Level III Fraction Unit
(Pilot Materials)
NSSAL
(Draft)
C. David Pilmer
2012
(Last Updated: October, 2013)
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This resource is the intellectual property of the Adult Education Division of the Nova Scotia
Department of Labour and Advanced Education.
The following are permitted to use and reproduce this resource for classroom purposes.
 Nova Scotia instructors delivering the Nova Scotia Adult Learning Program
 Canadian public school teachers delivering public school curriculum
 Canadian non-profit tuition-free adult basic education programs
The following are not permitted to use or reproduce this resource without the written
authorization of the Adult Education Division of the Nova Scotia Department of Labour and
Advanced Education.
 Upgrading programs at post-secondary institutions (exception: NSCC)
 Core programs at post-secondary institutions (exception: NSCC)
 Public or private schools outside of Canada
 Basic adult education programs outside of Canada
Individuals, not including teachers or instructors, are permitted to use this resource for their own
learning. They are not permitted to make multiple copies of the resource for distribution. Nor
are they permitted to use this resource under the direction of a teacher or instructor at a learning
institution.
Acknowledgments
The Adult Education Division would like to thank Dr. Genevieve Boulet (MSVU) for reviewing
this resource and providing valuable feedback.
The Adult Education Division would also like to thank the following ALP instructors for piloting
this resource and offering suggestions during its development.
Eileen Burchill (IT Campus)
Lynn Cuzner (Marconi Campus)
Carissa Dulong (Truro Campus)
Krys Galvin (Truro Campus)
Barbara Gillis (Burridge Campus)
Nancy Harvey (Akerley Campus)
Barbara Leck (Pictou Campus)
Suzette Lowe (Lunenburg Campus)
Shelly Meisner (IT Campus)
Alice Veenema (Kingstec Campus)
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Table of Contents
Introduction (for Learners) ………………………………………………………………..
Introduction (for Instructors) ………………………………………………………………
Prerequisite Knowledge ……………………………………………………………………
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iv
v
Part 1: Understanding Fractions …………………………………………………………
Introduction to Fractions: Area Models ………………………………………..…………..
The Online Fraction Sorter …………………………………………………………………
Comparing Fractions Investigation A …………………….……………………………….
Comparing Fractions Investigation B …………………….………………………………..
More Comparing Fractions …………………………………………………………………
Area Models for Fractions Greater Than 1 …………………………………………………
Comparing Proper Fractions and Mixed Numbers …………………………………………
From Mixed Numbers to Improper Fractions, and Vice Versa …………………………….
Area Models of Different Shapes and Sizes ………………………………………………..
Equivalent Fractions, Part 1 ………………………………………………………………..
Equivalent Fractions, Part 2 ………………………………………………………………..
Equivalent Fractions and the Number Line ………………………………………………..
Measuring and Fractions …………………………………………………..………………
1
1
7
9
12
20
24
31
34
43
47
49
58
60
Part 2: Operations with Fractions ………………………………………………………..
Estimating the Addition and Subtraction of Fractions ……………………………………..
Adding Fractions, Part 1 ……………………………………………………………………
Adding Fractions, Part 2 ……………………………………………………………………
Subtracting Fractions, Part 1 ………………………………………………………………..
Subtracting Fractions, Part 2 ………………………………………………………………..
Multiplying Fractions, Part 1 ……………………………………………………………….
Multiplying Fractions, Part 2 ……………………………………………………………….
Dividing Fractions, Part 1 …………………………………………………………………..
Dividing Fractions, Part 2 …………………………………………………………………..
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81
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95
103
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Answers …………………………………………………………………………………….
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Introduction (for Learners)
Many adult education math books on fractions spend all or most of their time looking at
operations with fractions (addition, subtraction, multiplication, and division). Unfortunately
these books tend to spend little or no time on having learners understand what a fraction is and/or
comparing the sizes of fractions. This is a problem.
Consider this. People generally understand whole numbers. For example, when people are
given the whole number 24, they might think a number between 20 and 30, two dozen, twenty
plus four, three sets of eight, thirty minus six, or a wide variety of other things. These people
truly understand the quantity of 24. Having this knowledge is very useful when they start
working with operations that involve the number 24 (e.g. 24  2 , 20  4 , 3  8 , 24  6 …).
Understanding whole numbers helps us understand operations with whole numbers.
Unfortunately many people do not have a similar understanding of fractions. For example, if
5
learners do not understand what the fraction
is and where it is on a number line, then how can
8
they add, subtract, multiply or divide with this fraction? We need to understand fractions before
we start doing operations with fractions. This math resource starts by focusing on this
understanding of fractions, spending much of the time looking at the size of specific fractions.
These are important concepts for all learners to grasp before looking at operations with fractions.
It is also important to note that learners are expected to complete almost all of the work in this
unit without the use of a calculator.
Introduction (for Instructors)
In most math resources, both area models and set models are used as pictorial representations of
fractions. This is not the case in this resource. We have only used area models, and limited them
to square area models. The reason stems from the fact that these square area models can be used
to illustrate all of the operations with fractions; circular area models, the other most common
area model, do not easily allow for this. Set models are not used because they can lead to
confusion regarding the magnitude of fractions because some learners view the numerator and
denominator of the fraction as separate entities when these models are used.
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Prerequisite Knowledge
The expectation for this unit is that learners are comfortable with the multiples, divisibility,
halves, and division that include remainders.
Multiples:
e.g. What are the multiples of 3?
Answer: 3, 6, 9, 12, 15, 18, …
Answer: 10, 20, 30, 40, 50, 60, …
e.g. What are the multiples of 10?
Divisibility:
e.g. Is 76 divisible by 2?
Answer: Yes (Reason: Even numbers are
divisible by 2.)
e.g. Is 85 divisible by 5?
Answer: Yes (Reason: Numbers, whose
ones digit is 5 or 0, are divisible by
5.)
e.g. Is 54 divisible by 3?
Answer: Yes (Reason: If the sum of the
digits of a number are divisible by
3, then the original number is
divisible by 3.)
e.g. What is 12 divisible by?
Answer: 1, 2, 3, 4, 6, and 12
e.g. What is 30 divisible by?
Answer: 1, 2, 3, 5, 6, 10, 15, and 30
Halves:
e.g. What is half of 8?
Answer: 4
e.g. What is half of 30?
Answer: 15
Division with Remainders:
e.g. What is 28  9 ?
Answer: 3 with a remainder of 1
e.g. What is 34  6 ?
Answer: 5 with a remainder of 4
e.g. What is 14  3 ?
Answer: 4 with a remainder of 2
Prime Factors:
e.g. Express 9 as a product of prime factors.
Answer: 9  3  3
e.g. Express 20 as a product of prime factors. Answer: 20  2  2  5
e.g. Express 30 as a product of prime factors. Answer: 30  2  3  5
Try these questions. Do not use a calculator.
(a) What are the multiples of 4?
______, ______, ______, ______, ______, …
(b) What are the multiples of 9?
______, ______, ______, ______, ______, …
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(c) What are the multiples of 6?
______, ______, ______, ______, ______, …
(d) What is half of 12?
______
(e) What is half of 50?
______
(f) What is half of 80?
______
(g) Is 76 divisible by 2?
Yes
No
(h) Is 67 divisible by 2?
Yes
No
(i) Is 107 divisible by 5?
Yes
No
(j) Is 730 divisible by 5?
Yes
No
(k) Is 48 divisible by 3?
Yes
No
(l) Is 81 divisible by 3?
Yes
No
(m) Is 92 divisible by 3?
Yes
No
(n) What is 9 divisible by?
______, ______, and ______
(o) What is 6 divisible by?
______, ______, ______, and ______
(p) What is 15 divisible by?
______, ______, ______, and ______
(q) What is 16 divisible by?
______, ______, ______, ______, and ______
(r) What is 21  9 ?
______ with a remainder of ______
(s) What is 25  6 ?
______ with a remainder of ______
(t) What is 10  7 ?
______ with a remainder of ______
(u) What is 44  8 ?
______ with a remainder of ______
(v) What is 41  7 ?
______ with a remainder of ______
(w) Express 15 as a product of prime factors.
___________________
(x) Express 18 as a product of prime factors.
___________________
(y) Express 24 as a product of prime factors.
___________________
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Introduction to Fractions; Area Models
Introduction
You may not realize it, but you’ve been working with fractions for years. Every time you handle
money, you are dealing with fractions.

Consider a quarter. It’s worth 25 cents and takes
4 quarters to make 1 dollar. As a fraction, you
1
would say that one 25 cent coin is
of a dollar.
4

Consider a 50 cent coin. This is a fairly rare coin
that is only made to commemorate special events.
It takes 2 fifty cent coins to make 1 dollar. As a
1
fraction, you would say that one 50 cent coin is
2
of a dollar
Those of you who have worked in trades (carpentry, electrical, plumbing…) have
also been exposed to fractions. In the trades, most measurements are made using
Imperial measure (inches, feet, pounds, ounces, cups, gallons…). Fractions are
frequently used in imperial measure. A carpenter would not ask for a piece of
wood to be cut to a length of 6.75 inches, rather the carpenter would state the
3
measurement as 6 inches.
4
If you need to learn about fractions, the first place to start is with area models. Area models are
pictures that represent fractions. For this course we will mostly focus on square area models.
Example 1
Draw an area model for the fraction
3
.
4
Answer:
This fraction is called "three-fourths" or "three-quarters." However, it is not called "three
3
over four." The fraction
has a numerator of 3 and a denominator of 4. The
4
denominator tells us to divide our square into 4 equal parts. The numerator tells us to shade
3 of those parts. All of the area models below are acceptable representations of this fraction.
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Example 2
(a) What fraction is represented by the area model on the right?
(b) What fraction of the area model is not shaded?
Answer:
2
("two-fifths") because the square is
5
divided into 5 equal parts and 2 of those parts are shaded in.
2
3
(b) If
of this square is shaded, then ("three-fifths) of the square is not shaded.
5
5
(a) The area model to the right represents the fraction
Example 3
(a) What fraction is represented by the area model on the right?
(b) What fraction of the area model is not shaded?
Answer:
5
(which is equal to 1) because the
5
square is divided into 5 equal parts and all 5 of those parts are shaded in.
5
0
(b) If ("five-fifths") of this square is shaded, then
("zero-fifths") of the square is not
5
5
shaded.
(a) The area model to the right represents the fraction
Example 4
2 5 1
5
, , , and . After doing, place each fraction by its
3 12 9
6
appropriate arrow on the number line below.
Create area models for the fractions
0
1
1
2
Answer:
Area models can be useful when you are trying to compare the magnitude (size) of different
fractions. We will start by drawing the four area models for the four fractions.
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2
3
5
12
1
9
5
6
2
1
is somewhere between
and 1 because more than half of the square is shaded but the
2
3
square is not fully shaded or nearly completely shaded.
5
1

is close to pretty close to, but slightly less than
because almost half the square is
12
2
shaded.
1

must be the smallest number and close to 0 because only a very small portion of the
9
square is shaded.
5

must be the largest number and close to 1 because almost the entire square is shaded.
6

