Iulia & Teodoru Gugoiu
The Book of
Fractions
Copyright © 2006 by La Citadelle
www.la-citadelle.com
Iulia & Teodoru Gugoiu
The Book of Fractions
ISBN 0-9781703-0-X
© 2006 by La Citadelle
4950 Albina Way, Unit 160
Mississauga, Ontario
L4Z 4J6, Canada
www.la-citadelle.com
info@la-citadelle.com
Edited by Rob Couvillon
All rights reserved. No part of this book may be
reproduced, in any form or by any means,
without permission in writting from the
publisher.
Content
Page Topic
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
Understanding fractions
The graphical representation of a fraction
Reading or writing fractions in words
Understanding the fraction notation
Understanding the mixed numbers
Reading and writing mixed numbers in words
Understanding mixed number notation
Understanding improper fractions
Understanding improper fraction notation
The link between mixed numbers and improper fractions
Conversion between mixed numbers and improper fractions
Whole numbers, proper fractions, improper fractions and mixed numbers
Understanding the addition of like fractions
Understanding the addition of like fractions (II)
Adding proper and improper fractions with like denominators
Adding mixed numbers with like denominators
Adding more than two like fractions
Understanding equivalent fractions
Finding equivalent fractions
Simplifying fractions
Checking fractions for equivalence
Equations with fractions
Adding fractions with unlike denominators
Adding fractions with unlike denominators using the LCD method
Understanding the subtraction of fractions with like denominators
Subtracting fractions with like denominators
Subtracting mixed numbers with like denominators
Subtracting fractions with unlike denominators
Subtracting fractions with unlike denominators using the LCD method
Order of operations (I)
Multiplying fractions
More about multiplying fractions
The order of operations (II)
Reciprocal of a fraction
Dividing fractions
Division operators
Order of operations (III)
Order of operations (IV)
Raising fractions to a power
Order of operations (V)
Converting fractions to decimals
Converting decimals to fractions
Order of operations (VI)
Time and Fractions
Canadian coins and fractions
Fractions, ratio, percent, decimals, and proportions
Fractions and Number Line
Comparing fractions
Solving equations by working backward method
Final Test
Answers
Preface
“The Book of Fractions" presents one of the primary concepts of middle and high school mathematics: the
concept of fractions. This book was developed as a workbook and reference useful to students, teachers, parents,
or anyone else who needs to review or improve their understanding of the mathematical concept of fractions.
The structure of this book is very simple: it is organized as a collection of 50 quasi-independent worksheets and
an answer key. Each worksheet contains:
· a short description of the concepts, notations, and conventions that constitute the topic of the worksheet;
· step-by-step examples (completely solved) demonstrating the techniques and skills the student should
gain by the end of each worksheet; and
· an exhaustive test to be completed independently by the students.
The concept of fractions and the relations between fractions and other types of numbers, like many abstract
mathematical concepts, is not always easy to understand. Bearing this in mind, the authors of this book introduce
each topic gradually, starting with the basic concepts and operations and progressing to the more difficult ones.
Geared specifically to help the beginners, the first part of the book contains graphical representations of the
fractions.
The techniques for solving both simple and complex equations implying fractions are explained. As well,
complete worksheets are provided, starting with very simple and basic equations and progressing to extremely
complex equations requiring the application of a full range of operations with fractions.
"The Book of Fractions" also presents the link between fractions and other related mathematical concepts, such
as ratios, percentages, proportions, and the application of fractions to real life concepts like time and money.
The importance of the concept of fractions comes both from its link to natural numbers and its link to more
complex mathematical concepts, like rational numbers. As such, the concept of fractions is a milestone in the
mathematical evolution of a student, being a concept that is simultaneously concrete (as a part of a whole) and
abstract (as a set of two numbers and a hidden division operation).
The concept of equivalent fractions is an essential part of understanding fractions, and a full range of techniques
is presented, starting with graphical representations (suitable for students in lower grades) and progressing to
advanced uses, like the factor tree method of finding the LCD.
The order of operations is also presented, gradually, after each main operation with fractions: addition,
subtraction, multiplication, and division; using multi-term expressions; expressions containing grouping symbols
of one or more levels; and more complex operations with fractions, like powers with positive and negative
exponents.
Single-step questions (requiring a basic knowledge and understanding of the topic presented in the worksheets)
and multi-step questions (requiring a complete understanding of all of the concepts presented in the worksheets
to that point) are presented throughout the entire book.
Combining more than 15 years of academic studies and 30 years of teaching experience, the authors of this book
wrote it with the intention of sharing their knowledge, experience and teaching strategies with all the partners
involved in the educational process.
Iulia & Teodoru Gugoiu,
Toronto, 2006
The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding fractions
1. A fraction represents a part of a whole.
Example 1.
2. The corresponding fraction is:
3
4
The whole is divided into four equal parts.
Three part are taken (considered).
The numerator represents how many parts are taken.
Fraction line or division bar
The denominator represents the number of
equal parts into which the whole is divided.
F01. Write the fraction that represents the part of the object that has been shaded:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
0
u)
© La Citadelle
v)
w)
x)
5
1
y)
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The Book of Fractions
Iulia & Teodoru Gugoiu
The graphical representation of a fraction
1. A fraction represents a part of a whole.
Example 1.
2. A corresponding graphical representation
(diagram) is:
3
4
The whole is divided into four equal parts.
Three part are taken (considered).
The numerator represents how many parts are taken.
Fraction line or division bar
The denominator represents the number of
equal parts into which the whole is divided.
F02. Draw a diagram to show each fraction (use the images on the bottom of this page):
a)
1
2
b)
1
3
c)
1
4
d)
2
5
e)
1
6
f)
2
4
g)
0
3
h)
2
9
i)
5
6
j)
2
12
k)
9
10
l)
1
1
m)
3
3
n)
4
6
o)
3
4
p)
4
12
q)
5
10
r)
4
9
s)
2
4
t)
8
12
u)
5
13
v)
5
16
w)
1
8
x)
7
49
y)
37
100
z)
11
18
0
© La Citadelle
1
6
www.la-citadelle.com
The Book of Fractions
Iulia & Teodoru Gugoiu
Reading or writing fractions in words
1. You can use words to refer to a part of a whole.
So one whole has:
2 halves
3 thirds
4 quarters
5 fifths
6 sixths
7 sevenths
8 eighths
9 ninths
10 tenths
11 elevenths
12 twelfths
13 thirteenths
20 twentieths
30 thirtieths
50 fiftieths
Example 1.
100 hundredths
1000 thousandths
1000000 millionths
1000000000 billionths
The fraction
3
4
can be written in words as:
three quarters
F03. Write the following fractions in words:
a)
2
3
b)
3
100
c)
1
10
d)
1
2
e)
3
7
f)
3
20
g)
1
1000
h)
4
5
i)
8
30
j)
8
13
k)
8
9
l)
5
6
m)
5
8
n)
7
1000
o)
3
50
p)
2
5
q)
21
100
r)
6
12
s)
7
11
t)
11
50
u)
11
1000000
v)
2
9
w)
7
10
x)
11
12
y)
2
50
z)
9
1000000000
F04. Find the fraction written in words:
a)
one third
b)
one half
c)
one sixth
d)
two fifths
e)
four sevenths
f)
seven eighths
g)
eleven fiftieths
h)
seven twentieths
i)
five twelfths
j)
eight ninths
k)
six tenths
l)
nine thousandths
m)
fifteen millionths
n)
eight sixths
o)
three fiftieths
p)
eleven billionths
q)
twenty-three hundredths
r)
seven thirteenths
s)
eleven twelfths
t)
three billionths
u)
thirteen thirtieths
v)
one fifth
w)
one eleventh
x)
eight ninths
y)
six tenths
z)
six twelfths
© La Citadelle
7
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding the fraction notation
1. A fraction also represents a
quotient of two quantities:
Example 1. The dividend (numerator) is 3.
3
The divisor (denominator) is 4.
The fraction in words is three quarters.
4
divident
divisor
2. The dividend (numerator) represents how many parts
are taken.
The divisor (denominator) represents the number of
equal parts into which the whole is divided.
A possible graphical representation of this fraction is:
F05. Fill out the following table:
Fraction
a)
2
3
b)
Numerator
(Dividend)
Denominator
(Divisor)
2
3
1
4
c)
The fraction written in words
Graphical representation
two thirds
three fifths
d)
e)
f)
3
5
2
2
g)
5
3
h)
.......... quarters
i)
five ..........
3
j)
k)
l)
© La Citadelle
4
.......... sixths
5
three ..........
8
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding the mixed numbers
fraction part
1. A mixed number is an addition of wholes and a part of a whole.
Example 1.
whole-number part
(the number of
complete wholes)
There are one complete whole and
three quarters of the second whole
The numerator indicates how many parts
are taken from the last whole.
The denominator represents the number
of equal parts into which the whole is
divided.
3
1
4
F06. Find the mixed number that corresponds to the shaded region:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
u)
v)
w) 0
1
2
0
x)
2
1
F07. Find a possible graphical representation of each mixed number:
a)
1
2
3
b)
3
1
2
c)
2
5
6
d)
3
4
5
e)
3
3
7
f)
1
h)
3
1
6
i)
2
2
9
j)
2
2
5
k)
2
2
10
l)
1
3
20
m)
o)
3
5
8
p)
1
7
16
q)
2
7
10
r)
2
4
5
s)
5
1
1
t)
v)
1
1
6
w)
5
1
2
x)
3
7
9
y)
2
3
8
z)
6
3
4
© La Citadelle
9
5
8
g)
4
5
11
6
6
10
n)
1
3
6
2
5
6
u)
3
7
10
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3
The Book of Fractions
Iulia & Teodoru Gugoiu
Reading and writing mixed numbers in words
1. You can use words to refer to a part of a whole.
So one whole has:
2 halves
3 thirds
4 quarters
5 fifths
6 sixths
7 sevenths
8 eighths
9 ninths
10 tenths
11 elevenths
12 twelfths
13 thirteenths
20 twentieths
30 thirtieths
50 fiftieths
Example 1.
The fraction 2
100 hundredths
1000 thousandths
1000000 millionths
1000000000 billionths
3
4
can be written in words as:
two wholes and three quarters or
two and three quarters
F08. Write the following mixed numbers in words:
a) 1
1
2
b)
h) 1
5
9
i)
o) 2
7
50
p) 2
3
17
w) 4 3
v) 2
1
3
c)
1
3
10
j)
1
3
100
q)
3
2
2
14
x)
1
4
d)
2
2
11
k)
3
9
1000
r) 2
2
5
15
y)
3
5
e) 1
5
6
f)
2
3
7
g)
3
5
12
l) 1
2
15
m) 3
7
20
n)
2
t)
7
19
u)
7
3
s) 1
1000000
40
2
1
60
z)
2
2
5
8
9
30
3
5
16
3
90
F09. Find the mixed numbers written in words:
a)
two and two thirds
b)
three and one half
c)
five and five sixths
d)
two and one third
e)
four and five sevenths
f)
seven and five fiftieths
g)
two and three quarters
h)
three and two ninths
i)
six and seven hundredths
j)
nine and one half
k)
eight and eleven fiftieths
l)
one and five billionths
m)
one and two elevenths
n)
eight and five sixths
o)
three and two twelfths
p)
five and three millionths
q)
twenty and three hundredths
r)
six and four fifteenths
s)
eleven and four thirtieths
t)
eight and seven tenths
u)
four and one third
v)
one and two fifths
w)
three and two elevenths
x)
eight and six ninths
y)
five and nine tenths
z)
one and eleven twelfths
© La Citadelle
10
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding mixed number notation
1. A mixed number is represented by the expression:
wholes
Example 1.
numerator
denominator
2
3
5
This mixed number written in words is two wholes and three fifths.
A possible graphical representation of this mixed number is:
The whole-number part is 2
(the number of complete wholes).
The numerator is 3.
The denominator is 5. 3
The fraction part is:
5
F10. Fill out the following table:
Mixed
Number
a)
2
3
5
b)
2
1
3
c)
Number
of wholes
Numerator
The mixed number
in words
Denominator
2
3
5
1
3
4
Graphical representation
two and three fifths
d)
e)
2
f)
3
g)
3
5
2
2
3
5
2
three and a half
h)
6
i)
j)
3
k)
l)
© La Citadelle
two and four ...........
.......... and three fifths
2
four and .......... thirds
2
.......
11
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding improper fractions
1. For an improper fraction the number of parts taken (the numerator) is equal to or greater than the number of
parts the whole is divided into (the denominator).
Example 1.
5
3
This is a possible graphical representation of this improper fraction:
F11. Find the improper fraction that corresponds to the shaded region:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
p)
0
1
2
0
1
2
3
o)
q)
r)
s)
t)
u)
v)
w)
x)
F12. Find a possible graphical representation of each improper fraction:
a)
3
2
b)
4
3
c)
5
2
d)
7
3
e)
7
5
f)
11
4
g)
8
6
h)
10
8
i)
20
9
j)
12
5
k)
22
10
l)
25
12
m)
30
16
n)
40
24
o)
21
6
p)
28
13
q)
11
3
r)
40
18
s)
55
49
t)
25
6
u)
17
9
v)
30
12
w)
15
4
x)
17
5
y)
24
10
z)
13
3
© La Citadelle
12
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding improper fraction notation
1. An improper fraction is represented by the expression:
Example 1.
numerator
denominator
5
3
The numerator is 5
The denominator is 3
The improper fraction in words is five thirds. A possible
graphical representation of this improper fraction is:
where the numerator is equal to or greater than the
denominator.
F13. Fill out the following table:
Fraction
a)
5
4
b)
Numerator
Denominator
5
4
7
4
c)
The fraction in words
Graphical representation
five quarters
seven fifths
d)
e)
f)
16
5
........
12
7
g)
5
11
h)
.......... quarters
i)
seven ..........
........
13
j)
k)
l)
© La Citadelle
13
.......... sixths
5
thirteen ..........
13
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The Book of Fractions
Iulia & Teodoru Gugoiu
The link between mixed numbers and improper fractions
1. There is a direct link between a mixed number and an improper fraction. A mixed number is a short way to write
the sum of a whole number and a fraction.
