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Dividing Fractions and Mixed
Numbers
Focus on…
After this lesson, you
will be able to…
φ divide two
φ
fractions or mixed
numbers
solve problems
involving the
division of
fractions or mixed
numbers
Russia and Canada are the two countries with the largest areas in the
world. How are the fraction and the mixed number in the following two
statements related?
3 of the area of Russia.
The area of Canada is about __
5
2 times the area of Canada.
The area of Russia is about 1__
3
How can you divide a fraction by a fraction?
Work with a partner.
1.
When you divide 8 by 4, you determine how many 4s there are in 8.
8 ÷ 4 = 2.
1
a) Show how you could determine how many __s there are in 3.
2
1
b) Copy and complete the equation 3 ÷ __ = .
2
1 s there are in 2.
Show how you could determine how many __
4
1
b) Copy and complete the equation 2 ÷ __ = .
4
2. a)
3.
1 s there are in __
Show how you could determine how many __
4
4
3 1
b) Copy and complete the equation __ ÷ __ = .
4 4
3. a)
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2.
1 s are in __
Show how you could determine how many __
6
3
2 1
b) Copy and complete the equation __ ÷ __ = .
3 6
4. a)
5. a)
Copy the table. Complete the divisions in the first column.
Division
3 ÷ __
1=
__
4 4
2 ÷ __
1=
__
3 3
8 ÷ __
2=
__
9 9
2 ÷ __
1=
__
3 6
1 ÷ ___
1 =
__
2 12
3 ÷ __
3=
__
4 8
Division With Equal Denominators
3 ÷ __
1=
__
4 4
2 ÷ __
1=
__
3 3
8 ÷ __
2=
__
9 9
4 ÷ __
1=
__
6 6
6 ÷ ___
1 =
___
12 12
6 ÷ __
3=
__
8 8
How are the divisions in the second column related to the
divisions in the first column? Explain.
c) Complete the second column.
d) Write a rule for dividing two fractions using common denominators.
b)
6. a)
Copy the table. Complete the divisions in the first column.
Division
3 ÷ __
1=
__
4 4
2 ÷ __
1=
__
3 3
8 ÷ __
2=
__
9 9
2 ÷ __
1=
__
3 6
1 ÷ ___
1 =
__
2 12
3 ÷ __
3=
__
4 8
Multiplication
3 × __
4=
__
4 1
3=
2 × __
__
3 1
8 × __
9=
__
9 2
6=
2 × __
__
3 1
1 × ___
12 =
__
2
1
3 × __
8=
__
4 3
Strategies
Look for a Pattern
Complete the multiplications in the second column.
c) How are the multiplications in the second column related
to the divisions in the first column? Explain.
d) Write a rule for dividing by a fraction using multiplication.
b)
Reflect on Your Findings
7.
Which method do you prefer to use to divide a fraction by a fraction?
Why?
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Example 1: Divide Using Diagrams
2 ÷ __
1.
Determine __
3 4
Solution
Use diagrams to determine how
1 s are in __
2.
many __
4
3
The diagram shows that the number
When you
divide a number
by a proper fraction,
the quotient is
greater than the
original number.
8 __
2
__
>
3 3
1 s in __
2 is between 2 and 3.
of __
4
3
2
1 and __
A common denominator for __
4
3
is 12. So use a rectangle divided
into twelfths.
8
1 = 2__
2 or __
2 ÷ __
__
3 4
3
3
8
In ___, there are 2 whole
12
3
2
groups of ___, plus __
12
3
of another group.
Determine using diagrams.
3 1 b) 1__
3 c) ___
1 ÷ __
1 ÷ __
1
a) __ ÷ __
4 3
4 8
10 5
Example 2: Divide Using a Rule
Estimate and calculate.
7 1
1
3
a) __ ÷ __
b) 2__ ÷ 3__
2
4
8 4
reciprocal
• the multiplier of a
number to give a
product of 1
3
2
• __ is the reciprocal of __,
2
3
3 2
because __ × __ = 1
2 3
• the result of switching
the numerator and
denominator in a
fraction
224
Solution
a) The diagram shows that the number
7 is between 3 and 4.
1 s in __
of __
4
8
You can calculate the quotient using either of the following methods.
Method 1: Divide Using a
Method 2: Divide Using
Common Denominator
Multiplication
To divide fractions, write them
To divide by a fraction,
with a common denominator
multiply by its reciprocal .
and divide the numerators.
