# Mulitplying by Whole Number

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In this pdf we will learn about Multiplication of whole numbers. Multiplication of whole numbers is the sort way to do repeated addition. The number by which any number is multiplied is known as the multiplicand.
1. Multiplying a Whole Number
2.1
Student book pages 46–50
by a Fraction
You will need Use repeated addition to multiply fractions
• counters by whole numbers.
You can use grids and counters to model fractions.
What fraction does this diagram represent?
• Cutout 2.1
numerator
___________ number of counters on the grid
_______________________ 5
___
denominator
 number of squares in the grid

8
terms You can use grids and counters to model fraction addition.
numerator
denominator
A model of __56  __36 is shown.

The denominator There are 8 counters in
tells the number the 2 grids.
of equal parts in
1 whole. 5
__ 3
__ 8
___
6
 6
 6
The numerator tells
8
__
the number of equal 6
is an improper fraction.
parts that the fraction
represents. Write __86 as a mixed number.
mixed number Redraw the 8 counters in
a number made up the grids so that the first 
of a whole number
grid is full.
and a fraction
There is 1 full grid, plus
improper fraction 2 counters in the second grid.
a fraction in which
the numerator is
greater than the 5
__ 3
__ 6
__ 2
___ 2
___ 2
___
denominator 6
 6
 6
 6
 1  6
 1 6
2 1
__
6
 ___
3
, so you can write 1__26 as 1__13 if you want to.
32 Lesson 2.1: Multiplying a Whole Number by a Fraction Copyright © 2009 Nelson Education Ltd.
2. Multiplication and repeated addition are equivalent.
For example, 3  5  15 is equivalent to 5  5  5  15 .
3  5 can be read as “3 sets of 5 .”
Use repeated addition to model 3  __34 .
Draw counters on the grids to show 3 sets of __34 .
  There are 9 counters in the 3 grids.
3
__ 3
__ 3
__ 3
__ 9
___
4
 4
 4
 3  4
 4
Draw the same number of counters, but this time
  fill up as many whole grids as you can.
1
3  __34  2__
4
PROBLEM Six pitchers of lemonade are each __38 full.
How many pitchers of lemonade are there?
Use Cutout 2.1 and counters to model 6  __38 .
18
___
Write the number of pitchers as an improper fraction. 8
Move the counters to fill as many grids as you can.
2
__ 1
__
Rewrite your answer as a mixed number. 2 8 or 2 4
63 4  5 20
6  __38  _____
8
 18
__
8
. So, 4  5
__
6
 _________
6
 ___
6
.
Hint Reflecting
When you add  Use these words to complete the statements below.
fractions with the
same denominator, numerator denominator
the denominator When you multiply a whole number by a fraction, the
stays the same. denominator stays the same.
To multiply a whole number by a fraction, multiply the
whole number by the numerator of the fraction.
Copyright © 2009 Nelson Education Ltd. Lesson 2.1: Multiplying a Whole Number by a Fraction 33
3. Practising
3. Multiply. Write your answer as a fraction and, if it is
greater than 1, as a mixed number or whole number.
Use a model and show your work.
a) 2  __13
 
1 1 2
Hint 2  __13  ___
3
 ___
3
 ___
3
A fraction is  1 if
the numerator is Is 2  __13 greater than 1? No
greater than the b) 5  __35
denominator.
Draw 5 sets of __35 .
   
53 15
5  __35  _____
5
 ___
5
Draw the same number of counters, but this time
fill up as many whole grids as you can.
   
5  __35  3
Note: Part c) is c) 4  __25
Part d) in Student Draw 3 more fifths grids.
Book
Draw counters on the grids to show 4 sets of __25 .
4  2 ___8
4  __25  _______
5
 5
8
___ 5
__ 3
___ 3
___
5
 5
 5
 1 5
34 Lesson 2.1: Multiplying a Whole Number by a Fraction Copyright © 2009 Nelson Education Ltd.
4. Note: Part d) is 3  7 21
d) 3  __76  __________
6
 ___
6
Part e) in Student
Book
21 __6 ___
___ 6 6
___ 3
___
Hint
6
 6
 6
 6
 6
3
__
6
and __12 are
equivalent fractions. 3 1
 3 ___
6
or 3 ___
2
21
3 Try this method to write __
6 3
6
as a mixed number.
21
__ 6 2 1
6
 21  6
1
2 – 1 8
Complete the division.
21  6 3
 3 remainder 3
Hint
21
So, __
6
 3__36 or 3__12 .
Write your answer 5. Art class is __12 of an hour each school day. How many
as a mixed or whole hours of art does a student have in 5 days?
number.
5  1  __
5  __12  _______ 5
2 2
5  __
__ 2  __
2  __
1  2__
1
2 2 2 2 2
1
2__
The student has 2 hours of art in 5 days.
6. Jason needs __23 of a cup of flour to make 1 batch of
bannock. How many cups of flour will he need if he
decides to make 6 batches of bannock?
6 2 6  2 12
___
 ___  ______
3

