working with ratios
discover
resources
2F Working with ratios
Deep Learning Kit
•
discover
2F discover card 1
›
Equivalent ratios
Additional skill practice
We have seen that when we compare the ratio of a part to its total, the ratio can be expressed as a fraction.
Are there other similarities between ratios and fractions?
Assess
Damian and Georgia need to construct pool enclosures with the
following dimensions.
2F eTutor
2F Guided example
2F Test yourself
example 1
➜ think
➜ write
Write 20 seconds
to 2 minutes
15 seconds as a
ratio in simplest
form.
1 Write the quantities in the order given.
2 Convert the quantities to the same unit.
To keep as whole numbers, express both
values in the smaller unit (seconds).
Remember that 1 minute = 60 seconds.
3 Write the comparison as a ratio now
that the quantities are in the same unit.
4 To simplify, divide each number in the
ratio by the HCF which is 5. (20 ÷ 5 = 4
and 135 ÷ 5 = 27.)
20 seconds to 2 minutes 15 seconds
= 20 seconds to 135 seconds
6m
3m
discover task
4m
8m
Enclosure B
Enclosure A
➜ Focus: To discover that ratios can
be written as equivalent ratios, using
strategies similar to those used when
finding equivalent fractions
•
•
78
•
example 2
Pool enclosure
Length
Width
Perimeter
A
4m
3m
14 m
B
8m
6m
28 m
C
12 m
9m
42 m
D
16 m
12 m
56 m
Use your understanding of equivalent ratios to find the value of a if 3:5 = 21:a.
2 For each enclosure, write as a ratio the length of the rectangle compared to the perimeter of the rectangle.
➜ think
➜ write
1 Equivalent ratios are formed by multiplying or dividing each part of a ratio
by a whole number. Find the number that when multiplied to each part of
the ratio 3:5 gives 21:a. The number to multiply by is 7 as 3 × 7 = 21.
2 Identify the value for a.
3:5
= (3 × 7):(5 × 7)
= 21:35
a = 35
POTENTIAL DIFFICULT Y
A 4:14, B 8:28, C 12:42, D 16:56
3 Look for a pattern in your answers to question 2. Write the next three ratios you would expect. 20:70, 24:84, 28:98
4 Use your answers to question 2 to write each ratio as a fraction in simplest form. Each ratio simplifies to 27 .
key ideas
➜ Numbers in a ratio can be multiplied or divided by the same value to create an equivalent ratio.
➜ To express a ratio in simplest form, divide the values in the ratio by the highest common factor (HCF).
➜ Equivalent ratios can be used to find an unknown value.
5 Compare the fractions you have obtained. What is special about them? They are the same.
20 24 28
, , ;
70 84 98
6 Write the three ratios you have listed in question 3 as fractions. What is this type of fraction called? equivalent
fractions
7 The ratios listed in questions 2 and 3 are called equivalent ratios. Explain how you obtained the equivalent
ratios in question 3. First value in ratio increases by 4, second value increases by 14.
now try these
8 Consider another pool enclosure which measures
12 m by 8 m. Write the length compared to the
perimeter as a ratio. 12:40
66
1 Simplify each of these ratios.
a 7:84 1:12
g 38:34 19:17
in
9 a Write two equivalent ratios that contain
n
larger numbers than those obtained in
question 8. Some possible answers are: 24:80, 36:120.
b Explain how you created these
equivalent ratios. Multiplied both numbers in
e1
b 9:12 3:4
h 17:102 1:6
11 Write a sentence explaining how to createe an equivalent ratio.
Equivalent ratios can be created by multiplying or dividing numbers in a ratio by the same value.
e2
•
students to do this independently.
Direct students to the examples.
Example 1 shows how to write a ratio
in simplest form. In this example,
one of the values is converted to a
different unit, so that the units of
the values within the ratio match.
The components of the ratio are then
simplified, by dividing them by the
HCF. In example 2 students are shown
how to use equivalent ratios to find an
unknown value, which is represented by
a pronumeral.
•
Q1 involves the simplification of
ratios. Students may like to use their
calculator to find the HCF (greatest
common divisor) as discussed in the
Preview section for 2A Understanding
percentages. If students do not use the
f
a 5:9 = 25:a 45
b 15:50 = 3:b 10
c 9:2 = c :16 72
d 8:36 = d:9 2
f f :4 = 42:24 7
g 2:g = 22:66 6
h 20:h = 5:2 8
67
4 Fill in the gaps to complete the equivalent ratios.
a 2:5 =
:20 = 20:
2:5 = 8:20 = 20:50
d 225:
•
➜ Focus: To use understanding
of equivalent ratios to write ratios in
simplest form and to find unknown
values
•
e 3.78 tonnes to 158.5 kg 7560:317
2
__
hour to 0.8 hour 5:6
3
e e:55 = 19:11 95
Direct students to the key ideas. You
may like them to copy this summary.
now try these
c 92 cents to $17.28 23:432
3 Use your understanding of equivalent ratios to find the value of each letter.
= 9:12 =
225:300 = 9:12 = 3:4
• Demonstrate 2F eTutor, or direct
b 17 mm to 2.34 m 17:2340
d 3 hours 15 min to 55 min 39:11
ratio by same value.
c 106:36 53:18 d 22:55 2:5
e 108:72 3:2
f 45:95 9:19
i 16:24:42 8:12:21 j 75:50:125 3:2:5 k 36:60:84 3:5:7 l 18:42:48 3:7:8
2 Write each of these as a ratio in simplest form.
a 65 mL to 3.2 L 13:640
in
10 a Write two equivalent ratios that contain
smaller numbers than those obtained in
question 8. Some possible answers are: 6:20, 3:10.
b Explain how you created these
equivalent ratios. Divided both numbers in ratio by same value.
➜ Focus: To consolidate student
understanding of the properties of
equivalent ratios
After students complete the Discover
task, consolidate their understanding.
Ensure students understand that:
− the numbers within a ratio can be
multiplied or divided by the same
number to create an equivalent ratio
− when an operation is performed on
one value within a ratio it must also
be performed on the other value/s
in the ratio to create an equivalent
ratio
− a ratio can be simplified by dividing
all of the values within the ratio by
the HCF
− equivalent ratios can be used to
calculate unknown values.
•
1 Copy and complete this table for each pool enclosure.
whole class
•
= 4:27
2F Working with ratios
•
Students are guided to explore the
relationship between the length of a
pool enclosure and the perimeter of the
enclosure. Once students have identified
the pattern, they are guided to write the
next three ratios in the pattern.
Students discover that, when the ratios
within the pattern are written as
fractions, the fractions are equivalent.
It then follows that the ratios are also
equivalent.
Students are guided to discover that
equivalent ratios can be found using
the same strategies which were used
to find equivalent fractions. (See 1C
Understanding fractions.)
16 m
Enclosure D
12 m
Enclosure C
= 20:135
•
b 6:7 =
:4
e
:35 = 54:
6:7 = 30:35 = 54:63
:104 = 68:
c
= 17:13
136:104 = 68:52 = 17:13
HCF in the simplification, they may
need to complete the simplification
using a number of steps.
Q2 requires students to write ratios
which compare amounts. One of the
values in the ratio will need to be
converted into a different unit, so that
the units of the values within the ratio
match. The components of the ratio
can then be simplified, by dividing the
values by the HCF.
In Q3 students use the concept of
equivalent ratios to find the value of
an unknown.
f
:360 = 70:60 = 7:
420:360 = 70:60 = 7:6
:48 = 75:
= 15:4
180:48 = 75:20 = 15:4
•
•
•
In Q4 students complete equivalent
ratios by finding the values required to
fi ll the spaces provided.
For those students experiencing
difficulty with these questions,
intervention through small group
teaching of the concept may be
necessary.
Some students may experience difficulty
in finding the HCF. If students are
unable to find the HCF they can be
advised to simplify over a few steps,
dividing through by factors such as 2,
3 or 5. Students may also need to be
When looking at a series of equivalent
ratios it is important that students always
refer back to the first, original ratio and
seek a multiplication or division pattern.
Equivalent ratios can be created either by
multiplying all parts or dividing all parts
by the same factors.
Deep Learning Kit
2F discover card 1
Equivalent ratios
Focus: To review the concept of equivalent
ratios and to use equivalent ratios to find
missing values
After reviewing simplification of ratios,
students are guided to consider that the
units for all components of a ratio need to
be the same. When they are not the same, a
conversion needs to be completed. Students
are taken through the steps of a scaffolded
process in which they use equivalent ratios
to calculate a missing value. Extra practice
questions similar to now try these Q1–4
are provided.
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
12 m
9m
reminded that they can use trial and
error when finding the HCF.
If students are finding it difficult to find
the value of the unknown, they may
benefit from completion of a structured
sequence of working. For example:
2:25 = __:75
25 and 75 are a pair (they are both
second in their ratio)
Divide 75 by 25 to find the factor. (3)
Multiply 2 by 3 to find the missing
number. (6)
2:25 = 6:75
For extra practice, direct students to
2F discover card 1.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
79
whole class
Some possible questions are:
• Which operations can be used to create
equivalent ratios? (multiplication and
division)
• What do we divide by when simplifying
ratios? (the HCF)
• Find the HCF for the following ratios
and simplify.
…overmatter
2 Percentages, ratios and rates
2F Working with ratios
explore
resources
•
explore
10 The ratio of blue pens to black pens to red pens in a classroom is 7:4:1.
If there are 24 black pens in the classroom, how many blue pens and
red pens are there? 42 blue, 6 red
2F explore card 1
›
Percentage – a special ratio
Problem solving task
1 This box of chocolates contains a mix of milk chocolates,
white chocolates and dark chocolates (in cones).
Write each of these as a ratio in simplest form.
2F explore card 2
›
a white chocolates to square chocolates 5:2
Motorbike dilemmas
Problem solving task
8000mL
12 A two-stroke fuel mixture is made by mixing oil and petrol in the ratio
of 1:25.
d heart chocolates to round chocolates 4:13
2 Monday night fitness classes at a local community
centre are very popular, with 64 people taking a
Zumba class and 38 taking a spin cycle class.
What is the ratio, in simplest form, of Zumba to
spin cycle participants? 32:19
4 a Write two ratios that are equivalent to 12:48, using larger numbers.
•
80
•
•
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
Q1 involves writing ratios to compare
quantities of chocolates in a gift
box. Once written the ratios can be
simplified to their simplest form.
In Q2 students write a ratio to compare
the number of people taking Zumba
classes in comparison to spin cycle
classes. The ratio needs to be simplified,
by dividing both values by the HCF.
Q3 requires that the students read the
question carefully to see what is being
asked. They will need to calculate the
amount still owing for the holiday.
Once the students have the two values
required to write the ratio, they will
see that they include decimal values,
15 Imelda purchased a pair of shoes that had been discounted by 60%. The shoes were originally priced at $850.
Find the ratio, in simplest form, of:
c Repeat part b if 50 pancakes are required.
For 50 pancakes (10 × 5), multiply each number in ratio by 5 to
keep in proportion. 300:60 = 1500:300
6 Determine which of the following pairs of ratios are
equivalent. Explain your reasoning.
a the original price to the sale price 5:2
b the discount compared to the original price 3:5
c the discount compared to the sale price. 3:2
Pancakes
f 1:5 and 15:30
d 25:d = 10:14 35
e 48:32 = 18:e 12
f 33:44 = f :12 9
g 33:54 = g :72 44
h 15:20 = 36:h 48
a If the shortest side length is 17.4 cm, find the length of the longest side. 27.84 cm
500 mL full-cr
eam milk
300 g self-raisi
ng flour
2 eggs
60 g butter
2 tablespoon
s sugar
b b:52 = 20:13 80
c 13:c = 65:5 1
16 The side lengths of a scalene triangle are in the ratio 7:8:5.
(serves 10)
7 Find the value of the letter in these equivalent
ratio statements. (Hint: you may like to simplify
the complete ratio in the statement first.)
68
•
•
Students will need to multiply both
values by a suitable multiple of 10 to
make the values whole numbers, before
dividing through by the HCF.
Q4 requires students to find equivalent
ratios for given ratios.
Q5 explores ratios as part of a recipe.
Students consider which of the
quantities are measured in like units
and, therefore, which can be compared
using ratios. They then use their
knowledge of equivalent ratios to adapt
the recipe to make different quantities
of pancakes.
•
•
•
•
a Discuss in a small group whether Leo’s suggestion is valid.
69
b
f
no
no
c no
19 Write your own ratio problem and swap with a classmate.
Discuss any differences in your answers.
reflect
useful?
•
•
the missing values. For students
experiencing difficulty with this
concept, strategies are discussed in the
small group section that follows.
Q11–13 require students to write
equivalent ratio statements and then use
the information provided to calculate
the value of an unknown.
Q14–16 require students to perform
extra calculations to find values to be
used in the ratios. In Q14 students need
to calculate the profit made per share,
in Q15 students need to calculate the
•
4 and 24 are a pair (they are both second in
their ratio).
Divide 24 by 4 to find the factor. (6)
Multiply 7 by 6 to find the missing number.
(42)
Therefore there are 42 blue pens.
We then find another equivalent ratio pair
to calculate the number of red pens:
How are equivalent ratios
d no
Q6 involves checking pairs of ratios and
identifying those which are equivalent.
Q7 involves finding the value of
unknowns represented by pronumerals
in ratios.
Q8 and Q9 require students to write
equivalent ratio statements and then use
these statements to calculate the value
of an unknown.
Q10 involves a three part ratio.
The students are required to use
the information provided to write
equivalent ratio statements and find
To find the number of blue pens we use the
equivalent ratios:
7:4 = x:24.
7:4 = 42:24
52
50
18 Compare the ratios 4:5 and 10:13 and state which is larger. Explain why. 4:5 is larger; 65 is larger than 65
9 The ratio of rainy days to sunny days in a month was 2:3. If there were 18 sunny days, how many rainy days
were there? 12
yes, second ratio obtained by multiplying first ratio by 5
yes, second ratio obtained by dividing first ratio by 3
b Find the perimeter of the scalene triangle. 69.6 cm
17 Angela wondered whether ratios such as 3:5 and 10:14 could be compared. She also wanted to know whether
they could be ranked. Leo knew that each ratio could be written as a fraction then expressed as an equivalent
fraction and compared.
b See if your group can come up with an alternative method for comparing ratios. Share your ideas with
the class. Express each ratio as a fraction. Then compare each fraction by rewriting as equivalent fractions with the same denominator.
