4. fractions 2. displaying data

iii

Introduction

1. Whole numbers

Home page:

Can you beat this? 1

1.1 Number systems 2

Laugh Zone

51.2 Whole number

problems 6

Investigation:

Palindromes 91.3 Magic squares 101.4 Number pyramids and cross-number

totals 111.5 Estimating and rounding 121.6 Order of operations 16

Activity:

The four 4s puzzle; Puzzling year; What number am I? 18

Investigation:

The multiplication target game 191.7 Mental maths strategies 201.8 Number sentences 22

Investigation:

Odds and evens 24

Maths in Action:

Calculating the Great Wall 25

Chapter 1 review:

Personal Learning Activity 1

27

Review questions 1

27

2. Displaying data

Home page:

Counting the catch 31

2.1 Surveys and frequency tables 32

2.2 Averages 35

Laugh Zone

38

Investigation:

Using averages 39

Computer investigation:

Collecting statistics 412.3 Bar graphs 422.4 Line graphs 45

Maths in Action:

The mystery of the Incas 492.5 Divided bar and sector graphs 522.6 The best statistical graph 53

Investigation:

Surveys 54

Chapter 2 review:


55

Review questions 2

55

3. Number patterns

Home page:

The truth is out there in the numbers 59

3.1 Multiples 603.2 Divisibility 62

Activity:

What number am I? 63

3.3 Factors 63

Investigation:

The sieve of Eratosthenes 653.4 Prime and composite numbers 66

Investigation:

Goldbach’s conjecture 683.5 Square and cube numbers 69

Laugh Zone

71

Maths in Action:

Keeping it secret 723.6 Other special numbers 74


Fibonacci and other number patterns 77

Chapter 3 review:


78

Review questions 3

78

4. Fractions

Home page:

Fractions give you rhythm 81

4.1 Equivalent fractions 82

Investigation:

Fraction wall 84

4.2 Improper fractions and mixed numbers 85

4.3 Key percentages 864.4 Comparing fractions 874.5 Probability 894.6 Adding and subtracting fractions 91

Activity:

Unit fractions 94

Maths in Action:

Egyptian fractions 954.7 Multiplying fractions 97

Investigation:

Ideal fractions 1004.8 Dividing fractions 101

Activity:

What fractions are we? 102

Laugh Zone

103

Chapter 4 review:


104

Review questions 4

104

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5. Decimals

Home page:

Richter scale a bit shaky 107

5.1 Place value 108

Activity:

Donna’s decimal 110

5.2 Comparing decimals 110

5.3 Rounding off 1125.4 Converting decimals to fractions 1145.5 Addition of decimals 1155.6 Subtraction of decimals 117

Activity:

Ten dollars and eighty-nine cents 118

Laugh Zone

1195.7 Multiplication of decimals by whole

numbers 1205.8 Multiplication of decimals by multiples

of 10 1215.9 Multiplication of decimals by other

decimals 1235.10 Division of decimals by whole

numbers 125

Investigation:

Diving scores 1275.11 Division of decimals by multiples

of 10 1295.12 Division of decimals by other

decimals 130

Maths in Action:

Decimal drinks 132

Chapter 5 review:


134

Review questions 5

134

6. Measurement

Home page:

Speaking volumes about Archimedes 137

6.1 Metric units 138

Activity:

Metric measures 143

6.2 Perimeter 144

Laugh Zone

1466.3 Area 147

Activity:

Removing matchsticks 148

Investigation:

How many squares on a chessboard? 149

6.4 Area of a rectangle 1506.5 Area of a triangle 152

Maths in Action:

From fingers to feet to metres 154

6.6 Volume 157

6.7 Volume of rectangular prisms 1586.8 Time 160

Chapter 6 review:


162

Review questions 6

162

7. Relationships

Home page:

Bubble algebra bursts onto the scene 165

7.1 Algebra rules 1667.2 Finding a

formula 1687.3 Pronumerals 169

Laugh Zone

1717.4 Describing patterns algebraically 172

Investigation:

Cups and counters 175

Activity:

Cutting string 176

Activity:

The handshake problem 1777.5 Grid references 178

Investigation:

Chess piece tours 180

Maths in Action:

