Fractions

(1)

Fractions

SERIES SERIES

H

Curriculum Ready

ACMNA: 152, 153, 154, 155

(2)

(3)

First edition printed 2009 in

First edition printed 2009 in Australia.Australia.

A catalogue record for this book

A catalogue record for this book is available from 3P Learning Ltd.is available from 3P Learning Ltd.

ISBN

ISBN 978-1-921861-40-6978-1-921861-40-6

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(4)

5

5 H

H

SER

SERIES IES TOPTOPICIC

1

1 Fractons

Fractons

Mathletics

Mathletics © © 3P 3P Learning Learning LtdLtd

Q

Q For For one one parcular parcular school:school:

There are

There are256256 students in Yearstudents in Year 77.. The Year

The Year88,,99 and and 1010 groups all have half the number of students than the year just below them. groups all have half the number of students than the year just below them.

How many students are there at this school in Years

How many students are there at this school in Years 77 to to 1010??

G

Giivveetthhiissaaggoo!!

Write down two occasions where you have had to split

Write down two occasions where you have had to split something up evenly between family memberssomething up evenly between family members

or friends. Describe how you made sure this was done fairly each me.

(5)

Fractions

How

does it work

?

Proper fractions

Proper fracons represen

Proper fracons represent parts of a whole t parts of a whole number or object.number or object.

(ii)

(ii) Shade these Shade these to matto match the ch the fracon:fracon:

Let’s look at some equally sized shaded

Let’s look at some equally sized shaded shapes.shapes.

Split into

Split into22 equal parts equal parts

Two halves

Two halves= 1= 1wholewhole

(i)

(i) Write Write a fra fracon facon for the or the shaded parts shaded parts of the of the squares belosquares below:w:

Split into

Split into33eqequaual l papartrts s SpSplilit t inintoto44 equal parts equal parts

1

1 whole square whole square

The number of equal parts shaded

(iii)

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

1

1 ₌₌ ₁₁

The total number of equal

The total number of equal partsparts

4 4 2 2 3 3 2 2 2 2 1 1 8 8 3 3 12 12 5 5 25 25 16 16 8 8 5 5 numerator numerator denominator denominator

number of equal parts

number of equal partsyou haveyou have

total

total number of equal parts number of equal parts

The numerator is always smaller than or equal to the denominator in proper fracons.

Shaded parts

T

Total equal otal equal partsparts

Larger denominator

=

= smaller equal parts smaller equal parts

2

1

(6)

5 H

SERIES TOPIC 3 Fractons Mathletics © 3P Learning Ltd

Proper fractions

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

b c

a

e f g h

d

Shade these to match the given fracon:

b c d a e 12 5 8 8 b c d a e 16 11 10 4 7 3 25 9 10 3 10 7 4 1 6 5 2

(7)

How does it work

?

Your Turn

Fractions

Proper fractions

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.

P R OP E R F _R A _C T I O N S P R O P E R F R A C T I O N S

..../.../ 2 0

...

4 5 b c d (ii) (i) (ii) (i) (i) (ii) (ii) (i) 5 3 9 2 d 8 5 7 4 4 a

(8)

5 H

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

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

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

8 4

Fracon of training groups wearing striped (or plain) shirts in each session.

Thegroups change sizebut the total number of people training remains the same

8 4 4 2 2 1 ` = = = Equivalent fracons

We nd equivalent fracons by dividing/mulplying the numerator and denominator by the same number.

Simplify = Find the smallest

equivalent fracon.  (ii) (i) 3₉ 9 3 24 18 3 1

` is the simplest equivalent fracon to

4 3

` is the simplest equivalent fracon to

9 3 3 1 3 3 ` ' ' = 24 18 6 4 3 6 ` ' ' = 24 18

HCF for 3 and 9 is: 3 HCF for 18 and 24 is: 6

2 1 4

2

HCF: the largest number that divides into both exactly

(ii) (i) 5 3 ₃ 5 3 15 9 3 3 # # # =

6 @

5 3 15 9

` and = equivalent fracons

32 12 4 4 32 12 ₄ 8 3 ' ' ' =

6 @

32 12 8 3

(9)

How does it work

?

