Add and Subtract Fractions
With Unlike Denominators
Focus on…
1
After this lesson, you
will be able to...
φ add and subtract
φ
φ
fractions
with unlike
denominators
solve problems
involving the
addition and
subtraction of
fractions
check that your
answers are
reasonable using
estimation
1
–
6
1
–
3
1
–
2
How could you use pattern blocks to model addition and subtraction?
How can you add and subtract fractions with unlike
denominators?
1 and __
1?
What two pattern blocks would you use to represent __
2
3
1+1
__ is using these two pattern
b) Can you tell what the answer to __
2 3
blocks? Explain.
1. a)
__ and 1
__. What fraction does
Use the green triangles to represent 1
2
3
each green triangle represent?
1 + __
1 is now? Explain.
b) Can you tell what the answer to __
2 3
2. a)
1?
__ and __
What pattern blocks would you use to represent 1
2
6
1 − __
1 is using these two
b) Can you tell what the answer to __
2 6
pattern blocks? Explain.
3. a)
• pattern blocks
• coloured pencils
__.
Use the green triangles to represent 1
2
1 − __
1 is now? Explain.
b) Can you tell what the answer to __
2 6
c) How many green triangles are left?
4. a)
Reflect on Your Findings
5.
How can you use pattern blocks to help you add and subtract
fractions with unlike denominators?
7.2 Add and Subtract Fractions With Unlike Denominators • MHR
237
Example 1: Add Fractions With Unlike Denominators
Add. Write the answer in lowest terms.
1
__ + 1
__
3 6
Is the sum
closest to
,
,
0
1
–
2
or
?
1
Solution
Method 1: Use Fraction Strips
1+1
__
__
3 6
+
To add, you need to use
+
parts that are the same size.
__. Count the parts.
Each part represents 1
6
1+1
3
__
__ = __
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
MHR • Chapter 7
Method 3: Use a Common Denominator
__ is 3.
The denominator of 1
3
Multiples of 3 are 3, 6, 9, 12, …
__ is 6.
The denominator of 1
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 = __
2+1
1 + __
__
__
3 6 6 6
+ 1 Add the numerators.
_____
=2
6
3
= __
6
Write the answer in lowest terms.
÷3
3 = __
1
__
6 2
÷3
Check:
Use pattern blocks.
Web Link
closer to
Is
–1 + –1
3 6
or
–1
2
?
1
1. The answer is reasonable.
It equals __
2
To learn more about
adding fractions, go to
www.mathlinks7.ca
and follow the links.
Literacy
Add. Write each answer in lowest terms.
1 4
1 +4
__ + 2
__
__
a) 1
b) ___
c) __ + __
2 8
6 3
10 5
Link
When the numerator
equals the
denominator, the
fraction is equal to 1.
16
3=1
__
___ = 1
16
3
7.2 Add and Subtract Fractions With Unlike Denominators • MHR
239
Example 2: Subtract Fractions With Unlike Denominators
Subtract.
1−2
__
__
2 5
Is the
difference
closer to
0
or
?
1
–
2
Solution
Method 1: Use Fraction Strips
1−2
__
__
2 5
__, you need
To subtract 2
5
parts that are the same size.
–
–
Subtract.
1−2
1
__
__ = ___
2 5 10
Check:
1 closer to 0 or 1
__?
Is ___
10
2
0
5
1
__ = ___
0 = ___
10
2 10
0 , or 0.
1
___ is a little more than ___
10
10
Method 2: Draw a Diagram
1
__ − 2
__
2 5
__, you need
To subtract 2
5
parts that are the same size.
Subtract.
1−2
1
__
__ = ___
2 5 10
Method 3: Use a Common Denominator
1 is 2.
The denominator of __
2
Multiples of 2 are 2, 4, 6, 8, 10, …
__ is 5.
The denominator of 2
5
Multiples of 5 are 5, 10, 15, 20, …
240
MHR • Chapter 7
Compare this to the
estimate you made
before you
subtracted.
–
The first multiple divisible by both 2 and 5 is 10.
A common denominator is 10.
Write equivalent fractions with 10 as the denominator.
5 − ___
2 = ___
4
1
__ − __
2 5 10 10
−4
_____
Subtract the numerators.
=5
10
1
= ___
10
Subtract. Write each answer in lowest terms.
3 1
2 __
__ − 1
__
a) 3
b) __ − 1
c) __ − __
3 4
4 8
4 2
• 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+1
3+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
1, or 1.
comparing fractions to 0, __
2
1.
1 and 2
__ alike? How are they different?
How are __
3
6
__ + 1
__?
How would you use diagrams to calculate 1
4 2
b) Compare your answer with a classmate’s.
2. a)
3.
4?
__ − 1
__ without changing 1
__ to __
Why is it difficult to calculate 1
2 8
2 8
7.2 Add and Subtract Fractions With Unlike Denominators • MHR
241
8.
For help with #4 to #7, refer to Example 1 on
pages 238–239.
4.
5.
a)
+
Write each addition statement
shown by the fraction strips.
Estimate and then add.
a)
+
b)
+
c)
+
For each diagram, write an addition
statement. Then add.