3 3
12  12  3  4
___
3
Jason needs 4 cups of flour for 6 batches
of bannock.
Copyright © 2009 Nelson Education Ltd. Lesson 2.1: Multiplying a Whole Number by a Fraction 35
5. Exploring Calculating a
2.2
Student book page 51
Fraction of a Fraction
You will need Represent one fraction as part of another
• a ruler fraction.
• Cutout 2.2 You can use a fraction strip tower to compare fractions.
Use the edge of a ruler to identify fractions that are equal
in length.
A __12 strip is the same length as a __24 strip, so __12  __24 .
term List some other fractions that are equivalent to __12 .
equivalent fractions 3 , __
__ 6
5 , ___
4 , ___
fractions that are 6 8 10 12
equal in value
1
1 1
2 2
1 1 1
3 3 3
1 1 1 1
4 4 4 4
1 1 1 1 1
5 5 5 5 5
1 1 1 1 1 1
6 6 6 6 6 6
1 1 1 1 1 1 1 1
8 8 8 8 8 8 8 8
1 1 1 1 1 1 1 1 1 1
10 10 10 10 10 10 10 10 10 10
1 1 1 1 1 1 1 1 1 1 1 1
12 12 12 12 12 12 12 1
12 12 12 12 12
You can also use a fraction strip tower to represent one
fraction as part of another fraction.
1
__
4
fits into __12 two times. 1
2
So, __14 is half of __12 . 1 1
1 4 4
1
__
2
of __12  __
4
36 Lesson 2.2: Exploring Calculating a Fraction of a Fraction Copyright © 2009 Nelson Education Ltd.
6. Aaron is playing a fraction game with his friends.
The game board is a fraction strip tower.
Each player picks a card and colours in the fraction that the
card represents.
3 __
Aaron coloured __ , 1 , __
8 6 5
2 1
, and ___
12
.
Match the fractions he coloured with the cards he picked.
Use Cutout 2.2 and the edge of a ruler.
Aaron picked
these cards. 1
1
__
3
of 1
__
2
Which fraction fits into __12 three times? __
1 of 1 6
3 2 1 1
1
__
3
of __12  __
= 6 6
1
__ 1
of __ 1
Which fraction fits into __13 four times? ___
4 3 12
1 1
4 of 3 1
1
__ of __13  ___
= 1 4 12
12
3
__ 1
of __ 1
Which fraction fits into __12 four times? __
4 2 8
3 of 1 1 1 1
__
4 2 __ of __ 
3 4 2 8
= 8
3
__
4
of __12  3  __41 of __12
1
 3  __
8
3
__
 8
2
__ 3
of __ 6
What is an equivalent fraction for __35 ? ___
3 5 10
2 3
3 of 5 2
What is __13 of this equivalent fraction? ___
= 2 10
5
4
So, what is __23 of this equivalent fraction? ___
10
2
__ 4
of __35  ___
3 10
Which fraction is equivalent to __ 4 2
? __
10 5
2 .
So, __23 of __35 is also equal to __
5
Copyright © 2009 Nelson Education Ltd. Lesson 2.2: Exploring Calculating a Fraction of a Fraction 37
7. 2.3
Student book pages 52–56
Multiplying Fractions
Multiply two fractions less than 1.
To multiply 2  3, you can draw a 2 -by- 3
grid and determine its area. 2  3  6
Hint Use an area model to multiply fractions < 1.
 __23 means the PROBLEM Calculate __12  __23 .
same as __12 of __23 . Use a 2  3 grid.
1
Each row is ___ of the grid.
2
1
Each column is ___ of the grid.
3
One grid square is __12 of __13 of the grid.
2
__
3
of the grid is 2 columns.
Shade __12 of __23 of the grid.
1
__ 2
__ 2
___ Area of shaded part (square units)
2
 3

