Remedial Maths Series: Fractions

Ready-Ed Publications

Ages 10+ r o e t s Bo r e p ok u S

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Remedial Maths Series:

Fractions

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

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For students requiring assistance with fraction concepts.

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Written by Jane Bourke. Illustrated by Rod Jefferson. © Ready-Ed Publications - 2001 Published by Ready-Ed Publications, P.O. Box 276, Greenwood ,WA, 6024 Email: info@readyed.com.au Website: www.readyed.com.au COPYRIGHT NOTICE Permission is granted for the purchaser to photocopy sufficient copies for noncommercial educational purposes. However this permission is not transferable and applies only to the purchasing individual or institution.

ISBN 1 86397 175 0

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Contents

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Teachers’ Notes ....................................................................................................................... 4 Fraction Chart .......................................................................................................................... 5 Introduction to Fractions ........................................................................................................ 6 Equivalent Fractions 1 ............................................................................................................ 7 Fractions as Parts of a Whole................................................................................................. 8 Equivalent Fractions: Exercises ............................................................................................ 9 Equivalent Fractions 2 ............................................................................................................ 10 Equivalent Fractions 3 ............................................................................................................ 11 Matching Fractions .................................................................................................................. 12 Building Up Fractions ............................................................................................................. 13 Comparing Fractions ............................................................................................................... 14 Equivalent Fractions ............................................................................................................... 15 Fraction Inequalities................................................................................................................ 16 Review 1: Equivalent Fractions ............................................................................................. 17 Simplifying Fractions 1 ............................................................................................................ 18 Simplifying Fractions 2 ............................................................................................................ 19 Simplifying Fractions 3 ............................................................................................................ 20 Addition of Fractions ............................................................................................................... 21 Improper Fractions 1 ............................................................................................................... 22 Improper Fractions 2 ............................................................................................................... 23 Mixed Numerals 1 .................................................................................................................... 24 Mixed Numerals 2 .................................................................................................................... 25 Addition of Fractions: Exercises 1 ......................................................................................... 26 Addition of Fractions: Exercises 2 ......................................................................................... 27 Addition of Fractions: Exercises 3 ......................................................................................... 28 Subtraction of Fractions 1 ...................................................................................................... 29 Subtraction of Fractions 2 ...................................................................................................... 30 Subtraction of Fractions 3 ...................................................................................................... 31 Review 2: Addition and Subtraction ...................................................................................... 32 Decimal Introduction ............................................................................................................... 33 Decimal and Fraction Relationship ........................................................................................ 34 Expressing Fractions as Decimals ........................................................................................ 35 Place Value 1 ............................................................................................................................ 36 Place Value 2 ............................................................................................................................ 37 Fractrion and Decimal Inequalities ........................................................................................ 38 Decimals and Fractions 1 ........................................................................................................ 39 Decimals and Fractions 2 ........................................................................................................ 40 Decimals and Fractions 3 ........................................................................................................ 41 Decimals and Equivalent Fractions ....................................................................................... 42 Review 3: Expressing Fractions as Decimals ...................................................................... 43 Calculating Decimals ............................................................................................................... 44 Percentages 1 .......................................................................................................................... 45 Percentages 2 .......................................................................................................................... 46 The Relationship between Decimals, Fractions and Percentages ................................... 47 Calculating Percentages ........................................................................................................ 48 Ratios ........................................................................................................................................ 49 Mixed Problems 1 .................................................................................................................... 50 Mixed Problems 2 .................................................................................................................... 51 Answers .................................................................................................................................... 52-55

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Teachers’ Notes Mathematics education encompasses a wide range of topics and concepts, many of which are only briefly dealt with in the classroom due to time constraints. It is important that these fundamental concepts are understood before students move onto the next mastery level.

Students often fail to grasp all concepts and are unable to catch up to the level at which the rest of the class are working. It is here that the real difficulty for these students begins as they will sometimes withdraw from activities and miss further valuable concepts, simply because they had not mastered the prerequisite skills.

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Remediation for many students is frequently associated with a reduced self esteem as students are aware that they are working behind the rest of the class, especially when text books and worksheets for lower grades are used to help them to catch up.

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This remediation series is designed to provide upper primary students with the necessary skills and knowledge of mathematical concepts required for their year level and can be used both in the classroom and as a "take-home" package for extra consolidation of concepts.

The reading and content level is appropriate to the age of the student, even though many of the remedial activities are focused on previous stages of the maths syllabus. It is hoped that this series will boost students’ self esteem as they realise that they are able to successfully complete the maths activities in the book. In addition, students will not feel as if they are doing “baby” work as is the case when maths sheets for 8 year olds are given to 12 year old students.

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For best results the series should be used to complement a remedial maths programme for a small group or for individual students who need to catch up. Many of the worksheets explain the mathematical concepts and provide examples, however, it is assumed that this is not the student's first experience with the concept. Each book in the series follows the same format and is directed at a particular age group, yet can be used in the secondary school if required.

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The Challenge question at the bottom of most pages tests the child's knowledge of the mathematical concept for that particular page. The Challenge is usually presented as a word problem in a real world context so as to highlight the need for the skill.

This book explains the basic concepts of fractions, including the relationship between fractions, decimals and percentages. The activities are sequenced in line with standard syllabus structures, covering a number of stages as opposed to a straight year level, and are basically designed to provide students with the opportunity to catch up on much needed mathematical skills.

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The Fraction Chart on page 5 is referred to in a number of activities. This sheet can be photocopied onto card, allowing students to colour the strips and cut them out.

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Fraction Chart 1 10

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Introduction to Fractions A fraction is a part of a whole. It is used to describe how much of something is left.

Look at the first circle below.

6

= 6

=

4 4

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We can say that 6⁄6 are equal to a whole. The number on top is known as the numerator and tells us how many parts we have. The bottom number is known as the denominator and tells us exactly how many parts the whole has been divided into.

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The second circle has four pieces left. How can we represent this as a fraction?

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2. What fraction is shared in each of these pictures? a. b. c.

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1. Circle the denominator in each of these fractions.

4

3 9

d.

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• .................. ................... ...................

3. Shade the fraction for each of these: 4

2

6

4

6

8

7

6

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3

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4. Complete the number sentences below by shading in the correct amount.

The first one has been done for you.

2 4

=

3 6

=

4 8

=

2

4

=

5

=

=

1

10

3

=

6

=

Challenge: What should I order if I’m really hungry: a pizza cut into ten pieces or twelve pieces?

E Understanding the concept of numerators and denominators.

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Equivalent Fractions 1 Equivalent fractions have the same value. Look at the pizza below.

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In the first picture half of the pizza has been eaten. We can say that two quarters of the second pizza has been eaten. In the third picture we can see that three pieces or 3⁄6 of the pizza has been eaten.

Each picture shows that half the pizza has been eaten. The only difference is that each pizza has a different number of pieces left. Write the number of pieces left as a fraction under each pizza.

We can say that these three fractions are equivalent as they represent the same value.

1. Shade these equivalent fractions in the shapes below.

2

3

4

2

5

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2. Circle the fractions that represent the amount shaded in the picture.

There may be more than one answer.

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2

1

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b.

2

1

1

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1

2

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Challenge: Anthony has 1⁄3 of his chocolate bar left and Mel has a quarter.

Who has the most chocolate left?

E Carrying out activities involving the equivalence of fractions.

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Fractions as Parts of a Whole Already we know that a fraction is part of a whole. For example, a cake cut into three even pieces equals one whole. Two pieces of the cake are eaten leaving only one piece left or one third.

This can be represented as 3⁄3 - 2⁄3 = 1⁄3. Only a fraction of the whole cake remains.

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2. Shade in the amount represented as a fraction for each picture below.

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3

5

4

3

6

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1. Write the fraction of food that remains in each drawing below.

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3. In the shapes below shade in the pairs of equivalent fractions.

Write the fractions underneath.

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=

=

1

3

4 6

=

=

Challenge: Joshua has 20 jellybeans and eight of them are blue. What fraction of the jellybeans are not blue? Can you simplify this fraction?

E Carrying out activities that demonstrate: i) part/whole nature of fractions; ii) subset/set nature of fractions.

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Equivalent Fractions: Exercises 1. Write an equivalent fraction for each of these: 2 = ................

4 2 6 = ................

1 3 = ................

6 = ................

8 1 4 = ................

2 3 = ................

3 2 = ................. = ................

9 10 4 6 4 = ................. 9 = ................

4 8 = .................

6 10 = ................

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3 = .................

5 4 8 = .................

4 5 = .................

2. Place = or ≠ in the box.

3 4

1 3

1 2

2 4

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2 3

6 8

2 5

3 6

2 3

3 4

4 6

2 3

2 5

3 3

6 6

4 5

5 6

7 8

7 9

2 2

9 9

7 9

2 5

2 6

3 8

3 6

3 6

4 8

3. Use <, >, or = to make these correct.

1 3

4.

3 4

4 4

5 6

1 2

7a 1y 1 dP 2 u 3l 7a 3 n 1 4 ©27 Re d E b i c t i o s 8 5 4 3 4 7 4 1 9 •f orr evi ew pur posesonl y• Shade the correct number of pieces for each picture.

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5. Now write the fractions above in order starting with the smallest.

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Challenge: Michelle and Andy collect stamps. Michelle’s album is 3⁄4 full while Andy’s is 3⁄8 full.

Who has collected the most stamps?

E Ordering fractions with unlike denominators.

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Equivalent Fractions 2 1. Look at the fraction in the column and circle the equivalent fractions. The first one has been done for you.

2 3

3 6

1 b. 2 1 3

1 d. 4

8 12

6 9

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e.

6 8

2 4

5 10

4 9

6 12

7 12

3 6

4 12

3 9

2 8

4 6

2 8

3 6

2 5

3 7

3 12

2 10

1 4

3 15

4 20

3 6

2. Shade in the correct amount to match each fraction with its diagram.

1 4

2 3

3 6

5 6

1 3

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3. Order the above fractions starting with the smallest.

(Hint: Use the Fraction Chart on page 5 to help you.)

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Challenge: Maree and Paul’s mum made each of them a blueberry pie. Maree cut her pie into eight pieces and Paul cut his pie into ten pieces. Both of the children then ate half of their pies.

How many pieces would each child have left?

E Matching fractions with unlike denominators.

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Equivalent Fractions 3 We know that there are many ways of writing a half. For example, if a cake has been cut into four even pieces then two pieces will be equal to a half. We would write this as 2⁄4 and say that 2⁄4 = 1⁄2.

We now know that there are two 1⁄4 in a 1⁄2.

Fraction Chart 1 10

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Look at the fraction chart and the way that it has been divided up. Use the chart to answer these.

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How many: 6 ’s in 8 ?..............................................

1 ’s in 3 ?..............................................

2 ’s in 3 ?..............................................

5 ’s in 10 ?..............................................

1 10 ’s in 1 8 ’s in 1 2 ’s in 1 3 ’s in

1 5 1 4

?............................................... ?...............................................

3

?...............................................

4 6

?...............................................

Challenge: Is 3⁄50 greater than or less than 3⁄500?

E Finding fractions of a fraction, using a fraction chart.

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Matching Fractions 1. Fill in the missing numerators for these equivalent fractions.

1 4

=

4 5

=

8

1 3

=

10

3 5

=

1 = 10 20

9

1 2

=

10

1 3

=

12

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=

3

1 5

=

10

3 9

=

4 8

3 5

=

1

4 = 12

12

3 4

=

2 4

=

4 6

=

12

8

2. Write the missing denominators into the boxes.

2

2 3

2

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4 = 10

=

=

4

3 = 10

6

2

1 3

5 = 10

1

3 9

=

=

6

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1 5

1

9 = 12

3

3. Write these fractions in the simplest form. The first one has been done for you.

9 3 = 12 4

8 10 = ............

