Fractions

Fractions

H SERIES

Fractions

Curriculum Ready ACMNA: 152, 153, 154, 155

 www.mathletics.com

Copyright © 2009 3P Learning. All rights reserved. First edition printed 2009 in Australia. A catalogue record for this book is available from 3P Learning Ltd.

ISBN

978-1-921861-40-6

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Fracons allow us to split things into smaller equal sized amounts. Write down two occasions where you have had to split something up evenly between family members or friends. Describe how you made sure this was done fairly each me.

is Give th

Q 

a go!

For one parcular school: There are 256 students in Year 7. The Year 8, 9 and 10  groups all have half the number of students than the year just below them. 10 groups How many students are there at this school in Years 7 to 10?

Fractons Mathletics

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1

How  does  does it work ?

  Fractions

Proper fractions Proper fracons represent represent parts of a whole number or object. 1 2

numerator denominator

number of equal parts you have total number of equal parts

The numerator is always smaller than or equal to the denominator in proper fracons. Let’s look at some equally sized shaded shapes. (i) Write a fracon fracon for for the shaded parts of the squares below: below:

Split into 2 equal parts

1 whole square

Split into 3 eq equa uall pa part rtss

Spli Sp litt in into to 4 equal parts

The number of equal parts shaded 1 = 1 1

1 2

2 3

2 4

The total number of equal parts

 Larger denominator = smaller equal parts

(ii) Shade these to match match the fracon: 3 8

5 12

Shaded parts Total equal parts

(iii) Include at least two half-shapes half-shapes when shading these to match the fracon: 5 8

16 25

Two halves = 1 whole

2

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Fractons Mathletics

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How does it work ?

 Your Turn

Fractions

Proper fractions 1

2

What fracon of these equal-sized shapes have been shaded? a

b

c

d

e

f 

g

h

Shade these to match the given fracon: a

3

5 12

b

8 8

3 7

c

d

11 16

e

4 10

e

1 4

Shade these to match the given fracon, including at least one pair of half-shapes: a

9 25

b

3 10

5 6

c

Fractons Mathletics

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d

7 10

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3

How does it work ?

 Your Turn

Fractions      P R  OP ER F    R  

A  

Proper fractions 4

Draw and shade diagrams with equal sized shapes to represent each of these fracons: (i) Shading whole shapes only. (ii) Including at least one pair of half-shaded shapes. 3 5

a

c

4

4  

2 9

b

(i)

(i)

(ii)

(ii)

5 8

4 7

d

(i)

(i)

(ii)

(ii)

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Fractons Mathletics

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C     T     

5  

I      O      N      

...  P ..../...../ 2 0

       S       N        O

   R       I    O       T    P       C     E     A      R      F     R

 S 

How does it work ?

  Fractions

Equivalent proper fractions These are fracons with dierent numbers that represent the same amount. For example, two tness teams do three sessions of training in the same park. Session 1: Grouped in pairs

Session 2: In groups of four

Session 3: Grouped as a whole team

1 2

2 4

4 8

Fracon of training groups wearing striped (or plain) shirts in each session. The groups change size but the total number of people training remains the same `

4 2 1 = = = Equivalent fracons 8 4 2

We nd equivalent fracons by dividing/mulplying the numerator and denominator by the same number. Write an equivalent fracon for each of these using the mulplicaon or division given in square brackets (i) 3 6# 3 @ (ii) 12 6' 4 @ 5 32 3#3 9 12 ' 4 3 = = 15 32 ' 4 8 5#3 `

3 and 9 = equivalent fracons 5 15

`

12 and 3 = equivalent fracons 32 8

Simplify these fracons by dividing the numerator and denominator by the highest common factor (HCF)

Simplify = Find the smallest equivalent fracon. 

(i)

3 9

(ii) 18 24

HCF for 3 and 9 is: 3

HCF: the largest number that divides into both exactly

HCF for 18 and 24 is: 6

`

3 '3 1 = 9 '3 3

`

18 ' 6 3 = 24 ' 6 4

`

1 is the simplest equivalent fracon to 3 3 9

`

3 is the simplest equivalent fracon to 18 24 4

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How does it work ?

 Your Turn

Fractions

Equivalent proper fractions 1

Write the equivalent fracons represented by these equally-sized shaded areas: a

b

=

2

3

a

1 6# 5 @ 4

b

8 6' 2 @ 10

c

3 6# 3 @ 5

d

12 6' 6 @ 24

6

=

=

Simplify these fracons by dividing the numerator and denominator by the highest common factor (HCF). 16 20

b

8 32

b

16 24

Simplify these two fracons. a

5

=

Write an equivalent fracon for each of these using the mulplicaon or division given in square brackets:

a

4

=

14 21

Are the fracons 14  and 16  from queson 4 equivalent fracons? Briey explain your answer. 21 24

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How does it work ?

