AUSTRALIAN CURRICULUM MATHEMATICS YEAR 7 · At Year 6 level Make connections between equivalent...

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AUSTRALIAN CURRICULUM

MATHEMATICS YEAR 7

Percentages


MATHEMATICS YEAR 7

Percentages

Student’s name: ________________________________

Teacher’s name: ________________________________

First published 2012

ISBN 9780730744382 SCIS 1564078

© Department of Education WA 2012 (Revised 2020)

Requests and enquiries concerning copyright should be addressed to:

Manager Intellectual Property and Copyright Department of Education 151 Royal Street EAST PERTH WA 6004

Email: [email protected]

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year andsubject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.

creativecommons.org/licenses/by-nc-sa/3.0/au/

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This product will be registered through the National Copyright Unit for use in all Australian schools without remuneration.

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Year 7 Mathematics Percentages

© Department of Education WA 2012 – MATHSAC018 Page 1

Contents

Signposts ....................................................................................................................................2

Introduction...............................................................................................................................3

Curriculum details....................................................................................................................4

1. What is a percentage?.........................................................................................................7

2. Fraction and decimal equivalents....................................................................................13

3. Finding a percentage of a quantity..................................................................................19

4. Finding a percentage of a quantity again .......................................................................25

5. Writing one quantity as a percentage of another...........................................................31

6. Problem solving.................................................................................................................37

7. Solving harder problems..................................................................................................43

8. Summary............................................................................................................................47

9. Review tasks ......................................................................................................................49

Solutions...................................................................................................................................57

Percentages Year 7 Mathematics

Page 2 © Department of Education WA 2012 – MATHSAC018

Signposts Each symbol is a sign to help you.

Here is what each one means:

The recommended time you should take to complete this section.

An explanation of key terms, concepts or processes.

A written response. Write your answer or response in your journal.

Correct this task using the answers at the end of the resource.

Calculators may not be used here.

Make notes describing how you attempted to solve the problem. Keep these notes to refer to when completing the Self-evaluation task. Your teacher may wish you to forward these notes.



Introduction This resource should take you approximately two weeks to complete. It comprises seven learning sections, a summary section and a review task section.

The learning sections have the following headings:

Key wordsThese are the main words that you need to understand and use fluently to explain yourthinking.

Warm-upWarm-up tasks should take you no longer than 10 minutes to complete. These are skillsfrom previous work you are expected to recall from memory, or mental calculations thatyou are expected to perform quickly and accurately. If you have any difficulties inanswering these questions, please discuss them with your teacher.

ReviewSome sections have reviews immediately after the warm-up. The skills in these reviewsare from previous work and are essential for that section. You will use these to developnew skills in mathematics. Please speak to your teacher immediately if you are havingany trouble in completing these activities.

Focus problemFocus problems are designed to introduce new concepts. They provide examples of thetypes of problems you will be able to solve by learning the new concepts in this resource.Do not spend too long on these but do check and read the solutions thoroughly.

Skills developmentThese help you consolidate new work and concepts. Most sections include skillsdevelopment activities which provide opportunities for you to become skilled at usingnew procedures, apply your learning to solve problems and justify your ideas. Pleasemark your work after completing each part.

Correcting your work

Please mark and correct your work as you go. Worked solutions are provided to show how you should set out your work. If you are having any difficulty in understanding them, or are getting the majority of the questions wrong, please speak to your teacher immediately.

Journal

Please keep an exercise book to record your notes and to summarise your learning. At the end of each section, write definitions for the key words that were introduced for that section.



Curriculum details

Content Descriptions This resource provides learning and teaching to deliver the Australian Curriculum: Mathematics for the following Year 7 Content Descriptions.

Connect fractions, decimals and percentages and carry out simple conversions (ACMNA157)

Find percentages of quantities and express one quantity as a percentage of another, with and without digital technologies. (ACMNA158)

Content Descriptions 1 2 3 4 5 6 7 R

ACMNA157

ACMNA158

Indicates the content description is explicitly covered in that section of the resource.

Previous relevant Content Descriptions

The following Content Descriptions should be considered as prior learning for students using this resource.

At Year 6 level

Make connections between equivalent fractions, decimals and percentages (ACMNA131)

Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies (ACMNA132)

Proficiency strand statements at Year 7 level At this year level:

Understanding includes describing patterns in uses of indices with whole numbers, recognising equivalences between fractions, decimals, percentages and ratios, plotting points on the Cartesian plane, identifying angles formed by a transversal crossing a pair of lines, and connecting the laws and properties of numbers to algebraic terms and expressions

Fluency includes calculating accurately with integers, representing fractions and decimals in various ways, investigating best buys, finding measures of central tendency and calculating areas of shapes and volumes of prisms

Problem Solving includes formulating and solving authentic problems using numbers and measurements, working with transformations and identifying symmetry, calculating angles and interpreting sets of data collected through chance experiments



Reasoning includes applying the number laws to calculations, applying known geometric facts to draw conclusions about shapes, applying an understanding of ratio and interpreting data displays

General capabilities

General capabilities 1 2 3 4 5 6 7 R

Literacy

Numeracy

Information and communication technology (ICT) capability

Critical and creative thinking

Personal and social capability

Ethical behaviour

Intercultural understanding

Indicates general capabilities are explicitly covered in that section of the resource.

Cross-curriculum priorities

Cross-curriculum priorities 1 2 3 4 5 6 7 R

Aboriginal and Torres Strait Islander histories and cultures

Asia and Australia’s engagement with Asia

Sustainability

Indicates cross-curriculum priorities are explicitly covered in that section of the resource.

This resource contains extracts from The Australian Curriculum Version 3.0 © Australian Curriculum, Assessment and Reporting Authority 2012. ACARA neither endorses nor verifies the accuracy of the information provided and accepts no responsibility for incomplete or inaccurate information. In particular, ACARA does not endorse or verify that:

the content descriptions are solely for a particular year and subject

all the content descriptions for that year and subject have been used

the author’s material aligns with the Australian Curriculum content descriptions for the relevant year and subject.

You can find the unaltered and most up to date version of this material at www.australiancurriculum.edu.au. This material is reproduced with the permission of ACARA.

creativecommons.org/licenses/by-nc-sa/3.0/au/





1. What is a percentage?When you complete this section you should be able to:

understand what a percentage is of a whole.

Keywords

percentage

Warm-up 1

1. Is 4 a factor of 20? ___________

2. 8 + 4 = ___________

3. What is the missing number?

a = ___________

4. Circle the greater fraction.2

3or

6

6

5. What is a half of 18? ___________

6. 6 – 4.3 = ___________

7.

5 . 4

5

8. Write 0.333… as a fraction. ___________

9. Complete the next number: 5, 10, 15, 20 ___________

10. The truck is at (1, 4).

If the truck moves 2 units right,

where will it then be?

x1 2 3 4 5

y

12345

30-3a



Review 1

Example

Write each of these numbers as fractions with denominators of 100.

(a) 0.23 (b) 0.7

(c)9

50(d)

56

200

Solution

(a)23

100(b) 0.7 = 0.70 =

70

100

(c)9 9 2 18

50 50 2 100

(d)

56 56 2 28

200 200 2 100

1. Write each of these numbers as fractions with denominators of 100.

(a) 0.09 ______________ (b) 0.88 ______________ (c) 0.2 _____________

(d) 12

50_______________(e)

12

200______________

2. Write the answers as fractions with a denominator of 100.

(a)5 87

100 100 ___________________________________________________

(b)13 8 29

100 100 100 ___________________________________________________

(c)51 15

100 100 ___________________________________________________

(d) 0.25 + 0.72 ___________________________________________________

(e) 0.2 + 0.37 ___________________________________________________

(f)3 4

10 50 ___________________________________________________

In (c), 50 × 2 = 100 so the numerator and

denominator had to both be multiplied by 2.



