Fractions
Fractions
Curriculum Ready
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SERIES
Copyright Š 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
978-1-921861-40-6Â
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Fractions 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 time.
is Give th
Q
a go!
For one particular 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?
Work through the book for a great way to do this
Fractions Mathletics
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How does it work?
Fractions
Proper fractions Proper fractions represent parts of a whole number or object. numerator denominator
number of equal parts you have total number of equal parts
1 2
The numerator is always smaller than or equal to the denominator in proper fractions. Let’s look at some equally sized shaded shapes. (i) Write a fraction for the shaded parts of the squares below:
1 whole square
Split into 2 equal parts
Split into 3 equal parts
Split into 4 equal parts
The number of equal parts shaded 1 = 1 1
1 2
2 4
2 3
The total number of equal parts
Larger denominator = smaller equal parts
(ii) Shade these to match the fraction: 3 8
5 12
Shaded parts Total equal parts
(iii) Include at least two half-shapes when shading these to match the fraction: 5 8
16 25
Two halves = 1 whole
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Fractions Mathletics
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How does it work?
Your Turn
Fractions
Proper fractions 1
What fraction of these equal-sized shapes have been shaded? a
b
c
d
e
f
g
h
2 Shade these to match the given fraction: a
5 12
b
8 8
3 7
c
d
11 16
e
4 10
e
1 4
3 Shade these to match the given fraction, including at least one pair of half-shapes: a
9 25
b
3 10
5 6
c
Fractions Mathletics
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d
7 10
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3
How does it work?
Your Turn
Fractions
Proper fractions
c
4
2 9
b
(i)
(i)
(ii)
(ii)
5 8
4 7
d
(i)
(i)
(ii)
(ii)
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Fractions Mathletics
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S ON
..../...../20...
PER FRACTI RO
3 5
a
5
P
4 Draw and shade diagrams with equal sized shapes to represent each of these fractions: (i) Shading whole shapes only. (ii) Including at least one pair of half-shaded shapes.
4
NS TIO AC
PROPER FR
How does it work?
Fractions
Equivalent proper fractions These are fractions with different numbers that represent the same amount. For example, two fitness 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
2 4 4 8
1 2
Fraction 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 fractions 8 4 2 We find equivalent fractions by dividing/multiplying the numerator and denominator by the same number. Write an equivalent fraction for each of these using the multiplication 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 fractions ` 12 and 3 equivalent = 32 8
fractions ` 3 and 9 equivalent = 5 15
Simplify these fractions by dividing the numerator and denominator by the greatest common factor (GCF) Simplify = Find the smallest equivalent fraction. ď Š
(i)
3 9
(ii) 18 24
GCF: the largest number that divides into both exactly
GCF for 3 and 9 is: 3
GCF for 18 and 24 is: 6
` 3 '3 = 1 9 '3 3 ` 1 is the simplest equivalent fraction to 3 3 9
` 18 ' 6 = 3 24 ' 6 4 ` 3 is the simplest equivalent fraction to 18 24 4
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How does it work?
Your Turn
Fractions
Equivalent proper fractions 1 Write the equivalent fractions represented by these equally-sized shaded areas: a
2
3
4
=
a
1 4 6# 5 @
b
8 6' 2 @ 10
c
3 6# 3 @ 5
d
12 6' 6 @ 24
=
=
Simplify these fractions by dividing the numerator and denominator by the greatest common factor (GCF). 16 20
b
8 32
b
16 24
Simplify these two fractions. a
=
Write an equivalent fraction for each of these using the multiplication or division given in square brackets:
a
b
=
14 21
Are the fractions 14 and 16 from question 4 equivalent fractions? Briefly explain your answer. 21 24 5
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Fractions Mathletics
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How does it work?
Your Turn
Fractions
Equivalent proper fractions M atch the pair of equivalent fractions below by joining them with a straight line. Solve the puzzle by matching the letter with the number each straight line passes through.
4 5
IONS ACT FR
27 72
.../20.
..
EQ
2 5
10 35
..../..
IONS ACT FR
EQ
LENT PROPE VA R UI
LENT PROPE VA R UI
6
8 12 3 4
6 10
6 15
2 7
6 24
1 4 8 14
1 2 5 15
6 30 1 5
2 3 4 7
1 3 3 5
3 8
15 30
Fractions Mathletics
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12 16
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How does it work?
Fractions
Improper fractions and mixed numbers An improper fraction has a bigger numerator (top) than denominator (bottom). 3 5 Improper fractions 2 4 numerator 2denominator Mixed numbers have a whole number and a proper fraction. Mixed numbers 11 11 2 4 a “mix” of whole numbers and proper fractions
2 means “bigger than”
Mixed numbers are simplified improper fractions. Simplify these: Improper fractions to mixed numbers (i) 5 3
5 = 5'3 3
numerator = numerator 'denominator denominator
= 1r 2
Whole number answer
(ii) 14 4
remainder
= 12 3
same denominator
14 = 7 = 7 ' 2 Simplify if possible 4 2 = 3r1 1 = 32 Whole number answer
remainder same simplified denominator
Mixed numbers to improper fractions +
(i) 1 2 3
1 #
(ii) 2 1 5
+
2 #
8
2 = 3 # 1+ 2 3 3 = 5 same denominator 3
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1 = 5 #2+1 5 5 same denominator = 11 5
Fractions Mathletics
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picture form
How does it work?