Your Turn

Fractions

Equivalent proper fractions

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

b

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

a

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

a

4 Simplify these two fracons.

a

5 Are the fracons

21 14 and

24

16 from queson 4 equivalent fracons? Briey explain your answer.

b b b a = = = = = 4 1

₆

_#₅

_@

10 8 ₂ '

6 @

c d 5 3

₆

_#₃

_@

24 12

_{6 @}

_'₆ 20 16 32 8 21 14 24 16

(10)

5 H

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. E Q U I

V A L E NT P R O P _E R F R A C T I O N S E Q U I V A L E N T P R O P E R F R A C T I O N S .... / ...._{. / 2 0 ...} 35 10 5 4 12 8 4 3 15 6 24 6 2 1 10 8 15 5 5 1 16 12 30 15 8 3 5 3 3 1 3 2 30 6 14 8 4 1 7 2 72 27 5 2 10 6 7 4

(11)

How does it work

?

Fractions

How does it work

?

Fractions

Mixed numerals to improper fracons Improper fracons to mixed numerals

2 3

2 means ‘bigger than’

Improper fracons numerator 2denominator

Mixed numerals are simplied improper fracons.

4 5

1 2

1 _{Mixed numerals}

A ‘mix’ of whole numbers and proper fracons. 1 4 1 (i) 3 5 (ii) 4 14 (i) 1 3 2 (ii) 2 5 1 r 3 5 _{5 3} 1 2 1 3 2 ' = = = numerator

denominator = numerator 'denominator

remainder

same denominator Whole number answer

r 4 14 2 7 _{7 2} 3 1 3 2 1 ' = = = = remainder

same simplied denominator Whole number answer

Simplify if possible picture form

1 3 2 3 3 1 2 3 5 # = + = same denominator 2 5 1 5 5 2 1 5 11 # = + = same denominator # + # +

Improper fractions and mixed numerals

An improper fracon has a bigger numerator (top) than denominator (boom)

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

(12)

5 H

SERIES TOPIC

9

Fracons

Mathletics Passport © 3P Learning

I M P R O P E R F R A C T I O N S A N D M I X E D N U M BE _R S

.... /.

.... / 2

0...

Improper fractions and mixed numerals

b

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

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

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

5 Write the equivalent improper fracon for these mixed numerals aer rst simplifying the fracon parts. a b a = 5 12 3 14 _c 2 23

1 Write the mixed numerals represented by these shaded diagrams:

b 9 15 14 21 _c 16 18 b 1 2 1 ₂ 4 3 _c ₄ 5 4 b 4 12 2 ₂ 24 6 _c ₂₅ 72 24 c d e f = = = = = I M P R O P E R F R A C T I O N S A N D M I X E D

N U M ER A L S

.... /.

.... / 2

0...

(13)

How does it work

?

Fractions

Fractions on the number line

Proper fracons represent values between 0 and 1 on a number line.

(ii) Display the fracons and on these number lines:

Mixed numerals

2

1 number of equal steps taken between 0 and 1

total number ofequalsteps between 0 and 1

Mark equal-sized steps matching the denominator between 0 and 1, then plot the fracon using the numerator.

2 1 , 2 1 5 3 3 2

2 equal steps between 0 and 1 1 step taken

5 3

5 equal steps between 0 and 1 3 steps taken

3 2

3 equal steps between 0 and 1 2 steps taken

= ₀ ₁ = 0 1 0 1

For mixed numerals, plot the fracon between the given whole number and the next whole number. 3

2

1 number of equal steps towards the next whole number ‘4’

totalnumber of equal steps between ‘3’ and the next whole number ‘4’

Start from this whole number

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

4 2 1

2 equal steps between 4 and 5

1 step taken towards 5 ₂

5 2

5 equal steps between 2 and 3 2 steps taken towards 3

2 3 7 3 1 =

3 equal steps between 2 and 3 1 step taken towards 3

5 4 2 3 3 2 2 1 Start (i) (ii) (i)