5
2
2
= 1 = 1+
3
3
3
Example 1:
9
3
3
= 1 = 1+
6
6
6
Example 2:
F14. Find the mixed number and the improper fraction that correspond to each picture:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
0
m)
1
n)
0
1
2
3
o)
2
p)
q)
r)
s)
t)
u)
v)
w)
x)
© La Citadelle
14
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The Book of Fractions
Iulia & Teodoru Gugoiu
Conversion between mixed numbers and improper fractions
1. To convert a mixed number to an improper fraction,
use the formula:
3. To convert an improper fraction to a mixed number,
divide the numerator n into the denominator d to obtain
the quotient q and the remainder r. Then write:
n w×d + n
=
d
d
3 2 × 5 + 3 10 + 3 13
Example 1. 2 =
=
=
5
5
5
5
w
n
r
=q
d
d
2. Fractions that have a denominator of 0 are not
defined.
9
1
=2
4
4
Example 2.
F15. Write each mixed number as an improper fraction:
a) 1
1
2
b)
2
3
c)
3
h) 5
3
4
i)
7
30
j)
o) 2
3
50
p) 4
4
5
v) 0
2
9
w) 2
0
10
2
2
3
4
d)
3
1
2
e) 2
3
7
f)
5
3
20
g)
5
1
5
13
k)
2
8
9
l) 3
5
6
m)
2
5
8
n)
12
q)
2
21
100
r)
2
1
12
s) 3
2
11
t)
2
49
50
u)
x)
3
2
0
y)
3
3
3
z) 2
3
10
7
100
2
11
100
15
10
F16. Write each improper fraction as a mixed number:
a)
3
2
b)
4
3
c)
5
4
d)
9
2
e)
13
7
f)
h)
14
5
i)
8
3
j)
80
13
k)
80
9
l)
50
6
m)
o)
13
5
p)
22
5
q)
21
10
r)
16
12
s)
70
11
t)
v)
0
9
w)
7
0
x)
11
7
y)
20
15
z)
70
9
© La Citadelle
15
33
20
g)
125
10
60
8
n)
17
10
111
50
u)
111
10
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The Book of Fractions
Iulia & Teodoru Gugoiu
Whole numbers, proper fractions, improper fractions and mixed numbers
1. Although written in fraction notation,
some numbers are actually whole numbers.
Example 1:
10
=5
2
Example 2: 2
3
=3
3
2. A whole number can be converted into a fraction.
This conversion is not unique.
Example:
5=
n
d
where n < d
Example:
An improper fraction is: n
d
where n ³ d
Example:
3. A proper fraction is:
A mixed number is:
w
A mixed number
in standard form is:
w
n
d
Example:
2
3
7
3
4
5
3
2
n
where n £ d Example: 5
d
3
4. Fractions that have a denominator of 0 are not defined.
50
10
3
30
14
=4 =4 =2 =3
10
10
3
10
7
F17. Convert fractions to whole numbers. Identify the expressions that are not defined.
1
1
b)
h) 1
1
1
i)
2
o)
0
1
p)
2
0
a)
2
2
c)
3
3
j)
q)
2
7
7
d)
4
2
e)
9
3
f)
24
3
g)
24
12
k)
3
9
3
l) 4
55
5
m) 2
16
8
n)
0
0
r)
0
5
5
s) 2
0
9
t)
0
6
u)
0
100
10
2
300
100
0
0
0
F18. Convert whole numbers to fractions (the conversion is not unique, so give at least
two solutions):
a)
1
b)
3
c)
7
d)
4
e) 2
f)
5
g)
10
h)
0
i)
25
j)
100
k)
11
l)
m)
13
n)
17
8
F19. Identify each of the following expressions as a whole number, a proper fraction,
an improper fraction, a mixed number, or a not defined expression:
a)
3
2
b)
1
c)
5
4
d)
2
9
e) 2
h)
4
0
i)
0
3
j)
0
k)
80
9
o)
2
3
3
p)
22
5
q)
30
r)
v)
2
3
5
w)
22
5
x)
10
11
y)
© La Citadelle
1
7
f)
l)
5
16
m)
16
2
s)
70
11
t)
16
2
z)
7
1
16
33
20
g)
2
25
10
3
5
8
n)
2
17
10
1
11
50
u)
0
10
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding the addition of like fractions
Two fractions with the same denominators are called like fractions. When you add two fractions, you add the parts
of the whole they represent.
1 2 3
+ =
4 4 4
Example 1.
So, by adding 1 quarter and 2 quarters you get 3 quarters.
F20. Add the fractions that correspond to the shaded regions:
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
0
u)
© La Citadelle
v)
w)
17
1
x)
y)
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding the addition of like fractions (II)
1. Sometimes when you add two like fractions, the number of parts you add exceeds a whole. The result is an
improper fraction or a mixed number.
Example 1.
=
+
Or, in mathematical symbols:
3 3 6
2
+ = =1
4 4 4
4
F21. Add the fractions that correspond to the shaded regions. Express the result both as an
improper fraction and as a mixed number.
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)
© La Citadelle
18
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Iulia & Teodoru Gugoiu
Adding proper and improper fractions with like denominators
1. To add proper or improper fractions with like
denominators (called like fractions), add the numerators
and keep unchanged the denominator, according to the
rule:
n1 n2 n1 + n2
+ =
d d
d
1 2 3
+ =
5 5 5
Example 1.
2. If the result is an improper fraction, you can
change it to a mixed number.
Example 2.
3 2 5
1
+ = =1
4 4 4
4
F22. Add the fractions:
a)
2 1
+
4 4
b)
1 1
+
3 3
c)
2 1
+
5 5
d)
3 5
+
11 11
e)
1 2
+
6 6
f)
2 3
+
7 7
g)
5 7
+
19 19
h)
20 10
+
100 100
i)
6 23
+
35 35
j)
2 10
+
41 41
k)
0 11
+
13 13
l)
21 11
+
54 54
m)
2 3
+
10 10
n)
5 4
+
12 12
o)
3 10
+
17 17
p)
0 0
+
4 4
q)
3 2
+
13 13
r)
7
3
+
20 20
s)
7 11
+
25 25
t)
5 2
+
9 9
u)
2 5
+
19 19
v)
2 3
+
10 10
w)
7 13
+
30 30
x)
3 8
+
13 13
y)
15
25
+
100 100
F23. Add the fractions. Write the result as a mixed number in standard form.
a)
2 2
+
3 3
b)
3 2
+
4 4
c)
3 4
+
5 5
d)
3 5
+
6 6
e)
3 5
+
7 7
f)
5 7
+
8 8
g)
7 8
+
9 9
h)
7 7
+
10 10
i)
8 10
+
11 11
j)
9 5
+
12 12
k)
10 11
+
15 15
l)
4 19
+
20 20
m)
10 20
+
25 25
n)
44 44
+
50 50
o)
99 11
+
100 100
p)
7 6
+
9 9
q)
8 9
+
10 10
r)
33 19
+
40 40
s)
4 9
+
3 3
t)
15 9
+
10 10
u)
12 5
+
9 9
v)
20 12
+
10 10
w)
22 77
+
50 50
x)
13 12
+
3 3
15 34
+
y) 10 10
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Iulia & Teodoru Gugoiu
Adding mixed numbers with like denominators
Example 1. 2 3 + 3 4 = ( 2 + 3) 3 + 4 = 5 7 = 6 2
1. To add mixed fractions with like denominators, add
separately the wholes and separately the
numerators, and keep the denominator unchanged:
5
5
5
5
5
2. In the same way you can add whole and mixed numbers.
n
n
n +n
w1 1 + w2 2 = ( w1 + w2 ) 1 2
d
d
d
Example 2. 2 + 3
1
0
1
0 +1
1
= 2 + 3 = (2 + 3)
=5
4
4
4
4
4
F24. Add the mixed numbers. Write the result as a mixed number in standard form or as
a whole number:
1
1
1
1
1
2
1
3
3 2
a) 1 + 2
b) 1 + 2
c) 1 + 2
d) 1 + 3
e) 2 + 1
2
2
3
3
4
4
5
5
6 6
3
2
f) 2 + 3
7
7
2
3
g) 2 + 3
8
8
2
1
h) 3 + 0
9
9
3 7
k) 2 +
5 5
4
6
l) 4 + 2
7
7
m) 2
2
8
p) 5 + 2
9
9
q) 2
22
11
+5
10
10
v) 2
10
41
+5
30
30
u) 2
21 12
+1
19 19
i) 2
5 3
+
10 10
j) 3
5
6
+2
15
15
1
12
+3
10
10
n) 2
12
11
+5
20
20
o) 3
8
8
+3
11 11
r) 3
11
22
+2
30
30
7
9
s) 3 + 2
3
3
t) 2
15
25
+3
10
10
w) 3
2
22
+1
23 23
x) 4
33
22
+2
35
35
y) 1
15
95
+2
100
100
F25. Add the wholes and the mixed numbers. Write the result as a mixed number in standard form
or as a whole number.
1
2
1
2
5
a) 2 + 1
b) 1+ 2
c) 1+ 3
d) 2 + 5
e) 5 + 2
2
3
4
5
6
f)
5+3
9
7
g)
5
7 +6
8
k)
9
3 +2
5
l)
5 +1
p)
9
19
+9
9
q)
20 + 10
u)
1+ 9
19
19
v)
4
© La Citadelle
h)
8
7
m) 1
100
10
14
+5
30
10
9
i)
1
11
+3
10
n)
11
30
40
23
3+
r)
11 + 3
w)
2+2
20
1
+ 11
10
j)
0+2
0
15
2+2
22
20
o)
13 + 13
s)
9+3
10
3
t)
4
x)
1+ 1
37
35
y)
111 + 1
13
11
4
+4
10
111
100
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Iulia & Teodoru Gugoiu
Adding more than two like fractions
1. To add more than two fractions or whole numbers, start to add in order, from left to right.
2 5 æ
2ö 5
2 5
7
3
1 + 3 + = ç1 + 3 ÷ + = 4 + = 4 = 5
4 4 è
4ø 4
4 4
4
4
Example 1.
2. Because the addition is a commutative operation, the order in which you add the fractions is not important.
So, group them conveniently.
2
3 1
2
2ö
æ 2 1ö æ 3
+ 2 +1 + 2 +1 = 2 + ç +1 ÷ + ç 2 +1 ÷ = 2 + 2 + 4 = 8
3
5 3
5
3
3
5
5
è
ø è
ø
Example 2.
F26. Add the fractions:
a) 1 + 3 + 3 1
2
f)
2
1 2 3
1+ + +
3 3 3
1 2 3
+ +
4 4 4
g)
2
7
3
17
+ +2
10 10
10
3
5
4
5
l)
1 5
2
1 + +1
9 9
9
1
9
7
9
q)
1
v)
1
3
5
3 + 2 +1
4
4
4
k) 1 + + 1
p) 1 + + 2
1
9
b)
4
9
u) 0 + + 1
8
9
1
3
5
+ +2
10 10
10
e)
1
2
3
4
+1 + 2 + 3
5
5
5
5
3
7
3
+2 +
50
50 50
j)
1
3
5
7
1 +2 +3 +4
6
6
6
6
3 5
7
+ +1
12 12 12
o)
1 3 5 7 9
+ + + +
11 11 11 11 11
21
31
+3
100
100
t)
2
1
3
5
+1 + 0
10 10
10
y)
1 2 3 4 5 6
+ + + + +
7 7 7 7 7 7
c)
3 5
1
1 + +2 +2
4 4
4
d)
1
1
1
3 + 2 +1
2
2
2
h)
5
3
2 +3+
2
2
i)
2
3
3
n)
2
1
3
5
1 +2 +5
2
2
2
s)
1+ 2
x)
2
1
3
2
3
m) 1 + 2 + 3
r)
3
3
2
3
w) 3 + 2 + 1
1
3
5
10 15
20
+1 + + 3
10 10 10
10
F27. Add the whole and the mixed numbers. Write the result as a mixed number in standard form
or as a whole number.
a)
1+
1 3
+
2 2
b)
f)
1+
2
4
+2+
3
3
g) 1 +
2
5
k) 1 + 1 + 2
l)
2
1
+2+
4
4
7
+2
10
2
3
2 +1 + 3
9
9
p)
7
8
2 +1 +1
9
9
q) 1 + 2
u)
2
7
0 + 2 +1
9
9
v) 3 5 + 1 5 + 2
© La Citadelle
4
5
7
+3
10
10
4
c) 1 + 1 + 3 + 2
d)
1 1
+ +2
2 2
h) 7 + 2 + 5
i)
1
4
4
2
2
4
3
2
3
7
11
+3+
50
50
n) 1 5 + 1 7 + 2
m) 2 + 1 + 2
12
12
11
22
+ 2 + +1
100
100
r) 1 3 + 2 5 + 5 7 + 1
s)
w) 2 4 + 1 1 + 2
x) 1 9 + 2 7 + 0
2
3
2
2
3
21
3
10
10
e)
3
3
+1+ + 2
5
5
9
6
7
6
j) 1 + + + 2
o) 1 2 + 2 + 3 4 + 2 + 1 8
11
t) 1
11
11
4
7
+3+ + 2
10
10
y) 2 9 + 1 7 + 1 + 3 5
10
10
10
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding equivalent fractions
1. Two fractions are considered equivalent if they represent the same part of the whole.
We’ll see later that equivalent fractions are equal in value and correspond to the same decimal number.
Example 1.
The shaded region can be expressed as:
So these fractions are considered equivalent (equal),
and you can write:
1 or 2 or 3
6
1 2 3 6
or
= = =
2
4
6
12
2
4
6
12
F28. For each image find the equivalent fractions that correspond to the shaded part:
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)
© La Citadelle
22
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Iulia & Teodoru Gugoiu
Finding equivalent fractions
There are two methods to find equivalent fractions.
Method 1. Multiply both the numerator and the
denominator by the same number, according to the
formula:
Method 2. Divide both the numerator and the
denominator by a common factor, according to the
formula:
n n¸a
Example 1.