 Your Turn

Fractions

Equivalent proper fractions 6

Match the pair of equivalent fracons below by joining them with a straight line. Solve the puzzle by matching the leer with the number each straight line passes through. T P RO       L E N P  E      V  A  R      I    U F        Q  R        E A    

2 5

10 35

27 72

.... /   .....  /  2  0  .. .

     S

4 5

 S

       I

       T

       C

     A

     R        F      R      E

8 12

6 15 6 24

1 4 8 14

1 2

5 15

6 30 1 5

2 3 4 7

1 3

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8 10

12 16

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  E

   Q

  U

   I   V

    A       P     L      O     E      R     N      P     T

2 7

15 30

I      

 O      

 N

      O

3 4

3 8

T      

     N

6 10

3 5

C     

7

How does it work ?

  Fractions

Improper fractions and mixed numerals An improper fracon has a bigger numerator (top) than denominator (boom) 3 2

5 4

Improper fracons numerator 2 denominator

2  means

‘bigger than’

Mixed numerals have a whole number and a proper fracon. 11 2

Mixed numerals 11 4 A ‘mix’ of whole numbers and proper fracons.

Mixed numerals are simplied improper fracons. Simplify these Improper fracons to mixed numerals (i)

5 3

5 = 5 '3 3

numerator  = numerator denominator

' denominator

= 1 r  2 remainder

2 = 1 3 Whole number answer (ii) 14 4

same denominator

14 7 = = 7 ' 2 Simplify if possible 4 2 = 3 r  1 remainder

1 = 3 2 Whole number answer

same simplied denominator

Mixed numerals to improper fracons +

(i) 1 2 3

2 3 # 1+ 2 = 3 3

1 #

=

(ii) 2 1 5

same denominator

+

1 5 #2+1 = 5 5

2 #

=

8

5 3

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11 5

same denominator

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picture form

How does it work ?

 Your Turn

Fractions

Improper fractions and mixed numerals 1

Write the mixed numerals represented by these shaded diagrams: a

b

=

=

c

d

=

=

Make sure you write the fracon in simplest form where possible.

f 

e

=

= B E L R  A  R S  

 E  U M    U  M    N   N

   D    D    E

2

a

3

b

14 3

    M

c

23 2

      D       D       N

      N      A

     A

 I

  P    R   O   P    E    E     R     R

  0...   2   / . . . .. . /    F       I       I ..      R       T .     A       T       C     F       C       R      A

       S        S       N       N       O       O

15 9

b

21 14

c

18 16

c

44 5

Write the equivalent improper fracon for these mixed numerals. a

5

12 5

     I     M

Write these fracons in simplest form rst, then change to the mixed numerals. a

4

   E    X     X     I

Simplify these improper fracons by wring them as mixed numerals.

11 2

b

23 4

Write the equivalent improper fracon for these mixed numerals aer rst simplifying the fracon parts. a

4 2 12

b

2 6 24

c

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25 24 72

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  M

How does it work ?

  Fractions

Fractions on the number line Proper fracons represent values between 0 and 1 on a number line. number of equal steps taken between 0 and 1 total number of equal steps between 0 and 1

1 2

Mark equal-sized steps matching the denominator between 0 and 1, then plot the fracon using the numerator. Display the fracons 1 , 3 and 2 on these number lines: 2 5 3 3 steps taken

1 step taken

1 = 2

3 = 5

1

0

2 equal steps between 0 and 1

2 steps taken

0

2 = 3

1

0

5 equal steps between 0 and 1

1

3 equal steps between 0 and 1

For mixed numerals, plot the fracon between the given whole number and the next whole number. number of equal steps towards the next whole number ‘4’ total number of equal steps between ‘3’ and the next whole number ‘4’

31 2

Start from this whole number

Display and read these fracons on a number line: Mixed numerals

(i) 4 1 2

(ii) 2 2 5

1 step taken towards 5 Start

1 2

4

2 steps taken towards 3 Start

5

2 5

2

3

5 equal steps between 2 and 3

2 equal steps between 4 and 5

Improper fracons – simply change to the equivalent mixed numeral rst then show on the number line

(i) 7 = 2 1 3 3

(ii) 18 = 3 = 1 1 12 2 2

1 step taken towards 3

Start

1 3

2

1 step taken towards 2

Start

3

1

1 2

2

2 equal steps between 1 and 2

3 equal steps between 2 and 3

Write down the fracon displayed on these number lines (i)

 

1

0

2 steps taken towards 1 1

0

4 equal steps between 0 and 1

2 1 Simplest form = 4 2

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(ii)

  3

(iii) 4

2 steps taken towards 4 4

3

3 equal steps between 3 and 4

32 3

4

5

4 steps taken towards 5 4

5

6 equal steps between 4 and 5

4 4 = 4 2 Simplest form 6 3

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How does it work ?