8 The ratio of girls to boys in a class is 5:4. If there are 15 girls in the class,
how many boys are there? 12
6 a
e
2F Working with ratios
•
b the original price to the profit made. 41:163
Flour to butter is 300:60 = 5:1. (Butter to flour is 60:300 = 1:5.)
a a:24 = 20:30 16
There are 24 black pens, and the students
must calculate the number of blue pens and
red pens. Black pens are represented by the
4 in the ratio.
a the original price to the selling price 41:204
Flour and butter; quantities are in the same units.
e 15:9 and 5:3
blue pens:black pens:red pens
14 Gordon bought shares in a Telco company for $8.20 and sold them for $40.80. Find the ratio, in simplest
form, of:
b Write a ratio for the ingredients identified in part a.
d 2:3 and 22:34
15, 10
d If he only used half a bag of cement, how many bags of sand and screenings are required? 1, 1.5
a Which ingredients can be expressed as a ratio?
Clearly explain your choices.
c 63:42 and 9:7
a If he purchases four bags of sand, how many bags of cement and screenings are required? 2, 6
c If he purchases five bags of cement, how many bags of screenings and sand are required?
Divide both numbers by a number greater than 1; for example, 25:14.
5 Jasmine is helping her mother make pancakes for herr
d
class end-of-term breakfast. The ingredients are listed
on the right and feed 10 people.
b 16:25 and 4:5
13 Deano has prepared concrete in the ratio 1:2:3, that is one part cement,
two parts sand and three parts screenings.
b If he purchases nine bags of screenings, how many bags of cement and sand are required? 3, 6
Multiply both numbers by a number greater than 1; for example, 24:96.
b Write two ratios that are equivalent to 50:28, using smaller numbers.
explore questions
➜ Focus: To apply understanding of
equivalent ratios to simplify ratios and
to also find unknown values
b How many millilitres of oil should be added to 15 L of petrol to make
the fuel mixture? 600 mL
3 Eric paid a deposit of $704.70 for his overseas trip, which cost a total of $4698. Write the ratio, in simplest
form, of the deposit paid to the amount owing. 3:17
a 2:9 and 10:45
Students may experience difficulty with
questions in which they are provided with
one value and a ratio, and they are required
to calculate the quantities represented by
the other components of the ratio. They
may need to be shown that they can write
equivalent ratio statements using part of the
original ratio. For example, in Q10, the ratio
is 7:4:1 and this represents:
a How many litres of petrol should be added to 850 mL of oil to make
the fuel mixture? 21.25 L
sale price before writing the ratio and
in Q16 students need to recognise they
are required to find two more lengths
before calculating the perimeter of the
triangle. Encourage students to read
the questions carefully and highlight
the information relevant to each
calculation.
In Q17 and 18 students are required to
compare ratios. Students may need to be
reminded that ratios can be written as
fractions. Once written in this format,
a common denominator can be found
4:1 = 24:x
4 and 24 are a pair (they are both first in
their ratio).
Divide 24 by 4 to find the factor. (6)
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
1.05 L
b How many millilitres of detergent are required if 28 L of water are used?
c milk chocolates to white chocolates 14:5
Remind students of what was learnt in the
Discover section. Some prompts are:
• Find the HCF for the following ratios
and simplify the ratios.
100:50:5 (HCF is 5, simplified ratio is
20:10:1)
49:14:56 (HCF is 7, simplified ratio is
7:2:8)
12:36:24 (HCF is 12, simplified ratio is
1:3:2)
• A student has partially simplified this
ratio. Can you finish it?
56:36 = 28:18 (Simplified ratio is 14:9)
• Is there another way that the ratio in the
point above could have been simplified?
(yes, by using the HCF of 4 rather than
starting with dividing by 2)
➜ Focus: To develop a strategy
that can be used to calculate unknown
quantities in a given ratio
a How many litres of water are required if 300 mL of detergent are used?
b chocolate cones to heart chocolates 3:4
whole class
small group
11 For optimal results, the instructions on a new cleaning product
recommend the ratio of detergent to water be 2:7.
➜
Deep Learning Kit
and the larger fraction, hence the larger
ratio can be identified.
Q19 is an open-ended task in which
students can write their own ratio
problem and swap with a classmate.
Multiply 1 by 6 to find the missing number.
(6)
4:1 = 24:6
81
Therefore there are 6 red pens.
Deep Learning Kit
2F explore card 1
Percentage – a special ratio
Focus: To use percentages as ratios to
determine the solution of various problems
Students solve simple everyday questions
involving percentages as a special case of
ratios. These questions do not require the
…overmatter
2 Percentages, ratios and rates
2G Dividing a quantity in a given ratio
discover
resources
2G Dividing a quantity in a given ratio
Deep Learning Kit
discover
2G discover card 1
›
example 2
Dividing a quantity in a given ratio
Additional skill practice
Laura, Sarah and Amelia decide to purchase a $36 Lotto
ticket for Saturday’s $30 million draw. They decidee to pay
equal shares for the ticket and share any winnings equally.
Assess
4 The number of parts in the simplified ratio that match Laura’s
contribution is 1. To write the number of parts for Laura’s contribution as a fraction of the total number
of parts, we can write __13.
•
82
whole class
➜ Focus: To consolidate student
understanding of dividing a quantity in
a given ratio
•
After students complete the Discover
task, consolidate their understanding.
Ensure students understand that:
b In a similar way, write a fraction for Amelia’s contribution.
5 The girls win a prize of $14 270.40. As they are sharing the winnings equally, how much will each
person receive? $4756.80
key ideas
6 Explain how the amount each girl contributed and won could have been calculated using ratios?
(Hint: use your answers to question 4.) 1 of $14 270.40 = $4756.80
➜ When dividing a quantity in a given ratio, follow these steps.
3
1 Find the total number of parts in the ratio.
2 Express each part of the ratio as a fraction of the total number of parts.
3 Multiply each fraction by the quantity and simplify.
➜ Remember to include the same units with your answer.
➜ One way to check that you have divided the quantity correctly is to add the individual amounts
and see that the result is the same as the original quantity.
Bruno, Xavier and Hector also decide to purchase a $36 Lotto ticket, and they contribute $18, $10 and $8,
respectively. They decide to share any winnings in the same ratio as their contribution.
7 Write Bruno, Xavier and Hector’s contribution for the ticket as a ratio in simplest form. 9:5:4
8 Add all the numbers (parts) in this simplified ratio. What is the total number of parts in the ratio? 18
9 The number of parts in the simplified ratio that match Bruno’s contribution is 9. To write the number
9
or __12.
of parts for Bruno’s contribution as a fraction of the total number of parts, we can write ___
18
a In a similar way, write a fraction for Xavier’s contribution.
5
18
4
18
and __14 of $4800
3 48001200
1 48001200
= ×
= ×
1
1
14
14
= 3 × 1200
= 1 × 1200
= 3600
= 1200
The ratio 12:4 divides $4800
into $3600 and $1200.
5 Answer the question and include the appropriate units.
Check: $3600 + $1200 = $4800.
1
3
now try these
e1
10 If the boys win $14 270.40, how much will each person receive? Bruno $7135.20, Xavier $3964, Hector $3171.20
1 Calculate the total number of parts for each ratio.
a 2:7 9
b 3:1 4
c 5:11 16
g 3:4 7
h 2:17 19
i 21:19 40
11 Discuss with a classmate how ratios can be used to calculate the boys’ winnings.
2 For each ratio in question 1, write each part of the ratio as a fraction of the total number of parts.
b In a similar way, write a fraction for Hector’s contribution.
2
or 9
d 14:9 23
j 3:2:9 14
e 13:15 28
k 6:12:11 29
2 a
3 Divide $300 into parts using the ratio 4:6.
d
a Find the total number of parts in the ratio. 10
g
b Express each part of the ratio as a fraction of the total number of parts.
4 6
i
:
c Multiply each fraction from part b by the quantity ($300).
10 10
$120; $180
k
Check that your answers add to $300.
example 1
70
Calculate the total number of parts for each ratio.
a 2:7
b 3:1:6
➜ think
➜ write
a Add together the numbers in the ratio to find the total
number of parts.
b Add together the numbers in the ratio to find the total
number of parts.
a 2 + 7 = 9 parts
The ratio 2:7 has 9 parts in total.
b 3 + 1 + 6 = 10 parts
The ratio 3:1:6 has 10 parts in total.
e2
4 Divide $9200 in each given ratio.
a 3:2
b 5:3
•
•
•
− answers can be checked by adding
the portions together and seeing if
the original quantity is reached.
Demonstrate 2G eTutor, or direct
students to do this independently.
Direct students to the examples.
Example 1 shows how to calculate
the total number of parts for ratios.
Example 2 demonstrates how to divide
a dollar amount into different portions
based upon the given ratio.
Direct students to the key ideas. You
may like them to copy this summary.
3
1
b 4 and 4
13
15
e 28 and 28
h
j
l
2
17
and 19
19
3 2
,
and
14 14
5 7
, and
21 21
c
f
h
e 21:25
f 2:8
d 4:5
e 12:3
f 1:1
6 Divide $240 in each given ratio.
a 1:2
b 9:1
d 10:6
e 3:1:4
f 7:2:1
$5750 and $3450
15 000 km and 21 000 km
16 000 km and 20 000 km
$80 and $160
$150 and $90
b
e
b
e
$6900 and $2300
c 2:10
$4600 and $4600
31 500 km and 4500 km
28 800 km and 7200 km
$216 and $24
$90, $30 and $120
c
f
c
f
now try these
•
Q1 involves calculating the number of
parts for given ratios. In Q2 students
represent each portion of the ratio as a
fraction comparing part to whole.
In Q3 students divide a quantity in a
given ratio. Students are guided through
the steps required with scaffolded parts
to this question.
11
and 16
3
and 11
and
19
14
71
$1840 and $7360
20 000 km and 16 000 km
18 000 km and 18 000 km
$40 and $200
$168, $48 and $24
•
➜ Focus: To use strategies developed
for using ratios to divide quantities into
given ratios
•
$4200 and $5000
5
16
8
11
21
40
9
14
9
21
d 1:1
$5520 and $3680
c 3:1
2
7
and 9
9
14
9
and 23
23
3
4
and 7
7
21
19
and 40
40
6 12
11
,
and
29 29
29
f 8:3 11
l 5:7:9 21
5 Divide 36 000 km in each given ratio.
a 5:7
b 7:1
c 10:8
5 a
d
6 a
d
− ratios should be written in simplest
form
− as a first step, the total number of
parts represented by the ratio must
be calculated
− a fraction is written to compare each
part to the whole
− a fraction is used to calculate
the portion of the total quantity
allocated to each share represented
by the ratio
− units should be included with
answers where appropriate
•
3
__
of $4800
4
•
•
In Q4–6 students complete a number
of calculations in which different
quantities are divided into given ratios.
In Q4 and Q5 all of the ratios specified
are composed of two values. In Q6
students complete the calculations using
ratios composed of three values.
For those students experiencing
difficulty with these questions,
intervention through small group
teaching of the concept may be
necessary.
Some students may experience difficulty
in visualising the number of parts in
•
•
POTENTIAL DIFFICULT Y
Students should be strongly encouraged
to check their answers, as shown above,
so that they can identify their calculations
are correct. It is important that students
take care when counting the number
of parts represented in a ratio. If they
are unsure, encourage them to use the
model described above to ensure that
they have the correct number to use in
the denominator of their fraction.
…overmatter
Professional Support
•
a In a similar way, write a fraction for Sarah’s contribution.
1
3
12:4
= 3:1
3 + 1 = 4 parts
3
__
and __41
4
2G Dividing a quantity in a given ratio
•
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
3 Add all the numbers (parts) in this simplified ratio.
What is the total number of parts in the ratio? 3
➜ Focus: To discover a strategy
which applies the use of ratios to
questions in which a quantity needs to
be divided in a specific way
➜ write
1 Express the ratio in simplest form. Divide each number in the
ratio by the HCF which is 4. (12 ÷ 4 = 3 and 4 ÷ 4 = 1.)
2 Add the number of parts in the simplified ratio.
3 Express each part of the ratio as a fraction of the total number
of parts.
4 Multiply each fraction by the quantity to be divided ($4800)
and simplify.
2 Write each person’s contribution for the ticket as a ratio
in simplest form. 1:1:1
discover task
Students are guided to consider
the scenario in which three people
purchased equal shares in a lottery
ticket, this being expressed by the ratio
1:1:1. Students consider the outcome
if the ticket was a winning ticket and
the winnings would be shared as equal
thirds; that is, each person would
receive 13 of the prize money.
Students consider the scenario in which
the people purchasing the ticket did not
each pay an equal share. The amount
they paid was in the ratio 18:10:8. In this
scenario people would receive different
quantities of the prize money won.
Students write the ratio and simplify
by dividing all values by the HCF.
18:10:8 = 9:5:4
Students are guided to count the total
number of parts represented by the
ratio (there are 18). They are guided
to represent each person’s share as a
fraction because each person’s share
is a comparison of part to whole, for
9.
example: 18
Students use these fractions to calculate
each person’s share of the prize money.
➜ think
1 How much does each person contribute? $12
2G eTutor
2G Guided example
2G Test yourself
•
Divide $4800 in the ratio 12:4.
a given ratio. This can be modelled
using counters. For example, for Q1a,
represent the ratio 2:7 using counters.
Ask the students to count the total
number of counters. (9)
Explain that the first value represents
2 out of a total of 9, and explain how
this information can be used to form
a fraction ( 29 ). Demonstrate that the
second value in the ratio represents
7 out of a total of 9, again explaining
how this can be written as a fraction ( 79 ).
It may be beneficial to refer students
to 1D Operations with fractions.