Example 2.
n n´a
=
d d ´a
d
1 1´ 2 2
1 1´ 3 3 3 ´ 5 15
=
=
=
= =
=
2 2´ 2 4
2 2 ´ 3 6 6 ´ 5 30
So: 1 = 2 = 3 = 15
2 4 6 30
=
d ¸a
6
6¸2 3 3¸3 1
=
= =
=
12 12 ¸ 2 6 6 ¸ 3 2
6 3 1
So:
= =
12 6 2
F29. Find at least three equivalent fractions by using the method 1:
a)
1
2
b)
1
3
c)
2
3
d)
1
4
e)
3
4
f)
1
5
g)
3
5
h)
1
6
i)
5
6
j)
1
7
k)
2
7
l)
1
10
m)
1
100
n)
5
12
o)
2
11
p)
2
9
q)
3
7
r)
5
11
s)
2
15
t)
4
5
u)
5
7
v)
1
30
w)
3
50
x)
3
200
y)
3
1000
F30. Find equivalent fractions by using the method 2:
a)
2
4
b)
3
9
c)
4
16
d)
5
25
e)
6
36
f)
10
100
g)
4
12
h)
20
25
i)
4
6
j)
15
20
k)
30
45
l)
25
40
m)
n)
60
90
o)
60
150
p)
20
35
q)
75
120
r)
66
154
s)
56
420
t)
u)
27
81
v)
63
77
w)
22
121
x)
225
625
64
y) 1024
© La Citadelle
18
60
23
32
128
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Iulia & Teodoru Gugoiu
The Book of Fractions
Simplifying fractions
To simplify (or reduce) a fraction means to find the
equivalent fraction having the simplest form (in lowest
terms).
Method 1. You can simplify a fraction by repetitive
division of the numerator and the denominator by a
common factor.
n n¸a
=
= ... where a is a common divisor of n and d
d d ¸a
Method 2. You can simplify a fraction by division of the
numerator and denominator by the Greatest Common Factor
(GCF).
n n ¸ GCF
=
= fraction in lowest terms
d
d ¸ GCF
To find the GCF build the factor trees for the numerator and
denominator.
24 = 23 ´ 31
Example 2.
24
Example 1.
12
2
2
15
2
3
GCF = 2 2 ´ 31 = 12
30
2
6
2
24 24 ¸ 2 12 12 ¸ 2 6
6¸3 2
=
=
=
= =
=
60 60 ¸ 2 30 30 ¸ 2 15 15 ¸ 3 5
60 = 2 2 ´ 31 ´ 51
60
3
5
24 24 ¸ 12 2
=
=
60 60 ¸ 12 5
F31. Write each fraction in lowest terms by using the method 1:
a)
6
72
b)
18
42
c)
50
75
d)
32
128
e)
60
80
f)
32
160
g)
54
90
h)
22
132
i)
200
240
j)
21
147
k)
36
126
l)
75
350
m)
105
135
n)
512
4096
o)
24
132
p)
50
225
q)
48
112
r)
60
264
s)
48
360
t)
u)
180
252
v)
126
270
w)
264
600
x)
45
300
y)
336
480
135
270
F32. Write each fraction in lowest terms by using the method 2:
a)
18
24
b)
48
60
c)
54
81
d)
90
120
e)
56
72
f)
28
98
g)
32
60
h)
48
120
i)
28
52
j)
64
80
k)
30
45
l)
25
40
m)
18
60
n)
60
75
o)
60
150
p)
20
45
q)
75
120
r)
66
154
s)
56
420
t)
u)
27
81
v)
630
770
w)
22
121
x)
225
625
y)
© La Citadelle
24
32
128
64
1024
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The Book of Fractions
Iulia & Teodoru Gugoiu
Checking fractions for equivalence
Given two or more fractions, you can check whether or not they are equivalent (equal).
2. For two fractions you can use the cross-multiplication
method. If the cross-products you get are equal then the two
fractions are equivalent.
1. Express each fraction in lowest terms and then
compare them. If you get the same fraction, the
original fractions are equivalent.
Example 1.
24
32
In lowest terms
they are:
So:
24 15
=
32 20
3
4
and 15
and
20
and
24 21
¹
32 30
3
4
and
21
30
a
b
7
10
and
c
d
are equivalent if
a´d = b´c
Example 2.
3
4
15 21
¹
20 30
and
21
are not equivalent because 3 ´ 30 ¹ 4 ´ 21
30
F33. Check if the fractions are equivalent (use the lowest terms method):
a)
2 3
;
4 6
b)
2 3
;
6 9
c)
2 6
;
3 8
d)
4 8
;
6 12
e)
5 6
;
10 18
f)
30 48
;
75 120
g)
30 20
;
35 24
h)
24 15
;
56 35
i)
9 15
;
25 40
j)
40 30
;
70 54
k)
25 15
;
80 50
l)
100 30
;
220 66
m) 14 ; 56
24 96
n)
15 40
;
40 104
o)
25 40
;
45 75
p)
10 16 20
;
;
15 24 30
q)
6 18 25
;
;
10 30 40
r)
25 5 40
; ;
30 6 48
s)
10 6 12
; ;
21 15 35
t)
60 15 25
;
;
96 24 40
u)
1 4 7 5
; ; ;
2 8 14 10
v)
3 8 10 12
; ; ;
4 10 12 14
w) 33 ; 21 ; 15 ; 3
44 28 20 4
x)
1 2 4 8
; ; ;
2 4 8 16
y)
1 2 3 4
; ; ;
2 3 4 5
F34. Check if the fractions are equivalent (use the cross-multiplication method):
a) 1 ; 2
2 3
b)
2 4
;
4 8
c)
6 4
;
12 10
d)
12 15
;
16 20
e)
12 4
;
35 12
5 6
;
4 5
g)
9 3
;
12 4
h)
4 12
;
10 30
i)
5 7
;
7 5
j)
30 48
;
35 56
k) 40 ; 72
45 80
l)
24 36
;
40 60
m) 40 ; 49
64 81
n)
5 2
;
20 8
o)
4 9
;
9 20
p) 1 3 ; 9
4 5
q)
125 200
;
15 24
r)
11 13
;
13 15
s)
65 40
;
85 51
t)
105 15
;
133 19
u) 3 ; 9 ; 15
4 12 20
v)
4 5 6
; ;
5 6 7
w) 4 ; 8 ; 12
8 12 16
x)
12 18 30
;
;
20 30 50
y)
10 25 40
;
;
14 35 56
f)
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25
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The Book of Fractions
Iulia & Teodoru Gugoiu
Equations with fractions
Method 1. You can solve simple equations with
fractions if you use two properties of equivalent
fractions:
n n¸a
=
d d ¸a
Method 3. This is the best method for solving simple
equations with fractions:
To get a term of a fraction, multiply the adjacent terms and
divide by the opposite term.
n n´a
=
d d ´a
Example 1:
x b
=
a c
Solution:
3
?
=
16 48
3
3´ 3
9
=
=
16 16 ´ 3 48
a´b
c
Example 3 (the result is a whole number):
Method 2. When applying the method 1, sometimes
you need first to express the fraction in the lowest
terms: :
Example 2:
Solution:
4 ?
=
8 10
x=
Û
x 2
=
15 3
Û
x=
15 ´ 2 30
=
= 10
3
3
Example 4 (the result is a fraction):
x 3
=
2 7
4 1 1´ 5 5
= =
=
8 2 2 ´ 5 10
Û
x=
2´3 6
=
7
7
F35. Find the unknown factor of the fraction using the method 1:
a)
f)
k)
1
=
2 8
6 30
=
40
5
=
25
75
b)
3
g)
l)
=
6
9
12
4
6 18
=
18
=
c)
h)
m)
2
=
4 32
8
2=
d)
3
=
10 100
n)
i)
3
e)
3 12
=
5
j)
8 40
=
12
9
=
16 128
o)
11 12
=
5
=
3=
12
16
5
F36. Find the unknown factor of the fraction using the method 2:
a)
2
=
4 6
b)
3 7
=
9
c)
8 10
=
12
d)
12
=
28 21
e)
f)
9
=
12 20
g)
12 24
=
15
h)
12 16
=
42
i)
=
18
48
j)
72
3
=
60
5
10
=
40
56
F37. Solve for x using the method 3 (see example 3 above):
a)
1 x
=
3 6
b)
2 10
=
3 x
c)
3 12
=
x 20
d)
4 20
=
5 x
e)
3 15
=
x 35
f)
5=
15
x
g)
4 3
=
x 6
h)
x 15
=
16 20
i)
20 15
=
x 18
j)
16 24
=
x 45
k)
45 40
=
72 x
l)
6=
x
6
m)
30 80
=
x 48
n)
40 x
=
25 30
o) 28 = x
20 25
F38. Solve for x using the method 3 (see example 4 above):
a)
1 x
=
2 3
b)
2 x
=
5 3
c)
4 5
=
3 x
d)
1 5
1 =
2 x
e)
9 x
=
16 3
f)
6 x
=
7 8
g)
12 x
=
16 5
h)
20 3
=
24 x
i)
16 x
=
64 2
j)
2
© La Citadelle
26
20 x
=
35 15
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The Book of Fractions
Iulia & Teodoru Gugoiu
Adding fractions with unlike denominators
To add fractions with different denominators, you must first replace them with equivalent fractions having the same
denominators.
Method 1. This is a general method that works for
two fractions and uses cross-multiplication
according to the formula:
Method 2. First, express the fractions in lowest terms and
then write down equivalent fractions by multiplication until you
get the lowest common denominator:
Example 2:
2 3
a c a´d + b´c
+ =
b d
b´d
+ =?
4 15
2 1 3 4 5
= = = =
= ...
4 2 6 8 10
Example 1:
2 3 2 ´ 4 + 5 ´ 3 8 + 15 23
3
+ =
=
=
=5
5 4
5´ 4
4
4
4
So:
3 1 2
= =
= ...
15 5 10
2 3
5 2
7
+ = + =
4 15 10 10 10
F39. Add the fractions using the method 1:
a)
1 1
+
2 3
b) 2 + 3
3 4
c)
1 2
+
4 5
d) 1 + 1
5 6
e) 2 + 3
5 7
f)
3 5
+
7 8
g) 3 + 5
8 9
h)
5 3
+
9 10
i)
3 1
+
10 11
j)
1 2
+
10 11
k)
3 4
+
10 15
l) 10 + 3
3 10
m)
5 2
+
9 3
n)
1 2
+
10 15
o)
3 4
+
15 20
p)
1 2
1 +
2 3
q) 2 + 2 4
3
5
r)
5
3
1 +2
3
6
s) 110 + 11
11 10
t) 1 2 + 2 1
5
4
u)
1
4
+
20 16
v) 1 3 + 8
40 25
w)
5
3
+1
12 4
x) 2 1 + 1 3
6 8
y) 1 1 + 3
16 32
e)
6 1
+
10 7
F40. Add the fractions using the method 2:
a)
1 2
+
2 3
b) 2 + 1
6 4
c)
3 2
+
4 10
d) 2 + 5
5 6
f)
8 3
+
14 8
g) 4 + 3
8 9
h)
5 5
+
9 10
i)
2 3
+
10 9
j)
5 2
+
10 12
k)
2 5
+
10 15
l) 10 + 3
3 12
m)
5 2
+
9 3
n)
1 2
+
10 15
o)
3 6
+
15 20
p)
1 12
1 +
2 18
q) 1 2 + 2 3
3
5
r)
4
3
1 +2
3
6
s) 1 2 + 8
3 10
t) 1 3 + 2 3
5
4
u)
4
4
+
20 16
v) 1 5 + 5
40 25
w)
4
4
+1
12 16
x) 2 2 + 1 6
6 8
y) 1 5 + 5
16 48
© La Citadelle
27
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The Book of Fractions
Iulia & Teodoru Gugoiu
Adding fractions with unlike denominators using the LCD method
1. First find the Least (Lowest) Common Denominator
(LCD), and then add fractions.
Example 1:
2. You can use the LCD method to add more than two
fractions.
Example 2:
5
2
+
=?
24 30
24 = 23 ´ 31
1 1
1
+ +
=?
12 45 40
1
1
24
30
12
2
15
2
2
3
6
2
5
3
1
30 = 2 ´ 3 ´ 5
12 = 2 2 ´ 31
120 ¸ 24 = 5
120 ¸ 30 = 4
LCD = 23 ´ 31 ´ 51 = 120
12
2
45
2
15
3
6
3
3
45 = 32 ´ 51
40
20
2
10
2
5
2
40 = 23 ´ 51
LCD = 23 ´ 32 ´ 51 = 360
5
2
5´ 5
2´ 4
25
8
33 11
+
=
+
=
+
=
=
24 30 24 ´ 5 30 ´ 4 120 120 120 40
1 1
1
1´ 30
1´ 8
1´ 9
47
+ +
=
+
+
=
12 45 40 12 ´ 30 45 ´ 8 40 ´ 9 360
F41. Add the fractions using the LCD method (see example 1 above):
1
3
a) 1 + 5
b) 2 + 3
c)
d) 4 + 1
+
4 6
6 16
16 20
15 12
e)
3 3
+
10 14
f)
3 5
+
14 4
g)
3 5
+
16 18
h)
5 2
+
12 9
i)
3 1
+
10 22
j)
1 2
+
6 33
k)
3 4
+
10 15
l)
7
3
+
12 20
m)
5 5
+
9 30
n)
1
2
+
10 25
o)
2 4
+
15 25
p)
1
q)
1
3
+2
30
50
r)
1
s) 1 1 + 1
15 50
t)
1
u)
3
3
+
20 16
v)
1
7
3
+
40 25
w)
7
3
+1
12 40
x) 2 1 + 1 3
14 84
1
3
1
+
y) 48 64
1
7
+
25 30
5
5
3
+2
36
16
3
1
+2
50
45
F42. Add the fractions using the LCD method (see example 2 above):
a)
1 2 3
+ +
2 3 4
b)
1 5 3
+ +
4 6 8
c)
1 3 7
+ +
6 8 10
d)
1 5 7 3
+ + +
4 6 12 16
e)
3 3 1
+ +
4 10 12
f)
3 1 1
+ +
10 15 30
g)
5 2 2
+ +
12 9 14
h)
1 1 1 1 1
+ + + +
2 3 4 5 6
i)
3 4 3
+ +
10 15 16
j)
1
3 2
+
+
12 20 15
k)
1 1
1
+ +
9 30 36
l)
1 1 5 2 3
+ + + +
4 6 12 15 20
m)
1 5 3
+ +
3 6 10
n)
3 1 1
+ +
10 25 6
o)
1
2
2
+ +
15 25 35
p)
1 3 1
5
3
+ +
+ +
15 4 20 32 50
q)
1 1 1 1
+ + +
2 3 4 5
r)
1 1 1 1
+ + +
3 4 5 6
s)
1 3 7
3
+ + +
5 10 15 20
t)
1 3 1
3
1
+ + +
+
5 10 15 20 25
© La Citadelle
28
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The Book of Fractions
Iulia & Teodoru Gugoiu
Understanding the subtraction of fractions with like denominators
1. When you subtract two fractions, you subtract the parts of the whole they represent.
Example 1.