We can use the above information to approximate the locations of our four fractions on the
1
number line. The benchmark numbers of 0, , and 1 were already on the number line to
2
provide us with some guidance.
1
9
5
12
5
6
2
3
0
1
1
2
Questions
1. For each area model, figure out the fraction that the model represents.
(a)
(b)
(c)
(d)
Answer: ____
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Answer: ____
Answer: ____
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Answer: ____
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2. In question 1, which one is the smallest fraction? Which one is the largest fraction?
Smallest Fraction: _______
Largest Fraction: _______
3. Look back at the area models in question 1. What fraction of these models is not shaded?
(a) Answer:
(b) Answer:
(c) Answer:
(d) Answer:
4. Do all of the following area models below represent the fraction
5
? Why or why not?
6
5. For the area models in question 4, what fraction of the model is not shaded?
6. Does the following area model represent the fraction
______
1
? Why or why
3
not?
7. Determine the fraction that is represented by each of these area models. Then take those
fractions and place them by the appropriate arrow on the number line.
Answer: ____
0
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Answer: ____
Answer: ____
Answer: ____
1
1
2
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4 5 3 7
1
,
, , , and in the space provided below.
4 10 8 8
6
After doing this, place each fraction by its appropriate arrow on the number line below.
8. Draw the area models for the fractions
4
4
5
10
3
8
0
7
8
1
6
1
1
2
9. Put the numbers in the blanks so that the statement makes sense. Please note that there are
four numbers and three blanks therefore one number should not be used.
(a) Taralee took the pizza and cut into ____ slices, all of the same size. If she took ____
slices for herself, then that meant that she took ____ of the pizza.
3
8
3
8
3
8
(b) Massato took the cake and cut into 15 pieces of the same size and
took ____ pieces for himself. That means that he took ____ of
the cake and that ____ of the cake was left for everyone else.
3
13
15
2
2
15
(c) A square is drawn, and then its two diagonals are drawn in. If ____ of the resulting
triangles within the square is shaded, then the shaded portion represents ____ of the
square. If an additional triangle is shaded, then the shaded portions represent ____ of the
square
1
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(d) Tanya has a piece of farm land that she wants to divide evenly amongst her ____
children. If she has 2 boys and ____ girls, it means that the boys would end up owning
____ of the land.
3
5
3
2
5
5
(e) Two brothers are sharing a pizza but not equally. The pizza is cut
into 12 slices of the same size but the older brother takes more. If
the older brother takes ____ slices, then he ends up getting ____ of
the pizza, and the younger brother gets ____ of the pizza.
7
12
5
7
5
12
10. (a) Write the fraction five-sevenths.
______
(b) Write the fraction seven-tenths.
______
(c) Write the fraction ten-twelfths.
______
(d) Write the fraction one-ninth.
______
(e) Write the fraction two-fifteenths.
______
(f) Write the fraction eleven-twenty-thirds.
______
11. (a) Use words to write out the fraction
1
.
6
(b) Use words to write out the fraction
5
.
9
(c) Use words to write out the fraction
13
.
16
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The Online Fraction Sorter
In this section you are going to use an interactive activity found on the internet. Using Google,
search the following key words: Shodor Interactive Activities. Once you are on the site, scroll
down and find the activity Fractor Sorter. It's a game where you are asked to create area
models for two fractions, and then figure out which one is the larger fraction.
This is what you will encounter when you play the game.
The activity will start by giving you
two fractions. In this case, we were
3
10
given the fractions and
. The
5
14
first thing we have to do is create the
area models for these two fractions.
The "-Col" and "+Col" buttons allow
you to subtract or add columns to your
models. The "+Row" and "-Row"
buttons allow you to add or subtract
rows to your area model. With the first
model, we created 5 rows; this divided
the model into five equal parts. With
the second model, we created 7 rows
and 2 columns; this divided the model
into 14 equal parts.
Now we need to shade the appropriate
regions of the models. Using your
mouse, click on the regions you wish
to shade. Once this is done, click on
the "Check" buttons for each model
and the program will tell you if your
area models are correct.
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In the last part of the activity, you are
asked to sort the fractions from
smallest to largest by dragging them to
3
the appropriate box below. Since is
5
smaller, it is dragged to the first box.
10
Because
is larger, it is dragged to
14
the second box. Notice that lines are
drawn to show the position of each
fraction on the number line.
Questions
1. Play three rounds of Fraction Sorter and print off each of the final answers. Please make sure
that in the upper right hand portion of the screen you are working with "Squares" (shape of
area model) and "2" (number of fractions). Record your results in the chart below.
Smaller Larger
Fraction Fraction
Round One
Round Two
Round Three
2. Play another round of fraction sorter
but in this case, select "3" for the
number of fractions (as shown on the
right). Again you are asked to print
off your final answer. Record your
final answer below.
Fractions (Smallest to Largest):
______, ______, ______
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Comparing Fractions; Investigation A
In this investigation we are going to be comparing fractions so that we can order them from
smallest to largest.
Part 1 - Same Number in the Denominator
(a) Four fractions and their corresponding area models have been supplied below. Based on this
information, order the fractions from smallest to largest in the space provided below.
3
4
2
4
4
4
1
4
Order: ______, ______, ______, ______
(b) Four fractions and their corresponding area models have been supplied below. Based on this
information, order the fractions from smallest to largest in the space provided below.
8
9
1
9
6
9
3
9
Order: ______, ______, ______, ______
(c) Complete the following statement using your own words.
"When asked to compare two fractions that have the same number in the denominator
(bottom), the larger fraction is the one with …"
(d) With each pair of fractions, circle the larger fraction.
(i)
5
6
(iv)
9
10
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6
8
10
(ii)
1
8
5
8
(iii)
7
16
5
16
(v)
1
3
2
3
(vi)
5
12
10
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Part 2 - Same Number in the Numerator
(a) Four fractions and their corresponding area models have been supplied below. Based on this
information, order the fractions from smallest to largest in the space provided below.
3
4
3
12
3
9
3
6
Order: ______, ______, ______, ______
(b) Four fractions and their corresponding area models have been supplied below. Based on this
information, order the fractions from smallest to largest in the space provided below.
5
9
5
6
5
18
5
12
Order: ______, ______, ______, ______
(c) Complete the following statement using your own words.
"When asked to compare two fractions that have the same number in the numerator (top), the
larger fraction is the one with …"
(d) With each pair of fractions, circle the larger fraction.
(i)
1
2
(iv)
3
16
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3
3
4
(ii)
2
3
2
5
(iii)
3
10
3
8
(v)
7
8
7
10
(vi)
5
20
5
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Questions
1. With each pair of fractions, circle the larger fraction.
(a)
3
10
2
10
(b)
1
5
(d)
4
5
4
7
(e)
13
16
(g)
7
16
7
8
(h)
3
8
(j)
3
8
3
12
(k)
1
12
1
3
9
16
1
8
1
16
(c)
4
9
7
9
(f)
4
8
4
4
(i)
6
10
7
10
(l)
6
32
6
16
2. Order the fractions from smallest to largest in the space provided.
(a)
5 15 11 3
,
,
,
16 16 16 16
_______, ______, ______, ______
(b)
5 5 5 5
,
, ,
5 16 8 10
_______, ______, ______, ______
(c)
7 5 1 3
, , ,
8 8 8 8
_______, ______, ______, ______
(d)
2 2 2 2
, , ,
12 3 8 16
_______, ______, ______, ______
(e)
7 9 4 1
,
,
,
10 10 10 10
_______, ______, ______, ______
(f)
7
7
7 7
,
,
,
7 100 32 8
_______, ______, ______, ______
3. A number line has been provided. There are four arrows that identify the approximate
8 3 1
5
locations of the fractions , , and . Figure out where each fraction would
8 8 8
8
approximately be on the number line.
0
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Comparing Fractions Investigation B
In the last section we compared fractions that had the same number in the denominator (e.g.
8
9
4
3
3
), and compared fractions that had the same number in the numerator (e.g. and
).
9
8
10
How do we compare fractions where the number differ in the numerator and denominator
1
7
(e.g. and
)? That is what you are going to learn in this investigation.
8
10
and
Investigation
Below you are supplied with a variety of area models. Your mission is to identify the fraction
1
that corresponds to this area model and then decide if that fraction is closest to 0, , or 1. The
2
area model helps you figure this out. If the fraction is close to 0, then a small portion of the area
model is shaded. If it’s close to 1, then a large portion of the area model is shaded. If it is close
1
to
, approximately half of the area model is shaded.
2
Area Model
Fraction
Closest
To:
Area Model
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
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Fraction
Closest
To:
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Area Model
Fraction
Closest
To:
Area Model
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
(r)
Fraction
Closest
To:
In the charts on the previous page and above, you were given 18 area models. You determined
1
the corresponding fraction and decided if that fraction was closest to 0, , or 1. In the table
2
below you are to write down the fractions you encountered in this investigation in their
appropriate column.
Fractions Closest to 0
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13
1
2
Fractions Closest to 1
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Complete these four statements with the numbers 0,
1
, or 1.
2
(a) When the number in the numerator of a fraction is close to the number in the denominator of
7 9 15
the fraction (examples: , , ), then the fraction is closest to the number ___.
8 11 18
(b) When the number in the numerator of a fraction is very small compared to the number in the
1 2 3
denominator of the fraction (examples: , , ), then the fraction is closest to the number
8 11 18
___.
(c) When the numerator of the fraction is about half the size of the denominator of the fraction
4 6 14
(examples: , , ), then the fraction is closest to the number ___.
9 11 30
(d) When the numerator of the fraction is the same number as the denominator of the fraction
3 8 11
(example: , , ), then the fraction is equal to ___.
3 8 11
1
, and 1 are called benchmark numbers because most people understand these
2
quantities. Comparing fractions to these benchmark numbers is useful when trying to understand
the size of fractions.
The numbers 0,
Example 1
Order the fractions
15 3 10 1
from smallest to largest.
, ,
,
16 7 10 32
Answer:
In this question, we will compare the four fractions to the benchmark numbers 0,
15
is close to 1
16
3
1
is close to
2
7
10
is equal to 1
10
1
is close to 0
32
Therefore the correct order is
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, and 1.
2
1 3 15 10
, ,
,
.
32 7 16 10
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Example 2
Order the fractions
2 11 1 8 15
from smallest to largest.
,
,
, ,
16 12 16 8 32
Answer:
1
, and 1 to do this question but we will also use
2
techniques that we learned in the previous section of this unit.
We will use the benchmark numbers 0,
2
is close to 0
16
11
is close to 1
12
8
is equal to 1
8
15
1
is close to
2
32
1
is close to 0
16
2
1
and
are both close to 0. In the previous section, we learned that when
16
16
comparing two fractions that have the same number in the denominator (bottom), the larger
fraction is the one with the larger number in the numerator (top). Based on this we can
2
1
conclude that
is larger than
.
16
16
The fractions
Therefore the correct order is
1 2 15 11 8
,
,
,
, .
16 16 32 12 8
Example 3
Order the fractions
8 4 9 8 3
from smallest to largest.
, ,
,
,
9 4 16 10 32
Answer:
We will use more than one strategy to do this question.
8
is close to 1
9
4
is equal to 1
4
8
is close to 1
10
3
is close to 0
32
9
1
is close to
2
16
8
8
and
are both close to 1. In the previous section, we learned that when
9
10
comparing two fractions that have the same number in the numerator (top), the larger fraction
is the one with the smaller number in the denominator (bottom). Based on this we can
8
8
conclude that is larger than
.
9
10
The fractions
Therefore the correct order is
NSSAL
©2012
3 9 8 8 4
,
,
, , .
32 16 10 9 4
15
Draft
C. D. Pilmer
Questions
1. Without drawing an area model, determine if each fraction is closest to 0,
Fraction
(a)
1
18
17
32
8
15
5
11
3
100
(d)
(g)
(j)
(m)
Closest to:
Fraction
(b)
Closest to:
(e)
(h)
(k)
(n)
Fraction
(c)
24
25
1
14
2
30
18
19
7
12
1
, or 1.
2
Closest to:
5
11
2
20
28
30
37
40
13
15
(f)
(i)
(l)
(o)
2. A number line has been provided. There are four arrows that identify the approximate
3 4 1
11
locations of the fractions , , and
. Figure out where each fraction would
3 7 10
12
approximately be on the number line. Be able to explain why.
1
2
0
1
3. For each question, a pair of fractions has been provided. Circle the larger number. A few of
these questions require you to use a strategy other than benchmark numbers.
9 12
4
1
(a)
(b)
10 12
5
12
(c)
3
8
5
8
(d)
1
16
7
16
(e)
1
7
1
10
(f)
3
8
2
25
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C. D. Pilmer
(g)
3
7
5
7
(h)
7
8
1
7
(i)
9
10
9
16
(j)
4
9
4
10
(k)
6
7
1
4
(l)
2
11
5
9
(m)
11
12
5
11
(n)
5
6
2
16
(o)
1
30
6
7
(p)
5
7
5
6
(q)
11
12
7
16
(r)
7
32
7
16
(s)
17
32
16
16
(t)
7
12
3
12
4. In each case, take the fractions and place them in order from smallest to largest
Smallest
Largest
(a)
3 5 1 7
, , ,
7 7 7 7
____
____
____
____
(b)
13 6 7 1
, ,
,
14 6 15 10
____
____
____
____
(c)
3 1 1 4
, ,
,
5 5 20 5
____
____
____
____
(d)
5 20 2 6
,
,
,
12 20 16 7
____
____
____
____
(e)
1 1 1 1
, , ,
7 2 3 9
____
____
____
____
(f)
4 1 9 9
,
, ,
7 12 9 11
____
____
____
____
(g)
2 2 2 2
, , ,
3 9 5 7
____
____
____
____
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C. D. Pilmer
Smallest
Largest
(h)
4 11 3 1
,
, ,
9 12 3 11
____
____
____
____
(i)
9 3 1 5
, ,
,
10 8 12 8
____
____
____
____
(j)
1 1 6 11
, , ,
6 8 6 20
____
____
____
____
(k)
5 1 7 2
, , ,
9 9 9 9
____
____
____
____
(l)
1 7 1 7
, , ,
21 9 5 13
____
____
____
____
(m)
2 9 1 7
,
,
,
19 10 19 12
____
____
____
____
(n)
3 10 11 7
,
,
,
11 11 11 11
____
____
____
____
(o)
19 5 1 17
,
, ,
20 12 5 20
____
____
____
____
(p)
30 8 31 3 16
,
,
,
,
32 14 32 32 16
____
____
____
____
____
(q)
1 7 25 1
8
, ,
,
,
9 8 25 20 17
____
____
____
____
____
(r)
8 7
3 8
2
,
,
,
,
9 15 32 15 32
____
____
____
____
____
5. A number line has been provided. There are five arrows that identify the approximate
9 1 3 1
5
locations of the fractions , , , and . Figure out where each fraction would
9 6 7 4
9
approximately be on the number line. Be able to explain why.
0
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©2012
1
2
1
18
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C. D. Pilmer
6. State all the fractions between 0 and
1
that have a denominator of 12.
2
1
3
or
, is larger. Her answer and
2
16
explanation is shown below. Her answer is incorrect. Explain to her how she should have
worked out the answer. You can include diagrams in your explanation.
7. Chantelle was asked to determine which fraction ,
Chantelle’s Answer and Explanation:
3
1
3
is larger than
The 3 in the numerator of the fraction
is larger than the 1 in the
16
2
16
1
numerator of the other fraction, . The 16 in the denominator of
2
3
the fraction
is larger than the 2 in the denominator of the other
16
1
3
1
fraction, . Based on this, I figure that
is larger than .
2
16
2
Your Explanation:
Open-ended Questions (There is more than one correct answer for each of these questions.)
8. Find two fractions that are between 0 and
9. Find two fractions that are between
1
.
2
_____
_____
_____
_____
1
and 1.
2
10. Find a fraction that is very close to, but slightly larger than
1
.
2
_____
11. Find a fraction that is very close to, but slightly smaller than
1
.
2
_____
NSSAL
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C. D. Pilmer
More Comparing Fractions
In the last section we used the benchmarks 0,
1
, and 1 to compare the size of fractions. In this
2
1
, specifically fractions that are slightly less than, slightly
2
more than, and exactly equal to this bench mark.
1
 Fractions that are equal to the benchmark .
2
2 3 4 5 6 7 8
e.g. , , ,
,
,
,
4 6 8 10 12 14 16
section we focus on the benchmark
Notice that the number in the numerator (top) is exactly half of the number in the
3
denominator (bottom). For example, with the fraction
, the 3 in the numerator is
6
exactly half of the 6 found in the numerator.
 Fractions that are slightly less than the benchmark
5
.
12
5 4 8 7 5 6 9
, ,
,
,
,
,
12 9 18 15 11 13 20
Notice that the number in the numerator (top) is slightly less than half of the number in
the denominator (bottom). This is easier to see when the number in the denominator is
an even number (e.g. 2, 4, 6, 8, 10,…), rather than an odd number (e.g.1, 3, 5, 7, 9, …).
5
- Consider the fraction
, which has the even number 12 in the denominator. Half of
12
12 is 6, and 5 in the numerator is slightly less than 6. Therefore it makes sense that
5
1
the fraction
is slightly less than .
12
2
4
- Now we will consider the fraction , which has the odd number 9 in the
9
denominator. We know that half of 8 is 4, and half of 10 is 5. Therefore half of 9 is
1
4
4.5 or 4 . The 4 in the numerator is less than 4.5, therefore
must be slightly less
2
9
1
than .
2
1
 Fractions that are slightly more than the benchmark .
2
5 7 10 8 6 7 11
,
,
,
,
,
e.g. ,
9 12 18 15 11 13 20
Notice that the number in the numerator (top) is slightly more than half of the number in
the denominator (bottom).
e.g.
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C. D. Pilmer
Questions
1. Look at each fraction and check off the most appropriate column. Three examples have been
completed to provide guidance.
Close to 0
e.g.
e.g.
e.g.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
NSSAL
©2012
11
20
1
16
8
8
10
20
7
8
3
32
7
12
6
14
3
3
4
8
5
9
9
18
1
100
19
20
9
17
14
16
Slightly
less
1
than
2
Equal to
1
2
Slightly
more
1
than
2
Close to 1
Equal to 1



21
Draft
C. D. Pilmer
Close to 0
(n)
Slightly
less
1
than
2
Equal to
1
2
Slightly
more
1
than
2
Close to 1
Equal to 1
3
7
50
50
15
30
52
100
2
25
10
19
(o)
(p)
(q)
(r)
(s)
2. A number line has been provided. There are six arrows that identify the approximate
12 1
4 5 50
9
locations of the fractions
. Determine where each fraction
,
,
, ,
, and
12 40 10 6 100
16
would approximately be on the number line.
1
2
0
1
3. With each pair of fractions, circle the larger fraction.
(a)
5
12
(d)
8
9
(g)
7
16
(j)
5
12
NSSAL
©2012
7
14
(b)
6
10
(e)
5
7
6
10
(h)
9
16
7
12
(k)
5
8
3
8
10
22
(c)
3
7
(f)
6
12
5
5
3
6
(i)
6
11
1
10
6
13
(l)
6
7
5
8
22
4
7
8
14
Draft
C. D. Pilmer
(m)
4
7
6
13
(n)
7
14
9
19
(0)
1
7
9
15
4. Order the fractions from smallest to largest in the space provided.
(a)
5 4 30 7 1
, ,
,
,
5 8 32 16 16
_______, ______, ______, ______, ______
(b)
5 7 11 5 11
,
,
,
,
9 100 11 10 12
_______, ______, ______, ______, ______
(c)
6 5 14 3 2
, ,
, ,
14 8 15 6 21
_______, ______, ______, ______, ______
(d)
15 4 6 14 7
, ,
,
,
16 7 12 16 16
_______, ______, ______, ______, ______
(e)
5 19 4 7 8
,
,
, ,
10 20 10 7 14
_______, ______, ______, ______, ______
(f)
8 1 28 11 1
,
,
,
,
15 20 30 22 30
_______, ______, ______, ______, ______
(g)
9 30 15 9 7
,
,
,
,
10 60 15 11 12
_______, ______, ______, ______, ______
(h)
9 5 7 2 2
,
,
,
,
16 11 14 19 50
_______, ______, ______, ______, ______
(i)
13 9
3 40 9
,
,
,
,
14 17 50 40 18
_______, ______, ______, ______, ______
(j)
6 5 12 7 12
, ,
,
,
14 9 13 14 14
_______, ______, ______, ______, ______
Important Note:
In the last few sections we have been taking an intuitive approach to comparing fractions, rather
than relying on the formal approach of creating common denominators. The intuitive approach
generally makes more sense to learners, is easier to remember, and takes far less time to execute.
For these reasons, we have chosen to focus on this intuitive approach. This being said, there are
some limitations to this approach.
NSSAL
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C. D. Pilmer
Area Models for Fractions Greater Than 1
In the previous section we learned how area models can be used to represent fractions. We
worked with fractions that were less than 1 (See area models below.). These are referred to as
proper fractions. They can be recognized quickly because the number in the numerator (top of
the fraction) is less than the number in the denominator (bottom of the fraction).
1
6
5
8
7
12
We also worked with fractions that were equal to 1. In these cases, all sections of the area model
were shaded.
6
1
6
8
1
8
12
1
12
How do we use area models to represent fractions that are greater than 1? These are referred to
as improper fractions and can be recognized quickly because the number in the numerator (top
of the fraction) is more than the number in the denominator (bottom of the fraction).
e.g.
11
("eleven-eighths")
8
e.g.
17
("seventeen-sixths")
6
All improper fractions can be written as a mixed number, which contain both a whole number
and a fraction.
e.g. 1
3
("one and three-eighths")
8
e.g. 2
5
("two and five-sixths")
6
Example 1
Use an area model to represent each of these improper fractions. After doing so, express the
improper fraction as a mixed number.
11
9
(a)
(b)
6
8
9
11
(c)
(d)
4
3
NSSAL
©2012
24
Draft
C. D. Pilmer
Answers:
11
6
5
(a)
is make up of
(a completely shaded square) and (a partially shaded square).
6
6
6
Appropriate Area Model
Use the same line of thinking to convert the improper fraction
11
to a mixed number.
6
11 6 5
 
6 6 6
5
 1
6
5
1
 Mixed Number "one and five-sixths"
6
9
8
1
is make up of (a completely shaded square) and (a partially shaded square).
8
8
8
(b)
Appropriate Area Model
Use the same line of thinking to convert the improper fraction
9
to a mixed number.
8
9 8 1
 
8 8 8
1
 1
8
1
1
 Mixed Number "one and one-eighth"
8
NSSAL
©2012
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C. D. Pilmer
(c)
9
4
4
is make up of
(a completely shaded square),
(another completely shaded square)
4
4
4
1
and (a partially shaded square).
4
Appropriate Area Model
Use the same line of thinking to convert the improper fraction
9
to a mixed number.
4
9 4 4 1
  
4 4 4 4
1
 11
4
1
2
 Mixed Number "two and one-fourth"
4
(d)
11
3
3
is make up of (a completely shaded square), (another completely shaded
3
3
3
3
2
square),
(another completely shaded square) and (a partially shaded square).
3
3
Appropriate Area Model
Use the same line of thinking to convert the improper fraction
11
to a mixed number.
3
11 3 3 3 2
   