Focus problem 1

Dan and Melinda were watching some games of basketball.

They knew there were one hundred players altogether.

Of the hundred players, they counted 40 female players, 64 blond haired players, 15 players over 185 cm tall and 23 wearing a green uniform.

Melinda said it would be easy to write these as percentages since a percentage is the same as a fraction with the denominator of 100.

Dan started to change them to percentages. He preferred to use the shorthand symbol for a percentage, %.

He knew that of the players, 40 out of the 100 were females so 40% were females. Melinda then noted that if that were the case then the percentage of males players would be 60%.

Help Dan and Melinda by completing the following table of percentages for players with the different characteristics.

Female players 40%

Male players

Blond haired players

Players without blond hair

Players over 185 cm tall

Players not over 185 cm tall

Players wearing green uniforms

Players wearing other colour uniforms apart from green

All the players

Check your work before continuing.

In the 15th century when writing a percent the shortened pc was used after the number. By the 17th century with some flourishes in the writing it had evolved to the symbol o

oand this eventually changed to the symbol we use today, %.



Skills development 1

One whole can be written as the fraction 100

100 or as the percentage 100%.

Example

If 42% of a class of students is boys, what percentage are girls?

Solution

The percentage of girls is 100% – 42% = 58%.

1. A grid has 100 squares. 18 of the squares are coloured red, 26 are coloured green and 41are coloured yellow. The remainder are uncoloured.

Write the percentage of each of the following.

(a) green squares ________________________________________

(b) uncoloured squares ________________________________________

(c) squares that are not red ________________________________________

(d) squares that are not yellow or red ________________________________________

(e) purple squares ________________________________________

2.

(a) On the grid above shade:

(i) 10% of the squares using sloped lines (////)

(ii) 25% of the squares using vertical lines (|||||)

(iii) 4% of the squares using a solid colour. ( )

(b) What percentage of squares are now:

(i) shaded ________________________________________

(ii) unshaded? ________________________________________



3. Answer the questions below using the following grid.

(a) How many squares are in the grid? _ ____________________________

(b) What percentage of the grid is one square? ____________________________

(c) What percentage of the grid is:

(i) unshaded ____________________________

(ii) shaded with sloped lines (\\\\) ____________________________

(iii) shaded with a solid colour ____________________________

(iv) shaded with cross hatches ( ) ____________________________

4. How many squares need to be shaded if you were asked to shade 20% of a grid that haseach of the following number of squares?

(a) 100 squares

__________________________________________________________________

(b) 50 squares

__________________________________________________________________

(c) 25 squares

__________________________________________________________________

(d) 200 squares

_________________________________________________________________

(e) 1000 squares

__________________________________________________________________






2. Fraction and decimal equivalentsWhen you complete this section you should be able to:

convert between fractions, decimals and percentages.

Warm-up 2

1. Is 4 a common factor of 24 and 32? ___________

2. 13 – 5 = _________

3. The temperature was 1 degree but it dropped 4 degrees.

What is the new temperature? _________

4. Insert <, > or = to make the following sentence true.

1 2

2 4

5. 1

210 = _________

6. Round 6.9 to a whole number. _________

7. 8.4 4 = _________

8. Write 1

21 as a decimal. _________

9. Complete the next number. 0.7, 1.4, 2.1, 2.8, ________

10.A six-sided die is rolled.

Express, as a fraction, the probability that it lands on an odd number. _________

0

5

10

20

°C



Review 2

Example

(a) Write the following as fractions with a denominator of 100.

(i)13

50(ii) 0.93 (iii)

425

500(iv) 0.6

(b) Use your calculator to find the equivalent decimals to the following fractions.

(i)27

50(ii)

17

20(iii)

5

8

Solution

(a) (i) 13 2 26

50 2 100

(ii)

93

100(iii)

425 5 85

500 5 100

(iv)

600.60

100

(b) (i) 27 ÷ 50 = 0.54 (ii) 0.85 (iii) 0.625

1. (a) Write the following as fractions with a denominator of 100.

(i)47

50 ___________________________________________________________

(ii) 0.19 ___________________________________________________________

(iii) 0.3 ___________________________________________________________

(iv)176

400 ___________________________________________________________

(v)7

25 ___________________________________________________________

(b) Use your calculator to find the equivalent decimals to the following.

(i)13

20 ___________________________________________________________

(ii)102

300 ___________________________________________________________

(iii)9

16 ___________________________________________________________

(iv)35

56 ___________________________________________________________



Focus problem 2

Leith, Hugh and Sam were talking about their love of music. They decided to compare what proportion of their pocket money they spent on music last month.

Leith said she spent 1

4, Hugh said 0.27 and Sam said he spent

22% of his pocket money on music.

It would be easier to compare the proportions if they were all in the same form.

Percentages have been used for a long time to easily compare values so the friends decided to use percentages for their comparisons.

Since percentages are equivalent to fractions out of 100 the idea would be to first convert the decimal and the fraction to fractions with a denominator of 100. Once they have the denominator of 100 the percentage value could be written easily.

(a) (i) Complete the following calculations.

1 %

4 100 0.27 %

100

(ii) Now write the three percentages in order from the smallest to the largest.

_______________________________________________________________________

(iii) Who had the largest percentage value? _______________.

Jemma came in later and she said her proportion was 11

40 because she spent $11 on music out

of her $40 pocket money. This was harder to change quickly to a fraction out of 100 and hence a percentage.

Sam suggested she change it to a decimal using her calculator.

To change 11

40 to a decimal using a calculator do the division 11 ÷ 40.

(b) (i) Write down the decimal equivalent of 11

40 from the calculator.

_______________

(ii) How do you change the decimal to a percentage?

______________________________________________________________________

______________________________________________________________________



Check the following working.

1111 40 0.275

400.275 100 27.5%

11So 27.5%

40or

11(11 40 100)% 27.5%

40

(c) To find the percentage of pocket money spent on music in each of the followingcases, write down the calculation you would perform on your calculator then writethe percentage answer.

Complete the calculation on your calculator to find the percentage answer.

Pocket money Amount spent on music Calculation Percentage

$10 $2.70

$25 $7.50

$32 $28.00





Converting fractions, decimals and percentage

Fraction or decimal to percentage:

Percentage to fraction or decimal:

Example

1 Change each of the following to percentages.

(a) 0.78 (b) 0.323 (c)4

5(d)

5

8

2. Change these percentages to simple fractions.

(a) 15% (b) 2.5%

3. Change these percentages to decimals.

(a) 85% (b) 2% (c) 1

333 %

Solution

1. (a) (0.78 × 100)% = 78% (b) (0.323 × 100)% = 32.3%

(c)4

(4 5 100)% 80%5 (d)

5(5 8 100)% 62.5%

8

2. (a) 15 3

100 2015% (b)

2.5 25 1

100 1000 402.5%

3. (a) 85% = 85 ÷100 = 0.85 (b) 2% = 2 ÷100 = 0.02

(c) 1

333 % 33.3... 100 0.333...

fraction

decimal

percentagenumerator ÷ denominator × 100

fraction

decimal

percentage

write the percent with a denominator

of 100, then simplify if possible

÷ 100



1. Change each of the following to percentages. Show your working.

(a) 0.61 ____________________________________________________________

(b) 0.04 ____________________________________________________________

(c) 1.13 ____________________________________________________________

(d)2

5____________________________________________________________

(e)5

16____________________________________________________________

(f)2

75____________________________________________________________

2. Change these percentages to simple fractions.

(a) 45% ____________________________________________________________

(b) 0.5% ____________________________________________________________

(c) 125% ____________________________________________________________

3. Change these percentages to decimals.

(a) 23% ____________________________________________________________

(b) 1.5% ____________________________________________________________

(c) 2

366 % ____________________________________________________________

4. In each of the following cases change all the numbers to the same form: fraction, decimalor percentage. Then arrange the original numbers in order from the smallest to thelargest.