3 2 1
- =
4 4 4
So, by taking 2 quarters away from 3 quarters you get 1 quarter.
F35. Subtract the fractions that correspond to the shaded regions using the pattern:
all shaded parts - all light shaded parts=all dark shaded parts
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
q)
r)
s)
t)
0
u)
© La Citadelle
v)
w)
29
1
x)
y)
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The Book of Fractions
Iulia & Teodoru Gugoiu
Subtracting fractions with like denominators
1. To subtract proper or improper fractions with like
denominators, subtract the numerators and keep the
denominators unchanged.
Example 1.
2. To subtract mixed numbers with like denominators, first
change mixed numbers to improper fractions, and then
subtract the numerators. Keep the denominators
unchanged.
n1 n2 n1 - n2
- =
d d
d
3 2 1
- =
5 5 5
1 2 7 5 2
2 -1 = - =
3 3 3 3 3
Example 2.
F44. Subtract the fractions (write the results in lowest terms):
a)
2 1
4 4
b)
4 2
3 3
c)
4 1
5 5
d)
10 3
11 11
e)
5 2
6 6
f)
5 2
7 7
g)
13 11
19 19
h)
19 17
10 10
i)
23 13
35 35
j)
7
3
40 40
k)
5 1
12 12
l)
7
1
54 54
m)
7 1
9 9
n)
11 7
32 32
o)
23 9
7 7
p)
3 1
4 4
q)
15 3
16 16
r)
13 7
24 24
s)
19 7
40 40
t)
5
1
1000 1000
v)
7
1
30 30
w)
17 7
50 50
x)
13 2
33 33
y)
13 7
60 60
u)
13
3
100 100
F45. Subtract the mixed numbers (write the results in lowest terms):
1 2
a) 1 3 3
b)
1 3
2 2 2
5
f) 3 - 1
7
g)
2-
l)
3
7 3
14 14
q)
5
15 5
16 16
v)
5
7
1
-2
30
30
k) 2
5 7
12 12
1 3
p) 2 4 4
u) 3
13
17
-2
100
100
© La Citadelle
11
19
d)
3
1
3
-2
11
11
e)
5
2
5 -3
6
6
i)
2
3 13
35 35
j)
4
1
5
m) 5 - 2
9
9
n)
2
11 17
-1
32 32
o)
2
9
3 -2
7
7
r) 1
3
7
24 24
s)
3
19
9
-1
40 40
t)
2
w) 3
13
7
-2
50
50
x)
3
1
4
-2
33
33
c) 2 4 - 1 3
5 5
h) 10
1 11
10 10
30
3
5
-2
40
40
5
3
-1
1000 1000
13
7
-2
60
60
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y)
5
The Book of Fractions
Iulia & Teodoru Gugoiu
Subtracting mixed numbers with like denominators
Method 1. To subtract mixed numbers with like
denominators, subtract separately the wholes, and
separately the numerators. Keep the denominator
unchanged, according to the rule:
Method 2. If the numerator of the first fraction is less than
the numerator of the second fraction, you must rewrite the
first fraction as an improper fraction by exchanging one
whole.
n1
n
n -n
- w2 2 = ( w1 - w2 ) 1 2
d
d
d
w1
Example 2:
1 2
4 2
4-2
2
3 - 1 = 2 - 1 = (2 - 1)
=1
3 3
3 3
5
5
Example 1:
3 1
3 -1
2
3 - 1 = (3 - 1)
=2
5 5
5
5
F46. Subtract the mixed numbers using the method 1 (write the results in lowest terms).
2 1
a) 2 - 1
3 3
b)
3
1
5 -3
2
2
5
2
f) 5 - 2
7
7
g)
6
l)
k) 4
7
5
-1
12 12
3
1
p) 5 - 2
4
4
u) 5
47
27
-3
100 100
c)
3
1
10 - 2
5
5
11
1
-5
19
19
h)
5
5
7
3
-3
14 14
7
1
m) 3 - 2
9
9
q)
7
15
7
-3
16 16
r)
4
v)
9
17
11
-7
30
30
w)
5
10
5
-5
11
11
e)
4
1
4 -2
6
6
17
12
-2
35
35
j)
5
o)
5 2
4 -1
7 7
27
17
-5
40
40
t)
5
37
17
-2
1000
1000
11 5
33 33
y)
7
37
17
-5
60
60
d)
11
i)
3
n)
10
17
5
-2
24
24
s)
7
11 5
50 50
x)
2
9
1
-2
10
10
17
9
-7
32
32
17
7
-3
40
40
F47. Subtract the mixed numbers using the method 2 (write the results in lowest terms):
a)
1 2
2 -1
3 3
b)
1
3
5 -2
7
7
c)
3
4
5 -2
5
5
f)
1 5
3 -1
9 9
g)
2-
11
19
h)
10
k)
2
5 7
12 12
l)
6
9
2
-1
14 14
2
5
m) 7 - 5
9
9
p)
5 -1
3
4
q)
5
7 17
16 16
r)
2
u)
6
v)
9
1
7
-3
30
30
w)
5
13
17
-2
100
100
© La Citadelle
d)
4
2
5
-1
11 11
e)
2
5
7 -2
6
6
i)
2
3 13
35 35
j)
4
n)
7
11
27
-5
32
32
o)
3
6
8 -5
7
7
13 17
-1
24 24
s)
5
39
29
-2
40
40
t)
3
17
7
-1
1000 1000
3 17
50 50
x)
2
11
23
-1
33 33
y)
4
7
15
-1
60 60
1 11
10 10
31
3
5
-2
40
40
www.la-citadelle.com
The Book of Fractions
Iulia & Teodoru Gugoiu
Subtracting fractions with unlike denominators
To subtract fractions with unlike denominators, first you must replace the fractions with equivalent fractions having
like denominators.
Method 1. This is a general method that works for
two fractions and uses cross-multiplication
according to the formula:
Method 2. First, express the fractions in lowest terms and
then write down equivalent fractions by multiplication until you
get the lowest common denominator:
2 3
- =?
4 15
Example 2:
a c a´d -b´c
- =
b d
b´d
2 1 3 4 5
= = = =
= ...
4 2 6 8 10
Example 1:
3 2 3 ´ 5 - 4 ´ 2 15 - 8 7
- =
=
=
4 5
4´5
20
20
So:
3 1 2
= =
= ...
15 5 10
2 3
5 2
3
- = - =
4 15 10 10 10
F48. Subtract the fractions using the method 1:
a)
1 1
2 3
b) 3 - 2
4 3
c)
2 1
5 4
d) 1 - 1
5 6
f)
5 3
8 7
g) 5 - 3
9 8
h)
5 3
9 10
i)
3 1
10 11
j)
2 1
11 10
k)
3 4
10 15
l)
m)
2 5
3 9
n)
2 1
15 10
o)
3 4
15 20
p)
1 2
1 2 3
q) 2 4 - 2
5 3
r)
5
3
1 -2
3
6
s) 110 - 11
11 10
t) 2 1 - 1 2
4 5
u)
4 3
16 20
v) 1 3 - 8
40 25
w)
3 5
1 4 12
x) 2 1 - 1 3
6 8
1 3
y) 116 - 32
e)
6 3
10 7
2 3
3 10
e) 3 - 2
7 5
F49. Subtract the fractions using the method 2:
a)
2 1
3 2
b) 2 - 1
6 4
c)
3 2
4 10
d) 5 - 2
6 5
f)
8 3
14 8
g) 4 - 3
8 9
h)
5 5
9 10
i)
3 2
9 10
j)
5 5
10 12
k)
5 2
15 10
l)
m)
2 5
3 9
n)
3 2
10 15
o)
6 3
20 15
p)
1 12
1 2 18
q) 3 2 - 2 3
3
5
r)
4 3
2 -1
3 6
s) 1 2 - 8
3 10
t) 3 3 - 2 3
5
4
u)
7
4
20 16
v) 1 5 - 5
40 25
w)
2
x) 2 2 - 1 6
6 8
5 5
1
y) 16 - 48
© La Citadelle
2 5
3 12
4
4
-1
12 16
32
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The Book of Fractions
Iulia & Teodoru Gugoiu
Subtracting fractions with unlike denominators using the LCD method
1. First find the Least (Lowest) Common Denominator
(LCD), then subtract fractions.
Example 1.
24
5
2
=?
24 30
3
1
1
1
30
12
2
2. You can use the LCD method to add or subtract more
than two fractions.
Example 2.
2
3
6
24 = 2 ´ 3
2
1 1
1
+ =?
12 45 40
15
2
12
5
2
45
15
3
6
2
40
3
3
20
2
2
3
1
30 = 2 ´ 3 ´ 5
2
1
12 = 2 ´ 3
120 ¸ 24 = 5
120 ¸ 30 = 4
LCD = 23 ´ 31 ´ 51 = 120
10
2
5
2
1
3
45 = 3 ´ 5
5
1
40 = 2 ´ 5
LCD = 23 ´ 32 ´ 51 = 360
5
2
5´ 5
2´ 4
25
8
17
=
=
=
24 30 24 ´ 5 30 ´ 4 120 120 120
1 1
1
1´ 30
1´ 8
1´ 9
29
+ =
+
=
12 45 40 12 ´ 30 45 ´ 8 40 ´ 9 360
F50. Subtract the fractions using the LCD method (see the example 1 above):
3 1
a) 5 - 1
b) 2 - 3
c)
d) 4 - 1
e) 3 - 3
6 4
6 16
20 16
15 12
10 14
f)
5 3
4 14
g)
5 3
18 16
h)
5 2
12 9
i)
3 1
10 22
j)
1 2
6 33
k)
3 4
10 15
l)
5 3
12 20
m)
4 5
9 30
n)
1 2
10 25
o)
4 2
25 15
p)
1
q)
2
1
3
-1
30 50
r)
1
s)
2
t)
1
u)
5 3
20 16
v)
1
3
3
40 25
w)
1
3
12 40
1
7
25 30
5 1
36 16
1
1
-2
15
50
x) 2 1 - 1 3
14 84
3
2
-1
50 45
y) 1 1 - 3
48 64
F51. Add and subtract the fractions using the LCD method (see the example 2 above):
a)
1 2 1
+ 2 3 4
b)
3 1 3
- +
4 6 8
c)
5 3 3
- 6 8 10
d)
3 1 7 3
- + 4 6 12 16
e)
3 3 1
- +
4 10 12
f)
3 1 1
- 10 15 30
g)
1 1 1
+ 12 9 14
h)
1 1 1 1 1
- + - +
2 3 4 5 6
i)
3 2 3
- +
10 15 16
j)
5 3 2
- 12 20 15
k)
1 1
1
+ 9 30 36
l)
1 1 5 2 3
- + - +
4 6 12 15 20
m)
2 1 3
- 3 6 10
n)
3 2 1
- 10 25 6
o)
2 3
1
- +
15 25 35
p)
13 3 1
1
7
- - - 15 4 20 32 480
q)
1 1 1 4
2 - -1 2 3 4 5
r)
1 1 1 1
+ - 3 4 5 6
s)
2 3 7 3
- + 5 10 15 20
t)
2 3 1
3
2
- - +
5 10 15 20 25
© La Citadelle
33
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The Book of Fractions
Iulia & Teodoru Gugoiu
Order of operations (I)
1. Additions and subtractions are operations of the
same priority, so the order in which they are done is
not important.
By convention, these operations are done one by
one from left to right.
Example 1.
3. By convention the order in which the brackets appear is
{ [ ( ) ] }.
Example 3:
3 ìï 4 é æ 1 1 öù üï 3 ì 4 é
1 ùü
+ í - ê1 - ç - ÷ú ý = + í - 1 ý=
2 ïî 3 ë è 3 4 øû ïþ 2 î 3 êë 12 úû þ
=
3 1 1 æ 3 1ö 1 7 1
5
- + = ç - ÷ + = + =1
2 3 4 è 2 3 ø 4 6 4 12
2. To change the order of operations, use:
parentheses ( ), brackets [ ] or braces { }. An inner
bracket has a greater priority.
Example 2:
3 ì 4 11 ü 3 5
11
+í - ý= +
=1
2 î 3 12 þ 2 12
12
4. One single type of brackets is enough to change the order
of operations.