6 Area of whole 2-by-3 grid (square units)
PROBLEM Use a grid to calculate __25  __
1
10
.
Use this 5-by-10 rectangle to represent 1 whole.
1
There are 5 rows. Each row is ___ of the grid.
5
1
There are 10 columns. Each column is ___ of the grid.
10
1
10
of the grid.
2
__ 1
__ 2
___
Area of shaded part (square units)
5
 10

50 Area of whole 5-by-10 grid (square units)
8. Use a procedure to multiply fractions.
Look back at your solution to __12  __23 .
Area of the part of the grid Numerator of the Product of the numerators
1 2 1 2
of __
3
product of __
2
ⴛ __
3
2 square units 1 2 2 1 2 2
12 __  __  ____ ____  ____  ____
2 3 6 2 3 6
Area of the whole Denominator of the Product of the
1 2
2-by-3 grid product denominators of __
2
ⴛ __
3
23 6 square units 1
__ 2
 __23  ____ 1  ____
____ 2  ____
2
2
6 2 3 6
1
__
2
 __23  __26 • Circle the 2 numbers you multiply to get the numerator
of the product.
• Underline the 2 numbers you multiply to get the
denominator of the product.
PROBLEM Calculate __35  __23 .
Multiply the numerators. 3  2  6
Multiply the denominators. 5  3  15
3
__ 2
__ 6 Product of the numerators
5
 3
 ___
15 Product of the denominators
PROBLEM Calculate __13  __34 .
1
__ 3
__ 3
___
Product of the numerators
3
 4

12 Product of the denominators
Reflecting
 Which method for multiplying 2 fractions less than 1 do
you prefer—the area model or the procedure? Explain.
Answers may vary, e.g., I prefer the procedure because it
is easy to remember and quick to do.
9. Practising
5. Draw a model for each multiplication expression.
Determine the product.
a) __38  __12
The denominators of the 2 fractions are 8 and
2 units wide. Draw this rectangle on the grid.
Inside this rectangle, shade a rectangle __38 of the
length and __12 of the width.
What fraction of the whole is shaded? ___ 3
16
3
__ 3
 __12  ___
8 16
b) __45  __13
Draw a 5 -by- 3 rectangle on the grid.
Shade a rectangle that is __45  __13 .
4
__ 4
 __13  ___
5 15
c) __16  __25
Draw a 6 -by- 5 rectangle on the grid.
Shade a rectangle that is __16  __25 .
1
__  2
__  2 or ___
___ 1
6 5 30 15
7. a) Draw a picture to show why __25 × __38 = __
6
40
.
To model __2  __3 , use a 5 -by- 8 rectangle
5 8
to represent 1 whole.
Draw this rectangle on the grid.
Area of rectangle = 40 square units
Inside this rectangle, shade a __25  __38 rectangle.
What fraction of the whole is shaded? ___ 6
40
2
__  3
__ = 6
___
5 8 40
10. 6
b) List 2 other pairs of fractions with a product of __
40
.
Write pairs of numbers that are factors of the
6
numerator and denominator of __ 40
.
Pair A Pair B
question 7 b) may 3 × 2 =6 6 × 1 =6
vary. 4 × 10 = 40 20 × 2 = 40
3 ___2 6 6 ___1 6
___ ×  __
40
___ ×  __
40
4 10 20 2
8. Matthew’s bed takes up __13 of the width of his bedroom
and __35 of the length.
What fraction of the floor area does the bed use up?
Solution:
Use the procedure to determine __13 of __35 .
Hint
Multiply the numerators and the denominators.
To write a fraction in
lower terms, divide 1  3
1
__ 3
__ _________
the numerator and 3
 5
 