 Your Turn

Fractions T HE   N U M    BER  LIN E  F   R   A       S  C       N  T       O  I       I  O       T  N       C  S       A      R  O        N      I      F      E      L     R      E      B  N        M       U       N     E      H      T      N      O

Fractions on the number line 1

What proper fracon do the following points on the number line represent? a

0

2

b

1

1

0

1

1

1 4

c

3 3

0

b

0

1

8 15

1

c

0

1

0

1

Write the mixed numeral and equivalent improper fracon for the dots ploed on these number lines: a

2

c

b

3

1

5

2

=

6

=

=

Display these improper fracons on the number line: a

27 = 10

b

2 6

1

Write and display the fracon of equal shapes shaded on a number line for these diagrams: a

5

0

b

0

4

c

0

Display these fracons on a number line: a

3

..../ ...../ 20  .  ..

11 = 2

3

22 = 5

c

4

5

6

Display these on the number line aer changing to equivalent fracons in simplest form rst. a

42 = 15

1

b

=

63 = 18

2

=

3

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c

110 = 25

=

4

5

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How does it work ?

  Fractions

Reciprocal fractions 2 5

Original fracon

5 2

swap

Reciprocal fracon

Write the reciprocal of these fracons (i)

(ii)

3 4

18 8

3 4

3 4

swap

4 3

Reciprocal fracon

18 9 = 8 4

9 4

swap

4 9

Reciprocal fracon

Simplied

For mixed numerals, change to an improper fracon rst then write the reciprocal. 7 2

31 2

Mixed numeral

7 2

2 7

swap

Reciprocal fracon

Improper fracon

Whole/mixed number examples: Always write as a fracon rst. Write the reciprocal of these 3 = 3 1

(i) 3

3 1

1 3

Reciprocal fracon

4 11

Reciprocal fracon

swap

Whole number as a fracon

3 (ii) 2 4

23 4

11 4

11 4

swap

Improper fracon

9 (iii) 4 15

4 9 15 Simplied fracon

43 5

23 5

23 5

swap

5 23

Reciprocal fracon

Improper fracon

We will see why we nd the reciprocal a lile later on in this booklet. It is used when dividing an amount by a fracon.

12

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How does it work ?

 Your Turn

Fractions    L F RA  C  T     C A    O I       R O        P N         I S          C      E

Reciprocal fractions 1

Write the reciprocal for these fracons:

     R

 3

4

 R

4

 3

.... /  .       N

       S

a

2

6 19

d

c

12 18

d

      O

15 4

.... /  2   0  ...

      I       T       C      A      A      R       F     L

6 10

b

14 8

25 10

1 5

b

1 9

c

2

d

4

21 3

b

32 5

c

15 9

Write the reciprocal of these mixed numerals aer rst wring as a fracon: a

6

c

Write the reciprocal of these mixed numerals: a

5

11 7

Write the reciprocal of these: a

4

b

  C   I   P     R    O     C

Write the reciprocal, then simplify these fracons: a

3

2 3

  E

3 2 10

b

1 10 12

c

2 9 21

c

15 115

Write the reciprocal of these fracons as a simplied mixed numeral: a

10 48

b

12 66

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 Where does it work ?

  Fractions

Comparing fractions This is where we see which fracons are larger than others. 1 2

1 3

or

Write equivalent fracons by changing the denominators to their LCM. 1#3 2#3

3 6

2 6

or

Lowest Common Mulple (LCM) The smallest value that appears in both mes tables

1#2 3#2

Since they have the same denominator, now just compare their numerators. bigger

smaller       1

`

3 6

2

2 6

1 2

2

1 3

Compare the size of these fracons (i)

2 , 3 and 5 3 4 12

2#4 3#4

2 3

,

3 4

8 12

,

9 12

`

5 12

1

8 12

5 12

1

2 3

and

1

1

5 12

3#3 4 #3 9 12

LCM of denominators = 12

and

5 12 Compare numerators

3 4

Order fracons by size

If comparing improper fracons, use the same method but leave the answer as a mixed numeral. (ii)

11 and 13 4 5

11 # 5 4#5

11 4

,

13 5

55 20

,

52 20

`

14

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LCM of denominators = 20

13 # 4 5#4

55 20

2

52 20

Compare numerators

23 4

2

23 5

Write in simplest form

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 Where does it work ?