Where in the world? 1827.6 The Cartesian plane 185


Scatterplots 1867.7 Latitude and longitude 189

Chapter 7 review


191

Review questions 7

191

8. Angles

Home page:

In the steps of the dinosaur 195

8.1 Measuring angles 196

Activity:

Count the angles 200

8.2 Drawing angles 200

Investigation:

Dot paper angles 202

Maths In Action:

Billiard ball bounces 2038.3 Describing angles 205

Investigation:

Line designs in angles 2068.4 Complementary and supplementary

angles 208

Laugh Zone

210

Activity:

Ella’s angles 2118.5 Angles in a revolution 2118.6 Vertically opposite angles 214

Chapter 8 review


216

Review questions 8

216

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9. Equations

Home page:

Soaking up the gravy equation 219

9.1 Number sentences 220

9.2 Using pronumerals in equations 222

9.3 Using a flowchart 224

Activity:

The mindreader game 2259.4 Building expressions with

flowcharts 2269.5 Solving equations using

backtracking 227

Maths in Action:

Pedal-powered flight 231

Investigation:

Guess, check and improve 233


Solving equations using substitution 234

9.6 Solving problems with equations 235

Activity:

Algebraic puzzles 238

Laugh Zone

239

Chapter 9 review:


240

Review questions 9

240

10. Shapes

Home page:

Bringing shapes to life 243

10.1 Triangles 244

Investigation:

Polyiamonds 246

Investigation:

Angle sum in a triangle 247

10.2 Angle sum in a triangle 24810.3 Quadrilaterals 250

Activity:

Quadrilateral quandaries 25310.4 Angle sum in a quadrilateral 253

Laugh Zone

25610.5 Polygons 257

Investigation:

Angle sum in a polygon 26010.6 Compass constructions 262

Maths@Work:

Graphic designer 26510.7 Plane shapes with curves 26710.8 Transformations 269


Using Microworlds or LOGO 270

10.9 Solids 272

Chapter 10 review:


274

Review questions 10

274

Rich tasks

275

Workbook and textbook answers

280

Glossary and index

324

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31

he Tasmanian Aquaculture and Fisheries Institute is undertaking a statistically-

based research project to help predict the number of abalone at any particular time. Until now the catch limit has been calculated using imprecise methods, based on the reports by divers as to how many abalone are in an area. While divers’ reports will still be valuable information, the new project will use a capture sampling process to get some real data related to the abalone population. With this industry worth more than $?25 million per year to the Australian economy, it is easy to see why it needs to be managed well.

T

hi.com.au

e

Word search 2

w

HMZQX_ch02.fm Page 31 Thursday, September 14, 2006 2:10 PM

http://www.hi.com.au/mathszoneqld/Hotlinks1.asp#007

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In this topic we talk a lot about

data

. Data is simply information that has been collected in some way.

One of the easiest ways to

collect

data is to conduct a

survey

. In a survey only a fraction of the total

population

is questioned or observed as opposed to a

census

which involves all of the population. Having collected the data we must then present it in some way. A

frequency table

is often used for this.If we surveyed a class of Year 7 students about the number of brothers and

sisters they have, the results might look like this: 2, 3, 5, 0, 1, 1, 2, 0, 3, 1, 4, 0, 7, 1, 3, 2, 2, 2, 1, 1, 0, 1, 1, 2, 4.

To summarise the data, we first draw up a table with three columns. The first column shows what is being surveyed.The second column is the tally column, where we count the number of

times each category occurs. Notice that is used to represent the number 5.

The third column is the frequency column, where we enter the tallies as numbers. We can add up the frequency column to check that we did not miss any numbers.Frequency table:

Sometimes the data we collect is so spread out that we need to group the results so that we only have between five and ten rows in our frequency table. In the following data the values range from 40 to 84:73, 84, 68, 45, 52, 44, 45, 52, 66, 42, 43, 40, 53, 47, 82, 76, 42, 57, 65, 81, 80, 40, 56, 72, 74, 83, 41, 66, 76, 75, 68, 81, 82, 79, 58, 81, 78, 80, 78, 76.

We could use

class intervals

of 40–49, 50–59, 60–69, 70–79 and 80–89, or smaller ones of 40-44, 45–49, 50–54, 55–59, etc. up to 80–84.