denominator by a 3 5
common factor. 3 or ___
 ___ 1
15 5
1
__
Matthew’s bed takes up 5 of the floor area.
Some examples of __23 : 13. Describe a situation where you might multiply __35  __23 .
Use one of these or your own ideas to describe a
• a pitcher of
lemonade that situation where you might calculate __35 of __23 .
is __23 full
2 of a project still
• __23 of a project still to 3
do 3 of the remaining
to do. That day, he completed __
5
• __23 of a class of
work. How much of the job was left to do then?
students
11. Exploring Estimating Fraction
2.4
Student book page 57
Products
Estimate to predict whether a fraction
1 , or 1.
product is closer to 0, _
2
Brian and Preston are playing a spinner game.
They spin twice and multiply.
They score 1 point if the product is closest to 0, 1 point if it
is closest to 1, and 2 points if it is closest to __12 .
1
Hint Predict whether each product is closer to 0, __ 2
, or 1.
What is the 3 2  3 6
simplest fraction
2
__
3
 __
4
 _________
  ___
3 4 12
that describes the
in lowest terms. 1
__
12 2
1
Is __23  __34 closest to 0, __12 , or 1? __
2
1
__ 3
__ 1  3
_________ 3
___
5
 4
  
5 4 20
Write fractions equivalent to 0, __12 , and 1 with a common
denominator of 20.
0 1 1  10 10 20
0  __
20
__
2
 _______
2
 ___
20
1  __
20
10
numerators of the equivalent fractions for 0, __12 , and 1.
Is __1  __3 closest to 0, __1 , or 1? 0
5 4 2
How do you know?
3 is closer to 0 than to __
___ 1 because 3 is closer to
20 2
0 than to 10.
42 Lesson 2.4: Exploring Estimating Fraction Products Copyright © 2009 Nelson Education Ltd.
12. 2  9 18
2
__
3
 9
___
10
 _________
  ___
3 10 30
Write equivalent fractions with a common denominator of 30.
0 1 1  15 15 30
0  ___
30
__
2
 _______
2
 ___
30
1  ___
30
15
9
Is __23  __ closest to 0, 1
__, or 1? 1
__
10 2 2
3
__ 9
___ 3  9
_________ 27
___
4
 10
  