4 6 = ............

8 12 = ............

20 40 = ...........

3 6 = ............

100 5 14 18 160 200 = ............ 10 = ............ 20 = ............ 20 = ............ 200 = ...........

40 40 = ............

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Wor d Pr oblems: ord Problems:

1. Jessica has three bread rolls and has cut them into quarters.

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50 5 15 30 100 = ............ 15 = ............ 30 = ............ 60 = ............

How many quarters does she have altogether? .................................

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2. Simon has four red marbles and six blue marbles. He gave two of the red marbles to Fiona and three of the blue marbles to John.

What fraction of each colour did Simon give away? ...........................

3. Natalie has six pieces of fruit and two pieces are bananas.

A third of the pieces are apricots and the rest are pears.

How many of each fruit does she have? ............................................

4. There are ten students in the art class. One fifth of the class are painting and two tenths of the class are drawing. Six tenths are making pottery.

How many students are doing each activity?

Painting: ............................ Drawing: ........................... Pottery: ...............................

E Finding denominators and numerators to complete number sentences.

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Building Up Fractions If we multiply both the denominator and the numerator by the same number we will find an equivalent fraction. For example, 1 x 3 = 3 2 x 3 = 6 We know that both 1⁄2 and 3⁄6 represent a half.

1. Using the method above write five equivalent fractions for a. ......................

1 : 3

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b.......................

1 2. Now try the same for 4:

c. ......................

d. .....................

e. .......................

a. ......................

c. ......................

d. .....................

e. .......................

b.......................

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3. Colour 1⁄3 of each of the circles below and write the equivalent fraction underneath each one.

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4. Colour 1⁄4 of each of the rectangles below:

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5. Colour 2⁄5 of each of the shapes below:

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Challenge: Scott has 24 photos in his sports scrapbook. Half of the photos are of footballers, a third of the photos are of basketballers and the rest are of tennis players. How many photos are there of each of the different types of sports stars?

E Looking at other methods for finding equivalent fractions.

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Comparing Fractions

The grid above has been divided into 100 squares. Shade in the following fractions according to the key.

© ReadyEdPubl i cat i ons How many did youv shade in each colour? ...............................................................

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1.

1 = red 5

green

= 30 squares. We can say that

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= ...................

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blue

yellow

= ...................

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squares. squares.

1 = blue 4

3 10 1 5 1 4 4 20

=

4 = yellow 20

30 100

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squares.

= ....................

2. What fraction of your grid remains unshaded? ......................................................................

3. Write two equivalent fractions for each of these: 60 25 20 100 ...................................... 100 ....................................... 100 ........................................

30 40 10 100 ...................................... 50 ....................................... 50 ........................................

Challenge: Matt has 1000 stamps in his collection. 450 are from Africa and the rest are from Europe, Asia and South America. What fraction of the stamps do not come from Africa?

Express your answer in the simplest form.

E Comparing fractions with unlike denominators by converting them to fractions with the same denominators.

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Equivalent Fractions 1. What fractions of the shapes below are shaded? Simplify your answer.

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.................................

.................................

..................................

1 3

2 6

3 9

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2. Shade these amounts below.

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1 3

2 6

2 8

4 16

3 12

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2 9

2 6

2 3

2 8

3 12

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3. Complete the following using = or ≠. You might like to refer back to the fraction chart on page 5.

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4. Use =, < or > to make these true.

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1 3

4 9

2 7

2 5

2 6

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1 6

1 8

2 4

2 3

5 6

7 8

3 8

3 7

9 10

9 9

4 4

3 3

1

2 3

5 5

2 2

1 6

1 9

Challenge: Emily and Sarah are painting the fence in the backyard. Sarah has painted 1⁄3 of the fence posts and Emily has painted 3⁄8 of the fence posts. Who has painted the most posts?

E Experience with equivalent fractions.

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Fraction Inequalities 1. Circle the greater fraction in each pair. The first one has been done for you.

1 5

3 10

2 6

3 6

1 2

2 5

4 9

1 2

1 3

6 7

4 6

4 5

5 10

5 8

4 8

3 9

2 6

2 5

7 8

8 9

1 2

1 10

5 5

3 4

1 2

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2. Place = or ≠ in the boxes below.

4 10

14 20

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1 3

3 9

2 4

5 10

3 6

6 9

4 6

8 10

4 5

9 9

3. Complete the following by adding a fraction of your own.

2 3

<

4 5

>

6 > 12

<

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8 < 12 4.

9 9

>

7 7

4 6

3 7

<

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6 > 8

<

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5. Match the fractions on the left with a fraction on the right. a. 2 50 b. 4 60 3 100 6 90 4 8 7 8 9 18 8 12 3 8

40 60

5 10

6 16

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3 9 2 8

4 16 2 5

3 4

75 100

4 10

2 6

2 3

14 16

8 12

4 6

Challenge: Murray and Sue each recieve the same amount of pocket money each week. Murray has spent 3⁄7 of his pocket money and Susie has spent 4⁄5. Which child has spent the most money?

E Ordering fractions with unlike denominators.

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Review 1: Equivalent Fractions 1. Order these fractions from smallest to largest.

1 3

4 6

1 5

1 2

2 5

3 3

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2. Place these fractions in order from largest to smallest.

1 6

10 10

3 9

1 4

1 7

2 5

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5 6

1

1

3 6

4

2 6

2 12

1

7 8

1

1 9

3

7 9

5 10

7

2 3

4

5

2

7 10

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3. Use < or > in the boxes below to make the number sentences true.

2 3 1 2

2

5 6

1 3

4

3 4

5

1 3

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4. Use <, > or = to complete these.

7 7

1 6 7

3 4

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4

1 3

4 12

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5 9

9

1 5 6

1 7

2

1 7

15 5

8

3

2 3

15 15

3 12

9 3

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3 3

4

2 4

6 6

5. Colour the amounts shown and complete the number sentence using =, < or >.

5 8

3 6

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3 4

7 8

Challenge: Joanne gets 80¢ a page for typing. If she types 3⁄4 of a page how much money will she receive?

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Simplifying Fractions 1 The fractions below can all be expressed as 1 because they represent all parts of a whole.

3 3

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9 9

7 7

50 50

6 6

20 20

1. Write four fractions below that are equivalent to a whole.

d. ..............................

2. Shade the shapes below according to the fraction.

2 2

4 4

12 12

20 20

3. Use the pictures and circle the amounts shown below.

4 3 2

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a. .............................. b. .............................. c. ..............................

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4. All of the amounts you have shown above are equal to ............................................................

5. Use a red pen to circle the fractions below that represent a third.

Use a blue pen to circle the fractions equivalent to a half.

4

4

3

8

12

9

20

5

50

2

5

6

100

40

10

100

6

15

12

200

Challenge: Kathy was checking her netball goal scoring for the last three games. In the first game she scored 15 of the 28 goals for the game. In the second game a total of 36 goals were scored by her team of which Kathy scored 18 and in the third game she scored 12 out of 26 goals. In which game did Kathy score exactly half the goals?

E Simplifying fractions by converting them to equivalent fractions.

Page 18

R eady-Ed Publications

Simplifying Fractions 2 We can simplify fractions in another way. By looking at the fraction chart on page 5 we can see that 6⁄8 is equivalent to 3⁄4. If we look at the fraction itself we can see that for 6⁄8 to become 3⁄4 we divide both the numerator and the denominator by 2.

6 8

÷ ÷

2 2

= =

3 4

1. Simplify these fractions by dividing by two.

8 = 12

r o e t s Bo r e p ok u S

4 = 10

2 6

2 4

=

=

4 = 12

9 ÷ 3 = 3 12 ÷ 3 = 4

5 = 20

80 = 100

3 = 18

4 = 12

ew i ev Pr

Teac he r

2. We can divide by other numbers to simplify fractions. Write simplified fractions for these:

Sometimes it is possible to divide the numerator into the denominator to simplify a fraction. For example, with 3⁄9 we have 3 as the numerator and 9 as the denominator. We know that 3 goes into 9 exactly three times and we can simplify the fraction to 1⁄3 because 3 goes into 3 once and into 9 three times.

3. Use this method to simplify the fractions below.

4 u 4a 5 © ReadyEdP b l i c t i o n s 8 = ............ 16 = ........... 20 = ............

•f o r ev40i ew pu pose nl y 2 r 7 • 50r 6so

4 3 10 12 = ............ 15 =............. 20 = ............

20 200 = ............ 10 =............. 80 = ............ 100 = ...........

18 = ...........

14 = ............

w ww

4 6 9 3 = ............ =............. = ............

20 30 18 27 = ............

. te

8 10 2 15 80 = ............ 100 =............. 12 = ............ 20 = ...........

m . u

4. Express these fractions in the simplest form.

5 30 = ...........

4 10 = ............

3 12 = ...........

8 10 = ............

o c . che e r o t r s super

5. Match a fraction on the top line with a fraction on the bottom line.

3 4

7 8

14 16

6 8

9 20

200 1000

3 50

6 200

3 100

4 20

6 100

18 40

Challenge: In his last maths test Aaron scored 75 out of a possible 100. What is the simplest way of expressing the fraction of the sums that Aaron correctly answered?

E Simplifying fractions by division using whole numbers.

R eady-Ed Publications

Page 19

Simplifying Fractions 3 1. Simplify these by dividing by ten or a hundred.

1000 50 700 9000 80 20 1000 =........... 60 = ........... 800 = ............ 1000 = .......... 100 =............ 100 = ..........

800 200 60 20 800 =........... 500 = ........... 100 = ............ 50

10 50 = .......... 100 =............ 200 = ..........

r o e t s Bo r e p ok u S

2. Simplify these fractions.

25 75

40 75 120 480 48 =........... 80 = ........... 100 = ............ 200 = .......... 2000 =............ 50 = ..........

Teac he r

54 25 =........... 60 = ........... 30

85 = ............ 90

25 = .......... 40

30 =............ 90 = ..........

Wor d Pr oblems: ord Problems:

Where possible express your answers in the simplest form.

1. Mark has sixteen apples and Tony has eight. Mark has eaten eight of his apples and Tony has eaten three.

What fraction does Mark have left? .............................................

© ReadyEdPubl i cat i ons Sandra has twelve socks in her draw. One third of the socks are red and the rest are blue.

•f orr evi ew pur posesonl y•

What fraction does Tony have left? ..............................................

2.

ew i ev Pr

35 40

How many blue socks does Sandra have? .........................................

m . u

3. Donelle is sharing her birthday cake with four friends. She has cut the cake into ten

w ww

pieces. What fraction will each person receive?...........................................

4. Tarlie has counted twenty students in her class and has noticed that five of the students

. te

wear glasses. What fraction of the students do not wear glasses? .........................................

o c . che e r o t r s super

5. Fiona and Peter played twelve games of chess. If Fiona won nine games, what fraction did Peter win? ........................................

6. Steve sells hotdogs at the football. He had enough supplies for 100 hotdogs. At the end of the day he had twenty hot dogs left over. What fraction did Steve sell? ...................................

7. In Jamie's maths exam there were 30 questions of which he answered twenty five correctly. What fraction did Jamie answer incorrectly? ..............................................

8. Jason is captain of his cricket team and there are twelve boys in the team. Four boys were injured in the last game. What fraction of the team were injured? ............................................

E Simplifying fractions by cancelling.

E Exploring the concept of fractions in a real world context.

Page 20

R eady-Ed Publications

Addition of Fractions Look at the pies below. What fraction of each pie has been left?

8

8

r o e t s Bo r e p ok u S

How many pieces are there altogether?........................................

We can say ⁄8 + ⁄8 = ⁄8.