 Your Turn

Fractions

Comparing fractions 1

Compare the size of these fracons: a

2 and 1 5 3

b

3 and 5 4 7

 NG  F  R    R I A C      A     P T        M I        O O         C N     

1

2

Compare the size of the fracons in each of these groups:

S      

2

       S

a

3

3 , 1 and 11 5 2 20

b

9 , 2 and 5 12 3 6

1

      N.

... /    .... . /    2  0  .. .

3 

  C

      O   O       I    M       T   P        C     A      A     R      R     I       F     G      N

Compare the size of these improper fracons: a

9 and 16 4 7

b

14 , 15 and 21 3 4 8

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 Where does it work ?

  Fractions

 Adding and subtracting fractions with the same denominator +

1 4 one quarter

=

+

2 4

=

3 4

and

two quarters

equals

three quarters

=

-

2 3

1 3

-

two thirds

less

one third

1 3

= equals

one third

If the denominator (boom) is the same, just add or subtract the numerators (top). Simplify these fracons with the same denominator

(i)

2 5 + 9 9

2 5 2 5 + = + 9 9 9 =

(ii) 6 - 2 7 7

7 9

6 2 6 2 = 7 7 7 =

(iii) 2 + 5 3 3

=

answers in simplest form

3 1 4 3 1 4 - + = - + 5 5 5 5 =

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Fractons Mathletics

Simplify

Subtract/add the numerators only

6 5

1 = 1 5

16

Add the numerators only

7 3

1 = 2 3

(iv) 3 - 1 + 4 5 5 5

Subtract the numerators only

4 7

2 5 2 5 + = + 3 3 3

Always  write

Add the numerators only

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Simplify

 Where does it work ?

 Your Turn

Fractions

 Adding and subtracting fractions with the same denominator 1

Simplify these without the aid of a calculator: a

d

1 1 + 3 3

8 6 11 11

b

3 1 5 5

c

5 2 + 9 9

e

11 4 15 15

f 

3 5 + 8 8    H    T    I     W

 E S A  M E  D    T  H E  N   

O   

M   

I     N     A    

    S

    N

T      

     O

O       R

       I

       T

      C

2

3

  D   D

     R

  A

... .        G . /   .

        F

  I    N ... /  2  0      G ...     N     A       T       C     A      R      T     S    D

      N       I

a

1 4 + 2 2

b

8 2 5 5

c

2 5 + 3 3

d

10 1 4 4

e

11 4 + 7 7

f 

15 8 2 2

     U      B

Simplify these without the aid of a calculator, remembering to write the answer in simplest form: a

4

      A

Simplify these without the aid of a calculator:

11 5 4 4

b

13 19 + 6 6

c

9 13 + 8 8

Simplify these without the aid of a calculator: a

4 1 2 + + 9 9 9

b

20 10 4 3 3 3

c

1 1 1 + 2 2 2

d

1 4 2 + 5 5 5

e

8 4 6 - + 7 7 7

f 

13 11 9 + 6 6 6

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 Where does it work ?

  Fractions

 Adding and subtracting fractions with a different denominator +

1 4

+

one quarter

and

=

1 2 one half

=

?

equals

?

+

1#2 2 = 4 2#2

=

1 4

+

2 4

=

one quarter

and

two quarters

equals

3 4 three quarters

Simplify these expressions which have fracons with dierent denominators

(i)

2 1 +   3 5

For

`

Mulply top and boom by the number used to make the denominator equal to the LCM

(ii) 7 - 1 + 3   8 2 4

2 1 and 3 5

Denominators are dierent

2 1 2#5 1#3 + = + 3 5 3#5 5#3 =

10 3 + 15 15

Equivalent fracons with LCM denominators

=

10 + 3 15

Add the numerators only

=

13 15

For 7 , 1 and 3 8 2 4

Denominators are all dierent

7 1 3 7 1#4 3#2 - + = + 8 2 4 8 2#4 4#2 =

7 4 6 - + 8 8 8

=

7- 4+ 6 8

=

9 8

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The LCM of all the denominators is 8

Equivalent fracons with LCM in the denominators

Simplify the numerator

1 = 1 8

18

The LCM of the denominators is 15

Simplify to mixed numeral

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 Where does it work ?