7 × __
7 ÷ __
1 = __
4
__
4
1
8
8
7
7
1
2
__ ÷ __ = __ ÷ __
28
8 4 8 8
= ___
1 __
2
__
8
=
7
1
4 8
= __ or 3__
7
1
__
2
2
= or 3__
2
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b)
You can estimate the quotient by dividing the
whole numbers closest to the mixed numbers.
3≈3÷4
1 ÷ 3 __
2__
2
4
3
≈ __
4
You can calculate the quotient using either of the following methods.
Method 1: Divide Using a
Common Denominator
3 = __
5 ÷ ___
15
1 ÷ 3 __
2__
2
4 2
4
15
10
___
÷ ___
=
4
4
10
___
=
15
__
=2
3
Estimate and calculate.
3
4
2 5
a) __ ÷ ___
b) __ ÷ __
5 10
9 6
c)
Method 2: Divide Using
Multiplication
3 = __
5 ÷ ___
15
1 ÷ 3__
2__
2
4 2
4
5
4
__
___
= ×
2 15
___
= 20
30
__
=2
3
15
4
___
is the reciprocal of ___.
15
4
When you
divide a number
by an improper
fraction or mixed
number, the quotient
is less than the
original number.
1
2
__
< 2__
3
2
1 ÷ 1__
2
3__
6
3
Example 3: Apply Division With Fractions
The baby teeth, or milk teeth, that develop in childhood are replaced
5 as
by larger teeth as people mature. A full set of teeth for a child has __
8
many teeth as a full set of teeth for an adult. There are 20 teeth in a
full set for a child. How many teeth are there in a full set for an adult?
Solution
5 to determine the number of adult teeth.
Divide 20 by __
8
20
5
5
__
___
÷ __
20 ÷ =
8
1
8
Instead of multiplying,
20
8
___
__
you could divide using
=
×
5
1
common denominators.
5 20 5
160
20 ÷ __ = ___ ÷ __
= ____
1
8
8
5
160 5
= ____ ÷ __
8
8
= 32
There are 32 teeth in a full set for an adult.
Check:
Use multiplication to check the division.
5 × 32 = ____
160
__
8
8
= 20
A beaver, like other
rodents, has only one
set of teeth in its
lifetime. A beaver’s
front teeth can grow
by over a metre a
year. The gnawing
that the beaver does
to cut trees and to eat
wears down its teeth
and keeps them the
right length.
5
20 is __ of the
8
number of adult
teeth, so there are
more than 20 adult
teeth.
5
Dividing 20 by __ will
8
result in a quotient
greater than 20.
5
1
20 ÷ __ ≈ 20 ÷ __
8
2
≈ 40
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1 of a tray of lasagna, how many servings are in
If one serving is __
6
three trays of lasagna?
• You can estimate and determine the quotient of two fractions
using diagrams.
3 __
__
÷ 1 is between 2 and 3.
4
3
3 __
1 or __
9
__
÷ 1 = 2__
4
3
4
4
• You can estimate the quotient of two improper fractions or mixed
numbers by dividing the whole numbers closest to them.
1 ÷ 3__
1≈5÷4
5__
4
2
1
5 or 1__
≈ __
4
4
• To divide two fractions, you can write them with a common
denominator and divide the numerators.
7
3
5
7
3 = ___
1 = __
7 ÷ __
6
___
1__ ÷ 2__
÷ __
÷ ___
10 5 10 10
4
2 4 2
7 ÷ ___
10
7
1
__
__
= __
= or 1
6
6
4
4
7
___
=
10
• To divide by a fraction, you can multiply by its reciprocal.
7
7
5
5
3
3 = ___
1 = __
7 ÷ __
___
÷ __
× __
1__ ÷ 2__
10 5 10 3
4
2 4 2
2
7 × __
35
= __
= ___
30
4 5
7
1
14
__
__
___
= or 1
=
6
6
20
7
___
=
10
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1.
2.
3.
5 by __
5.
1 and stated that the quotient is ___
Anna divided __
6
2
12
a) What was Anna’s mistake?
b) What is the correct quotient? Use a diagram to show how you know.
3 ÷ __
2 as follows.
Mike carried out the division __
4 3
3 ÷ __
2 = __
4 × __
2
__
4 3 3 3
8
= __
9
Do you agree with Mike’s method and answer? Explain.
3 ÷ __
1 is
Explain how the diagram shows that the quotient of __
5 7
between 4 and 5.
1 ÷ 1__
1 equal 1__
1 ÷ 2__
1?
Does 2__
2
2
2
2
b) If the quotients in part a) are not equal, how are they related?
4. a)
For help with #7 to #10, refer to Example 2 on
pages 224–225.