4

2

6

4 + 2 = 6 8 + 8 = 8

You will notice the denominator does not change when fractions are added.

1. Complete these sums.

2.

2 a 25 u 1i 3 n 2 © Re d y E d P b l c a t i o s + = + = + 4 4 6 6 7 7 = • f omethod rr e i e wfractions. puThe r p o se o nl y • Use the written tov add these first one hass been done for you. +

1 2 = 3 3

3 2 + = ............

8 8

1 1 + = ............

2 2

2 5 + = ............

9 9

2 1 + = ............

4 4

4 5 + = ............

9 9

25 30 = ............ + 10 10

20 25 = ............ + 50 50

w ww

1 2 1 + = 3 3 3

. te

12 5 = ............ + 20 20

m . u

1 3

ew i ev Pr

Teac he r

To work this out we simply added 4 and 2.

4 2 + = ............

8 8

23 56 = ............ + 100 100

o c . che e r o t r s super

3. Complete the following.

1 1 1 + + 3 3 3

= ........................................

2 4 3 + + = ........................................

10 10 10

2 1 1 + + 5 5 5

= ........................................

15 1 2 + + = ........................................

20 20 20

Challenge: Michael cut a pizza into twelve pieces. He gave three pieces to Stephanie, four pieces to Simone and then gave two slices to Bill. What fraction of the pizza did Michael give away?

E Adding fractions with like denominators up to 10 and multiples of 10.

R eady-Ed Publications

Page 21

Improper Fractions 1 Look at the three pies below. Each has been cut into quarters.

r o e t s Bo r e p ok u S

How many quarters are there altogether? .....................................

12 We can express this as . This is known as an improper fraction.

4

+

+

ew i ev Pr

Teac he r

1. Use the shapes below to complete the following. The first one has been done for you.

2 4 + = ....................... = .......................

5 5

3+3 6 2 = =1 4 4 4

adyEdPubl cat i ons +© Re +i

•f orr evi ew pur posesonl y•

8 4 + = ....................... = .......................

10 10

w ww +

. te

+

o c . che e r o + t r s super

5 7 = ....................... = ....................... + 9 9

+

m . u

3 5 + = ....................... = .......................

6 6

6 4 = ....................... = ....................... + 7 7

4 4 + = ....................... = .......................

5 5

5 7 = ....................... = ....................... + 8 8

Challenge: Joe had six apples which he cut into quarters. Express the number of quarters Joe has as an improper fraction.

E Expressing fractions as improper fraction through addition sums.

Page 22

R eady-Ed Publications

Improper Fractions 2 Follow this example:

3 2 + = 4 4

3+2 5 1 = =1 4 4 4

⁄4 is known as an improper fraction because the numerator is greater than the denominator. This means that there is more than 1 whole. We know that 4⁄4 is equal to one whole so 5⁄4 must be equal to 1 whole and 1⁄4 or 11⁄4 .

5

r o e t s Bo r e p ok u S

1. Express the answers to these sums as improper fractions.

8 4 + = ............

10 10

3 6 + = ............

7 7

4 3 + = ............

5 5

4 7 + = ............

8 8

10 10 + = ............

15 15

12 15 + = ............

20 20

Teac he r

2 5 + = ............

6 6

2. Simplify these fractions. The first one has been done for you.

1 28

=

2 69

= ..................

4

4

+

2 29

= ..................

2 68

= ..................

5 108

= ..................

3 126

= ..................

3 46

= ..................

2 48

= ..................

1 24

+

1 24

= ............

4 56

+

2 16

= ............

© ReadyEdPubl i cat i ons ⁄ = 1 so 1 ⁄ is equal to 2. Complete the following by adding the fractions.

• f orr evi e w pu r po sesonl y• 3 4 2 5 4

4

2 5

1 36

+

4

5 = ............

1 79

= ............

w ww

6

+

6 = ............

9

+

7 109

+

1 101 = ............

2 3

+

9 = ............

1 13

= ............

m . u

3.

1 14

ew i ev Pr

6 5 + = ............

9 9

4. Jessica had four whole oranges and half of an orange and Ben had one orange and a

. te

half. If they put them together how many oranges would there be? ......................................

o c . che e r o t r s super

5. Hamish divided six bananas into thirds. How many thirds did he now have? Express your answer as an improper fraction. ............................................................................................

6. Suzanne had four sets of pencils each with eight colours and half a set with four colours. Drew gave her three pencil sets and four extra pencils. Write a number sentence to show how many pencils Suzanne has. ...............................................................................

E Adding fractions with mixed numerals.

R eady-Ed Publications

Page 23

Mixed Numerals 1 When there is more than one whole the fraction is expressed as a mixed numeral.

For example, the shaded area in the circles below represents 41⁄3.

1

+

r o e t s Bo r e p ok u S 1

+

1

+

1

+

2 15

ew i ev Pr

Teac he r

1. Shade in the correct amount below.

1 3

4 23

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

3 18

w ww

We can also express the above mixed numerals as improper fractions.

For example,

. te

2 15

is equal to

5 5

5

1

11

+ 5 + 5 = 5

o c 3c 2 6. e her r o t s 2 sup r 2 3 e

2. Express these mixed numerals as improper fractions.

4 6

= ..................

4 34

= ..................

5 23

= ..................

1

m . u

1 36

2 4

= ..................

5 6

9 10

= ..................

5 6

8 45

= ..................

= ..................

1 2

= ..................

= ..................

4 7

= ..................

45 3 100 = ..................

55 1 200 = ..................

Challenge: At a party Mark cut 5 cheesecakes into 10 pieces each. At the end o fthe party the following amounts were left on each plate; 3 pieces, 2 pieces, 6 pieces, 1 piece and 4 pieces. Express the amount left over as a mixed numeral.

E Representing mixed numerals in diagrams.

E Converting mixed numerals to improper fractions.

Page 24

R eady-Ed Publications

Mixed Numerals 2 To convert improper fractions into a mixed numeral divide the denominator into the numerator.

For example, 3⁄2 = 11⁄2. There are three halves and we know that two halves make a whole. One half is left over so the answer is 11⁄2.

1. Change these improper fractions into mixed numerals.

5 2 =.............

7 3 = ............

8 2 = ............

9 7 = ...........

10 6 = ............

12 6 = ............

9 2 =.............

15 8 3 = ............ 5 = ...........

20 6 = ...........

20 9 = ............

85 32 25 16 = ............ =............. = ............

10 5 6 3 = ............

39 7 = ...........

100 20 = ............

r o e t s Bo r e p ok u S

2. Change these mixed numerals into improper fractions.

ew i ev Pr

Teac he r

4 3 = ............

3 14

= ..................

5 23

= ..................

6 26

= ..................

2 37

= ..................

9 12

= ..................

1 7 10

= ..................

8 48

= ..................

7 46

= ..................

3.

3 =d .................. =b .................. =s .................. © R5ea yEd4Pu l i cat i on Place • these fractions in order starting the smallest. f orr e vi ewwithp u r posesonl y• 4 5

= ..................

3 18

1 8

60 100

3 20

20 8

6 8

4 28

8 8

2 58

13 8

m . u

9 89

11 6

w ww

....................................................................................................................................................

5 7

75

21 3

3

4. Use <, > or = to make these true.

1 16

. te

o c . che 1 1 e r o t r s super 2 3

3 2

4 5

9 5

12 2

4 3

4

13 6

3

4 5

3 6

18 6

5 2

2 15

5 10

10 5

24 6

16 4

Challenge: Nicole cut some oranges into quarters for her netball team to eat at half time. At the end of the game there were 17 quarters left on the tray. Express the fraction of oranges on the tray as a mixed numeral.

E Expressing improper fractions as mixed numerals.

E Ordering combinations of fractions.

R eady-Ed Publications

Page 25

Addition of Fractions: Exercises 1 1. Add these fractions and express the answer as a mixed numeral.

7 4 10

For example:

+

=

4

=

10 +

=

10 + 1 11

=

11

6 10 3 5

Teac he r

=

+

4

4 5

=

3 13

+

=

2

7

9

+ 10 + 10

16 10

r o e t s Bo r e p ok u S 3

6

+ 10

5 6

+

4 6

=

2

7 9

+

4

2 9

=

5 38 =

+

6

7 8

©R eadyEd= Publ i cat i ons = = •f orr evi ew pur posesonl y•

2 3

6

3 7

+

5 7

3

5 8

+

4

5 8

2. Add the amounts shown in the shapes below.

w ww

. te

+

3 38

+

5 8

m . u

+

6

=

ew i ev Pr

2 5

9 6 10

o + c . che + e r o t r s super

..............................................

............................................

..............................................

Challenge: Jerry had three bread rolls and 2⁄5 of a roll and Kelly had five rolls and 4⁄5 of a roll.

What fraction of bread rolls did they have together?

E Adding fractions with mixed numerals with like denominators.

Page 26

R eady-Ed Publications

Addition of Fractions: Exercises 2 1. Simplify the answers to these sums.

9 4 20

+

9 3 20

7 15 50

=

21 50

+

1 20 30

=

3 10 30

+

3 15 30

2 15 40

+

Teac he r =

1

3 5

6

+

4 10

+ 4

=

3

2 4

2

+

5 6

4

1 2

=

1

=

5

+

5 34

8 10

+

3 5

+

3 45

=

1 6

2 47

+ 9 10

+

ew i ev Pr

2 34

3 4

=

=

5

+

8

=

=

4 5

2 20 30

6r 3 5 o e t s B r e oo p u k S 3 15 40

=

+

4 7

=

5 17

4 10

=

3 5 10

w ww 3 9

+

7 9

= .................

. te

2

7 8

+

2 38

8 1 10

+

7 2 10

m . u

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

= .................

o c . che e r o t r s super

= .................

2

5 6

+

1 46

= .................

Challenge: Sam and Chloe weeded gardens in their spare time. On Monday they weeded two gardens and three quarters of a garden. On Tuesday they weeded the quarter from Monday plus two more gardens and on Thursday they weeded one and a quarter gardens. What fraction of gardens did they weed during the week?

E Simplifying mixed numerals.

E Addition of mixed numerals with like denominators.

R eady-Ed Publications

Page 27

Addition of Fractions: Exercises 3 So far we have added thirds to thirds and quarters to quarters and so on. We can say that these fractions have like denominators. Sometimes we need to add fractions with unlike denominators.

Look at the diagrams below and write the fraction of the remaining portions of pie underneath.

...............................

ew i ev Pr

Teac he r

................................

r o e t s Bo r e p ok u S

If we combine the leftovers, what fraction will we have? To find the answer we need to express the fractions so that the denominator is the same. This is known as finding the lowest common denominator.

Example 1:

1 4

To add

we convert

to 1 = 4 2+5 = 8

Example 2:

5 8 7 8

=

2 3

1 9

+

6+1 9 7 9

+

3 8

1 3

+

5 6

2 3

+

2 9

1 5

w ww

2. Add these fractions and simplify the answer.

2 6

+

1 3

3 4

+

5 8

. te

2 4

2 6

1

+ 10

m . u

3 4

5 8

© ReadyEdPub=l i cat i ons Complete these sums by using equivalent fractions to find the lowest common denominator.

•f orr evi ew pur posesonl y• so sum is

1.

+

o c . che e r o t r s super +

4 8

4 10

+

2 5

1 9

+

2 3

3 6

+

1 2

1 2

+

2 3

6

+ 10

Challenge: Darren played in three quarters of his basketball match this Saturday. Last Saturday he played half a game, and the Saturday before that he was on for only half of the first quarter. What fraction of games has Darren played in the last three weeks?

E Addition of fractions with unlike denominators to 10.