 Your Turn

Fractions

 Adding and subtracting fractions with a different denominator 1

Fill in the spaces for these calculaons: a

1 1 The LCM of the denominators is: + 3 6 `

1 1 1# + = 3 6 3#

=

+

+

b

4 1 The LCM of the denominators is: 7 5

1 6

`

1 6

=

=

=

2

5 1 5# = 7 5 7#

-

1#7 5#7

-

=

simplest form

simplest form

Simplify these without the aid of a calculator: a

1 1 + 3 2

b

5 1 6 2

c

2 1 5 4

d

1 3 + 6 4

e

6 2 7 3

f 

3 3 + 5 8

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 Where does it work ?

 Your Turn

Fractions

 Adding and subtracting fractions with a different denominator 3

Simplify these expressions without the aid of a calculator, remembering to write the answer in simplest form. a

1 4 + 2 5

b

13 3 8 5

   S

    N

 THE D IF     F      T  H E    R  E    N    

   W  I

    O      I      T

T     D     

E     

     C      A       R

N       O      

 I 

 M 

       F

       G

      N        I        T

..../ ...../      A 20  .  .   I.   D   D      S     N

       C

     A      R       T      B       U

20

c

1 3 1 + 2 8 4

d

3 3 3 + 5 10 4

e

2 1 5 - + 3 4 6

f 

7 1 11 - + 12 3 24

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     D      N      A    G

 N    A   T   O    R

 Where does it work ?

 Your Turn

Fractions

 Adding and subtracting fractions with a different denominator The same rules apply for quesons with a mix of whole numbers and fracons. Here are some examples: Simplify these expressions which have a mix of whole numbers and fracons (i) 3 + 1 4

3+ 1 = 3 1 4 4

Write the fracon aer the whole number

(ii) 1 - 2 5

1- 2 = 5 - 2 5 5 5

Write the whole number as a fracon wit h same denominator

= (iii) 4 - 2 7

3 5

Subtract the numerators only

4 - 2 = 28 - 2 7 7 7 =

Write the whole number as a fracon wit h same denominator

26 7

5 = 3 7

4

Simplify the fracon

Simplify these expressions: a

2+ 1 2

b

1+ 3 4

c

1- 2 3

d

1- 3 8

e

2- 3 5

f 

4- 1 4

g

3- 5 3

h

5- 5 2

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  Fractions

Multiplying and dividing fractions To mulply fracons, just remember: Mulply the numerators (top) and the denominators (boom) of = ‘ # ’

1 of  2 = 3 5

1 3

#

2 5

=

1#2 2 = 15 # 3 5

To divide an amount by a fracon, just remember: ip the second fracon then mulply

1 3

'

2 1 = 5 3

#

5 2

=

1#5 3#2

=

5 6

Change the ‘ ' ’ to a ‘ # ’

Only ip the second fracon

Remember: A ipped fracon is called the reciprocal fracon

Simplify these: We can use shaded diagrams to calculate the mulplicaon of two fracons (i)

2 of  4 3 5

Draw a grid using the denominators as the dimensions

3 5 4 2

3

Use the numerators to shade columns/rows

5 =

`

2 3

#

8 15

4 8 = 5 15

Write where they overlap as a fracon

If whole numbers are involved, write them as a fracon (ii) 28 ' 2 7

`

28 ' 2 7

7 = 28 # 2 =

28 1

=

196 2

=

98 1

#

7 2

Flip the second fracon and change sign to ‘ # ’ Write the whole number as a fracon

Simplify

= 98

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Fractions

Multiplying and dividing fractions 1

Calculate these fracon mulplicaons by shading the given grids: a

1 of  3 5 4

b

2 of  4 3 7

3 5 7 4 `

c

`

2 of  4 = 3 7

1 of  3 = 5 4

4 of  4 5 5

d

2 of  3 5 8

5

5

5

`

8

4 of  4 = 5 5

`

2 of  3 = 5 8

= simplied

e

3 of  7 4 9

f 

3 of  5 4 6

4

4

9 `

3 of  7 = 4 9

6 =

`

simplied

= simplied

Fractons Mathletics

3 of  5 = 4 6

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 Your Turn

Fractions

Multiplying and dividing fractions 2

Simplify these without the aid of a calculator: a

1 2

c

` 23 j

#

2

24

1 3

psst: this is just

2 3

#

b

3 5

d

` 53 j

#

1 4

2

2 3

e

1 3

'

3 2

f 

2 11

g

5 6

'4

h

3 4

'

1 4

'8

i

10 # 4 5

j

24 # 3 8

k

12 ' 3 5

l

2' 2 13

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  N  D D      A  I       N G  I V        Y I      I P L   T  L D       U  M  I      ' N            S G            N      O F            I R           T A           C C           A .  T           R. . .  I           F  /     .  O     .  N      .        G . .  /     S           N       I 2         M 0        U      D     L .   #    T . .    Y    I       I    P       V    I    L    N    G       I    N      D    D    A

 Where does it work ?