4 10 40
Write equivalent fractions with 40 in the denominator.
0 1 1  20 20 40
0  ___ __
2
 _______
2
 ___ 1  ___
40 20 40 40
9
Is __34  __ 1
closest to 0, __12 , or 1? __
10 2
1  9 9
1
__
5
 9
___
10
 _________
  ___
5 50 10
0 1 1  25 25 50
0  ___
50
__
2
 _______
2
 ___
50
1  ___
50
25
9
Is __15  __
10
closest to 0, __12 , or 1? 0
1  1 1
1
__
5
 1
__
5
 _________
  ___
5 25 5
Is __15  __15 closest to 0, 1
__
2
, or 1? 0
9 9 9  9 81
___
10
 ___
10
 _________
  _____
10 100 10
9 9
Is __
10
 __
10
closest to 0, __12 , or 1? 0
What happens when you multiply 2 fractions close to 0?
The product is close to 0.
What happens when you multiply 2 fractions
close to 1?
The product is close to 1.
Copyright © 2009 Nelson Education Ltd. Lesson 2.4: Exploring Estimating Fraction Products 43
13. Multiplying Fractions
2.5
Student book pages 58–63
Greater Than 1
Multiply mixed numbers and improper
fractions.
1
1 Use an area model to multiply fractions > 1.
2
You can use a grid to model 2__12  1__12 .
1 whole
5
2__12  __
2
Use a grid with 5 rows.
1
2 3
2 1__12  __
2
Use a grid with 3 columns.
A 2-by-2 rectangle represents 1 whole.
So, each grid square represents __14 .
15 Number of shaded grid squares
2__12  1__34  ___
4 Fraction each grid square represents
PROBLEM Use a grid to calculate 2__13  1__14 .
7 Use a grid with 7 rows.
11 2__13  ___
4 3 3 rows represent 1 whole.
5 Use a grid with 5 columns.
1__14  ___
4 4 columns represent 1 whole.
3 Shade a 7-by- 5 rectangle on the grid.
Label the sides of the rectangle 2__13 and 1__14 .
Outline a 3-by- 4 rectangle to show 1 whole. There
are 12 grid squares inside this rectangle, so each
grid square represents ___1 .
12
35 Number of shaded grid squares
2__13  1__14  ___
12 Fraction each grid square represents
44 Lesson 2.5: Multiplying Fractions Greater Than 1 Copyright © 2009 Nelson Education Ltd.
14. Write each product you calculated as a mixed number.
15
__ 35
__
4
 15  4  3 remainder 3
12
 35  12  2 remainder 11
15 3 35 11
So, __
4
 3 ___
4
So, __
12
 2 ___
12
Use a procedure to multiply fractions > 1.
Calculate __34  2__35 .
Step 1: Write 2__35 as an improper fraction. Step 2: Multiply.
Here are 3 methods you can use. 3 3 3 13
__  2 __  __  ___
4 5 4 5
Shade the fraction strip to show 2__35 .
3  13
 _____
45
13
2__35  ___
5
39
___
 20
OR
Write 2 as an improper fraction. Then add __35 . Step 3: Write the product as a
221 2__35  2  __35 mixed number.
39
___
10 10 13  39  20
 2  __55  ___  ___  __35  ___ 20
5 5 5
 1 R 19
39 19
OR So, ___
20
 1 ___
20
Combine the steps in the procedure above.
(5  2)  3 13
2__35  _________
5
 ___
5
Hint Reflecting
If a fraction is < 1,  How can you tell that the product of 2 fractions less
its numerator is less than 1 will always be less than 1?
than its denominator.
The numerators will be smaller than the denominators, so
when you multiply the tops and bottoms of the fractions,
the top part of the product will always be smaller.
Copyright © 2009 Nelson Education Ltd. Lesson 2.5: Multiplying Fractions Greater Than 1 45
15. Practising
4. Calculate each product.
a) __23  2__14
9
Write 2__14 as an improper fraction. __
4
2 9
__ 18
__
3  4  ___
12
3 1
 __ OR 1__
2
2
b) __58  1__12
5 3
__ 15
__  2  ___
8 6
3 1
5. Use the grid to model 1__
4
 2__
3
.
Note: Question 5
has been modified. Then calculate the product.
Solution:
7
1__34  ___
4
, so use a grid with 7 rows.
7
2__13  ___
3