3 3 3 3 3
2
 111
3
2
3
 Mixed Number "three and two-thirds"
3
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C. D. Pilmer
Example 2
We have been provided with a variety of area models. Write the fraction that corresponds to
each model. If the fraction is greater than 1, write it as both an improper fraction and as a mixed
number. Then transfer the mixed number to its appropriate position on the number line. Arrows
and benchmark fractions have been placed on the number line to provide some assistance.
Model A
Model B
Model C
Model D
Model E
Model F
Model G
Model H
Model I
0
1
2
4
3
Answers:
Model A:
29
5
or 2
12
12
Model B:
5
2
or 1
3
3
Model C:
13
1
or 3
4
4
Model D:
5
6
Model E:
31
7
or 3
8
8
Model F:
11
3
or 2
4
4
Model G:
7
1
or 3
2
2
Model H:
7
1
or 1
6
6
Model I:
1
8
1
8
0
NSSAL
©2012
5
6
1
1
1
6
1
2
3
2
2
5
12
2
3
4
3
3
27
1
1
3
4
2
3
7
8
4
Draft
C. D. Pilmer
Questions:
1. Express each area model as an improper fraction and as a mixed fraction.
(a)
Improper Fraction:
Mixed Number:
(b)
Improper Fraction:
Mixed Number:
(c)
Improper Fraction:
Mixed Number:
2. Create an area model to represent each of the following.
2 7
1 9
(a) 1 or
(b) 2 or
5 5
4 4
(c) 1
3
11
or
8
8
(d) 2
3. Circle the improper fractions in the following list.
9
3
3
7
43
9
1
7
5
2
12
4
8
10
NSSAL
©2012
28
1
7
or
3
3
3
16
Draft
C. D. Pilmer
4. We have been provided with a variety of area models. Write the fraction that corresponds to
each model in the space provided. If the fraction is greater than 1, write it as both an
improper fraction and as a mixed number. Then transfer the mixed number to its appropriate
position on the number line. Arrows and benchmark fractions have been placed on the
number line to provide some assistance.
Model A
Model B
Model C
Answer:
Answer:
Answer:
Model D
Model E
Model F
Answer:
Answer:
Answer:
Model G
Model H
Model I
Answer:
Answer:
Answer:
0
1
2
3
4
5. When dealing with fractions that are greater than 1, do you find it easier to understand the
size of a fraction when it is written as an improper fraction or when it is written as a mixed
number? Explain.
NSSAL
©2012
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C. D. Pilmer
6. (a) Write the fraction fifteen-sevenths.
______
(b) Write the fraction three and one-tenth.
______
(c) Write the fraction ten-sixths.
______
(d) Write the fraction two and one-twelfth.
______
(e) Write the fraction twenty-five-fifteenths.
______
(f) Write the fraction one and eleven-twenty-seconds. ______
7. (a) Use words to write out the fraction
18
.
7
1
(b) Use words to write out the fraction 2 .
5
(c) Use words to write out the fraction
13
.
8
(d) Use words to write out the fraction 3
NSSAL
©2012
9
.
16
30
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C. D. Pilmer
Comparing Proper Fractions and Mixed Numbers
1 5 11
, , ) are less than 1. We also
10 9 12
learned a variety of strategies to compare the sizes of proper fractions so that we can order them
from smallest to largest.
In previous sections we learned that proper fractions (e.g.
3
7 4
, 3 , 2 ), which can also be
10
9 7
expressed as improper fractions. Mixed numbers are greater than 1. In this section we will learn
how to use the previously-learned strategies to order both proper fractions and mixed numbers.
In the last section we learned about mixed numbers (e.g. 1
Example 1
4 1
3 3 8 3
Order the following numbers 2 ,
, 2 , 1 , , 1 from smallest to largest.
7 20
7 6 9 5
Answer:
We can start by sorting the numbers into three groups.
1. Numbers Less Than 1 or Equal to 1 (i.e. proper fractions):
1
8
and
20
9
3
3
and 1
5
6
4
3
3. Numbers Between 2 and 3: 2 and 2
7
7
2. Numbers Between 1 and 2: 1
Now compare the numbers in each of the three groups using previously-learned strategies.
1
8
1
8
1.
is close to 0.
is close to 1. Therefore
is smaller than .
20
9
20
9
3
3
1
2. The , which is the fraction portion of the mixed number 1 , is equal to .
6
2
6
3
3
1
The , which is the fraction portion of the mixed number 1 , is slightly more than .
5
5
2
3
3
Therefore 1 is smaller than 1 .
5
6
4
4
3
3. The , the fraction portion of the mixed number 2 , is larger than , the fraction
7
7
7
3
3
4
portion of the mixed number 2 . Therefore 2 is smaller than 2 .
7
7
7
The correct order is
NSSAL
©2012
1 8 3 3
3
4
, ,1 ,1 , 2 , 2
20 9 6 5
7
7
31
Draft
C. D. Pilmer
Example 2
Order the following numbers 1
9
4 3
1
9
6 4
4
, 2 ,
,1 ,1 ,1 ,
,2
from smallest to
10
18 32 9 16 12 32
20
largest.
Answer:
We can start by sorting the numbers into three groups.
1. Numbers Less Than 1 or Equal to 1 (i.e. proper fractions):
3
4
and
32
32
3 4
,
32 32
9
1
9
6
2. Numbers Between 1 and 2: 1 , 1 , 1 , and 1
10
12
9 16
Order these two numbers from smallest to largest:
1 6 9 9
,
, ,
9 12 16 10
1
6
9
9
We can now state the order of these mixed numbers: 1 , 1 , 1 , 1
9 12 16 10
4
4
3. Numbers Between 2 and 3: 2
and 2
18
20
4 4
Order just the fractional portions of each of these mixed numbers:
,
20 18
4
4
We can now state the order of these mixed numbers: 2 , 2
20 18
Order just the fractional portions of each of these mixed numbers:
The correct order is
3 4
1
6
9
9
4
4
,
, 1 , 1 ,1 , 1 , 2 , 2
32 32 9 12 16 10
20 18
Questions
1. Place the following numbers by the appropriate arrow on the number line below.
5 5 10 7
7 1 5
7
11 1
1
1 , , 2 ,
, 2 ,1 ,
, 2 ,1 ,
,2
10 5
8 8 12 16 12 16
18 12
32
0
NSSAL
©2012
1
2
32
3
Draft
C. D. Pilmer
2. Place the following numbers by the appropriate arrow on the number line below.
1 3 1
4
7 1
31
1 10
5 15
, 2 ,1 , , 2 ,1 ,
, 2 ,
2 ,1 ,
5 7 32
8 12 7
6 30
32 10 10
0
1
2
3
3. With each pair of numbers, circle the larger one.
(a)
1
1
12
(d)
1
3
8
(g)
2
7
16
2
(j)
3
7
12
3
4
7
(m) 1
1
2
7
14
(b)
1
1
8
(e)
2
7
12
(h)
5
9
5
12
(k)
1
1
16
(n)
1
8
16
1
6
13
9
10
2
1
16
(c)
3
7
7
10
2
3
10
(f)
2
(i)
24
25
9
15
15
16
5
5
6
12
2
1
1
32
1
(l)
3
6
7
3
(0)
2
1
7
2
7
8
8
16
8
14
15
16
4. Order the fractions from smallest to largest in the space provided.
(a)
5 4 30
7
1
, , 1 , 1 ,1
5 8 32 16 16
_______, ______, ______, ______, ______
(b)
5
7 11
5 11
,2
,
, 1 ,1
9 100 11 10 12
_______, ______, ______, ______, ______
(c)
2
6
5 14 3 7
,1 ,
,1 ,1
14 8 15 8 8
_______, ______, ______, ______, ______
(d)
1
15
4
6
14
7
,2 ,1 ,1 ,2
16
7 12 16 16
_______, ______, ______, ______, ______
(e)
3
5
19
4 7
8
,1 ,3 , ,3
10 20 10 7 14
_______, ______, ______, ______, ______
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From Mixed Numbers to Improper Fractions, and Vice Versa
In the last two sections we learned that when dealing with fractions that are greater than 1, it is
easier to understand their size if they are expressed as mixed numbers, rather than improper
fractions. Based on this, it makes sense that we should be able to convert between these two
representations of fractions. We will learn how to convert from mixed number to improper
fractions, and vice versa, in this section.
In the first two examples we will learn about two methods for converting from mixed numbers to
improper fractions.
Example 1
2
Write 3 as an improper fraction.
5
Answer:
We have shown two methods to handle this. You choose the one you prefer.
Method 1 (More Efficient Method)
Method 2
Step 1: Multiply the whole number part by Step 1: Take the whole number portion of
the denominator of the fraction.
the mixed number and express it as
Step 2: Add that number to the numerator.
a series of 1's.
Step 3: Then write the result as the
Step 2: Those 1's are then expressed as
numerator of the improper
fractions with the same
fraction. The denominator will
denominator as the fraction portion
remain the same.
of the mixed number.
Step 3: Finally add the fractions; we can
do this because they have a
2 3  5  2
3 
common denominator. If we have
5
5
5 fifths plus 5 fifths plus 5 fifths
15  2

plus 2 fifths, then the answer must
5
be 17 fifths.
17

5
2
2
3  111
5
5
5 5 5 2
   
5 5 5 5
17

5
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Example 2
Write the following mixed numbers as improper fractions.
5
5
1
(a) 2
(b) 1
(c) 4
4
8
6
Answers:
Method 1 (More Efficient Method)
1 2  4   1
(a) 2 
4
4
8 1

4
9

4
(d) 5
3
10
Method 2
1
1
2  11
4
4
4 4 1
  
4 4 4
9

4
Note: 4 fourths plus 4 fourths plus 1
fourth will give us 9 fourths.
(b)
5 1 8   5
1 
8
8
85

8
13

8
5
5
1  1
8
8
8 5
 
8 8
13

8
Note: 8 eighths plus 5 eighths will give
us 13 eighths.
(c)
(d)
5  4  6  5
4 
6
6
24  5

6
29

6
5
5
5
4  1111
6
6
6 6 6 6 5
    
6 6 6 6 6
29

6
3  5 10   3

10
10
50  3

10
53

10
5
3
3
 11111
10
10
10 10 10 10 10 3
     
10 10 10 10 10 10
53

10
Converting from improper fractions to mixed numbers is a little more challenging. We will look
at these types of questions in our next two examples.
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Example 3
16
Write
as a mixed number.
5
Answer:
We have shown two methods to handle this. You choose the one you prefer.
Method 1 (More Efficient Method)
Method 2
Step 1: Divide the numerator by the
Express the number in the numerator as the
denominator.
sum of as many of the numbers in the
Step 2: Write down the whole number
denominator as possible plus what is left
answer.
over to reach the number in the numerator.
Step 3: Then write down any remainder
16 5  5  5  1

above the denominator.
5
5
Break the fraction up into multiple
When we divide 16 by 5, we get 3 with a
fractions, all with the same denominator.
remainder of 1. Write down the 3, and then
16 5 5 5 1
   
write down the remainder of 1 over the
5 5 5 5 5
denominator of 5.
When the numerator and denominator are
16
equal, the fraction is equal to 1. Make all
 16  5
5
of those necessary changes.
 3 with a remainder of 1
16
1
 111
1
5
5
3
Add
the
1s
together.
5
16
1
3
5
5
Example 4
Write the following improper fractions as mixed numbers.
11
31
7
(a)
(b)
(c)
4
8
6
Answers:
Method 1 (More Efficient Method)
(a) When we divide 11 by 4, we get 2 with a
remainder of 3. Write down the 2, and
then write down the remainder of 3 over
the denominator of 4.
11
 11  4
4
 2 with a remainder of 3
3
2
4
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(d)
68
10
Method 2
11 4  4  3

4
4
4 4 3
  
4 4 4
3
 11
4
3
2
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C. D. Pilmer
Method 1 (More Efficient Method)
When we divide 31 by 8, we get 3 with a
remainder of 7. Write down the 3, and
then write down the remainder of 7 over
the denominator of 8.
31
 31  8
8
 3 with a remainder of 7
7
3
8
Method 2
31 8  8  8  7

8
8
8 8 8 7
   
8 8 8 8
7
 111
8
7
3
8
(c)
When we divide 7 by 6, we get 1 with a
remainder of 1. Write down the 1, and
then write down the remainder of 1 over
the denominator of 6.
7
 76
6
 1 with a remainder of 1
1
1
6
7 6 1

6
6
6 1
 
6 6
1
 1
6
1
1
6
(d)
When we divide 68 by 10, we get 6 with
a remainder of 8. Write down the 6, and
then write down the remainder of 8 over
the denominator of 10.
68
 68  10
10
 6 with a remainder of 8
8
6
10
68 10  10  10  10  10  10  8

10
10
10 10 10 10 10 10 8
      
10 10 10 10 10 10 10
8
 111111
10
8
6
10
(b)
Example 5
Sort the numbers
13 1 5 23
, 3 , ,
from smallest to largest.
5
8 9 6
Answer:
It is generally easier to compare fractions that are greater than 1 when they are expressed as
mixed numbers. Therefore we will start by converting the two improper fractions to mixed
numbers.
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13
23
 13  5
 23  6
5
6
 2 with a remainder of 3
 3 with a remainder of 5
3
5
2
3
5
6
We can now re-write the original question using the mixed numbers we just worked out.
3 1 5 5
" Sort the numbers 2 , 3 , , 3 from smallest to largest."
5 8 9 6
Now it is easy to put the fractions in the proper order.
5
3 1 5
5 13 1 23
Proper Order:
or
, 2 , 3 ,3
,
, 3 ,
9
5 8 6
9 5
8 6
Example 6
7 11 15 4 1
3 6
19
Place the numbers 2 ,
and
by their appropriate arrow on the
,
,1 , ,2 ,
8 5 16 5 9
7 12
16
number line below.
0
1
2
3
Answer:
Start by converting the two improper fractions to mixed numbers.
11
19
 11  5
 19  16
5
16
 2 with a remainder of 1
 1 with a remainder of 3
1
3
2
1
5
16
We are now able to place all the numbers by the appropriate arrow on the number line.
1
9
0
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6
12
15
16
19
16
or
3
1
16
1
1
11
5
or
1
2
5
4
5
2
38
2
3
7
2
7
8
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C. D. Pilmer
Questions
1. Write the following mixed numbers as improper fractions. Show your work.
1
7
9
(a) 3
(b) 2
(c) 1
4
8
10
(d) 4
1
6
(e) 2
4
7
(f) 3
2
5
(g) 7
1
2
(h) 1
5
16
(1) 4
3
8
2. Write the following improper fractions as mixed numbers. Show your work.
6
19
11
(a)
(b)
(c)
5
8
9
(d)
19
6
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(e)
7
3
(f)
39
24
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(g)
27
10
(h)
28
5
(i)
19
4
3. Order the numbers from smallest to largest.
1
1 11 5 1
(a) 2 , 1 ,
______, ______, ______, ______, ______
,2 ,
16 8 12 6 32
(b) 1
7 9
7
3
1
,
,2 ,2 ,1
16 10 12 8 10
______, ______, ______, ______, ______
(c)
9
1 3
3 4
,2 ,
,1 ,
5 7 12 6 12
______, ______, ______, ______, ______
(d)
7
5 7
9 9
,2 , ,1 ,
8 8 9 10 4
______, ______, ______, ______, ______
6
1 10
5 1
,1 ,
(e) 1 , 2 ,
7 32 9
10 4
______, ______, ______, ______, ______
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7 1 17 13 1
(f) 1 ,
,
,
,
8 16 8 10 32
______, ______, ______, ______, ______
3 17 9
4
1
(g) 1 ,
, ,1 ,3
4 16 2 10 7
______, ______, ______, ______, ______
9 8 13 3 4 10
, ,
, ,1 ,
10 3 4 8 7 7
______, ______, ______, ______, ______, ______
5
1 32 1 19 1
,2 ,
,1 ,
,
4 18 10 3 9 15
______, ______, ______, ______, ______, ______
(h) 2
(i)
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4 3 5 1 1 11
9 10 7
1
4. Place the numbers 1 , , 2 , 1 ,
and 2
by their appropriate
,
,2 ,
,
8 3 8 6 10 6
10 4 12
16
arrow on the number line below.
0
1
2
3
7 8 12 1 8
3 25
6 7 29
4
and 2 by their
, ,
,2 ,
,
,
,1 , ,
16 7 4
8 16 32 8
10 4 32
5
appropriate arrow on the number line below.
5. Place the numbers 2
0
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2
42
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Area Models of Different Shapes and Sizes
Area models are not restricted to just squares. Below we have used a square, rectangle and a
3
circle to represent the fraction . Size and shape do not matter; as long as the region is divided
4
into 4 equal parts, and 3 of those four parts are shaded, then we have the area model representing
3
the fraction .
4
Example 1
What fraction is represented by the following area model?
Answer:
6
1
7
We have
(or 1) plus . That means we have the improper fraction , which can be
6
6
6
1
expressed as the mixed number 1 .
6
Example 2
What fraction is represented by the area model on the right?
Answer:
Sometimes an area model can appear incomplete. Such is the case with this area model. The
model has been into six parts but they are not of equal size; therefore this does not represent
3
the fraction .
6
When the model is properly divided into portions of the same size, we
3
can see that model represents the fraction .
8
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Example 3
Lei creates an area model for
3
2
. Thomas creates an area model for . Which fraction is
4
5
bigger? Explain.
Lei's Model:
Thomas' Model:
Answer:
3
2
is bigger than
4
5
This can be challenging to see because the area model of
2
is physically larger than the
5
3
,
4
three of its four sections are shaded. Since a greater portion of the first model is shaded, then
3
the fraction
is bigger.
4
other model, however only two of its five sections are shaded. With the area model for
Questions:
1. Do all four of these area models represent the same fraction? Explain.
2. Create an area model for each of the fractions below using the supplied geometric figures.
2
3
(a)
(b)
7
5
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(c)
2
3
(d)
11
12
(e)
1
8
(f)
6
6
(g)
3
1
or 1
2
2
(h)
7
3
or 1
4
4
3. (a) Which of the fractions in question 2 is equal to 1?
Answer: _____
(b) Which of the fractions in question 2 is slightly less than 1?
Answer: _____
(c) Which of the fractions in question 2 is closest to 0?
Answer: _____
(d) Which is the largest fraction in question 2?
Answer: _____
4. Determine the fraction that is represented by each of these area models.
(a)
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C. D. Pilmer
(c)
(d)
(e)
(f)
(g)
(h)
5. Maurita creates an area model for
1
1
. Tanya creates an area model for . Which fraction is
2
3
bigger? Explain.
Maurita's Model:
Tanya's Model:
Important Note:
With the exception of this section, we will only focus on area models that are squares. We only
wanted to show you other area models in case you see them in other math resources.
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Equivalent Fractions, Part 1
Explanation:
Consider the following area models.
1
2
2
4
3
6
9
18
In all four cases, the same part of the whole has been shaded. When this happens you can say
that the fractions are equivalent fractions.
1 2 3 9
  
2 4 6 18
Equivalent fractions initially appear different but when you look more closely, you discover that
they are equal. For example:
 If you cut a pizza into 6 pieces of the same size and give yourself 3 pieces, you end up
with half of the pizza.
 If you took that same pizza and cut it into 8 pieces of the same size and gave yourself 4
pieces, you still end up with half of the pizza.
3
4
 The fractions
and
are equivalent.
6
8
Questions:
1. The area model for
2
is shown on the right.
3
(a) Circle the area models below that show a fraction equivalent to
2
?
3
(b) For each of the area models that you circled in question 1(a), state the corresponding
fraction.
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2. The area model for
3
is shown below.
4
Create three area models that are equivalent to the area model for
3
. Also state the
4
corresponding fraction for each of your new models.
3. You are going to use an interactive activity found on the internet. Use the link below or
Google search NCTM Illuminations Equivalent Fractions.
http://illuminations.nctm.org/activitydetail.aspx?id=80
In this game, you are supplied with an area model for a particular fraction and its position on
the number line. Your mission is two create two equivalent fractions to the one supplied.
Play four successful rounds of the game and record your results for in the table below.
Equivalent
Fractions
Equivalent
Fractions
First Successful
Round
Second Successful
Round
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Round
Fourth Successful
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Equivalent Fractions Part 2
Creating Equivalent Fractions
To create an equivalent fraction, multiply or divide the numerator and denominator of a fraction
by the same number.
Example 1
Create three equivalent fractions for
Answer:
6  3 18

10  3 30
6
.
10
62 3

10  2 5
6  5 30

10  5 50
Example 2
3
Create three equivalent fractions for 2 .
9
Answer:
3 2
6
3 5
15
2
2
2
2
9 2
18
95
45
2
33
1
2
93
3
Determining if Fractions are Equivalent
There are two methods that can be used to figure out if fractions are equivalent.
Method 1
This method involves changing fractions to their simplest form and comparing them. A fraction
is written in its simplest form when the numerator and denominator have no common factors
other than one.
3
 The fraction
is written in its simplest form because the only number that divides into
4
both 3 and 4 without leaving a remainder is 1 (i.e. 1 is the only common factor).
6
 The fraction
is not written in its simplest form because the number 3 divides into the
15
numerator, 6, and the denominator, 15 without leaving a remainder (This means 3 is a
common factor of 6 and 15). When the fraction is simplified, the equivalent fraction is
2
. This is accomplished by dividing the numerator and denominator by 3.
5
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Example 3
Determine if these fractions are equivalent.
4
10
21
12
(a) and
(b)
and
7
35
35
20
(c)
8
16
and
3
6
Answers:
4
10
(a) and
7
35
4
is in its simplest form because the only number that divides into 4 and 7 without
7
leaving a remainder is 1.
10
needs to be put in its simplest form. The number 5 divides into both the numerator
35
and denominator without leaving a remainder (This means 5 is a common factor of 10
and 35).
10 10  5

35 35  5
2

7
Since
4
2
4
10
is not equal to
, then and
are not equivalent fractions.
7
35
7
7
21
12
and
35
20
21
needs to be put in its simplest form. The number 7 is a common factor of 21 and 35.
35
21 21  7

35 35  7
3

5
(b)
12
needs to be put in its simplest form. The number 4 is a common factor of 12 and 20.
20
12 12  4

20 20  4
3

5
Since
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3 3
21
12
 , then the fractions
and
are equivalent.
5 5
35
20
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(c)
8
16
and
3
6
8
is in its simplest form because the only number that divides into 8 and 3 without
3
leaving a remainder is 1.
16
needs to be put in its simplest form. The number 2 divides into both the numerator
6
and denominator without leaving a remainder (This means 2 is a common factor of 16
and 6).
16 16  2