Your Turn
Fractions
Improper fractions and mixed numbers 1
Write the mixed numbers represented by these shaded diagrams:
a
b
=
=
c
d
=
=
Make sure you write the fraction in simplest form where possible.
f
=
=
23 2
.
RO
../20..
. ..../..
NS
c
N U M B E RS
R F RA C T I O
Write these fractions in simplest form first, then change to mixed numbers. 21 15 18 b a c 14 9 16 3
Write the equivalent improper fraction for these mixed numbers. b a 11 23 4 2 4
c
44 5
Write the equivalent improper fraction for these mixed numbers after first simplifying the fraction parts. 2 25 24 b c a 4 2 6 12 72 24 5
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Simplify these improper fractions by writing them as mixed numbers. 12 14 b a 5 3 2
A N D M IX E
D
PE
e
9
How does it work?
Fractions
Fractions on the number line Proper fractions 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 fraction using the numerator. 1 ,3 2 Display the fractions and 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 numbers, plot the fraction between the given whole number and the next whole number. Start from this 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
Display and read these fractions on a number line Mixed numbers 1 step taken towards 5 (i) 4 1 2 Start
1 2
4
(ii) 2 2 5
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 fractions – simply change to the equivalent mixed number first then show on the number line 1 step taken towards 3 1 step taken towards 2 (i) 7 = 2 1 (ii) 18 = 3 = 1 1 12 2 2 3 3 Start
1 3
2
Start
3
1
1 2
2
2 equal steps between 1 and 2
3 equal steps between 2 and 3
Write down the fraction 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
4
2 steps taken towards 4 4
3
3 equal steps between 3 and 4
32 3
(iii)
4
5
4 steps taken towards 5 4
5
6 equal steps between 4 and 5
4 4 = 4 2 Simplest form 6 3
Fractions Mathletics
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How does it work?
Your Turn
Fractions FRACTIONS ON
T HE NUMBE R LINE
What proper fraction do the following points on the number line represent? a
2
0
0
0
1
1 4
3 3
0
c
b
1
0
1
8 15
1
c
0
1
2
b
3
1
2
=
5
c
0
1
Write the mixed number and equivalent improper fraction for the dots plotted on these number lines: a
b
1
0 4
1
Write and display the fraction of equal shapes shaded on a number line for these diagrams: a
0
Display these fractions on a number line:
a
3
b
1
THE NUMB ER LINE
1
..../...../20...
FRACTIONS ON
Fractions on the number line
c
5
6
=
=
Display these improper fractions on the number line:
27 a = 10
b
2
a
42 = 15
1
11 = 2
3
4
b
=
63 = 18
2
5
=
3 Fractions Mathletics
22 5 =
c
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c
6
110 25 =
=
4
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How does it work?
Fractions
Reciprocal fractions
2 5
Original fraction
5 2
swap
Reciprocal fraction
Write the reciprocal of these fractions (i) 3 4
(ii) 18 8
3 4
3 4
swap
4 3
Reciprocal fraction
18 = 9 8 4
9 4
swap
4 9
Reciprocal fraction
Simplified
For mixed numbers, change to an improper fraction first then write the reciprocal. 7 2
31 2
Mixed number
Improper fraction
7 2
2 7
swap
Reciprocal fraction
Whole/mixed number examples: Always write as a fraction first. Write the reciprocal of these 3 = 3 1
(i) 3
3 1
1 3
Reciprocal fraction
4 11
Reciprocal fraction
swap
Whole number as a fraction
3 (ii) 2 4
23 4
11 4
11 4
swap
Improper fraction
(iii) 4 9 15
4 9 15 Simplified fraction
43 5
23 5
23 5
swap
5 23
Reciprocal fraction
Improper fraction
We will see why we find the reciprocal a little later on in this booklet. It is used when dividing an amount by a fraction.
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How does it work?
Your Turn
Fractions
5
1 23
d
c
12 18
d
15 4
../20...
b
14 8
25 10
b
1 9
c
2
d
4
b
32 5
c
15 9
Write the reciprocal of these mixed numbers after first writing as a fraction:
2 a 3 10
6
6 19
Write the reciprocal of these mixed numbers: a
c
Write the reciprocal of these:
1 a 5
4
11 7
Write the reciprocal, then simplify these fractions:
6 a 10
3
b
..../...
S ON
2 3
3 4
F CAL RACTI RO
a
2
4 3
Write the reciprocal for these fractions:
REC IP
1
REC IP
Reciprocal fractions
NS IO
F CAL RACT RO
b
1 10 12
c
2 9 21
c
15 115
Write the reciprocal of these fractions as a simplified mixed number:
a 10 48
b
12 66
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Where does it work?