Page 28

R eady-Ed Publications

Subtraction of Fractions 1 Jerry cut a pie into four quarters and decided to eat a piece. What fraction now remains?

4 4 1 . can be written as – 4 4 4 Later he ate another piece. What fraction is left?

4–1 4

He started with

3 – 4

1 = 4

3–1 4

=

=

3 4

2 1 = 4 2

r o e t s Bo r e p ok 1 u S

1. Use the method above to complete the following.

6 6

3 6

4 5

= ......................

2

1 6

–

= ......................

2 5

–

3 7

–

6 7

= ......................

= ......................

9 10

10 –

4

– 10

= ......................

3 8

= ......................

5 9

= ......................

2. Subtract the shaded amount from the shapes below.

3 7

–

ew i ev Pr

Teac he r

–

5 9

–

= ......................

w ww

................................

................................

3. Now try these. Remember to simplify your answers.

2 37

1

– 7

–

3

2 6

.................................

. te 4 6 10 o c . che e r o t r s 4 4 sup r 3 2 2 e1

= ......................

9 56

m . u

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

= ......................

3 4

1

5 8

– 4

= ......................

5 8

–

2 8

= ......................

4

6 7

– 8

= ......................

9 10

–

2 10

= ......................

6

– 7

= ......................

6 7

–

5 7

= ......................

Challenge: Hannah had five bunches of flowers, each made up of nine flowers. She gave two bunches and a third of a bunch to her sister and another two bunches to her Mum. How many bunches does she have left?

E Subtraction of fractions including mixed numerals with like denominators.

R eady-Ed Publications

Page 29

Subtraction of Fractions 2 To subtract mixed numerals, first convert the fractions to improper fractions. For example: 1 5 11– 8 5 3 5

2 = =

–

1

3 5

1 – 8 25 – 19 8 6 8 3 4

3 = =

2

3 8

r o e t s Bo r e p ok u 2 1 6 5 S3 =

1. Convert these fractions to improper fractions to find the answer.

1 6

4

– 6

= ......................

4 9

7

4 10

– 9

= ......................

8 10

–

= ......................

3 7

–

4 7

ew i ev Pr

Teac he r

2

= ......................

2. Express these whole numbers as improper fractions. The first one has been done for you.

18 9

3

4

6

2

=

4

3.

© ReadyEdPubl i cat i ons Subtract the fractions from the whole numbers by changing to improper fractions.

Write your answer as a mixed numeral.

•f orr evi ew pur posesonl y•

8

–

1

3 4

= ......................

2 10 20

–

= ......................

=

7

–

4

–

. te

=

2

6

=

3 10

4

–

= ......................

w ww

3

5

6

75 100

= ......................

5

3

2 3

= ......................

6

7 30

–

= ......................

7

3

–

2

1 5

= ......................

9

–

6 7

= ......................

2

=

10

3

–

2 6

= ......................

2

1 9

–

= ......................

o c 10c 5 3. 2 e her r o t s super

4. Subtract these fractions.

4 5

=

m . u

=

=

–

2 3

= ......................

4

–

7 20

= ......................

2 6

3 6

–

2 5

–

= ......................

= ......................

3 509

8

–

11 2 50

= ......................

–

3

4 5

= ......................

Challenge: During the term Steve had to read 5 books. So far he has read 2 books and 4⁄6 of a book.

What fraction of the books does he still have to read?

E Subtraction of fractions from a whole number, and a mixed numeral.

Page 30

R eady-Ed Publications

Subtraction of Fractions 3 1. Complete the following and simplify answers where possible.

For example:

5 – 8 37 – 31 8 6 8 3 4

4

1. =

=

Teac he r

2 9

= ......................

10

= =

5 9

–

4 7

= ......................

3 48

–

=

4 7

6 7

1

–

25 – 13 7 12 7 5 7

9

3 5

–

= ......................

= ......................

6 25

1

3 45

3 9

5 7

2 8

= ......................

4 5

2

–

2

–

= ......................

7 10

3

= ......................

7 9

1 78

–

ew i ev Pr

3 16

–

3

2.

r o e 1 t s B r e oo p u k S4 3 6 4 3 =

5 26

7 8

3

4 6

1 56

–

= ......................

8 20

2 7

3 67

© R=e adyEd=P ubl i cat i ons ...................... ...................... = ......................

•f orr evi ew pur posesonl y• Fill in the square to make each number sentence true.

–

–

4

–

5

–

= ......................

7

6

3 5

2

5

1 7

–

2 5

–

1 5

–

–

5 6

2 3

2

=

w ww

2

=

3

.

2 te 2 5

=

=

7

1 6

1 5

25 4 100

1 3

–

–

5 2 100

=

5 13

4 45

–

=

2

4 38

–

3

5

=

1 25

2

8

=

1 78

m . u

2.

4 o c . che e r o t r s super 10

–

9

=

9

3 9

5

–

2 6

=

6

Challenge: Jason has five bags of marbles and each bag contains eight marbles. He gave two bags and three quarters of the marbles in another bag away to David.

What fraction does Jason still have? ................................................

How many marbles did David receive from Jason? ................................................

E Subtraction of mixed numerals from other mixed numerals and whole numbers.

R eady-Ed Publications

Page 31

Review 2: Addition and Subtraction 1. Add the following fractions.

1 3

+

1 3

1 4

= ......................

1

2 3

2

+

2

2

2 4

+

2

1 4

= ......................

5 7

2 8

2

1

+ 3

= ......................

1 5

–

1 3

= ......................

2

+

3

2 5

4

4 5

= ......................

7 8

= ......................

+

3

9 10

= ......................

3 4

–

1 4

= ......................

6 9

–

2 9

4 5

8

3 10

+

4 5

= ......................

1

= ......................

+

= ......................

3 4

+

2 5

= ......................

2. Subtract the fractions below.

2 3

+

2 3

ew i ev Pr

= ......................

1 3

r o e t s Bo r e p o2k 2 5 4 u S 1

+ 3

Teac he r

3 7

1 4

= ......................

= ......................

2

+

2

– 3

= ......................

© ReadyEdPubl i cat i ons = ...................... = ...................... = ...................... = ......................

•f orr evi ew pur posesonl y• 1

2 5

–

–

1

1 5

5 6

= ......................

1 3

–

2 3

4

10

–

3 5

2

3

= ......................

2 5

–

–

2

3 5

6

1

2 3

6

= ......................

3 9

–

–

4

4 9

3

4 5

= ......................

w ww

Wor d Pr oblems: ord Problems:

m . u

2

1. Sarah had nine chocolate biscuits and five of them were iced. She cut each biscuit in half.

. te

Express the total number of iced biscuits as a fraction. .....................................................

o c . che e r o t r s super

2. Lucy had 4 pears which she cut into quarters. She ate two pears and a quarter.

How much does she have left? .......................................................

3. Amanda has made some soup and is serving dinner for the family.

She has given half of the soup to her Dad, a quarter to her Mum, an eighth to her younger sister and an eighth to herself.

What fraction of the soup remains? .....................................................

4. Julian makes model aeroplanes and buys each piece separately.

Each aeroplane had 10 pieces. He has seven completed aeroplanes and 2⁄5 of another one.

How many pieces does he have in his collection? .....................................................

5. Brad is 14⁄10 of a metre tall.

How many centimetres tall is Brad? .........................................................

Page 32

R eady-Ed Publications

Decimal Introduction We know that a fraction represents part of a whole. A decimal is a number where a decimal point seperates the whole number and the fraction. For example, 13⁄10 = 1.3

25

⁄100 = 0.25

In the above example the 3 represents the three tenths.

r o e t s Bo r e p ok u S

What does the 25 in the second mean? ........................................

1. Complete the following table by writing the numbers below into the correct unit place.

The first one has been done for you.

Tens

Ones

.

1/tenth

2

5

.

3

25.34 12.3 1.56

2.458

13.256 0.98

1/hundredth 1/thousandth

ew i ev Pr

Teac he r

Hundreds

4

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

2. Using the table below write down each number. Remember, if there is no number in the column a zero is used to represent the place.

a.

b. c.

. te 1

d. e.

Tens

Ones

.

2

5

.

6

.

1

.

7

.

6

.

1/tenth

1/hundredth 1/thousandth

m . u

w ww

Hundreds

3

o c . che e r o t r s super 3

2

2

4

6

5

4

a. .........................................

b. ............................................

d. .........................................

e. ............................................

2

4 3

c. .........................................

Challenge: Which is greater: 200.2 or 200.002?

E Introduction to decimals exploring the place value.

R eady-Ed Publications

Page 33

Decimal and Fraction Relationship 1. Shade in the correct amounts below:

r o e t s Bo r e p ok u S

0.23

0.7

0.26

0.69

0.6

0.54

© ReadyEdPubl i cat i ons 0.24 0.63 0.12 0.89 •f orr evi ew pur posesonl y• Write the decimal that shows the shaded area below.

w ww

. te

........................

m . u

2.

0.5

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Teac he r

0.1

o c . che e r o t r s super .........................

..........................

..........................

3. Write the fraction and the decimal for each of the shaded areas below.

0.6 = 6⁄10

................................

................................

.............................

Challenge: Decimals are used to show amounts of money. How would you express a dollar coin and a twenty cent piece as a decimal?

E Exploring the relationship between decimals and fractions as parts of wholes.

Page 34

R eady-Ed Publications

Expressing Fractions as Decimals Fractions and decimals can be used to express the same amounts. We use fractions for some objects and decimals for others. Consider the objects below and circle the way you would describe them.

1 2

or 0.5 a glass of orange juice; 3

r o e t s Bo r e p ok u S

0.3 of a metre or 10 of a metre; 1 4

of a sandwhich or 0.25 of a sandwich; 3

1. Express the decimals below as fractions.

0.2 = ............. 0.5 =.............

0.6 = ............. 0.23 = ........... 0.98 = ........... 0.47 = ..........

2. Express these fractions as decimals.

1 10

7 10

= ..............

= .............

ew i ev Pr

Teac he r

0.75 or 4 of a job finished.

10 100

34

= ............. 100 = .............

28 100

567

= ............. 1000 = ...........

© ReadyEdPubl i cat i ons For example: = r = 0.6 •f or e vi ew pur posesonl y•

Fractions need to be expressed with denominations of 10, 100 or 1000 before being expressed as decimals.

1 2

6 10 5

= 10 = 0.5

3. Express these fractions as decimals.

w ww

2 5

= ..............

4 10

10 20

= ..............

40 200

= ............. 600 = ............

= .............

. te 20 50

300

= .............

8

m . u

3 5

1 5

= ............. 20 = .............

6 20

= ............. 50 = ...........

2 5

= .............

3 4

1 4

= ............. 20 = ...........

200 200

= ............. 40 = ...........

= .............

10 15

o c . che e r o t r s super 20 2000

150

= ........... 200 = .............

20

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

0.5 4 5

4 10

0.45

6 10

6.0

8 100

0.08

90 100

0.9

8 1000

0.08

Challenge: Miles has collected 36 sports cards this year. If there are 200 in the set, what fraction has he collected so far? Express this fraction as a decimal.

E Expressing fractions with denominators up to 10 and multiples of 10 as decimals.

R eady-Ed Publications

Page 35

Place Value 1 In the examples below, the underlined number represents a different value even though it is the same digit.

5432 = 4 x 100 = 400 62.45 = 4 x 1⁄10 = 4⁄10 1. What value does each underlined number represent below?

r o e t s Bo r e p ok u S

9675 ................... 29.38 .................. 1.987 .................. 135.3 .................. 209.08 ...............

24.34 .................. 147.2 .................. 100.333 .............. 24.24 .................. 999.99 ...............