Alternatively, where students are
experiencing difficulty with the
arithmetic of multiplying a whole
number by a fraction, they can complete
the multiplication on their scientific
calculator. For example, for Q4a:
Divide $9200 in the ratio 3:2
There are 3 + 2 = 5 parts
Type in 35 , using the fraction key: q.
Then key in V9200.
Press < to see the portion
represented by the first value in this
ratio. Remember to include a unit.
($5520)
Type in 25 , using the fraction key: q.
Then key in V9200.
Press < to see the portion
represented by the second value in
this ratio, remember to include a unit.
($3680)
Check: Does $5520 + $3680 = $9200?
(yes)
For extra practice, direct students to
2G discover card 1.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
83
2 Percentages, ratios and rates
2G Dividing a quantity in a given ratio
explore
Students may be asked to apply their
understanding and effectively work
backwards to find the ratio for a given
scenario. For example, in Q7, the prize
winnings are in the ratio of:
resources
explore
10 The fat, protein and carbohydrate content in one of Al’s specialty
hamburgers is in the ratio 8:6:7. List the amount of fat, protein
and carbohydrate in the 420-g specialty hamburger.
2G explore card 1
›
1 Morning Sunshine fruit juice drink contains fruit juice and water
in the ratio 1:3.
A fair distribution
Exploration
The farmer’s dilemma
Investigative task
›
3 The angles in a triangle are in the ratio 3:4:2. What is the size of each angle? (Hint: the sum of angles in any
triangle is 180°.) 60°, 80°, 40°
4 Nadia plans to allocate her pocket money for April in the ratio 3:2:1 to cover a gift purchase, entertainment
expenses and savings. Calculate the amount she has allocated for entertainment
expenses if her total pocket money for April is $84. $28
whole class
5 Shane has set himself a goal of completing a 210 km bike ride in three days. Over the
three days, the number of kilometres he plans to cycle each day is in the ratio 3:5:2.
b What is the difference between the longest and shortest distance? 63 km
b Miriam’s history and science test results are in the ratio 10:11. If her combined
score is 168, did she perform better or worse than Stanley in science? history 80, science 88; better
7 Katie, Mike and Sam contributed different amounts to a $5 raffle ticket. When they won, they distributed the
winnings in the same ratio as their contribution. If Katie won $450, Mike won $200 and Sam won $350, find:
8 Sarah, Cameron and Hunter bought a $5 raffle ticket. If Sarah contributed $2.50, Cameron contributed $1.50
and Hunter contributed $1.00, find:
a the ratio of their contribution 5:3:2
b the individual winnings if the ticket won:
$25:$15:$10
•
•
iii $700
$350:$210:$140
iv $2000
$1000:$600:$400
v $10 000.
$5000:$3000:$2000
a If she wants 25 mL of paint, how much yellow and blue paint should she use? yellow: 10 mL; blue: 15 mL
b What is the total amount of paint she has now? 28 mL
c What is the ratio of yellow to blue paint in this mixture? 10:18 = 5.9
d How much yellow paint should Gloria add in order to get her favourite shade of green? (Hint: 18 mL
of blue paint represents 3 parts of the ratio.) 2 mL
e How much paint in total does Gloria have now? 30 mL
f How much excess paint did Gloria make? 5 mL
catching up with friends: 40%; outdoor activities: 25%; doing homework/chores: 10%
•
•
450:200:350 HCF = 50
9:4:7
(total parts = 20)
b Hence find the percentage of time she will be relaxing at home. 25%
c Write these percentages as a ratio in its simplest form. 8:5:2:5
•
•
•
•
The amount each person contributed to
the purchase of the ticket can then be
calculated.
a Calculate the width and height of the screen. width 72 cm, height 42 cm
b Calculate the area of the screen. 3024 cm2
14 Knowledge of ratios can be used to divide a line into various lengths. Draw a line segment of length 8 cm
in your workbook and label it as shown.
A
B
C
D
E
F
G
H
I
Copy and complete these sentences. Part a is completed for you.
a Point D divides the line segment AI in the ratio 3:5. The length of AD is __38 of the length of AI.
7
of the length of AI.
b Point H divides the line segment AI in the ratio 7:1 . The length of AH is 8
1
of the length of AI.
c Point E divides the line segment AI in the ratio 1:1 . The length of AE is
2
3
of the length of AI.
d Point G divides the line segment AI in the ratio 3:1 . The length of AG is
4
15 Using question 14 as a guide, draw a line segment of length 15 cm in your workbook. Mark dots at 1-cm
intervals and label as A to P. Place a point on the line segment to divide it in each of the given ratios.
a 7:8 H
b 11:4 L
c 4:1 M
d 2:3 G
e 2:1 K
f 1:2 F
16 Which point divides the line segment AJ in the given ratios below?
A
a 5:4 F
e 2:7 C
B
C
b 1:2 D
f 1:8 B
D
E
F
G
H
I
c 8:1 I
g 2:1 G
17 How do the questions in this topic highlight the difference
between a ratio that shows the relationship of part to part,
and a fraction that shows the relationship of part to whole?
Why is it important?
of 10 to remove the decimals from
the ratio. For students experiencing
difficulty with this concept, strategies
are discussed in the small group section
that follows.
Q9 explores the real-life scenario of
mixing paint to make different colours.
Q10 involves the calculation of the
quantity of different components found
in a hamburger.
In Q11 and Q12 students are given the
fractions used to divide the quantity
and need to write the ratio.
In Q13 students are given the perimeter
of a computer screen and the ratio
•
comparing length and width. They
are asked to calculate the length and
the width of the screen and then are
required to calculate the area of the
screen. They will need to recall the
formula to calculate the area of a
rectangle:
A=l×w
Q14 involves students using a line
diagram, and the representation of
different letters on the line, to complete
sentences which describe the length of
the different line segments. An example
is completed for students to follow. Q15
and Q 16 are extra questions in which
73
d 4:5 E
h 7:2 H
•
The raffle ticket cost $5.00, so check that the
different contributions add to cover the cost
of the ticket:
$2.25 + $1.00 + $1.75 = $5.00
This indicates that the calculations are
correct.
In Q8, the contributions to the purchase of
the ticket are in the ratio of:
The dollar values need to be represented in
the same order:
$2.50:$1.50:$1.00
reflect
What other practical
applications of ratios are there?
Katie: × $5.00 = $2.25
Mike: × $5.00 = $1.00
Sam: × $5.00 = $1.75
Sarah:Cameron:Hunter.
J
The ratio that shows relationship of part to part compares one part to another. For example, in 2:3 there is a
2
3
total of 5 parts. First number represents 5 of total and second number represent
ntts 5 of total. The ratio that shows
relationship of a part to a whole is better expressed as a fraction as it compares o
one number to the total.
to calculate portions of a given amount.
Students may need to be reminded to
include units in their answer.
In Q6 students use a given ratio to
calculate results for two students in two
different subjects. They then compare
the results of these students.
Q7 and Q8 require students to work
backwards and find the ratio, given
the winnings for each person. Q7
works with whole dollar values and
Q8 incorporates the use of decimals.
Students may need to be reminded to
multiply by an appropriate multiple
The ratio is found by simplifying the ratio
comparing the dollar amounts:
a Find the percentage of time that Bella will catch up with friends, do outdoor activities and homework
or chores. (Hint: for each fraction find an equivalent fraction with a denominator of 100.)
She mixes in 10 mL of yellow paint and then accidentally puts 18 mL of blue paint in.
• In Q3–5 students use three part ratios
$450 + $200 + $350 = $1000
students can use a line diagram to write
ratios.
In Q17 students consider the difference
between a ratio describing the
relationship between part and part, and
a fraction comparing part to whole.
Students consider the importance of
this difference.
small group
➜ Focus: To develop a strategy
which can be used to work backwards,
given the portions, to find the ratio
The total amount contributed is $5.00. The
ratio comparing contributions is found by
simplifying the ratio comparing the dollar
amounts:
2.50:1.50:1.00
Multiply by 10 to remove the decimal
values:
25:15:10 HCF = 5
5:3:2
(total parts = 10)
Professional Support
84
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
Q1 explores the ratio representing the
amount of fruit juice compared to the
amount of water used to create a fruit
drink. Students calculate the quantity
of fruit juice and the quantity of water
in 2 L of juice. This question could be
extended into an exploration about the
quantity of fruit juice and the quantity
of water in the juices that are popular
with the members of the class.
In Q2 students consider a two part ratio
and use this to calculate the number of
males at a football training session.
$75:$45:$30
9 Gloria is painting and prepares to mix up her favourite shade of green, using yellow and blue paint
in a ratio of 2:3.
➜ Focus: To apply understanding of
ratios and understanding of dividing a
quantity into ratios expressed in their
simplest form
•
ii $150
12 Bella finds it difficult to organise her time effectively during the holidays. She decides that she will spend __25 of
1
of her time doing her homework
her time catching up with friends, __14 of her time doing outdoor activities, ___
10
and chores, and the remainder of her time relaxing at home.
b the ratio of their contribution 9:4:7
c how much they each contributed to the ticket. Katie: $2.25; Mike: $1.00; Sam: $1.75
i $50
The total amount won is calculated by
adding the values:
c Find how much Madeleine spent on each category. $5:$15:$10:$20
13 The perimeter of a rectangular computer screen is 228 cm with a width to height ratio of 12:7.
6 a Stanley’s history and science test results are in the ratio 22:19. If his combined
score is 164, what result did he obtain for each subject? history 88, science 76
a the total amount won $1000
$450:$200:$350
b Write a ratio representing the money Madeleine spent on lollies, movie ticket, necklace and iTunes
credit respectively. 1:3:2:4
d Assuming that for the two weeks she is on holiday she has 10 ‘usable’ hours a day, find the number of hours
that Bella spends on each activity. 56 hours:35 hours:14 hours:35 hours
a What distance does Shane plan to cycle each day? 63 km, 105 km, 42 km
72
2
=5
2G Dividing a quantity in a given ratio
Remind students of what was learnt in the
Discover section. Some prompts are:
• Simplify each of the following ratios and
identify how many parts are represented
in each ratio.
a 4:20 (simplified ratio is 1:5, 6 parts)
b 77:11 (simplified ratio is 7:1, 8 parts)
c 900:100 (simplified ratio is 9:1,
10 parts)
d 50:25:100 (simplified ratio is 2:1:4,
7 parts)
• Simplify these ratios and then divide
$42 000 by each ratio.
a 2:8 (simplified ratio is 1:4,
$8400 and $33 600)
b 10:25 (simplified ratio is 2:5, $12 000
and $30 000)
4
10
Dollar values need to be represented in the
same order:
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
2 The ratio of female to male fans at a particular football training session is 5:2. If there are 224 fans at the
training session, how many are male? 64
On the ropes
Investigative task
explore questions
a What fraction of her birthday money did she spend in the iTunes store?
b How much fruit juice is in 600 mL of Morning Sunshine? 150 mL
2G explore card 3
Katie:Mike:Sam
1
___
on lollies,,
10
11 Madeleine gets $50 for her birthday and decides to spend
3
2
___
on a movie ticket, ___
on a new necklace and the remainder on iTunes
es credit.
10
10
a How much water (in mL) is in a 2-L container of Morning
Sunshine? (Hint: 1 L = 1000 mL.) 1500 mL
2G explore card 2
›
fat 160 g, protein 120 g, carbohydrate 140 g
➜
Deep Learning Kit
85
This ratio can then be used to calculate the
winnings for each person. Remind students
to check that their shares add to the total
quantity won.
Deep Learning Kit
2G explore card 1
A fair distribution
Focus: To use ratios to determine a fair
distribution of the assets of a couple among
their family members
…overmatter
2 Percentages, ratios and rates
2H Understanding rates
discover
•
2H Understanding rates
resources
•
discover
Deep Learning Kit
2H discover card 1
›
Quantities and their units
Intervention task
Alexis and Kym arrange to meet at the running track in
preparation for the upcoming athletics carnival. They both
decide to cycle to the track and leave their homes at the
same time.
2H discover card 2
›
Writing rates
Additional skill practice
yes, as quantities are of same kind (time)
Assess
Alexis travels 2800 m, Kym travels 6120 m. Kym travels more than twice as far as Alexis.
4 Write the distance travelled by Alexis compared to Kym’s as a ratio in simplest form. Remember to make sure
that the distances are in the same units. 70:153
key ideas
7 The girls were curious to find out who was faster. Alexis thought she was, but Kym was not convinced.
distance .
a Calculate each girl’s speed using the rule: speed = _______
Alexis 350 m per min, Kym 340 m per min
time
b Who was the fastest? How do you know? Alexis, as she travelled a greater distance per minute
travelling at a speed of 100 km/h.
➜ Order is important when writing a rate.
➜ For a rate to be in simplest form, the second of the two quantities being compared must have a value of 1.
c Comment on the unit required for speed. Explain why this unit was chosen.
Speed is distance (m) per one unit of time (min) so speed unit is m per min or m/min.
8 A rate compares the change in one quantity with respect to another. Speed is an example of a rate.
What is speed comparing? change in distance over time
9 In your own words, explain the difference between a ratio and a rate. A ratio compares quantities of same kind. A rate describes
now try these
e1
change in one quantity with respect to another.
10 Can Alexis’ speed rate be compared with Kym’s and expressed as a ratio? Explain your answer and,
if it exists, write the ratio in simplest form. yes, comparing quantities of same kind (speed) and in same units (m/min); 350:340 = 35:34
11 Copy and complete these sentences.
Ratios compare quantities of the
same
kind.
Rates compare quantities of a different kind.
example 1
➜ think
➜ write
Write this statement
as a rate with the
appropriate unit.
We can write this statement as a rate, as we are
comparing two different quantities (distance and
time). Show the number of the first quantity (100)
for one unit of the second quantity. The word ‘per’
(meaning ‘for each’) can be replaced by the symbol /.