Write down the fracon displayed on these number lines 12 18 2 3 ₁ 2 1 = =

2 equal steps between 1 and 2 1 step taken towards 2

2 1 2 1 Start 3 1

6 equal steps between 4 and 5 4 steps taken towards 5 1 0 3 4 4 5 4 2 2 1 = 1 0 3 4 4 5

(i) (ii) (iii)

Simplest form 3 3 2 ₄ 6 4 ₄ 3 2 = Simplest form Start 5 2 Start =

(14)

5 H

Fractions on the number line

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

2 Display these fracons on a number line:

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

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

a

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

4 1 15 42 10 27 1 0 a b c 1 0 0 1 1 0 a b c 1 0 0 1 3 3 15 8 b c 3 2 b 1 2 c 6 5 a 2 11 5 22 b c

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

a ₌ ₌ b c 18 63 ₌ ₌ 25 110 = = 2 1 3 4 5 3 2 4 5 6 = = = = = = 1 0 F R A C T I O N S

O NT H E N U M BER LIN E F R A C T I O N S O N

..../ .../ 2 0 ...

T H E N U M B E R L I N E 1 0 0 1

(15)

How does it work

?

Fractions

Reciprocal fractions

Original fracon 5 2 _swap 2 5 _{Reciprocal fracon}

Write the reciprocal of these fracons (i) 4 3 (ii) 4 3 4 3 3 4

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

Whole/mixed number examples: Always write as a fracon rst.

Reciprocal fracon 8 18 swap 8 18 4 9 4 9 9 4

= swap Reciprocal fracon

Simplied

Write the reciprocal of these

(i) 3 (ii) 3 1 3 1 3 3 1 = Reciprocal fracon 2 4 3 swap 2 4 3 4 11 4 11 11 4

swap Reciprocal fracon

Improper fracon Mixed numeral 3 2 1 2 7 2

7 _swap _{Reciprocal fracon}

Whole number as a fracon

(iii) 4 15 9 4 15 9 ₄ 5 3 5 23 5 23 23 5

swap Reciprocal fracon

Improper fracon Simplied 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.

Improper fracon

7 2

(16)

5 H

Reciprocal fractions

1 Write the reciprocal for these fracons:

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

3 Write the reciprocal of these:

4 Write the reciprocal of these mixed numerals:

a

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

6 Write the reciprocal of these fracons as a simplied mixed numeral: a a a a a b c d 3 2 7 11 19 6 4 15 b c d 10 6 8 14 18 12 10 25 b c 2 d 4 5 1 9 1 b c 2 3 1 3 1 5 2 9 5 b c 3 1 2 10 2 12 10 21 9 b c 48 10 66 12 115 15 R E C I P R O C A L F R A C T I O N S R E C I P R O C A L F R A _C T _I O N S .... / ..._{/ 2 0 ...} 4 3 4 3

(17)

Fractions

Where

does it work

?

Comparing fractions

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

1

3 1

or

Write equivalent fracons by changing the denominators to their LCM.

Since they have the same denominator, now just compare their numerators. 2 1 6 3 3 3 # # 6 2 3 1 2 2 # # or

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

bigger 1smaller 6 3 6 2 2 1 3 1 2 2

Compare the size of these fracons

(i) , 3 2 4 3 12 5 (ii) 4 11 5 13 and , 3 2 4 3 12 5 and LCM of denominators = 12 Compare numerators

Order fracons by size

3 2 12 8 4 4 # # 12 9 4 3 3 3 12 5 # # , _and 12 5 12 8 12 9 12 5 3 2 4 3 1 1 1 1 `

If comparing improper fracons, use the same method but leave the answer as a mixed numeral. and , 4 11 5 13 4 11 5 5 20 55 # # 5 13 4 4 20 52 # # , 20 55 20 52 2 4 3 ₂ 5 3 2 2 ` LCM of denominators = 20 Compare numerators

Write in simplest form

(18)

5 H

Comparing fractions

1 Compare the size of these fracons:

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

3 Compare the size of these improper fracons: a a a b b b 4 9 7 16 , 5 3 2 1 20 11 and and , 12 9 3 2 6 5 and , 3 14 4 15 8 21 and 5 2 3 1 and 4 3 7 5 and C O M P A R I N G F R A C T I O N S C O M P A R I NG F _R A C T I O N S .... / .._{... / 2 0 .} .. 1 2 3 1

(19)

Where does it work

?