Fractions
Comparing fractions This is where we see which fractions are larger than others. 1 2
1 3
or
Write equivalent fractions by changing the denominators to their LCM. 1#3 2#3
3 6
2 6
or
Least Common Multiple (LCM) The smallest value that appears in both times tables
1#2 3#2
Since they have the same denominator, now just compare their numerators. 3 2 2 6 6
bigger 1 smaller
` 1 2 1 2 3 Compare the size of these fractions (i) 2 , 3 and 5 4 12 3
2#4 3#4
`
2 3
,
3 4
8 12
,
9 12
and 5 12
3#3 4#3
LCM of denominators = 12
and
5 12
5 1 8 1 9 12 12 12
Compare numerators
5 1 2 1 3 12 3 4
Order fractions by size
If comparing improper fractions, use the same method but leave the answer as a mixed number. (ii) 11 and 13 4 5
11 # 5 4#5
11 4
,
13 5
55 20
,
52 20
LCM of denominators = 20
13 # 4 5#4
55 2 52 20 20
Compare numerators
` 23 2 23 4 5
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Write in simplest form
Fractions Mathletics
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Where does it work?
Your Turn
Fractions
Comparing fractions 1
Compare the size of these fractions: a
2 and 1 5 3
b
3 and 5 4 7
9 , 2 and 5 12 3 6
..../.
ONS TI
b
1 3
..../2
0...
CO M
3 , 1 and 11 a 2 20 5
3
1 2
Compare the size of the fractions in each of these groups:
RING FRAC PA
2
ONS TI
CO M
RING FRAC PA
Compare the size of these improper fractions:
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
two thirds
1 3
less
one third
1 3
= equals
one third
If the denominator (bottom) is the same, just add or subtract the numerators (top). Simplify these fractions with the same denominator (i) 2 + 5 2 + 5 = 2 + 5 9 9 9 9 9
Add the numerators only
= 7 9 (ii) 6 - 2 6 - 2 = 6 - 2 7 7 7 7 7
Subtract the numerators only
= 4 7 (iii) 2 + 5 2 + 5 = 2 + 5 3 3 3 3 3 Always write answers in simplest form
(iv) 3 - 1 + 4 5 5 5
Add the numerators only
= 7 3 = 21 3
3 - 1 + 4 = 3 - 1+ 4 5 5 5 5
Simplify
Subtract/add the numerators only
= 6 5 = 11 5
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Fractions Mathletics
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Simplify
Where does it work?
Your Turn
Fractions
Adding and subtracting fractions with the same denominator Simplify these without the aid of a calculator: a
1+1 3 3
b
3-1 5 5
c
5+2 9 9
d
8 - 6 11 11
e
11 - 4 15 15
f
3+5 8 8
a
1+4 2 2
d
10 - 1 4 4
CTIONS FRA WI
NG
c
2+5 3 3
e
11 + 4 7 7
f
15 - 8 2 2
..
Simplify these without the aid of a calculator, remembering to write the answer in simplest form:
11 5 a 4 4
4
b
8-2 5 5
.../20.
SUBTRAC TI AND
3
..../..
Simplify these without the aid of a calculator:
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
Fractions Mathletics
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ADD IN
2
SAME D THE EN
ATOR IN OM
TH
G
1
Where does it work?
Fractions
Adding and subtracting fractions with a different denominator +
=
1 4
+
1 2
=
?
one quarter
and
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 fractions with different denominators: For 2 and 1 3 5
(i) 2 + 1 3 5
Denominators are different
` 2 + 1 = 2 #5 + 1#3 3 5 3#5 5#3
Multiply top and bottom by the number used to make the denominator equal to the LCM
= 10 + 3 15 15
Equivalent fractions with LCM denominators
= 10 + 3 15
Add the numerators only
= 13 15
(ii) 7 - 1 + 3 For 7 , 1 , and 3 8 2 4 8 2 4 7 - 1 + 3 = 7 - 1#4 + 3#2 8 2 4 8 2#4 4#2 = 7-4+6 8 8 8
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The LCM of the denominators is 15
Denominators are all different The LCM of all the denominators is 8 Equivalent fractions with LCM in the denominators
= 7- 4+ 6 8 = 9 8
Simplify the numerator
= 11 8
Simplify to mixed number
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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 calculations: a
1 + 1 The LCM of the denominators is: 3 6 1` 11 1 1 # 1 # 3 +36+ =6 3=# 3 #
=
b
4 - 1 The LCM of the denominators is: 7 5
5# `55-- 11 = 77 55 = 7 #
1+ 1 +6 6
+1 6
=
=
=
2
1# 7
- 15##77 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 Simplify these expressions without the aid of a calculator, remembering to write the answer in simplest form. 1+4 2 5
b
13 - 3 8 5
HE TH T DIFFE R WI
..../...../20.
..
ENOMINATOR TD EN
a
NG FRACTIO NS CTI A R
3
G AND SU DIN BT AD
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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Fractions Mathletics
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Where does it work?