Number ofbrothers and

sisters Tally Frequency

01234567

48632101

25


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DATA

33

Questions

7

–

10

can be done by the class as a whole with results being recorded on the blackboard.

7

Find out how many pets each of the students in your class owns by surveying your class. Draw up a frequency table of your results.

(a)

How many members of your class have no pets?

(b)

How many members of your class have one pet?

(c)

What number of pets do the majority of the members of your class have?

8

Find out how many hours of television each student in your class watches on a normal week night by surveying your class. Draw up a frequency table of your results. Round your answers to the nearest half hour.

(a)

How many of your class members watch no television on a normal week night?

(b)

How many of your class members watch less than 1 hour of television on a normal week night?

(c)

How many of your class members watch 3 or more hours of television on a normal week night?

(d)

How much television do most of your classmates watch on a week night?

(e)

Do you watch more or less television than the majority of the members of your class? Or do you watch about the same amount as the majority of the members?

9

Find out the favourite school subject of each of the students in your class by surveying your class. Draw up a frequency table of your results.

(a)

What is the favourite subject of the members of your class?

(b)

What is the second favourite subject of the members of your class?

(c)

How many students had no favourite subject?

10

Find out the hair colour of each of the students in your class by surveying your class. Draw up a frequency table of your results.

(a)

What are your categories?

(b)

Why was this harder to do than the other surveys?

(c)

What was the most common hair colour?

(d)

What was the second most common hair colour?

exercise 2.1 Surveys and frequency tablesEx 2.1 Q1–6w Worksheet C2.1e hi.com.aue


http://www.hi.com.au/mathszoneqld/Hotlinks1.asp#008

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11

Draw up a frequency table that has a total of 20 pieces of data shared between the possible outcomes 1, 2, 3, 4, 5, and 6. No individual frequency is to be greater than 4.

12

Look at the following data set:20 32 42 53 63 21 33 42 53 6522 34 43 57 61 24 34 48 50 6124 39 40 50 62 29 31 41 52 62

(a)

Draw up a frequency table using class intervals of 20–29, 30–39, etc.

(b)

Describe what you find in your frequency table.

13

Sixty 12-year-old students were tested to find their pulse rate when resting. The following figures were obtained (beats per minute):

70 68 68 76 79 68 76 55 55 6060 94 72 65 64 93 71 62 67 8276 65 77 82 81 59 74 74 67 6878 76 63 82 81 82 74 70 66 6384 81 69 84 79 71 70 54 68 6478 58 84 61 75 72 73 71 91 66

Draw up a frequency table using class intervals of 51–60, 61–70, etc.

14

The following is a list of birth weights (in grams) of 30 babies:2900 2805 2925 2010 2720 3125 2670 2555 2963 29723151 2515 3529 3098 2126 2417 3000 3254 2997 29862719 2842 3519 3509 3218 3002 2437 2222 2019 2113

Draw a frequency table for this data using the class intervals 2000–2249, 2250–2499, etc.

11


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DATA

35

One of the most important things that statistics helps us to do is to work out

averages

. The use of the word ‘average’ in statistics is a bit different from in everyday life. In mathematics, we have to be more precise about the meaning of words.

Usually when people speak about averages they mean one value that is typical or representative of a whole group of values. Often we have a vague idea that the average is somewhere in the middle of the group of values.Look at the following maths test results (out of 10) of a group of seven students:

9, 4, 5, 7, 8, 7, 2Just by looking at the results (don’t do any

calculations), what do you think the average result would be?

There are actually

three

different types of averages in statistics, and they are calculated in different ways.

Mean

This is the type of average you have possibly come across before. To find the

mean

, we add up all the values and divide the sum by the number of values.

Median

The

median

is the

middle

value when the data is placed in ascending or descending order. If there is an even number of values, we use the mean of the middle two values; that is, we add them together and then divide by 2.