Fractions

Adding and subtracting fractions with the same denominator

one quarter and two quarters equals three quarters 4

1

4 2

two thirds less one third equals one third

Simplify these fracons with the same denominator

(i) 9 2 9 5 + 9 2 9 5 9 2 5 9 7 + = + = (ii) 7 6 7 2 -7 6 7 2 7 6 2 7 4 - = -= (iii) 3 2 3 5 + 3 2 3 5 3 2 5 3 7 2 3 1 + = + = = (iv) 5 3 5 1 5 4 - +

Add the numerators only

Subtract the numerators only

Simplify

Subtract/add the numerators only

Simplify

+ =

4 3

If the denominator (boom) is the same, just add or subtract the numerators (top). 3 2 -3 1 + = = 5 3 5 1 5 4 5 3 1 4 5 6 1 5 1 - + = - + = = Always write answers in simplest form -= 3 1

(20)

5 H

Adding and subtracting fractions with the same denominator

1 Simplify these without the aid of a calculator:

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

a b c 3 1 3 1 + 5 3 5 1 -9 5 9 2 + d e f 11 8 11 6 -15 11 15 4 -8 3 8 5 + a b c 2 1 2 4 + 5 8 5 2 -3 2 3 5 + d e f 4 10 4 1 -7 11 7 4 + 2 15 2 8 -a b c 4 11 4 5 -6 13 6 19 + ₈9 + 13₈ a b c 9 4 9 1 9 2 + + 3 20 3 10 3 4 - -2 1 2 1 2 1 + -d e f 5 1 5 4 5 2 + -7 8 7 4 7 6 - + 6 13 6 11 6 9 + - A D D I N G A N D S U B T R A C T I N G F R A C T I O N S W I T H T H E S A _M E D _E N _O M I N A T O R .... / ..._{/ 2 0 ...}

(21)

Where does it work

?

Fractions

Adding and subtracting fractions with a different denominator

one quarter and one half equals ?

4 1

2 1

one quarter and two quarters equals three quarters

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

(i) 3 2 5 1 + For 3 2 5 1 (ii) 8 7 2 1 4 3 - + For , 8 7 2 1 4 3

Denominators are dierent

The LCM of the denominators is 15

Equivalent fracons with LCM denominators

Denominators are all dierent

The LCM of all the denominators is 8

Equivalent fracons with LCM in the denominators

Simplify the numerator

Simplify to mixed numeral

4 1 4 2 4 3 + = = 3 2 5 1 3 2 5 1 3 15 10 15 3 15 10 3 15 13 5 5 3 # # # # + = + = + = + = + and and 2 1 4 3 4 4 2 2 8 7 2 1 4 3 8 7 8 7 8 4 8 6 8 7 4 6 8 9 11 # # # # - + = - + = - + = - + = = ? 2 1 2 2 4 2 # # =

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

+ =

(22)

5 H

Adding and subtracting fractions with a different denominator

1 Fill in the spaces for these calculaons:

a b 3 1 6 1 + 7 4 5 1 -a b c 3 1 2 1 + 6 5 2 1 -5 2 4 1 - d e f 6 1 4 3 + 7 6 3 2 -5 3 8 3 +

The LCM of the denominators is: The LCM of the denominators is:

3 1 6 1 3 1 6 1 # # ` + = + = = = 7 5 5 1 7 5 5 1 7 7 # # # # ` - = -= = -simplest form simplest form 6 1 +

(23)

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 b c 2 1 5 4 + 13₈ - ₅3 2 1 8 3 4 1 + - d e f 3 2 4 1 6 5 - + 5 3 10 3 4 3 + -12 7 3 1 24 11 - + A D D I N G A N D S U B T R A C T I N G F R A C T I O N S W I T H THE _D I F F _E R E _N T D E N O M I N A T O R

..../ .../ 2 0 ...