Rate is distance per time.
rate = 100 km in 1 hour
= 100 km per hour
= 100 km/h
100 km in each hour
c four quarters in each game
four quarters/game
f five patients each hour
2 Write each statement as a rate in simplest form.
a 200 metres in 4 minutes
b 6 L in three bottles
50 m/min
2 L/bottle
d 135 students with nine teachers e 468 seats in 18 rows
15 students/teacher
26 seats/row
g 68 runs in 10 overs
h $20 for eight muffins
c
f
i
$16/h
6.8 runs/over
12 With a partner, make a list of similarities and differences between rates and ratios. Share your list with
the class.
74
e2
1 Write each statement as a rate with the appropriate unit.
a 60 km in each hour
b 591 L in each minute
60 km/h
591 L/min
d $16 for each hour
e 45 cm each year
45 cm/year
$2.50/muffin
➜ Focus: To consolidate student
understanding of rates as a way of
describing the relationship between two
variables, and the changes within the
relationship
• After students complete the Discover
task, consolidate their understanding.
Ensure students understand that:
− ratios compare quantities of the
same kind, for example apples and
oranges (both are fruit)
•
− rates compare quantities of different
kind, for example distance and time
− a rate contains units, defined by the
variables within the comparison
and incorporating the word per, for
example metres per second
− as in ratios, the order is important
when writing a rate
− rates can be simplified.
If students are experiencing difficulty
with these questions, or if they require
more support, refer to 2H discover
card 1.
80 cents for 20 seconds
100 newspapers in four bundles
25 newspapers/bundle
102 goals in 15 games
4 cents/s
6.8 goals/game
3 Copy and complete these sentences.
a Five metres of fabric cost $214.95 so the rate is $ 42.99 per metre.
b Mani worked a total of 18 hours and earned $283.50 so the rate is $ 15.75 per hour.
c The Australian cricket team made 156 runs in 30 overs so the rate is 5.2 runs per over.
d A leaky tap dripped a total of 1780 mL over 16 hours so the rate is 111.25 mL per hour.
e The temperature rose by 18°C over an 8-hour period so the rate is 2.25 °C per hour.
f The most valuable player on the basketball team scored 576 points over 22 games so the rate is 26.18 points
per game.
•
•
•
Demonstrate 2H eTutor, or direct
students to do this independently.
Direct students to the examples.
Example 1 shows how to write a rate
which describes distance per time. In
example 2 students are shown how to
write different statements as rates, and
how to simplify the rates by dividing by
the HCF so that the second variable is
represented by the number 1.
Direct students to the key ideas. You
may like them to copy this summary.
75
0.93 sheep/min
now try these
➜ Focus: To use understanding of
rates to write rates with the appropriate
unit, and to write rates in their simplest
form
•
•
•
•
4 Write each of these statements as a rate in simplest form.
a Bobby earns $182.40 working 12 hours.
b The cost of 600 g of breakfast cereal is $9.54.
$15.20/hour
1.59 cents/gram
c Tess typed 3000 words in 65 minutes.
d 56 sheep are sheared in 60 minutes.
46.15 words/min
whole class
five patients/h
Q1 involves writing statements as
rates with the appropriate unit. In Q2
students also write rates, but each rate
needs to be written in simplest form.
In Q3 students complete sentences
requiring them to calculate the cost
per unit.
POTENTIAL DIFFICULT Y
Students may find it difficult to
recognise the units of rates as involving
a comparison between two variables.
Explicit use of per when discussing
rates, and when reading questions,
will assist students in developing this
understanding. Encourage students to
read the solidus (/) in the written unit as
the word per.
Deep Learning Kit
2H discover card 1
Quantities and their units
Focus: To identify the units that relate to a
given quantity
Resources: ruler, coloured pencils or
highlighters
…overmatter
Professional Support
•
➜ A rate compares two quantities that are of a different kind.
➜ A rate contains units.
➜ The unit of a rate is two units separated by the word ‘per’ (meaning ‘for each’) or the symbol /; for example,
2H Understanding rates
•
b rate = $54 per 40 L
$54
40 L
= ____ per ____
40
40
= $1.35 per 1 L
= $1.35 per L
The rate is $1.35/L.
3 Write your answer.
units) are compared, order is important, whole numbers only.
➜ Focus: To discover the
relationship between variables which
can be used to describe rates; in
particular, the relationship between
distance and time used to describe speed
86
3 Write your answer. The word ‘per’ can be
replaced by /.
b 1 Write the two quantities as a rate statement.
•
5 Write three important things you remember about ratios. Some possible answers are: quantities of same kind (in same
discover task
•
a rate = 30 m per 5 seconds
5s
30 m per ___
= _____
5
5
= 6 m per 1 s
= 6 m per s
The rate is 6 m/s.
2 For the rate to be in simplest form, the
second quantity needs be 1. To achieve this,
divide both quantities by 5.
2 For the rate to be in simplest form, the
second quantity needs be 1. To achieve this,
divide both quantities by 40.
6 Alexis was able to cover a distance of 2800 m in 8 minutes. Can these values be written as a ratio?
Explain your answer clearly. No, as quantities are not of same kind (distance and time).
•
a 1 Write the two quantities as a rate statement.
3 Compare the distances travelled by each girl. What do you notice?
2H eTutor
2H Guided example
2H Test yourself
Students consider the exploits of two
students who ride different distances in
different times. They consider how the
information related to these students
can be compared.
Students compare distances travelled
and recognise that they are expressed in
different units, meaning that one must
be converted before comparisons can be
made. Once the distances are expressed
in the same unit they can be written in
ratio format and simplified.
Students consider whether distance
and time can be written as a ratio, and
are guided to recognise that ratios and
rates have a specific difference, which is
summarised by students at the end of
the task.
Students and are guided to discover the
.
rate formula: speed = distance
time
Students discover that a rate compares
the change in one quantity with respect
to another. When exploring the concept
of speed, the rate considers the change
in distance over time.
Write each
statement as a rate
in simplest form.
a 30 m in
5 seconds
2 Write the time taken by Alexis to reach the track compared to the time taken by Kym as a ratio
in simplest form. 49
Reproducible master
•
➜ write
b $54 for 40 L
of petrol
1 Can the time taken by Alexis to get to the track be compared with Kym’s time? Explain your answer.
File 2.01: Conversion charts
➜ think
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
he
It takes Alexis 8 minutes to travel 2800 m from her home to the
track. Kym takes 18 minutes, after travelling a distance of 6.12 km.
Professional Support Online
example 2
In Q4 students write statements as rates
in simplest form.
For those students experiencing
difficulty with these questions,
intervention through small group
teaching of the concept may be
necessary.
Some students may experience difficulty
in converting between units. File 2.01:
Conversion charts can be provided to
students. It contains conversion charts
which students may find helpful for
these questions.
Students may experience difficulty
when a rate needs to be found ‘per
unit’. Students will need to be shown
a sequence of steps which can be
followed to complete these types of
calculations. For example, in Q3a, 5 m
of material costs $214.95. Students need
to recognise that they need to calculate
the cost per metre. Encourage students
to read the question as follows:
5 m cost $214.95, so 1 m will cost
$214.95 ÷ 5.
Calculations can be completed on
calculators. The important thing that
students need to recognise is that they
will need to complete a division to
calculate the rate per unit.
For extra practice, direct students to
2H discover card 2.
Direct students to complete the eTutor,
Guided example and Test yourself
either in class or at home.
87
2 Percentages, ratios and rates
2H Understanding rates
explore
considers which of two plans provides
the best deal for phone calls of differing
lengths.
resources
explore
›
Rating antipasto
Problem solving task
1 Write a rate (in simplest form) to represent each of these situations.
b The cost of 15 mangoes is $29.85. $1.99/mango
›
c A bath fills up to 54 L in 4 __12 minutes. 12 L/min
Solution rates
d
Challenging problem
›
Dealing with dosages
Problem solving task
km2.
2 A particular rate has the unit $/kg where the quantities represented are money and mass. What quantities are
represented by these rates?
d tonnes/year
e $/m
f wickets/over
g m/s
h $ per line
i 55 min 17.1875 km
ii 60 min 18.75 km
iii 65 min 20.3125 km
i $/litre
j
k mm per year
l
v 1 hour 25 min
vi 1.5 h 28.125 km
vii 1.75 h 32.8125 km
distance, time
percentage, time
wickets, overs bowled
distance, time
sheep/hectare
sheep, area
88
•
•
distance, volume
a If a 15-minute phone call costs $11.70, how much will an 8-minute phone call cost? $6.24
b The Bosancic family use 1170 kilowatt hours (kWh) of electricity in 90 days. How much will they use in
58 days? 754 kWh
f Shane Warne bowling wickets/over or km/h
g downloading songs on to an iPod KB/s or GB/min
h driving on a freeway km/h
4 The following food was consumed over a two-week period at the Australian Open.
pasta: 5000 kg/day, fish: 500 portions/
79 000 Australian sausages day, sushi: 5714.3 pieces/day,
noodles: 714.3 kg/day, sausages:
35 tonnes of chips
5642.9 sausages/day, chips:
9000 schnitzels
2.5 tonnes/day, schnitzels:
642.9 schnitzels/day
Calculate the rate of the amount of food consumed per day for each item listed.
5 Max’s team scored 230 runs in 40 overs. William’s team scored 280 runs in 50 overs.
a Calculate each team’s run rate per over. Max 5.75 runs/over, William 5.6 runs/over
b Which team had a better run rate? Max’s team
6 On Thursday, Mauro paid $65.32 for 46 L of petrol. The next day Joy paid $61.09 for 41 L.
a How much did each person pay for one litre of petrol? Mauro $1.42/L, Joy $1.49/L
b Comment on the answers obtained and why you think this occurred. Joy paid higher price
76
on Friday. Prices vary
across week.
c If a 5.7 kg parcel costs $14.25 to post, how much will it cost to send a 750 g parcel? $1.88
a Comment on which car is more economical. Provide calculations to support your answer.
Hani’s car (11.5 km/L compared to 11.2 km/L)
b How much petrol would be consumed by each car to travel 830 km?
Illias 74.1 L, Hani: 72.2 L
13 At a convenience store, milk is available in two sizes. Explain to a friend how you
one with cheaper
could decide which of the two containers of milk is the better buy. the
price per litre
a 5 kg of potatoes for $4.95 or 7.5 kg of potatoes for $7.15
7.5 kg of potatoes ($0.95/kg compared to $0.99/kg)
b a packet of four batteries for $3.99 or a packet of 10 batteries for $9.89
packet of 10 batteries ($0.989/battery compared to $0.9975/battery)
c a 500 g tin of coffee for $9 or a 150 g jar for $2.99
500 g tin of coffee ($0.018/g compared to $0.01993/g)
d a 440 g tin of peaches for $1.85 or a 650 g tin of peaches for $2.56
650 g tin of peaches ($0.0039/g compared to $0.0042/g)
15 A phone company offers its customers two plans.
Plan A: connection fee of 40 cents and then 40 cents per minute or part thereof
c Write this new rate in its simplified form and find how long it will take Peter to make 30 tarts now.
24 mins/tart; 3 days
• In Q5 students compare the run rate of
two different cricket teams, to identify
which of the two teams had the best
run rate. This can be extended further
if cricket season is occurring at the time
of teaching. Students could further
explore the run rate per innings of
different teams and could also discuss
the Australian cricketers in terms of
their ability to score runs per over.
In Q6–8 students explore the use of
rates in reference to different real-life
scenarios.
•
•
In Q9 students compare the running
rate of Usain Bolt with that of other
athletes and also with the running rate
of a cheetah. If this unit is being taught
at the time of an Olympic games, this
concept could be further extended by
asking the students to compare the
speeds of different athletes in different
events.
Q10 involves the calculation of a rate to
describe the distance covered over time
by a runner. Once the rate is found,
students calculate the distance that the
Rating antipasto
Focus: To determine the different costs of an
antipasto mix
Resources: calculator
b Calculate the cost of a 3-minute call for each of the phone plans. Which plan offers the best deal?
c Calculate the cost of a 5-minute call for each of the phone plans. Which plan offers the best deal?
Plan A: 240 c, Plan B: 240 c; both plans
2H explore card 2
d Calculate the cost of a 6-minute 23-second call for each of the phone plans. Which plan offers the best deal?
Plan A: 320 c, Plan B: 316 c; Plan B
Solution rates
a Calculate the cost of a 1-minute call for each of the phone plans. Which plan
offers the best deal? Plan A: 80 c, Plan B: 88 c; Plan A
If Peter can buy the pastry readymade, he can increase his tart-making rate to five tarts every two hours.
2H explore card 1
Students determine the quantities and costs
of different mixes of antipasto from various
food outlets and identify both the best buy
and a pricing schedule.
b How many minutes does it take for Leandro’s heart to beat 1512 times? 27 min
b If Peter cooks for 4 hours a day, how many days will it take him to make 30 tarts? 4 days
Deep Learning Kit
14 Compare the options in each pair of grocery items and determine which option
is the better value for money.
Plan B: connection fee of 50 cents and then 38 cents per minute or part thereof
a Write this as a simplified rate in minutes per tart. 30 mins/tart
Providing the students with the
opportunity to compare best buys and value
for money is a valuable learning experience
which highlights the use of rates in a reallife context. There are several shopping
websites that would be good for this sort
of task.
12 Illias’ car uses 65 L of petrol on a trip of 728 km while Hani’s car uses 48 L travelling 552 km.
a Heart rate is often expressed in beats per minute (bpm). Write Leandro’s heart rate in bpm. 56 bpm
8 Peter is making his famous lemon tarts for a stall. He figures that he can make three large lemon tarts in
1.5 hours.
•
11 Calculate each of these.
d the cost of tiling the bathroom $/m2
e filling a bath with water L/min
70 000 kg of pasta
7000 portions of fish
80 000 pieces of sushi
10 000 kg of noodles
26.5625 km
iv 78 min 24.375 km
viii 1__45 h 33.75 km
2H Understanding rates
•
km/litre
b Usain Bolt sprinting 100 m m/s
7 Leandro’s resting heart rate is 14 beats in 15 seconds.
•
money, line
length, time
a a cheetah chasing its prey m/s
c the cost of 10 apples cents/apple or $/kg
➜ Focus: To apply understanding of
rates and the simplification of rates to
application questions
It may be appropriate for some students
not to complete all the Explore
questions but to work on one or more
discover or explore cards to develop and
deepen their understanding.