 Your Turn

Fractions

Multiplying and dividing fractions 3

Simplify these without the aid of a calculator, remembering to write the answer in simplest form. 2

a

c

4

3 8

` 28 j

b

3 4

#

3 2

5 4

d

2 3

'

5 3

f 

3 4

#

2 3

h

1 2

'4 '

'

e

9 10

g

2 5

'

#

8 5

3 6

#

1 3

Is 2 of  4 exactly the same as 2 3 6 3

'

#

1 psst: same as the others! 2

1 psst: work le to right! 2

12 ? Explain your answer. 8

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  Fractions

Operations with mixed numerals Just change to improper fracons then use the same methods as shown earlier. Simplify these calculaons involving mixed numerals Addion and subtracon

(i) 1 2 + 2 1 3 6

1 2 + 2 1 = 5 + 13 3 6 3 6 =

10 13 + 6 6

=

23 6

5 = 3 6

Or just add the whole numbers and the fracons separately. 2 1 5 1+ 2 = 3 + = 3 6 6

Equivalent fracons with LCM denominators

Simplify to mixed numeral

4 1 - 1 1 = 21 - 3 5 2 5 2

(ii) 4 1 - 1 1 5 2

Change to improper fracons

=

42 15 10 10

=

27 10

= 2 7 10

Change to improper fracons

Equivalent fracons with LCM denominators

Simplify to mixed numeral

Mulplicaon and division

(iii) 1 3 4

#

21 3

13 4

#

21 = 7 3 4 =

#

7 3

49 12

Mulply tops and booms together

1 = 4 12 (iv) 1 1 6

'2

11 6

Remember 2 3 etc 2 = ,3 = , 1 1

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'2

Change to improper fracons

Simplify to mixed numeral

=

7 6

'

2 1

Change to improper fracons

=

7 6

#

1 2

Flip second fracon and change to mulply

=

7 12

Mulply numerators and denominators together

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 Your Turn

Fractions   H      O NS W IT M      T I I    X       R  A  E       E D       P     O N     #  +

Operations with mixed numerals

U     

     S      L

M    

E      R     

     A

1

2

Simplify these addions and subtracons without the aid of a calculator: a

21 +42 4 3

b

21 -12 4 5

c

53 -21 5 2

d

31 +11 6 5

b

43 7

#

d

51 3

#

b

12 3

'3

d

51 2

'1

c

4 #1 2 5

13 4

#

31 2

 S

....   O  /    ..   P       D    E       E ... /        R       X 2  0  ..    T    A       I       M .    N    O     I       H       T      I       W     S

     M       U       N

#  +

2

14 5

Simplify these divisions without the aid of a calculator: a

3 '2 1 2

c

22 5

'1

1 2

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 L

 A

Simplify these without the aid of a calculator: a

3

     R       E

2 3

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Fractions

Combining all the operations Earn yourself an awesome passport stamp by trying these trickier quesons without using a calculator. 1 63 1   Simplify + 7 5

1 #1 2

 E  O M  *

psst: remember your order of operaons

  E S    W

   A

       *

*

..    A .   W   2  0   / . .     E . .     S . .    O ...  /      M

    *     E

2

Simplify 4 1 2

3

Simplify this shaded diagram into a single fracon. psst: write as fracons, then work le to right

'3

5 6

#

5 1 psst: work le to right… so do the division rst 3

+

#

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  Fractions

Fractions of an amount How many links are there in 2  of a chain made using a total of 20 links? 5

Simplied, this queson is just: Find 2 of  20 . 5 `

2 of  20 2 = 5 5

#

=

2 5

#

=

40 5

Remember: ‘of’ = ‘ # ’

20 20 1

= 8 `

2 of the 20 links = 8 links 5

Here are some quesons that calculate fracons of an amount. (i) Juliet lost 1  of the 116 songs she had downloaded when a computer virus infected the les. 4 How many songs were not aected by the virus? 1 of 116 songs 1 = 4 4

#

1 3  are gone, then remain. 4 4 3 So nding of 116 will get 4 the same answer. Try it!