(a) 47% 0.4512

25 _______________________________________________________________________

_______________________________________________________________________

(b) 19

20 92%

15

160.975 94.8%

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________




3. Finding a percentage of a quantityWhen you complete this section you should be able to:

find a given percentage of a quantity.

Warm-up 3

1. Circle the composite number. 5, 7, 9, 11

2. 9 7 = _________


a = __________

4. Locate 5

10 on the number line.

5. What is a quarter of 20? ________

6. Estimate the difference by first rounding to whole numbers. 8.9 – 4.1 _______

7. 1.2 4 = _________

8. Write 20

100 as a percentage. _________

9. Write the next number.1 3 5

8 8 81 , 1 , 1 , _________

10.

Which shape is at (4, 3)?

_______________

0 1

-3-4 -1 0a

x1 2 3 4 5

y

1

2

3

4

5



Review 3

Finding a fraction or a decimal of a quantity is revised in these examples.

Example

Find the following.

(a) 1

5 of 35 km (b)

3

5 of 35 km (c) 0.3 of 35 km

Solution

(a) 1

5 of 35 km = (35 × 1 ÷ 5) km = 7 km

(b) 3

5 of 35 km = (35 × 3 ÷ 5) km = 21 km

(c) 0.3 of 35 km = (35 × 0.3) km = 10.5 km

1. Find the following fractions or decimals of 72 litres of water.

(a)1

8________________________________________________________________

(b)3

4________________________________________________________________

(c) 0.6 ________________________________________________________________

(d) 0.09 ________________________________________________________________

2. Find the following fractions or decimals of the given quantity.

(a) 0.15 of $70

______________________________________________________________________

(b) 2

3 of 51 kg

______________________________________________________________________

(c) 0.06 of 150 minutes

______________________________________________________________________

(d) 11

25of 94.5 km

______________________________________________________________________

Notice that the word 'of' means

to multiply.



Focus problem 3

A nursery was growing a number of different trees. It was known that for fruit trees 5% of the seeds that were planted wouldn’t sprout.

1. Calculate the following for fruit trees.

(a) How many seeds out of 100 wouldn't sprout? _________

(b) How many seeds out of 200 wouldn't sprout? _________

(c) How many seeds out of 300 wouldn't sprout? _________

(d) How many seeds out of 260 wouldn't sprout? _________

(e) How many seeds out of 480 wouldn't sprout? _________

When growing wattles all the seeds sprouted but 2% of the

young trees died before maturity.

2. Calculate the following for wattle trees.

(a) How many young trees out of 100 died before maturity? _________

(b) How many young trees out of 200 died before maturity? _________

(c) How many young trees out of 50 died before maturity? _________

(d) How many young trees out of 550 died before maturity? _________

3. Explain how you worked out your answers for:

(a) the fruit trees

___________________________________________________________________

___________________________________________________________________

(b) the wattles.

___________________________________________________________________

___________________________________________________________________


Wattles

Wattles are the most widely spread of any of the native plants in Australia. There are over 700 species throughout the country. At any time of the year you can find a wattle flowering somewhere. They are especially suited to our harsh climate.




To find a percentage of a quantity do one of the following.

• Multiply the quantity by the percentage number and then divide by 100. This is thesame as multiplying by the fraction equivalent with a denominator of 100.

• Multiply the quantity by the percentage written as a decimal.

• Multiply the quantity by the fractional equivalent of the percentage.

Example

Find the following.

(a) 15% of $340

(b) 20% of 45 litres

Solution

(a) 15% of $340 = $(15 × 340 ÷ 100) = $51

or

15% of $340 = $(0.15 × 340) = $51

(b) By using the known fact that 20% is equivalent to the fraction 1

5 the calculation

will be finding 1

5of 45 litres.

1

5 of 45 litres = 9 litres

1. Find 20% of $35 by:

(a) writing the percentage as a decimal first

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(b) writing the percentage as a fraction with a denominator of 100.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________



2. Calculate each of the following percentages. Show the calculations you used.

(a) 15% of 27 litres

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(b) 25% of 472 people at a concert

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(c) 9% of 15 km

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(d) 1.2% of $2300

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

3. A property had an area of 540 hectares. The house and sheds occupied 1.1% of theproperty and 28% was used for cattle grazing. A further 2% was fenced off to protectsome rare bush orchids and 17% was uncleared scrub. Find how many hectares wereused for each of the purposes.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________






4. Finding a percentage of a quantityagain

When you complete this section you should be able to:

solve harder problems involving finding a percentage of a quantity.

Warm-up 4

1. What is the next number? 2, 3, 5, 8, 12, __________

2. 42 7 = _________

3. The temperature was minus 2 degrees but it went up 4 degrees.

What is the new temperature? __________

4. Express the value of w as a fraction.

__________

5. 1

330 = _________

6. 5201 1000 = _________

7. 4 8.24 _________

8. Write 50% as fraction. __________

9. Find the next number. 75, 64, 53, ________

10. Determine the size of the missing angle.

__________

10

w

?40°



Review 4

Example

1. Find 2

3 of each of the following amounts.

(a) 69 kg (b) 18 hours (c) $22

2. Find 12% of each of the following amounts.

(a) 120 km (b) $5.00 (c) 95 litres of fuel

Solution

1. (a) (69 ×2

3) = (69 × 2 ÷ 3) kg = 46 kg

(b) (18 × 2 ÷ 3) hours = 12 hours

(c) $(22 × 2 ÷ 3) = $14.67 (to the nearest cent)

2. (a) (120 × 0.12) km = 14.4 km

(b) $(5 × 0.12) = $0.60

(c) (95 × 0.12) litres = 11.4 litres

1. Find 3

5 of each of the following amounts.

(a) 75 kg ________________________________________________________

(b) 105 minutes ________________________________________________________

(c) $13 ________________________________________________________

2. Find 7% of each of the following amounts.

(a) 500 hectares ________________________________________________________

(b) 56 kg ________________________________________________________

(c) $4.20 ________________________________________________________

Notice that in the money calculation the answer has been rounded to the nearest

cent.



Focus problem 4

Lance wants to buy a new bike but the shop says he needs to make a

deposit of1

333 % of the asking price of $264 before they order it from

the manufacturer.

How can Lance find 1

333 % of the price?

It is hard to write that percentage as a fraction with a denominator of 100. The decimal value is a repeating decimal so you can't enter that easily on the calculator without doing some rounding.

1. How would you suggest he calculates the deposit?

_____________________________________________________________________

_____________________________________________________________________

_____________________________________________________________________

2. Now think back to earlier work. Where did you see 1

333 % before?