For help with #5 and #6, refer to Example 1 on
page 224.
5.
6.
Determine each quotient using diagrams.
5 1
1 1
a) __ ÷ __
b) __ ÷ __
8 4
4 3
5
1
2
1 ÷ __
__
__
c) 1 ÷
d) 2__
2 3
3 6
Use diagrams to determine each quotient.
9
1
1 3
a) ___ ÷ __
b) __ ÷ __
10 5
4 8
3 2
2
1
__
__
c) 1 ÷
d) 2__ ÷ __
3 2
4 3
7.
Divide using a common denominator.
9
3
5
1 5
1
a) __ ÷ ___
b) 1__ ÷ __
c) 3__ ÷ 1__
5 10
2 6
3
6
8.
Divide using multiplication.
5
3
1
1
1
a) ___ ÷ __
b) 4__ ÷ 1__ c) 10 ÷ 2__
12 4
2
4
2
9.
Divide.
3 4
a) __ ÷ __
4 5
b)
5
2 ÷ 2__
1__
3
6
c)
3
12 ÷ __
4
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10.
Divide.
8
4
1
1
a) 1___ ÷ 2__ b) ___ ÷ __
12
2
11 5
c)
3 ÷ 2__
3
1__
8
4
For help with #11 to #13, refer to Example 3 on
page 225.
11.
12.
1 scoops of flour to make one
It takes 2__
2
cake. How many cakes do 15 scoops of
flour make?
Three quarters of a can of apple juice fills
six glasses. How many glasses will a
whole can of apple juice fill?
14.
An incandescent light bulb uses about
1 times as much energy as a compact
4__
2
fluorescent light bulb to produce the same
amount of light. What fraction of the
energy used by the incandescent bulb does
the fluorescent light bulb use?
228
17.
In a comedy review, each performer has
1 -h slot. How many performers are
a __
4
there in a 2-h show?
13.
15.
16.
18.
3 is in
Of all the land on Earth, about ___
10
3
___
is in South America.
Asia and about
25
How many times as big as South America
is Asia?
4
The average wind speed in Calgary is __
5
of the average wind speed in Regina.
The average wind speed in Calgary is
16 km/h. What is the average wind speed
in Regina?
Use examples to explain your answer to
each of the following.
a) Can the reciprocal of a proper fraction
be a proper fraction?
b) Can the product of two proper
fractions be greater than 1?
c) Can the quotient of two proper
fractions be greater than 1?
The world’s longest river is the Nile
in Africa, with a length of 6825 km.
5 times a long as the
This is about 1__
8
Mackenzie River, which is Canada’s
longest river.
a) How long is the Mackenzie River?
1
b) The Mackenzie River is about 2___
10
times as long as the Columbia River.
How long is the Columbia River?
19. a)
Shana and Zack painted their rooms
using paint in cans of the same size. Shana
3 cans
1 cans of paint. Zack used 2__
used 1__
2
4
of paint. How many times as much paint
did Zack use as Shana?
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20.
21.
22.
1 of the Earth’s
Russia covers about ___
30
surface. The area of Russia is about
2 times the area of Canada. What
1__
3
fraction of the Earth’s surface does
Canada cover?
Suppose a friend knows how to
divide by whole numbers, but not
by fractions.
a) How could you use the following
pattern to show your friend how
1?
to calculate 4 ÷ __
2
1
__
4÷8=
2
4÷4=1
4÷2=2
4÷1=4
1=
4 ÷ __
2
b) Make up a pattern to show your
1.
friend how to calculate 9 ÷ __
3
3 min to ski up a slope
It took Svend 9__
4
on a cross-country ski trail and only
1 min to ski back down the same slope.
2__
4
How many times as fast did he ski down
the slope as he skied up it?
23.
The three largest islands in Canada are all
north of the Arctic Circle. Baffin Island
1 times the area of Victoria
has about 2__
3
3 times
Island. Baffin Island has about 2__
5
the area of Ellesmere Island. What fraction
of the area of Victoria Island is the area of
Ellesmere Island?
24.
Write a word problem that you can solve
3 ÷ 2__
1.
using the expression 3__
4
4
MATH LINK
The Prairies ecozone includes the Manitoba Plain and the
grasslands of southwest Saskatchewan and southeast Alberta.
The wettest part of this ecozone is the Manitoba Plain, which has
an average annual precipitation of about 70 cm. This amount of
4
precipitation is 2__ of the amount in the dry grasslands.
5
What is the average annual precipitation in these grasslands?
The Prairies ecozone contains much of Canada’s farmland, but it is
vulnerable to droughts.
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