Your Turn
Fractions
Adding and subtracting fractions with a different denominator The same rules apply for questions with a mix of whole numbers and fractions. Here are some examples: Simplify these expressions, expressions which whichhave haveaamix mixof ofwhole wholenumbers numbersand andfractions fractions: (i) 3 + 1 4
3+ 1 = 3 1 4 4
Write the fraction after the whole number
(ii) 1 - 2 5
1- 2 = 5 - 2 5 5 5 = 3 5
Write the whole number as a fraction with same denominator
4 - 2 = 28 - 2 7 7 7 = 26 7 = 35 7
Write the whole number as a fraction with same denominator
(iii) 4 - 2 7
4
Subtract the numerators only
Simplify the fraction
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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Where does it work?
Fractions
Multiplying and dividing fractions To multiply fractions, just remember: Multiply the numerators (top) and the denominators (bottom). (bottom) of of = = “‘ # ’”
1 of 2 = 1 3 5 3
#
2 = 1#2 = 2 5 15 3#5
To divide an amount by a fraction, just remember: flip the second fraction, then fraction then multiply multiply. 1 '2 = 1 3 5 3
#
5 2
Onlyonly Flip flip the second fraction fraction.
= 1#5 3#2
Change Changethe the“'” ‘'’to toaa“‘##”’
Remember: Remember: AA flipped flipped fraction fraction isis called called the the reciprocal reciprocal fraction. fraction
= 5 6
Simplify these: We can use shaded diagrams to calculate the multiplication of two fractions. fractions (i) 2 of 4 3 5
Draw a grid using the denominators as the dimensions
3 5 4 3
2
Use the numerators to shade columns/rows
5 = 8 15 `22##44 == 88 33 55 15 15
Write where they overlap as a fraction
If whole numbers are involved, write them as a fraction. fraction (ii) 28 ' 2 7
` 28 ' 2 7
= 28 # 7 2 = 28 # 7 1 2 = 196 2 = 98 1
Flip the second fraction and change the signsign to ‘ #to’ “ # ” Write the whole number as a fraction
Simplify
= 98
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Fractions Mathletics
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Where does it work?
Your Turn
Fractions
Multiplying and dividing fractions 1
Calculate these fraction multiplications by shading the given grids: a
1 of 3 5 4
b
2 of 4 3 7
3 5 7 4
` 2 of 4 = 3 7
` 1 of 3 = 5 4
c
4 of 4 5 5
d
2 of 3 5 8
5
5
5
8 ` 2 of 3 = 5 8
` 4 of 4 = 5 5
= simplified
e
3 of 7 4 9
f
3 of 5 4 6
4
4 9 ` 3 of 7 = 4 9
6 ` 3 of 5 = 4 6
= simplified
simplified
Fractions Mathletics
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Your Turn
Fractions
2
24
....
Simplify these without the aid of a calculator: a
1 2
c
2 2 2 ` 3 j psst: this is just 3
e
#
1 3
b
3 5
#
1 4
d
3 2 `5j
1 '3 3 2
f
2 '1 11 4
g
5 '4 6
h
3 '8 4
i
10 # 4 5
j
24 # 3 8
k
12 ' 3 5
l
2' 2 13
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#
2 3
Fractions Mathletics
ONS ACTI R F ING IVID
Multiplying and dividing fractions
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/...
../2
0...
S TION FRAC ING IVID
ANDD YING L P I LT ' MU
ANDD ING IPLY MULT
Where does it work?
#
Where does it work?
Your Turn
Fractions
Multiplying and dividing fractions 3
4
Simplify these without the aid of a calculator, remembering to write the answer in simplest form. a
2 2 `8j
b
3 4
c
3 '5 8 4
d
2 '5 3 3
f
3 4
h
1 ' 4 ' 1 psst: work left to right! 2 2
e
9 '8 10 5
g
2 5
#
3 6
#
1 3
#
#
3 2
2 3
#
1 psst: same as the others! 2
Is 2 of 4 exactly the same as 2 ' 12 ? Explain your answer. 3 6 3 8
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Where does it work?
Fractions
Operations with mixed numbers Just change to improper fractions, fractions then thenuse usethe thesame samemethods methodsas asshown shownearlier. earlier. Simplify these calculations involving mixed numbers: numbers Addition and subtraction (i) 1 2 + 2 1 3 6
1 2 + 2 1 = 5 + 13 3 3 6 6 = 10 + 13 6 6
Change to improper fractions Equivalent fractions with LCM denominators
= 23 6 = 35 6
Or just add the whole numbers and the fractions separately. 1+ 2 = 3 2 + 1 = 5 3 6 6
(ii) 4 1 - 1 1 5 2
Simplify to mixed number
4 1 - 1 1 = 21 - 3 5 2 5 2 = 42 - 15 10 10
Change to improper fractions Equivalent fractions with LCM denominators
= 27 10 = 2 7 10
Simplify to mixed number
Multiplication and division (iii) 1 3 4
#2
1 3
(iv) 1 1 ' 2 6
13 4
#2
1 = 7 4 3
#
Multiply tops and bottoms together
= 4 1 12
Simplify to mixed number
1 1 '2 = 7 ' 2 6 6 1
Remember etc 2 = 2 , 3 = 3 , etc. 1 1
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Change to improper fractions
= 49 12
= 7 6
26
7 3
#
1 2
= 7 12
Flip second fraction and multiply change to multiply Multiply numerators and denominators together
Fractions Mathletics
Change to improper fractions
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Your Turn
Fractions
a
21 +42 4 3
b
21 -12 4 5
c
53 -21 5 2
d
31 +11 6 5
b
43 7
#2
d
51 3
#1
BERS NUM
..../2
0...