Write each addition statement shown by
the pattern blocks. Then add.
b)
+
For help with #9 to #12, refer to Example 2 on
pages 240–241.
9.
a)
+
b)
+
10.
c)
+
Write each subtraction statement
shown by the fraction strips.
Estimate and then subtract.
a)
–
b)
–
c)
–
For each diagram, write a subtraction
statement. Then subtract.
a)
–
6.
7.
242
b)
Add. Write your answers in lowest terms.
5+1
1
2 + ___
__
a) __
b) __
5 10
8 4
1 __
5
1 + ___
c) __
d) __ + 3
4 5
3 12
3 __
1
1
__
+ __
e)
f) __ + 1
2 5
8 6
Determine the sum. Write your answers
in lowest terms.
3
1 + __
1 +5
__
a) __
b) ___
12 6
2 8
4
2 + __
__ + 2
__
c) ___
d) 1
10 5
3 9
2 + __
1
1+3
__
e) __
f) __
5 2
6 4
MHR • Chapter 7
–
c)
–
11.
Subtract. Write your answers in lowest
terms.
3
5 − __
1
__ − ___
a) 3
b) __
5 10
6 2
1
1
__ − ___
__ − __
c) 1
d) 7
2 10
8 2
5 − ___
5
2
__ − __
e) 2
f) __
3 5
8 12
12.
13.
Determine the difference. Write your
answers in lowest terms.
3−1
11 − 5
__
__
a) __
b) ___
12 6
4 8
2−1
1
__
__ − __
c) __
d) 1
3 2
6 9
5 − ___
11
__ − 1
__
e) 2
f) __
5 4
6 15
15.
16.
A friend shows you the following work for
an addition problem.
1 = __
2
1
__ + __
4 3 7
a) Explain the error in your friend’s work.
b) Use a diagram to show the correct
answer.
17.
An airplane was loading in Pond Inlet for
its flight to Iqaluit, Nunavut. The plane
1 full of passengers and 1
__ full of
was __
6
3
cargo. How much space was left?
18.
You can use a number line to show
9.
1 = ___
2
__ + ___
3 12 12
Write each subtraction statement shown
by the pattern blocks. Then subtract.
a)
–
b)
–
14.
The students made
muffins in cooking
class. They get to
take some muffins
home. There are
12 muffins in a
muffin tray.
1 of a tray.”
a) John says, “I’m taking __
4
__ of a tray.”
Katie says, “I’m taking 1
3
What fraction of a tray are John and
Katie taking altogether?
__ of a tray.”
b) Marjoe says, “I’m taking 1
6
1 of a tray.”
Sandeep says, “I’m taking ___
12
What fraction of a tray are Marjoe and
Sandeep taking altogether?
5
Zach was leading in a swimming race by __
8
__ a length.
of a length. He won the race by 1
2
By how much did the second-place
swimmer catch up by the end of the race?
2 8
– = ––
3 12
1
––
12
8 9
–– ––
12 12
0
1
Draw number lines to add the fractions.
1 1
3 + __
3
1 + __
1
a) __ + __
b) __
c) ___
4 4
2 8
10 5
19.
You can use a number line to show
7.
1 = ___
2
__ − ___
3 12 12
2
8
– = ––
3 12
0
1
––
12
7 8
–– ––
12 12
1
Draw number lines to subtract the fractions.
1 − ___
1
1
__
__ − __
__ − 1
a) 1
b) __
c) 5
2 8
4 12
6 4
7.2 Add and Subtract Fractions With Unlike Denominators • MHR
243
22.
20.
The ancient Egyptians thought the fractions
in the Eye of Horus added up to 1. Were
they correct? Show your work.
The sum of each row, column, and diagonal
in this magic square must equal 1. Copy the
square and fill in the blanks.
–1
8
–1
4
–1
2
1
––
64
21.
1
––
16
23.
1
––
32
Water is pumped into a pool. After
__ of the pool is filled.
one hour, 1
5
a) After 3 h, how full is the pool?
b) How long does it take in total to fill
the pool?
5
___
7
___
12
1
__
4
1
__
3
12
A tangram is a square
puzzle that is divided
into seven shapes.
__.
a) Suppose piece A is 1
4
What are the values
of pieces B, C, D, E,
F, and G?
The Egyptians of 3000 B.C.E. used only unit fractions. These are fractions with a
1, 1
__, and 1
__. They wrote all other fractions as sums of unit
numerator of 1, such as __
4
23
fractions. For example,
+
+
=
3=1
__
__ + 1
__
4 2 4
5
__ = 1
__ + 1
__
6 2 3
These sums are called Egyptian fractions.
a) Add the unit fractions. Use diagrams to show your work.
1
1
1
__ + __
__ + 1
__
4 8
3 9
b) Which one of the two sums in a) is greater? By how much?
5
12
c) How would ancient Egyptians have written ___ as the sum of two unit fractions?
244
MHR • Chapter 7
D
E
F
B
A
What is the sum of A and B?
c) Subtract the value of D from the whole.
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.
b)
MATH LINK
=
C
G