In Q1 students write rates to represent
different situations. The rates need to
be expressed in simplest form. Students
may need to be reminded that simplest
form is when the second variable is
represented by the number 1 and the
rate is described as being per unit.
In Q2 students identify each of the two
variables written in rate format.
Q3 involves writing rate units which
could be used in different real-life
situations.
In Q4 students calculate the rate of
different foods consumed over a twoweek period at the Australian Open.
This could be extended into a whole
class activity in which the rate of
different food sold in the school canteen
over a specified time period could
be calculated.
mass, time
3 What rate unit could be used in each of these situations?
explore questions
d
ld he
h cover in these
h times.
b If Victor is able to maintain this rate, what distance
would
c % per annum
money, volume
Remind students of what was learnt in the
Discover section. Some prompts are:
• Write each of the following statements
as a rate in simplest form:
a 100 metres in 25 seconds (4 metres/
second)
b 5 L in five bottles (1L/bottle)
c $20 for 10 minutes ($2/minute)
d $49 for 7 tickets ($7/ticket)
e 72 runs in 9 overs (8 runs/over)
a What distance is he able to cover per minute?
ute? 0.3125 km (312.5 m)
b km/h
money, distance
whole class
10 Victor takes 48 minutes to run 15 km.
a words per minute
words, time
•
(Write your answer correct to two decimal
Students can explore this concept by
exploring the supermarket or, alternatively,
an online shopping website. It is law for
stores to provide a comparison per unit.
Students could be asked to compare the
prices of different items, purchased in
different sizes and compare the value, by
comparing the rates per unit.
c a cheetah that was able to cover a distance of 600 m in
37.5 seconds. 16 m/s
77
Plan A: 160 c, Plan B: 164 c; Plan A
e Use your answers from parts a–d to explain when it
would be best to use each plan.
reflect What is the difference
between a ratio and a rate?
Use Plan A for calls shorter than 5 min, use Plan B for calls longer
than 5 min, use either plan for calls of exactly 5 min.
•
•
runner could cover, given that he is able
to maintain his rate.
In Q11 students consider rates
charged for the use of utilities. This
is generally a contentious issue and is
discussed frequently on current affairs
programs. This could be extended into
a research project in which students are
encouraged to compare the rates for
different providers and select the most
affordable.
In Q12 students consider fuel
consumption by two cars. Again, this
•
•
•
concept provides opportunity for
project work by the students.
Q13 involves consideration of which
of two options gives the best value
for money. For students experiencing
difficulty with this concept, strategies
are discussed in the small group section
that follows.
In Q14 students use the skills discussed
in the small group section that follows
to calculate the ‘best buy’.
In Q15 students consider two different
options for phone plans. The question
Focus: To apply the skills and techniques
of combining rates to solve concentration
problems
Resources: calculator, bucket (optional),
stopwatch (optional)
In this more challenging problem students
are guided through various questions to
determine the required times, rates and
quantities to achieve a required chemical
concentration.
2H explore card 3
Dealing with dosages
Focus: To use rates and percentages to
determine drug dosage quantities for
different people
Resources: calculator
…overmatter
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
ding a time of
b the US 4 × 100 m relay team of 1992 recording
37.4 seconds 10.70 m/s
A runner takes 2 __34 hours to complete a 42.2 km marathon. 15.4 km/h
e Victoria’s population is 5 547 527 and its area is 237 629
places.) 23.35 people/km2
2H explore card 3
➜ Focus: To develop a strategy to
determine which of two items provides
the best value for money
or the
a Michael Johnson’s time of 19.32 seconds for
200 m sprint 10.35 m/s
a A car uses 18 L of petrol to travel 252 km. 14 km/L
2H explore card 2
small group
9 Usain Bolt recorded a world-record time of
9.58 seconds for the 100 m sprint. Compare his rate
with that of: Usain Bolt’s rate is 10.4384 m/s.
2H explore card 1
➜
Deep Learning Kit
89
2 Percentages, ratios and rates
revise
c
choose
2 Percentages, ratios and rates
1 Answer: D.
15 × 4 = 60 = 60%
25 4 100
2D
summarise
Create a summary of the unit using the key terms below. You may like to write a paragraph, create a concept map
or use technology to present your work.
2H 16 Write each statement as a rate with the
appropriate unit.
b
a a 10% mark-up on $950 $1045
a 100 km per hour 100 km/h
c
b a 20% discount on $78 $62.40
b 120 L in 40 seconds 3 L/s
c a 44% mark-up on $628 $904.32
c $140 for 8 hours of work $17.50/h
d a 28% discount on $1255 $903.60
d 26 books read in 6 months 4.333 books/month
equivalent fraction
commission
percentage increase
fraction
decimal
profit
percentage discount
percentage
percentage of an amount
loss
ratio
denominator
mark-up
selling price
equivalent ratio
numerator
discount
wholesale price
rate
2D
8 Calculate the percentage profit or loss in each
situation.
2H 17 Express each statement as a rate in simplest form.
a the cost of 45 L of petrol was $62.55
2A
1 Consider this figure,
which has shaded and
unshaded sections.
2A
b a 375 mL can of soft drink costs $1.80
b a book bought for $35 and sold for $21
c a football team scored 180 points in
120 min of play 1.5 points/min
40%
c shares bought for $2.98 and sold for $4.52
The percentage of this
figure that is shaded is:
2B
A 10%
B 15%
C 40%
D 60%
2D
2E
2F
2G
= $326.40
2H
5:6
c Express the fraction in part b as an equivalent
35
fraction with a denominator of 100. 100
D 95%
C 162%
D 315
2B
B $326.40
C $448
D $633.60
2B
B 37:3
C 37:180 D 180:37
2B
C 15
2C
a 35 kg to 47 kg 35:47
b 7 weeks to 31 weeks 7:31
3 Write each of these decimals as a percentage.
b 7.62
762%
c 31.625 d 0.0003
3162.5%
3
__
75%
4
b
7
___
70%
10
c
17
___
85%
20
B
C
A: Has incorrectly divided 35 by
the multiplication factor, instead of
multiplying the factor by 3.
B: Has incorrectly multiplied the
numerals in the first ratio together and
has subtracted the result from 35.
D: Has incorrectly multiplied by the
factor of 7, not 5.
7 Answer: B. There are four parts in total.
1 × $3600:3 × $3600
4
4
$900:$2700
A: Incorrectly used a total of 3 parts;
1 × $3600:2 × $3600.
3
3
D 5 m/s
a 12:44 3:11
b 64:108 16:27
c 16:32:96 1:2:6
d 15:75:105 1:5:7
2F 13 Calculate the value of the pronumeral in each
ratio statement.
b 13:4 = 52:b 16
c c:48 = 3:12 12
d 15:d = 3:7 35
d 58 as a percentage of 125. 46.4%
A
d 45 cents to $1.25 9:25
2F 12 Simplify each ratio.
a 6:7 = a:49 42
D $1200 and $2700
1
__
5 m/s
15
___
46.875%
32
c 8 months to 3 years 2:9
b 24 as a percentage of 150 16%
B $900 and $2700
24
___
120 m/s
d
5 Express:
C $1500 and $2100
120
___
24 m/s
0.03%
4 Write each of these fractions as a percentage.
A $1200 and $2400
2C
2E 11 Express each of these as a ratio.
d 291.6% 2.916
c 300 as a percentage of 800 37.5%
8 The statement 120 m in 24 seconds expressed
as a rate in simplest form is:
b the number of yellow sections to the total
number of sections 18
c 16.52% 0.1652
a 20 as a percentage of 50 40%
7 When $3600 is divided in the ratio 1:3, the
parts are:
2E 10 Write each comparison as a fraction.
a the number of green sections to the total
number of sections 163
b 62% 0.62
a
D 21
b the number of yellow sections to green
sections 2:3
a 23% 0.23
32%
6 The value of a in the ratio statement
3:7 = a:35 is:
B 14
2 Write each of these percentages as a decimal.
a 0.32
5 Expressed as a ratio, 3 hours to 37 minutes is:
9 Write each comparison as a ratio.
a the number of red sections to blue sections
d Express the shaded amount as a percentage. 35%
B 198
A $153.60
A 7
78
C 20%
4 After a 32% discount, the selling price of an
item originally marked at $480 is:
A 3:37
2E
b Write the total number of shaded sections as
a fraction of the total number of sections. 207
3 45% of 360 is:
A 162
Questions 9 and 10 refer to this
coloured grid.
a 3:5
6 Calculate each of these.
8
a 10% of 340
b 25% of 62
c 150% of 150 000
d 37% of 2400
34
225 000
2G 14 Calculate the total number of parts for each ratio.
15.5
888
C: Incorrectly chosen at random but able
to see that $1500 + $2100 = $3600.
D: Incorrectly used a total of 3 parts
for the first part of the calculation only;
1 × $3600:3 × $3600.
3
4
8 Answer: D
120 m/24 seconds = 5 m/second (divide
both values in the rate by 24)
A: The rate is correctly written but the
can be simplified to 51 = 5.
fraction 120
24
B: The rate is incorrectly written as an
unsimplified fraction, the order of the
quantities has also been incorrectly
swapped.
b 12:5
17
c 4:5:8
17
d 6:10:15
31
2G 15 Divide $1500 in each of the given ratios.
a 2:3
b 3:7
c 3:4:8
d 4:5:11
$600, $900
$450, $1050
$300, $400,
$800
$300, $375,
$825
C: The rate is incorrectly written, but
has been simplified; the order of the
quantities has also been incorrectly
swapped.
answer
1 a 7 shaded sections
7
b 20
7 × 5 = 35
c 20
5 100
d 35% is shaded
2 To convert a percentage to a decimal,
divide by 100.
a 23% = 0.23
b 62% = 0.62
$0.0048/mL or 0.48 c/mL
d a worker earned $218.40 for 13 hours of work
$16.80/h
analyse
A market stall holder purchased a variety of different
coloured ‘hoodies’ to sell at her market stall. Each
hoodie cost the stall holder $20 and she intends to sell
them to the public at a mark-up of 65%.
a Express the percentage mark-up value as a:
i fraction in simplest form 13
ii decimal. 0.65
20
b Calculate the value of the mark-up on each hoodie. $13
c What price will the stall holder sell each hoodie to
the public? $33
d If the stall holder sold 35 hoodies on the first day,
what was her profit that day? $455
15 × 100 = 37.5%
= 40
b Loss = $35 − $21 = $14
loss
× 100
Percentage loss = original
price
22APercentages,
Understanding
ratios
percentages
and rates
B: May have incorrectly subtracted 45%
from 100%, and incorrectly calculated
55% of 360.
C: Has correctly calculated 162, but
the calculation result is a value not a
percentage.
D: Has incorrectly subtracted 45
from 360.
4 Answer: B.
100% − 32% = 68%
B 15%
31.57%
a What is the total
number of shaded
sections? 7
2 The fraction __15 written as a percentage is:
A 10%
2C
d a television bought for $1400 and sold
for $958
1 Consider this figure,
which has shaded and
unshaded sections.
$1.39/L
a a radio bought for $40 and sold for $55
37.5%
answer
d
6 a 10% of 340 = 0.1 × 340 = 34
b 25% of 62 = 0.25 × 62 = 15.5
c 150% of 150 000 = 1.5 × 150 000
= 225 000
d 37% of 2400 = 0.37 × 2400 = 888
7 a 110% × $950 = 1.1 × 950 = $1045
b 80% × $78 = 0.8 × 78 = $62.40
c 144% × $628 = 1.44 × 628 = $904.32
d 72% × $1255 = 0.72 × 1255= $903.60
8 a Profit = $55 − $40 = $15
profit
× 100
Percentage profit = original
price
In an attempt to improve the sale of the hoodies for her
second day at the market, the stall holder decided to
offer a 10% discount on the selling price.
e What is the discount amount for each hoodie? $3.30
f What is the new selling price for each hoodie now? $29.70
g What profit does the stall holder make on the sale of
each hoodie? $9.70
h The stall holder sold 55 hoodies at the discounted
amount. Express the profit from the sale of these as
a percentage of the cost she paid for them. 48.5%
i Of the hoodies sold in part h, 20 were blue, 25 were
red and the rest were white. Write the number of blue,
79
red and white hoodies as a ratio in simplest form. 4:5:2
On the first day at the market, the stall holder spent
5 hours selling the hoodies while she spent 8 hours
selling them on the second day.
j Express the sales for each day as a rate in simplest
form? day 1: 7 hoodies/h, day 2: 6.875 hoodies/h
k Which day represents the best rate of sale? day 1
c 16.52% = 0.1652
d 291.6% = 2.916
3 To convert a decimal to a percentage,
multiply by 100.
a 0.32 = 32%
b 7.62 = 762%
c 31.625 = 3162.5%
d 0.0003 = 0.03%
4 To convert a fraction to a percentage,
multiply by 100. Alternatively: divide the
numerator by the denominator as a short
division.
= 31 × 25
= 75%
a 34 × 100
1
1
b
7 × 100 = 7 × 10 = 70%
10
1
1 1
14 × 100 = 40%
= 35
c
Profit = $4.52−$2.98 = $1.54
profit
× 100
Percentage profit = original
price
1.54 × 100 = 51.68%
= 2.98
d Loss = $1400 − $958= $442
loss
× 100
Percentage loss = original
price
442 × 100 = 31.75%
= 1400
9 a red:blue = 5:6
b yellow:green = 2:3
3
10 a 16
b
11 a
b
c
d
12 a
b
c
d
13 a
2 =1
16 8
35:47
7:31
8:36 = 2:9
45:125 = 9:25
12:44 = 3:11
64:108 = 16:27
16:32:96 = 1:2:6
15:75:105 = 1:5:7
6:7 = a:49
Multiplication factor = 7
a = 6 × 7 = 42
b 13:4 = 52:b
Multiplication factor = 4
b = 4 × 4 = 16
c c:48 = 3:12
Multiplication factor = 4
c = 3 × 4 = 12
d 15:d = 3:7
Multiplication factor = 5
d = 7 × 5 = 35
…overmatter
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
per cent
choose
= 162
= 162
1
90
7 Calculate the price to be paid after each
mark-up or discount.