# +
Simplify these without the aid of a calculator: a
c
3
..../.
ED
Simplify these additions and subtractions without the aid of a calculator:
OP E
2
# +
BERS NUM
1
OP
Operations with mixed numbers
ED
W IONS ITH M IX AT R E
NS WITH MI TIO X RA
Where does it work?
4 #1 2 5
13 4
#3
1 2
4 5
Simplify these divisions without the aid of a calculator: a
3 '2 1 2
b
1 2 '3 3
c
2 2 '1 1 5 2
d
5 1 '1 2 2 3
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Where does it work?
Your Turn
Fractions
Combining all the operations Earn yourself an awesome passport stamp by trying these trickier questions without using a calculator.
ME * SO E W
*A
1 psst: remember your order of operations #1 2
ME * SO 2
Simplify 4 1 ' 3 5 2 6
3
Simplify this shaded diagram into a single fraction. psst: write as fractions, then work left to right
#5
1 psst: work left to right‌ so do the division first 3
+
#
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*A
/20...
.... ..../.
WE
1
Simplify 1 + 6 3 7 5
What else can you do?
Fractions
Fractions of an amount How many links are there in 2 of a chain made using a total of 20 links? 5
Simplified, this question is just: Find 2 of 20 . 5 Remember: ‘of’ = ‘ # ’
` 2 of 20 = 2 # 20 5 5 = 2 # 20 5 1 = 40 5 = 8 ` 2 of the 20 links = 8 links 5 Here are some questions that calculate fractions of an amount.
(i) Juliet lost 1 of the 116 songs she had downloaded when a computer virus infected the files. 4 How many songs were not affected by the virus? 1 of songs 116 = 1 # 116 4 4 = 1 + 116 4 1 = 116 4
If 1 are gone, then 3 remain. 4 4 So finding 3 of 116 will get 4 the same answer. Try it!
= 29 ` 116 - 29 = 87 songs not affected by the virus (ii) How long is 7 of 1 hour? 10
7 = 7 of 60 minutes 10 of 1 hour 10 = 7 # 60 10 1 = 420 10
Answer the question
Change to smaller units if possible
= 42 ` 7 of 1 hour = 42 minutes 10
Fractions Mathletics
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Answer the question
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Your Turn
Fractions FRACTI
2 of 24 3
d
5 of 42 6
.... /... ../2 0...
AM OU NT
c
AN OF
3 of 32 4
ONS
b
T
1 of 20 5
UN
a
AM O
2
Calculate the amount for each of these, showing all working:
AN
1
ONS
OF
Fractions of an amount
FRACTI
What else can you do?
Calculate the amount for each of these by first making the mixed number an improper fraction: 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 fractions of quantities, 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
Fractions Mathletics
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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 kilometers? 5 1 km = 1000 meters
In an orchestra of 60 musicians, 1 were in the brass section. How many brass section players were there? 5 psst: remember to include a statement answering the question at the end
5
Krista and her team mates each receive 1 of a $900 prize for winning a competition. 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 6
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? 7
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Your Turn
Fractions
Fractions of an amount Here is what to do if you already have the fraction amount and need to find 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 multiply 250 g by the reciprocal. 250 g # 5 = 625 g 2
2 of the total ingredients used = 250 g Divide the amount by the numerator 5 ` 1 of the total ingredients = 250 g ' 2 5 = 125 grams Multiply 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 letter ‘e’ is used twenty one times in this question. If this is 1 of all the letters used to write 8 this question, work out how many letters there are in total (show all working and check your amount by counting). 8
9
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
10
32
y 3:00 pm, Juan had been waiting 30 minutes for his bus to arrive. If this is 5 of the total time he B 6 spent waiting, at what time did his bus arrive?
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What else can you do?
Fractions
Two amounts as a fraction Sometimes we need to compare two amounts by writing one as a fraction of the other. For example, write 14 out of 36 as a fraction in simplest form First amount Second ‘out of’ amount
14 = 14 ' 2 36 36 ' 2 = 7 18 ` 14 out of 36 = 7 18
Divide the amount by the GCF Simplest form
These examples show how both amounts must be in the same units before writing as a fraction. Write these amounts as fractions 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 fraction of the total weight is being carried? 40 kg = 40000 g
Need both weights in the same smaller units
` 2000 = the fraction of the total weight being carried 40 000 Simplify fraction = 1 1 kg = 1000 g 20 ` the box being carried is 1 of the total weight Answer question 20 (ii) What fraction of 2 1 hours is 15 minutes? 2 2 1 hours = 2 1 2 2
# 60
minutes
= 150 minutes First smaller amount total amount
15 = 1 150 10
1 hour = 60 minutes Need both times in the same smaller units Simplify fraction
` 15 minutes is 1 of 2 1 hours 10 2
Answer question
(iii) If one ream of paper contains 500 sheets, what fraction is 240 sheets out of 4 reams? 4 reams = 4 # 500 sheets = 2000 sheets First smaller amount total amount
240 = 3 2000 25
Need both amounts in the same smaller units Simplify
` 240 sheets is 3 of 4 reams 25
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Your Turn
Fractions AM TWO ON TI C A
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)
3
After 100 test rolls, a die displayed the number 5 sixteen times. What fraction of the rolls were 5?
Francesca fired 27 arrows at a target and hit the bullseye 6 times. What fraction of arrows missed the bullseye? 4
A biscuit recipe contained 500 g of flour, 175 g of sugar and 125 g of butter. What fraction of the recipe is butter?
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AM
5 out of 20
In a school of 800 students, 240 were in Year 7. What fraction of the school are Year 7 students?
34
TWO
a
2
5
..
Write the first amount as a fraction of the second for each of these in simplest form
.../20.
1
..../..
Two amounts as a fraction
ON TI AC
AS A NTS FR OU
AS A FR NTS OU
What else can you do?
What else can you do?
Fractions
Word problems with fractions While on a shopping trip, Xieng spent two-fifths two fifths of her money on clothes and one one-third third on on cosmetics. cosmetics. What fraction of her money did Xieng have left? 2 + 1 = fraction of Xieng’s money spent on shopping 5 3 = 6+ 5 15 11 Add the numerators together = 15 ` 15 - 11 = 4 15 15 15 Fraction for all of Xieng’s money
Fraction spent
Fraction of money Xieng has left
` Xieng Xieng still still has has4 4ofof her her money money after after shopping shopping 1515 Here are some other word problem examples: examples (i) I n a group of eighteen friends, one-third one third are girls and one one-sixth sixth of of these these girls girls have have blonde blonde hair. hair. How many blonde girls are in the group? ` 1 of 1 of 18 = number of blonde girls in the group 6 3 = 1 # 1 # 18 6 3 1 = 18 18 = 1 ` There is 1 blonde girl in the group of friends. (ii) D uring one night, possums ate two-fifths two fifths of the fifty fifty-five five fruits fruits on on aa tree. tree. IfIf one one-eleventh eleventh of ofthe the eaten fruits fruit grew grewback, back,how howmany manyfruits fruitsare arenow nowon onthe thetree? tree? 2 # 55 = Number of fruits eaten 5 = 110 5 = 22 1 # 22 = Number of fruits that grew back 11 = 22 11 = 2 Number of of fruits now onon thethe tree ` Number fruits now tree = 55 - 22 + 2 = 35 pieces of fruit
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What else can you do?
Your Turn
Fractions
Word problems with fractions A t a recent trivia night, one table of competitors answered five-eighths five eighths of the fifty fifty-six six questions questions correctly. correctly. How many questions did they get incorrect?
2
Co Tin usually takes approximately sixty and one-quarter 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 two third hours each day?
WO RD
FR AC T IO NS
..../...../20...
PR OB LE MS
WO RD
PR OB L EM S
3
A vegetable garden has one-third one third carrots, one one-sixth sixth pumpkins, pumpkins, one one-quarter quarter herbs, herbs.and Thethe restrest areare potato potato plants. How many potato plants are in this garden of eighty plants?
4
A class of twenty-four twenty four students compared eye colours colors on onaachart. chart.Two-thirds Two thirdsofofthe theclass classhad hadbrown browneyes, eyes, and three-eighths three eighths of those brown-eyed students were boys. How many girls had brown eyes?
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WI TH
WI TH
FR AC TI ON S
1
What else can you do?
Your Turn
Fractions
Word problems with fractions 5
For one particular school: There are 256 students in Grade Year 7.7. The Year8,8,9,9and Grades and10 10all groups have all half have the half number the number of students of students as the year thanjust thebelow year just them. below them. How many students are there at this school in Grades Years 7 7 toto 10? 10?
Rememb
er me?
6
Five students in a class have a combined total of ninety friends profileson added a web-based as friendssocial on a web-based network site. social network site.of the ninety profiles are shared by all five of the students. These shared profiles represent Three-fifths Three fifths one-sixth ofof thethe total ninety number profiles of different are shared profiles by alladded five of as thefriends students. by all These the students shared profiles in the class. represent one sixth How manyofdifferent the totalprofiles numberare of linked different to students profiles added from this as friends class? by all the students in the class. How many different profiles are linked to students from this class?
7
Five-sevenths ive sevenths of the fifty fifty-six six images images used used as as backgrounds backgrounds on on Meagan’s Meagan’s touchpad touchpad were were photos photos she she took took herself. .After five-eighths moving five eighths of theseofphotos these were photos deleted, to another whatcomputer, fraction ofwhat the background fraction of the images background now are images not photos nowtaken are not by photos her? taken by her?
Fractions Mathletics
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What else can you do?
Your Turn
Fractions
Reflection Time Reflecting on the work covered within this booklet: 1 What useful skills have you gained by learning about fractions?
2
Write about one way you think you could apply fractions to a real life situations.
3
I f you discovered or learnt about any shortcuts to help with fractions or some other cool facts, jot them down here:
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Fractions Mathletics Passport
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Cheat Sheet
Fractions
Here is a summary of the things you need to remember for fractions Proper fractions Represent parts of a whole number or object. The numerator is smaller than or equal to the denominator. numerator number of equal parts you have 1 2 denominator total number of equal parts Equivalent proper fractions These are fractions with different numbers that represent the same amount. `44 == 22 == 11 == Equivalent Equivalent fractions fractions 44 22 88 Improper fractions and mixed numbers 5 3 Improper fractions 4 2 numerator > denominator Fractions on the number line 1 0 3
4
Start
number of equal steps taken between 0 and 1 total number of equal steps between 0 and 1
1 2
number of equal steps towards the next whole number total number of equal steps between start and next whole number
31 2
Reciprocal fractions Original fraction
Mixed number
11 11 Mixed numbers 2 4 AA “mix” ‘mix’ of of whole whole numbers numbers and and proper proper fractions. fractions.
31 2
2 5
7 2
7 2
5 2
Reciprocal fraction 2 7
Reciprocal fraction
Comparing fractions Write equivalent fractions by changing the denominators to their LCM, then compare the numerators 1#2 1#3 3 2 3#2 6 2 6 2#3 Adding and subtracting fractions If the denominators (bottom) are the same, then simply add or subtract the numerators (top). (tops). If the denominators are different, the different, change change to equivalent to equivalent fractions fractions withwith the same the same denominators denominators using using the the LCM. LCM. ThenThen add add or subtract or subtract the the numerators numerators of the of the newnew fractions. fractions. Multiplying and dividing fractions To multiply fractions, just remember: Multiply the numerators (top) and the denominators (bottom). To divide an amount by a fraction, just remember: flip the second fraction (reciprocal), (reciprocal) then thenmultiply. multiply. Fractions of an amount “of” amount ‘of’ means means‘ #“ #’ .” Find 22 of of22means meanscalculate calculate 22 ` Find 55 55
## 22
Two amounts as a fraction 2 out of 5 as a fraction is 2 . If the two amounts are in different units, change the larger amount into 5 the smaller units. So Eg,200 g 200 gout outof of2 kg 2 kgbecomes becomes200 g 200 gout outof of2,000 g. 2000 g. Fractions Mathletics Passport
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Answers
Fractions
Proper fractions
Proper fractions
1. a 1 3
b
5 8
c
3 4
d
9 19
1 2
f
11 18
g
3 5
h
5 9
e
2. a
4.
c
(i)
d
(i)
b
(ii)
(ii)
Equivalent proper fractions c
d
2 4 16 1. a 3 = 6 = 24
b
1 3 6 2 2 = 6 = 12 = 4
1# 5 5 2. a 4 # 5 = 20
b
8'2 4 10 ' 2 = 5
3#3 = 9 5#3 15
d
12 ' 6 2 24 ' 6 = 4
16 ' 4 4 3. a 20 ' 4 = 5
b
8'8 1 32 ' 8 = 4
14 ' 7 2 4. a 21 ' 7 = 3
b
16 ' 8 2 24 ' 8 = 3
e
c
3. a
b
5. Yes, becasue they both simplify fully to give 2 . 3 6. c
2 5
d
27 72
10 35
4 5
8 12
6 10
3 4
2 7
6 15
1 4
e
6 24
8 14
1 2
6 30
4.