Mode

The

mode

is the value that occurs

most often

. For the data 9, 4, 5, 7, 8, 7, 2:

mean median mode

First find the sum:9

+

4

+

5

+

7

+

8

+

7

+

2

=

42Then count the values: 7

mean

=

=

=

6

First place the values in order, then find the middle one:

2 4 5 7 8 9median

=

7

There are more 7s than any other value so:mode

=

7

On average I’m warm.

eTutoriale

eTutoriale

eTutoriale

sum of valuesnumber of values----------------------------------------------

427-----

7


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4

(a)

Write down five test results that have their mean, median and mode all equalling 6. Don’t have the results all the same.

(b)

Write down five test results where the mean and the median are 7 and the mode is 9.

(c)

Write down five test results where the mean is 7 and the median and mode are 6.

5

The Korea–Japan 2002 World Cup in soccer was played in more stadiums than ever before. The following table shows the city and capacity of the different stadiums.

Sometimes a set of results can have more than one result which occurs most frequently. If there are two values which occur most frequently we say the results are bimodal. If there are more than two values which occur most frequently we usually say the results have no mode.

exercise 2.2 AveragesEx 2.2 Q1–3w Worksheet C2.2e Worksheet C2.3e e Interactive

eTestere eQuestionse eQuestionse eQuestionse4


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37

(a)

Find the

(i)

mean and

(ii)

median capacity for the stadiums in Korea.

(b)

Which stadium is closest to the mean capacity in Korea?

(c)

Find the

(i)

mean and

(ii)

median capacity for the stadiums in Japan.

(d)

Which stadium is closest to the mean capacity in Japan?

(e)

How do the sizes of the stadiums in the two countries compare?

(f)

Find the

(i)

mean and

(ii)

median capacity for all twenty stadiums.

(g)

Which stadium is closest to the mean capacity for all twenty stadiums?

6

A matchbox indicates that the box contains 50 matches. The matches in each of 30 boxes were counted and the results obtained appear in the following table.

(a)

What is the modal number of matches per box?

(b)

Write the 30 values out in order from smallest to largest, i.e. 48, 48, 48, 49, … 52, and hence find the median number of matches per box.

(c)

What is the total number of matches in the 30 boxes?

(d)

Find the mean number of matches per box.

7

A small company has a manager, an assistant manager, two office workers and ten factory workers. The manager is paid $70 000 per year, the assistant manager $55 000, the office workers $35 000 each, and the factory workers $32 500 each.

(a)

Find the mean, median and mode for the annual income of all the people in the company.

(b)

Suggest a use for each of the three values obtained.

(c)

Which figure do you think does the best job of describing the ‘average’ income at the factory?

Explain your answer.

Korea Capacity Japan Capacity

SeoulIncheonSuwonDaejeonDaeguJeonjuGwangjuUlsanBusanSeogwipo

63 96152 17943 18840 40768 01442 39142 88043 55055 98242 256

SapporoMiyagiNiigataIbarakiSaitamaShizuokaKobeOsakaOitaYokohama

42 00049 00042 30042 00063 00050 60042 00050 00043 00070 000

Number of matches

48 49 50 51 52

Number of boxes

3 7 17 2 1

Homework 2.1e


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The data displays below have many numbers missing. Find the number that each letter represents and then arrange the letters in the order given by the corresponding answers to find the cartoon caption. Show all working in your book.51, 2, 65, 4, 62, 3, 2, 54, 27, 39, 42, 17, 24,28, 37, 18, 36, 46, 12, 22, 36, 39, 35, 28,39, 47, 17, 24, 5, 53, 1, 57, 2, 59, 6Use the numbers above to complete the following frequency table.

Score Frequency

0–9R–19L–A

30–3940–4950–5960–69

KWCUOTY

Total 35

4 35 20 20 2 3 7 81 3 27 2 27 29 5 40 13

40 3 27 15 4 3 10 8 5 3 1 135 360 40 5

,

1 3 , 35 5 4 3 7 20 81 1 ’ 5 9 15

10 35 36 40 5

,

4 15 20 20 , 2 3 7 6 3 7 20 81 5 10 2

For the data 9, 9, 13, 20, 24

mean = E median = S mode = B

For the data 27, 27, 27, 39, 41, 100, 187, 200

mean = D median = H mode = M

For the data 1, 1, 5, 10, 60, 69, 70, 72

mean = G median = I mode = N

‘?’