51.68%
A: May have incorrectly multiplied the
numerator by 10.
B: May have incorrectly multiplied the
numerator by 15.
D: May have incorrectly subtracted the
value of the denominator from 100.
3 Answer: A.
45 × 360 = 45 × 36 = 45 × 18 = 9 × 18
100
1
10 1
5
1 1 1
A: Has answered with the value of the
discount, not the new selling price.
C: Has incorrectly subtracted 32 from the
original price.
D: Has incorrectly added the value of the
discount, $153.60, to the original price.
5 Answer: D.
3 hours = 3 × 60 minutes = 180 minutes
180:37
A: Has not converted the hours into
minutes.
B: Has not converted the hours into
minutes, and has written the ratio in the
incorrect order.
C: Has written the ratio in the incorrect
order.
6 Answer: C.
3:7 = a:35
The multiplication factor is 5 (because
7 × 5 = 35)
a = 3 × 5 = 15
5 a
revise
A: May have incorrectly written the
number of unshaded squares with a
percentage symbol.
B: May have incorrectly written the
number of shaded squares with a
percentage symbol.
C: May have incorrectly calculated the
percentage of the figure that is unshaded.
2 Answer: C.
1 × 20 = 20 = 20%
5 20 100
68 × 480 = 68 × 48 = 68 × 24 = 1632
100
1
10 1
5
1
5
d
17 × 100 = 17 × 5 = 85%
20
1
1 1
15 × 100 = 15 × 25 = 46.875%
32
1
8
1
20 × 100 = 20 × 2 = 40 = 40%
50
1
1 1 1
24 × 100 = 24 × 2 = 48 = 16%
150
1
3 1 3
300 × 100 = 3 × 100 = 3 × 25 = 37.5%
800
1
8
1
8 1
58 × 100 = 58 × 4 = 232 = 46.4%
125
1
5 1
5
91
2 Percentages, ratios and rates
connect
2 connect card 2
resources
Deep Learning Kit
2 Percentages, ratios and rates
Mathematics with Mojo
connect
Focus: To investigate the use of percentages,
ratios and rates when purchasing clothes,
shoes and accessories
This table displays the membership prices for 2012 and
the proposed prices for 2013.
2 connect card 1
Don’t mess with the menace
Problem solving task
Let’s get physical
After the festive season, the start of each New
Year usually sees people make New Year’s
resolutions. Many join a gym in order to get fit
and lose weight. Regular exercise and a wellbalanced diet are key features to staying fit,
healthy and strong.
2 connect card 2
›
Mathematics with Mojo
Problem solving task
Professional Support Online
›
›
›
File 2.06: Let’s get physical
Assessment rubric
File 2.07: Don’t mess with the menace
Assessment rubric
File 2.08: Mathematics with Mojo
Assessment rubric
92
•
$240
$340
$440
$580
$950
$1200
• calculate the number of male and female members
at the fitness centre
• compare the nutrition panels of two products by
converting the amount of nutrients to percentages
• calculate the total number of members at the fitness
centre
Resources: calculator, access to the Internet
(optional)
A further task is to analyse the nutritional information
on the panel shown and make comparisons
between nutrients found in the cereal
and sweet biscuits.
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
2013
Students explore different shopping
possibilities and outcomes using the skills
with percentages, ratios and rates they have
developed throughout the unit.
• calculate the percentage change in male and female
members at the fitness centre each year
In 2010, the gym had a total of
860 members. Two-thirds
of these were male. In the
following year, the percentage
of females and males at the gym
increased by 20%, and then in
2012 there were 432 males and
252 females.
An assessment rubric can be downloaded
from the Professional Support Online site
(see File 2.08: Mathematics with Mojo).
• calculate the percentage change in the total
number of members at the fitness centre each
year
• predict future trends in gym memberships
Costs
Fiona owns The Definitive Body Fitness Centre. Her
job involves comparing membership numbers from
one year to the next, reviewing membership costs,
helping clients with their exercise
programs and nutritional queries.
• calculate the ratio of current membership
costs to proposed costs
• calculate the percentage change in
membership costs
• rank membership costs in ascending order
Steven, a new member at the gym who has not
exercised for many years, decides to measure his
heart rate before using the treadmill. He finds his
pulse and counts 23 beats in 15 seconds.
The target heart rate chart provides an estimate of
the maximum heart rate (MHR) for various activities
based on a person’s age.
Maximum Heart Rate (MHR) = 220 − age
200
180
160
140
120
100
80
Membership
80
f MHR
00% o
90%–1
MHR
e
n
o
z
90% of
e
e 80%–
red-lin
n
o
z
ld
R
sho
MH
–80% of
bic thre
HR
anaero aerobic zone 70%
–70% of M
%
60
e
n
t zo
en
em
ag
an
of MHR
weight m
50%–60%
eart zone
healthy h
• express a variety of membership costs as monthly
rates
• determine the most economical gym membership
Nutrient
Fitness measures
• calculate a person’s heart rate per minute
• calculate a person’s maximum heart rate
• read a target heart rate chart
• calculate the various zones
of a target heart rate chart
• calculate the heart rate
for a given zone of the
target heart rate chart
Cereal
Mass per
40 gram serve
Sweet biscuits
Mass per
22 gram serve
4.9
2.0
22.4
1.7
1.2
14.9
3.4
7.3
6.8
0
Protein
Fat
Carbohydrates
(total)
Sugars
Dietary fibre
Besides recording all working and answers to the task
in your Student Progress Book, you may like to present
your findings as a report.
Your report could be in the form of:
• a poster
• a brochure
81
• an Excel spreadsheet
60
70
60
50
40
30
• a PowerPoint presentation
20
• other (check with your teacher).
Age
As additional preparation for the unit test, complete
the Review on pages 20–1 of the Student
Progress Book.
• As an extension, students could
•
consider researching similar categories
at a local health and fitness centre.
Students could prepare a range of
questions and make an appointment to
meet with and survey the owner of the
business to collect real data and then
compare this data to the data provided
for the task in the book.
Sample answers are provided to the
Connect task (see over the page).
•
•
An assessment rubric can be
downloaded from the Professional
Support Online site (see File 2.06: Let’s
get physical).
Students can undertake an alternative
or extra Connect task using the
investigations provided in the Deep
Learning Kit. Direct them to 2 connect
card 1: Don’t mess with the menace
and/or 2 connect card 2: Mathematics
with Mojo.
Deep Learning Kit
2 connect card 1
Don’t mess with the menace
Focus: To use percentages, ratios and rates
to examine the menacing meningococcal
disease
Resources: calculator, access to the Internet
(optional)
Students are guided to examine certain
facts related to meningococcal disease
and to use their mathematical skills to
determine how this disease might impact
them in different scenarios. Although a
relatively rare disease, meningococcal
disease can strike quickly, causing
fatality and long-term disability. This
task raises awareness of this disease and
mathematically also enables students
to interpret medical information and
outcomes. An assessment rubric can be
downloaded from the Professional Support
Online site (see File 2.07: Don’t mess with
the menace.)
Professional Support
•
$85
$115
2012
Heart rate
•
Twelvemonth
membership
2 Percentages, ratios and rates
•
Sixmonth
membership
Complete the Connect section on pages 18–19 of your
Student Progress Book to show all your working and
answers to this task.
➜ Focus: To use a familiar context
to connect the key ideas of percentages,
ratios and rates
Students analyse different aspects of a
health and fitness business.
The task requirements are expressed
using everyday language so that
students need to recognise when each
of the different types of calculation is
required. Students will need to identify
when to use their understanding of
percentages, ratios or rates.
You may like students to discuss the
task requirements in small groups to
identify each of the contexts described
underneath the given headings:
– membership
– costs
– fitness measures
– nutritional information.
Direct students to complete the
appropriate section of Connect in the
obook. This section provides scaffolding
for the task, to guide students through
the problem solving process. Students
can use this as a foundation for
presenting their findings in a report.
Encourage students to be creative in
presenting their reports but stress that
correct calculations with appropriate
reasoning should be shown. They need
to justify their findings and include any
assumptions they have made.
Threemonth
membership
How do calculations involving percentages, ratios and
rates relate to joining a gym, fitness and diet?
connect task
•
Onemonth
membership
Nutritional information
➜
›
Your task is to help analyse different aspects of the
business.
93
2 Percentages, ratios and rates
connect
2 Percentages, ratios and rates
connect
3 Calculate and comment on the yearly percentage
change in the total number of members from 2010
to 2012.
Percentage increase =
=
increase
× 100
original number
172
×
100
=
20%
increase
860
Total membership 2011 to 2012: Difference = 1032 – 684 = 348
Percentage decrease =
=
Fiona owns The Definitive Body Fitness Centre
and each year she reviews the number of members
at her gym.
For this task you will need to refer to the information
on pages 80–81 of your Student Book.
1 Read the information about membership numbers
and use it to complete this table.
240 ÷ 3 = $80/month
Cost per month in 2013
340 ÷ 3 = $113.33/month
6 months
440 ÷ 6 = $73.33/month
580 ÷ 6 = $96.67/month
12 months
950 ÷ 12 = $79.17/month
1200 ÷ 12 = $100/month
4 If the trend in male and female members displayed
in 2012 continues, will the number of females
exceed the number of males? Explain your
reasoning and show calculations to support your
answer.
8 Compare the rates obtained in question 7 for 2012
and 2013. Comment on the most economical option
for someone requiring a 12-month membership.
Number
of male
members
Number
of female
members
Total
number of
members
2010
573
287
860
2011
688
344
2012
432
252
1032
684
5 Use the table of membership prices to compare
and express each type of membership as a ratio in
simplest form.
One-month 85:115 = 17:23
Three-month 240:340 = 12:17
Six-months 440:580 = 22:29
Twelve-month 950:1200 = 19:24
Males: Difference = 688 – 573 = 115
increase
× 100 =
Percentage increase = original
number
Females: Difference = 344 – 287 = 57
increase
× 100 =
Percentage increase = original
number
115
573
57
287
× 100 = 20.1% increase
× 100 = 19.86% increase
174
180
× 100 = 96.67%
If exercising at 174 beats/min, Steve is exercising in the red line zone.
Six-monthly membership is the most economical buy when comparing
cost per month.
There is less difference in the number of females discontinuing their
gym membership, as compared to the number of males. The number of
females remains more constant than the number of males. If the trend
from 2012 continued, there would be more female members than male
members by 2017.
Costs
13 After walking on the treadmill at a moderate pace
for 5 minutes, Steven’s heart rate is 174 beats per
minute. Calculate this heart rate as a percentage of
the MHR and state what zone he is exercising in.
9 Calculate Steven’s heart rate per minute before he
starts on the treadmill.
23 beats in 15 seconds = 23 × 4 = 92 beats per minute
14 As Steven is just starting to exercise again after
being inactive, it is recommended that his heart
rate does not go beyond the ‘aerobic zone’. What
heart rate range would this correspond to?
70% × 180 = 126 beats/min
80% × 180 = 144 beats/min
Steve should have a heart rate of 126–144 beats/min if exercising in
the ‘aerobic zone’
Nutritional information
10 Steven is 40 years old. Use the target heart rate
chart to calculate Stephen’s MHR per minute.
15 What percentage of carbohydrates is sugar in the
cereal compared with that in the sweet biscuits?
MHR = 220 – age = 220 – 40 = 180 beats per minute
6 Calculate the change in each membership cost as
a percentage of the respective 2012 cost. Rank the
percentage change in ascending order.
Difference in one-month membership = $30
increase
× 100 = 30
× 100 = 35.3% increase
Percentage increase = original
number
85
2011 to 2012: The rate of decrease in membership was more significant
males than for females.
Difference in three-month membership = $100
increase
× 100 = 100
× 100 = 41.7% increase
Percentage increase = original
number
240
Males: Difference = 688 – 432 = 256
decrease
× 100 =
Percentage decrease = original
number
256
688
× 100 = 37.21% decrease
Difference in six-month membership = $140
increase
× 100 = 140
× 100 = 31.8% increase
Percentage increase = original
number
440
Females: Difference = 344 – 252 = 92
decrease
× 100 =
Percentage decrease = original
number
92
344
× 100 = 26.74% decrease
Difference in twelve-month membership = $250
increase
× 100 = 250
× 100 = 26.3% increase
Percentage increase = original
number
950
Ranking: twelve-months, six-month, one-month, three-month
11 If Steven wants to work in the ‘healthy heart zone’,
he needs to elevate his heart rate so that it beats at
50–60% of his MHR. Calculate the heart rate range
these percentages relate to.
50% × 180 = 90 beats/min
60% × 180 = 108 beats/min
For Steve the ‘healthy heart zone’ is 90–108 beats/min
3.4
22.4
× 100 = 15.18% in cereal
6.8
14.9
× 100 = 46.64%% in sweet biscuit
16 Using the nutritional information, comment on
other interesting comparisons between the cereal
and sweet biscuits.
Cereal has 12.25% protein, compared to the sweet biscuit which
has 7.73%.
Cereal has 5% fat, compared to the sweet biscuit which has 5.45%.
Cereal has 8.5% sugar, compared to the sweet biscuit which has 30.9%.
Cereal contains fibre, but the sweet biscuit has no fibre.
Professional Support
Year
2010 to 2011: The rate of increase in membership was similar for males
and females.
18
Cost per month in 2012
3 months
decrease
2 Calculate and comment on the yearly percentage
change in both male and female members from
2010 to 2012.
94
Time period
2 Percentages, ratios and rates
.
2
2010: 3 × 860 = 573.3 ≈ 573 males
860 – 573 = 287 females
2011: 120% × 573 = 687.6 ≈ 688 males
120% × 287 = 344.4 ≈ 344 females
Total: 688 males + 344 females
= 1032 members
decrease
× 100
original number
348
×
100
=
33.72%
1032
60% × 180 = 108 beats/min
70% × 180 = 126 beats/min
For Steve the ‘weight management zone’ is 108–126 beats/min
Converting to monthly rates
Fitness measures
Memberships
12 Calculate the heart rate range that the ‘weight
management zone’ percentages relate to.