(24)

5 H

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:

a b c 2 2 1 + 1 4 3 + 1 3 2 - d e f 1 8 3 -2 5 3 - 4 4 1

-Simplify these expressions which have a mix of whole numbers and fracons

4 Simplify these expressions:

g h (i) 3 4 1 + (ii) 1 5 2 -(iii) 4 7 2 -1 5 2 5 5 5 2 5 3 - = -= 4 7 2 7 28 7 2 7 26 3 7 5 - = -= =

Write the fracon aer the whole number

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

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

Simplify the fracon

3 3 5 - 5 2 5 -3 4 1 ₃ 4 1 + =

(25)

Where does it work

?

Fractions

Multiplying and dividing fractions

3 1 5 2 3 1 5 2 3 5 1 2 15 2 # # # = = =

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)

3 2

5 4

If whole numbers are involved, write them as a fracon (ii) 28 7 2 ' 3 28 7 2 ₂₈ 2 7 1 28 2 7 2 196 1 98 98 # # ' = = = = =

To mulply fracons, just remember: Mulply the numerators (top) and the denominators (boom)

3 2 1 5 3 1 5 2 3 1 2 5 6 5 # # # ' = = = Change the ‘'’ to a ‘#’ of 3 2 5 4 15 8 # ` = 5

Draw a grid using the denominators as the dimensions

Use the numerators to shade columns/rows

Write where they overlap as a fracon

Flip the second fracon and change sign to ‘#’

Write the whole number as a fracon

Simplify 3 5 2 4 15 8 = `

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

Only ip the second fracon

of

(26)

5 H

Multiplying and dividing fractions

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

a b 5 1 4 3 of 3 2 7 4 of c d e f 5 = 5 4 5 4 of 5 2 8 3 of 4 3 9 7 of 4 3 6 5 of simplied 5 4 5 4 of 5 2 8 3 of = 3 2 7 4 of ` ₌ ` 5 1 4 3 of ` ₌ 4 3 7 5 5 = ` 5 8 = of ` of = 9 = = ` 4 4 3 9 7 6 4 4 3 6 5 simplied simplied

(27)

Where does it work

?

Your Turn

Fractions

Multiplying and dividing fractions

a b c 2 1 3 1 # 5 3 4 1 # 3 2 2

` j

d e f 3 1 2 3 ' 11 2 4 1 ' 5 3 2

` j

g h i 6 5 ₄ ' 4 3 ₈ ' 10 5 4 # j k 12 l 5 3 ' 2 13 2 ' 24 8 3 # psst: this is just 3 2 3 2 # # ' M U L T I P L Y I N G A N DD I V I D I N G F R A C T I O N S M U L T I P L Y I N G A N D D I V I D I N G F R A C T I O N S . . . . _/ . . . . _. / 2 ₀ . . .

(28)

5 H

Multiplying and dividing fractions

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

a b c 8 2 2

` j

₄3 # ₂3 8 3 4 5 ' d e f 10 9 5 8 ' 4 3 3 2 2 1 # # 3 2 3 5 ' g h 5 2 6 3 3 1 # # 4 2 1 2 1 ' ' 4 Is 3 2 6

4 exactly the same as 3 2

8 12

' ? Explain your answer.

psst: same as the others!

psst: work le to right!

(29)

Where does it work

?

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 3 2 6 1 + (ii) 4 1 5 1 2 1

-Mulplicaon and division

(iii) 1 2 4 3 3 1 # (iv) 1 6 1 _'₂ 1 3 2 ₂ 6 1 3 5 6 13 6 10 6 13 6 23 3 6 5 + = + = + = = 4 5 1 ₁ 2 1 5 21 2 3 10 42 10 15 10 27 2 10 7 - = -= -= =

Change to improper fracons

Mulply tops and booms together

Flip second fracon and change to mulply

Mulply numerators and denominators together

1 4 3 ₂ 3 1 4 7 3 7 12 49 4 12 1 # = # = = 1 2 6 1 6 7 1 2 6 7 2 1 12 7 # ' = ' = = Remember , , 2 = 2 3 = 3 etc

Or just add the whole numbers and the fracons separately.