6
62
8

3
Since
8 8
8
16
are equivalent.
 , then the fractions and
3 3
3
6
Method 2
Any two fractions are equivalent if their cross products are equal. Most instructors are not keen
about this technique because:
1. It appears to be a trick that does not rely on truly understanding fractions.
2. Learners are often required to multiply large numbers. Since calculators are not
permitted in this unit, learners can make careless mistakes when multiplying these large
numbers.
For these reasons, most instructors will recommend that you use Method 1.
Example 4
Determine if these fractions are equivalent.
4
10
21
12
(a) and
(b)
and
7
35
35
20
(c)
8
16
and
3
6
Answers:
These are the same fractions we encountered in Example 3.
4
10
(a) and
7
35
4
10
7
35
4  35  140
7 10  70
4
10
Since the cross products are not equal, then and
are not equivalent fractions.
7
35
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21
12
and
35
20
21
35
(b)
12
20
21 20  420
35 12  420
Since the cross products are equal (i.e. both equal 420), then
21
12
are
and
35
20
equivalent fractions.
8
16
and
3
6
8
3
(c)
16
6
8  6  48
3 16  48
Since the cross products are equal (i.e. both equal 48), then
8
16
are
and
3
6
equivalent fractions.
Example 5
Change the following fractions to their simplest forms.
12
20
24
(a)
(b)
(c) 1
18
15
32
(d) 2
15
24
Answers:
In each case we need to find the largest number that divides evenly into the numerator and
denominator of the fraction. With the last two questions, we are dealing with mixed
numbers. With these questions, only focus on the fractional part of the mixed number.
(a)
12 12  6 2


18 18  6 3
(c) 1
24
24  8
3
1
1
32
32  8
4
(b)
20 20  5 4


15 15  5 3
(d) 2
15
15  3
5
2
2
24
24  3
8
Example 6
A work shift at a factory is 8 hours. What fraction of an employee’s work shift is represented by
6 hours? Express the fraction in its simplest form.
Answer:
6 62 3


8 82 4
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Questions
1. Create at least three equivalent fractions for each of the fractions supplied below.
5
5
(a)
(b)
7
6
(c)
2
10
(d)
4
3
(e)
12
8
(f)
4
12
(g)
5
4
(h)
30
12
(i) 1
4
10
(j) 3
5
6
(l) 1
(k) 2
3
8
12
16
2. Change each of these fractions to their simplest form.
6
8
(a)
(b)
12
10
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(c)
9
21
(d)
30
50
(e)
12
18
(f)
35
40
(g)
28
20
(h)
35
63
(i)
3
30
(j)
36
24
(k)
21
49
(l)
30
27
(m)
49
14
(n)
18
15
(o)
30
12
(p) 1
4
8
(q) 2
9
12
(r) 4
6
18
(s) 2
15
25
(t) 3
28
35
30
100
(v) 3
14
16
(u) 1
3. Determine whether the following pairs of fractions are equivalent or not.
(a)
6
9
and
12
18
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(b)
54
1
3
and
5
15
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C. D. Pilmer
(c)
2
6
and
14
36
(d)
6
9
and
14
21
(e)
6
24
and
5
20
(f)
6
8
and
27
36
(g)
12
10
and
21
16
(h)
20
5
and
8
3
4. With each of these questions, four fractions have been supplied. Three of the fractions are
equivalent fractions. Circle the one fraction that is not equivalent to the other three.
(a)
6
8
1
5
(c)
5
6
15
18
(e) 2
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8
12
3
4
4
5
2
10
15
9
12
10
12
2
2
3
2
5
6
(b)
6
12
(d)
10
7
(f) 1
55
12
16
1
2
4
8
5
3
1
2
3
25
15
3
4
1
10
6
8
12
1
6
8
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C. D. Pilmer
5. Place the appropriate symbol ( <, >, or = ) between each pair of fractions. In some cases you
may wish to simplify one or both of the fractions.
Note:
< means less than
> means greater than
= means equal to
(a)
2
5
4
10
(b)
2
6
4
5
(c)
7
6
6
7
(d)
2
8
1
4
(e)
3
4
15
20
(f)
3
8
8
9
(g)
1
5
1
6
(h)
2
6
7
21
(i)
5
6
9
8
(j)
4
10
6
15
(k)
8
6
12
9
(l) 2
7
9
2
(n) 3
9
12
3
(m) 1
6
9
1
8
20
6
11
16
20
6. Insert the numbers from the table below in the appropriate blanks.
Tom and Jorell purchased a pizza that had been cut into _____ pieces of the same size. Tom
ate _____ pieces. That meant he ate _____ of the pizza, which when expressed in its
simplest form is _____ of the pizza.
1
4
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8
2
2
8
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7. Insert the numbers from the table below in the appropriate blanks.
Nita and Meera purchased a _____ acre lot. Since Nita contributed more money to the
purchase, she took ownership of _____ acres. This left ____ acres for Meera. That means
Nita obtained _____ of the original lot, which when expressed in simplest form is _____ of
the lot. Meera obtained _____ of the original lot, which when expressed in simplest form is
_____ of the lot.
6
16
10
10
16
3
8
6
16
5
8
8. A work shift at a fast food restaurant is 8 hours. What fraction of an employee’s work shift is
represented by 4 hours? Express the fraction in its simplest form.
Answer:
____
9. There are 12 inches in a foot. What fraction of a foot is represented by 8 inches? Express
the fraction in its simplest form.
Answer:
____
10. There are 100 centimetres in a metre. What fraction of a metre is 20 cm? Express the
fraction in its simplest form.
Answer:
____
11. Concrete is made by mixing 1 part cement to 2 parts water to 9 parts gravel. For all of the
questions below, express the fractions in their simplest form.
(a) What fraction of the mixture is cement?
Answer:
____
(b) What fraction of the mixture is water?
Answer:
____
(c) What fraction of the mixture is gravel?
Answer:
____
(d) What fraction of the mixture is not gravel?
Answer:
____
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Equivalent Fractions and the Number Line
In this section we will continue to compare fractions but in some cases we will be dealing with
equivalent fractions.
Example
9 18 8 7 4 1
4 6
3
and 1 by their appropriate arrow on
,
,
, ,1 ,
,2 ,
10 10 16 6 5 32 7 12
18
the number line below.
Place the numbers 2
0
1
2
3
Answer:
You may have noticed that we were given more numbers than there are arrows on the
number line. This has to do with the fact that some of the fractions are equivalent fractions
and therefore share the same arrow.
Start by changing the appropriate fractions to their simplest form.
18 18  2
8
8 8
6
66



10 10  2
16 16  8
12 12  6
9
1
1



5
2
2
1
3
33
1
18 18  3
1
1
6
Now convert the improper fractions to mixed numbers.
18 9
4
7
1
 1
1
10 5
5
6
6
Now place the number (in their original forms) on the number line.
1
32
0
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3
18
and
7
6
8
16
and
6
12
4
5
and
18
10
1
1
1
2
2
58
4
7
2
9
10
3
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C. D. Pilmer
Questions
2 14
3
9 17 10 16 5
3
1. Place the numbers 1 , 2 ,
and by their appropriate arrow
,
,
,2 ,
,
8
21 100 15 8
15 5 4
5
on the number line below.
0
1
2
3
8 19 3 2 1
7 11 3 97
6
4
by their
,
, ,1 , ,2 ,
, ,
, 2 and 3
4 6 8 4 5 16 4 2 100 8
24
appropriate arrow on the number line below.
2. Place the numbers
0
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1
2
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Measuring and Fractions
In Imperial measure, length can be measured in inches, feet, yards, and miles. In this section
we will focus on measuring in inches, the smallest measure of the four measures mentioned
above. In order to work in inches, one must be familiar with equivalent fractions.
Examples:
2
4
1
inches  2 inches
16
4
6
3
1 inches  1 inches
8
4
3
8
1
inches  3 inches
16
2
10
5
inches  inches
16
8
The diagrams below show how the scale on an Imperial ruler or tape measurer is divided. In the
first diagram, the ruler is divided into inches.
1
2
3
This scale is then divided into halves.
1
2
3
It is then divided into quarters.
1
2
3
2
3
It is then divided into eighths.
1
Then it is finally divided in sixteenths.
1
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3
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Example 1
What is the length of the line segment?
1
2
3
Answer:
13
inches long.
16
How could we have arrived at this answer? Here are two different ways suggested by
learners. Both of these techniques are perfectly acceptable.
 Tom’s Response: “I know that every tick mark represents one sixteenth of an inch. I just
counted the number of tick marks to the right of the tick mark for 2 inches. I counted 13
13
tick marks so I knew that the segment went
beyond 2 inches. That means that the
16
13
line segment is 2
inches long.”
16
 Maurita’s Response: “Every tick mark is one sixteenth of an inch. This line segment is
3
16
3
short of 3 inches. If I take 3, which can be expressed as 2  , and remove
,I
16
16
16
13
am left with the answer, 2 .”
16
The line segment is 2
Example 2
What is the length of the line segment?
1
2
3
Answer:
3
inches long.
8
Below we have provided to acceptable techniques for arriving at this answer.
 Charlene’s Response: “I went six tick marks or 6 sixteenths to the right of 1. That means
6
that the line segment is 1
inches long. This fraction can be reduced because the
16
numerator and denominator are both divisible by 2. When I reduce the fraction, I get my
3
answer 1 .
8
The line segment is 1
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1
1
inches and 1 inches.
4
2
2
4
Another way to say this is the measure is half way between 1 inches and 1 inches.
8
8
3
Therefore the line segment must be 1 inches long.
8

Nashi’s Response: “The measure is half way between 1
In the metric system, length is most often measured in millimetres, centimetres, metres, and
kilometres. In this section we will focus on measuring in centimetres. When you pick up a ruler,
the scale is typically in centimetres.
When the ruler is in centimetres (cm), every tick mark represents one tenth of a centimetre.
1
2
3
4
5
6
7
cm
Example 3
What is the length of the line segment?
1
2
3
4
5
6
7
cm
Answer:
3
5
cm (or 5.3 cm)
10
Example 4
What is the length of the line segment?
1
2
3
4
5
6
7
cm
Answer:
8
4
3
cm (or 3 cm) (or 3.8 cm)
10
5
Most rulers have centimetres along one edge and inches along the other.
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Questions
1. Measure the length of each of the following line segments in both centimetres and inches.
(a)
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
Inches:
Centimetres:
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
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Estimating the Addition and Subtraction of Fractions
One of the biggest problems many learners encounter learning how to add and subtract fractions
is that they get wrapped up in the rules and don’t take the time to see if their final answer is
reasonable. That’s why you’re going to spend some time working on your estimation skills
rather than focusing on the rules. Estimation is the process of finding a reasonable
approximation for the final answer, without doing the formal calculation. In previous sections,
1
you spent a significant amount of time comparing fractions to familiar numbers like 0, , and 1.
2
This skill can be very useful when attempting to estimate the sum or difference of two fractions.
Example 1
Estimate each of these sums.
7 14
1
2
(a) 
(b)

6 15
30 45
(d)
10 8

9 17
Answers:
7 14
(a) 
6 15
1
2

30 45
(b)
(e) 3
(c)
7
3
2
15
50
1 5

16 9
(f) 1
5
8
2
12
15
7
14
is close to 1. The proper fraction
is also
6
15
7 14
close to 1. To estimate the  , think 1 + 1. This means that the
6 15
sum will be close to 2.
The improper fraction
Both of these fractions are very small numbers that are close to 0. To
estimate
1
2

, think 0 + 0. That means that the sum will be close
30 45
to 0.
(c)
1 5

16 9
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1
5
1
is close to 0. The fraction is close to . That
16
9
2
means that when you add these two fractions, the answer should be
1
close to .
2
The fraction
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C. D. Pilmer
(d)
10 8

9 17
7
1
7
1
is close to , then the mixed number 3
is close to 3 .
15
2
15
2
3
3
Since
is close to 0, then the mixed number 2
is close to 2.
50
50
1
1
Since 3  2  5 , then you can say that the sum of the two original
2
2
1
mixed numbers is close to 5 .
2
7
3
2
15
50
Since
5
8
2
12
15
The mixed number 1
(e) 3
(f) 1
10
8
is close to 1. The proper fraction
is
9
17
1
1
1
close to . Since 1   1 , then you can say that the sum of the
2
2
2
1
two original fractions is close to 1 .
2
The improper fraction
5
1
8
is close to 1 . The mixed number 2
is
12
2
15
1
1
1
close to 2 . Since 1  2  4 , then you can say that the sum of the
2
2
2
two original mixed numbers is close to 4.
Example 2
Estimate each of these differences.
12 2
13 5


(a)
(b)
13 37
12 9
Answers:
12 2

(a)
13 37
(b)
13 5

12 9
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(c) 3
1
1
1
15
20
(d) 4
2
11
1
31 20
12
2
is close to 1. The fraction
is close to 0. To
13
37
12 2
estimate
, think 1 - 0. Since 1  0  1 , then you know that the

13 37
difference between the two original fractions is close to 1.
The fraction
13
5
1
is close to 1. The fraction is close to . To
12
9
2
1
13 5
1 1
 , think 1  . Since 1   , then you know that
estimate
12 9
2 2
2
1
the difference between the two original fractions is close to .
2
The fraction
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C. D. Pilmer
(c) 3
1
1
1
15
20
The mixed number 3
1
1
is close to 3. The mixed number 1
is
15
20
close to 1. Since 3  1  2 , then you know that the difference between
these two mixed fractions is close to 2.
(d) 4
2
11
1
31 20
The mixed number 4
2
11
is close to 4. The mixed number 1
is
20
31
1
1
1
close to 1 . Since 4  1  2 , then you know that the difference
2
2
2
1
between these two original mixed fractions is close to 2 .
2
Questions
1. For each sum, check the column that is the best estimate.
Close to 0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
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©2012
Close to
1
2
Close to 1
Close of 1
1
2
Close to 2
7 15

8 14
1 4

20 9
1
1

12 15
21 7

20 13
9
5

17 11
8 11

9 10
2
1

31 25
12 1

13 32
8
9

17 10
8
1

15 40
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2. Estimate each of these sums or differences.
1
12
5
1
(a) 1  4
(b) 3  4
20
13
12
32
5
7
(c) 2  3
9
8
(e)
13 7

12 13
(d)
21 2

22 53
(f) 6
14
1
2
15
20
(g) 8
8
6
4
17
14
(h) 8
7
2
3
15
45
(i) 9
3
7
6
50
13
(j) 7
14
10
4
15
21
3. You are asked to check a fellow student’s work. This student was asked to add two fractions
together. He decided that all he had to do was add the numbers in the numerator, and add the
numbers in the denominator. He wrote down the following.
6 12 18


13 25 38
Demonstrate using your estimation skills, why this student’s answer has to be wrong.
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4. Put the numbers in the blanks so that the statement makes sense. Please note that there are
four numbers and three blanks therefore one number should not be used.
(a) Tanya had a piece of lumber that was _____ inches thick. She runs it through a planer
that removes a thin strip of wood from the thickness of the lumber. She set the planer to
remove _____ inches. She estimates that the lumber coming out of the planer will be
_____ inches thick.
1
32
1
1
2
2
1
8
1
5
8
(b) Manish initially measured out _____ cups of flour, which he estimated to
be pretty close to 2
1
cups. He then added another _____ cups. That
2
means that he measured out a little more than _____ cups of flour in total.
1
1
2
2
2
3
3
1
2
4
(c) Alex and Jacob ordered large pizzas from the same pizzeria but for different parties.
Alex had a lot more pizza left over from this party. Alex had _____ pizzas left. Jacob
had _____ pizzas left. Together they approximately had _____ pizzas.
2
1
2
6
1
2
3
4
3
7
8
(d) Colin puts _____ gallons of gasoline in his all-terrain vehicle, which was initially empty.
He then takes a trip in the woods and estimates that he used _____ gallons of gas, the
bulk of the gas that he initially added. Therefore approximately only _____ gallon
remains in the tank after his trip.
1
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2
1
2
1
2
3
1
8
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Adding Fractions, Part 1
Suppose you were given the following problem.
You walked 5 feet, and then walked 24 inches. How far did you walk?
The answer is not 29. The units of measure are not the same; one is measured in feet and the
other in inches. You need to change them to the same units of measure first. You have two
choices.
First Choice: Change Feet to Inches
If 1 foot equals 12 inches, then 5 feet equals 60 inches (note: 5 12  60 ).
Total distance  60 inches  24 inches
 84 inches
Second Choice: Change Inches to Feet
If 12 inches equals 1 foot, then 24 inches equals 2 feet (note:
Total Distance  5 feet  2 feet
 7 feet
24
 2 ).
12
So how does this apply to the addition and subtraction of fractions? Consider the next two
examples.
Example 1
3 2

7 7
5

7
Example 2
2 3

5 4
8 15


20 20
23

20
In this example, the two fractions have a common denominator. In this
case, the denominator is 7.
You have "3 sevenths" plus "2 sevenths." The answer will be "5
sevenths." It’s as if you were dealing with the same unit of measure;
sevenths. That’s why it’s easy to add these two fractions.
In this example, the two fractions have different denominators. This
makes the question more challenging because at this point we don’t know
how to add fifths and fourths. It’s like trying to add different units of
measure.
You need to create a common denominator. It’s similar to creating the
same units of measure. In this case, the common denominator is 20. The
2
8
3
15
fraction
is equivalent to
. The fraction
is equivalent to
. We
5
20
4
20
did not show you how we selected and created the common denominator;
we will show you later.
Now you have "8 twentieths" plus "15 twentieths." The answer will be
"23 twentieths."
The Big Question:
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How do you figure out a common denominator for two fractions with
different denominators?
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Example 3
Evaluate
1 1

3 2
Method 1: Using Area Models
1
1
and .
3
2
Notice that the first model is divided into columns, and
the second model is divided into rows. This is
deliberate. We can’t add these fractions in their present
form.
You start by drawing the area models for
+
+
Answer:
5
6
In the next step, we put two rows on the first model,
and put three columns on the second model. By doing
this, both models are now dealing with sixths, rather
2
than thirds and halves. The first model is , which is
6
1
3
equivalent to . The second model is
, which is
6
3
1
equivalent to .
2
You have "2 sixths" plus "3 sixths." The answer is "5
sixths."
Method 2: The Mathematical Procedure
The denominators (3 and 2) are different. List the multiples of 3 and 2
1 1

separately.
3 2
1 2 1 3
Multiples of Three: 3, 6, 9, 12, 15, …


3 2 2 3
Multiples of Two: 2, 4, 6, 8, 10, 12, 14, …
2 3
 
Notice that the least common multiple of 3 and 2 is 6. This means that the
6 6
common denominator will be 6. Now you write each fraction as an
5
equivalent fraction whose denominator is 6. We have to multiply the

6
1
numerator and denominator of
by 2 to create an equivalent fraction with
3
a denominator of 6. Similarly we have to multiply the numerator and
1
denominator of
by 3 to create an equivalent fraction with a denominator
2
of 6.
You have "2 sixths" plus "3 sixths." The answer is "5 sixths."
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Example 4
Evaluate
1 2

6 3
Method 1: Using Area Models
1
2
and .
6
3
Notice that the first model is divided into columns, and
the second model is divided into rows. This is
deliberate. We can’t add these fractions in their present
form.
You start by drawing the area models for
+
In the next step, we put three rows on the first model,
and put six columns on the second model. By doing
this, both models are now dealing with eighteenths,
3
rather than sixths and thirds. The first model is
,
18
1
12
which is equivalent to . The second model is
,
6
18
2
which is equivalent to .
3
+
Answer:
5
6
You have "3 eighteenths" plus "12 eighteenths." The
15
answer is "15 eighteenths." The answer
can be
18
5
changed to its simplest form, .
6
Method 2: The Mathematical Procedure
The denominators (6 and 3) are different. List the multiples of 6 and 3
1 2

separately.
6 3
1 2 2
Multiples of Six: 6, 12, 18, 24, 30, …
 
6 3 2
Multiples of Three: 3, 6, 9, 12, 15, 18, …
1 4
 
Notice that the least common multiple of 6 and 3 is 6. This means that the
6 6
common denominator will be 6. Now you write only the second fraction as
5
an equivalent fraction whose denominator is 6. We have to multiply the

6
2
numerator and denominator of
by 2 to make this happen.
3
You now have "1 sixths" plus "4 sixths." The answer is "5 sixths."
Important Note: The area model technique that was shown for this example didn’t create the
least common denominator, although it ultimately gave the correct answer.
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Questions
1. Solve each of the following. Note that all of these sums already involve common
denominators. At the end of your answer, you may have to put the fraction in its simplest
form and/or change it to a mixed number.
3
4
2 6
(a)
(b)


 
10 10
11 11
(c)
5
6


14 14
(d)
6 1
 
7 7
(e)
5
3


16 16
(f)
3 1
 
6 6
(g)
7
8


20 20
(h)
6 4
 
9 9
(i)
8 9
 
11 11
(j)
5 7
 
8 8
(k)
23 5


10 10
(l)
7 9
 
6 6
2. Two questions involving the addition of fractions have been solved using area models. For
each of the partial solutions,
- state the original question,
- state the second step where fractions with a common denominator have been created,
and
- state the final answer.
(a)
Original Question:
+
Second Step:
+
Final Answer:
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(b)
Original Question:
+
Second Step:
+
Final Answer:
3. Use area models to evaluate
1 2
 .
4 3
+
+
Final Answer:
4. (a) List the multiples of 6.
_____, _____, _____, _____, _____, . . .
(b) List the multiples of 4.
_____, _____, _____, _____, _____, . . .
(c) What is the least common multiple of 4 and 6?
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5. For each question, two numbers have been supplied. In the first column, list at least the first
five multiples of the first number. In the second column, list the first five multiples of the
second number. In the last column, state the least common multiple (LCM).
Multiples of the First Number
Multiples of the Second Number
LCM
(a) 5, 2
(b) 4, 10
(c) 2, 6
(d) 5, 3
(e) 6, 8
6. Fill in the missing pieces of the solution and explanation.
(a)
The multiples of 5 are 5, 10, 15, 20, 25, 30, …
3 1

5 4
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
3  4 1 5


5 4 45
The LCM of 5 and 4 is ____.