3

5

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Teac he r

2. Write these decimals in expanded form. The first one has been done for you.

24.35 = 20 + 4 + 10 + 100

1

1

(2 x 10) + (4 x 1) + (3 x 10 ) + (5 x 100)

a. 136.57 = ............................................................................................................................... .............................................................................................................................................

b. 26.987 = ...............................................................................................................................

© ReadyEdPubl i cat i ons 35.57 = .................................................................................................................................

•f orr evi ew pur posesonl y• .............................................................................................................................................

c.

.............................................................................................................................................

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d. 49.08 = .................................................................................................................................

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............................................................................................................................................. e. 765.297 = .............................................................................................................................

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.............................................................................................................................................

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3. Use < or > to make these true: 35.46

3.546

1.256

125.6

24.78

2.478

1000

1.000

2.002

2.2

860.086

860.068

2.3

3.2

56.65

65.56

980

9.8

12

0.12

154.3

134.5

264.1

264.9

Challenge: Andrew is putting petrol into Dad’s car. The litre gauge has stopped and reads 40.72 litres.

What value in litres does the 7 represent?

E Expressing decimals in an expanded form.

E Ordering decimals with unlike numbers of decimal places.

Page 36

R eady-Ed Publications

Place Value 2 You will need three dice for this activity.

1. Roll the three dice and record the largest possible number you can make from the dice faces. For example, if you roll a 2, 3 and a 5, the largest possible number would be 532.

Throw the dice five times and complete the table below.

r o e t s Bo r e p ok u S Tenths

1.

Thousandths

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Teac he r

2.

Hundredths

3.

4.

5.

2. Now roll each dice twice. Record the face values and then write the smallest possible number in the table below. Do this five times.

© ReadyEdPubl i cat i ons •100’s f orr e vi ew1’spur o seso nl y• 10’s .p 10ths 100ths 1000ths

w ww

2.

3.

4.

5.

. te

m . u

1.

o c . che e r o t r s super

Challenge: Charlie has measured five different distances around the house. The distance from his front door to the rubbish bin is 15.2 m. From the door to the pot plant is 0.152 m, from the door to the end of the verandah is 1.52 m and from the door to the letterbox is 15.22 m. He has also measured the distance between the door and the park and has found it to be 152 m.

Place the objects in order starting with the closet to Charlie’s front door.

E Ordering decimals according to place and face value.

R eady-Ed Publications

Page 37

Fraction and Decimal Inequalities 1. Use <, > or = to make the following true.

2 4

0.5

1 10

1.10

3 10

0.3

10 100

10 50

0.2

20 100

2

1 100

0.1

3 4

100 200

0.6

0.1 0.75

r o e t s Bo r e p ok u S 1 3

4 20

0.25

10 40

0.2

0.4

2. Place a number in the box to make the number sentences below true. The first one has been done for you.

Teac he r

3 5

0.8 =

0.25 =

5

10

0.5 =

= 0.

0.4 =

0.25 =

100 8

0.3 = 0.8 =

0.5 =

4

0.23 =

20 4

0.46 =

0.2 =

6

100 46

3 5

2 20 1 5

0.75 =

0.687 =

3. Complete the following by adding a decimal of your own.

5

ew i ev Pr

18 20

0.9 =

4

687

> ................. = ................... =i ................. <s ................... ©R eadyE dPubl cat i on > .................. < ................. = ................... ................. > ...................

•f or r evi ew pur po>s esonl y • < ..................

2 6

1 5

2 8

4 5

3 5

1 4

40 50

= ..................

> .................

20 50

3 10

80 100

140

90

= ................... 200 = ................. 1000 = ..................

w ww

m . u

4. Write 5 fractions equal to 0.1. ................................................................................................

5. Write 5 fractions equal to 0.4. ................................................................................................

6. Write the decimal that is equal to 3⁄5, 6⁄10, 600⁄1000 and 12⁄20. ........................................................

. te

o c . che e r o t r s1 super

7. Use = or ≠ to make the following true.

0.78

78 100

0.72

7 2

0.25

2 5

0.48

48 100

0.15

1.5

15 1000

5 10

Challenge: Jason and Steve were both in the long jump event. Steve jumped 2.50 metres and Jason jumped 1⁄4 of a metre more than Steve. Exactly how far did Jason jump?

E Comparing values of fractions and decimals.

Page 38

R eady-Ed Publications

Decimals and Fractions 1 1. The grids below have been divided into 100 units. Shade the amount shown underneath.

b. 0.45

c. 0.01

f. 0.05 g. 0.68 © Read yEdPub l i cat i onsh. 0.8 What fraction of the above grids have you shaded? Express in the simplest form.

The first one hasr been done you.

• f o r e vfori e w pur posesonl y• e. 0.96

2.

d. 0.86

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Teac he r

a. 0.2

r o e t s Bo r e p ok u S

20

1

b. .............................

c. ..............................

d. ................................

e. .............................

f. ..............................

g...............................

h. ................................

w ww

3. Complete these using = or ≠.

3 4 4 8

. te

0.75 0.6

2 6

0.4

1 3

0.3

8 10

0.8

4 8

0.6

m . u

a. 100 = 5

2 3

0.3

2 5

0.25

o c . che e r o t r s super 2

4. Use =, < or > to complete these.

1

3 4 8

9 10

1.75 9.8

4 8

5

6 100

2.4 6.5

Challenge: Bridget has painted 0.75 of the garage door.

What fraction does she still need to paint?

E Expressing decimals as fractions.

E Exploring inequalities.

R eady-Ed Publications

Page 39

Decimals and Fractions 2 1. Shade the amounts shown below.

r o e t s Bo r e p ok u S b. 3 4

How many squares are shaded in each box?

a.................................

By looking at the boxes we can see that 0.73 is less than

3 4

b. ..............................

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Teac he r

a. 0.73

.

2. Use the grids below to complete the amounts shown. Add =, < or > for each pair.

© ReadyEdPubl i cat i ons 0.84 0.47 ⁄ ⁄ •f orr evi ew pur posesonl y•

w ww

7

0.69

. te

4

2

⁄10

1

0.35

o c . ch e r ⁄ e o t r s super 3

0.15

8

0.4

20

5

m . u

1

⁄4

⁄20

3. Complete these number sentences using =, < or >. Use squared paper if needed.

3 10

0.13

6.32

6

2 5

Challenge: Which is longer: Page 40

2

5 6

5 20

0.5

7.85

7 20

17

3 5

2.65

0.35

2

3 5

2 5

3.16

0.4 16

3 20

sticks of liquorice or 2.56 sticks of liquorice?

R eady-Ed Publications

Decimals and Fractions 3 1. Express these decimals as simplified fractions. The first one has been done for you.

1

5

2.05 = 2 100 = 2 20 3.2 ...................

4.65 ..................

5.25 ..................

13.26 ...............

7.8 ......................

623.02 .............

0.5 ....................

4.04 ..................

6.008 ...............

22.22 ..................

7.75 .................

3.025 ................

12.6 ..................

10.42 ...............

17.017 ................

15 1000

....................

23 1000

....................

700 1000

....................

r o e t s Bo r e p ok u S

3 10

...................

50 100 ....................

30 100

...................

24 1000

..................

12 100

...................

57 100 ....................

350 1000 ...................

3 1000

..................

49 100

...................

23 100 ....................

7 10

70 100

..................

...................

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Teac he r

2. Convert these fractions to decimals.

3. Change these improper fractions to decimals. The first one has been done for you.

16 10 32 10

© ReadyEdPubl i cat i ons .................. .................. .....................

•f or................... r evi ew ................... pur pos esonl y •

= 4.35

39 10

..................

2795 100 ..................

25 10

143 100

...................

3423 1000 ...................

43 10

198 100

...................

9098 100 ...................

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4. Write five equivalent fractions for each decimal below.

. te

11 10

656 100

..................

264 100

.....................

3456 1000

..................

578 10

.....................

m . u

435 100

6.5 = .........................................................................................................

o c . che e r o t r s super

2.25 = .......................................................................................................

0.75 = .......................................................................................................

3.6 = .........................................................................................................

9.75 = .......................................................................................................

Challenge: Matthew is counting his savings and has calculated that he has 687 ¢.

Express this amount as a decimal and also as a fraction.

E Converting mixed numerals to decimals.

E Finding equivalent fractions for decimals.

R eady-Ed Publications

Page 41

Decimals and Equivalent Fractions So far the fractions we have changed to decimals have all had a denominator which is a multiple of 10, such as 10, 100 and 1000. Sometimes it is necessary to convert fractions that cannot evenly be divided into 100.

36

For example 60 needs to be divided by 6 so that the denominator is 10.

36 ÷ 6 = 6 60 ÷ 6 = 10

r o e t s Bo r e p ok u S

= 0.6

1. Try these by first simplifying the fraction so that the denominator is 10.

35

8

64

= .............

14 20

= .............

81

56

= ............. 60 = ............ 90 = ............. 70 = ............

48 80

= .............

36 40

= ............

63 90

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24 30

42

= 10 = 0.8 70 = ............. 80

Teac he r

32 40

= .............

28 70

= ............

2. Now try converting these fractions to a decimal. Some of them are quite tricky.

2 5

4

3 5

3 4

5

3

= ......... 25

= ............. 15

3

= ...........

10 20 = ......... 3 12 = ............. 4 20 = ............ 6 50

4

9

= ............

4

2 8

= ...........

13

3

2 5

= ...........

9

= ...........

2 100= ...........

= ...........

3

© ReadyEdPubl i cat i ons •f orr evi ew pur posesonl y•

3. Change these decimals to fractions by filling in the boxes below. The first one has been done for you.

70

700

14

35

10

100

1000

20

50

100

1000

5

w ww

7

b. 0.2 =

c. 0.8 =

d. 1.0 =

e. 1.5 =

10

. te

10

10

10

m . u

a. 0.7 =

20

o c . che e r o t r s super 100

1000

5

40

100

1000

5

9

100

1000

2

6

Challenge: Fiona buys her stamps in sets and there are twenty stamps to a set. She has five complete sets and another set that contains only 12 stamps.

Express the number of sets Fiona has as a decimal.

E Converting fractions to decimals by first expressing fractions with denominators with multiples of 10.

Page 42

R eady-Ed Publications

Review 3: Expressing Fractions as Decimals Answer the following quick questions.

1.

What is eight tenths as a decimal? ..............................................................................

2.

Write five and nine tenths as a decimal. .......................................................................

3.

What is seven point four as a fraction?.........................................................................

4.

Write two point two five as a fraction. ...........................................................................

5.

Which is greater: 0.60 or 4 ? ....................................................................................

Teac he r

3

If you have $10.00 pocket money and spend a quarter of it, how much would you have

ew i ev Pr

6.

r o e t s Bo r e p ok u S

left? .............................................................................................................................

7.

What is zero point two five as a fraction? .....................................................................

8.

True or False: Six and three tenths is less than six and four fifths. .................................

2 5

9.

of a dollar is equal to how many cents? ..................................................................

© ReadyEdPubl i cat i ons 11. True or False: Zero point four five is the same as forty five over a hundred. ...................

•f orr evi ew pur posesonl y• 12. What is a quarter of twenty? .........................................................................................

1

10. 4 of $2.00 is equal to how many cents? ....................................................................

13. Two thirds of nine is equal to ........................................................................................

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15. True or False:

3 4

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Problems: 1.

3

m . u

14. Express two and four fifths as a decimal. ..................................................................... is greater than 3.65. ....................................................................

o c . che e r o t r s super

Anne bought fifteen bananas from the shop and gave five to her brother.

What fraction does she still have? ................................................................................

2.

Steve rode 6.75 km on the weekend. Express this amount as a fraction. ......................

3.

Suzy received $6 pocket money and spent two thirds of it on a book.