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

(30)

5 H

Operations with mixed numerals

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

3 Simplify these divisions without the aid of a calculator:

a b c 2 4 4 1 3 2 + 2 1 4 1 5 2 -5 2 5 3 2 1 - d 3 1 6 1 5 1 + a b c 4 1 5 2 # 4 7 3 _#₂ 1 4 3 ₃ 2 1 # _d 5 1 3 1 5 4 # a 3 2 b 2 1 ' 1 3 2 ₃ ' 2 5 2 ₁ 2 1 ' 5 1 2 1 3 2 ' .... / .._{... / 2 0 .} .. # + O P E R A T I O N S W I T H M I X E D N U M E R A L S O P E R A T I O NSW I T H M _I X E _D N U M E R A L S # + c d

(31)

Where does it work

?

Your Turn

Fractions

Combining all the operations

Earn yourself an awesome passport stamp by trying these trickier quesons without using a calculator.

1 Simplify 6 1 7 1 5 3 2 1 # + 2 Simplify 4 3 5 2 1 6 5 3 1 # '

3 Simplify this shaded diagram into a single fracon.

psst: remember your order of operaons

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

psst: write as fracons, then work le to right

# + * A W E S O M E* .... /... / 2 0... * A W E S O M E *

(32)

5 H

Fractions of an amount

How many links are there in 5

2 of a chain made using a total of 20 links?

Here are some quesons that calculate fracons of an amount. (i) Juliet lost

4

1 of the 116 songs she had downloaded when a computer virus infected the les. How many songs were not aected by the virus?

(ii) How long is

107 of 1 hour? 4 1 ₁₁₆ 4 1 ₁₁₆ 4 1 1 116 4 116 # = = + = 10 7 10 7 10 7 1 60 10 420 # = = =

Change to smaller units if possible

Answer the queson

Simplied, this queson is just: Find 20 5 2 _of _. 5 2 ₂₀ 5 2 ₂₀ 5 2 1 20 5 40 8 # # = = = = of of songs If 4 1

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

Remember: ‘of’ = ‘#’

songs not aected by the virus

of 1 hour of 60 minutes 29 116 29 87 = - = 42 42 = = 10 7 ` _{of 1 hour} minutes 5 2

` _{of the 20 links = 8 links} `

(33)

What else can you do

?

Your Turn

Fractions

Fractions of an amount

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

2 Calculate the amount for each of these by rst making the mixed numeral an improper fracon:

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

a b c 20 5 1 d a b c d a b of 4 3 _of₃₂

How many hours is 4

3 of 1 day? How many grams is

103 of 2 kilograms?

1 day = 24 hours 1 kg = 1000 grams

2 3 2 _of ₄ ₂ 6 5 _of₄ 2 2 1 _of₄ ₁ 5 4 _of₁₅ 3 7 2 _of₁₄ ₄ ₃ 3 2 _of ₆

psst: the answers will be bigger than the given whole number

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

(34)

5 H

Fractions of an amount

4 In an orchestra of 60 musicians, 5

1 were in the brass secon. How many brass secon players were there?

psst: remember to include a statement answering the queson at the end

5 Krista and her team mates each receive

5

1 of a $900 prize for winning a compeon. How much does Krista receive?

6 Hank bought

7

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

3 How long is

6

5 of 2 hours? How far is

5

1 of 3 kilometres?

1 hour = 60 minutes 1 km = 1000 metres

c d

7 The lead in one brand of HB pencil is

118 graphite.

(35)

What else can you do

?

Your Turn

Fractions

Fractions of an amount

Amani used 250 g of one ingredient, which was 5

2 of the total ingredients used in the recipe she was following. What is the total weight of ingredients used in this recipe?

Here is what to do if you already have the fracon amount and need to nd the original whole amount.