, so use a grid with 7 columns.
Shade a 7-by- 7 rectangle on the grid.
The rows show fourths. 7 rows show __74 , so 4 rows
show __44 , or 1 whole.
The columns show thirds. 7 columns show __73 , so
3 columns show __3 , or 1 whole.
3
So, a 4 -by- 3 rectangle represents 1 whole.
Outline a rectangle that represents 1 whole.
There are 12 grid squares inside this 1
___
rectangle, so each grid square represents 12 .
49 Number of shaded grid squares
1__34  2 __13  ___
12 Fraction each grid square represents
1
Write the product as a mixed number. 4 ___
12
46 Lesson 2.5: Multiplying Fractions Greater Than 1 Copyright © 2009 Nelson Education Ltd.
16. 10. Tai calculated 3__13  4 __38 .
He multiplied the whole number parts together and
then the fraction parts together to get an incorrect
3.
product of 12__
24
a) Explain why estimation would not help Tai realize
To estimate, which whole numbers are close to 3__13
and 4 __38 ? 3 and 4 are close.
What is the product of your estimate? 12
Why would estimation not help Tai realize that he
Because my estimate is close to Tai’s incorrect answer.
b) How could you show Tai that his answer is
incorrect?
Write 3__13 and 4 __38 as improper fractions.
10
3__13  ___ 35
4__38  ___
3 8
10 35
3__13  4__38  ___  ___
3 8
350
 _____
24
Hint Divide the numerator and denominator of your
If a number is even, answer by a common factor to write the improper
it is divisible by 2. fraction in lower terms.
350  35 175
_____
___________ 
24  8 2
7
14 ___
Write the product as a mixed number. 12
15. Describe a situation at home in which you might
Hint 1 1
multiply 3__
2
by __
2
.
Think of situations
fractions, such as in 1 cups of ﬂour.
recipe books. and the original recipe asked for 3__
2
Copyright © 2009 Nelson Education Ltd. Lesson 2.5: Multiplying Fractions Greater Than 1 47
17. Dividing Fractions
2.6
Student book pages 68–71
by Whole Numbers
Use a sharing model to represent the quotient
of a fraction divided by a whole number.
Use grids and counters to divide a fraction.
9
You can think of dividing as sharing. __
20
 3 tells you the
9
share size if 3 people share __
20
of something.
9
You can use a grid and counters to model __
20
 3.
A 4-by-5 grid represents the denominator (20).
Place 9 counters on the grid to represent the numerator (9).
Circle the 9 counters to divide them into 3 equal groups.
Each person would have 3 counters out of 20.
9
__ 3
 3  ___
20 20
PROBLEM Calculate __23  4.
Draw counters on the 3-by-1 grid to represent __23 .
Can you divide 2 counters into 4 equal groups? No
Write a fraction equivalent to __23 , with a numerator that can
be divided into 4 equal groups.
24 8
_____
34
 ___
12
Draw counters on a 3-by-4 grid to represent this fraction.
Circle the counters to divide them into 4 equal groups.
2
Each of the 4 groups represents ___ of the grid.
12
2
__ 8
 4  __ 2
 4  ___
3 12 12
48 Lesson 2.6: Dividing Fractions by Whole Numbers Copyright © 2009 Nelson Education Ltd.
18. Multiply by a fraction to divide a fraction.
Divide. Multiply.
1
42 2 __
2
of 4  2 4  __12  2
1
62 3 __
2
of 6  3 6  __12  3
Dividing by 2 is the same as taking __12 of the number.
Divide. Multiply.
1
63 2 __
3
of 6  2 6  __13  2
1
93 3 __
3
of 9  3 9  __13  3
Dividing by 3 is the same as taking __13 of the number.
Multiply to divide.
9 9 1 2 1 4 __ 1
4  __
__  3  __  ____ __  4  __23  ____ __
5
2
20 20
3
3
4 5 2
4  1
 1 2 1  __________
 9________
20 
 ________
3