How much did the book cost? .......................................................................................

4.

Rebecca and Michael went fishing and caught 20 fish. Eight of the fish were undersized and so they threw them back.

What fraction do they have left? ....................................................................................

R eady-Ed Publications

Page 43

Calculating Decimals You will need a calculator for these activities.

So far we have looked at the methods for converting fractions into decimals.

3

In order to convert 4 to a decimal using a calculator we need to type in 3 ÷ 4.

1. Use your calculator to calculate decimals for the following: 4 5

..................

5 8

2 8

..................

9 20

45 60

..................

1 4

...................

18 20

..................

14 28

.....................

...................

28 50

...................

27 30

..................

15 40

.....................

...................

18 50

...................

50 200

..................

375 1000 .....................

ew i ev Pr

Teac he r

35 70

r o e t s Bo r e p ok u S ...................

2. Using your calculator find three different fractions that are equal to 0.75. ...........................................

3.

............................................

Find three fractions equal to 0.6. ...........................................

4.

............................................

............................................

............................................

© ReadyEdPubl i cat i ons •f orr evi ew pur pose sonl y• ........................................... ............................................ ............................................

Find three fractions equal to 0.25.

5. Sometimes when we calculate a decimal we end up with a recurring number.

m . u

1

w ww

Divide 1 by 3 ( 3). What answer did you get? ...........................................................

This can be recorded as

. te

1 3

y ≈ 0.33. It is known as a recurring number.

o c . che e r o t r s super

Express these fractions as recurring numbers.

2 3

≈ ....................

Challenge:

1 6

≈ ....................

1 9

≈ .....................

4 6

≈ ....................

7 9

≈ .....................

Divide fractions with a denominator of 9 ( 1⁄9, 2⁄9, 3⁄9, etc.) and record your answers.

What pattern can you find? ...............................................................................................

Find another pattern similar to this one and record the fractions you calculated.

E Using a calculator to find the decimal equivalent for fractions.

Page 44

R eady-Ed Publications

Percentages 1 So far we know that decimals and fractions can be used to represent the same amount. A percentage is another way of expressing a part of a whole. Per means ‘for every’ and cent means ‘100’ so it is easy to remember that percent means ‘for every 100’. For example, if we had 100 students playing on the school sports field and 56 of them were girls, we could say that 56 percent (56 out of 100) of the students playing are girls.

1. We use the symbol % to express percentage. Name three everyday places where you might see this symbol.

r o e t s Bo r e p ok u S

...........................................

............................................

............................................

.............................

..............................

...............................

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Teac he r

2. Look at the shaded amounts below. What percentage of each grid has been shaded?

...............................

© ReadyEdPubl i cat i ons = 50% • f orr evi ew pur posesonl y•

3. Convert these fractions to a percentage. The first one has been done for you.

5 10

50

= 100

..................

3 100

...................

15 100

...................

170 1000 ..................

8 10

.....................

1 100

..................

25 100

...................

4 10

...................

400 1000 ..................

2 5

.....................

w ww

4. Express these percentages as decimals and then simple fractions.

. te 75

3

75% = 0.75 = 100 = 4

m . u

5 10

o c . che e r o t r s super

a. 60% = ................... b. 80% = ................... c. 50% = ...................

d. 32% = ...................

e. 24% = ................... f. 40% = .................... g. 90% = ...................

h. 25% = ...................

5. Express these percentages as decimals.

35% ....................................

23% ....................................

89% ...........................................

67% ....................................

79% ....................................

100% .........................................

Challenge: Melanie scored 98% in a maths test. If there were fifty questions, how many questions must Melanie have answered correctly?

E Introduction percentages and exploring the relationship between decimals, fractions and percentages.

R eady-Ed Publications

Page 45

Percentages 2 In order to convert these fractions to a percentage, we need to find an equivalent fraction with a denominator of 10, 100 or 1000. Look at these examples and then convert these fractions to a percentage.

1. 200 = 100 = 20%

40

20 200

= ........................

30 50

= ........................

2 20

= ........................ 800 = ........................

40 60

= ........................

30 40 25 50

= ........................

= ........................

40

460 2000 = .........................

15 20

= .........................

= ........................ 400 = ........................

600 800

= .........................

5 25

9 20

= .........................

300

= ........................

Match the percentage on the left with the correct fraction and decimal. There may be more than two answers.

a) 25%

0.25

2.5

2 5

25 100

1 4

b) 55%

5.5

0.55

5 10

55 100

1 5

c) 32%

3.2

32 10

0.23

d) 80%

4 5

ew i ev Pr

2.

1 4

r o e t s Bo r e p ok u S

........................

Teac he r

300 3000 =

20

0.14

0.5

32 100

16 50

© Ready Ed8.0Pub0.45 l i cat i ons 0.08 0.8 •f orr evi e w pur po seso nl y• e) 15% 1.5 0.15 0.015 5.1 0.32

8 1000

w ww

Wor d Pr oblems: ord Problems:

15 100

m . u

1 5

1. Sophie spent 10% of her pocket money on a new pencil. What percentage of her pocket

. te

money does she have left? ............................................................................................

o c . che e r o t r s super

2. Taylor spent 4⁄5 of his spare time reading. What percentage of time is this?

.....................................................................................................................................

3. Marcelle collected snails in the garden. She found 50% of them near the rose bushes, 15% of them near the hose and 25% around the clothes line.

What percentage were found elsewhere? ......................................................................

4. Katie and Greg have 100 chocolates in a jar. If 37 of them have hard centres and the rest are soft centred, what percentage have soft centres?..............................................

5. Lilly had 10 000 Frequent Flyer points. She received a 10% bonus for reaching the 10 000 mark. How many points does she now have? ....................................................

E Finding percentages for fractions with denominators of multiples of 10.

Page 46

R eady-Ed Publications

The Relationship between Decimals, Fractions and Percentages 1. Complete the table below. The first one has been done for you.

Fraction 1 4 1 5

Per centage ercentage

r o e t s Bo r e p ok u S 0.25

25%

0.2

36%

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Teac he r

2 10

Decimal

0.42

4 5

64%

3 20

© ReadyEd Publ i cat i ons 0.95 •f orr evi ew pur posesonl y• 28 100

73%

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m . u

4 200 175 1000

. te

4.5

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20.5%

2. Use = or ≠ in the boxes below.

0.75

75%

3 4

0.8

80%

4 5

34%

1 4

25%

0.25

0.8

5 25

25%

7 25

2 4

0.28

50%

5.0

17 20

0.59

Challenge: Joey scored 65% on the science test, Shelley got 0.75 of the questions correct and Matt answered 4⁄5 of the test correctly. Which student received the highest mark for the test?

E Matching fractions with equivalent decimal values and percentages.

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Calculating Percentages We know that 10% means 10 for every 100, 20% means 20 for every 100 and so on.

We also know that 10% of 200 is 20, 10% of 1000 is 100, 10% of 750 is 75 and so on.

1. Find 10% of the following amounts: $30 .............. $60 ............... $130 ............

$290............. $400 ............

$2000 ............

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$25 .............. $48 ............... $120 ............

$293............. $498 ............

$450 ..............

20 ................ 30 ................. 100 ..............

150............... 200 ..............

1000 ..............

1200 ............ 64 ................. 65 ................

250............... 16 ................

17 ..................

44 ................ 28 ................. 350 ..............

12.6 .............. 25.8 .............

243.2..............

3. Find 20% of the following amounts: $3.00 ................

$2.30 ................. $1.20 ................. $4.80 ................ $9.60 ...................

©$25.50 Re ady Ed Pub l i c at i o ns............... ............... $16.30 ............... $29.80 .............. $320.00 f or r e vi ewbelow: pur posesonl y• Subtract• 20% from each of the amounts

$0.50 ................

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2. Find 50% of the following amounts:

$10.80 ............... $4.00 ................. $2.00 ................ $150 ....................

$1000 ...............

$3.50 ................. $6.00 ................. $7.20 ................ $9.50 ...................

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Wor d Pr oblems: ord Problems:

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$9.60 ................

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1. Rick received a 10% discount on his new basketball. Originally, the basketball cost

$50.00. How much did Rick pay for the ball? ...............................................................

2. In Ali’s class 5 of the students are away sick. If there are normally 20 students in the class, what percentage of the students is absent? .........................................................

3. Donelle sold 30 icecreams at the football. The following week she sold 10% more.

What amount did she sell? ............................................................................................

4. Tanya correctly answered 180 questions out of 200 in an exam. What percentage did she answer correctly? ...................................................................................................

E Finding percentages of whole numbers and money.

E Carrying out activities which give experience with simple everyday usage of percentages.

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Ratios A ratio describes the relationship of two numbers at the same time. For example, Sarah, Jane and Emily are sharing a cake which has been cut into nine pieces. For each person there are three pieces of cake. This is known as a one to three ratio and can be written as either 1⁄3 or 1:3.

Express the following ratios as fractions: 1. If each table has four chairs, what is the ratio of tables to chairs? ...........................................

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2. If each girl has five work files what is the ratio of girls to files? ................................................

3. If each student has four pencils what is the ratio of students to pencils? ..................................

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4. There are two tennis balls in the shed for every three students.

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What is the ratio of tennis balls to students? ...........................................................................

5. Steve, Louis and Mark each have a bike. What is the ratio of people to bikes? ......................

6. If the ratio of computers to students is 1:5 during computing lessons, how many students share a computer? .........................

7. Eight people are sharing five pies between them. What is the

© ReadyEdPubl i cat i ons Sixteen students have four basketballs between them.

•f orr evi ew pur posesonl y• How can this be represented as a fraction? ................................ fraction that each person will receive? ........................................

8.

What is the ratio of shirts to students? .........................................

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10. The ratio of cows to sheep in Farmer Joe’s field is 4:5.

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If there are 20 cows how many sheep must there be? .................

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9. There are sixty sports shirts to be shared amoung 20 students.

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11. The ratio of cauliflowers to lettuces in Farmer Joe’s garden is 1:7.

If there are two rows of cauliflowers how many rows are there of lettuces? ..............................

12. In Miss Take’s class one quarter of the students live on farms. Express this as a ratio. .............................................................................................................................................

13. The record of shipwecks at Rocks Harbour shows that one in every eight ships visiting was shipwrecked. Express this as a fraction. .........................................................................

14. In the library the ratio of students to chairs is 3:1. What fraction of the class can sit on a chair during library times? .....................................................................................................

E Exploring the relationship between ratios and fractions.

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Mixed Problems 1 1.

A bike is marked at $200.00. The retailer decides to mark the bike down by 10%.

What will be the new cost of the bike? ...................................................................................

2.

Each year Josh’s dad pays 10% of his salary into a superannuation fund. If he makes $35 000 a year, how much money will go into the fund each year?.........................................

3.

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Julia made a chocolate cake and flour made up 65% of the mixture. What percentage of the cake is not flour? .............................................................................................................

Sam is buying books on sale at the local newsagent. Each book is discounted by 20% of

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what the marked price states. If he wants to buy a book marked at $20.00, what will he

5.

Jarrad took thirty seconds to brush his teeth.

What fraction of a minute is this? ..................................

6.

Helen took 45 minutes to walk to the beach.

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actually pay after the discount? .....................................

© ReadyEdPubl i cat i ons What was totalr distance she rode?

•thef o r ev i e w.................................................................................. pur posesonl y• What percentage of an hour is this? ..............................

7.

Claudia rode 2 km in an hour. This was 25% of the total amount she rode all day.

8.

Billy had a bag of fruit. 50% of the fruit were bananas, 25% were apples, 10% were

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strawberries and the rest were peaches. What fraction of the bag did the peaches take

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up? .......................................................................................................................................