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
Total membership 2010 to 2011: Difference = 1032 – 860 = 172
Let’s get physical
7 Calculate the 3-month, 6-month and 12-month
membership costs for 2012 and 2013 as monthly
rates to the nearest dollar.
95
19
2 Percentages, ratios and rates
review
2 Percentages, ratios and rates
review
13 In how many minutes does Hayden complete the
run?
To answer each question, shade one bubble or write
your answer in the box provided.
1 Which arrow is pointing closest to 70% on this
number line?
7 Josie scored 35 out of 40 on her latest Maths test.
Which of these expresses 35 as a percentage of 40?
19 When $240 is shared in the ratio 1:3, what fraction
is the smallest share?
14%
B
C
D
75%
87.5%
14 Consider this diagram. What is the ratio of
non-shaded squares to shaded squares?
114.3%
0%
100%
20 Four students, Mikayla, Morgan, Chelsea and
Alexandra, collected money for a charity in the
ratio of 5:4:4:2. If the total amount collected was
$330, how much did Mikayla collect?
8 What is 15% of $250?
$265
2 What percentage of the
figure has been shaded?
$235
$37.50
$35
9 When an item originally priced at $340 has a
mark-up of 70%, what is the selling price of the
item?
$578
36%
10 Damien notices a sale advertising a discount of
40% for all goods in a store. How much would he
pay for a coffee machine originally priced at $380?
3 What is 42% as a decimal?
0.42
4.2
42.0
4200.0
4 Which of these is equivalent to
5
__
?
8
6
22 Which of these options best describes an example
of a rate?
driving 100 km to the beach
107:65
running as fast as you can to the canteen
$152
A loss of 3500% has been made.
A loss of 15.91% has been made.
A loss of 20.45% has been made.
Questions 12 and 13 refer to the following information.
7
128%, ___
, 36%, 95
15
Ben runs 10 km in 60 minutes. Hayden completes the
same run but does it 1.25 times faster.
55%
65:107
$400
1.07:65
25%, 7.1, __12, 5.2%
1 __
__
, 5, 0.905, 15.1%
3 6
11
What is ___
as a percentage?
20
15 Magan and Michael measure their pace lengths.
Magan’s pace length is 65 cm while Michael’s is
1.07 m. What is the ratio of Michael’s pace length to
Magan’s?
$228
A profit of $3500 has been made.
3.6%, 15%, __87, 1.1
12:13
$340
0.625
5 Which list of numbers is ordered from smallest
to largest?5
13:12
21 Tess, Alicia and Alexandra contributed money
to a lottery ticket. Tess contributed $2, Alicia
contributed $1.50 and Alexandra contributed
$2.50. If the lottery ticket won $1200, how much
money would Tess receive if they distributed their
winnings in the same ratio as their contribution?
65:1.07
11 A car is bought for $22 000 and sold six months
later for $18 500. Which one of these options is
correct?
5
___
%
800
13:25
$532
62.5
58%
$110
12:25
12 Write how much faster Hayden is than Ben, as a
percentage.
125%
earning $15.75 for every hour worked
16 Which of these ratios is not equivalent to 3:5:6?
observing the second hand on a clock for
one minute
9:15:18
15:25:30
30:15:25
30:50:60
17 Which of these represents the statement ‘__43 hour to
90 minutes’ when written as a ratio in simplest form?
23 A car travels 120 km and uses 10 L of petrol.
Which of these options correctly states the rate in
simplest form?
120 km/L
12 km/L
1:2
16
17
12 L/km
5:6
110 km/L
45:90
24 Which of these is the best value?
2:1
18 What is the value of x to complete this ratio
statement?
12:x = 132:55
21
8 chocolate bars for $7.00
3 chocolate bars for $2.50
5 chocolate bars for $4.00
13 chocolate bars for $11.00
5
A: Has incorrectly calculated
35 × 40.
100 1
B: Has incorrectly calculated the
percentage that represents 30 out of 40
D: Has incorrectly calculated
40 × 100.
35
1
Refer to 2C Percentage calculations.
8 Answer: C.
15 × 250 = 15 × 10 = 150 = $37.50
100
1
4
1
4
A: May have incorrectly added 15 to 250.
B: May have incorrectly subtracted 15
from 250.
D: May have incorrectly calculated 14%
of $250.
Refer to 2D Financial calculations.
9 Answer: $578.
100% + 70% = 170%
170 × 340 = 17 × 340
170% of $340 = 100
1
10
1
× 34
= 578
= $578
= 17
1
1
1
Refer to 2D Financial calculations.
10 Answer: C.
100% − 40% = 60%
60 × 380 = 3 × 380
60% of $380 = 100
1
5
1
= 228
= $228
= 31 × 76
1
1
A: Incorrectly calculated a 40% mark-up
140
rather than a 40% discount to obtain 100
.
× 380
1
B: Incorrectly subtracted 40 from $380.
D: Incorrectly calculated 40% of $380,
rather than 60% of $380.
Refer to 2D Financial calculations.
11 Answer: C.
Loss = $22 000 − $18 500 = $3500
loss
× 100
Percentage loss = original
price
3500 × 100 = 15.91%
= 22
000
A: This option can be discounted because
a loss has been made. (The loss is $3500.)
B: $3500 is the value of the loss, not the
percentage loss.
D: Incorrectly calculated
4500 × 100.
22 000
Refer to 2D Financial calculations.
12 Answer: 1.25 = 125%. Hayden runs 125%
faster than Ben.
Refer to 2B Percentages, decimals and
fractions.
13 Answer: 48 minutes
60 ÷ 1.25 = 48 minutes
Refer to 2C Percentage calculations.
14 Answer: C.
non-shaded squares:shaded squares =
13:12
A: Has incorrectly written the ratio
comparing the number of shaded squares
to the total number of squares.
B: Has incorrectly written the ratio
comparing the number of unshaded
18
19
squares to the total number of squares.
D: Has incorrectly written the ratio
comparing the number of shaded squares
to the number of non-shaded squares;
that is, the order has been reversed
Refer to 2E Understanding ratios.
Answer: D.
Remember that the units need to be the
same before the ratio can be written, and
also that ratios cannot contain decimals
so 1.07 m needs to be converted to
107 cm.
A: Has incorrectly written the ratio
comparing the pace length of Magan
to Michael; that is, the order has been
reversed.
B: Has incorrectly written the ratio
comparing the pace length of Magan
to Michael; that is, the order has been
reversed, and has forgotten to convert the
1.07 m to 107 cm.
C: Has written the ratio in the correct
order, but has forgotten to convert the
1.07 m to 107 cm.
Refer to 2E Understanding ratios.
Answer: C.
30:15:25 is simplified to 6:3:5.
The ratios in options A, B and D all
simplify to 3:5:6.
Refer to 2F Working with ratios.
Answer: A.
Remember to convert to the same units.
45 minutes to 90 minutes = 45:90 = 1:2
B: Simplified the components of the ratio
by different factors. 45 has been divided
by 9 and 90 has been divided by 15.
C: The units have been converted but this
ratio is not expressed in simplest form
D: The order of the ratio has been
reversed.
Refer to 2F Working with ratios.
Answer: x = 5
12:x = 132:55
Multiplication factor = 11
55 ÷ 11 = 5
Refer to 2F Working with ratios.
Answer: B. 14 is the smallest share.
A: This option is incorrect. The ratio 1:3
has 4 parts, therefore the denominator of
the fraction for each part should be 4.
C: This option is incorrect. The ratio 1:3
has 4 parts, therefore the denominator of
the fraction for each part should be 4.
D: This option is incorrect; because this
fraction represents the largest share.
Refer to 2G Dividing a quantity in a
given ratio.
…overmatter
Professional Support
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
A
20
15
1
__
3
1
__
4
2
__
3
3
__
4
48 minutes
2 Percentages, ratios and rates
96
Multiple choice options have been listed as
A, B, C and D for ease of reference.
1 Answer: C.
A: A is pointing to a value slightly less
than 50%.
B: 70% is approximately halfway between
50% ad 100%, line C is closer to half way
than line B.
D: D is pointing to a value which is close
to 100%.
Refer to 2A Understanding percentages.
9 = 36 = 36% is shaded.
2 Answer: 25
100
Refer to 2A Understanding percentages.
3 Answer: A. 42 ÷ 100 = 0.42
B: 42 has been incorrectly divided by 10,
rather than 100.
C: 42 has been incorrectly divided by 1,
rather than 100.
D: 42 has been incorrectly multiplied by
100, rather than divided by 100.
Refer to 2B Percentages, decimals and
fractions.
4 Answer: C. This can be calculated by
completing the short division 5 ÷ 8.
5 = 0.625
8
A: Has incorrectly constructed a
percentage using the numerator and the
denominator.
B: Has converted the fraction to a
percentage, by multiplying the numerator
by 100.
D: Has incorrectly multiplied the fraction
1 .
by 100
Refer to 2B Percentages, decimals and
fractions.
5 Answer: B is written in ascending order.
3.6%, 15%, 87.5%, 110%
Convert all of the values in each list
to the same type (fraction, decimal or
percentage). Once they are all in the
same form the values can be compared.
Suggestion: Convert to percentages by
multiplying by 100 each value given as a
fraction or decimal.
A: 25%, 710%, 50%, 5.2% These are not in
ascending order.
C: 28%, 46.67%, 36%, 950% These are not
in ascending order.
D: 33.34%, 83%, 90.5%, 15.1% These are
not in ascending order.
Refer to 2B Percentages, decimals and
fractions.
11 = 55 = 55%
6 Answer: 20
100
Refer to 2B Percentages, decimals and
fractions.
7 Answer: C.
35 = 100 = 35 × 10 = 35 × 5 = 175 = 87.5%
40
1
4
1
2 1
2
97
…overmatter pages 58-59
POTENTIAL DIFFICULT Y
Some students may experience difficulty
finding an equivalent fraction with a
denominator of 100. The following
strategy may assist them in finding the
multiplication factor.
1
For example: 4 = 100. To find the factor
used, divide 100 by 4. The factor is 25.
So 1 needs to also be multiplied by 25 to
find the equivalent fraction.
1
= 25 = 25%
4 100
Deep Learning Kit
2A discover card 1
2A discover card 4
Writing percentages
Focus: To express different quantities as
fractions and percentages
Students consider the meaning of the term
percentage. They complete questions in
which they identify what percentage of an
image is shaded, using their understanding
that percentages are representative of
quantities out of 100. Students also
use their understanding of equivalent
fractions to convert fractions so they have
a denominator of 100 and, thereby, write
the fractions as percentages. Extra practice
questions similar to now try these Q1–4
are provided.
whole class
Focus: To express the number of parts out of
a total as a fraction
Provide each student with File 2.03:
Percentage mix and match. Ask the
students to cut out the six cards. They will
need the first three today and the remaining
three tomorrow, so they will need to put
those remaining three cards somewhere
safe. On the first of the three cards, ask
each student to shade a number of squares
of their own choosing. On the second card
they need to write the shaded amount as a
fraction and on the third card they write
the amount as a percentage. These cards
are then used to play ‘Mix and match’ as
described below.
2A discover card 2
Finding the highest common factor
Focus: To find the highest common factor of
two numbers
Students review the terms factor and factor
pair; and are guided to identify factor
pairs for given numbers. They consider the
meaning of the term highest common factor
(HCF) and are taken through the steps of
a defined process to identify the HCF for
given numbers.
2A discover card 3
Equivalent fractions
Focus: To review that equivalent fractions
have the same value and to work to express
fractions in an equivalent form
Students are guided to identify shaded
sections of diagrams as representing the
same, or equivalent, amounts. They are
guided to review the process for calculating
equivalent fractions, where the same
operation (multiplication or division) is
applied to both the numerator and the
denominator to create a fraction which is
equivalent to the original.
whole class: reflect
Possible answer: It is important to be
able to visualise the fraction that a
percentage represents because fractions and
percentages are used interchangeably in
everyday life.
Direct students to complete the appropriate
section of My learning in the obook.
…overmatter pages 62-63
POTENTIAL DIFFICULT Y
When using the calculator, students
need to ensure that the cursor has
been moved to the end of the number
or fraction being keyed in. This can be
achieved by using the arrow keys.
Deep Learning Kit
2B discover card 4
Converting percentages, fractions and
decimals
Focus: To write percentages as fractions and
decimals and to also write fractions and
decimals as percentages
Students are guided to recognise the
relationship existing between fractions,
decimals and percentages and are guided
through the process required to convert
between all three forms representing the
same value. They complete questions
independently. If students are experiencing
difficulty, they may find it beneficial to use
their scientific calculator.
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
Writing fractions
Students review the terms numerator and
denominator and are guided to write a
fraction to describe the shaded part as a
fraction of the total number of parts. This
concept is extended in further questions in
which students write fractions to describe
different relationships.
98
…overmatter pages 60-61
2B discover card 1
Multiplying and dividing by 10, 100 and 1000
Focus: To explore the results when numbers
are multiplied or divided by 10, 100 and
1000
Resources: calculator
Students complete a number of
multiplication and division questions
using multiples of 10. They are guided to
recognise the relationship between the
number of zeros in the multiplication factor
or the divisor and the number of zeros in
the answer. Students are able to complete
extra questions independently to practise
this skill.
2B discover card 2
Rounding
Focus: To round values to the nearest whole
number or a specified number of decimal
places and apply the concepts of rounding
to money
Organise the students into groups of six
or eight. Ask them to jumble the cards
thoroughly and place them in a jumbled
pile on the table. Students then need to
match the sets of three cards. This could be
a race, and you may like to award prizes to
the fastest team. Groups could then swap
cards and the class could play another game
of ‘Mix and match’.
Resources: coloured pencils, highlighters
Direct students to complete the appropriate
section of My learning in the obook.
2B discover card 3
Students very briefly review the division
process required to convert fractions to
decimals. They then review the conventions
for rounding, and complete further
questions involving this skill independently.
Students consider the rounding conventions
for money and complete questions in which
they round monetary values.