‘ ’

’‘ ’


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2 ● displaying DATA 39

Using averagesYou will need: A calculator. Round off your answers to 2 decimal places.

1 Brookvale High’s mixed basketball team has had the following results in their matches for the season so far.

The next game is a crucial one, and it is against the traditional rival East Brookvale HS. Suppose you are the team statistician, and the coach has asked you to calculate some averages.

When people talk about averages to do with sport, they are nearly always talking about the mean.(a) What is the average Brookvale score?(b) What is the average score of Brookvale’s opponents?(c) What is the range of Brookvale’s scores? The range is the difference between the

largest value and the smallest value.(d) What is the range of Brookvale’s opponents’ scores?(e) The coach wants you to make a prediction, on the basis of your calculations, about the

next game. What do you think the final score against East Brookvale will be?(f) What are some problems with making these sorts of predictions?

2 Your coach has a theory that your team only wins if they play a team which is at least 5 cm shorter on average. He asks you to test his theory.

The Brookvale High basketball players, together with their heights, are shown opposite.(a) What is the mean height?(b) What is the median height?(c) What is the mode height?(d) Which one of the three types of

averages is probably the wrong one to use? Why?

(e) What is the range of the heights?

Brookvale HS 35 State Hill SC 12

Brookvale HS 17 Kingston HS 40

Brookvale HS 33 Wakefield HS 30

Brookvale HS 35 Fullerton SC 6

Tim Sue Rajina Jeff

163 cm 156 cm 156 cm 160 cm

Leanne Sam Chuck ‘Magic’

162 cm 159 cm 160 cm 156 cm


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3 You obtain the statistics of the teams you have played against so far. They are given in the table below.

(a) Calculate the means of the heights of each of these teams.(b) Compare these with the mean you worked out in Question 2(a). Do your results back

up your coach’s theory?

4 Your coach wants you to predict whether you will win your next game if his theory is true. You have managed to obtain some information on East Brookvale’s team. The East Brookvale players, together with their heights, are shown in the next diagram.(a) What is the mean height?(b) What is the median height?(c) What is the mode height?(d) Why is there such a big difference between the mean and the median?(e) What is the range of the heights? How does the range compare to Brookvale’s range?(f) Compare the means of Brookvale’s and East Brookvale’s players. According to your

coach’s theory, will Brookvale win?(g) You have to decide which average to use. Why is the average you choose crucial in this

case? Which one do you decide on? Why? What does this depend on?(h) Suppose you find out the day before that ‘Tiny’ is injured and won’t be playing. He is

going to be replaced with a player who is 151 cm tall. What do you predict will happen?

Team Heights (cm)

State Hill SCKingston HSWakefield HSFullerton SC

148, 151, 152, 160, 148, 148, 149, 158155, 158, 156, 155, 157, 160, 162, 161150, 146, 159, 152, 144, 158, 157, 158143, 154, 148, 151, 153, 152, 156, 148

Gerald Fleur Anders Con Seline John Christine ‘Tiny’149 cm 154 cm 152 cm 151 cm 154 cm 153 cm 164 cm 179 cm


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Collecting statisticsLet’s collect some data based on a survey to the question ‘In which month of the year were you born?’

1 Survey your classmates (or another group) and enter the results in column A of a spreadsheet. To make things simpler, enter only the number of the month, i.e. 9 for September. We’ll assume you have no more than 40 responses. Add the following headings to the sheet.

2 In E4 enter the formula =AVERAGE(A1:A40). In E5 enter =MODE(A1:A40) and enter similar formulae into E6, E7 and E8. In E9 enter =COUNT(A1:A40), which will count how much data you have collected.If you were to change the question and/or the data, these statistics would be produced for the new data.

3 To complete the frequency distribution table requires a new formula, COUNTIF. In H6 enter the formula =COUNTIF($A$1:$A$40,G6). This formula looks in cells A1 to A40 and COUNTs them IF they are like G6. That is, it will count how many 1s it finds. Go back to H6 and move the mouse to the bottom right-hand corner of the cell. The cursor should change into the black cross called the ‘fill handle’. Drag it down to H17 and the spreadsheet should complete the table. Why use the $ signs? They are to ‘fix’ where the data is.