Jeff spent 3⁄5 of his savings on some CDs. If each CD cost $20, and Jeff bought 3, how

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much money did Jeff start off with? .......................................................................................

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10. In the final exam, 30% of the students failed. If 140 students passed how many students

must have failed? ................................................................................................................

11. Karen ran 2.46 km on Wednesday. Express this distance as a fraction. ................................

12. The bank is offering interest rates of 5% on savings accounts. How many cents for each dollar will the bank pay? ........................................................................................................

13. At the local mine 85% of the miners were under 30 years of age. What fraction of the miners were over thirty? .........................

Express this amount as a decimal .....................

14. Justin, Thomas and James had six icecreams to share. What is the ratio of boys to icecreams? ..........................................................................................................................

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Mixed Problems 2 1.

In Mario’s class one quarter of the students wear glasses. If there are 24 students in the class how many wear glasses? .......................

2.

Leanna walked 3.56 km and Stephanie walked three and a half km. Who walked the furthest distance? ........................................

3.

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The ratio of beachgoers to umbrellas was six to one. What fraction

of beachgoers had an umbrella? .......................................................

The ratio of football spectators to raincoats was 20:1.

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Express this amount as a percentage. ...............................................

Katrina took 40 minutes to complete her homework.

What fraction of an hour is this? .........................................................

6.

Alex spent two hours a day practising the piano.

What fraction of the day is this? .........................................................

7.

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5.

Lara spent two hours on homework each night. If 1⁄4 is spent on maths and 50% is spent on

© ReadyEdPubl i cat i ons How are spent on maths? .........................................................................

•many f ominutes rr e v i ew pu r posesonl y•

history what is the deimal amount left for other subjects? ....................................................... a.

b. How many minutes are spent on history? ........................................................................

The football team won 80% of its matches during the last season. If twenty matches were

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played, what was the total number of games won? ................................................................

The rowing team has won 0.65 of its total races. What fraction of races did it lose?

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10. Bill and Ted are playing chess. Bill has beaten Ted 60% of the time. If they have played fifty games, how many has Bill won? ...........................................................................................

11. Sam and Tess are playing cards. They have played eighteen games and Tess has won two thirds of the games. How many games has Sam won? .........................................................

12. Maria invited 35 guests to her birthday party, however only 5⁄7 of the guests are able to come.

How many of the guests will be able to attend? .....................................................................

13. Marguerite spent $11.00 on a new hat. She now has 4⁄5 of her savings left. a. What percentage of her savings was spent on the hat? ................................................... b. How much money did Marguerite start off with? ..............................................................

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Fractions Answers

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Answers correspond to row order of questions.

Page 6 Introduction to Fractions 1. 4, 7, 8, 8, 10, 6,9; 2. 2⁄4, 5⁄9, 6⁄8, 2⁄4; 3. Check diagrams; 4. 2⁄4, 4⁄10, 2⁄6.

Challenge: They would both be equal to a whole.

Page 7 Equivalent Fractions 1 1. Check diagrams; 2a. 1⁄4, 2⁄8; b. 2⁄4, 1⁄2; c. 2⁄5, 4⁄10.

Challenge: Anthony.

Page 8 Fractions as Parts of a Whole 1. 1⁄4, 2⁄8, 3⁄6, 5⁄10, 6⁄9; 2. Check diagrams; 3. 6⁄8, 4⁄8, 4⁄12, 2⁄8, 2⁄4, 6⁄9.

Challenge: 12⁄20 = 3⁄5.

Page 9 Equivalent Fractions: Exercises 1. Answers will vary; 2. ≠, =, ≠, ≠, =, =, ≠, ≠, =, ≠, ≠, =; 3. <, <, >, >, <, =, =, <, <, <, >, >; 4. Check diagrams; 5. 3⁄7, 2⁄4, 5⁄8, 6⁄9, 4⁄6, 9⁄10.

Challenge: Michelle.

Page 10 Equivalent Fractions 2 1a. 8⁄12, 6⁄9; b. 2⁄4, 5⁄10, 6⁄12; c. 4⁄12, 3⁄9; d. 2⁄8, 3⁄12; e. 2⁄10, 3⁄15, 4⁄20; 2. Check diagrams; 3. 1⁄4, 1⁄3, 3⁄6, 3⁄5, 2 ⁄3, 3⁄4, 5⁄6, 4⁄4.

Challenge: Maree - 4, Paul - 5.

Page 11 Equivalent Fractions 3 1. 3, 2, 3, 2, 4, 6, 2, 2.

Challenge: Greater than.

Page 12 Matching Fractions 1. 2, 3, 2, 5, 4, 9, 8, 9, 1, 2, 4, 4; 2. 10, 6, 20, 4, 10, 12, 5, 6, 2, 3, 3, 4; 3. 4⁄5, 1⁄2, 1⁄2, 1⁄2, 1⁄2, 1⁄2, 1⁄3, 1 ⁄2, 1⁄2, 2⁄5, 3⁄4, 1⁄2, 1⁄2, 7⁄10, 9⁄10, 4⁄5, 1.

Word Problems: 1. 12; 2. red - 1⁄2, blue - 1⁄2; 3. 2 pieces each of bananas, apricots and pears; 4. Painting - 2, drawing - 2, pottery - 6.

Page 13 Building Up Fractions 1. Answers will vary; Check diagrams; a. 1⁄3, 2⁄6, 3⁄9, 4⁄12; b. 1⁄4, 2⁄8, 3⁄12, 5⁄20; c. 2⁄5, 4⁄10, 6⁄15, 8⁄20.

Challenge: Footballers - 12, Basketballers - 8, Tennis Players - 4.

Page 14 Comparing Fractions 1. Red - 20, blue - 25, yellow - 20; 2. Unshaded = 5⁄100; 3. Answers will vary.

Challenge: 55⁄100 = 11⁄20.

Page 15 Equivalent Fractions 4 1. 3⁄4, 1⁄4, 1⁄3, 3⁄5; 2. Check diagrams; 3. =, =, =, ≠, ≠, ≠, =, ≠; 4. <, >, <, <, >, <, <, =, >, >, =, >.

Challenge: Emily.

Page 16 Fraction Inequalities 1. 3⁄6, 1⁄2, 1⁄2, 6⁄7, 4⁄5, 5⁄8, 4⁄8, 2⁄5, 8⁄9, 1⁄2, 5⁄5; 2. ≠, =, =, =, ≠, =, =, =; 3. Answers will vary; 4. 3⁄4, 1⁄3, 4⁄5; 5a. 4⁄9 = 8⁄18, 3⁄8 = 6⁄16, 5⁄10 = 50⁄100; b. 4⁄6 = 8⁄12, 7⁄8 = 14⁄16, 3⁄4 = 75⁄100, 2⁄3 = 60⁄90; c. 3⁄9 = 2⁄6, 2⁄8 = 4⁄16, 4 ⁄10 = 2⁄5, 8⁄12 = 4⁄6.

Challenge: Susie.

Page 17 Review 1: Equivalent Fractions 1. 1⁄5, 1⁄3, 2⁄5, 1⁄2, 4⁄6, 3⁄3; 2. 10⁄10, 2⁄5, 3⁄9, 1⁄4, 1⁄6, 1⁄7; 3. <, >, >, >, >, <, >, <, >, <; 4. =, <, >, <, =, =, <, =, >, <, =, =; 5. >, <, <.

Challenge: 60 c.

Page 18 Simplifying Fractions 1 1. Answers will vary; 2 and 3. Check diagrams; 4. 1⁄2; 5. red: 4⁄12, 3⁄9, 2⁄6, 5⁄15; blue: 4⁄8, 20⁄40, 5⁄10, 50 ⁄100, 6⁄12, 100⁄200.

Challenge: Second game 18⁄36 = 1⁄2.

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Answers correspond to row order of questions.

Page 19 Simplifying Fractions 2 1. 4⁄6, 2⁄5, 1⁄3, 1⁄2, 2⁄6; 2. 1⁄4, 4⁄5, 1⁄6, 1⁄3; 3. 1⁄3, 1⁄5, 1⁄2, 1⁄2, 1⁄4, 1⁄4, 1⁄5, 1⁄5, 1⁄2, 1⁄2, 1⁄3, 1⁄2; 4. 1⁄5, 1⁄5, 1⁄2, 1⁄9, 1⁄6, 2 ⁄5, 1⁄10, 1⁄10, 1⁄6, 3⁄4, 1⁄4, 4⁄5; 5. 3⁄4 = 6⁄8, 7⁄8 = 14⁄16, 9⁄20 = 18⁄40, 200⁄1000 = 4⁄20, 3⁄50 = 6⁄100, 6⁄200 = 3⁄100.

Challenge: 3⁄4.

Page 20 Simplifying Fractions 3 1. 10⁄10, 5⁄6, 7⁄8, 9⁄10, 8⁄10, 2⁄10, 8⁄8, 2⁄5, 6⁄10, 2⁄5, 1⁄10, 5⁄20; 2. 7⁄8, 9⁄10, 5⁄6, 17⁄18, 5⁄8, 1⁄3, 1⁄3, 1⁄2, 3⁄4, 3⁄5, 6⁄25, 24⁄25; Word Problems: 1. Mark - 1⁄2, Tony - 5⁄8; 2. 8; 3. 1⁄5; 4. 3⁄4; 5. 1⁄4; 6. 4⁄5; 7. 1⁄6; 8. 1⁄3.

Page 21 Addition of Fractions 1. 4⁄4, 6⁄6, 5⁄7; 2. 5⁄8, 2⁄2, 6⁄7, 7⁄9, 3⁄4, 9⁄9, 6⁄8, 17⁄20, 55⁄10, 45⁄50, 79⁄100; 3. 3⁄3, 4⁄5, 9⁄10, 18⁄20.

Challenge: 3⁄4.

Page 22 Improper Fractions 1 1. 11⁄5, 22⁄6, 12⁄10, 13⁄9, 13⁄5, 13⁄7, 14⁄8.

Challenge: 24⁄4.

Page 23 Improper Fractions 2 1. 11⁄9, 7⁄6, 12⁄10, 9⁄7, 7⁄5, 11⁄8, 20⁄15, 27⁄20; 2. 11⁄2, 23⁄4, 54⁄5, 22⁄3, 31⁄2, 32⁄3, 21⁄2; 3. 1, 1, 1, 3, 4, 9, 2, 7; 4. 6; 5. 18⁄3; 6. 41⁄2 + 31⁄2 = 8.

Page 24 Mixed Numerals 1 1. Check diagrams; 2. 10⁄6, 14⁄4, 17⁄6, 13⁄2, 19⁄4, 29⁄10, 23⁄6, 18⁄7, 17⁄3, 44⁄5, 345⁄100, 255⁄200; Challenge: 13⁄5.

Page 25 Mixed Numerals 2 1. 11⁄3, 21⁄2, 21⁄3, 4, 12⁄7, 14⁄6, 2, 41⁄2, 22⁄3, 3, 32⁄6, 22⁄9, 85⁄10, 62⁄5, 41⁄6, 51⁄3, 54⁄7, 5; 2. 13⁄4, 17⁄3, 38⁄6, 17⁄7, 19 ⁄2, 71⁄10, 68⁄8, 46⁄6, 89⁄9, 29⁄5, 83⁄20, 360⁄100; 3. 1⁄8, 6⁄8, 8⁄8, 13⁄8, 20⁄8, 25⁄8, 31⁄8, 42⁄8; 4. >, <, >, >, <, =, >, <, =, >, =, =.

Challenge: 41⁄4.

Page 26 Addition of Fractions: Exercises 1 1. 51⁄5, 43⁄6, 7, 122⁄8, 6, 71⁄7, 82⁄8, 4; 2. 31⁄4 + 22⁄4 = 53⁄4; 32⁄3 + 12⁄3 = 51⁄3; 12⁄6 + 13⁄6 = 25⁄6.