If

116

=

1 116 + 4 1

=

116 4

= 29 `

(ii) How long is 7 of 1 hour? 10

116 - 29 = 87 songs not aected by the virus

7 7 = of 1 hour 10 10 of 60 minutes =

7 10

=

420 10

#

Answer the queson

Change to smaller units if possible

60 1

= 42 `

7 of 1 hour =   42 minutes 10

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Answer the queson

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Fractions F R A    CT   I   O NS  

   T    N    U    O    M    A

Fractions of an amount 1

2

Calculate the amount for each of these, showing all working: a

1 of  20 5

b

3 of  32 4

c

2 of  24 3

d

5 of  42 6

 .    A     0 . .  O    M     2     /  .   U  .  . . .   T  N     /  .  .      S       N      O      I  . .     A      T      C      R       F

  N   A   F   O

Calculate the amount for each of these by rst making the mixed numeral an improper fracon: psst: the answers will be bigger than the given whole number

3

a

2 1 of  4 2

b

1 4 of  15 5

c

3 2 of  14 7

d

4 2 of  36 3

Calculate these fracons of quanes, showing all working: a

How many hours is 3  of 1 day? 4

b

1 day = 24 hours

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How many grams is 3  of 2 kilograms? 10 1 kg = 1000 grams

 

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O    F    A    N   

 What else can you do?

 Your Turn

Fractions

Fractions of an amount 3

c

How long is 5  of 2 hours? 6

d

1 hour = 60 minutes

4

How far is 1  of 3 kilometres? 5 1 km = 1000 metres

In an orchestra of 60 musicians, 1  were in the brass secon. How many brass secon players were there? 5 psst: remember to include a statement answering the queson at the end

5

6

7

Krista and her team mates each receive 1  of a $900 prize for winning a compeon. 5 How much does Krista receive?

Hank bought 2  of the 28 towels that were on sale in a shop. How many towels were not bought by Hank? 7

The lead in one brand of HB pencil is 8  graphite. 11 How many grams of non-graphite material are there in this brand if every pencil contains 33 grams of lead?

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 What else can you do?

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Fractions

Fractions of an amount Here is what to do if you already have the fracon amount and need to nd the original whole amount. Amani used 250 g of one ingredient, which was 2  of the total ingredients used in the recipe she 5 was following. What is the total weight of ingredients used in this recipe?

A short-cut way is to mulply 250 g by the reciprocal. 5 250 g # = 625 g 2

8

9

10

32

2 of the total ingredients used Divide the amount by the numerator = 250 g 5 1 ` of the total ingredients = 250 g ' 2 5 = 125 grams Mulply answer by the denominator 5 (the total ingredients) ` = 125 g # 5 5 = 625 grams `  Total

weight of ingredients used in the recipe = 625 grams

The leer ‘e’ is used twenty one mes in this queson. If this is 1  of all the leers used to write 8 this queson, work out how many leers there are in total (show all working and check your amount by counng).

During a performance by Jolly Rob, 280 members of the audience found his jokes funny. If this was 4  of the total audience members at the performance, how many people were watching Jolly Rob? 7

By 3:00 pm, Juan had been waing 30 minutes for his bus to arrive. If this is 5  of the total me he 6 spent waing, at what me did his bus arrive?

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 What else can you do?

  Fractions

Two amounts as a fraction Somemes we need to compare two amounts by wring one as a fracon of the other. For example, write 14 out of 36 as a fracon in simplest form 14 14 ' 2 = 36 36 ' 2 7 = 18 7 ` 14 out of 36 = 18

First amount Second ‘out of’ amount

Divide the amount by the HCF Simplest form

These examples show how both amounts must be in the same units before wring as a fracon. Write these amounts as fracons of one another in simplest form (i) All the boxes in the picture weigh a total of 40 kg. The box being carried is 2000 g. What fracon of the total weight is being carried? 40 kg = 40000 g `

2000 = the fracon of the total weight being carried 40000 =

1 kg = 1000 g `  the

Need both weights in the same smaller units

box being carried is

1 20

Simplify fracon

1 of the total weight Answer queson 20

(ii) What fracon of 2 1  hours is 15 minutes? 2 2 1 hours = 2 1 2 2

#

60 minutes

= 150 minutes 15 1 = 150 10

First smaller amount total amount `

1 hour = 60 minutes Need both mes in the same smaller units Simplify fracon

15 minutes is 1  of 2 1  hours 10 2

Answer queson

(iii) If one ream of paper contains 500 sheets, what fracon is 240 sheets out of 4 reams? 4 reams = 4 # 500 sheets = 2000 sheets 240 3 = 2000 25

First smaller amount total amount `

Need both amounts in the same smaller units Simplify

240 sheets is 3  of 4 reams 25

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 What else can you do?