20

20
20
We have to multiply the numerator and denominator of
3
by 4 to make our common denominator.
5
We have to multiply the numerator and denominator of
1
by ____ to make our common denominator.
4
(b)
5 1

6 4
5
1


6
4
10 3


12 12

12
1
1
12
The multiples of 6 are ____, ____, ____, ____, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The LCM of 6 and 4 is ____.
We have to multiply the numerator and denominator of
5
by ____ to make our common denominator.
6
We have to multiply the numerator and denominator of
1
by ____ to make our common denominator.
4
We had to change the improper fraction ____ to the
1
mixed number 1
.
12
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(c)
2 3

3 5
2

3


The multiples of 3 are ____, ____, ____, ____, ____, …

3
5

15 15
The LCM of 3 and 5 is ____.
We have to multiply the numerator and denominator of
2
by ____ to make our common denominator.
3
15
1
The multiples of 5 are 5, 10, 15, 20, 25, 30, …
We have to multiply the numerator and denominator of
3
by ____ to make our common denominator.
5
15
We had to change the improper fraction ____ to the
mixed number ____ .
(d)
3 5

4 8
3

4
5
 
8 8

8
1
8
The multiples of 4 are 4, 8, 12, 16, 20, 24, …

5
8
The multiples of 8 are ____, ____, ____, ____, …
The LCM of 4 and 8 is ____.
We have to multiply the numerator and denominator of
3
by ____ to make our common denominator.
4
5
because it already has
8
the denominator that we need.
We don’t have to change the
We had to change the improper fraction ____ to the
mixed number ____ .
7. Figure out each of these sums. Show all your work.
1 1
1 2
(a)  
(b)  
2 5
4 5
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(c)
3 1
 
5 6
(d)
2 3
 
3 4
(e)
4 3
 
5 4
(f)
2 3
 
3 5
(g)
1 3
 
6 4
(h)
3 1
 
8 6
(i)
4 5
 
9 6
(j)
1 5


6 12
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(k)
2 1


3 12
(l)
4 8


5 15
(m)
7 3
 
8 4
(n)
1 3 5
  
2 4 8
8. A freight truck is carrying computer components and office furniture. There is
computer components and
1
ton of
2
5
ton of office furniture. What is the total weight of the load?
8
1
3
inch plywood onto
inch tongue and groove
8
4
boards. How thick is the flooring at this stage?
9. For the flooring in your kitchen, you nail
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1
3
1
cup of sour cream with
cup of salsa sauce and
cup of cooked ground
3
4
2
hamburger meat. How many cups of the mixture do you now have?
10. You mix
11. Angela is making U-shaped brackets by bending a straight piece of metal in two places.
How long is the piece of metal given the following measurements? All the measurements are
in inches.
7
8
3
4
7
8
12. A sales clerk at a candy shop mixes
of chocolate-covered raisins and
3
3
pound of chocolate-covered almonds with
pound
8
4
1
pound of chocolate-covered cherries. How much does
2
the mixture weigh?
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2 4
 . Her answer and explanation are shown below. Her
5 7
answer is incorrect. Explain to her why her answer is correct and how she should have done
the question.
13. Janice was asked to work out
Janice’s Answer and Explanation:
To add fractions, you just have to add the numbers in the
2 4 6
 
numerators of the original fractions to get the number in the
5 7 12
numerator of the new fraction. You then add the numbers in
1

the denominators of the original fractions to get the number
2
in the denominator of the new fraction. I also had to change
my answer to its simplest form by dividing the numerator and
denominator by 6.
Your Explanation:
14. Three students were given the same question but they all arrived at different answers. Which
student did the question correctly? Circle the first mistake made by each of the other two
students.
Student #1
Student #2
Student #3
4 1 4  3 1 5
4 1 4  3 1 5
4 1 4  3 1 5
 

 

 

5 3 53 35
5 3 5 3 3 5
5 3 5 3 3 5
7 6
12 5
12 5
 




8 8
15 15
15 15
13
17
17



8
15
30
5
2
1
1
8
15
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Open-ended Questions
There is more than one correct answer for each of the remaining questions. For this reason,
you will not find answers for these questions in the answer key at the back of this resource.
15. Two fractions add to
1
. What might the fractions be?
2
16. Create a word problem using
1 1
 .
2 3
17. Three fractions add to 1. What might the fractions be?
? 1
1
 must be less than , then what might the missing number be? Verify your answer
2
11 9
by working out the sum.
18. If
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Adding Fractions, Part 2
In the previous section we learned how to add fractions but we limited ourselves to proper
fractions. In this section we will focus on adding mixed numbers. As before, we still need
common denominators.
When adding mixed numbers, it is best to add the whole numbers together and add the fractions
together.
Example 1
Complete the addition.
1
2
1
5
(a) 4  3
(b) 6  2
5
5
8
8
5
8
(c) 12  2
9
9
(d) 1
7
11
3
16
16
Answers:
With each of the questions, the mixed numbers already have common denominators. For
example in question (a), the mixed numbers have a common denominator of 5. Already
having a common denominator makes the question much easier on us.
1
2
(a) 4  3
5
5
1
2
 Expressed mixed numbers as sum of a whole number and fraction.
4   3
5
5
1 2
 4  3      Grouped whole numbers together. Grouped fractions together
5 5
3
 Completed the addition of the grouped numbers.
7
5
3
 Final Answer
7
5
1
5
(b) 6  2
8
8
1
5
6 2
8
8
1
5
 6  2     
8 8
6
8
8
62
8
82
3
8
4
3
8
4
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 Expressed mixed numbers as sum of a whole number and fraction.
 Grouped whole numbers together. Grouped fractions together
 Completed the addition of the grouped numbers.
 Simplified fraction by dividing numerator and denominator by 2.
 Final Answer
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5
8
(c) 12  2
9
9
5
8
12   2 
9
9
5 8
12  2     
9 9
13
14 
9
4
 Changed the improper fraction to a mixed number.
14  1
9
4
 Final Answer
15
9
7
11
3
16
16
7
11
1  3 
16
16
7 11
1  3    
 16 16 
18
4
16
18  2
4
 Simplified fraction by dividing numerator and denominator by 2.
16  2
9
4
8
1
4 1
 Changed the improper fraction to a mixed number.
8
1
5
 Final Answer
8
(d) 1
With this last example, we simplified the fraction (Steps 4 and 5) and then changed the
improper fraction to a mixed number (Step 6). We could have done in this reverse and it
would not have changed our final answer.
Example 2
Complete the addition.
1
1
3
1
(a) 5  4
(b) 6  1
3
2
4 12
2
4
(c) 2  8
3
5
5
7
(d) 1  2
6
9
Answers:
With each of the questions, the mixed numbers do not have common denominators. We have
to create common denominators by looking at least common multiples.
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1
1
(a) 5  4
3
2
1
1
5  4
3
2
1 1
 5  4     
3 2
1 2 1  3 

 5  4   

 3 2 2  3 
2 3
 5  4     
6 6
5
9
6
5
9
6
3
1
(b) 6  1
4 12
3 1
 6  1    
 4 12 
3 3 1 
 
 6  1  
 4  3 12 
9 1
 6  1    
 12 12 
10
7
12
10  2
7
12  2
5
7
6
 The LCM of 2 and 3 is 6.
 The LCM of 4 and 12 is 12.
 Simplify fraction.
2
4
(c) 2  8
3
5
 2  8  
2 4
 
3 5
25 43 

 2  8  

 3 5 5 3 
10 12
 2  8    
 15 15 
22
10 
15
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 The LCM of 3 and 5 is 15.
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10  1
11
7
15
 Change from an improper fraction to a mixed number.
7
15
5
7
(d) 1  2
6
9
1  2   
5 7
 
6 9
5 3 7  2 

1  2   

 63 9 2 
15 14
1  2     
 18 18 
29
3
18
11
3 1
18
11
4
18
 The LCM of 6 and 9 is 18.
 Change from an improper fraction to a mixed number.
Questions
1. Add the following numbers.
3
2
(a) 1  2
7
7
4
2
(c) 6  5
9
9
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1
(b) 4  6
5
5
(d) 2
84
8
4
7
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8
2
(e) 4  3
9
9
6
3
(f) 6  2
7
7
7 3
(g) 3  1
8 8
(h) 1
5
1
(i) 3  2
6
6
(j)
2. Add the following numbers.
1
2
(a) 4  8
3
5
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9
7
5
10
10
3
1
2
8
8
3
1
(b) 1  2
7
2
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2
1
(c) 2  5
3
6
(d) 3
1
3
(e) 3  5
6
8
4
3
(f) 6  2
5
4
2 3
(g) 2  1
3 4
1
5
(h) 3  4
3
6
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1
1
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5
1
(i) 4  6
6
4
7
3
(j) 1  2
8
4
7 7

15 10
4
5
(l) 3  7
5
6
(k) 2
3. On the first month of his exercise and healthy eating program, Jim lost 2
second month he lost 3
1
pounds. On the
4
1
pounds. How much weight did he lose in total over those two
2
months?
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Subtracting Fractions, Part 1
As with the addition of fractions, we need a common denominator to subtract fractions.
Example 1
5 2

7 7
3

7
In this example, the two fractions have a common
denominator. In this case, the denominator is 7.
You have "5 sevenths" minus "2 sevenths." The answer will
be "3 sevenths."
If the original fractions don’t have a common denominator, you have to figure out an equivalent
fraction for each one such that a common denominator is created.
Example 2
1 1

2 3
1 3 1 2


2  3 3 2
3 2
 
6 6
1

6
The denominators (2 and 3) are different. List the multiples of 2
and 3 separately.
Multiples of Two: 2, 4, 6, 8, 10, 12, 14, …
Multiples of Three: 3, 6, 9, 12, 15, …
Notice that the least common multiple of 2 and 3 is 6. This
means that the common denominator will be 6.
Now you write each fraction as an equivalent fraction whose
denominator is 6.
You have "3 sixths" minus "2 sixths." The answer is "1 sixth."
Example 3
2 1

3 6
2 2 1


3 2 6
4 1
 
6 6
3

6
33

63
1

2
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The denominators (3 and 6) are different. List the multiples of 3
and 6 separately.
Multiples of Three: 3, 6, 9, 12, 15, 18, …
Multiples of Six: 6, 12, 18, 24, 30, …
Notice that the least common multiple of 3 and 6 is 6. This
means that the common denominator will be 6.
Now you write each fraction as an equivalent fraction whose
denominator is 6.
You have "4 sixths" minus "1 sixth." The answer is "3 sixths"
which can be simplified to "1 half."
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Questions
1. Solve each of the following. Note that all of these differences already involve common
denominators. At the end of your answer, you may have to put the fraction in its simplest
form.
7 2
5 4
(a)  
(b)  
9 9
6 6
(c)
7 3
 
8 8
(d)
7
3
 
12 12
(e)
10 1
 
12 12
(f)
13 3


25 25
(g)
7
5
 =
16 16
(h)
19 3


20 20
2. Fill in the missing numbers.
(a)
6 3

7 4
6  4 3 7


7 4 47


28

28
28
The multiples of 7 are ____, ____, ____, ____, ____, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The LCM of 7 and 4 is ____.
We have to multiply the numerator and denominator of
6
7
by 4 to make our common denominator.
We have to multiply the numerator and denominator of
3
4
by 7 to make our common denominator.
(b)
5 3

6 8
5
3


6
8
20 9


24 24

24
The multiples of 6 are ____, ____, ____, ____, ____, …
The multiples of 8 are ____, ____, ____, ____, ____, …
The LCM of 6 and 8 is ____.
We have to multiply the numerator and denominator of
5
6
by ____ to make our common denominator.
We have to multiply the numerator and denominator of
3
8
by ____ to make our common denominator.
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3. Determine each of these differences. Show all your work.
4 1
3 2
(a)  
(b)  
5 3
4 5
(c)
3 2
 
4 3
(d)
4 1
 
5 2
(e)
5 1
 
7 2
(f)
7 1
 
16 4
(g)
5 2
 
6 3
(h)
5 1
 
6 4
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(i)
1 1
 
6 9
(j)
5 1
 
8 6
(k)
5 1
 
6 2
(l)
9 1
 
10 2
Important Note:
Some of the remaining word problems require you to subtract fractions, while others require you
to add fractions. Make sure you think about this before working out the solution.
3
of an inch thick. When the wood is run repeatedly through a planer,
4
3
the planer shaves
of an inch off. Figure out the thickness of the planed lumber.
16
5. A piece of wood is
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6. Figure out the length of the following shaft.
3
4
"
7
8
"
1
2
"
7
1
of a gallon of gas in your lawn tractor. You ran the tractor and now only
of a
8
4
gallon of gas remains. How much gas did you use?
7. You had
8. Figure out the length of side A.
13
16
7
8
"
5
8
"
"
1
3
8
"
Side A
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9. A small piece of metal measuring
15
15
inch by
inch has to have a hole drilled
16
16
1
of an inch of metal between the hole and the
4
side of metal, what is the diameter of the hole?
in the center. If there must be
7
inch shaft be
8
11
reduced so that the diameter will be an
inch
16
shaft?
10. How much must the diameter of a
Open-ended Questions (There is more than one correct answer for each of these questions.)
11. When you subtract two fractions you get
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. What might the fractions be?
2
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12. Create a word problem that would be solved using
1 1
 .
2 3
? 1
1
but less than 1, then what might the missing number be?
 must be greater than
2
7 9
Verify your answer by working out the difference.
13. If
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Subtracting Fractions, Part 2
In this section, we will focus on subtracting mixed numbers. The process is very similar to the
one that we use for adding mixed numbers; we typically work with the whole number
components and the fractional components separately. The exception to this rule occurs when
borrowing (i.e. regrouping) has to occur to complete the subtraction. Regrouping comes down to
understanding the following.
e.g. 3
1
6
e.g. 4
1
6
6 1
 2 
6 6
7
 2
6
5
8
e.g. 9
5
8
8 5
 3 
8 8
13
 3
8
 2 1
 3 1
Example 1
Complete the subtraction.
4
1
5 3
(a) 9  7
(b) 6  1
5
5
8 8
5
8
(c) 14  2
9
9
2
3
2
3
3 2
 8 
3 3
5
8
3
 8 1
(d) 8
7
11
3
16 16
Answers:
With each of the questions, the mixed numbers already have common denominators.
4
1
(a) 9  7
5
5
4 1
 9  7       Grouped whole numbers together. Grouped fractions together.
 5 5
3
2
5
3
2
 Final Answer
5
5 3
(b) 6  1
8 8
 6  1  
5 3
   Grouped whole numbers together. Grouped fractions together.
8 8
2
8
22
5
82
1
5
4
5
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 Simplify fraction by dividing numerator and denominator by 2.
 Final Answer
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With the two remaining questions we have to borrow (i.e. regroup) to complete the
subtraction.
5
8
(c) 14  2
9
9
14  2   
5 8
 
9 9

13  2    1 
5 8
 
9 9

 9 5 8
13  2        
 9 9  9 
 Problem: How do we subtract
8
5
from ?
9
9
 Answer: Borrow 1 from 14 (i.e. regroup the 1 with the
5
)
9
13  2   
14 8 
 
 9 9
6
9
63
11 
93
2
11
3
11 
 Simplify fraction.
7
11
3
16 16
11
7
7 11
 Problem: How do we subtract
from
?
8  3    
16
16
 16 16 

7
11 
7
 7  3   1      Answer: Borrow 1 from 8 (i.e. regroup the 1 with the )
16
  16  16 
 16 7
11 
 7  3       
  16 16  16 
(d) 8
 7  3  
23 11 
 
 16 16 
12
16
12  4
4
16  4
3
4
4
4
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Example 2
Complete the subtraction.
2
1
3
1
(a) 9  4
(b) 6  1
3
2
4 12
1
4
(c) 9  2
3
5
1
7
(d) 17  8
6
9
Answers:
With each of the questions, the mixed numbers do not have common denominators. We have
to create common denominators by looking at least common multiples.
2
1
(a) 9  4
3
2
 9  4   
2 1
 
3 2
2  2 1 3 

 9  4   

 3 2 2  3 
4 3
 9  4     
6 6
1
5
6
3
1
(b) 6  1
4 12
3 1
 6  1    
 4 12 
3 3 1 
 
 6  1  
 4  3 12 
9 1
 6  1    
 12 12 
8
5
12
84
5
12  4
2
5
3
 The LCM of 2 and 3 is 6.
 The LCM of 4 and 12 is 12.
 Simplify fraction.
With the two remaining questions we have to borrow (i.e. regroup) to complete the
subtraction.
1
4
(c) 9  2
3
5
 9  2   
1 4
 