4 Change the survey question to collect some other numerical data and enter it. For example, which date in the month were you born? (Answer could be 27, for instance.)


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Personal Learning Activity 21 Write a list of the different types of graphs in this chapter. Think of at least

one example of when you could use each one, and give a reason for your choice.

2 Your friend wants to know what mean, median and mode are, how to find them and why they are all called ‘average’. Explain the answers to their questions. You could use numbers from an exercise in this chapter or make up your own numbers.

Review questions

7 A number of year 7 students were surveyed about their shoe size. The results were:

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

4 , 3, 4, 5, 5 , 2 , 2, 3 , 4 , 3, 1, 3 , 1 , 4, 3

(a) Draw up a frequency table to show this information.(b) Which shoe size occurred most frequently?(c) Which shoe size occurred least frequently?

8 A number of families were surveyed as to the number of TV sets in their house. The results were as follows.

(a) How many families were surveyed?(b) Find the mode of TV sets per family.(c) Write out the data as a list (that is, 0, 0, 0, 1, … 4) and hence find

the median number of TV sets per family.(d) Find the mean number of TV sets per family.

9 Twenty-five Year 7 boys and twenty-five year 7 girls had their height measured to the nearest centimetre. The results are as follows.Boys: 140 143 144 142 148 148 152 140 146 144

154 151 145 140 141 147 153 150 149 140147 152 148 151 149

Number of TV sets 0 1 2 3 4

Number of families 3 18 15 10 5

DIY Summary 2w Worksheet C2.8e

Review 2 Q1–6w2.1

12--- 1

2--- 1

2--- 1

2--- 1

2--- 1

2--- 1

2--- 1

2---

12--- 1

2--- 1

2--- 1

2--- 1

2--- 1

2--- 1

2--- 1

2---

2.2

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Girls: 142 151 157 138 145 148 143 141 146 154156 154 150 143 150 144 148 144 157 156146 157 154 145 152

(a) Draw up an ungrouped frequency table for the boys.(b) Find the (i) mode (ii) median and (iii) mean height for the boys.(c) Draw up an ungrouped frequency table for the girls.(d) Find the (i) mode (ii) median and (iii) mean height for the girls.(e) Describe the similarities and differences between the heights of the

boys and the girls.(f) Draw a combined ungrouped frequency table for boys and girls.(g) Find the (i) mode (ii) median and (iii) mean height for the

students.(h) Describe the height of the students.

10 Look at the following graph, which shows the average amount of rainfall (cm) in the city of Jakarta over one year.(a) What type of graph is this?(b) What is unusual about the position

of the vertical scale?(c) What is Jakarta’s second driest

month?(d) What is the highest average monthly

rainfall?(e) Which four months form the rainy season?

11 Look at the following graph, which shows the percentage of people who were out of work during the period from 1981 to 2000.

Average amount ofrainfall in Jakarta

cm35302520151050J F M A M J J A S O N D

2.3

2.4

1987 88Year

12

10

8

6

1981 821984 85 1990 91

% Unemployed

1993 941996 97

1999 00

Unemployment rate in Australia


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(a) What type of graph is it?(b) What does the jagged line on the vertical axis mean?(c) When did unemployment peak in the period from 1981 to 2000?(d) In what years was unemployment at its lowest?(e) What was the lowest rate of unemployment during the time shown

on the graph?(f) Approximately what percentage of people were unemployed in late

1996 and early 1997?

12 Look at the following information.

(a) Draw a bar chart showing the various expenditure types for dogs.(b) Draw a divided bar chart to show the total expenditure for the three

categories of pets.(c) In a sector graph that shows the various expenditure types for cats,

what sector would be bigger than half the pie?(d) Explain why a line graph could not be used for any of these sets of

data.

How much Australians spend on pets each year ($ million)

Expenditure type Dogs Cats Other Total

Food 560 431 90 1081

Vet charges and prescriptions 230 148 26 404

Pet care products/equipment 153 85 12 250

Pet services 116 35 – 151

Other expenses 109 20 12 141

Total 1168 719 140 2027

Source: BIS Shrapnel

2.3, 2.5, 2.6

Assignment 2eReplay 2w


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4. fractions 2. displaying data

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