…overmatter pages 64-65
of determining solutions. Students then
extend this concept to a game to be played
with a partner.
whole class: reflect
Possible answer: It is important to be able
to convert between fractions, decimals
and percentages because they are used
interchangeably in everyday life.
Direct students to complete the appropriate
section of My learning in the obook.
Extra practice questions similar to now try
these Q1–7 are provided.
whole class
A possible question is given below.
Convert the following percentages to:
i fractions in simplest form
ii decimals
a
27%
27 , 27)
(100
b 49%
49 , 0.49)
(100
c
32%
32 = 8 , 0.32)
(100
25
d 56%
56 = 14, 0.56)
(100
25
= 1 = 1, 1.00)
100% (100
100 1
Direct students to complete the appropriate
section of My learning in the obook.
e
Converting fractions to decimals
Focus: To write a fraction in its equivalent
decimal form
Students review in detail the division
process required to convert fractions to
decimals and are guided through two
examples, before practising this skill by
completing questions independently.
99
…overmatter pages 66-67
…overmatter pages 70-71
…overmatter pages72-73
to complete the multiplication of a fraction
by 100. Students are guided through the
process of cross-cancelling. If students fi nd
the process of cross-cancelling difficult,
they may find it beneficial to use their
scientific calculator.
whole class: reflect
What is:
Resources: calculator
Possible answer: Some examples of
situations that involve percentage
calculations are: calculating a test score,
calculating discounts and calculating
increases on prices such as for the GST.
a
2C discover card 2
Direct students to complete the appropriate
section of My learning in the obook.
Students are guided through a series of
questions involving the associated costs
and sale prices of running a small business
selling gobbledegook’s, and competing with
large businesses that have the benefit of
bulk orders.
Calculating percentages
Focus: To perform some percentage
calculations, such as, expressing one amount
as a percentage of another and calculating a
percentage of an amount
Students are taken through the steps of the
mathematical process for expressing one
value as a percentage of another and then
the process of converting this fraction to a
percentage. They are also guided through
the calculation of a percentage of an
amount. Extra practice questions similar to
now try these Q1–4 are provided.
whole class
Some possible questions are:
• In your own words explain how you can
recognise the difference between the
different calculations required. (Where
a question reads as ‘out of’ or ‘as a
percentage of’ the question requires you
to write a fraction and then convert this
to 100. Where the question reads ‘a per
cent of’, you are finding the percentage
of an amount.)
• Express these values as percentages:
a 18 as a percentage of 20
(90%)
b 34 as a percentage of 50
(68%)
c 27 as a percentage of 45
(60%)
d 300 as a percentage of 800 (37.5%)
• Calculate each of the following:
a 12% of 180
(21.6)
b 34% of 600
(204)
c 52% of 245 m (127.4 m)
d 95% of $1000 ($950)
Direct students to complete the appropriate
section of My learning in the obook.
100
…overmatter pages 68-69
a mark-up? (an amount added to the
original price.)
b a discount? (the difference between the
regular price and a lower price)
c
another name for the original price?
(the wholesale price)
d a profit (when the selling price is greater
than the wholesale price)
2D explore card 2
e
Focus: To investigate further percentage
discounts after original items have already
been discounted
…overmatter pages 74-75
Resources: ruler, coloured pencils or
highlighters
Students consider types of measurement
and the units used for each. They are
guided to convert between units of length,
discovering rules which can be applied
when converting from smaller to larger
units, and from larger to smaller units. File
2.01: Conversion charts could be provided.
Students complete extra questions
independently.
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
a loss? (when the selling price is less
than the wholesale price)
Direct students to complete the appropriate
section of My learning in the obook.
Taking a further percentage discount
Resources: calculator
Students explore different scenarios
which involve purchasing clothes and the
different discounts offered. Students are
then required to determine the best offer in
terms of being the purchaser and then again
as the business owner.
2D explore card 3
Just how much GST?
Focus: To determine the rules to calculate
the quantities involved in GST calculations
and obligations
Resources: calculator
Students investigate the mathematical
rules involved with determining the GST
obligations of transactions involving tax
invoices.
2D explore card 4
Inflating rates
Focus: To investigate how percentages are
applied in calculations involving Consumer
Price Index (CPI) and inflation
Resources: calculator, Internet access
(optional)
Students investigate the percentage change
in the prices of everyday items over several
years and mathematically link this to the
Consumer Price Index and the rate of
inflation.
whole class: reflect
Possible answer: Using percentages to
compare profit and loss is beneficial because
percentages are a standard measure; they
are all out of 100. If comparing profit and
loss in dollar terms, it cannot be assumed
that they are being compared in regards to
the same original dollar value.
Direct students to complete the appropriate
section of My learning in the obook.
description? (the amounts in the ratio
must be in the same order as the worded
description)
Direct students to complete the appropriate
section of My learning in the obook.
2E discover card 2
Comparing two quantities
Focus: To write a comparison of two
quantities and express it in ratio form
Students consider ratios describing part to
part relationships and also part to whole
relationships. Students may need to be
reminded that the order in which the ratio
is written is important. They also write
ratios to compare units of measurement,
and may need to be reminded to check that
all units in the ratio are the same.
2E discover card 3
Writing ratios
Focus: To write a comparison of two
quantities as a ratio in the correct order
Students are guided to consider that ratios
can only be written to compare similar
objects. They are taken through the steps
of the process to write a ratio in which the
units to be compared are not the same.
That is, they are taken through the steps
of the conversion process and then write
the correct comparison. Extra practice
questions similar to now try these Q1–4 are
provided.
whole class
Some possible questions are:
• Explain why you could not write a ratio
to compare cats and building blocks.
(they are not the same)
• What must be considered when writing
a ratio comparing measurements? (all
units must be expressed using the same
unit of measurement)
• Are units included when ratios are
written? (no, ratios only contain
numbers)
• Can ratios include fractions or
decimals? (no, whole numbers only)
• What consideration must be made when
writing a ratio to represent a worded
101
…overmatter pages 78-79
a
•
6:12 (HCF is 6, simplified ratio
is 1:2)
b 14:35 (HCF is 7, simplified ratio
is 2:5)
c 1000:200 (HCF is 200, simplified
ratio is 5:1)
d 100:50 (HCF is 50, simplified ratio
is 2:1)
What would happen if we used 10 as
the HCF for 80:20? (We would get 8:2,
which is not completely simplified
because both numbers can be further
divided by 2, to give 4:1.)
…overmatter pages 80-81
…overmatter pages 82-83
…overmatter pages 86-87
use of a calculator, but rather encourage a
mental approach to solving such questions.
Deep Learning Kit
Resources: calculator
2G discover card 1
2F explore card 2
Dividing a quantity in a given ratio
Students explore, and compare, the use
of ratios in different scenarios in which
a couple is dividing assets among their
family members.
Students consider types of measurement
and the units for each. They also consider
abbreviations for units. Students complete
extra questions in which the correct unit is
selected to represent a given measurement.
2G explore card 2
2H discover card 2
The farmer’s dilemma
Writing rates
Focus: To consider the relationship between
fractions and ratios and the implied
constraints when dividing a quantity in
a given ratio
Focus: To express two quantities as a rate
with the correct units
Motorbike dilemmas
Focus: To use ratios to determine the
fuel range possibilities when travelling by
motorbike between outback towns
Resources: ruler, map of Australia
(optional)
Focus: To review the parts of a ratio and
divide a quantity in a given ratio
Students review the process of calculating
the total number of parts in a ratio and
then are guided through an example in
which a fraction is written to describe the
relationship between each part of the ratio
and the total number of parts. Using this
fraction and completing a multiplication
calculation, the students divide an amount
in a given ratio. Extra practice questions
similar to now try these Q1–6 are provided.
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
Direct students to complete the appropriate
section of My learning in the obook.
Students use ratios and map scale factors to
determine the likely outcomes of travelling
in the outback with limited resources
of fuel.
whole class: reflect
Possible answer: Equivalent ratios are useful
because we can use the multiplication or
division factor to calculate unknown values
in equivalent ratio statements.
Direct students to complete the appropriate
section of My learning in the obook.
whole class
Some possible questions are:
• How many parts are represented in each
of the following ratios?
a 4:5
(9)
b 2:11 (13)
c 9:10 (19)
d 50:27 (77)
• Divide $5000 into each of the given
ratios
a 2:3 ($2000 and $3000)
b 10:1 ($4545.45 and $454.55)
• Describe how you can check that the
working is correct. (check to see if the
portions add to the total quantity that
you started with)
Direct students to complete the appropriate
section of My learning in the obook.
102
…overmatter pages 84-85
Resources: calculator
Students calculate and compare the
different scenarios which occur when a
dithering farmer leaves his large parcel of
land to be split among his five children.
2G explore card 3
Resources: calculator
Students consider the differences and
similarities between ratios and rates. They
are guided to identify the two quantities
compared in a given rate, and explore the
correct notation for the given rate. Also
considered is the simplification of rates.
Extra practice questions similar to now try
these Q1 and Q2 are provided.
On the ropes
Focus: To relate fractions and ratios and
consider their impact on the total quantity
Students calculate and compare the
different lengths of a long rope that is cut
according to certain requirements.
whole class: reflect
Possible answer: Ratios can also be used
to describe chance. For example, there
is a 50:50 chance of rain today. They are
also used to describe different quantities
in recipes and can be used to describe the
relationships in some sports, for example
gear ratios in cars and bikes.
Direct students to complete the appropriate
section of My learning in the obook.
whole class
Some possible questions are:
• How are rates different to ratios?
(A ratio compares two quantities of
the same type. A rate compares two
quantities which are of a different type.)
• Describe a similarity between rates
and ratios? (Both can be simplified, by
dividing through using the HCF.)
• What is important when writing rates
and ratios? (order)
• Write each of the following statements
as a rate:
a 50 km in each hour (50 km per hour,
or 50 km/h)
b five animals seen in each hour at
the vet (50 animals per hour, or
5 animals/h)
c 600 mL in each day (600 mL per day,
or 600 mL/day)
d $12 in each shif $12 per shift, or
$12/shift)
Direct students to complete the appropriate
section of My learning in the obook.
103
…overmatter pages 88-89
Students explore the different parameters
required to treat illness in people
requiring short-term and long-term drug
intervention, without exceeding drug
toxicity levels.
…overmatter pages 90-91
14 a
b
c
d
15 a
whole class: reflect
b
Possible answer: A ratio compares two
quantities of the same type. A rate
compares two quantities which are of a
different type.
c
d
Direct students to complete the appropriate
section of My learning in the obook.
16 a
b
c
d
17 a
b
c
d
3 + 5 = 8 parts
12 + 5 = 17 parts
4 + 5 + 8 = 17 parts
6 + 10 + 15 = 31 parts
2 × 1500: 3 × 1500
5
5
$600:$900
3 × 1500: 7 × 1500
10
10
$450:$1050
3 × 1500: 4 × 1500: 8 × 1500
15
15
15
$300:$400:$800
4 × 1500: 5 × 1500: 11 × 1500
20
20
20
$300 : $375 : $825
100 km/h
3 L/s
(120 ÷ 40)
$17.5/h
(140 ÷ 8)
4.3 books/month (26 ÷ 6)
$1.39/L
(62.55 ÷ 45)
0.48c/mL
(180 ÷ 375)
1.5 points/minute (180 ÷ 120)
$16.80/h
(218 ÷ 13)
analyse
× 100 = 8.5%
= 533.50
1100
i
j
104
20 Answer: Mikayla’s share = $110
Mikayla:Morgan:Chelsea:Alexandra
5:4:4:2
Total number of parts = 5 + 4 + 4 + 2 = 15
5
Mikayla’s share = 15
5 × $330 = $110
15
Refer to 2G Dividing a quantity in a given
ratio.
21 Answer:
Tess:Alicia:Alexandra
$2:$1.50:$2.50
200 cents:150 cents:250 cents
4:3:5
Total number of parts = 4 + 3 + 5 = 12
4.
Tess would receive 12
4 × $1200 = $400
12
Refer to 2G Dividing a quantity in a
given ratio.
22 Answer: C. Earning $15.75 for every hour
worked is an example of a rate.
A: Example of a distance measurement,
not a comparison of two quantities.
B: Example of a time measurement, not a
comparison of two quantities.
D: Example of an observation of time
passing, not a comparison of two
quantities.
Refer to 2H Understanding rates.
23 Answer: B.
120 km/10 L = 12 km/L
A: This rate is incorrectly stating that the
distance travelled per litre is 120 km
C: This rate has units that are written in
the incorrect order.
D: This rate is incorrectly stating that the
distance travelled per litre is 110 km. 110
may have been calculated by subtracting
10 from 120, rather than completing
120 ÷ 10.
Refer to 2H Understanding rates.
24 Answer: C. This choice represents the best
price per bar.
Price per bar = $4.00 ÷ 5 = $0.80
A: Price per bar = $7.00 ÷ 8 = $0.88
B: Price per bar = $2.50 ÷ 3 = $0.83
D: Price per bar = $11.00 ÷ 13 = $0.85
Refer to 2H Understanding rates.
There are two parallel unit tests
(Test A and B) available on
Professional Support Online.
D
E
T
C
E
S
R
F
R
O
O
O
C
R
N
P
U GE
A
P
65 = 13
i 65% = 100
20
ii 0.65
b 65% × $20 = $13 mark-up
c Selling price = $20 + $13 = $33
d Profit per hoodie = $13
Profit on sale of 35 hoodies = 35 × $13 =
$455
e Discount = 10% × $33 = $3.30
f New selling price = $33 − $3.30 = $29.70
g Profit based on new selling price = $29.70
− $20 = $9.70
h Profit on sale of 55 hoodies at new price =
55 × $9.70 = $533.50
Wholesale price of 55 hoodies = 55 × $20
= $1100
profit
× 100
Percentage profit = original
price
a
…overmatter pages 96-97
blue:red:white = 20:25:10 = 4:5:2
first day: sells 35 hoodies in 5 hours
rate is 7 hoodies/hour
second day: sells 55 hoodies in 8 hours
rate is 6.875 hoodies/hour
k The first day represents the best rate of
sale.
105