3 5
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1 5 4  3 


 3 5 5  3 
5 12
 9  2     
 15 15 

5
12 
8  2    1    
  15  15 
 9  2   
 The LCM of 3 and 5 is 15.
 Problem: How do we subtract
12
5
from
?
15
15
 Answer: Borrow 1 from 9 (i.e. regroup the 1 with the
5
)
15
 15 5  12 
  
  15 15  15 
8  2    
8  2   
20 12 
 
 15 15 
6
8
15
1
7
(d) 17  8
6
9
17  8  
1 7
 
6 9
1 3 7  2 

17  8  
  The LCM of 6 and 9 is 18.
 63 9 2 
14
3
3 14
 Problem: How do we subtract
from
?
17  8    
18
18
 18 18 

3
14 
3
16  8   1      Answer: Borrow 1 from 17 (i.e. regroup the 1 with the )
18
  18  18 
 18 3  14 
  
  18 18  18 
16  8   
16  8  
21 14 
 
 18 18 
8
7
18
Questions
1. Subtract the following numbers.
3
2
(a) 5  2
7
7
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(b) 6  1
5 5
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7
4
(c) 6  5
9
9
(d) 9
4
7
(e) 6  2
9
9
3
6
(f) 14  5
7
7
3 7
(g) 4  1
8 8
(h) 7
2. Subtract the following numbers.
2
2
(a) 8  4
3
5
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8
9
15
15
3 9

10 10
5 1
(b) 8  1
7 2
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3
1
(c) 6  5
4
6
(d) 3
2
3
(e) 9  5
5
4
1
5
(f) 11  2
6
8
2 5
(g) 9  1
5 6
1 5
(h) 11 
3 6
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1
1
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3
3
(i) 10  9
8
4
(j) 10  4
5
6
Important Note:
Some of the remaining word problems require you to subtract fractions, while others require you
to add fractions. Make sure you think about this before working out the solution.
1
5
inches long is to pass through a piece of lumber that is 1 inches
2
8
thick. How many inches does the bolt extend beyond the lumber?
3. A metal bolt that is 3
2
1
cups of flour and 2 cups of sugar into a large mixing bowl. At this stage,
3
4
how many cups of ingredients are in the bowl?
4. Paul pours 3
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5. A piece of lumber, which initially had a width of 7
that its width is reduced by 1
1
inches, is run through table saw such
2
3
inches. What is the new width of the lumber?
4
1
7
inches in the first month, and 1 inches in the second month. How
4
8
many inches in total did the plant grow in that two month period?
6. The plant grew 2
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Multiplying Fractions Part 1
You may not realize it, but most of you have been multiplying fractions for years. Consider the
following real world problems.
Half of the earnings are mine. If we earned $50, how much do I get?
The three of us collected recyclables. Therefore one third of the profit belongs to me. If
the profit was $90, how much should be coming to me?
Three quarters of a mixture is flour. If there are 8 cups of mixture, how many cups of
flour were used?
Many of you can already answer these questions. You probably know that half of 50 is 25. You
probably know that one-third of 90 is 30. You might know that three-quarters of 8 is 6. All of
these can be expressed as multiplication questions.
1
 50  25
2
1
 90  30
3
3
8  6
4
If 1 quarter of 8 is 2, then
3 quarters of 8 must be 6.
In all of these examples, we are multiplying a fraction by a whole number. What happens when
we multiply a fraction by another fraction? Let’s look at area models to help us out with this.
Example 1
1 1

3 2
The question could be presented in this manner.
1
1
What is of ?
3
2
Based on this, you would start with the area model for
In the next step you would divide the
1
.
2
1
into thirds.
2
In the next step, you would shade only one of those thirds.
In the last step, you look at that newly shaded region and ask
yourself what it represents in terms of the whole square. In this
case, the shaped region represents one-sixth of the whole square.
Therefore
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 
3 2 6
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Example 2
2 1

3 3
The question could be presented in this manner.
2
1
What is of ?
3
3
Based on this, you would start with the area model for
In the next step you would divide the
1
.
3
1
into thirds.
3
In the next step, you shade two of the thirds.
In the last step, look at that newly shaded region and ask yourself
what it represents in terms of the whole square. In this case, the
shaped region represents two ninths of the whole square.
Therefore
2 1 2
 
3 3 9
Questions, Part 1
1
of 8?
2
_____
(b) What is
1
of 9?
3
_____
(c) What is
1
of 20?
4
_____
(d) What is
1
of 60?
10
_____
(e) What is
1
of 10?
5
_____
(f) What is
3
of 16?
4
_____
(g) What is
2
of 6?
3
_____
(h) What is
2
of 20?
5
_____
(i) What is
2
of 12?
3
_____
(j) What is
3
of 100?
10
_____
1. (a) What is
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2. There are partially completed solutions below. Complete the solutions and state the final
answer.
(a)
1 3

2 4
The question could be presented in this manner.
What is
1
3
of ?
2
4
Final
Answer:
(b)
3 1

4 4
The question could be presented in this manner.
What is
3
1
of ?
4
4
Final
Answer:
(c)
1 1

3 4
The question could be presented in this manner.
What is
1
1
of ?
3
4
Final
Answer:
(d)
1 4

3 5
The question could be presented in this manner.
What is
1
4
of ?
3
5
Final
Answer:
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3 3

5 4
(e)
The question could be presented in this manner.
What is __________?
Final
Answer:
Rules for Multiplying Fractions
In first two examples, which we solved using area models, we obtained the following.
1 1 1
2 1 2
Example 1:
Example 2:
 
 
3 2 6
3 3 9
In question 2, you should have obtained the following.
1 3 3
3 1 3
(a)  
(b)  
2 4 8
4 4 16
1 4 4
 
3 5 15
(d)
(e)
(c)
1 1 1
 
3 4 12
3 3 9
 
5 4 20
What should we notice from these answers?
 The first thing we should notice is that we do not need common denominators to multiply
fractions; this is very different from the addition and subtraction of fractions where we
must have common denominators.
 The second thing is that we merely multiply the numbers in the numerators to get the new
numerator, and multiply the numbers in the denominators to get the new denominator.
Example 3
Multiply each of the following.
1 1
2 1
(a) 
(b) 
4 2
3 5
Answers:
1 1
(a) 
4 2
1 1
4 2
1

8

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(b)
(c)
2 1

3 5
5 8

3 7
(c)
2 1
3 5
2

15

106
5 8

3 7
58

3 7
40

21
19
1
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Sometimes multiplication questions require an additional step because we do not end up with
answer in its simplest form. Such is the case with the next few examples. In each case, we have
shown two different ways to solve the question.
 With the first method, we multiply the fractions and then simplify the resulting fraction.
 With the second method, we express the numbers in the numerators and denominators as
prime factors, divide the appropriate numbers in the numerators and denominators (i.e.
those that produce a quotient of 1), and finally multiply all the numbers in the numerators
and all the numbers in the denominators to produce the final answer.
Example 4
Complete the multiplication for each of the following.
4 1
3 5
12 10
(a) 
(b)
(c)


5 14
10 6
30 21
Answers:
Method 1:
(a) 4 1

5 14
4 1

5  14
4

70
42

70  2
2

35
(b) 3 5

10 6
3 5

10  6
15

60
15  15

60  15
1

4
Method 2:
4 1

5 14
2 2
1


5
27
2 2
1


5
2 7
2 1
=
5 7
2

35
3 5

10 6
3
5


25 23
3
5


2 5 2 3
1 1
=
2 2
1

4
(d)
10 15

9 14
 Prime factors of 4 and 14.
 The 2's cancel out (i.e. produce a
quotient of 1)
 Prime factors of 10 and 6.
 The 3's cancel out (i.e. produce a
quotient of 1). The 5's also cancel
out.
Many would simplify
by dividing by 5, and
then by 3; this is
perfectly acceptable.
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(c)
(d)
Method 1:
12 10

30 21
12  10

30  21
120

630
120  10

630  10
12

63
12  3

63  3
4

21
Method 2:
12 10

30 21
2 23 25


2  3 5 3 7
2  2 3 2 5


2  3  5 3 7
2 2
=
3 7
4

21
10 15

9 14
10  15

9  14
150

126
150  2

126  2
75

63
75  3

63  3
25

21
4
1
21
10 15

9 14
2  5 3 5


3 3 2  7
2 5 3 5


3 3 2 7
5 5
=
3 7
25

21
4
1
21
 Prime factors of 12, 10, 30, and
21.
 The 2's cancel out, the 3's
cancel out, and the 5's cancel
out (i.e. all produce quotients
of 1).
 Prime factors of 10, 15, 9, and
14.
 The 2's cancel out (i.e. produce
a quotient of 1). The 3's also
cancel out.
We have been shown two methods to solve these questions. You do not need to know both
techniques; choose the one you prefer.
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Example 5
2
Evaluate  7 .
3
Answer:
In this case we are multiplying a whole number by a fraction, but the question cannot be done
2
easily in your head like the question  9 (Answer: 6). The key to the original question is
3
7
remembering that the whole number 7 can be expressed as the fraction . Once you do this,
1
you simply follow the rules for multiplying two fractions.
2
2 7
7  
3
3 1
14

3
2
4
3
Important Note:
2
, you would handle it in the
3
7 2
same manner. The question could be expressed as  .
1 3
If the question had been 7 
Questions, Part 2
3. Multiply the following fractions.
1 2
(a)  
3 5
(b)
1 1
 
4 5
(c)
2 3
 
5 7
(d)
3 5
 
4 7
(e)
8 2
 
3 5
(f)
9 7
 
4 5
(b)
3 1
 
4 6
4. Multiply the following fractions.
2 1
(a)  
5 4
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(c)
2 3
 
3 5
(d)
1 2
 
4 3
(e)
3 5
 
5 6
(f)
4 3
 
5 8
(g)
8 5
 
15 12
(h)
10 7
 
21 8
(i)
3 15
 
10 8
(j)
4 9
 
3 8
(k)
8 15
 
9 6
(l)
25 14
 
21 5
(b)
2
5 
11
5. Multiply the following.
3
(a)  2 
7
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(c) 4 
1

10
(d)
1
6 
9
(e)
3
4 
5
(f)
5
5 
6
(g)
4
3 
9
(h)
4
7 
5
(i)
7
2 
6
(j)
16
5 
15
9
(k) 4  
8
(l) 6 
10

9
6. Multiple Choice
2 4
 ), the product:
3 5
(i) is always smaller than both of the original proper fractions.
(ii) is always larger than both of the original proper fractions.
(iii) is larger than one of the original proper fractions and smaller than the other.
(a) When you multiply two proper fractions (e.g.
4 7
 ), the product:
3 5
(i) is always smaller than both of the original improper fractions.
(ii) is always larger than both of the original improper fractions.
(iii) is larger than one of the original improper fractions and smaller than the other.
(b) When you multiply two improper fractions (e.g.
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(c) When you multiply a proper fraction by an improper fraction (e.g.
3 7
 ), the product:
5 2
(i) is always smaller than the original improper fraction.
(ii) is always larger than the original improper fraction.
(iii) may or may not be larger than the original improper fraction.
(d) When you multiply a proper fraction by a whole number (e.g.
3
 7 ), the product:
4
(i) is always smaller than the original whole number.
(ii) is always larger than the original whole number.
(iii) may or may not be larger than the original whole number.
(e) When you multiply an improper fraction by a whole number (e.g.
5
 8 ), the product:
3
(i) is always smaller than the original whole number.
(ii) is always larger than the original whole number.
(iii) may or may not be larger than the original whole number.
Some of the remaining word problems require you to multiply fractions, while others require you
to add or subtract fractions. Make sure you think about this before working out the solution.
3
cup of vinegar. If you wanted to
4
triple the recipe, how much vinegar would you need?
7. A recipe for a homemade cleaning solution requires
8. If the diameter of a hole is
7
inch, what is the radius of the
8
hole?
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112
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9. Three quarters of a pizza is left over from last night’s party. You want half of the remaining
pizza. How much of the original pizza does your piece represent?
10. The cross-sectional view of a pipe is provided.
Based on the information in the diagram, figure out
the outer diameter of the pipe.
11. A dog groomer uses a 36 gallon container to wash dogs. If she only fills it
1
4
"
7
8
"
1
4
"
3
full of water,
4
how many gallons of water does she use?
12. A recipe calls for
1
1
cup of sugar. How much sugar should be used if only of the recipe is
2
3
being made?
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13. A drawn line is
15
1
inch long. If you erase
inch from the end of the line, then how long is
16
4
the line now?
1
1
cup of sugar. At another stage, you add cup of sugar.
3
4
How much sugar has been added in total to the recipe?
14. At one stage in a recipe, you add
5
inch plywood. If he stacks 10 sheets of plywood on top of each other,
8
how high is the stack?
15. Brian is stacking
16. For each hour that an oil burner runs, it uses
for
3
gallon of fuel. If the burner is only running
4
2
hour, then how much fuel is used?
3
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Open-ended Questions (There is more than one correct answer for each of these questions.)
17. The product of two fractions is
5
. What might the fractions be?
6
18. Create a word problem that would be solved using
19. The product of three fractions is
1 1
 .
2 3
7
. What might the fractions be?
24
? 6
1
but less than 1, then what might the missing number be?
 must be greater than
2
5 7
Verify your answer by working out the product.
20. If
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Multiplying Fractions, Part 2
In this section we will continue to multiply fractions but now we will include mixed numbers.
These questions only involve one additional step; we must change the mixed numbers to
improper fractions before attempting the multiplication.
Example 1
Multiply each of the following.
2 3
4 1
(a) 1 
(b)  3
7 5
9
2
Answers:
(a) 1 2  3
7 5
9 3
 
7 5
27

35
(b)
4
1
3
9
2
4 7
 
9 2
28

18
28  2

18  2
14

9
5
1
9
1 2
(c) 2 1
5 3
(c)
1 2
2 1
5 3
11 5
 
5 3
55

15
55  5

15  5
11

3
2
3
3
2
(d) 2  5
3
(d)
2
2 5
3
8 5
 
3 1
40

3
1
 13
3
Questions
1. Complete the multiplication for each of the following.
2 4
2 3
(a) 1 
(b) 1
3 7
9 5
1
1
(c) 2  2
4
2
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1 5
(e) 1 
3 8
(f)
7 1
(g) 1 1
8 3
1
(h) 1  6
8
1 11
(i) 1  2
5 12
(j) 2 1
4 1
(k) 1  3
5 3
7
2
(l) 2  2
9
5
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2
1
2
5
7
3
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Answer the following questions. Not all of these questions involve multiplication; some involve
addition or subtraction.
2. In 1970, Chelsie's grandmother bought a china figurine for $30. Forty years later that same
1
figurine is worth 6 times what her grandmother initially paid. What is the new dollar
2
value of the figurine?
1
1
pounds of beef, and used 2 to make hamburger patties, how
4
2
much is left to make spaghetti sauce?
3. If Glenda initially had 3
1
2
cups of flour, but Angus only wants to make
of the
2
3
recipe. How many cups of flour does he need?
4. The original recipe called for 4
5
3
inch thick plywood by gluing a laminate that was
8
32
of an inch thick. What is the new thickness of the resulting material?
5. Hanna increased the thickness of the
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Dividing Fractions, Part 1
We’ll start with a question. How many one-quarters are in three? The first time you encounter
this type of question, you’ll find it easiest to answer using area models.
In the first step, you draw the area model for three and the area model for one-quarter.
In the next and final step, we want to fit as many of the area models for one-quarter into
the area model for three. You can fit 12 one-quarter area models into the area model for
three. Therefore the answer is 12.
Answer each of the following.
How many halves are in one?
How many halves are in two?
How many halves are in three?
_____
_____
_____
How many one-thirds are in one?
How many one-thirds are in two?
How many one-thirds are in three?
_____
_____
_____
Many of you can now answer these types of questions with little assistance. What you may not
know is that these are really division questions involving fractions.
Consider the question 6  2 . A past instructor probably told you that this question is asking you
to figure out how many 2’s are in 6. You would have answered 3 because there are three 2’s in
6. Based on this, when you are asked “How many one-thirds are in two?”, you now know that it
1
can be expressed as 2  .
3
If you reconsider the questions above, those questions and their answers can be expressed this
way.
1
1
1
1  2
2  4
3  6
2
2
2
1
1
3
3
2
1
6
3
3
1
9
3
You might see a pattern here. (If not, don’t worry about it.)
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2 1
 . This is
3 6
actually asking, “How many one-sixths are in two-thirds?” We’ll look at area models to solve
this.
First Step Draw the two area models.
What about questions where we divide a fraction by another fraction. Consider
Second Step - Figure out how many area models for one-sixth can be fit into the
area model for two-thirds.
There are 4 one-sixths in two-thirds.
2 1
Therefore:   4
3 6
How do we do these types of questions without using area models? In other words, what are the
rules for dividing fractions?
Dividing Fractions
To divide two fractions, multiply the first fraction by the reciprocal of the
second fraction.
reciprocals
Therefore:
a c a d
  
b d b c
ad

bc
Note:
b, c, and d are not equal to 0.
Let’s see if this works for two questions that we previously addressed using the area models.
2 1 2 6
  
3 6 3 1
12

3
4
It works!
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1 3 4
 
4 1 1
12

1
 12
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Let’s do a few more examples where the answers don’t work out to be whole numbers.
Example 1
Complete each of the division questions.
2 3
4 2
(a) 
(b) 
3 4
7 3
Answers:
2 3
(a) 
3 4
2 4
 
3 3
8

9
(b)
4 2

7 3
4 3
 
7 2
12

14
6

7
(c)
8 2

3 5
(c)
(d)
8 2

3 5
8 5
 
3 2
40

6
20

3
2
6
3
2
3
5
(d)
2
3
5
2 3
 
5 1
2 1
 
5 3
2

15
Questions
1. (a) How many one-fifths are in one?
_____
(b) How many one-fifths are in two?
_____
(c) How many one-sevenths are in four?
_____
(d) How many one-tenths are in three?
_____
(e) How many one-fourths are in five?
_____
(f) How many one-thirds are in six?
_____
2. Answer each of the division questions. Do not show any work. (Hint: Refer back to
question 1 to answer this question .)
1
1
1
(a) 3  
(b) 4  
(c) 6  
10
7
3
1
(d) 2  
5
1
(e) 1  
5
(f) 5 
3. In a brief sentence, how would you explain what the question 4 
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1

4
1
means?
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C. D. Pilmer
3 1
 .
4 8
Draw the two area models.
4. Use area models to help solve the question
First Step -
Second Step - Figure out how many area models for one-eighths can be fit into
the area model for three-fourths.
There are _____ one-eighths in three-fourths.
3 1
Therefore  
4 8
5. Solve each of the following.
3 1
(a)  
7 2
(b)
3 7
 