8 The leer ‘e’ is used twenty one mes in this queson. If this is

8

1 of all the leers used to write this queson, work out how many leers there are in total (show all working and check your amount by counng).

9 During a performance by Jolly Rob, 280 members of the audience found his jokes funny. If this was

7

4 of the total audience members at the performance, how many people were watching Jolly Rob?

10 By 3:00 pm, Juan had been waing 30 minutes for his bus to arrive. If this is

6

5 of the total me he spent waing, at what me did his bus arrive?

Divide the amount by the numerator

Mulply answer by the denominator

of the total ingredients used of the total ingredients

(the total ingredients)

250 250 2 125 125 5 625 # ' = = = = = 5 2 5 1 ` 5 5 `

` Total weight of ingredients used in the recipe = 625

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

(36)

5 H

SERIES TOPIC

33

Fracons

Two amounts as a fraction

(i) All the boxes in the picture weigh a total of 40 kg. The box being carried is 2000g.

What fracon of the total weight is being carried?

Write these amounts as fracons of one another in simplest form

the fracon of the total weight being carried 40 40000 40000 2000 20 1 = = = `

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

These examples show how both amounts must be in the same units before wring as a fracon. 36 14 2 2 36 14 18 7 36 18 7 ' ' = = =

Divide the amount by the HCF Simplest form

First amount Second ‘out of’ amount

14

` out of

kg Need both weights in the same smaller units

Simplify fracon Answer queson

1 hour = 60 minutes

Need both mes in the same smaller units Simplify fracon

Answer queson

Need both amounts in the same smaller units Simplify

Answer queson

(ii) What fracon of 2 2

1 hours is 15 minutes?

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

g

First smaller amount total amount

` the box being carried is

20

1 _{of the total weight}

2 2 1 ₂ 2 1 ₆₀ 150 150 15 10 1 # = = = ` 15 minutes is 101 of 221 hours minutes hours 4 4 500 2000 2000 240 25 3 # = = = reams ` 240 sheets is 253 of 4 reams sheets sheets

First smaller amount total amount 1 kg = 1000 g

(37)

What else can you do

?

Your Turn

Fractions

Two amounts as a fraction

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

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?

a b

c d

5 out of 20

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? e f g h e f g h 16 out of 22

35 marks out of a possible 40 2 hours of 1 day (1 day = 24 hours)

5 minutes of half-an-hour (1 hour = 60 minutes) 25 cents out of $2($1 = 100 cents)

300 seconds of 1 hour(1 hour = 3600 seconds) 23 days of March (31), April (30) and May (31)

. . _. . / _. . . _. . / ₂ 0 _. . _. _W T O A M O U N T S A S A F R A C T I O N T W O A M O U N T S A S _A F R _A C T I O N

(38)

5 H

Word problems with fractions

(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?

Here are some other word problem examples

`

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?

`_{Xieng sll has}

154 of her money aer shopping 5 2 3 1 15 6 5 15 11 15 15 15 11 15 4 + = = + = - =

Add the numerators together

(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?

fracon of Xieng’s money spent on shopping

Fracon of money Xieng has le Fracon spent

Fracon for all of Xieng ’s money

` There is 1 blonde girl in the group of friends.

6 1 3 1 ₁₈ 6 1 3 1 1 18 18 18 1 # # = = = =

number of blonde girls in the group

55 22 2 55 22 2 5 2 5 110 22 11 1 11 22 35 # # = = = = = = = - + =

Number of fruits eaten

Number of fruits that grew back

`_{Number of fruits now on the tree}

pieces of fruit

(39)

What else can you do

?

Your Turn

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 getincorrect?

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?

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?

..../ .../ 2 0 ...

W O R _D P R O B L E _M S W I T H F R A C T I O N S W O R D P R O B L E M S W I T H F R A C T I O N S

(40)

5 H

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?

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 arenot photos taken by her?

(41)

What else can you do

?

Your Turn

Fractions

Reflection Time

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

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 _{If you discovered or learnt about any shortcuts to help with fracons or some other cool facts,}

jot them down here: