 Contributed by: This includes:-
1. To find a common denominator and use it
to write equivalent fractions
2. To add and subtract fractions with unlike
denominators
3. To add and subtract mixed numbers
4. To solve fraction problems
Fractions have been a part of our lives for a long time. In ancient times,
Sumerians, Babylonians, Romans, Egyptians, Greeks, Chinese, and
Arabians all used fractions.
The Eye of Horus was used by ancient Egyptians to represent fractions.
It was a powerful symbol that they believed protected against evil.
According to an old myth, Horus’ eye was torn out, broken into pieces,
and then put back together.
What You Will Learn Key Words
φ to ﬁnd a common denominator and use it multiple
to write equivalent fractions common denominator
φ to add and subtract fractions with unlike improper fraction
denominators mixed number
φ to add and subtract mixed numbers
φ to solve fraction problems
Each part of the Eye of Horus was a –1
8
fraction. When all the parts were put
together, it was believed that the eye –1
4
1
was whole again. Ancient Egyptians –1 ––
2 16
thought that the parts had a combined
value of 1. Were they correct? In this
chapter, you will explore fractions 1 1
–– ––
through history and around the world. 64 32
228 NEL • Chapter 7
2. ;DA967A:H /
Make the following Foldable to organize
what you learn in Chapter 7.
Step 1 Staple seven sheets of notebook
paper together along the top edge.
Step 2 Make a line 9 cm
up from the
bottom of the
top page. Cut
across the entire
page at this
mark.
Step 3 Make a line 7.5 cm up from the
bottom of the second page. Cut
across the entire page at this mark.
Step 4 Cut across a line 6 cm up from the
bottom of the third page.
Step 5 Cut across a line 4.5 cm up from
the bottom of the fourth page.
Step 6 Cut across a line 3 cm up from the
bottom of the ﬁfth page.
Step 7 Cut across a line 1.5 cm up from
the bottom of the sixth page.
Step 8 Label the
tabs formed
as shown.
Chapter 7 Add and Subtract Fractions
Key Words
.
71
.
72
.
73
.
74
What I Need to Work On
As you work through Chapter 7, take
notes under the appropriate tab.
words, examples, and key ideas.
Chapter 7 • NEL 229
3. Common Denominators
Focus on…
After this lesson, you
will be able to...
φ ﬁnd a common
denominator for a
set of fractions
φ compare and
order positive
fractions
Jasmin and Tyler collect trading cards. Jasmin has collected __31 of a set.
Tyler has collected 1
__ of a set. They want to know who has more cards.
4
Jasmin and Tyler need to compare the fractions. It is easier to compare
fractions when the denominators are the same. So, Jasmin and Tyler
need to find a common denominator.
How can you determine a common denominator?
• coloured pencils 1. Fold a piece of paper
into 3 equal parts.
Shade 1__ of the paper red.
3
2. Fold the same piece of
paper into 4 equal parts
the other way.
230 NEL • Chapter 7
4. 3. a) How many equal parts is the paper divided into?
b) Count how many parts you shaded red. Name an equivalent
fraction for __
3
1 of the
4. Fold a different piece of paper into 4 equal parts. Shade __
4
paper blue.
5. Fold the piece of paper into 3 equal parts the other way.
6. Count how many parts you shaded blue. Name an equivalent
1.
fraction for __
4
7. a) What is the relationship between the denominators 3 and 4, and
the denominator 12? common
b) What is one method for determining a common denominator ? denominator
• a common multiple of
the denominators of a
Example: Determine a Common Denominator set of fractions
• a common
a) Determine a common denominator for _2_ and 1
__. denominator for _1_
3 2 4
2
_ _ 1
__ and 1__ is 12 because a
b) Determine equivalent fractions for and using the common 6
3 2 common multiple of
denominator from a).
4 and 6 is 12
Method 1: Use Paper Folding or Diagrams
a) Divide a rectangle into 3 equal parts. Either fold a piece
of paper, or draw a rectangle.
Fold the paper or divide the rectangle into 2 equal parts
the other way.
There are 6 parts in the rectangle.
A common denominator for __ 2 and 1
__ is 6.
3 2
b) Shade 2__ of the rectangle red.
3
4 of the 6 parts are red. 2=4
__ __
3 6
Turn the paper over, or draw another rectangle
and divide it as in step a).
Shade 1__ of this rectangle blue.
2
3 of the 6 parts are blue. 1=3
__ __
2 6
7.1 Common Denominators • NEL 231
5. Method 2: Use Multiples
1 is 2.
a) The denominator of __ You can use divisibility
2 rules to ﬁnd multiples.
multiple Multiples of 2 are 2, 4, 6, 8, 10, 12, … 6 is divisible by both 2 and 3.
• the product of a given
The denominator of 2 __ is 3. So, multiples of both 2 and 3
number and a natural 3 will be multiples of 6:
number like 1, 2, 3, 6, 12, 18, …
Multiples of 3 are 3, 6, 9, 12, 15, …
and so on
• for example, some The first multiple divisible by both 2 and 3 is 6.
multiples of 3 are 3, 6,
9, 12, and 15 A common denominator is 6.
You could use any multiple of 6 as the
common denominator, but the ﬁrst
multiple is often better to use. The
denominator will be a smaller number,
which is easier to work with.
b) Write equivalent fractions using 6 as the denominator.
×3 ×2
To determine equivalent fractions,
__ = 3
1 __ 2=4
__ __
multiply the numerator and denominator
2 6 3 6 by the same number. This process does
×3 ×2 not change the value of the fraction.
Check: 2
__ = 4
__
Strategies 3 6
Use pattern blocks.
Model It
Refer to page xvi.
=
–1 –3
2 6
=
–2 –4
3 6
Determine a common denominator for each pair of fractions. Then
use the common denominator to write equivalent fractions. Show
two different methods.
1
a) __ and _3_
5
b) __ and _1_
3 4 8 6
232 NEL • Chapter 7
6. • You can use paper folding, diagrams, or multiples to determine
a common denominator.
Paper Folding or Diagrams
5 of the 10 parts are blue. 6 of the 10 parts are red.
5
1 = ___
__ 3 6
__ = ___
2 10 5 10
Multiples
1 is 2. Multiples of 2 are 2, 4, 6, 8, 10, …
The denominator of __
2
3 is 5. Multiples of 5 are 5, 10, 15, 20, …
The denominator of __
5
A common denominator is 10.
• To write fractions with a common ×5 ×2
denominator, determine
equivalent fractions. 5
1 = ___
__ 3 = ___
__ 6
2 10 5 10
×5 ×2
1. Tina wanted to find a common denominator and equivalent
3 and 2
fractions for __ __. This is what she did:
5 3
a) Was she correct? If not, what was her error?
b) Draw diagrams to show what she should have done.
c) Discuss your diagrams with a classmate.
2. Ian says, “A common denominator for __ 3 and 5__ is 12.” Meko
4 6
says, “I think it is 10.” Do you agree with Ian or Meko? Why?
3. How can you use multiples to find a common denominator for
1, __
the fractions __ 3?
2, and __
2 5 4
7.1 Common Denominators • NEL 233
7. 7. Use a diagram to determine a common
denominator for each pair of fractions.
For help with #4 to #9, refer to the Example on Then write equivalent fractions using the
pages 231–232. common denominator.
4. Use the folded papers shown to determine a) _3_ and 1
__ b) _5_ and 3
__ c) _1_ and _1_
8 3 6 4 5 2
a common denominator and equivalent
fractions for each pair of fractions. 8. Use multiples to determine a common
a) denominator for each set of fractions.
Then write equivalent fractions using the
common denominator.
a) _1_ and _2_ b) _1_ and _1_ 5, __
c) __ 5
1, and ___
2 5 3 4 8 6 12
1
__ 2
__
4 3
9. Using multiples, determine a common
b) denominator for each set of fractions.
Then use the common denominator to
write equivalent fractions.
a) _3_ and _1_ b) _1_ and _1_ 1, __
c) __ 7
2, and ___
8 4 6 4 5 3 10
1
__ 3
__
2 4
5. Look at the diagrams to determine a
common denominator and equivalent
fractions for each pair of fractions.
10. Determine a common denominator for
a)
each pair of fractions. Which is the larger
fraction in each pair?
3 13
a) __, ___
5 36
b) __, ___
4 16 7 49
1
__ 3
__ 11 3
c) ___, ___
12
d) ___, _4_
3 5 30 10 27 9
b)
11. Draw a Venn diagram like the one shown
to list common denominators that are less
1 and __
than 50 for __ 1.
5
__ 1
__ 4 6
6 4
Multiples Multiples
6. Draw a diagram to determine a common of 4 of 6
denominator for each pair of fractions.
Then use the common denominator to
write equivalent fractions.
a) _1_ and _1_ b) _2_ and _1_ c) _1_ and _2_
2 3 3 5 6 5
234 NEL • Chapter 7
8. 12. Fill in the blanks to make equivalent 16. 5 of a schoolyard is taken up by grass.
___
12
fractions. 7 is the track. The rest is pavement.
___
1 � � � � � �
a) __ = __ = ___ = ___ = ___ = ___ = ___ 18
4 8 12 16 20 24 28 a) What common denominator could be
b) 1
__ = = 3 = __
2
__ __ 5 = __
4 = __ 7 = ___
11 used to compare these fractions?
5 � � � � � �
24 12 = __ 6 = __3 = ___
48 = __9 b) Does the grass or the track take up
c) ___ = ___
56 � � � � � more space?
d) 30
___ = ___ 10 = __
15 = ___ � = ___
5 = ___ �
48 � � � 96 32
13. Fill in each blank with a numerator to 17. a) Copy the shapes. For each shape,
make the statement true. Provide as many 3.
answers as possible. Use diagrams to show colour in __
8
1 �
a) __ < __ < _3_
4 2 4
1 �
b) __ < __ < _5_
3 6 6
2 �
c) __ < ___ < _4_
5 10 5
14. Determine a common denominator for b) Which shapes were more difficult to
the set of fractions. Use the common colour in? Which were easier? Explain.
denominator to write an equivalent c) Imagine you are using paper folding to
fraction for each fraction. Then list the determine a common denominator for
fractions in order from least to greatest. 3 2. Which of the shapes would it
__ and __
8 5
1 5, __
1, __
__, __ 3, _1_
2, __ be possible for you to use? Show the
3 4 6 3 4 2
work by drawing the fold lines on the
15. The ancient Greeks thought of numbers shapes.
as being represented by rectangles. They d) Compare your drawings with a
would have made a rectangle like this to classmate’s.
represent 6:
18. Write as many different proper fractions
in lowest terms as you can that have
denominators from 2 to 9 and numerators
that are positive numbers.
a) How could this rectangle be used to find
a common denominator for __ 1 and 1
__? 19. Which of the following fractions is closest
2 3 3?
Explain. to ___
10
b) Use a rectangle to find a common 1 B ____
21 9
A __ C ___ D 2__
denominator for __3 and __
1. 4 100 40 5
4 7
7.1 Common Denominators • NEL 235
9. 20. You have three beakers that are the same 21. The table shows the fraction of the total
size. _2_ of beaker 1 contains oil. _1_ of number of students at Maple Leaf
3 4 Elementary School that are in each grade.
beaker 2 contains water. Beaker 3 is
empty. When you pour the liquids into Kindergarten 7
___
beaker 3, the level of the combined liquids 40
corresponds exactly to one of the Grade 1 3
___
markings on the side of beaker 3. Which 20
of the following beakers is beaker 3? Grade 2 11
___
72
A B 5
36
____
180
____
180
___
C D 90
a) Which grade has the greatest number
of students?
b) Which grade has the least number
of students?
c) Which two grades have the same
number of students?
d) If there are 54 students in grade 1,
what is the total number of students
in the school?
8
a) Determine a common denominator for the fractions
in the Eye of Horus. Show your work. –1
4
1
–1 ––
b) Use this common denominator to determine an 2 16
equivalent fraction for each part in the eye.
1 1
–– ––
64 32
236 NEL • Chapter 7
With Unlike Denominators
Focus on… 1
1 1 1
– – –
After this lesson, you 6 3 2
will be able to...
fractions How could you use pattern blocks to model addition and subtraction?
with unlike
denominators
φ solve problems
involving the
subtraction of denominators?
fractions
φ check that your 1. a) What two pattern blocks would you use to represent _1_ and __
1?
2 3
b) Can you tell what the answer to __ 1 is using these two pattern
reasonable using 2 3
blocks? Explain.
estimation
2. a) Use the green triangles to represent _1_ and 1 __. What fraction does
2 3
each green triangle represent?
b) Can you tell what the answer to __1+1 __ is now? Explain.
2 3
3. a) What pattern blocks would you use to represent 1 1?
__ and __
2 6
• pattern blocks 1−1
b) Can you tell what the answer to __ __ is using these two
• coloured pencils 2 6
pattern blocks? Explain.
4. a) 1.
Use the green triangles to represent __
2
b) Can you tell what the answer to __1−1 __ is now? Explain.
2 6
c) How many green triangles are left?
fractions with unlike denominators?
7.2 Add and Subtract Fractions With Unlike Denominators • NEL 237
11. Example 1: Add Fractions With Unlike Denominators
1 1
__ + __
3 6
Solution
Is the sum
closest to Method 1: Use Fraction Strips
0 , 1 + __
__ 1 +
1 , 3 6

2
or
To add, you need to use
parts that are the same size. +
1 ?
1. Count the parts.
Each part represents __
6
1 __ = 3
__ + 1 __
3 6 6
Write the answer in lowest terms.
3 = __
__ 1
6 2
Method 2: Draw a Diagram
1 + __
__ 1
3 6
+
To add, you need to use parts that are the same size.
1+1
__ __ = 3
__
3 6 6
Strategies
Solve a Simpler
Problem
Refer to page xvii. +
=
Write the answer in lowest terms.
3
__ = 1
__
6 2
238 NEL • Chapter 7
12. Method 3: Use a Common Denominator
The denominator of __1 is 3.
3
Multiples of 3 are 3, 6, 9, 12, …
The denominator of 1 __ is 6.
6
Multiples of 6 are 6, 12, 18, 24, …
The first multiple divisible by both 3 and 6 is 6.
A common denominator is 6.
Write equivalent fractions with 6 as the denominator.
1 + __
__ 1 = __
2+1 __
3 6 6 6
=2 + 1 Add the numerators.
_____
6
3
= __
6
Write the answer in lowest terms.
÷3
3 = __
__ 1
6 2
÷3
Use pattern blocks.
Is closer to or ? adding fractions, go to
3 6 2
It equals 1
2
When the numerator
equals the
1 __ fraction is equal to 1.
a) 1
__ + 2
__ 1 +4
b) ___ __ c) __ + 4 3=1
__ 16
___ = 1
6 3 10 5 2 8 3 16
7.2 Add and Subtract Fractions With Unlike Denominators • NEL 239
13. Example 2: Subtract Fractions With Unlike Denominators
Subtract.
1 − __
__ 2
2 5
Solution
Is the Method 1: Use Fraction Strips
diﬀerence 1−2
__ __ –
closer to 2 5
0 To subtract 2__, you need
or 5 –
? parts that are the same size.
1

2 Subtract.
1 − __
__ 2 = ___
1
2 5 10
Check:
1?
1 closer to 0 or __
Is ___
10 2
0 = ___0 1 5
__ = ___ Compare this to the
10 2 10 estimate you made
___ 0 , or 0.
1 is a little more than ___ before you
10 10 subtracted.
Method 2: Draw a Diagram
1 2
__ − __
2 5

To subtract 2__, you need
5
parts that are the same size.
Subtract.
1 − __
__ 2 = ___
1
2 5 10
Method 3: Use a Common Denominator
The denominator of __1 is 2.
2
Multiples of 2 are 2, 4, 6, 8, 10, …
The denominator of 2 __ is 5.
5
Multiples of 5 are 5, 10, 15, 20, …
240 NEL • Chapter 7
14. The first multiple divisible by both 2 and 5 is 10.
A common denominator is 10.
Write equivalent fractions with 10 as the denominator.
1 5 − ___
2 = ___
__ − __ 4
2 5 10 10
5−4
= _____ Subtract the numerators.
10
1
= ___
10
Subtract. Write each answer in lowest terms.
a) 3 1
__ − __ 2 __
b) __ − 1
3 __
c) __ − 1
4 2 3 4 4 8
• When adding and subtracting fractions using models or diagrams,
show each fraction using parts of the whole that are of equal size.
Pattern Blocks Diagram
=
– =
1 + __
__ 3 + __
1 = __ 2 2 − __
__ 1=4 1
__ − __
2 3 6 6 3 6 6 6
• To add or subtract fractions with unlike denominators, use a
common denominator.
• You can estimate when adding or subtracting fractions by
comparing fractions to 0, 1
__, or 1.
2
1. How are _1_ and 2
__ alike? How are they different?
3 6
2. a) 1+1
How would you use diagrams to calculate __ __?
4 2
3. Why is it difficult to calculate __ 1 without changing 1
1 − __ 4?
__ to __
2 8 2 8
7.2 Add and Subtract Fractions With Unlike Denominators • NEL 241
15. 8. Write each addition statement shown by
For help with #4 to #7, refer to Example 1 on a)
pages 238–239.
+
shown by the fraction strips.
b)
a) + +
b) +
c) +
For help with #9 to #12, refer to Example 2 on
pages 240–241.
5. For each diagram, write an addition
statement. Then add. 9. Write each subtraction statement
a) shown by the fraction strips.
+ Estimate and then subtract.
a) –
b) –
b)
+ c) –
10. For each diagram, write a subtraction
c) statement. Then subtract.
+ a)

b)
5+1 –
a) _2_ + ___
1 b) __ __
5 10 8 4
5
1 + ___
c) __
1 __
d) __ + 3
3 12 4 5
c)
e)
1
_ _ 1
+ __ 3 __
f) __ + 1
2 5 8 6 –
3
1 + __ 1 +5 terms.
a) __ b) ___ __
2 8 12 6 3 − ___
a) __ 3 5−1
b) __ __
2 + __
c) ___ 4 d) 1
__ + 2
__ 5 10 6 2
10 5 3 9 1 − ___
c) __ 1 d) 7
__ − 1
__
2 + __
e) __ 1 __ + 3
f) 1 __ 2 10 8 2
5 2 6 4 e) 2
__ − 2
__ 5 − ___
f) __ 5
3 5 8 12
242 NEL • Chapter 7
16. 12. Determine the difference. Write your 15. Zach was leading in a swimming race by _5_
8
__
of a length. He won the race by a length.
3−1
a) __ __ 11 − 5
b) ___ __ 2
4 8 12 6 By how much did the second-place
2−1
c) __ __ d) 1__ − 1
__ swimmer catch up by the end of the race?
3 2 6 9
2−1
e) __ __ 5 − 11
f) __ ___
5 4 6 15 16. A friend shows you the following work for
13. Write each subtraction statement shown 1 1 = __
__ + __ 2
by the pattern blocks. Then subtract. 4 3 7
a) Explain the error in your friend’s work.
a)
b) Use a diagram to show the correct

its flight to Iqaluit, Nunavut. The plane

was _1_ full of passengers and _1_ full of
6 3
cargo. How much space was left?
18. You can use a number line to show
2 9.
1 = ___
__ + ___
14. The students made 3 12 12
muffins in cooking 2 8 1
– = –– ––
class. They get to 3 12 12
take some muffins
home. There are 0 8 9
–– ––
1
12 muffins in a 12 12
muffin tray. Draw number lines to add the fractions.
1 of a tray.” 1 __ 3 +3
a) John says, “I’m taking __ a) __ + 1 b) _1_ + 1
__ c) ___ __
4 4 4 2 8 10 5
Katie says, “I’m taking 1
__ of a tray.”
3 19. You can use a number line to show
What fraction of a tray are John and 7.
2 1 = ___
__ − ___
Katie taking altogether? 3 12 12
b) Marjoe says, “I’m taking _1_ of a tray.” 2 8
1
––
6 – = –– 12
Sandeep says, “I’m taking ___1 of a tray.” 3 12
12
What fraction of a tray are Marjoe and 0 7 8 1
–– ––
Sandeep taking altogether? 12 12
Draw number lines to subtract the fractions.
1−1
a) __ __ 1 − ___
b) __ 1 c) _5_ − 1
__
2 8 4 12 6 4
7.2 Add and Subtract Fractions With Unlike Denominators • NEL 243
17. 22. The sum of each row, column, and diagonal
in this magic square must equal 1. Copy the
20. The ancient Egyptians thought the fractions square and fill in the blanks.
in the Eye of Horus added up to 1. Were 5
� � ___
12
7 1
–1
___
12
__
3

8
1
__
4 � �
–1
4
–1 1
––
2 16 23. A tangram is a square G C
puzzle that is divided
D
1
into seven shapes. E
1 B
––
64
––
32 1.
a) Suppose piece A is __
4 F
A
What are the values
21. Water is pumped into a pool. After of pieces B, C, D, E,
one hour, 1
__ of the pool is filled. F, and G?
5
b) What is the sum of A and B?
a) After 3 h, how full is the pool?
c) Subtract the value of D from the whole.
b) How long does it take in total to fill
the pool? d) Which two tangram pieces add up to
the value of C?
e) Make a problem of your own using
tangram pieces. Have a classmate solve it.
The Egyptians of 3000 B.C.E. used only unit fractions. These are fractions with a
numerator of 1, such as __ 1, and __
1, __ 1. They wrote all other fractions as sums of unit
23 4
fractions. For example,
= + = +
3 1+1
__ = __ __ 5 = __
__ 1+1__
4 2 4 6 2 3
These sums are called Egyptian fractions.
__ 1
1 + __ 1+1
__ __
4 8 3 9
b) Which one of the two sums in a) is greater? By how much?
5
c) How would ancient Egyptians have written ___ as the sum of two unit fractions?
12
244 NEL • Chapter 7
Focus on…
After this lesson,
you will be able to...
numbers with
like and unlike
denominators After the class pizza party, there
φ solve problems
involving the were 1 _5_ cheese pizzas and 1 __
5 vegetarian pizzas
6 6
addition of left over. How many pizzas were left over in total?
mixed numbers
To find out, you need to add mixed numbers .
φ check that your
reasonable using
estimation
How do you add mixed numbers?
mixed numbers Example 1: Add Mixed Numbers With Like Denominators
a whole number and a 5 + 15
1 __ __
1
fraction, such as 2 __ 6 6
3
Solution
Method 1: Use Pattern Blocks
Strategies
Model It +
Refer to page xvi.
+
+
5 + 15
1 __ 5+5
__ = 1 + 1 + __ __
6 6 6 6
7.3 Add Mixed Numbers • NEL 245
19. Move 1 green triangle to the
hexagon with 5 green triangles
to make 1 whole hexagon.
There are now 3 whole hexagons and 4 green triangles.
1+1+1+4
__ = 3 _4_
6 6
Write the answer in lowest terms.
34
__ = 3 2
__
=
6 3
Method 2: Use an Addition Statement
Strategies
15 5=1+1+5
__ + 1 __ 5 Add the whole numbers.
__ + __
Solve a Simpler 6 6 6 6
Problem 5+5
= 2 + _____ Add the fractions.
Refer to page xvii. 6
10
= 2 + ___
6
improper fraction =2+6 __ + 4
__ Write the improper fraction as a mixed number.
6 6
• a fraction that has a
numerator greater = 2 + 1 + _4_
6 You can use diagrams.
than the denominator, 4
_ _
9 =3
such as __ 6 + + +
8
Write the answer in lowest terms.
÷2 5+5
1 + 1 + __ __ = 3 _4_
6 6 6
34
__ = 3 _2_
6 3
÷2
Check:
15 5≈2+2
__ + 1 __
6 6
2+2=4
32
__ is a little less than the estimate of 4. The answer is reasonable.
3
1 + 21
a) 1 __ 1 + 25
__ b) 3 __ __ 2+2
c) 3 __ __
3 3 6 6 3 3
246 NEL • Chapter 7
20. Example 2: Add Mixed Numbers With Unlike Denominators
How many pies are there in total?
1 apple pies.
1 __
2
1 apple pies
2 __
3
Method 1: Use Pattern Blocks
Strategies
+ Model It
Refer to page xvi.
__ + 2 _1_
2 3
+
Add the red trapezoid and blue rhombus.
+
1+2+1 __ + 1 1+1
__ = 3 + __ __
2 3 2 3
__ and 1
To add 1 __, you need to use pattern blocks that are the same size.
2 3
+
There are 5 green triangles altogether.
3 + __ 1=3+5
1 + __ __
2 3 6
=3 5
_ _
6
5
__
There are 3 pies.
6
7.3 Add Mixed Numbers • NEL 247
21. Method 2: Use an Addition Statement Use multiples to determine
Strategies
Solve a Simpler
1 + 2 __
1 __ 1=1+2+1 1
__ + __ a common denominator.
Problem
2 3 2 3 Multiples of 2 are 2, 4, 6, 8, …
Refer to page xvii. =1+2+3 2 Add the whole numbers.
__ + __ Multiples of 3 are 3, 6, 9, 12, …
6 6 Use 6 as a common
=3+3 __ + 2
__ denominator.
6 6
3+2
= 3 + _____ Add the numerators.
6
=3+5 __
6 You can use diagrams.
=3 5
_ _
6 +
There are 3 5
__ pies altogether.
1+2=3
The Arabs were the 6
ﬁrst people to use a
fraction bar:
1
__ Check:
fraction bar
5
11 1≈2+2
__ + 2 __
+
2 3
2+2=4 3+2
__ __ = _5_
6 6 6
35
__ is a little less than the estimate of 4.
6
1 + 41
a) 2 __ __ 2 + 13
b) 1 __ __ 3+1
c) 3 __ __
2 6 3 4 5 2
• When adding mixed numbers with like denominators, you can
• When adding mixed numbers with unlike denominators, you can
– determine a common denominator for the fractions
248 NEL • Chapter 7
22. 1. After dinner, 1 1
__ ham sandwiches and 2 __3 egg salad sandwiches are
2 4
left. Jeremy and his sister want to use 4 of the leftover sandwiches
for their lunches tomorrow.
a) Are there enough sandwiches, not enough sandwiches, or more
than enough sandwiches left over?
c) Solve the problem using another method.
2. Which method that you used in #1 did you prefer? Explain.
3. a) How would you use estimation to check your answer in #1?
b) Compare your estimate with a partner’s.
For help with #4 to #7, refer to Example 1 on a) 1 _1_ + 1 1
__ b) 3_1_ + 5 5
__ c) _3_ + 1 _1_
pages 245–246. 3 3 8 8 4 4
d) 2 1
___ +3 7
___ _2_
e) 3 + 1 4
_ _ f) 4 + 1 7
8
__ __
4. Write each addition statement shown. 10 10 5 5 9 9
a)
lowest terms.
b) 2 + 21
a) 1 __ 1 + 15
__ b) 3 __ __ c) 5 4
__ + 2
__
+ 5 5 8 8 9 9
1 + 23
d) 2 __ 3+4
__ e) 2 __ __ 7 + 6 11
f) 4 ___ ___
c) 4 4 5 5 12 12
+
For help with #8 to #11, refer to Example 2 on
pages 247–248.
5. For each of the following, write the 8. Write each addition statement shown.
+
a) +
b) +
b)
+
c)
+
c)
+
7.3 Add Mixed Numbers • NEL 249
23. 9. For each of the following, write an 14. The camp cook uses 1 _1_ dozen eggs to
2
addition statement. make pancakes. She uses another 3 _1_ dozen
a) 3
+ for scrambled eggs. How many dozen eggs
does she use altogether? Check your
b)
+
15. Chef Dimitri finished cutting 1 _1_ trays
4
c) of spinach pie before his break. After his
+ break he cut another 2 2 __ trays. How many
3
trays of pie in total did he cut? Include
3 + 1 ___
a) 2 __ 1 1 + 21
b) 3 __ __ 16. Jenny studied 1 __1 h for her math
5 10 2 6 3
c) 3
__ +2 5
_ _ 1
__
d) 5 + 3
___ test and 3
__ h for her science test. For
4 6 2 10 4
1
__
e) 4 + 3 5
___ 2 + 75
f) 2 __ __ how long did she study in total? Check
11. Determine the sum. Write your answers in 17. Jonas and Amy collect comic books. Jonas
lowest terms.
has 21___ boxes of Granite Guy comics and
1 + 21
a) 4 __ __ 1 + 32
b) 3 __ __ 10
3 6 2 5 2
__
2 boxes of Quest of Koko comics. Amy
1
__
c) 1 + 5 1
_ _ d) 4 4
___ 1
+ 5 ___ 3
5 4 15 10 has 2 5 __ boxes of Alpha Woman comics and
3 + __
e) 1__ 5 f) 6 __ 9
1 + ___ 6
4 6 2 10 3
__
1 boxes of Quest of Koko comics.
5
a) Who has the larger collection?
b) How many boxes of Quest of Koko
comics do Jonas and Amy own
12. 3 h and then walked for
Susie ran for 1 __
4 altogether?
7
21
__ h. For how long did she travel? c) Jonas trades ___
4 10
of a box of
13. 3 pages of homework
Kathleen did 1 __ comics to Amy
4 for her Granite
before dinner. After dinner, she did
Guy DVD. How
another 7
__ of a page. In total, how
many boxes of
8
many pages of homework did comics does she
Kathleen do? have now?
250 NEL • Chapter 7
24. 19. At the school’s spring fair they sold
3 pepperoni pizzas,
1 vegetarian pizzas, 6 __
5 __
18. Melissa is in training for a rowing 3 4
competition. She keeps track of the hours and 4 5__ cheese pizzas.
6
she practises. At the end of the week, she a) Draw diagrams to show how much of
totals her hours. each pizza was sold.
Hours Practised
b) Estimate, then calculate how much
Sun Mon Tues Wed Thurs Fri Sat
pizza was sold altogether.
2 3
__ 21
__ 1 _3_ 11
__ 11
__ 1 _1_
4 4 4 2 4 2
20. The movie started 2 h 12 min ago.
a) This week she had a goal to practise 5 h.
The movie will end in 1 __
for a total of at least 10 h. Estimate 6
whether she met her goal. a) What is the total length of the movie in
b) How many hours did she practise? hours written as a fraction?
c) Was your estimate reasonable? Explain. b) If the movie started at 2:15 p.m., when
did the movie end? Write the time as a
fraction.
Egyptian fractions can be useful today. Suppose you have 13 sacks
of rice to divide among 8 people. That means each person would
5 sacks.
get 1 __
8
How can you give each person 1 5 __ sacks of rice if you do not have a
8
calculator or scale?
First, give each person 1 whole sack. Then, use Egyptian
fractions to determine how to give each person __ 5 sack:
8 Strategies
5 = __
__ 1 + _1_
8 2 8 Solve a Simpler
The Egyptian fraction shows that you should give each person 1 __ sack, plus 1
__ sack. Problem
2 8
1 sack to each person, there will be 1 sack left. Refer to page xvii.
After you give __
2
You then divide this sack into 8 and give each person __ 1.
8
The diagram shows that each person gets
1 sack and 1
1 whole sack, plus __ __ sack.
2 8
How would you divide the following?
a) 7 sacks of potatoes among 4 people
b) 7 bags of ﬂour among 5 people
c) 9 loaves of bread among 5 people
7.3 Add Mixed Numbers • NEL 251
25. Subtract Mixed Numbers
Focus on…
After this lesson,
you will be able to...
φ subtract mixed
numbers with
like and unlike
denominators
φ solve problems
involving the
subtraction of
mixed numbers
φ check that your
reasonable using After Lucy worked on her art project, she had 2 __43 jars of paint left.
estimation
Later, she used 1 1
__ jars of paint to finish her painting. How much paint
4
is left now?
How do you subtract mixed numbers?
Example 1: Subtract Mixed Numbers With Like Denominators
Subtract. Write the answer in lowest terms.
3 − 11
2 __ __
4 4
Solution
Method 1: Use Fraction Strips
3 − 11
2 __ __
4 4
Subtract. –
252 NEL • Chapter 7
26. 2 fraction strips.
There are now 1 __
4
Write the answer in lowest terms.
__ = 1 _1_
4 2
Method 2: Use a Subtraction Statement Strategies
Subtract the whole numbers. Solve a Simpler
Problem
Refer to page xvii.
Subtract the fractions. You can use diagrams.
3 1=2
__ − __ __
4 4 4
23 1 = 1 _2_
__ − 1 __
4 4 4 2–1=1
Write the answer in lowest terms.
÷2
3
__ 1
__ 2
__
4–4=4
__ = 1 _1_
4 2
÷2
3 − 1 __
2 __ 1≈3−1
4 4
11__ is close to the estimate of 2.
2
Subtract. Write each answer in lowest terms.
2 1
a) 2 __ − 1 __
7 3
b) 3 __ − 1 __
3 __
c) 4 __ − 1
3 3 8 8 4 4
7.4 Subtract Mixed Numbers • NEL 253
27. Example 2: Subtract Mixed Numbers With Unlike Denominators
Subtract.
3 − 11
3 __ __
8 2
Solution
Method 1: Use Fraction Strips

3 - 11
3 __ __
8 2
To subtract 3
__ and 1
__, you need to use parts that are the same size.
8 2
You cannot subtract _4_ from 3__.
8 8
8.
Take 1 whole strip from 3 _3_ and make it the equivalent fraction __
8 8
Subtract.

7 strips left.
There are 1 __
8
33 1 = 1 _7_
__ − 1 __
8 2 8
Method 2: Use a Subtraction Statement and Regroup
Use multiples to determine a common denominator.
Multiples of 2 are 2, 4, 6, 8, …
Multiples of 8 are 8, 16, …
Use 8 as a common denominator.
33 3 − 1 _4_
1 = 3 __
__ − 1 __
8 2 8 8
You cannot subtract 4 3. You need to regroup.
__ from __
8 8
254 NEL • Chapter 7
28. Regroup 1 whole from 3 __ 3.
8
3=2+8
3 __ __ + 3
__ You can use diagrams.
8 8 8
= 2 11
___
8
3
__
3 −1 =24
__ 11
___ − 14__ Subtract the whole
8 8 8 8 numbers and subtract
= 1 _7_ the fractions.
8
11 – 1__
2___ 4 = 17
__
8 8 8
Method 3: Use a Subtraction Statement and Improper Fractions
Determine a common denominator.
3 − 1 __
3 __ 3 − 1 _4_
1 = 3 __
8 2 8 8
You cannot subtract 4 __ from 3__.
8 8 You can use diagrams.
You can change to improper
3 − 1 __
3 __ 4 = 27
___ − 12
___ Subtract.
8 8 8 8
15
= ___ 27 15
12 = ___
___ – ___
8 8 8 8
= 1 _7_ = 17
__
8 8
33 1 ≈ 3 __
__ − 1 __ 1 − 1 _1_
8 2 2 2
31 1=2
__ − 1 __
2 2
__ is a little less than the estimate of 2. The answer is reasonable.
8
Subtract. Write each answer in lowest terms.
3
a) 3 __ − 1 1
1 − 32
__ b) 4 __ __ 1 __
c) 4 __ − 7
8 2 4 5 4 8
Chinese fractions do not have a fraction bar. A symbol is used that represents the
1 is written or spoken as “1 part of 2.”
words “part of” or “parts of.” __
2
7.4 Subtract Mixed Numbers • NEL 255
29. • When subtracting mixed numbers with like denominators, you can
– subtract the whole numbers
– subtract the fractions
• When subtracting mixed numbers with unlike denominators, you can
– determine a common denominator for the fractions
– subtract the whole numbers
– subtract the fractions
• Sometimes, mixed numbers need to be regrouped or changed to improper
fractions before subtracting.
Regroup Change to Improper Fractions
43 5 = 3 ___
__ − 1 __ 11 − 1 5
__ 3 − 1 __
4 __ 5 = 35
___ − 13
___
8 8 8 8 8 8 8 8
=2 6
_ _ = 22
___
8 8
= 23
__ = 2 _6_
4 8
= 2 _3_
4
1. After Jack’s party, 2 3
__ bottles of pop were left. The next day, Jack’s
4
1
__
family drank 2 bottles. How much pop is left now? Discuss with a
4
partner how you would solve this problem.
2. a) What do you need to do before you can calculate 2 __ 5?
1 − 1 ___
6 12
5.
1 − 1 ___
b) Determine the answer to 2 __
6 12
c) Use estimation to check your answer. What method did you use?
d) With a partner, compare how you calculated the answer to
21 5 . Then compare the method you used to check your answer.
__ − 1 ___
6 12
256 NEL • Chapter 7
30. For help with #7 to #10, refer to Example 2 on
pages 254–255.
For help with #3 to #6, refer to Example 1 on
7. Write a subtraction statement for each
pages 252–253.
set of fraction strips.