_____________________________________________________________________

3. What is the simple fraction equivalent to 1

333 % ? __________

That is a good fraction equivalent to remember along with some others so that you can work quickly when finding percentages of quantities.

4. Complete this table of common fraction and percentage equivalents.

Percentage Fraction

100% 1

10% 1

10

5%

25%

1

8

1

3



5. Once you know 1

8 you may be able to recognise other relationships with fractions with a

denominator of 8. This should make some of your calculations easier to complete.

Make use of the percentage equivalent of 1

8to find the percentage equivalent to each of

the following fractions.

(a)3

8 _________________

(b)5

8 _________________

(c)7

8 _________________

6. You should now recognise that 1 1

3 333 % .

(a) What will be the percentage equivalent to 2

3? __________________

(b) What will be the percentage equivalent to 4

3? ___________________

7. How much was the deposit in dollars, that Lance had to make for the bike he wanted?

______________________________________________________________________

______________________________________________________________________





If the percentage required is over 100% it means there has been a percentage increase from the original value.

Example

Calculate:

(a) 1

333 % of 45 kg

(b) the value of an antique originally valued at $40 if it increased to 120% of the originalvalue.

Solution

(a) 1

333 % of 45 kg =

1

3 × 45 kg = 15 kg

(b) 120% of $40 = $40 1.2 $48

The value of the antique is now $48.

Check:

100% of $40 = $40

20% of $40 = 20 × 40 ÷ 100 = $8

120% of $40 = $40 + $8 = $48

1. Without the use of a calculator find 1

333 % of the following.

(a) 6 kg ____________________________________________________________

(b) 36 litres ____________________________________________________________

(c) 24 hours ____________________________________________________________

(d) 336 km ____________________________________________________________

2. A shop reduced all their prices by 10%. One jacket was originally priced at $60.

(a) Calculate the reduction in price. _________________________________________

(b) Calculate the reduced price. _________________________________________

(c) Calculate 90% of $60. _________________________________________

(d) Explain the relationship between your answers to (b) and (c).

_______________________________________________________________________

_______________________________________________________________________

1 133 %

3 3



3. A shop wanted to increase its prices by 15% to increase its profits.

(a) Calculate the price increase on a shirt priced at $80. __________________________

(b) Calculate the price of the shirt after the increase. ____________________________

(c) Calculate 115% of $80. ________________________________________________

(d) Explain the relationship between your answers to (b) and (c).

_______________________________________________________________________

_______________________________________________________________________

4. Another shop discounted all jacket prices by 15% and shoe prices by 22%. Whatpercentage of the original price will jackets and shoes be sold for?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

5. Due to a new government tax all the books on sale at a book shop had to have the priceincreased by 10% and all other items increased by 5%.

What percentage of the original prices will the new prices be?

_______________________________________________________________________

_______________________________________________________________________

6. Find the new value in each of the following cases.

(a) A person who weighed 90 kg reduced their weight by 15%.

_______________________________________________________________________

(b) During a rainstorm a water tank that had 800 litres in it had a 23% increase in itswater storage.

_______________________________________________________________________

(c) A concert venue had seating for 2600 and then it was extended to increase theseating capacity by 20%.

_______________________________________________________________________

(d) A metal bar that was 15 metres long contracted by 2.4% when placed in a freezer.

_____________________________________________________________________




5. Writing one quantity as apercentage of another

When you complete this section you should be able to:

write one quantity as a percentage of another.

Warm-up 5

1. Circle the triangular number. 4, 5, 6, 7, 8

2. 34 + 9 = _________


q = __________

4.1 2

5 5 __________

5. What is an eighth of 24? ___________

6. 6 m = __________ cm

7. 12 – 3 2 = _________

8. Write 1

10as decimal. ___________

9. Write the next number. 9.3, 8.9, 8.5, 8.1, ________

10.

At what point is the truck? __________

x1 2 3 4 5

y

12345

-6-8 -4 0q



Review 5

Example

Write each of the following as a fraction of $20.

(a) $10 (b) $2 (c) 40 cents (d) $17.50

Solution

(a)$10 1

$20 2 (b)

$2 1

$20 10

(c)$0.40 4 1

$20 200 50 (d)

$17.50 175 7

$20 200 8

1. Write each of the following as a simplified fraction of 36 kg.

(a) 18 kg ____________________________________________________________

(b) 4 kg ____________________________________________________________

(c) 22 kg ____________________________________________________________

2. Write 45 cm as a simplified fraction of the following.

(a) 55 cm ____________________________________________________________

(b) 81 cm ____________________________________________________________

(c) 1 metre ____________________________________________________________

Notice that the answer is a fraction and the units are

no longer used.



Focus problem 5

Lena is a keen tennis player. She has recently taken part in a number of tournaments.

In the first match on Saturday she counted that she served 100 first serves and that she got 53 of them in.

(a) What fraction of her first serves did she get in?

_____________________________________

(b) What percentage of her first serves did she get in during that match?

_____________________________________

In the second match she only got 36 first serves in out of a total number of 66 first serves.

(c) What fraction of her first serves did she get in during the second match?

_____________________________________

(d) What percentage of her first serves did she get in during the second match?

_____________________________________

(e) In which match was she serving better?

_____________________________________

(f) Why was it useful for Lena to know the percentage of first serves in rather than the actualnumber?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________





To write one quantity as a percentage of another first write the relationship as a fraction then convert the fraction to a percentage. The quantities must be in the same units.

Example

What percentage of 4 metres is each of the following?

(a) metres

(b) 40 cm

(c) 5 cm

Solution

(a)3

(3 4 100)% 75%4

(b)40 1

10%400 10

(c)15

(15 400 100)% 3.75%400

1. Write $60 as a percentage of the following.

(a) $100 ____________________________________________________________

(b) $1000 ____________________________________________________________

(c) $200 ____________________________________________________________

(d) 540 ____________________________________________________________

2. The apples in the orchard needed picking. Maria, Kevin and Helen offered to help outafter they finished their homework. Maria picked 38 kg, Kevin picked 27 kg and Helenpicked 15 kg of the apples.

What percentage of the total did each of them pick?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(b) was easy becauseyou should already

know that 1

10%.10



3. A 500 mL drink mix had 380 mLof water, 30 mL of flavouring and the remainder wasskim milk. Find what percentage of the drink each ingredient represents.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

4. The price of a mobile phone case increased from $16 to $22.

(a) What was the actual increase? ________________

(b) Find the increase as a percentage of the original price.

_______________________________________________________________________

_______________________________________________________________________

5. Find the reduction in price of each of the items for sale as a percentage of the originalprice shown below.

$15 caps drastically

reduced to $9

Trainers were $48

now just $32

Model planes

$28

$18

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________




It would be a really good time to summarise everything you have learned

about using percentages. This will be useful for solving the word problems in

the next two sections.



6. Problem solvingWhen you complete this section you should be able to:

solve simple problems involving percentages.

Warm-up 6

1. 10.8 10 = _________

2. 47 – 28 = _________

3. The temperature is minus 1 degree.

How much will it need to increase to get to 3 degrees? _________

4.7 3

8 8 _________

5. 1

816 = _________

6. 0.9 g = _________ mg

7. 10 5 2 = _________

8. Write 0.5 as a percentage. __________

9. Write the next number. 1 3 1

4 4 22 , 2, 1 , 1 ________

10. Determine the probability the spinner will land on a 2.

Express your answer as a percentage.