5 2
(c)
1 2
 
5 3
(d)
1 1
 
4 16
(e)
2 6
 
5 5
(f)
4 2
 
9 3
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(g)
3 2
 
4 3
(h)
3 2
 
5 7
(i)
9 2
 
10 5
(j)
5 5
 
6 8
(k)
2
3 
5
(l)
2
4
7
(m) 10 
5
6
(n) 8 
3
4
Look at the answers you got in question 5 when answering the next two multiple choice
questions.
6. When a smaller fraction is divided by a larger fraction, then the quotient is always:
(a) less than 0.
(b) less than 1
(c) greater than 1.
(d) greater than 2.
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7. When a larger fraction is divided by a smaller fraction, then the quotient is always:
(a) less than 0.
(b) less than 1
(c) greater than 1.
(d) greater than 2.
Some of the remaining word problems require you to divide fractions, while others require you
to add, subtract, or multiply fractions. Make sure you think about this before working out the
solution.
8. You have seven-eighths of a pound of hamburger meat that you are making into small
meatballs. If each meatball is supposed to weigh one-sixteenth of a pound, how many
meatballs can you make?
9. Three-fourths of a pizza are divided equally among five people. How much of the original
pizza does each of the five people get?
3
inch counter top is comprised of a
4
1
particle board core with
inch laminate glued
16
on top. How thick is the particle board core?
10. A particular
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4
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11. Figure out the number of sheets of
5
inch plywood in a stack 25 inches high?
8
7
of an inch. If they need to cut 40 of
8
these pieces, what should be the minimum length of the original strip of sheet metal?
12. A strip of sheet metal is to be cut into pieces every
13. A crank case on a motor has
1
2
quart of oil. If you add quart of oil, how much oil is now
4
3
in the crank case?
14. You have 12 pounds of flour in a bag. You are removing it from the bag using a container
2
that can hold
pound of flour. Assuming that you are filling the container each time, how
3
many times will you use the container to completely empty the bag?
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Dividing Fractions, Part 2
In this section we will continue to divide fractions but now we will include mixed numbers.
These questions only involve one additional step; we must change the mixed numbers to
improper fractions before attempting the division.
Example 1
Divide each of the following.
4 2
3 5
(a)  1
(b) 2 
5 7
4 8
Answers:
4 2
(a)
1
5 7
4 9
 
5 7
4 7
 
5 9
28

45
(b)
3 5
2 
4 8
11 5
 
4 8
11 8
 
4 5
88

20
88  4

20  4
22

5
2
4
5
1 1
(c) 4  1
6 9
(c)
1 1
4 1
6 9
25 10


6 9
25 9
 
6 10
225

60
225  5

60  5
45

12
45  3

12  3
15

4
3
3
4
1
(d) 5  10
3
(d)
1
5  10
3
16 10
 
3 1
16 1
 
3 10
16

30
16  2

30  2
8

15
Questions
1. Complete the division.
2 2
(a)  1
9 5
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1 1
(c) 2  1
3 4
3 5
(d) 1  1
4 6
1
2
5
1
(f) 1  3
9
3
1
(g) 6  5
4
2 3
(h) 3 
5 10
(e) 6  4
3
pizzas left after the party. The three people who hosted the party wanted to
4
divide it evenly between each other. How much does each person get?
2. There is 2
3. Solve the following. Please note that different questions involve different operations
(addition, subtraction, multiplication or division).
3 1
2 3
(a) 
(b) 
7 2
7 5
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(c)
4 1

5 3
(d)
5 3

7 2
(e)
8 2

9 3
(f)
4 5

3 2
(g)
5 1

6 4
(h)
4
6
3
3
1
(i) 1  2
7
5
5 1
(j) 2  1
6 4
3
1
1
10 6
1
3
(l) 3  6
4
8
(k)
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4
1
(m) 1  4
5
6
2 4
(n) 5  1
3 5
5
1
(o) 2  2
8
4
(p) 5
3
2
1
10 3
Some of the remaining word problems require you to divide fractions, while others require you
to add, subtract, or multiply fractions. Make sure you think about this before working out the
solution.
4
4. There are 225 beds in the local hospital. If
of those beds are occupied, how many patients
5
are using beds?
5. A copper pipe that is 6
1
feet long is cut into four pieces of equal length. How long is each
3
piece?
2
1
cups of flour in the pantry. If they used 3 cups to make
3
2
dinner rolls, how much flour remains?
6. The family originally had 5
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1
pounds of ground beef that he had break into separate packages, where each
2
3
package weighed
pounds. How many packages could he make?
4
7. Jacob had 4
5
inch thick plywood together to create a beam with a greater
8
thickness. What is the thickness of this beam?
8. Glen glued three strips of
Open-ended Questions (There is more than one correct answer for each of these questions.)
9. The quotient of two fractions is
1
. What might the fractions be?
2
1
10. The sum of two mixed numbers is 3 . What might the mixed numbers be?
2
1
11. Create a word problem that would be solved using 4  6 .
5
1 2
12. Create a word problem that would be solved using 3  .
4 3
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Answers
Prerequisite Knowledge (pages v to vi)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
(r)
(s)
(t)
(u)
(v)
(w)
(x)
(y)
4, 8, 12, 16, 20, …
9, 18, 27, 36, 45, …
6, 12, 18, 24, 30, …
6
25
40
Yes
No
No
Yes
Yes
Yes
No
1, 3, and 9
1, 2, 3, and 6
1, 3, 5, and 15
1, 2, 4, 8, and 16
2 with a remainder of 3
4 with a remainder of 1
1 with a remainder of 3
5 with a remainder of 4
5 with a remainder of 6
15  3  5
18  2  3  3
24  2  2  2  3
Introduction to Fractions; Area Models (pages 1 to 6)
1. (a)
1
6
2. smallest:
3. (a)
5
6
(b)
1
6
8
9
largest:
(b)
1
9
(c)
4
6
(d)
7
18
(c)
2
6
(d)
11
18
8
9
4. Yes because with each area model the region has been divided into six sections of equal size
and five of those sections are shaded.
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5.
1
6
6. No because the three sections are not of equal size.
7.
2
18
3
6
0
11
12
5
9
1
1
2
8.
1
6
3
8
5
10
0
7
8
4
4
1
1
2
9. (a) Taralee took the pizza and cut into 8 slices, all of the same size. If she took 3 slices for
herself, then that meant that she took
3
of the pizza.
8
(b) Massato took the cake and cut into 15 pieces of the same size and took 2 pieces for
himself. That means that he took
2
13
of the cake and that
of the cake was left for
15
15
everyone else.
(c) A square is drawn, and then its two diagonals are drawn in. If 1 of the resulting triangles
within the square is shaded, then the shaded portion represents
1
of the square. If an
4
addition triangle is shaded, then the shaded portions represent
2
of the square
4
(d) Tanya has a piece of farm land that she wants to divide evenly amongst her 5 children.
If she has 2 boys and 3 girls, it means that the boys would end up owning
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2
of the land.
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(e) Two brothers are sharing a pizza but not equally. The pizza is cut into 12 slices of the
same size but the older brother takes more. If the older brother takes 7 slices, then he
ends up getting
5
7
1
(d)
9
7
5
of the pizza, and the younger brother gets
of the pizza.
12
12
7
10
2
(e)
15
10
12
11
(f)
23
10. (a)
(b)
(c)
11. (a) one-sixth
(b) five-ninths
(c) thirteen-sixteenths
Comparing Fractions Investigation A (pages 9 to 11)
Part 1
1 2 3 4
(a) , , ,
4 4 4 4
(b)
1 3 6 8
, , ,
9 9 9 9
(c) "When asked to compare two fractions that have the same number in the denominator
(bottom), the larger fraction is the one with the larger number in the numerator (top)."
(d) (i)
(iv)
5
6
(ii)
5
8
(iii)
7
16
9
10
(v)
2
3
(vi)
10
12
Part 2
3 3 3 3
(a)
, , ,
12 9 6 4
(b)
5 5 5 5
,
, ,
18 12 9 6
(c) "When asked to compare two fractions that have the same number in the numerator (top), the
larger fraction is the one with the smaller number in the denominator (bottom)."
(d) (i)
(iv)
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2
(ii)
2
3
(iii)
3
8
3
4
(v)
7
8
(vi)
5
8
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Questions
1. With each pair of fractions, circle the larger fraction.
(a)
3
10
(b)
1
3
(c)
7
9
(d)
4
5
(e)
13
16
(f)
4
4
(g)
7
8
(h)
3
8
(i)
7
10
(j)
3
8
(k)
1
12
(l)
6
16
3 5 11 15
,
,
,
16 16 16 16
2. (a)
(b)
5 5 5 5
,
, ,
16 10 8 5
(c)
1 3 5 7
, , ,
8 8 8 8
(d)
2 2 2 2
,
, ,
16 12 8 3
(e)
1 4 7 9
,
,
,
10 10 10 10
(f)
7
7 7 7
,
, ,
100 32 8 7
3.
1
8
0
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2
8
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Comparing Fractions; Investigation B (pages 12 to 19)
Fraction
(a)
(c)
(e)
(g)
(i)
(k)
(m)
(o)
(q)
Closest
To:
8
9
2
18
10
18
2
16
5
12
7
16
17
18
5
9
9
16
Fraction
1
(b)
0
(d)
1
2
(f)
0
(h)
1
2
1
2
1
1
2
1
2
(j)
(l)
(n)
(p)
(r)
5
8
1
12
5
6
8
18
7
8
1
9
11
12
1
6
16
18
Closest
To:
1
2
0
1
1
2
1
0
1
0
1
1
2
Fractions Closest to 0
Fractions Closest to
2 1 2 1 1
,
,
, ,
18 12 16 9 6
5 8 5 7 5 9 10
,
,
,
, ,
,
8 18 12 16 9 16 18
Fractions Closest to 1
8 5 7 17 11 16
,, , ,
,
,
9 6 8 18 12 18
(a) When the number in the numerator of a fraction is close to the number in the denominator of
7 9 15
the fraction (examples: , , ), then the fraction is closest to the number 1.
8 11 18
(b) When the number in the numerator of a fraction is very small compared to the number in the
1 2 3
denominator of the fraction (examples: , , ), then the fraction is closest to the number
8 11 18
0.
(c) When the numerator of the fraction is about half the size of the denominator of the fraction
1
4 6 14
(examples: , , ), then the fraction is closest to the number .
9 11 30
2
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(d) When the numerator of the fraction is the same number as the denominator of the fraction
3 8 11
(example: , , ), then the fraction is equal to 1.
3 8 11
Questions
Fraction
1.
(a)
1
18
17
32
8
15
5
11
3
100
(d)
(g)
(j)
(m)
Closest to:
0
1
2
1
2
1
2
0
Fraction
(b)
Closest to:
24
25
1
14
2
30
18
19
7
12
(e)
(h)
(k)
(n)
1
0
0
1
1
2
Fraction
Closest to:
5
11
2
20
28
30
37
40
13
15
1
2
(c)
(f)
(i)
(l)
(o)
0
1
1
1
2.
1
10
0
4
7
11
12
1
2
1
12
12
(b)
4
5
(c)
5
8
(d)
7
16
(e)
1
7
(f)
3
8
(g)
5
7
(h)
7
8
(i)
9
10
(j)
4
9
(k)
6
7
(l)
5
9
3. (a)
NSSAL
©2012
3
3
136
Draft
C. D. Pilmer
(m)
11
12
(n)
5
6
(o)
6
7
(p)
5
6
(q)
11
12
(r)
7
16
(s)
16
16
(t)
7
12
1 3 5 7
, , ,
7 7 7 7
(b)
1 7 13 6
,
,
,
10 15 14 6
(c)
1 1 3 4
, , ,
20 5 5 5
(d)
2 5 6 20
,
, ,
16 12 7 20
(e)
1 1 1 1
, , ,
9 7 3 2
(f)
1 4 9 9
, , ,
12 7 11 9
(g)
2 2 2 2
, , ,
9 7 5 3
(h)
1 4 11 3
, ,
,
11 9 12 3
(i)
1 3 5 9
, , ,
12 8 8 10
(j)
1 1 11 6
, ,
,
8 6 20 6
(k)
1 2 5 7
, , ,
9 9 9 9
(l)
1 1 7 7
, ,
,
21 5 13 9
(m)
1 2 7 9
,
,
,
19 19 12 10
(n)
3 7 10 11
,
,
,
11 11 11 11
(o)
1 5 17 19
,
,
,
5 12 20 20
(p)
3 8 30 31 16
,
,
,
,
32 14 32 32 16
(q)
1 1 8 7 25
, ,
, ,
20 9 17 8 25
(r)
2
3 7 8 8
,
,
,
,
32 32 15 15 9
4. (a)
5.
1
6
1
4
0
6.
3
7
5
9
9
9
1
2
1
1 2 3 4 5
,
,
,
,
12 12 12 12 12
NSSAL
©2012
137
Draft
C. D. Pilmer
More Comparing Fractions (pages 20 to 23)
Close to 0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
NSSAL
©2012
10
20
7
8
3
32
7
12
6
14
3
3
4
8
5
9
9
18
1
100
19
20
9
17
14
16
3
7
50
50
15
30
52
100
Slightly
less
1
than
2
Equal to
1
2
Slightly
more
1
than
2
Close to 1
Equal to 1

















138
Draft
C. D. Pilmer
Close to 0
2
25
10
19
(r)
(s)
2.
Slightly
less
1
than 2
Equal to
1
2
Slightly
more
1
than 2
Close to 1

1
40

4
10
5
6
50 9
100 16
12
12
1
2
0
1
7
14
(b)
6
10
(c)
4
7
(d)
8
9
(e)
5
7
(f)
5
5
(g)
6
10
(h)
9
16
(i)
6
11
(j)
7
12
(k)
5
8
(l)
6
7
(m)
4
7
(n)
7
14
(0)
9
15
3. (a)
4. (a)
1 7 4 30 5
,
, ,
,
16 16 8 32 5
(b)
7
5 5 11 11
,
, ,
,
100 10 9 12 11
(c)
2 6 3 5 14
,
, , ,
21 14 6 8 15
(d)
7 6 4 14 15
,
, ,
,
16 12 7 16 16
(e)
4 5 8 19 7
,
,
,
,
10 10 14 20 7
(f)
1
1 11 8 28
,
,
,
,
30 20 22 15 30
(g)
30 7 9 9 15
,
,
,
,
60 12 11 10 15
(h)
2 2 5 7 9
,
,
,
,
50 19 11 14 16
NSSAL
©2012
Equal to 1
139
Draft
C. D. Pilmer
3 9 9 13 40
,
,
,
,
50 18 17 14 40
(i)
6 7 5 12 12
,
, ,
,
14 14 9 14 13
(j)
Area Models for Fractions Greater Than 1 (pages 24 to 29)
5
4
11
(b) Improper Fraction:
6
8
(c) Improper Fraction:
3
1
4
5
Mixed Number: 1
6
2
Mixed Number: 2
3
1. (a) Improper Fraction:
9
7
3.
3
2
43
10
Mixed Number: 1
9
4
5
1
or 2
2
2
Model B:
13
1
or 1
12
12
Model C:
13
5
or 1
8
8
Model D:
1
6
Model E:
27
3
or 3
8
8
Model F:
15
3
or 3
4
4
Model G:
33
9
or 2
12
12
Model H:
4
1
or 1
3
3
Model I:
9
1
or 2
4
4
4. Model A:
1
6
0
1
1 1
1
12 3
1
5
8
1
2
2
1
1
2
2
4
2
9
12
3
3
8
3
3
3
4
4
5. Most people say that they better understand the size of a fraction when it is written as a
mixed number.
6
15
7
1
(d) 2
12
(a)
7. (a) eighteen-sevenths
(c) thirteen-eighths
NSSAL
©2012
1
10
25
(e)
15
10
6
11
(f) 1
22
(b) 3
(c)
(b) two and one-fifth
(d) three and nine-sixteenths
140
Draft
C. D. Pilmer
Comparing Proper Fractions and Mixed Numbers (pages 31 to 33)
1.
1
16
5
12
0
7
12
5 1
1
5 8
1
5
10
1
11
1
2
12 32
1
2
7 10
2
16 18
2
2
7
8
3
2.
15
30
1 1
32 7
0
10 1
1
10 10
1
3
7
1
7 12
2
1
1
5
2
4
8
2
2
1
1
12
(b)
2
1
16
(c)
5
5
(d)
1
3
8
(e)
2
7
10
(f)
2
(g)
2
7
12
(h)
15
16
(i)
1
(j)
3
7
12
(k)
1
1
16
(l)
3
6
7
(m) 2
6
13
(n)
1
9
15
(0)
2
15
16
31
32
7
8
8
16
4. (a)
4 5
1
7 30
, , 1 , 1 ,1
8 5 16 16 32
(b)
5 11 5 11
7
,
,1 ,1 , 2
9 11 10 12 100
(c)
14 3 5 7
6
, 1 , 1 ,1 , 2
15 8 8 8 14
(d)
1
(e)
7 19
4
5
8
,1 ,3 , 3 ,3
7 20 10 10 14
141
2
3
3. (a)
NSSAL
©2012
5
6
6
14 15
7
4
,1 ,1 ,2 ,2
12 16 16 16
7
Draft
C. D. Pilmer
From Mixed Numbers to Improper Fractions, and Vice Versa (pages 34 to 42)
13
4
25
(d)
6
15
(g)
2
1. (a)
1
5
1
(d) 3
6
7
(g) 2
10
2. (a) 1
3. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
23
8
18
(e)
7
21
(h)
16
19
10
17
(f)
5
35
(i)
8
(b)
(c)
3
8
1
(e) 2
3
3
(h) 5
5
2
9
3
(f) 3
7
3
(i) 4
4
(b) 2
(c) 1
1 11 1
1
5
,
, 1 , 2 ,2
32 12 8 16 6
9
1
7
3
7
,1 ,1 ,2 ,2
10 10 16 8 12
3 4
3 9 
4 1
,
,1 ,
 or 1  , 2
12 12 6 5 
5 7
7 7
9 9 
1 5
, ,1 ,
 or 2  , 2
9 8 10 4 
4 8
1 10 
1
5
6
1
,
 or 1  , 1 , 1 , 2
4 9 
9  10 7
32
1 1 13 
3  7 17 
1
,
,
 or 1  , 1 ,
 or 2 
32 16 10 
10  8 8 
8
17 
1
4
3
1 9 
1
 or 1  , 1 , 1 , 3 ,
 or 4 
16 
16  10 4
7 2 
2
3 10 
3 4 8 
2
9 13 
1
,
 or 1  , 1 ,
 or 2  , 2 ,
 or 3 
8 7 
7 7 3 
3  10 4 
4
1 5 
1 1
1 19 
1  32 
2
,
 or 1  , 1 , 2 ,
 or 2  ,
 or 3 
15 4 
4  3 18 9 
9  10 
10 
NSSAL
©2012
142
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C. D. Pilmer
4.
1
10
3 1
1
3 6
7
12
0
1
11
6
or
5
1
6
4
8
1
2
10
4
or
5
2
2
2
8
4
1
16
2
2
9
10
3
5.
3
32
8
16
8
7
or
1
1
7
29
32
0
1
6
3
1
10
4
1
2
1
8
2
7
16
2
4
5
2
12
4
or
3
25
8
or
1
3
8
3
Area Models of Different Shapes and Sizes (pages 43 to 46)
2
. In every case, the region is divided into three
3
sections of the same size, and two of those sections are shaded. It does not matter that the
area models are of different sizes or shapes.
1. They all represent the same fraction
2. Answers will vary slightly.
3. (a)
6
6
(b)
11
12
(c)
1
8
(d)
7
3
or 1
4
4
4. (a)
3
10
(b)
2
5
(c)
3
4
(d)
3
8
4
1
or 1
3
3
(h)
5
1
or 1
4
4
(g)
NSSAL
©2012
143
(e)
5
8
(f)
3
16
Draft
C. D. Pilmer
5.
1
1
is bigger than
2
3
1
is physically larger than the other
3
1
model, however only one of its three sections is shaded. With the area model for , one of
2
its two sections is shaded. Since a greater portion of the first model is shaded, then the
1
fraction
is bigger.
2
This can be challenging to see because the area model of
Equivalent Fractions, Part 1 (pages 47 and 48)
1. (a) second, fourth and fifth area models
8 6
12
(b)
, , and
12 9
18
2. Answers will vary. Possible answers could include
6 9 12 15
18
,
,
,
, and
8 12 16 20
24
Equivalent Fractions, Part 2 (pages 49 to 57)
1. Five possible answers have been supplied for each. You only have to supply three. These
are not the only correct answers.
10 15 20 25 30
10 15 20 25 30
,
,
,
,
,
,
,
,
(a)
(b)
14 21 28 35 42
12 18 24 30 36
1 4
6
8 10
8 12 16 20 24
,
,
,
,
,
,
(c) ,
(d) ,
5 20 30 40 50
6 9 12 15 18
3 6 24 36 48
1 2 8 12 16
(e) , ,
(f) , ,
,
,
,
,
2 4 16 24 32
3 6 24 36 48
10 15 20 25 30
5 10 15 60 90
,
,
,
,
,
,
,
(g)
(h) ,
8 12 16 20 24
2 4 6 24 36
2
8
12 16 20
6
9
12 15 18
(i) 1 , 1 , 1 , 1 , 1
(j) 3 , 3 , 3 , 3 , 3
5 20 30 40 50
16 24 32 40 48
10 15
20
25 30
3 6 24 36 48
(k) 2 , 2 , 2 , 2 , 2
(l) 1 , 1 , 1 , 1 , 1
12 18
24 30 36
4 8 32 48 64
1
2
3
(c)
7
2. (a)
NSSAL
©2012
4
5
3
(d)
5
(b)
144
Draft
C. D. Pilmer
2
3
7
(g)
5
1
(i)
10
3
(k)
7
7
(m)
2
5
(o)
2
3
(q) 2
4
3
(s) 2
5
3
(u) 1
10
(e)
3. (a)
(c)
(e)
(g)
equivalent
not equivalent
equivalent
not equivalent
1
5
4
(c)
5
4. (a)
(e) 2
5. (a) =
(c) >
(e) =
(g) >
(i) <
(k) =
(m) >
(f)
(h)
(j)
(l)
(n)
(p)
(r)
(t)
(v)
(b)
(d)
(f)
(h)
7
8
5
9
3
2
10
9
6
5
1
1
2
1
4
3
4
3
5
7
3
8
equivalent
equivalent
equivalent
not equivalent
2
3
10
(d)
7
8
(f) 1
12
(b)
5
6
(b)
(d)
(f)
(h)
(j)
(l)
(n)
<
=
<
=
=
>
<
6. Tom and Jorell purchased a pizza that had been cut into 8 pieces of the same size. Tom ate 2
2
1
pieces. That meant he ate
of the pizza, which when expressed in its simplest form is
8
4
of the pizza.
NSSAL
©2012
145
Draft
C. D. Pilmer
7. Nita and Meera purchased a 16 acre lot. Since Nita contributed more money to the purchase,
10
she took ownership of 10 acres. This left 6 acres for Meera. That means Nita obtained
16
5
of the original lot, which when expressed in simplest form is
of the lot. Meera obtained
8
6
3
of the original lot, which when expressed in simplest form is
of the lot.
8
16
8.
1
2
9.
2
3
10.
1
5
1
12
3
(c)
4
1
6
1
(d)
4
11. (a)
(b)
Equivalent Fractions and the Number Line (pages 58 and 59)
1.
3
100
0
NSSAL
©2012
3
5
and
9
15
2
8
and
5
4
1
10
15
and
14
2
21
2
2
1
2
146
1
8
3
1
5
3
Draft
C. D. Pilmer
2.
2
4
and
3
2
1
1
5
97
100
3
8
0
8
4
1
2
2
7
16
11
4
and
6
2
8
19
6
and
4
3
24
3
Measuring and Fractions (pages 60 to 63)
1. Answers rounded to the nearest sixteenth when measured in inches, and rounded to the
nearest tenth when measured in centimetres. Please note that sometimes during the printing
of these documents, the images sizes change slightly. Your answers may not correspond to
the answers here for this reason. Please check with your instructor if you believe this is
occurring.
(a) Inches: 3
Centimetres: 7
6
3
or 7
10
5
(b) Inches: 2
6
3
or 2
16
8
Centimetres: 6
(c) Inches: 4
1
16
Centimetres: 10
(d) Inches: 2
3
10
Centimetres: 5
1
10
(e) Inches: 3
4
1
or 3
16
4
Centimetres: 8
3
10
(f) Inches: 2
3
16
Centimetres: 5
6
3
or 5
10
5
(g) Inches: 2
13
16
Centimetres: 7
2
1
or 7
10
5
(h) Inches: 4
9
16
Centimetres: 11
6
3
or 11
10
5
(i) Inches: 5
1
16
Centimetres: 12
9
10
NSSAL
©2012
147
Draft
C. D. Pilmer
(j) Inches: 2
8
1
or 2
16
2
Centimetres: 6
(k) Inches: 3
8
1
or 3
16
2
Centimetres: 9
4
2
or 6
10
5
Estimating the Addition and Subtraction of Fractions (pages 64 to 67)
1.
Close to 0
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
2.
(a)
(b)
(c)
NSSAL
©2012
7 15