3. For each set of fraction strips, write the a)
subtraction statement.
a) –

b)
b) –

c)
– c)

4. Write a subtraction statement to represent
each diagram.
a) –
8. For each diagram, write a subtraction
b) statement.
– a) –
c) b)

c)
using estimation.
2 − 1 _1_
a) 1 __ b) 6 7
__ − 5 _5_
5 5 8 8
1
__
c) 3 − 1 1
_ _ d) 3
1
e) 2 __ 5
1 − __ f) 4 − 1 _1_ using estimation.
6 6 7
7
a) 6 ___ − 3 2 __ 1 1
b) 4 __ − __
6. Determine the difference. Write your 10 5 2 5
7 − 31
c) 7 ___ __ 5
d) 5 __ − 2 2__
15 6 3
5
a) 4 __ − 3 1
__ b) 2 1
__ − 2 _1_ 4
__ 2
_ _ 3
___ 6
__
9 9 3 3 e) 1 − 1 f) 2 −
5 3 14 7
2
__
c) 5 − 1 __ 4 d) 4
3
___ − 2 ___9
5 5 10 10
7
e) 5 − 4 ___ f) 3 5
__ − 2 _7_
12 8 8
7.4 Subtract Mixed Numbers • NEL 257
31. 10. Determine the difference. Write your 15. Mark and Lin race to see who can collect
answers in lowest terms. the most hockey cards. Mark has collected
3
2 − 1 ___ 1 − 11 3 sets. Who has
5 _1_ sets. Lin has collected 4 __
a) 3 __ b) 1 __ __
3 4
5 10 3 4
5 1 1 5 collected more sets? How much more?
c) 7 __ − 5 __ d) 4 __ − 2 ___
9 6 4 12
e) 3 __ 3
1 − __ 3
f) 2 __ − 1 4
__ 16. Alex has just completed 2 _3_ h of a
6 4 4 5 4
babysitting course. He must complete
13 1
__ h to get his certificate.
2
a) How many more hours does he need?
Karen goes to swimming practice for 1 __1h
3
each day. In the morning she has 2
__ h of
3 17. 5 laps. Mei
For gym class Ben ran 1 ___
practice. How many hours of practice 12
does she have in the afternoon? ran 18
___ laps. Who ran farther and by
12
how much?
12. A large Thermos™ has enough water to
fill 9 3
__ water bottles for a team of soccer 18. You can subtract a mixed number and
4 an improper fraction. Determine each
players. Halfway through practice,
difference.
the players drink 4 __1 bottles of water.
2 a) 3 3
__ − 3
__ 7 −6
b) 2___ 1−7
__ c) 5__ __
How much water is left for the rest of the 4 2 10 5 3 4
practice? 3 trays of dinner rolls are for sale in the
19. 1 __
4
bakery window. A customer comes and
buys 5__ of a tray. How much is left?
6
20. Daniel spends 9 __1 h sleeping. He spends
3 h to complete the
13. It takes Ria 3__ 2
4 61__ h at school.
1 h ago.
marathon. The race started 1 __ 4
2 a) How much more time does he spend
a) How much longer will Ria be running? sleeping than at school?
b) Check your answer using estimation. b) How much time does he spend at
school and sleeping altogether?
14. 1 packages of
A pie recipe calls for 3 __
2 c) How much time is left in the day to do
Saskatoon berries. Julia has 1 1
__ packages. other things?
3
How much more does she need? Include
258 NEL • Chapter 7
32. 21. Diana is allowed to use the computer for a) How much more paper does Bella use?
3 h each weekend. She used it for _1_ h on b) How much paper do Bella and Shelly
2 use in total?
Saturday morning, 1 1__ h on Saturday
4
3
__
night, and h on Sunday morning. 23. There are 12 golf balls in a package. The
4 2 packages. Cindy
a) For how much time can Diana use the Takeda family has 2 __
3
computer on Sunday night? takes 1
__ package, her dad takes 1 package,
2
b) Show how you would check your and her brother takes 4 golf balls.
a) What fraction of a package is left?
22. Bella uses 4.1 pieces of construction paper b) How many golf balls is this?
to make an art project. Shelly uses 3 _1_
4
pieces. For each of the following
only fractions. Then calculate using only
The Babylonian system of numbers was based on 60, not 10.
3 , ___
2 , ___
Babylonian fractions were expressed as numbers out of 60, e.g., ___ 5 , 12
___ .
60 60 60 60
Many things we use today come from the Babylonian times.
Our clock is based on the number 60.
10.
The time can be written as a fraction out of 60 min. For example, 9:10 a.m. = 9 ___
60
a) Write each time as a fraction out of 60.
8:10 p.m. 9:20 a.m. 7:48 a.m. 12:12 p.m.
b) The time now is 2:15 p.m. What was the time 1 h and 12 min ago?
c) The time now is 4:30 p.m. What will be the time 2 h and 36 min from now?
1
d) Amanda studied for __ of an hour. She started studying at 9:15 a.m. At what time
3
did she ﬁnish studying?
e) How much time passed between 1:07 p.m. and 3:42 p.m.? between 5:45 p.m. and
9:20 p.m.?
7
f) Sam started reading the newspaper at 9:45 a.m. and ﬁnished reading it in ___ h.
12
1 h more to read the paper than Sam did. She started at 10:30 a.m. At
Mila took __
4
what time did she ﬁnish reading the paper?
7.4 Subtract Mixed Numbers • NEL 259