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Fractions  A S  A    T S F       U  N R  

  O

   M

Two amounts as a fraction

A   .  .   C     .   T     .   I        /          .   O      .  .   N      .   .     /          2         0          T .   .   .    W

    A     O

     W

      T

1

Write the rst amount as a fracon of the second for each of these in simplest form

      N       O       I       T        C

5 out of 20

b

16 out of 22

c

35 marks out of a possible 40

d

2 hours of 1 day (1 day = 24 hours)

e

5 minutes of half-an-hour (1 hour = 60 minutes)

f 

25 cents out of $2 ($1 = 100 cents)

g

300 seconds of 1 hour (1 hour = 3600 seconds)

h

23 days of March (31), April (30) and May (31)

     A      R        F

     A     S       A    S

2

In a school of 800 students, 240 were in Year 7. What fracon of the school are Year 7 students?

3

Aer 100 test rolls, a die displayed the number 5 sixteen mes. What fracon of the rolls were 5?

4

Francesca red 27 arrows at a target and hit the bullseye 6 mes. What fracon of arrows missed the bullseye?

5

A biscuit recipe contained 500 g of our, 175 g of sugar and 125 g of buer. What fracon of the recipe is buer?

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   O    U     N    T

a

 O

   A

 What else can you do?

  Fractions

 Word problems with fractions While on a shopping trip, Xieng spent two hs of her money on clothes and one third on cosmecs. What fracon of her money did Xieng have le?

`

Fracon for all of Xieng ’s money `  Xieng

2 1 + =  fracon of Xieng’s money spent on shopping 5 3 6 5 = + 15 11 Add the numerators together = 15 15 11 4 = 15 15 15 Fracon spent

Fracon of money Xieng has le

sll has 4  of her money aer shopping 15

 Here are some other word problem examples (i) In a group of eighteen friends, one third are girls and one sixth of these girls have blonde hair. How many blonde girls are in the group? `

1 of  1 of  18 =  number of blonde girls in the group 6 3 =

1 6

=

18 18

#

1 3

#

18 1

= 1 `  There

is 1 blonde girl in the group of friends.

(ii) During one night, possums ate two hs of the y ve fruits on a tree. If one eleventh of the eaten fruit grew back, how many fruits are now on the tree? 2 5

#

55 =  Number of fruits eaten 110 5 = 22 =

1 11

#

22 = Number of fruits that grew back 22 11 = 2 =

`  Number

of fruits now on the tree = 55 - 22 + 2 = 35 pieces of fruit

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 What else can you do?

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Fractions

 Word problems with fractions 1

At a recent trivia night, one table of competors answered ve eighths of the y six quesons correctly. How many quesons did they get incorrect?

2

Co Tin usually takes approximately sixty and one quarter steps every minute when walking. How many steps does he expect to take when he exercises by walking for one and two third hours each day?  S W    O  R     N  O D    I P  R     T  C O  B    A   R L  E     F M      H S     T  W  I  I   W  T   S  H   M   E  F   L  R   B  A   O  C   R  T   P  I   D  O   R  N   O   W  S

..../ ...../ 2 0 ..

.

3

A vegetable garden has one third carrots, one sixth pumpkins, one quarter herbs, and the rest are potato plants. How many potato plants are in this garden of eighty plants?

4

A class of twenty four students compared eye colours on a chart. Two thirds of the class had brown eyes, and three eighths of those brown-eyed students were boys. How many girls had brown eyes?

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 What else can you do?

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Fractions

 Word problems with fractions 5

For one parcular school: There are 256 students in Year 7. The Year 8, 9 and 10 groups all have half the number of students than the year just below them. How many students are there at this school in Years 7 to 10? ber Remem

me?

6

Five students in a class have a combined total of ninety proles added as friends on a web-based social network site. Three hs of the ninety proles are shared by all ve of the students. These shared proles represent one sixth of the total number of dierent proles added as friends by all the students in the class. How many dierent proles are linked to students from this class?

7

Five sevenths of the y six images used as backgrounds on Meagan’s touchpad were photos she took herself. Aer moving ve eighths of these photos to another computer, what fracon of the background images now are not photos taken by her?

Fractons Mathletics

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 What else can you do?

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Fractions

Reflection Time Reecng on the work covered within this booklet: 1

What useful skills have you gained by learning about fracons?

2

Write about one way you think you could apply fracons to a real life situaons.

3

38

If you discovered or learnt about any shortcuts to help with fracons or some other cool facts,  jot them down here:

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