3 4 5 2
9 2  4
___
 ___  ___ 10
60 12
93
 _____ 2 2 4  2
60  3  __________
 _______
122 10  2
3
 ___
20 1  2
__
 __
6 5
Reflecting
 Use __15  2 to explain how a division of a fraction by a
whole number can be done as a multiplication.
To divide __ 1 by __
1 by 2, you can multiply __ 1.
5 5 2
Copyright © 2009 Nelson Education Ltd. Lesson 2.6: Dividing Fractions by Whole Numbers 49
19. Practising
a) __89  4
Use a grid and counters to represent __89 .
Draw a grid to represent 1 whole __99 .()
3  3  9 Draw a grid this size.
Draw 8 counters on the grid to represent __8 . 9
Circle the counters to divide them into 4 equal groups.
There are 2 counters in each group.
2
Each of the 4 groups represents ___ of the grid.
9
8
__  4  2
__
9 9
b) __29  4
Can you divide 2 counters into 4 groups? No
Write a fraction equivalent to __29 , with a numerator
that can be divided into 4 equal groups.
2 2
_______ 4
9
 ___
2 18
The denominator of a fraction shows the number
of parts in 1 whole.
Draw a grid to represent 1 whole.
Draw counters on the grid to represent the
equivalent fraction.
4
To calculate __
18
 4 , you can think of sharing 4
counters out of 18 between 4 people.
Note: Dimensions
1
of grids drawn may Each person would have ___ of the counters.
vary. 18
8
__  4  1
___
9 18
50 Lesson 2.6: Dividing Fractions by Whole Numbers Copyright © 2009 Nelson Education Ltd.
20. 6. Kevin used __56 of a can of paint to cover 4 walls.
How much of a can did he use for each wall?
Solution:
Write a division sentence to represent this problem.
5
___  4 ?
6
1
To divide by 4, you can multiply by ___ .
4
5
__ 1
___ 51
  _____
6
4 64
5
 ___
24
5
___
Kevin used 24 of a can of paint for each wall.
9. a) Create a problem you might solve by dividing __32 by 4.
Hint 2 of my book left to
Answers will vary, e.g., I have __
Think of something 3
you could have __23 of. read, and I have to ﬁnish in 4 days. How much must
Divide it between
4 people or things. I read each day?
2  4  __
__ 2  __
1
3 3 4
21
 ______
34
2 or __
 ___ 1
12 6
1 of my book each day.
6
Copyright © 2009 Nelson Education Ltd. Lesson 2.6: Dividing Fractions by Whole Numbers 51
21. Estimating Fraction
2.7
Student book pages 72–75
Quotients
Interpret and estimate the quotient of fractions
less than 1.
The fraction of students in a school who participate in
school sports has increased from __18 to __25 .
Is __25 closer to double __18 or triple __18 ?
Participants
last year Fit one fraction into the other fraction.
You can divide to find out how many times __18 fits into __25 .
Estimate __25  __18 .
Shade __25 and __18 on the fraction strips.
Participants
this year
About how many times does __18 fit into __25 ? 3 times
So, __25  __18 is close to 3 .
Is __2 about double __1 or triple __1 ? __________________
5 8 8
Compare fractions using equivalent fractions.
3 .
2 . Triple __1 is 3  __1  __
Double __18 is 2  __18  __
8 8 8 8
Hint Which of the fractions above is closer to __25 ?
To find a common To compare __28 , __38 , and __25 , rewrite the fractions using a
denominator, common denominator.
compare the
multiples of the The denominators of __82, __83, and __52 are 8 , 8 , and 5 .
denominators. Circle the lowest common denominator of 5 and 8.
5, 10, 15, 20, 25, 30, 35, 40, 45, …
8, 16, 24, 32, 40, 48, 56, 64, 72, …
52 Lesson 2.7: Estimating Fraction Quotients Copyright © 2009 Nelson Education Ltd.
22. Write equivalent fractions with a common denominator.
5  5  8
2
__ 10
___ 3
__ 15
___ 2
__ 16
___
8
= 40 8
= 5
=
40 40
5  5  8
10
Is __ or __ 15 16
closer to __ 15 is closer.
? ___
40 40 40 40
3 is closer.
So, is __28 or __38 closer to __25 ? __
8
3
___
8
is close to __25 , so __18 fits into __25 about 3 times.
2
__
5
 __18 is close to 3 .
PROBLEM Estimate __79  __14 .
Shade __79 and __14 on the fraction strips.
1
__
4
fits into __79 about 3 times.
So, __79  __14 is close to 3 .
PROBLEM Estimate __35  __14 using common denominators.
 4  5 One common denominator is 5  4  20
䉳 Write equivalent fractions.
12 5
__ = ___ 1
__ = ___ About how many times does __14 fit into __35 ? Compare
5 4
20 20
the numerators of the equivalent fractions.
 4  5 5 fits into 12 about 2 times, so __1 fits into __3
4 5
3
__ 1
__
about 2 times. So, 5  4 is close to 2 .
Hint
Reflecting
abc 2 1
 __ __
5 8
is about 3. The quotient, 3, is greater than 1.
dividend divisor quotient
1 2
__
85
__ is about __13 . The quotient, __13 , is less than 1.
When the dividend
is greater than the When will a quotient be less than 1?
divisor, the quotient
When the dividend is less than the divisor.
is less than 1.
Copyright © 2009 Nelson Education Ltd. Lesson 2.7: Estimating Fraction Quotients 53
23. A useful fact … Example: __46  __26  4  2  2
 The quotient of 2 fractions with the
same denominator is the same as
the quotient of the numerators.
a b Think of it this way: 2 fits into 4 the
__ __
nnab same number of times as __26 fits into __46 .
Practising
5. Estimate each quotient as a whole number.
11 3
a) __
12
 __
12
The denominators are the same, so
11
__ 3
12
 __
12
 11  3
11  3 is close to 12  3  4 .
11 3
So, __
12
 __
12
is close to 4 .
11
b) __
12
 __16
 2 Circle a common denominator of 6 and 12.
6, 12, 18, … 12, 24, …
11
__ 1
__ 2
___
12
and 6
= 䉳 Write a fraction equivalent to __16 using the
12
common denominator that you circled.
 2 Compare the numerators of the equivalent fraction
11
and __
12
. 2 fits into 11 about 5 times,
Note: Part c) is so 12  __16 is close to
11
__ 5 .
Part d) in Student c) __34  __
1
Book 10
Circle a common denominator of 4 and 10.
 5  2
4, 8, 12, 16, 20, … 10, 20, 30, …
__ 15
___ 1
__ 2
___ 䉳 Write equivalent fractions with this denominator.
= 10
=
20 20 Compare the numerators.
2 fits into 15 about 7 times, so __3  __ 1
is
4 10
 5  2 close to 7 .
54 Lesson 2.7: Estimating Fraction Quotients Copyright © 2009 Nelson Education Ltd.