a b
2 3
(ii)
(i) (i)
5 15 1 5 4 7
1 3
(ii)
3 5
3 8
15 30
8 10
S I M P L E S T
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12 16
F O R M
Answers
Fractions
Improper fractions and mixed numbers numerals
Fractions on the number line 14 4 6. a 42 15 = 5 = 2 5
1. a 2 2 3
b
31 4
c
32 5
1 14
e
1 22
f
33 4
2. a 2 2 5
b
42 3
c
1 11 2
1
3. a 1 2 3
b
1 12
c
1 18
Reciprocal fractions
3 4. a 2
11 b 4
25 5. a 6
b
d
c
9 4
76 3
c
2
2. a 1 4 b
c
b
8 4 14 = 7
1 18 3 12 = 2 = 1 2
d
10 = 2 25 5
c
1 2
b
17 32 5 = 5
b
LCM = 28
1 = 7 4. a 2 3 3
1
c
3 3
0
8 15
0
8 2 12 = 3
7 12
0
1
0
1
or
3 7
14 15 9 = 9
d
4 15
d
1 4 5 17
9 14 10 5 32 = 16
b
10 = 22 1 12 12
12 6 22 = 11
c
9 = 51 2 21 21
17 = 7 7 17
5 6. a 10 = 24 48
24 = 4 4 5 5
b
12 = 2 66 11
11 = 5 1 2 2
c
15 = 3 23 115
1
0
1
b
7 5. a 27 10 = 2 10
b
3
9 =9 1
b
2 = 32 5. a 3 10 10 1
8 4. a 2 2 3 = 3
2
7 11
b
1
0 c
5
2 5 2. a 10 6 = 3 =13
3. a 5 1 =5
3 8
c
0
3. a 2 5 b
4 5
4
19 6
Fractions on the number line b
3
c
c
1. a 1 3
63 = 7 = 3 1 2 2 18
110 = 22 = 4 2 25 5 5
3 1. a 2
24 c 5
b
1 = 5 14 4
c
11 = 5 1 2 2 4
c
5
23 = 7 2 3 3
Comparing fractions
3 = 28 55 5
1. a LCM = 15
22 = 4 2 5 5
1 2 3 1 5 2. a LCM = 20
6
1 11 3 2 1 20 1 5
Fractions Mathletics
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5 1 3 7 4 b
LCM = 12
2 1 9 1 5 3 6 12
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Answers
Fractions
Comparing fractions 3. a LCM = 28
Adding and subtracting fractions with a different denominator b
LCM = 24 2. a LCM = 6
2 3 25 8 134 143
1 122 24 7
Adding and subtracting fractions with the same denominator b
2 5
c
7 9
2 11
e
7 15
f
8 =1 8
1 2. a 2 2
b
1 15
c
1 24
e
1 27
1 3. a 1 2
b
7 4. a 9 3 5
1. a 2 3 d
d
d
5 6
b
LCM = 6
1 3
c
LCM = 20
3 20
d
LCM = 12
11 12
e
LCM = 21
4 21
f
LCM = 40
39 40
3 1 10
b
LCM = 40
1 1 40
3. a LCM = 10 c
LCM = 8
5 8
d
LCM = 20
3 20
1 23
e
LCM = 12
1 14
f
LCM = 24
17 24
f
1 32
1 4. a 2 2
b
3 14
c
1 3
d
5 8
1 53
c
3 24
e
12 5
f
3 34
g
1 13
h
1 22
b
2
c
1 2
e
13 7
f
1 22
Multiplying and dividing fractions 3 20
1. a 5
Adding and subtracting fractions with a different denominator 1 + 1 = 1#2 + 1 3 6 3#2 6 = 2+1 6 6 = 3 6 1 = 2
1. a LCM = 6
b
4
LCM = 35
b
8 21
3 7
16 25
c
5
5 - 1 = 5#5 - 1#7 7 5 7#5 5#7 = 25 - 7 35 35 = 18 35
5
6 3 40 = 20
d
5
8
42
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Answers
Fractions
Multiplying and dividing fractions
Fractions of an amount 21 = 7 36 12
1. e
1. a 4
b
24
c
16
d
35
2. a 10
b
27
c
46
d
168
b
600 grams
d
600 meters metres
4 9
3. a 18 hours
15 5 24 = 8
f
c
4
100 minutes
4. There were 12 players in the brass section.
6
1 2. a 6
b
3 20
c
4 9
d
9 25
5. Krista received $180.
e
2 9
f
5 24
g
8 11
h
3 32
6. 20 of the towels were not bought by Hank.
i
8
j
9
k
20
l
13
3. a
1 16
b
1 18
c
3 10
d
2 5
e
9 16
f
1 4
g
1 15
h
1 4
7. 9 grams of the lead was made of non-graphite materials. 8. There are 168 letters in total used in the question.
4. 2 of 4 is exactly the same as 2 ' 12 3 6 3 8
9. 490 people were there to watch Jolly Rob perform. 10. The bus arrived at 3:06 pm.
Operations with mixed numbers numerals 11 1. a 6 12
b
17 20
c
1 3 10
d
11 4 30
3 2. a 5 5
b
86 7
c
1 68
d
93 5
1 3. a 1 5
b
5 9
c
13 5
d
Two amounts as a fraction 1 1. a 4
b
8 11
c
7 8
d
1 12
1 6
f
1 8
g
1 12
h
1 4
e
3 3 10 2.
3 of the school are Year 7 students. 10
3.
4 of the rolls were 5. 25
Combining all the operations 3 1. 10 70
4. 7 of the arrows missed the bullseye. 9
6 2. 6 23
5.
3. 7 43 96
5 of the recipe is butter. 32
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Fractions
Word problems with fractions 1. The table got 21 questions incorrect. 2. CoTin would expect to walk 6 025 steps every 1 2 3 hours. 3. There are 20 potato plants in the garden. 4. 10 girls had brown eyes. 5. There are 528 students in Years 7 to 10.
6. There are 324 different profiles linked to students from this class. 7. 16 of the photos were not taken by Meagan. 31
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