Challenge: 32⁄5 + 54⁄5 = 91⁄5.

Page 27 Addition of Fractions: Exercises 2 1. 79⁄10, 718⁄25, 41⁄3, 141⁄2, 65⁄6, 53⁄4, 101⁄5, 92⁄5; 2. 4, 2, 2, 3, 1, 4; 3. 11⁄9, 41⁄2, 51⁄4, 41⁄2.

Challenge: 23⁄4 + 21⁄4 + 11⁄4 = 55⁄4 = 61⁄4.

Page 28 Addition of Fractions: Exercises 3 1 ⁄4, 5⁄8; 1. 11⁄8, 11⁄6, 8⁄9, 3⁄10; 2. 2⁄3, 1, 4⁄5, 7⁄9, 13⁄8, 1, 1, 1, 1⁄10.

Challenge: 13⁄8 of a game.

Page 29 Subtraction of Fractions 1 1. 3⁄6 (1⁄2), 2⁄5, 4⁄7, 3⁄7, 15⁄6, 5⁄10 (1⁄2), 95⁄8, 0; 2. 5⁄8, 4⁄9, 1⁄2; 3. 22⁄7, 41⁄2, 61⁄8, 10, 63⁄6 (1⁄2), 13⁄8, 17⁄10, 1⁄7.

Challenge: 1⁄3.

Page 30 Subtraction of Fractions 2 1. 13⁄6 (1⁄2), 26⁄9 (2⁄3), 6⁄10 (3⁄5), 6⁄7; 2. 18, 20, 12, 4, 10, 18, 21; 3. 71⁄4, 67⁄10, 31⁄3, 94⁄6, 10⁄20, 325⁄100 (1⁄4), 523⁄30, 18⁄9; 4. 13⁄5, 91⁄3, 45⁄6, 12⁄5, 81⁄7, 313⁄20, 48⁄50 (24⁄25), 41⁄5.

Challenge: 21⁄3.

Page 31 Subtraction of Fractions 3 1. 21⁄6, 2⁄3, 16⁄7, 13⁄8, 61⁄2, 64⁄5, 13⁄10, 15⁄6, 23⁄5, 5⁄9, 33⁄5, 13⁄7; 2. 1, 5, 2, 3, 20⁄100, 4, 3, 6, 4, 8.

Challenge: 21⁄4, 22.

Page 32 Review 2: Addition and Subtraction 1. 2⁄3, 2⁄4, 2, 33⁄5, 21⁄3, 33⁄4, 81⁄5, 131⁄10, 51⁄7, 11⁄8, 93⁄4, 53⁄5; 2. 1⁄3, 2⁄4, 4⁄9, 1, 11⁄5, 22⁄3, 14⁄5, 18⁄9, 1⁄6, 92⁄5, 1 ⁄3, 21⁄5.

Word Problems: 1. 5⁄9; 2. 13⁄4; 3. none; 4. 74; 5. 140 cm.

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Answers correspond to row order of questions.

Page 33 Decimal Introduction 25 hundredths; 1. Check table; 2a. 25.034; b. 136.206; c. 1.05; d. 27.404; e. 6.023.

Challenge: 200.2.

Page 34 Decimal and Fraction Relationship 1. Check diagrams; 2. 0.07, 0.3, 0.5, 0.95; 3. 5⁄10 = 0.5, 6⁄10 = 0.6, 3⁄10 = 0.3.

Challenge: 1.2.

Page 35 Expressing Fractions as Decimals 1. 2⁄10, 5⁄10, 6⁄10, 23⁄100, 98⁄100, 47⁄100; 2. 0.1, 0.7, 0.1, 0.34, 0.28, 0.567; 3. 0.4, 0.4, 0.2, 0.4, 0.3, 0.2, 0.5, 0.4, 0.4, 0.75, 0.25, 0.75, 0.2, 0.5, 0.1, 0.75, 1, 0,5; 4. >, <, =, <, =, <.

Challenge: 0.18 Page 36 Place Value 1 1. 7 x 10, 8 x 1⁄100, 1 x 1, 3 x 10, 0 x 1⁄10, 4 x 1⁄100, 4 x 10, 3 x 1⁄1000, 4 x 1⁄100, 9 x 1⁄10; 2a. (1 x 100) + (3 x 10) + (6 x 1) + (5 x 1⁄10) + (7 x 1⁄100); b. (2 x 10) + (6 x 1) + (9 x 1⁄10) + (8 x 1⁄100) + (7 x 1⁄1000); c. (3 x 10) + (5 x 1) + (5 x 1⁄10) + (7 x 1⁄100); d. (4 x 10) + (9 x 1) + (0 x 1⁄10) + (8 x 1⁄100); e. (7 x 100) + (6 x 10) + (5 x 1) + (2 x 1⁄10) + (9 x 1⁄100) + (7 x 1⁄1000); 3. >, <, >, >, <, >, <, <, >, >, >, <.

Challenge: 7⁄10 of a litre.

Page 37 Place Value 2 Challenge: Pot plant, verandah, rubbish bin, letterbox, park.

Page 38 Fraction and Decimal Inequalities 1. =, <, =, =, =, <, <, =, <, >, =, <; 2. 6, 1, 3, 1, 4, 25, 6, 23, 3, 20, 20, 5, 100, 1000; 3, 4 and 5.

Answers will vary; 6. 06; 7. =, ≠, ≠, ≠, =, =.

Challenge: 2.75.

Page 39 Decimals and Fractions 1 1. Check diagrams; 2b. 9⁄20; c. 1⁄100; d. 43⁄50; e. 24⁄25; f. 1⁄20; g. 34⁄50; h. 4⁄5; 3. =, ≠, =, ≠, ≠, =, ≠, ≠; 4. =, >, =, <.

Challenge: 1⁄4.

Page 40 Decimals and Fractions 2 1. Check diagrams; a. 73; b. 75; 2. <, >, <, >, =, =; 3. >, <, >, =, <, =, >, <.

Challenge: 25⁄6.

Page 41 Decimals and Fractions 3 1. 31⁄5, 413⁄20, 51⁄4, 1313⁄50, 74⁄5, 6231⁄50, 1⁄2, 41⁄25, 61⁄125, 2211⁄50, 73⁄4, 31⁄40, 123⁄5, 1021⁄50, 1717⁄1000; 2.

0.3, 0.5, 0.3, 0.024, 0.015, 0.12, 0.57, 0.35, 0.35, 0.003, 0.023, 0.49, 0.23, 0.7, 0.7, 0.7; 3. 1.6, 3.9, 1.43, 1.98, 1.1, 3.2, 2.5, 4.3, 6.56, 2.64, 27.95, 3.423, 90.98, 3.456, 57.8; 4. Answers will vary.

Challenge: 6.87, 687⁄100.

Page 42 Decimals and Equivalent Fractions 1. 0.5, 0.8, 0.7, 0.9, 0.8, 0.8, 0.7, 0.6, 0.9, 0.7, 0.4; 2. 0.4, 3.75, 0.2, 0.2, 4.25, 9.4, 4.6, 10.2, 3.75, 4.65, 6.06, 2.03; 3b. 2, 20, 200, 1, 4; c. 8, 80, 800, 4, 32; d. 10, 100, 1000, 5, 9; e. 15, 150, 1500, 3, 9.

Challenge: 5.6.

Page 43 Review 3: Expressing Fractions as Decimals 1. 0.8; 2. 5.9; 3. 72⁄5; 4. 21⁄4; 5. 3⁄4; 6. $7.50; 7. 1⁄4; 8. True; 9. 40 c; 10. 50 c; 11. True; 12. 5; 13. 6; 14. 2.8; 15. True.

Problems: 1. 2⁄3; 2. 63⁄4; 3. $4.00; 4. 3⁄5.

Page 44 Calculating Decimals 1. 0.8, 0.625, 0.25, 0.9, 0.5, 0.25, 0.45, 0.56, 0.9, 0.375, 0.75, 0.5, 0.36, 0.25, 0.375; 2, 3 and 4. y y y y y Answers will vary; 5. 0.66, 0.166, 0.11, 0.66, 0.77. y Challenge: The decimal rises by 0.11 for each fraction.

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Answers correspond to row order of questions.

Page 45 Percentages 1 1. Shop windows, Bank advertisements, newspaper advertisements, statistics; 2. 25%, 30%, 50%, 97%; 3. 50%, 3%, 15%, 17%, 80%, 1%, 25%, 40%, 40%, 40%; 4a. 0.6 = 3⁄5; b. 0.8 = 4⁄5; c. 0.5 = 1⁄2; d. 0.32 = 8⁄25; e. 0.24 = 6⁄25; f. 0.4 = 2⁄5; g. 0.9 = 9⁄10; h. 0.25 = 1⁄4; 5. 0.35, 0.23, 0.89, 0.67, 0.79, 1.

Challenge: 49 Page 46 Percentages 2 1. 10%, 25%, 60%, 10%, 5%, 75%, 66.6%, 75%, 75%, 75%, 10%, 50%, 20%, 45%; 2b. 0.55, 55 ⁄100, c. 0.32, 32⁄100, 16⁄50, 4⁄5, 0.8; e. 0.15, 15⁄100.

Word Problems: 90%, 80%, 10%, 63%, 11 000.

Page 47 The Relationship between Decimals, Fractions and Percentages Fraction Decimal Percentage 1 ⁄4 0.25 25% 1 ⁄5 0.2 20% 2 ⁄ 10 0.2 20% 36 ⁄ 100 0.36 36% 42 ⁄ 100 0.42 42% 4 ⁄5 0.8 80% 16 ⁄ 25 0.64 64% 3 ⁄ 20 0.15 15% 95 ⁄ 100 0.95 95% 28 ⁄ 100 0.28 28% 73 ⁄ 100 0.73 73% 4 ⁄200 0.02 2% 175 ⁄1000 0.175 17.5% 450 ⁄ 100 4.5 450% 205 ⁄ 100 0.205 20.5% 2. =, ≠, =, ≠, ≠, =, =, ≠, =, ≠.

Challenge: Matt.

Page 48 Calculating Percentages 1. $3, $6, $13, $29, $40, $20, $2.50, $4.80, $12.00, $29.30, $49.80, $45; 2. 10, 15, 50, 75, 100, 500, 600, 32, 32.5, 125, 8, 8.5, 22, 14, 175, 6.3, 12.9, 121.6; 3. 60%, 46 c, 24 c, 96 c, $1.92, 10 c, $5.10, $3.26, $5.96, $64; 4. $7.68, $8.64, $3.20, $1.60, $120, $800, $2.80, $4.80, $5.76, $7.60.

Word Problems: $45.00, 25%, 33, 90%.

Page 49 Ratios 1. 1:4; 2. 1:5; 3. 1:4; 4. 2:3; 5. 1:1; 6. 5; 7. 5⁄8; 8. 1⁄4; 9. 3:1; 10. 25; 11. 14; 12. 1:4; 13. 1⁄8; 14. 1⁄3.

Page 50 Mixed Problems 1 1. $180.00; 2. $3 500; 3. 35%; 4. $16; 5. 1⁄2; 6. 75%; 7. 8 km; 9. $100; 10. 60; 11. 233⁄50; 12. 5; 13. 3⁄20, 0.15; 14. 1:2.

Page 51 Mixed Problems 2 1. 6; 2. Leanna; 3. 1⁄6; 4. 5%; 5. 2⁄3; 6. 1⁄12; 7. 0.25; a. 30; b. 60; 8. 16; 9. 7⁄20; 10. 30; 11. 6; 12. 25; 13a. 20%, b. $55.00.

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