Example 4:
1 æ æ 1 æ 1 1 ööö 1 æ æ 1 1 öö
+ ç1 - ç - ç - ÷ ÷ ÷ = + ç1 - ç - ÷ ÷ =
2 çè çè 2 è 3 4 ø ÷ø ÷ø 2 çè è 2 12 ø ÷ø
3 æ 1 1 ö 3 7 11
-ç + ÷ = - =
2 è 3 4 ø 2 12 12
=
1 æ
5ö 1 7
1
+ ç1 - ÷ = + = 1
2 è 12 ø 2 12 12
F52. Solve each exercise by following the proper order of operations:
a) 1 + 1 + 1 + 1
2 3 4 5
d) 1 - 1 + 1 - 1 + 1
2 3 4 5 6
2 1 1
g) 2 - æç + 1 ö÷
3 è 3 3ø
1
2 1
5 1
j) 1 - æç + ö÷ - æç - ö÷
2 è 3 2ø è6 2ø
b) 1 - 1 + 1 - 1
2 3 4 5
c) 1 + 2 1 - 1 1 + 4 1 - 3 1 + 5
2 3
4
5
7 æ 3ö
e) 5 - çè1 - 5 ÷ø
1 æ1 1ö æ1 1ö
h) 2 - çè 3 - 4 ÷ø - çè 5 - 6 ÷ø
k)
æ æ 1 1 1 öö
l) 2 - ç1 - çç - æç - ö÷ ÷÷ ÷
ç
÷
è è 2 è 3 4 øøø
1 é 1 æ 1 1 öù
- -ç - ÷
2 êë 3 è 4 5 øúû
3
4 é
3 5 ù
m) æç 8 - 2 ö÷ - ê5 - æç 3 - 1 ö÷ú
7 ø ë è 4 7 øû
è 7
3
2 3
f) æç 2 - ö÷ - æç1 - ö÷
4ø è 4 4ø
è
æ 1 1ö æ 1 1 1ö
i) 1 + çè 2 - 3 ÷ø - çè 4 + 5 - 6 ÷ø
n)
1 éæ 1 1 ö æ 1 öù
1 + 2 - êç1 + ÷ - ç1 + 1÷ú
2 ëè 3 4 ø è 5 øû
3 é7
1 1 ù ì 1 é 1 æ 4 3 öù ü
+ ê - ( + )ú - í - ê - ç - ÷ú ý
5 ë10 2 5 û î 2 ë10 è 5 4 øû þ
p)
2 ìæ 2 5 ö éæ 2 5 ö æ 5 1 öù ü
2 - íç + ÷ - êç - ÷ - ç - ÷ú ý
9 îè 3 6 ø ëè 3 9 ø è 12 6 øû þ
ì é
3 1
7 3 ù 1ü 3
q) í3 - ê2 - æç1 - ö÷ + - æç - ö÷ú + ý +
î ë è 8 ø 2 è 10 5 øû 10 þ 4
r)
ìé 3 æ 8
3 öù æ 3 öü æ 1 7 ö
í ê - ç - ÷ ú + ç1 - ÷ ý - ç - ÷
îë 5 è 25 20 øû è 5 øþ è 4 50 ø
o)
æ1 æ4 æ3
4 3 2ö 1ö 1ö æ 1
2 1 ö
s) 1 - ç - ç - çç - æç - ö÷ - ÷÷ - ÷ - ÷ + çç1 - - æç - ö÷ ÷÷
ç 2 ç 3 è 4 è 5 4 ø 3 ø 2 ÷ 5 ÷ è 2 è 3 4 øø
è
ø
è
ø
ì 1 é
2 1 ù
1 ü ì é2 3 1 ù é
2 ù 3ü
t) í2 - ê1 - æç - ö÷ú - æç1 - ö÷ý - í1 - ê - æç - ö÷ú - ê1 - æç1 - ö÷ú - ý
î 2 ë è 3 2 øû è 3 øþ î ë 5 è 5 2 øû ë è 5 øû 10 þ
ìé
1
1 1 ù é 1 1
1 1 ù ü ìé 1 1
1 1 ù é 1 1
1 1 ùü
u) íêæç1 - ö÷ - æç - ö÷ú - êæç - ö÷ - æç - ö÷ú ý - íêæç - ö÷ - æç - ö÷ú - êæç - ö÷ - æç - ö÷ú ý
îëè 2 ø è 2 3 øû ëè 3 4 ø è 4 5 øû þ îëè 2 3 ø è 3 4 øû ëè 4 5 ø è 5 6 øû þ
© La Citadelle
34
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The Book of Fractions
Iulia & Teodoru Gugoiu
Multiplying fractions
1. To multiply two proper or improper fractions, you
multiply the numerators first and then the
denominators, according to the rule:
2. To multiply mixed numbers, first convert them to
improper fractions.
Example 2:
1 5 3 5 3 ´ 5 15
7
1 ´ = ´ =
= =1
2 4 2 4 2´ 4 8
8
n1 n2 n1 ´ n2
´ =
d1 d 2 d1 ´ d 2
Example 1:
3. To multiply a whole and a fraction, rewrite the whole as
a fraction.
Example 3:
4 2 4 2´ 4 8
3
2 4 2´ 4 8
´ =
=
3 5 3 ´ 5 15
2´
5
= ´ =
= =1
1 5 1´ 5 5
5
F53. Multiply the proper or improper fractions (write the results in lowest terms):
a) 1 ´ 3
2 4
b)
1 2
´
2 3
c) 1 ´ 3
4 5
d) 2 ´ 15
5 4
e) 3 ´ 1
2 4
4 3
´
3 2
g)
1 2
´
2 4
h) 1 ´ 1
2 4
i) 3 ´ 5
4 6
j)
k) 1 ´ 8
2 9
l)
3 12
´
6 15
m) 5 ´ 22
11 5
n) 4 ´ 21
7 5
o) 3 ´ 2
4 6
f)
1 5
´
10 20
F54. Multiply the mixed numbers or fractions (write the results in lowest terms):
a) 1 1 ´ 1
2 2
b)
f) 1 1 ´1 3
5 12
k) 1 1 ´ 2 2
2
3
p) 1 1 ´ 2 2
2
9
1
1
´2
2
3
c) 1 1 ´ 5
5 3
d) 2 2 ´ 1
5 3
e) 2 ´1 3
3 4
g) 1 1 ´ 2 1
2
2
h) 1 2 ´ 2 2
3
5
i) 1 1 ´ 2 3
4
2
j)
l)
5
2
1 ´2
6
5
m) 1 1 ´ 5
10 11
n) 1 3 ´ 14
7 5
o) 2 1 ´1 2
4 6
q)
1 3
2 ´
6 13
r) 1 4 ´ 4 2
11
5
s) 1 3 ´ 14
7 5
t)
2
2
1
´1
11 12
1 5
2 ´1
4 3
F55. Multiply the wholes and fractions (write the results in lowest terms):
a) 3´ 1
4
b)
1
´2
2
c) 2´ 2
5
d) 2 ´ 5
5
e) 3´ 5
4
f) 5´ 3
15
g)
1
´6
2
h) 3´ 2
9
i) 3 ´ 6
4
j) 15´ 4
20
k) 1 1 ´ 3
2
l)
5´ 2
m) 3´ 2 5
6
n) 1 4 ´14
7
o) 2´1 1
6
© La Citadelle
2
15
35
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The Book of Fractions
Iulia & Teodoru Gugoiu
More about multiplying fractions
1. Although the most common multiplication operator is
“x”, there are two other acceptable operators: “of” and “×”
Example 1:
2. When multiplying fractions, first try to cancel out the
common factors between numerators and denominators.
Example 3:
2
1
3 1 3 1´ 3 3
of = ´ =
=
2
5 2 5 2 ´ 5 10
Example 2:
1 4 1´ 2 2
´ =
=
2 5 1´ 5 5
1
3. This process of cancellation can
be applied more than one time.
Example 4:
1 3 1× 3 3
× =
=
2 5 2 × 5 10
1
2 5
4 15 2 ´1 2
´ =
=
9 10 3 ´1 3
3 2
1
F56. Multiply the fractions (write the results in lowest terms):
a) 2 of 1
3
2
b)
2
3
of
3
4
c)
1
1
of
5
5
d)
2
4
of
3
5
e)
1
1
of
5
3
f) 5 of 3
2
4
g)
3
3
of
2
2
h)
4
5
of
5
4
i)
1
2
of
2
4
j)
2
9
of
3
6
k) 2 of 90
3
l)
10 of
m) 2 of 3
n)
1
1
1 of
3
4
o) 2 1 of 1 1
5
2
1
2
F57. Multiply the fractions (write the results in lowest terms):
a) 1 × 2
2 3
b)
2 3
×
3 4
c)
3 5
×
5 3
d)
1 2
1 ×
2 5
e) 1 1 × 2 1
2 3
f) 3 × 14
7 6
g)
1 5
×
10 2
h)
4 15
×
5 20
i)
3 12
×
5 9
j)
1 1
×
4 5
k) 1 × 2
2 1
l)
1 1
×
2 2
m)
5 18
×
12 15
n)
10 22
×
12 30
o)
6
× 30
21
F58. Cancel out the common factors and then multiply the fractions:
a) 1 ´ 15
5 4
b)
2 3
´
3 2
c)
2 10
´
5 4
d)
3 14
´
7 9
e)
5 6
´
3 10
f) 5´ 3
15
g)
2
´8
6
h)
3 12
´
4 9
i)
33 40
´
4 44
j)
4 121
´
22 20
k) 1 1 ´ 2
2 3
l)
2
6
1 ´1
3 15
m) 15 ´ 2 4
7
5
n)
1
1 ´ 21
7
o) 1 1 ´ 2 2
2
3
p) 1 1 ´ 2 2
2
9
q)
1 3
2 ´
6 13
r)
4 5
´
10 8
s)
1
© La Citadelle
1
36
3 25
´
20 23
t)
2
1
4
´2
40 18
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The Book of Fractions
Iulia & Teodoru Gugoiu
The order of operations (II)
1. Because multiplication is a commutative operation,
the order in which you multiply them is not important.
By convention, the order is from left to right.
Example 1:
4
2 4 1 æ 2 4ö 1 8 1 4
´ ´ = ç ´ ÷´ = ´ =
3 5 2 è 3 5 ø 2 15 2 15
1
2. Do not forget to cancel out
the common factors, before
multiplying the fractions .
Example 2:
3. If the expression contains addition, subtraction and
multiplication, do the multiplication first.
Example 3:
1 3 5 1 æ 3 5ö 1 5 9
1
+ ´ = +ç ´ ÷ = + = =1
2 4 6 2 è 4 6ø 2 8 8
8
4. If the expression contains brackets, replace the
brackets with the result of the operation(s) inside the
brackets.
Example 4:
1 1
1 3 2
2 15 6 1
´ ´ =
5 9 8 2
1 3 4
1 2
1
æ 1 3 ö 5 5 5 25
=1
ç + ÷´ = ´ =
24
è 2 4 ø 6 4 6 24
F59. Multiply the fractions (write the results in lowest terms):
a) 1 ´ 2 ´ 3
b) 1 ´ 2 ´ 3
c) 4 ´ 3 ´ 3
d) 1 ´12 ´ 3
2 3 4
3 5 2
9 8 4
3
4
e) 2 ´ 1 ´ 2
5 3 4
f) 1 1 ´ 2 ´ 2 3
2 3
4
j) 1 1 ´ 2 ´ 15
5 3 8
k) 1 ´ 4 ´ 10 ´ 15
2 5 16 8
g) 2 1 ´ 2 ´1 3
2 5 4
l)
h) 2 1 ´1 2 ´ 3
2 3 5
1 2 3 4 5
´ ´ ´ ´
2 3 4 5 6
i) 2 1 ´ 2 ´ 2
4
9
m) 1 ´ 4 ´ 8 ´ 10 ´ 14
2 6 10 12 16
n)
1 3 5 7 9
´ ´ ´ ´
2 4 6 8 10
F60. Find the value of each expression (write the results in lowest terms):
a) 1 + 1 ´ 4
b) 1 ´ 2 + 5
c) 2 - 1 ´ 1
d) 1 ´ 2 - 1
e) 4 ´ 1 ´ 2 - 1
2 2 5
4 3 6
3 2 3
2 3 6
5 3 4 10
f) 1 + 1 ´ 2 - 1
3 2 3 2
g) 1 - 1 ´ 2 + 1
2 2 3 3
h) 2 + 1 - 1 ´ 2
3 2 2 3
i) 3 ´ 15 + 5 - 2
5 9 6 3
j)
2 3 3 5
´ + ´
3 4 5 6
k) 1 ´ 2 - 2 ´ 3
2 3 3 8
l)
1 3 2 2
´ ´ +
2 4 3 3
m) 2 - 1 ´ 1 ´ 3
3 2 3 4
n) 1 1 ´ 2 + 6 ´1 2
2 3 5 3
o) 1 - 2 ´1 3 + 1 3 ´ 15
7 4 5 8
F61. Find the value of each expression (write the results in lowest terms):
a) æç 1 + 1 ö÷ ´ 4
è2
f)
3ø 5
æ1 1ö 3 1
ç + ÷´ è3 2ø 5 2
k) 1 1 ´ æç 1 - 1 ´ 2 ö÷
4 è3
p)
2 5ø
b)
1 æ2 1ö
´ç + ÷
5 è 3 6ø
c)
æ2 1ö 3
ç - ÷´
è 3 2ø 5
d) 3 ´ æç 1 - 1 ö÷
g)
1 1 æ2 1ö
- ´ç + ÷
2 7 è 3 2ø
h)
1 1ö 2
æ2
ç +1 - ÷´
3
2 6ø 3
è
i)
l)
3 æ 3 2 1ö
´ç ´ + ÷
5 è 4 3 3ø
m) æç 2 - 1 ´ 2 ö÷ ´ 3
n)
1 1 é 1 æ 2 3ö 1 1ù 2
1 ´ - ê1 ´ ç - ÷ + 1 ´ ú ´
3 4 ë 2 è 3 4 ø 2 4û 3
© La Citadelle
è3
q)
2 3ø 4
e)
æ 3 2ö æ 3 2ö
ç + ÷´ç - ÷
è 4 3ø è 4 3ø
4 æ1 1ö 1
´ç + ÷ 7 è3 4ø 5
j)
2 æ 1 1ö 6
´ç + ÷´
5 è 2 3ø 7
5 æ 2 6 2ö
´ç + ´ ÷
11 è 3 5 3 ø
o)
æ 2ö æ 1 3ö 1
ç1 - ÷ ´ ç + ÷ ´ 1
è 5ø è 2 3ø 9
2 è2
3ø
ìï 3 é 2 1 æ 1 1 ö 1 ù é 5 æ 3 1 ö 1 ù üï 2
í ´ ê1 + 1 - ç + ÷ ´ 3 ú - ê ´ ç - ÷ - ú ý ´
3 è 4 5 ø 3 û ë11 è 5 2 ø 4 û ïþ 7
ïî 2 ë 2
37
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The Book of Fractions
Iulia & Teodoru Gugoiu
Reciprocal of a fraction
1. The reciprocal of a fraction is the fraction obtained by
interchanging its numerator and denominator.