8 14
1 4

20 9
1
1

12 15
21 7

20 13
9
5

17 11
8 11

9 10
2
1

31 25
12 1

13 32
8
9

17 10
8
1

15 40
Sum or
Difference
1
12
1 4
20
13
5
1
3 4
12
32
5
7
2 3
9
8
Close to
1
2
Close to 1
Close of 1
1
2
Close to 2










Estimate
1+5=6
1
1
3 47
2
2
1
1
2 46
2
2
148
Draft
C. D. Pilmer
Sum or
Difference
21 2

22 53
13 7

12 13
14
1
6 2
15
20
8
6
8 4
17
14
7
2
8 3
15
45
3
7
9 6
50
13
14
10
7 4
15
21
(d)
(e)
(f)
(g)
(h)
(i)
(j)
3.
Estimate
1-0=1
1
1 1

2 2
7-2=5
1
1
8 4  4
2
2
1
1
8 3  5
2
2
1
1
96  2
2
2
1
1
84  3
2
2
1 1
1
 equals 1, not
2 2
2
4. (a) Tanya had a piece of lumber that was 1
5
inches thick. She runs it through a planer that
8
removes a thin strip of wood from the thickness of the lumber. She set the planer to
remove
1
1
inches. She estimates that the lumber coming out of the planer will be 1
32
2
inches thick.
(b) Manish initially measured ot 2
2
2
cups of flour, which he estimated to be pretty close to
3
1
1
cups. He then added another 1 cups. That means that he measured out a little
2
2
more than 4 cups of flour in total.
(c) Alex and Jacob ordered large pizzas from the same pizzeria but for different parties.
Alex had a lot more pizza left over from this party. Alex had 3
2
NSSAL
©2012
7
pizzas left. Jacob had
8
1
1
pizzas left. Together they approximately had 6 pizzas.
2
2
149
Draft
C. D. Pilmer
(d) Colin puts 3
1
gallons of gasoline in his all-terrain vehicle, which was initially empty.
8
He then takes a trip in the woods and estimates that he used 2
1
gallons of gas, the bulk
2
of the gas that he initially added. Therefore approximately only
1
gallon remains in the
2
tank after his trip.
Adding Fractions, Part 1 (pages 69 to 80)
1. (a)
(c)
(e)
(g)
(i)
(k)
7
10
11
14
1
2
3
4
6
1
11
4
2
5
8
11
(d) 1
2
3
1
(h) 1
9
1
(j) 1
2
2
(l) 2
3
(f)
1 2

2 3
3 4

6 6
7
1
or 1
6
6
2. (a)
(b)
(b)
5 1

6 2
10 6

12 12
16
4
or 1
12
12
4
1
or 1
3
3
NSSAL
©2012
150
Draft
C. D. Pilmer
3.
11
12
4. (a) 6, 12, 18, 24, 30, …
(b) 4, 8, 12, 16, 20, …
(c) 12
5. (a) 5, 10, 15, 20, 25,…
2, 4, 6, 8, 10, …
LCM: 10
(b) 4, 8, 12, 16, 20, …
10, 20, 30, 40, 50, …
LCM: 20
(c) 2, 4, 6, 8, 10, …
6, 12, 18, 24, 30, …
LCM: 6
(d) 5, 10, 15, 20, 25, …
3, 6, 9, 12, 15, …
LCM: 15
(e) 6, 12, 18, 24, 30, …
8, 16, 24, 32, 40, …
LCM: 24
6. (a) The LCM of 5 and 4 is 20.
We have to multiply the numerator and denominator of
1
by 5 to make our common
4
denominator.
(b) The multiples of 6 are 6, 12, 18, 24, 30,…
The LCM of 6 and 4 is 12.
We have to multiply the numerator and denominator of
5
by 2 to make our common
6
denominator.
We have to multiply the numerator and denominator of
1
by 3 to make our common
4
denominator.
We had to change the improper fraction
13
1
to the mixed number 1
.
12
12
(c) The multiples of 3 are 3, 6, 9, 12, 15, …
The LCM of 3 and 5 is 15.
We have to multiply the numerator and denominator of
2
by 5 to make our common
3
denominator.
We have to multiply the numerator and denominator of
3
by 3 to make our common
5
denominator.
We had to change the improper fraction
Final Answer: 1
NSSAL
©2012
19
4
to the mixed number 1 .
15
15
4
15
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C. D. Pilmer
(d) The multiples of 8 are 8, 16, 24, 32, 40, …
The LCM of 4 and 8 is 8.
We have to multiply the numerator and denominator of
3
by 2 to make our common
4
denominator.
We had to change the improper fraction
Final Answer: 1
5 2

10 10
18 5
(c) Hint:

30 30
16 15
(e) Hint:

20 20
2 9
(g) Hint:

12 12
5
(i) 1
18
3
(k)
4
5
(m) 1
8
7. (a) Hint:
3
8
7
10
23
Answer:
30
11
Answer: 1
20
11
Answer:
12
Answer:
5
8

20 20
8 9
(d) Hint:

12 12
10 9
(f) Hint:

15 15
9
4
(h) Hint:

24 24
7
(j)
12
1
(l) 1
3
7
(n) 1
8
(b) Hint:
13
20
5
Answer: 1
12
4
Answer: 1
15
13
Answer:
24
Answer:
1
8
8. 1
9.
11
3
to the mixed number 1 .
8
8
7
8
10. 1
7
12
11. 2
1
2
12. 1
5
8
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13. If we use our estimation techniques, we realize the answer should be close to 1, not
correct answer is
1
. The
2
34
.
35
14. Correct Answer: Student Two
Adding Fractions, Part 2 (pages 81 to 87)
5
7
2
(c) 11
3
1
(e) 8
9
1
(g) 5
4
1. (a) 3
(i) 6
 1 5 2  3 
2. (a) Hint:  4  8  


 3 5 5  3 
 3  2 1 7 
(b) Hint: 1  2   


 7 2 2 7 
 23 1 
(c) Hint:  2  5  
 
 3 3 6 
 5 1 4 
(d) Hint:  3  1   

 16 4  4 
 1 4 3  3 
(e) Hint:  3  5  


 6 4 8 3 
 4  4 3 5 
(f) Hint:  6  2   


 5 4 4 5 
 2  4 3 3 
(g) Hint:  2  1  


 3 4 4  3 
 1 2 5 
(h) Hint:  3  4   
 
 3 2 6 
 5  2 1 3 
(i) Hint:  4  6   


 6 2 43 
 7 3 2 
(j) Hint: 1  2    

 8 4 2 
NSSAL
©2012
4
5
4
9
5
2
9
7
3
7
5
1
2
2
(b) 10
(d)
(f)
(h)
(j)
11
15
13
3
14
5
7
6
9
4
16
13
8
24
11
9
20
5
4
12
1
8
6
1
11
12
5
4
8
Answer: 12
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
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C. D. Pilmer
 7 2 73 
(k) Hint: 2  


 15  2 10  3 
 4 6 55 
(l) Hint:  3  7   


 5 6 6 5 
3. 5
1
6
19
Answer: 11
30
Answer: 3
3
pounds
4
Subtracting Fractions, Part 1 (pages 88 to 94)
5
9
1
(c)
2
3
(e)
4
1
(g)
8
1
6
1
(d)
3
2
(f)
5
4
(h)
5
1. (a)
(b)
2. (a) The multiples of 7 are 7, 14, 21, 28, 35, …
The LCM of 7 and 4 is 28.
3
Final Answer:
28
(b) The multiples of 6 are 6, 12, 18, 24, 30, …
The multiples of 8 are 8, 16, 24, 32, 40, …
The LCM of 6 and 8 is 24.
We have to multiply the numerator and denominator of
5
by 4 to make our common
6
denominator.
We have to multiply the numerator and denominator of
3
by 3 to make our common
8
denominator.
Final Answer:
7
15
1
(c)
12
3. (a)
NSSAL
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11
24
7
20
3
(d)
10
(b)
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C. D. Pilmer
3
14
1
(g)
6
1
(i)
18
1
(k)
3
(e)
5.
9
16
6. 2
7.
3
16
7
(h)
12
11
(j)
24
2
(l)
5
(f)
1
8
5
8
8. 1
7
16
9.
7
16
10.
3
16
Subtracting Fractions, Part 2 (pages 95 to 102)
1
7
1
(c) 1
3
2
(e) 3
3
1
(g) 2
2
1. (a) 3
 25 23 
2. (a) Hint:  8  4   


 3 5 5 3 
 5  2 1 7 
(b) Hint:  8  1  


 7 2 27 
NSSAL
©2012
(b) 5
(d)
2
5
1
3
4
7
2
(h) 6
5
(f) 8
4
15
3
Answer: 7
14
Answer: 4
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Draft
C. D. Pilmer
 3  3 1 2 
(c) Hint:  6  5  


 43 6 2 
 9 1 4 
(d) Hint:  3  1   

 16 4  4 

8  15 
(e) Hint:  8  5   1    
  20  20 

4  15 
(f) Hint: 10  2    1    
  24  24 
  12  25 
(g) Hint:  8  1   1    
  30  30 
 2  5 
(h) Hint: 10   1    
 6  6 
 3  6 
(i) Hint:  9  9    1    
 8  8 
 5
(j) Hint:  9  4   1  
 6
3. 1
7
inches
8
4. 5
11
cups
12
5. 5
3
inches
4
6. 4
1
inches
8
7
12
5
Answer: 2
16
Answer: 1
Answer: 3
13
20
Answer: 8
13
24
Answer: 7
17
30
Answer: 10
Answer:
1
2
5
8
Answer: 5
1
6
Multiplying Fractions, Part 1 (pages 103 to 115)
1. (a) 4
(c) 5
(e) 2
(b) 3
(d) 6
1
3
of 16 is 4, then
of 16 must be 12  i.e. 3  4  .
4
4
Answer: 12
1
2
(g) Hint: If of 6 is 2, then
of 6 must be 4  i.e. 2  2  .
3
3
Answer: 4
(f) Hint: If
NSSAL
©2012
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C. D. Pilmer
1
2
of 20 is 4, then
of 20 must be 8  i.e. 2  4  .
5
5
Answer: 8
(i) 8
(j) 30
(h) Hint: If
3
8
1
(c)
12
9
(e)
20
2. (a)
2
15
6
(c)
35
1
(e) 1
15
3. (a)
4. (a)
(c)
(e)
(g)
(i)
(k)
1
10
2
5
1
2
2
9
9
16
2
2
9
6
7
2
(c)
5
5. (a)
2
5
1
(g) 1
3
1
(i) 2
3
(e) 2
NSSAL
©2012
3
16
4
(d)
15
(b)
1
20
15
(d)
28
3
(f) 3
20
(b)
(b)
(d)
(f)
(h)
(j)
(l)
(b)
(d)
(f)
(h)
(j)
157
1
8
1
6
3
10
5
12
1
1
2
1
3
3
10
11
2
3
1
4
6
3
5
5
1
5
3
Draft
C. D. Pilmer
(k) 4
6. (a)
(b)
(c)
(d)
(e)
7. 2
(i)
(ii)
(i)
(i)
(ii)
(l) 6
2
3
is always smaller than both of the original proper fractions.
is always larger than both of the original improper fractions.
is always smaller than the original improper fraction.
is always smaller than the original whole number.
is always larger than the original whole number.
1
4
8.
7
16
9.
3
8
10. 1
1
2
3
8
11. 27
12.
1
6
13.
11
16
14.
7
12
15. 6
16.
1
4
1
2
Multiplying Fractions, Part 2 (pages 116 to 118)
1. (a)
NSSAL
©2012
20
21
(b)
158
16
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Draft
C. D. Pilmer
(c) 5
(e)
5
8
17
21
6
7
3
6
4
3
2
4
2
6
3
(d) 3
5
6
(f)
1
2
1
(i) 3
2
(g) 2
(h)
(j)
(k) 6
(l)
2. $195
3.
3
pounds
4
4. 3 cups
5.
23
inches
32
Dividing Fractions, Part 1 (pages 119 to 125)
1. (a) 5
(c) 28
(e) 20
(b) 10
(d) 30
(f) 18
2. (a) 30
(c) 18
(e) 5
(b) 28
(d) 10
(f) 20
3. How many one-thirds are in four?
4. 6
6
7
3
(c)
10
1
(e)
3
1
(g) 1
8
5. (a)
NSSAL
©2012
(b)
6
35
(d) 4
(f)
2
3
(h) 2
159
1
10
Draft
C. D. Pilmer
1
4
2
(k)
15
1
3
1
(l)
14
(i) 2
(j) 1
(m) 12
(n) 10
2
3
6. (b) less than 1
7. (c) greater than 1
8. 14
9.
3
20
10.
11
16
11. 40
12. 35
13.
11
12
14. 18
Dividing Fractions, Part 2 (pages 126 to 130)
10
63
13
(c) 1
15
1
(e) 1
3
1
(g) 1
4
1. (a)
2.
5
14
21
(d)
22
7
(f)
15
1
(h) 11
3
(b) 2
11
of a pizza
12
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13
14
7
(c)
15
5
(e) 1
9
7
(g)
12
1
(i) 3
7
9
(k)
35
1
(m) 7
2
1
(o) 1
6
3. (a)
(b)
(d)
(f)
(h)
(j)
(l)
(n)
(p)
6
35
10
21
1
3
3
2
9
7
1
12
5
9
8
7
7
15
19
3
30
4. 180 patients
5. 1
7
feet
12
6. 2
1
cups
6
7. 6 packages
8. 1
7
inches
8
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©2012
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C. D. Pilmer