_________

1

23

4



Review 6

Example

(a) Write 35% as a simplified fraction and as a decimal.

(b) Find 35% of $40.

(c) If a price was reduced by 35% what percentage of the original price is the new price?

(d) What percentage of $70 is $14?

Solution

(a)35 7

35%100 20

and 35% = 0.35

(b) 35% of $40 = 0.35 × $40 = $14 or1

735% of $40

20 $ 40

2$14

(c) New price is (100 – 35)% or 65% of the original price.

(d) Percentage = 14

( 100)% 20%70

1. (a) Write 60% as a simplified fraction and as a decimal.

_______________________________________________________________________

(b) Find 60% of 900 kg.

_______________________________________________________________________

(c) If a seating capacity was increased by 60% what percentage of the original seating isthe new capacity?

_______________________________________________________________________

(d) What percentage of 60 kg is 42 kg?

_______________________________________________________________________

_______________________________________________________________________

2. Find 1

212 % of the following.

(a) 80 kg ________________________________________________________

(b) 9.6 metres ________________________________________________________



Focus problem 6

The local shire council decided to increase the rates by 10%.

The Warner family had a rates bill of $1200 before the increase.

What will their new rates bill be?

_________________________________________________________________________

_________________________________________________________________________

Because the Warner family paid their rates within the first week of receiving the notice they got a discount of 10% of the total.

How much will they have to actually pay?

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

You should have found that the second answer is not equal to the $1200 that you started with. Write an explanation of why this has happened.

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________

_________________________________________________________________________





1. On Friday Jimmy calculated that he had spent 8 hours sleeping, 6 hours at school, 3hours on the computer, 2 hours eating and the remainder at basketball training. Whatpercentage of his day did he spend on each activity?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

2. (a) The main road builder had completed 45 km of a new section of highway. It isplanned to increase this by 12% next year. How long will the section of highway be altogether after they have extended it?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(b) A major veterinary chain has 325 employees. Due to cost cutting they have toreduce that number by 8%. How many employees will they have after the reduction?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

When solving word problems you need to: Read the question carefully.

Decide on the methods you will need to use.

Show as much working as possible to makeyour calculations clear.

Write only mathematically correct statements.

Include words to explain your answers.



3. Stamp duty is a government tax that is collected when buying and selling large items likecars, houses and land. In Western Australia in a particular year the stamp duty for buyinga car was listed as 2.75% for a value from $0 to $25 000.

Find the amount of stamp duty payable for cars with the following values.

(a) $2500 ____________________________________________________________

(b) $12 000 ____________________________________________________________

(c) $24 999 ____________________________________________________________

Cars with a value over $50 000 were considered to be in the luxury market and the stamp duty was 6.5% of their value.

Find the amount of stamp duty payable for cars with the following values.

(d) $54 000 ____________________________________________________________

(e) $140 000 ____________________________________________________________

(f) $550 000 ____________________________________________________________

4. The stamp duty calculations on cars in NSW at the same time were 3% for the first

$45 000 of value then a further 5% for any value above $45 000.

Calculate the stamp duty in NSW for the cars with the following values.

(a) $24 999 ____________________________________________________________

(b) $140 000 ____________________________________________________________

5. Compare your answers from question 4 to (c) and (e) in question 3. Comment onwhich state it was better to buy a car in at that time.

_______________________________________________________________________

_______________________________________________________________________






7. Solving harder problemsWhen you complete this section you should be able to:

solve harder problems involving percentages.

Warm-up 7

1. 44.5 10 = _________

2. 17 5 = _________

3. The temperature is 8 degrees.

How much will it need to decrease to get to minus 2 degrees? _________

4.6 1

10 2 _________

5. 1

436 = _________

6. 0.4 L = ________ mL

7. (12 – 5) 3 = _________

8. Find 10% of $60. ____________

9. Describe the rule for the following pattern.

5, 10, 15, 20, 25, …

________________________________________________________________

10.

Determine the size of the missing angle.

_________

62°?



Focus problem 7

A retailer pays $45 each for some new MP3 players from the manufacturer.

She then adds a 32% profit margin to the price before advertising them.

Raylene comes to buy one and finds that they have been discounted by 10%. How much does Raylene have to pay? Show your working and reasoning in the space below.

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

Terry wants to buy one too. He has saved up $48.50. His mum said that if he had saved at least 90% of the sale price she would pay the rest. Will Terry be able to buy the MP3 player?

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________

__________________________________________________________________________





1. Increase the prices of the following items by 15% and then find the sale price if adiscount of 12% of the new price is given.

(a) $200 watch

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

(b) $36 backpack

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

(c) $15 book

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

2. Explain why the answer to each of the parts in question 1 cannot be simply found bysaying: 15% – 12% = 3% so find a 3% increase on the original price.

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________

3. The price of a music DVD was $25 but it was on sale for $17.50. What was thepercentage discount?

__________________________________________________________________

__________________________________________________________________

__________________________________________________________________



4. A new chocolate bar is advertised as being 20% bigger for the same price. If the originalbar was 120 g, what is the weight of the new one? Include a check to be sure youranswer is correct.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

5. The Brooks family left for their holidays. They had to travel 948 km altogether.

(a) They hoped to travel 35% of the distance on the first day. How far should that be?

_______________________________________________________________________

_______________________________________________________________________

(b) The second day they wanted to travel a further 237 km. What percentage of the totaldid they want to travel on the second day?

_______________________________________________________________________

_______________________________________________________________________

(c) What percentage of the trip did they plan to cover on the third (and last) day?

_______________________________________________________________________

_______________________________________________________________________

6. Jean had 19 pens and six of them were red. Frank had 25 pens and seven of themwere red. Who had the larger percentage of red pens? Show the working you need tofind the answer.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________




8. Summary A percentage is equivalent to a fraction with a denominator of 100.

1 whole = 100

100%100

To change a decimal to a percentage multiply the decimal by 100 and add a percentsign to your answer.

To change a fraction to a percentage find the decimal equivalent by dividing thenumerator by the denominator then multiply the decimal by 100 and add a percentsign to your answer.

To change a percentage to a fraction write the number with a denominator of 100then simplify.

To change a percentage to a decimal divide the percentage number by 100, that is,move the decimal point two places to the left.

To find a percentage of a quantity choose one of the following.

Multiply the quantity by the percentage number and then divide by 100. This is the same as multiplying by the fraction equivalent with a denominator of 100.

Multiply the quantity by the percentage written as a decimal.

Multiply the quantity by the fractional equivalent of the percentage.

To find a percentage increase of a quantity choose one of the following.

Find the increase and add it to the original value.

Add the percentage to 100% and then calculate the increased percentage of the quantity.

To find a percentage decrease of a quantity choose one of the following.

Find the decrease and take it from the original value.

Take the percentage from 100% and then calculate the decreased percentage of the quantity.

To write one quantity as a percentage of another first write the relationship as afraction then convert the fraction to a percentage. The quantities must be in the sameunits.

When solving word problems you need to do the following.

Read the question carefully.

Decide on the methods you will need to use.

Show as much working as possible to make your calculations clear.

Write only mathematically correct statements.

Include words to explain your answers.





9. Review tasksThe following tasks will assist you to consolidate your learning and understanding of the concepts introduced in this resource, and assist you to prepare for assessments.

Task A

Name: _____________________________ Suggested time: 45 minutes

Actual time taken: __________

Instructions

Complete this work on your own.

You may use a calculator, but show how you got your answer.

Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

1. Complete the following.

(a) Write 0.065 as a percentage.