So, the reciprocal of
n
d
Example 1:
The reciprocal of
3
4
d
n
is
2. To find the reciprocal of a mixed number, express it as
an improper fraction and then interchange the numerator
and the denominator (invert the fraction).
Example 2. The reciprocal of
3=
Example 3. The reciprocal of
4
3
is
3. To find the reciprocal of a whole number, express
it as an improper fraction and then interchange
(invert) the numerator and the denominator.
3 7
1 =
4 4
is
3
1
is
1
3
4. If you multiply a fraction by its reciprocal, the
product is always 1. Thus, the reciprocal of a fraction
is called also the multiplication inverse of that
fraction.
Example 4:
3 4
´ =1
4 3
4
7
F62. Find the reciprocal of each fraction:
a)
1
2
b)
2
3
c)
5
4
d)
7
3
e)
0
10
f)
7
6
g)
13
3
h)
2
4
i)
3
5
j)
3
0
F63. Find the reciprocal of each mixed number:
a) 1
1
2
b) 2
2
3
c)
2
2
5
d) 1
1
7
e) 3
2
3
f) 3
3
3
g) 2
2
11
h) 2
5
7
i) 1
1
4
j) 3
3
4
F64. Find the reciprocal (multiplication inverse) of each whole number:
a) 1
b) 2
c) 5
d) 100
e) 0
F65. Check if the pair of fractions are reciprocal:
a)
1
2
and
2
1
1
3
f) 3 and
3
11
1
2
b) 1 and
2
4
c)
7
1
and 2
3
7
d)
1
and 3
3
e)
1
10
h)
3
5
and
5
2
i)
3
6
and
9
2
3
15
j) 1 and
5
24
g) 10 and
F66. Find the unknown fraction (f):
1
1
a) f ´ = 1
b)
c)
´ f =1
4
2
f)
f´
2
=1
7
© La Citadelle
g) 7 ´ f = 1
h)
f´
2
=1
5
2
f ´1 = 1
9
38
3
2
and 2
5
5
d) f ´ 5 = 1
e)
5
f ´1 = 1
4
3
i) 1 ´ f = 1
4
j)
f ´2
4
=1
5
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The Book of Fractions
Iulia & Teodoru Gugoiu
Dividing fractions
1. To divide two proper or improper fractions, change the
divisor to its reciprocal and then multiply, according to
the rule:
2. To divide mixed numbers, first change them to
improper fractions.
Example 2:
1 2 5 5 5 3 3
1
2 ¸1 = ¸ = ´ = = 1
2 3 2 3 2 5 2
2
n1 n2 n1 d 2
¸
= ´
d1 d 2 d1 n2
3. To divide a whole number and a fraction, rewrite the
whole as a fraction.
Example 3:
In short, invert the divisor and multiply.
Example 1:
2 4 2 5 5
¸ = ´ =
3 5 3 4 6
2¸
4 2 4 2 5 5
1
= ¸ = ´ = =2
5 1 5 1 4 2
2
F67. Divide the proper or improper fractions (write the results in lowest terms):
a) 1 ¸ 3
2 4
f)
4 12
¸
3 5
k) 1 ¸ 8
3 9
b)
1 1
¸
2 3
c)
2 4
¸
3 9
d)
2 8
¸
5 15
e) 3 ¸ 6
2 5
g)
5 3
¸
2 4
h)
1 1
¸
2 4
i)
10 5
¸
4 6
j)
l)
3 12
¸
6 15
m)
5 25
¸
11 22
n)
4 16
¸
7 21
o) 3 ¸ 6
4 16
1 3
¸
10 20
F68. Divide the mixed numbers or fractions (write the results in lowest terms):
a) 2 1 ¸ 1
2 2
b) 1 1 ¸ 2 1
2
4
c)
1
1
1 ¸1
5 15
d)
2 3
2 ¸
5 5
e) 2 ¸ 1 1
3 6
f) 1 1 ¸ 1 3
5 15
g) 1 1 ¸ 2 1
2
4
h)
2 1
1 ¸1
3 9
i)
1
3
1 ¸6
4
2
j) 2 3 ¸ 1 3
11 22
k) 1 1 ¸ 2 2
2
8
l)
5
2
1 ¸3
6
3
m) 1 2 ¸ 3
10 25
n)
3 10
1 ¸
7 21
o) 2 1 ¸ 1 5
4 16
p) 1 1 ¸ 2 2
3
9
q)
1
1
2 ¸3
6
4
r)
2 5
¸
13 26
s)
3 8
1 ¸
7 35
t) 2 1 ¸ 2 1
4
3
1
F69. Divide the wholes and fractions (write the results in lowest terms):
a) 1¸ 1
2
b)
1
¸2
3
c)
4¸
f) 5 ¸ 2 1
7
g)
1
¸2
3
h)
3¸2
k) 1 1 ¸ 3
2
l)
5¸2
© La Citadelle
3
11
4
5
2
5
m) 13 ¸ 3 5
7
39
d)
5
¸ 15
2
e) 32 ¸ 5 1
3
i)
3
¸6
4
j) 15 ¸ 6 2
3
n)
4
1 ¸ 44
7
o) 5 ¸ 3
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The Book of Fractions
Iulia & Teodoru Gugoiu
Division operators
There are four operators used to express the division operation between two numbers or fractions.
1. The operator is ¸
Example 1.
2. The operator is :
Example 2.
3. The operator is
(the division line)
3 9 3 10 2
¸ = ´ =
5 10 5 9 3
_
1
2 = 1´4 = 2
3 2 3 3
4
Example 3.
4. The operator is
3 6 3 21 3
: = ´ =
7 21 7 6 2
/
3 9 3 25 5
2
/
= ´
= =1
5 25 5 9 3
3
Example 4.
F70. Divide the fractions (write the results in lowest terms):
a) 2 ¸ 4
3 5
b) 1 1 ¸ 2
2 3
c)
2
1
1 ¸2
3
2
2
7
d)
2¸
e) 4 ¸ 5
d)
2
:4
5
e) 3 : 7
d)
3
5 =
1
2
10
1
2 =
e)
3
2
4
F71. Divide the fractions (write the results in lowest terms):
a) 1 : 1
2 3
b) 1 1 : 1
2 4
c)
1
1
1 :2
5 10
F72. Divide the fractions (write the results in lowest terms):
a)
f)
2
3=
4
6
2
=
4
6
2
k) 3 =
4
b)
g)
l)
4
5 =
8
15
5
=
10
3
5
2=
4
2
9 =
4
6
1
c)
h)
3
=
1
2
4
4
5=
10
i)
j)
n)
1
3=
3
o)
1
m)
3
=
1
3
1
5
=
2
1
3
2
5=
4
2
F73. Divide the fractions (write the results in lowest terms):
a) 1 / 2
2 3
b)
2 3
/
3 5
c)
5 3
/
3 5
d)
1 3
/
4 8
e) 5 / 3 1
3
f) 3 / 2 1
4
g)
1
/2
3
h)
1
2 /3
4
i)
2/5
j) 1 2 / 3 3
3 4
k) 5 / 15
7 14
l)
2 1
4 /1
3 6
m)
3 /1
1
3
n)
2
2 /4
7
o) 7 / 2
© La Citadelle
40
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The Book of Fractions
Iulia & Teodoru Gugoiu
Order of operations (III)
1. Division is not a commutative operation, so the order
in which you divide numbers (or fractions) is important. If
an expression contains more than one division
operation, then by convention the divisions must be
done, one by one, from left to right.
Example 1:
2. Division and multiplication are considered operations
of equal priority. If an expression contains both division
and multiplication, then by convention the operations
must be done, one by one, from left to right.
Example 2:
1 2 5 æ1 2ö 5 3 5 3
¸ ¸ =ç ¸ ÷¸ = ¸ =
2 3 4 è 2 3ø 4 4 4 5
4 1 2 æ4 1ö 2 2 2 3
´ ¸ =ç ´ ÷¸ = ¸ =
5 2 3 è5 2ø 3 5 3 5
3. Division and multiplication are considered operations of greater priority than addition and subtraction. If an
expression contains all these kinds of operations, do the multiplication and division first, then the addition and
subtraction.
Example 3:
3 2 6 3 6
3 æ 2 6 ö æ3 6ö
3 2 1
1
1 - ´ + ¸ =1 -ç ´ ÷+ç ¸ ÷ =1 - +
=1
5 3 10 5 1
5 è 3 10 ø è 5 1 ø
5 5 10
10
F74. Use the correct order of operations to find the value of each expression (see example 1):
a) 1 ¸ 2 ¸ 4
2 3 5
b) 2 ¸ 3 ¸ 5
c) 1 ¸ 2 ¸ 3 ¸ 4 ¸ 5
d)
3 1
1
2 ¸ ¸4
5 15
3
e) 1 ¸ 3 ¸ 5 ¸ 7
2 4 6 8
f) 1 ¸ 1 ¸ 3 ¸ 2 ¸ 4
2
3
g) 1 ¸ 2 ¸ 3 ¸ 4 ¸ 5
2 3 4 5 6
h)
1
2
3
4
1 ¸2 ¸3 ¸4
2
3
4
5
j) 1 2 ¸ 2 1 ¸ 3 3 ¸ 4 1
3
4
5
6
k) 1 1 ¸ 6 ¸ 4 ¸ 6
11 22 9
l)
3
6 5 3
¸1 ¸ ¸1
7 21 6 5
i)
2 4 2 15
¸ ¸ ¸
3 3 5 4
F75. Use the correct order of operations to find the value of each expression (see example 2):
a) 1 ¸ 3 ´ 5
2 4 6
b) 2 3 ´ 2 ¸ 4
4 5 15
c)
e) 1 ´ 2 ¸ 3 ¸ 4
5 5 5 5
f) 1 1 ´ 2 3 ¸ 3 3 ¸ 4 4
2
3
4
5
g) 3 ¸ 2 ´ 3 ¸ 9
2 3 16 8
i)
1 3 5 1 3 6
¸ ´ ´ ¸ ´
2 4 6 2 4 5
j)
1 3 5 7 9
´ ¸ ´ ¸
2 4 6 8 10
2 3 12 3 3 3
¸ ´ ¸ ¸ ´
5 5 5 10 4 20
d)
1´ 2 ¸ 3 ´ 4 ¸ 5
h)
1 1 1 1 1
1¸ ´ ¸ ´ ¸
2 4 8 16 32
k) 3 ¸ 6 ´ 9 ¸ 3 ´ 15 ¸ 12 ´ 3
2 4 6 2 10 8 2
F76. Use the correct order of operations to find the value of each expression (see example 3):
a) 1 + 3 - 5 ´ 2 ¸ 1
2 4 6 15 9
b) 1 + 2 ´ 6 - 3 ¸ 6
3 8 5 10
c) 1 2 - 3 ´ 10 ¸ 3 + 2 ¸ 1 - 1
15 5 12 2 5 3 2
d) 5 + 1´ 2 ¸ 3 - 4 + 5 ´ 6 ¸ 7
e) 1 1 ´ 2 ¸ 2 + 1 ¸ 1 ´1 1
2 3
2 6 4
f) 1 ¸ 1 - 2 ´ 3 + 1 ¸ 1
2 3 4 4 2
g) 1 1 ´ 1 - 1 ¸ 2 - 1 ´ 1 + 1 ¸ 1 + 1 ¸ 4
2 2 3
4 2
2 6
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h)
41
1
1 3 3 1 3 1 1 1
1 6 1
+1 ´ ¸ + ¸ ´ - ´ - 2 ¸ +
12 2 4 4 2 4 2 2 2
2 2 12
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The Book of Fractions
Iulia & Teodoru Gugoiu
Order of operations (IV)
2. When an expression contains fraction (division) lines, the
numerators and the denominators must be calculated
independently.
Example 2:
1. You can change the order of operations using
brackets.
Example 1:
1
æ1 2ö æ1 1ö 7 7 1 7 1
ç + ÷´ç - ÷ ¸ = ´ ¸ =
è 2 3ø è 2 4ø 3 6 4 3 8
-
2
2¸
1
1
1 2 2 4 1 8 25
7
= 6 + 1 = ´ + ´ = + =
=2
1 1
3 1 1 6 3 1 3 9 3
9
9
+ ´
+
2 2 2
2 2 4
3+
4 1
3
2
2
F77. Solve each exercise by following the proper order of operations (see example 1):
a)
15 æ 1 2 ö æ 3 2 ö 5
´ç + ÷ -ç - ÷ ¸
3 è 2 5 ø è 4 3 ø 12
d)
11 2 æ 1 2 ö é 1 æ 2 3 ö 6 ù
+ ´ç + ÷ ¸ ê + ç - ÷ ¸ ú
25 17 è 6 5 ø ë 3 è 5 10 ø 5 û
e)
f)
é1 æ 3 1 ö 8ù æ 2 1 ö
ê + ç - ÷´ ú ¸ç + ÷
ë2 è 4 6 ø 7û è 3 2 ø
b)
e)
3 æ 10 5 ö 2 5
´ ç ¸ ÷ - ´ ¸ (1 ¸ 3)
5 è 9 27 ø 25 6
c)
ì 1 1 éæ 1 1 ö 1 1 ù 1 1 ü é 1 3 æ 3 2 ö ù
í2 - ¸ êç - ÷ ¸ - ú ´ - ý ¸ ê + ¸ ç - ÷ ¸ 6ú
î 4 2 ëè 2 3 ø 2 4 û 3 6 þ ë 2 4 è 4 3 ø û
éæ 1 ö æ 1 3 öù 5 éæ 3 1 ö 3 1 2 ù æ 1 1 ö 6 1
êç 2 + 1÷ ¸ ç 2 + 4 ÷ú ´ 4 ¸ êç 4 + 2 ÷ ¸ 4 ´ 2 - 6 ú ´ ç 2 - 3 ÷ ¸ 5 + 12
ø è
øû
ø
ø
ëè
ëè
û è
1 é 2 1 æ 3 1 3 ö 1 2 ù é1 3 æ 2 3 ö 2 ù æ 5 2 8 ö
´ ê + ´ç - ¸ ÷ ¸ + ú - ê + ¸ç - ÷´ ú´ç - ¸ ÷
2 ë 5 2 è 2 5 10 ø 2 3 û ë 5 10 è 3 5 ø 5 û è 6 3 9 ø
F78. Solve each exercise by following the proper order of operations (see example 2):
a)
æ1ö
ç ÷
è 2ø
æ 3ö
ç ÷
è 4ø =
5
6
b)
æ 3ö
ç ÷
è 4ø
5 =
2
5
f)
2 3 2
+ ´
3 4 9
3 5
3¸ 4 3
j)
3 1 8
1 - ´
4 2 3 ¸ 4 ¸ 5 +1 ´ 9
1 5
3 2 13 15
´ +
2 12
5 9 15
© La Citadelle
c)
g) 2 ´ 3 - 4 ¸ 3
1¸ 2´ 3 + 2
æ 2ö
ç ÷
è 3ø
æ 3ö
ç ÷
è5ø =
6
d)
3
æ1ö
ç ÷
è 2ø =
æ1ö
ç ÷
è 3ø
æ 2ö
ç ÷
è 3ø
2 3 9 1
¸ ´ h) 3 2 8 3
5 15 2 1
¸ - ´
3 6 3 2
k)
42
e)
æ 3ö
ç ÷
3
è 5 ø ¸æ 3 ö´
=
ç ÷
2
4
æ ö è ø
æ 2 ö
8¸ç
ç ÷
÷
è 3ø
è 3 + 1/ 2 ø
i)
2 15 1 2 2
´ - ¸ +
5 8 2 3 3
1 3 9 3 1
+ ¸ ´ 2 2 4 4 3
3 1 5
2 10 2
- ¸
´ +
5 2 4+ 5 6 3 ¸
1 5 1 1 1 4
¸ +
¸ ´ -1
2 4 5 2 3 3
1 3 1
¸ 5 10 3
1 9 2
+ ´ -1
2 4 3
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The Book of Fractions
Iulia & Teodoru Gugoiu
Raising fractions to a power
1. If you multiply a fraction by itself k times, you can use
exponential notation to make the expression more
compact (simpler):
n n
n ænö
´ ´ L´ = ç ÷
d4
d 244
d èdø
1
4
3
n1´42
n ´L
´ n = nk
43
k times
3. You can use also the following rule to calculate the
value of a fraction raised to a power:
n1´42
n ´L
´n
43
k
n n
n ænö
nk
k times
´ ´ L´ = ç ÷ = k =
d
d 244
d èdø
d
d1´42
d ´L
´d
14
4
3
4
43
4
k
k times
The left side is the expanded notation and the right side
is the exponential notation.
Example 1:
3 ´ 3 ´ 3 ´ 3 = 34
Example 2:
1 1 1 æ1ö
´ ´ =ç ÷
2 2 2 è 2ø
Example 4:
3
4
3
3´ 3 9
æ 3ö
=
ç ÷ == 2 =
4
4 ´ 4 16
è 4ø
2 2 2 2 16
æ 2ö
ç ÷ = ´ ´ ´ =
3 3 3 3 81
è 3ø
ænö
ædö
ç ÷ =ç ÷
èdø
ènø
-3
3
2
5
53 125
5
Example 5: æç ö÷ = æç ö÷ =
=
= 15
3
8
8
è5ø
è 2ø 2
5. By convention, any number (including a
fraction) raised to the power of 0 equals 1.
Example 6:
F79. Write each expression in exponential notation:
a) 1 ´ 1 ´ 1 ´ 1
b) 3 ´ 3
c) 3 ´ 3 ´ 3 ´ 3 ´ 3
2 2 2 2
4 4
2 2 2 2 2
g) 4 ´ 4
f) 1 2 ´1 2 ´1 2
3 3 3
5 5
F80. Write each expression in expanded notation:
e) 2 ´ 2 ´ 2 ´ 2
a) æç 3 ö÷
è 4ø
e) æç 3 ö÷
è7ø
2
2
b) æç 5 ö÷
è 3ø
2
4. If the exponent is negative, replace the fraction with its
reciprocal raised to a positive exponent according to the
rule:
-k
k
2. To calculate the value an expression written in
exponential notation, rewrite it in expanded notation and
do the multiplication.
Example 3.
k times
2
k times
4
f) æç 3 ö÷
è 2ø
c) æç 1 ö÷
è5ø
3
d)
3 3 3 3
´ ´ ´
2 2 2 2
h)
1 1 1 1 1
´ ´ ´ ´
4 4 4 4 4
3
g) æç 5 ö÷
è 4ø
0
æ 4ö
ç ÷ =1
è5ø
d)
æ 2ö
ç ÷
è 3ø
h)
æ 4ö
ç ÷
è5ø
d)
æ 3ö
ç ÷
è5ø
h)
æ 3ö
ç ÷
è 2ø
d)
æ 2ö
ç ÷
è 3ø
h)
æ 2ö
ç ÷
è5ø
-1
-2
3
F81. Calculate the value of each expression:
a) æç 3 ö÷
è7ø
0
b) æç 3 ö÷
è 2ø
f) æç 1 ö÷
è 3ø
e) 23
3
c) æç 2 ö÷
è5ø
4
g) æç 2 ö÷
è1ø
2
4
3
4
F82. Calculate the value of each expression:
b) æç 5 ö÷
è1ø
a) 3-2
e) æç 3 ö÷
è 4ø
-2
© La Citadelle
-2
f) æç 5 ö÷
è7ø
c) æç 1 ö÷
è 2ø
-1
g) æç 4 ö÷
è 3ø
43
-4
-2
-3
-3
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The Book of Fractions
Iulia & Teodoru Gugoiu
Order of operations (V)
1. If the base of a power contains other operations,
replace the base with the result of that operations, then
raise the new base to the exponent.
2
2. If the exponent of a power contains other operations,
replace the exponent with the result of that operations,
then raise the base to the new exponent.
2
25
æ 1 1ö æ 5ö
ç + ÷ =ç ÷ =
36
è 2 3ø è 6ø
Example 1.
1+ 2´2
æ 2ö
ç ÷
è 3ø
Example 2.
5
32
æ 2ö
=ç ÷ =
243
è 3ø
3. If an expression contains addition, subtraction, multiplication, division, and powers, do the powers first.
2
1 16 6 æ 2 ö
1 16 6 4
1 16 6 9
1
1
+ ´ ¸ ç ÷ -1 = + ´ ¸ -1 = + ´ ´ -1 = + 3 -1 = 2
2 9 8 è 3ø
2 9 8 9
2 9 8 4
2
2
Example 3.
4. The order of operations can be changed by brackets.
-1
2
-1
-1
1 æç 1
1 æ 1
9ö
1 æ1ö
1 4
1 1
æ 3 ö ö÷
+ 2 - 4´ç ÷
¸8 = + ç2 - 4´ ÷ ¸8 = + ç ÷ ¸8 = + ¸8 = + = 1
2 çè 2
2 è 2
16 ø
2 è 4ø
2 1
2 2
è 4 ø ÷ø
Example 4.
F75. Calculate the value of each expression (see example 1):
2
a)
æ1 1ö
ç + ÷
è3 4ø
e)
æ1 3ö æ1 1ö
ç ´ ÷ ´ç ¸ ÷
è3 2ø è3 2ø
3
b)
æ 1 15 8 ö
ç ´ ´ ÷
è5 4 7ø
f)
æ 2 1 4ö
ç - ´ ÷
è5 4 5ø
2
2
2
c)
æ1 6 4ö
ç ´ ¸ ÷
è3 5 5ø
g)
æ2 5 1ö
ç + - ÷
è 3 6 2ø
-1
2
2
1
2
d) æç 1 - 1 ö÷ + æç 1 + 1 ö÷
è 2 3ø è 2 3ø
-2
h) æç 1 - 1 ö÷ + æç 2 - 1 ö÷
è 2 3ø è 3 4ø
F76. Calculate the value of each expression (see example 2):
3´1- 4´
1
2
a)
æ 2ö
ç ÷
è 3ø
e)
æ 2 2 ö2
ç + ÷
è 3 3ø
1 1
2
1 1
¸ -1
6
b)
æ 3 ö2
ç ÷
è5ø
f)
æ 1 3 ö3
ç + ÷
è 2 4ø
1
1 1
¸
6
2 1
´6 + ¸
3 3
c)
æ 1 ö3
ç ÷
è 2ø
g)
æ 1 3 ö3
ç ´ ÷
è3 2ø
d) æç 2 ö÷
è5ø
2 1
¸
6
1
3 3
( ´3+ ) ¸
2
2 2
3
h) æç 3 - 2 ö÷ 2
è 4 3ø
2
2
´( ´ 2 - )
3
3
F77. Calculate the value of each expression (see example 3):
-1
0
1
-2
0
2
1
a) æç 1 ö÷ + æç 1 ö÷ + æç 1 ö÷
è 2ø
è 2ø è 2ø
b) æç 2 ö÷ ´ æç 2 ö÷ ´ æç 2 ö÷
è 3ø
è 3ø è 3ø
2
2
3 ö é 4 3 æ 3 öù
æ
ç
÷ú
d) ç ÷ ´ ê + ¸
ç
÷
4
3
2
2
è ø ëê
è øûú
e) 1 + æç 2 ö÷ ´ 25 - 3 + æç 1 ö÷ ´ 2
5 è 2ø
è5ø
2
2
3
æ1ö æ 3ö æ 3ö
ç ÷ +ç ÷ ¸ç ÷
è 2ø è 4ø è 2ø =
g)
é 3 æ 3 ö -1 ù
2´ ê - ç ÷ ú
êë 4 è 2 ø úû
3
2
2
æ 2ö
ç ÷
33
j) è 3 2ø ´
=
2
2
æ 3ö
ç ÷
è 4ø
© La Citadelle
1
3
5
2
f) 1 + æç 3 ö÷ ¸ æç 1 ö÷ ´ æç 1 ö÷ - 1
è 4ø è 2ø è 3ø 2
2
æ1 1 3ö
æ1ö 2
ç + ´ ÷
ç ÷ +
4
2
2
3
9
è
ø
h)
´ è ø
=
2
2
æ1ö 3
æ1 2 4ö
ç ÷ +
ç + ¸ ÷
è 2ø 4
è6 9 3ø
3
2
2
c) æç 1 ö÷ + æç 2 ö÷ ´ æç 3 ö÷ ¸ æç 1 ö÷
è 2ø è 3ø è 4ø è 3ø
2
æ4 1 ö
-1
ç + ÷
i) ç 5 10 ÷ ¸ 3 - æç 1 ö÷ =
ç 3 ÷ 4 è 2ø
ç
÷
è 5 ø
-2
2
2
é
ù é
ù
k) ê12 ´ æç 1 ö÷ ´ æç1 - 1 ö÷ - 2 - 2 ú ¸ ê 3 ¸ 1 - æç 1 ö÷ ú ¸ 52 =
è 2ø è 4ø
ëê
ûú êë 4 2 è 2 ø ûú
2
3
ìïæ 3 2 ö é 7 æ 1 1 öù -1 æ 1 ö 2 üï éæ 1 ö 2 æ 1 ö -3 æ 1 ö 2 ù æ 3 ö 2
l) íçè 4 - 3 ÷ø ´ ê 6 - çè 2 + 3 ÷øú + çè 2 ÷ø ý ¸ êçè 2 ÷ø ´ çè1 - 2 ÷ø - çè 2 ÷ø - 1ú ´ çè 2 ÷ø =
ïî
ïþ ëê
ë
û
ûú
44
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The Book of Fractions
Iulia & Teodoru Gugoiu
Converting fractions to decimals
1.212...
3. If the denominator of the fraction written in
lowest terms has factors other than 2 or 5,
33 40.000
the fraction can be written as a non33
terminating repeating decimal.
Example 2:
70
1. The fraction line is a division operation, so to write a
fraction as a decimal, divide the numerator (dividend) by
the denominator (divisor).
0.375
2. Usually we are dealing with the
8 3.000
decimal system (the base is 10). The
single prime factors of 10 are 2 and 5.
24
If the denominator of the fraction written
60
in lowest terms has only 2 or 5 as prime
56
factors, then the fraction can be written
40
as a terminating decimal.
40
Example 1:
8 = 2´ 2´ 2
3
= 0.375
8
40
= 1.2121... = 1.21
33
33 = 3 ´ 11
66
40
33
70
66
4...
4. The part of the decimal that repeats is
called the period. The period is 21 in
example 2. The number of repeating digits is
the length of the period. The length of the
period is 2 digits in example 2.
00
F86. Build the factor tree for the denominator and classify each fraction as
a terminating or a non-terminating decimal:
a)
1
2
b)
2
3
c)
3
4
d)
4
5
e)
5
6
f)
2
7
g)
7
9
h)
3
10
i)
8
11
j)
2
15
k)
11
20
l)
3
6
m)
3
40
n)
13
100
o)
25
120
d)
1
5
e)
3
20
i)
3
500
j)
1
625
F87. Write each fraction as a terminating decimal:
1
3
5
a)
b)
c)
2
4
8
f)
11
80
g)
13
100
h)
1
125
F88. Write each fraction as a non-terminating repeating decimal. Use a bar over the repeating
decimals:
a)
1
3
b)
1
6
c)
11
36
d)
7
9
e)
2
11
f)
1
15
g)
7
30
h)
2
7
i)
2
13
j)
1
27
F89. Write each fraction as either a terminating or non-termonating repeating decimal. Use a
bar over the repeating decimals:
a)
1
1
2
b)
6
15
c)
1
f)
1
5
4
g)
5
64
h)
k)
3
250
l)
1
7
m)
© La Citadelle
7
21
d)
1
75
e)
7
3
1
32
i)
3
64
j)
1
128
20
14
n)
5
12
o)
2
45
3
13
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The Book of Fractions
Iulia & Teodoru Gugoiu
Converting decimals to fractions
1. To convert a terminating decimal to a fraction, use
the place value of each digit of the decimal part.