_____________________________________________________________________

(b) Find the percentage equivalent to 5

.16

_____________________________________________________________________

(c) Write each of the following percentages as a simple fraction.

(i) 25% __________ (ii) 10% __________

(iii) 1

212 % __________ (iv)

1

333 % __________

(d) Find the decimal equivalent to 1.2%. ___________________________________

2. Find the following values.

(a) 14% of 25 km

_____________________________________________________________________

_____________________________________________________________________



(b) 1

333 % of 42 litres

_____________________________________________________________________

_____________________________________________________________________

3. (a) The price of a pair of shoes was increased by 5%.

(i) What percentage of the original price is the new price? __________________

(ii) What is the new price of the shoes if the original price was $120?

____________________________________________________________________

____________________________________________________________________

(b) The capacity of a water tank was reduced by 3.5% because of silt in the bottom ofthe tank.

(i) What percentage of the capacity remains? ___________________

(ii) What is the capacity of the tank now if it originally held 6000 litres of water?

____________________________________________________________________

____________________________________________________________________

4. The birth weight of a baby was 3.2 kg. After one month the baby weighed 4 kg. Whatpercentage of its birth weight was the increase in the first month?

_______________________________________________________________________

_______________________________________________________________________

5. Last year the population of the town of Needham was 5650 people. This year thepopulation increased to 6200 due to the opening of a new research farm. Theneighbouring town of Kilton had its population increase from 4225 to 4800 at the sametime.

Which town had the bigger percentage increase in population? Show all necessaryworking.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________



6. The following table shows the suggested number of grams of protein youngpeople should be consuming per day on a healthy diet.

Ages Grams of protein

Males 9 to 13 35

Males 14 to 18 52

Females 9 to 13 34

Females 14 to 18 45

Edward is a 12 year old male. His protein intake on Saturday was from a bowl of cereal (3 g) with milk (6 g) for breakfast and a burger (28 g) for lunch.

Leith is a 15 year old female. Her protein intake on Saturday was from a boiled egg (6 g) with a slice of toast (2 g) for breakfast and salad with cheese (14 g) for her

lunch.

What percentage of their daily protein requirements have they already met? Approximately what percentage of the suggested protein requirement will they need to eat for dinner to meet the requirement?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________



Task B

Name: _____________________________ Suggested time: 40 minutes

Actual time taken: __________

Instructions

Complete this work on your own.

You may use a calculator, but show how you got your answer.

Attempt every question. Take as long as you need and record the time in the space provided above after you have finished.

1. Percentages are used by banks to calculate interest. You can get a percentage of yoursavings paid to you in interest for allowing the bank to use your money.

One type of bank account is a term deposit. You can put a sum of money in there and getpaid the interest at the end of a chosen time period.

The interest can be paid to you or put back into the account to increase the amount in theaccount balance.

Annie has put $10 000 into a one year term deposit at an interest rate of 5%.

(a) How much interest will she get at the end of the first year?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

(b) If she decides to pay the interest back into the account what will her new accountbalance be?

_______________________________________________________________________

Annie continues with this new amount for another one year term deposit at an interest rate of 5%.

(c) How much will the interest be for the second year?

_______________________________________________________________________

_______________________________________________________________________



(d) If she continues doing the same thing for another two years, how much will heraccount balance be after four years? Show your working.

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

(e) How much interest has Annie gained altogether over the four years?

_______________________________________________________________________

2. Many items that are bought, especially for businesses, suffer from depreciation. Thismeans that the value at the end of the year has decreased by a certain percentage of thevalue at the beginning of the year.

A company buys a computer for $500 and the depreciation is set at 10%, for taxpurposes. This means that the value of the computer at the end of the year is 10% lessthan the value at the start of the year.

Work through the necessary steps to find out how much the computer is worth after threeyears.

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________



_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________

_______________________________________________________________________

______________________________________________________________________



Self-evaluation task Please complete the following.

How well did you manage your own learning using this resource?

Always Usually Rarely Not sure

Each section took approximately 45 minutes to complete.

I needed extra help.

I marked and corrected my work at the end of each section.

I made the journal entries and summaries when asked.

I have kept to my work schedule.

How much mathematics have you learnt using this resource?

Always Usually Rarely Not sure

Understanding I can recognise the equivalent fractions, decimals and percentages.

I understand how to calculate a percentage of a quantity.

Fluency

I can write one quantity as a percentage of another.

I can calculate a percentage of a quantity

Problem Solving I can write the correct calculations needed to solve a word problem.

I can solve word problems involving percentages.

Reasoning

I can choose the easiest method to solve a problem.

I can compare percentage changes and decide on the best.



Write a list of topics for which you need additional assistance. Discuss these with your teacher.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________



Solutions

1. What is a percentage?

Solutions to Warm-up 1

1. Yes, 4 is a factor of 20.

2. 12

3. a = (-6)

4.6

6 should be circled.

5. 9

6. 1.7

7. 27.0 or 27

8.1

3

9. 25

10. (3, 4)

Solutions to Review 1

1. (a) 9

100

(b)88

100

(c)20

100

(d)24

100

(e)6

100

2. (a) 5 87 92

100 100

(b)13 8 29 50

100 100

(c)51 15 36

100 100

(d)97

0.97100



(e)57

0.20 0.37 0.57100

(f)3 4 30 8 38

10 50 100 100 100

Solution to Focus problem 1

You were asked to complete the table of percentages for categories of basketball players.

Female players 40%

Male players 60%

Blond haired players 64%

Players without blond hair 36%

Players over 185 cm tall 15%

Players not over 185 cm tall 85%

Players wearing green uniforms 23%

Players wearing other colour uniforms apart from green 77%

All the players 100%

Solutions to Skills development 1

1. (a) 26%

(b) 100 – 18 – 26 – 41 = 15. 15% of the squares are uncoloured.

(c) 100 – 18 = 82. 82% of the squares are not red.

(d) 100 – (18 + 41) = 41. This means that 41% of the squares are not red or yellow.

(e) There are no purple squares so the percentage of purple squares is 0%.

2. (a)

(b) (i) Percentage shaded = 39%

(ii) Percentage unshaded = 61%



3. (a) 50 squares

(b) One square is 1

50 of the grid so it is 2%.

(c) (i) Percentage unshaded = 24 × 2% = 48%

(ii) Ten squares are shaded with \\\\, this is 20% of the grid.

(iii) Four squares are shaded with a solid colour. This is 8% of the grid.

(iv) Twelve squares are crosshatched which is 24% of the grid.

4. (a) 20

(b) 10

(c) 5

(d) 40

(e) 200

2. Fraction and decimal equivalents


1. Yes, 4 is a common factor.

2. 8

3. (-3) degrees

4.1 2

2 4

5. 5

6. 7

7. 2.1

8. 1.5

9. 3.5

10.3 1

6 2


1. (a) (i) 94

100

(ii)19

100

(iii)30

100



(iv)176 4 44

400 4 100

(v)7 4 28

25 4 100

(b) (i) 0.65

(ii) 0.34

(iii) 0.5625

(iv) 0.625


(a) (i) 1 25

25%4 100 and

270.27 27%

100

(ii) Smallest to largest percentages are 22%, 25% then 27%.

(ii) Hugh had the largest percentage.

(b) (i) 11

0.27540

(ii) To change 0.275 to a percentage you move the decimal point two places right(or multiply by 100). You get 27.5%.

(c)

Pocket money Amount spent on music Calculation Percentage

$10 $2.70 (2.7 ÷ 10 × 100) 27%

$25 $7.50 (7.5 ÷ 25 × 100) 30%

$32 $28.00 (28 ÷ 32 × 100) 87.5%


1. (a) 0.61 = (0.61 × 100)% = 61%

(b) 0.04 = (0.04 × 100)% = 4%

(c) 1.13 = (1.13 × 100)% = 113%

(d)2

5 = (2 ÷ 5 × 100)% = 40%

(e)5

16 = (5 ÷ 16 × 100)% = 31.25%

(f)2

75 = (2 ÷ 75 × 100)% = 2.666…%

2. (a) 45% = 45 9

100 20



(b) 0.5% = 0.5 1

100 200

(c) 125% = 125 25 1

1 1100 100 4

3. (a) 23% = 23 ÷ 100 = 0.23

(b) 1.5% = 1.5 ÷ 100 = 0.015

(c) 2

366 % 66.666... 100 0.666...

4. (a) All written as decimals gives 0.47, 0.45, 0.48.

In order the values are 0.45, 47%, 12

.25

(b) All written as percentages as follows.

19(19 20 100)% 95%

20 92% and 94.8% stay the same

15(15 16 100)% 93.75%

16 0.975 = (0.975 × 100)% = 97.5%

Arranged in order from the smallest to the largest: 92%, 15

16, 94.8%,

19

20, 0.975.

3. Finding a percentage of a quantity


1. 9 should be circled.

2. 63

3. a = (-2)

4.

5. 5

6. 9 – 4 = 5

7. 4.8

8. 20%

9. 7

81

10. Rectangle

0 1




1. (a) 1

8of 72 litres = (72 × 1 ÷ 8) litres = 9 litres

(b)3

4of 72 litres = (72 × 3 ÷ 4) litres = 54 litres

(c) 0.6 of 72 litres = (72 × 0.6) litres = 43.2 litres

(d) 0.09 of 72 litres = (72 × 0.09) litres = 6.48 litres

2. (a) 0.15 of $70 = $(70 × 0.15) = $10.50

(b)2

3of 51 kg = (51 × 2 ÷ 3) kg = 34 kg

(c) 0.06 of 150 min = (150 × 0.06) min = 9 minutes

(d)11

25of 94.5 km = (94.5 × 11 ÷ 25) km = 41.58 km


1. (a) 5

(b) 5 × 2 =10

(c) 5 × 3 =15

(d)5

100× 260 = 13

(e)5

100 × 480 = 24

2. (a) 2

(b) 4

(c) 1

(d) 11

3. (a) Multiply the number of trees by 5 then divide by 100 or divide the number of treesby 100 then multiply by 5.

(b) Multiply the number of trees by 2 then divide by 100 or divide the number of treesby 100 then multiply by 2.


1 (a) 20% = 0.2. 20% of $35 = $(0.2 × 35) = $7

(b) 20% of $35 = 20

$35 $(35 20 100) $7100

2. (a) (27 × 0.15) litres = 4.05 litres



(b)1

4724 people = 118 people

(c) (15 × 9 ÷ 100) km = 1.35 km

(d) $(2300 × 1.2 ÷ 100) = $27.60

3. House and sheds: 1.1% of 540 hectares = 5.94 hectares

Cattle grazing: 28 % of 540 hectares = 151.2 hectares

Rare orchids: 2% of 540 hectares = 10.8 hectares

Uncleared scrub: 17% of 540 hectares = 91.8 hectares

4. Finding a percentage of a quantity again


1. 17

2. 6

3. 2 degrees

4.2

3w

5. 10

6. 5.201

7. 2.06

8.1

2

9. 42

10. 140°


1. (a) (75 × 3 ÷ 5) kg = 45 kg

(b) (105 × 3 ÷ 5) mins = 63 mins

(c) $(13 × 3 ÷ 5) = $7.80

2. (a) (500 × 0.07) ha = 35 ha (b) (56 × 0.07) kg = 3.92 kg

(c) $(4.20 × 0.07) = $0.29 (to the nearest cent)


1. Answers will vary. Check with your teacher.

2. In Skills development 2 the percentage 1

333 % is written as a decimal.



3. 1 1

3 333 %

4.

Percentage Fraction

100% 1

10% 1

10

5% 1

20

25% 1

4

1

212 %

1

8

1

333 %

1

3

5. (a)1

2

312.5% 3 37.5% or 37 %

8

(b) 1

2

512.5% 5 62.5% or 62 %

8

6. (a)2 2

3 366 %

(b) 4 1 1

3 3 31 133 %

7. Deposit = 1

3$264 $88


1. (a) 1

3 of 6 kg = 2 kg

(b)1

3 of 36 litres = 12 litres

(c)1

3 of 24 hours = 8 hours

(d)1

3 of 336 km = 112 km



2. (a) 10% of $60 = 1

$60 $610

(b) Reduced price = $60 – $6 = $54

(c) 90% of $60 = 0.9 $60 $54

(d) The original price would be 100% of the price so when you reduce it by 10% it isthe same as finding 90% of the original price.

3. (a) 15% of $80 = 15

$80 $12100

(b) Increased price = $80 + $12 = $92

(c) 115% of $80 = 1.15 $80 $92

(d) The original price would be 100% of the price so when you increase it by 15% it isthe same as finding 115% of the original price.

4. Jackets will be sold for 85% of the original price because 100% – 15% = 85%.Shoes will be sold for 78% of the price because 100% – 22% = 78%.

5. New price for books = 110% of the old price.New price for other items = 105% of the old price.

6. (a) 15% reduction gives 85% of the original. 85% of 90 kg = 0.85 × 90 kg = 76.5 kg

(b) 123% of 800 litres = 1.23 × 800 L = 984 L

(c) 120% of 2600 seats = 1.2 × 2600 seats = 3120 seats

(d) 100% – 2.4% = 97.6%97.6% of 15 metres = 0.976 × 15 metres = 14.64 metres

5. Writing one quantity as a percentage of another


1. 6 is a triangular number (1 + 2 + 3 = 6)

2. 43

3. q = (-2)

4.3

5

5. 3

6. 600

7. 6

8. 0.1

9. 7.7

10. (5, 2)




1 (a) 18 1

36 2 (b)

4 1

36 9 (c)

22 11

36 18

2. (a) 45 9

55 11 (b)

45 5

81 9 (c)

45 9

100 20


(a)53

100

(b) 53%

(c)36 6

66 11

(d) (6 ÷ 11 × 100)% = 54.5454…% = 54.5% (1dp)

(e) She was serving better in the second match.

(f) With the percentages she could easily compare howwell she served in each match.


1 (a) 60

60%100

(b)6 0

100 06%

(c)60 30

30%200 100

(d)60

(6 54 100)% 11.11...% 11.1% (1dp)540

2. Total weight picked = (38 + 27 + 15) kg = 80 kg

Percentage Maria picked = (38 ÷ 80 × 100)% = 47.5%

Percentage Kevin picked = (27 ÷ 80 × 100)% = 33.75%

Percentage Helen picked = (15 ÷ 80 × 100)% = 18.75%

3. Percentage of water = 380

76%500

Percentage of flavouring = 30

6%500

Percentage of skim milk = 100% 76% 6% 18%

Note that a check is often a good idea in a question like this.

Note that in 2(c) the metre was changed to

100 cm.

Notice that if the percentage has a repeating decimal in

the answer it is usually rounded.

Unless the answers are really easy to find you should be using your calculator for these

answers.



Check: Skim milk is 90 mL because (500 – 380 – 30) mL = 90 mL.

Percentage of skim milk = 90

18%500

4. (a) Increase = $(22 – 16) = $6

(b) Increase as a percentage = 6

15100 % 37.5%

5. Reduction in price of caps as a percentage = 6

15100 % 40%

Reduction in price of trainers as a percentage = 1

3

16

48100 % 33 %

Reduction in price of model planes as a percentage = 10

28100 % 35.7% (1dp)

6. Problem solving


1. 108

2. 19

3. 4 degrees

4.4 1

8 2

5. 2

6. 900

7. 4

8. 50%

9. 1

41

10. 25%

Solutions to Review 6.

1. (a) 60% = 60 3

100 5 and 60% = 0.6

(b) 60% of 900 kg = 0.6 × 900 kg = 540 kg

(c) The new seating capacity is 160% of the old seating.

(d)42 7

70%60 10



2. (a) 1

2

112 %

8 and

1

8of 80 kg = 10 kg.

(b)1

8 of 9.6 metres = 1.2 metres


New rates bill: 110% of $1200 = 1.1 × $1200 = $1320

With a 10% discount the family will pay 90% of $1320.

90% of $1320 = 0.9 × $1320 = $1188

The 10% reduction for paying quickly was 10% of the larger amount, $1320, so it was more than 10% of $1200.


1. There are 24 hours in a day

Activity Working Percentage

Sleeping 8 1

24 3

1%

333

School 6 1

24 4 25%

Computer 3 1

24 8

1

212 %

Eating 2 1(1 12 100)%

24 12

1

38.333...% 8 %

Basketball training 5(5 24 100)%

24 20.8% (1dp)

2. (a) 12% increase mean that the total will be 112% of 45 km. 112% of 45 km = 1.12 × 45 km = 50.4 km

(b) After they have the reduction they will have 92% of the original number ofemployees.92% of 325 = 0.92 × 325 = 299 employees

3. (a) 2.75% of $2500 = 0.0275 × $2500 = $68.75

(b) 2.75% of $12 000 = 0.0275 × $12 000 = $330

(c) 2.75% of $24 999 = 0.0275 × $24 999 = $687.47 (to the nearest cent)

(d) 6.5% of $54 000 = 0.065 × $54 000 = $3510

(e) 6.5% of $140 000 = 0.065 × $140 000 = $9100

(f) 6.5% of $550 000 = 0.065 × $550 000 = $35 750

4. (a) 3% of $24 999 = $749.97



(b) 3% of $45 000 + 5% of $(140 000 – 45 000)

= 0.03 × $45 000 + 0.05 × $95 000

= $1350 + $4750

= $6100

5. Possible comment: Cheaper cars have lower stamp duty in Western Australia but it isbetter to buy expensive cars in NSW because the stamp duty is lower.

7. Solving harder problems


1. 4.45

2. 85

3. 10 degrees

4.6 5 1

10 10 10

5. 9

6. 400 mL

7. 21

8. $6

9. Add five to get the next number.

10. 62°


Advertised price for the MP3 player = 132% of $45

= 1.32 × $45

= $59.40

Raylene's price = 90% of $59.40 = $53.46

There are several ways of finding if Terry can buy one too. Get your working checked if it differs from the answer below.

The easiest way to find the answer is to find 90% of $53.46 and see if Terry's $48.50 is enough.

90% of $53.46 = 0.9 × $53.46 = $48.11 (to nearest cent). Terry will be able to buy the player as he has saved up at least 90% of the sale price.


1. (a) 115% of $200 = 1.15 × $200 = $23088% of $230 = 0.88 × $230 = $202.40



(b) 1.15 × $36 = $41.400.88 × $41.40 = $36.43 (nearest cent)

(c) 1.15 × $15= $17.250.88 × $17.25 = $15.18

2. The percentage discount is on a different amount than the percentage increase so theyhave to be calculated separately.

3. Discount = $25 – $17.50 = $7.50

Discount percentage = 7.5

25100 % 30%

4. The new chocolate bar will be 120% of the original.120% of 120 g = 1.2 ×120 g = 144 gThe new bar weighs 144 g

Check: 20% of 120 g = 1

120 g5 = 24 g and 120 g + 24 g = 144 g

5. (a) 35% of 948 km = 0.35 × 948 km = 331.8 km

(b) 237

948100 % 25%

They wanted to travel 25% of the trip on the second day.

(c) Percentage of the trip on the third day = (100 – 35 – 25)% = 40%

6. Percentage of red pens for Jean = 6

19100 % 31.6% (1dp)

Percentage of red pens for Frank = 7

25100 % 28%

Jean had the larger percentage of red pens.

Solutions to Review tasks

Solutions to Task A

1. (a) 6.5%

(b) 31.25% or 1

431 %

(c) (i) 1

4

(ii)1

10

(iii)1

8

(iv)1

3



(d)1.2

0.012100

2. (a) 14% of 25 km = 0.14 × 25 km = 3.5 km

(b) 1

333 % of 42 litres =

1

3 × 42 litres = 14 litres

3. (a) (i) 105%

(ii) 105% of $120 = 1.05 × $120 = $126

(b) (i) 96.5%

(ii) 96.5% of 6000 litres = 0.965 × 6000 litres = 5790 litres

4. Change in weight of baby = (4 – 3.2) kg = 0.8 kg

Percentage change = 0.8

3.2100 % 25%

The weight of the baby increased by 25% in the first month.

5. Change in population for Needham = 6200 – 5650 = 550

Percentage change = 550

5650100 % 9.7%

(1dp)

Change in population for Kilton = 4800 – 4225 = 575

Percentage change = 575

4225100 % 13.6%

(1dp)

Kilton had the bigger percentage increase in population. 6. Total of protein for Edward = (3 + 6 + 28) g = 37 g

Percentage of daily requirements = 37

35100 % 105.7%

(1dp)

Total of protein for Leith = (6 + 2 + 14) g = 22 g

Percentage of daily requirements = 22

45100 % 48.9%

(1dp)

Edward has already consumed more protein than the daily requirement. Leith will need to consume approximately 51.1% for dinner to meet the daily

requirement.

Solutions to Task B

1. (a) 5% of $10 000 = 0.05 × $10 000

= $500

Annie will get $500 interest at the end of the first year.

(b) New balance = $10 500

(c) 5% of $10 500 = 0.05 × $10 500

= $525

The interest for the second year is $525.



(d) New balance = $10 500 + $525 = $11 025

5% of $11 025 = 0.05 × $11 025

= $551.25

Balance for third year = $11 025 + $551.25 = $11 576.25

5% of $11 576.25 = 0.05 × $11 576.25

= $578.81 (nearest cent)

Final balance = $12 155.06 (nearest cent)

(e) Interest over four years = $12 155.06 – $10 000 = $2155.06

2. Year 1: Value at start of year = $500

10% of $500 = 1

10× $500

= $50

Value at the end of the year = $500 – $50

= $450

Year 2: Value at start of year = $450

10% of $450 = 1

10× $450

= $45

Value at the end of the year = $450 – $45

= $405

Year 3: Value at start of year = $405

10% of $405 = 1

10× $405

= $40.50

Value at the end of the year = $364.50

The computer will have a value of $364.50 after three years.


MATHSAC018 PERCENTAGES

ISBN: 9780730744382

Date post:	22-May-2020
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