Example 1:
1.25 = 1 +
2
5
1 1
5
1
+
= 1+ +
=1 =1
10 100
5 20
20
4
1000
3. To convert a non-terminating repeating decimal to
a fraction, you can use algebra.
Example 3:
x = 0.666... = 0.6
10 x = 6.666...
10 x - x = 6.666... - 0.666...
9x = 6
x=
2. A faster method to convert a terminating decimal to a fractions
is:
a) the numerator is the number without the decimal point
b) the denominator is 1 followed by a 0 for each digit of the
decimal part
1625 13
5
1.625 =
= =1
Example 2:
6 2
=
9 3
8
8
4. A faster method to convert a non-terminating repeating
decimal to a fraction is the following:
1) the numerator is a mixed number:
a) the whole is the decimal written without the decimal
point and the period
b) the numerator is the period
c) the denominator is one 9 for each digit of the period
2) the denominator is 1 followed by one 0 for each digit between
the decimal point and the period
45
Example 4:
6123
6.12345 =
99 = 6 679
1000
5500
F90. Write each terminating decimal as a fraction in lowest terms (see the example 1):
a)
0 .1
b)
0 .5
c)
1 .4
d) 1.25
e)
0.75
f)
0.035
g)
2.125
h)
10.125
i)
j)
100.725
5.075
F91. Write each terminating decimal as a fraction in lowest terms (see the example 2):
a)
0.625
b)
1 .5
c)
0.125
d) 0.4
e)
2.16
f)
0.275
g)
0.24
h)
0.35
i)
j)
0.640625
2.45
F92. Write each non-terminating decimal as a fraction in lowest terms (see the example 3):
a)
0.3
b)
1.2
c)
0.12
d) 1.21
e)
4.025
f)
0.23
g)
1.25
h)
2.012
i)
j)
1.23456
0.123
F93. Write each non-terminating decimal as a fraction in lowest terms (see the example 4):
a)
0.7
b)
1.3
c)
2.53
d) 1.32
e)
1.129
f)
0.12
g)
3.01
h)
1.312
i)
j)
1.0123
© La Citadelle
46
6.12345
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The Book of Fractions
Iulia & Teodoru Gugoiu
Order of operations (VI)
2. If an expression contains operations with non-terminating
repeating decimal numbers, convert decimal numbers to
fractions and do all the operations with fractions.
Example 2:
1
1. If an expression contains only decimal numbers, you
can do all the operations using decimal numbers.
Example 1:
3
1 25
533
1.3 ´1.25 - 0.31 = 1 ´1 - 9 = 1
@ 1.359
3 99 10
1485
1 .2 + 0 .5 ´ 1 .6 - 2 .4 ¸ 2 = 1 .2 + 0 .8 - 1 .2 = 0 .8
3. If an expression contains both fractions and decimal numbers, it is recommended that you first convert the decimal
numbers to fractions, and do all the operations with fractions.
Example 3:
1 .2 +
2
4 3
6 2 1 4 3 3 7
- 0 .5 ´ + ¸ 1 .5 = + - ´ + ¸ = = 1 .4
5
5 10
5 5 2 5 10 2 5
F94. Find the value of each expression (only use operations with decimals):
a) 0.25 + 0.15 =
b)
0.50 - 0.15 =
c)
d)
2 .4 ´ 0 .5 =
0.10 ¸ 0.05 =
e) 0.5 2 + 0.25 =
1.2 ¸ 0.75 =
e) 1 - 0.4 2 =
F95. Find the value of each expression by converting decimals to fractions:
a) 0.75 + 1.25 =
b) 0.60 - 0.25 =
c)
d)
0 .5 ´ 1 .4 =
F96. Find the value of each expression by converting repeating decimals to fractions:
a) 1.3 + 0.6 =
b) 2.4 - 0.45 =
c)
d)
0.5 ´ 2.2 =
0.28 ¸ 0.01 =
e) 0.6 2 - 0.32 =
F97. Find the value of each expression by converting fractions to decimals:
a) 2 + 0.5 ´ 1 ¸ 0.2 - 0.2
5
2
b)
2 1
1 .5 ´ ¸ - 0 .5 2
5 2
c)
4
- 0.3 ¸ 1.5 + 0.82 d)
5
1ö 3
æ
2
ç 0 . 1 + ÷ ´ - 0 .2
5 ø 10
è
F98. Find the value of each expression by converting decimals to fractions:
a) 1 + 0.20 ´ 3
4
2
b)
5
2
0.25 ¸ - 0.15 ´
6
3
c)
2
1
¸ 0.25 ´ - 0.3
3
4
3
d)
æ1ö
0.52 - ç ÷ ´ 0.8
è 2ø
F99. Find the value of each expression by converting decimals to fractions:
a) 1 + 0.5 ´ 3 - 3 ¸ 1.5 ´ 2
2
5 4
5
æ 2 5 3ö
ç1.5 ´ - ÷ ¸ 24
6 8ø
d) è
3
0.4 ´ - 0.25
4
b) 1.25 ¸ 5 ´ 0.5 + 0.5 ´ 0.752 ´ 2
4
3
2
e)
1ö
3
æ
ç 0.25 + ÷ +
2 ø 16
è
´ 0.5
æ4
ö
ç - 0.3 ÷ ´ 3
è5
ø
2
5
é 2
ù
3
2
g) êæç + 1.3 ö÷ ¸ 5 - 0.4ú ´ æç1.6 - ö÷ + 0.2 ¸
5ø
3
ø
êëè 3
úû è
i)
© La Citadelle
æ3
ö æ3
ö
ç - 0.1÷ ¸ ç - 0.5 ÷
ø è4
ø ´3
f) è 5
1ö æ1
æ
ö 4
ç1.25 - ÷ ´ ç + 1.25 ÷
2ø è 4
è
ø
3
2
h) é2.5 ¸ æç 4 + 0.7 ö÷ + 1 ù ¸ éæç1.8 - 3 ö÷ ´ 5 ù + 1
ê
ú ê
5 ø 2 úû 9
è5
ø 3 û ëè
ë
2
3
4ù
é
2
1
êë2.4 - 9 úû ´ 0.5 0.6 ´ 0.3 +
9 ¸ 52
¸
2
2ö
æ
1ö
æ
2.4 - ç1.2 + ÷
1.5 + ÷
ç
5ø
è
9ø
è
c) 4 ´ 0.5 ´ 5 - 0.3 ´ 1 + 0.25 ´ 1 ´ 4
5
3
2
3
j)
3
æ1ö
0 .2 - ç ÷ + 0 .6 ´
-1
2
0.1´ 0.25 ¸ 0.4
5
æ1ö
è5ø
-5
+
7
¸
ç ÷ ´2
-1
2
æ
ö
è 3ø
é 1 1 æ 1 öù
1 .6 ´ ç 0 .4 - ´ 0 .5 ÷
ê 2 ´ 5 ¸ ç 2 ÷ú
5
è
ø
è øû
ë
47
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The Book of Fractions
Iulia & Teodoru Gugoiu
Time and Fractions
1. One hour has 60 minutes. This relation can be written
in two ways:
1 h = 60 min
1 min =
4. To make conversions between minutes and seconds,
use the formulas above.
Example 3:
1h 1
=
h
60 60
0.25 min =
2. To make conversions between hours and minutes,
use the formulas above.
Example 1:
Example 4:
1
4
4
1 h = h = ´ 60 min = 80 min
3
3
3
100 s = 100 ´
1
25
5
h=
h=
h
60
60
12
4500 s = 4500 ´
3. One minutes has 60 seconds. This relation can be
written in two ways:
1 min = 60 s
1s =
1
100
5
min =
min = min
60
60
3
5. The conversion between seconds and hours is a twostep task.
Example 2:
25 min = 25 ´
3
3
min = ´ 60 s = 45 s
4
4
1
1
1
min = 75 min = 75 ´
h =1 h
60
60
4
1 min 1
=
min
60
60
F100. Convert hours to minutes. Write the results as mixed numbers in lowest terms:
a) 0.5 h
b) 1 h
c) 1.75 h
d) 2 3 h
e) 0.1 h
3
4
5
7
h) 3 h
g) 2.25 h
i)
j) 1.23 h
h
h
6
8
15
F101. Convert minutes to hours. Write the results as mixed numbers in lowest terms:
f)
a)
b)
5 min
c)
10 min
d)
15 min
25 min
e)
50 min
3
j)
2.5 min
min
4
F102. Convert minutes to seconds. Write the results as mixed numbers in lowest terms:
a) 1.5 min
b) 1 2 min
c) 0.25 min
d) 1 2 min
e) 1.15 min
3
5
f)
70 min
g)
36 min
h)
250 min
i)
f)
7
min
12
g)
1.75 min
h)
5
min
12
i)
11
min
45
j)
7.3 min
F103. Convert seconds to minutes. Write the results as mixed numbers in lowest terms:
a)
b)
12 s
10 s
1
g) 0.75 s
s
2
F104. Do the required conversions:
f)
10
a) 500 s = ? h
b) 1500 s = ? h
g) 2 h 10 min = ? s
© La Citadelle
c)
45 s
h)
15
3
s
4
c) 9000 s = ? h
h) 15 min 5 s = ? s
d)
d)
90 s
e)
200 s
i)
100
s
3
j)
20.6 s
0 .2 h = ? s
i) 0.5 h 15 s = ? min
48
e) 1.75 h = ? s
f) 0.15 h = ? s
j) 2 h 10.5 min 10 s = ? min
3
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The Book of Fractions
Iulia & Teodoru Gugoiu
Canadian coins and fractions
1. One dollar (loonie or $) has 100 cents (pennies):
Example 1:
1 $ = 100 cents
1
1
125 cents = 125 ´
$ =1 $
100
4
2. One twonie (or toonie) has 200 cents:
Example 2:
1.75 twonies =
Example 4:
25 cents = 25 ´
5. One quarter has 25 cents:
Example 5:
1 twonie
1
=
twonie
200
200
1 nickel = 5 cents
1 cent =
1 nickel 1
= nickel
5
5
1 dime = 10 cents
1 cent =
1 dime 1
=
dime
10
10
1 quarter = 25 cents
1 cent =
1 quarter 1
=
quarter
25
25
1
6
1 nickels = ´ 5 cents = 6 cents
5
5
4. One dime has 10 cents:
1
1
dimes = 2 dimes
10
2
1$
1
=
$
100 100
1 twonie = 200 cents 1 cent =
175
´ 200 cents = 350 cents
100
3. One nickel has 5 cents:
Example 3:
1 cent =
2
2
quarters = ´ 25 cents = 10 cents
5
5
F105. Convert dollars to cents. Write the results as mixed numbers in lowest terms:
a) 0.12 $
b)
1
$
125
d) 1 2 $
75
c) 1.02 $
e) 0.07 $
f)
7
$
150
F106. Convert cents to dollars. Write the results as mixed numbers in lowest terms:
a) 25 cents
b) 20 cents
c) 45 cents
d) 160 cents
e) 450 cents
f) 5.5 cents
F107. Convert twonies to cents. Write the results as mixed numbers in lowest terms:
a) 0.02 twonies
b) 0.95 twonies
c)
3
twonies
100
d)
11
twonies
150
e) 1 3 twonies
10
f) 1.2 twonies
25
F108. Convert cents to twonies. Write the results as mixed numbers in lowest terms:
a) 125 cents
b) 250 cents
c) 40 cents
d) 120 cents
e) 500 cents
f) 10.2 cents
F109. Convert nickels to cents. Write the results as mixed numbers in lowest terms:
a) 0.2 nickels
b) 1.2 nickels
c) 7 nickels
20
d) 7 nickels
5
e) 2 3 nickels
20
f) 10.5 nickels
2
F110. Convert cents to nickels. Write the results as mixed numbers in lowest terms:
a) 15 cents
b) 25 cents
c) 75 cents
d) 4 cents
e) 1.5 cents
f) 0.5 cents
F111. Convert quarters to cents. Write the results as mixed numbers in lowest terms:
a) 0.2 quarters
b) 1.6 quarters
c) 3 quarters
25
d)
9
quarters
125
e) 1 3 quarters
5
f) 1.2 quarters
5
F112. Convert cents to quarters. Write the results as mixed numbers in lowest terms:
a) 50 cents
b) 125 cents
c) 100 cents
d) 20 cents
e) 0.5 cents
f) 55 cents
F113. Do the required conversions:
a) 3 quarters = ? nickels b) 1.5 dimes = ? nickels c) 2.5 dimes = ? quarters d)
e) 8 nickels = ? dimes
© La Citadelle
f) 15 nickels = ? dimes
0.2 quarters = ? dimes
g) 1 quarters 2 dimes = ? nickels h) 5 quarters = ? dimes
49
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Iulia & Teodoru Gugoiu
Fractions, ratio, percent, decimals, and proportions
1. A fraction is a comparison between a part and the whole. For example 40/100
2. A ratio is a comparison between two numbers. For example 40:100
3. A percent is a comparison between a number and 100. For example 40%
4. A decimal is a comparison between a number and 1. For example 0.40
5. The same number can be written as a fraction, ratio, percent, or decimal. To convert the number from one
expression to another, use this two-step algorithm:
1) Express the number as a fraction
Ex. 1
2 7
1 =
5 5
Ex. 2
7 to 5 = 7 : 5 =
7
5
Ex. 3
140 % =
140 7
=
100 5
Ex. 4
1 .4 =
14 7
=
10 5
2) Convert the fraction using a proportion and the cross-multiplication rule
Ex. 5
7
2
=1
5
5
Ex. 6
7
x
7 ´ 35
= x to 35 = ; x =
= 49; so
5
35
5
Ex. 7
7
= 1 .4
5
Ex. 8
7
x
7 ´100
=
; x=
= 120; so
5 100
5
7
= 49 to 35
5
7
= 120%
5
F114. Fill out the table:
Fraction
Ratio
a)
1/ 4
... to 20
b)
1/ 3
... out of 21
c)
3/ 2
12 to ...
d)
5/6
15 out of ...
e)
3/ 5
42 out of ...
Percent
Decimal
f)
2 out of 5
g)
10 to 45
h)
3 out of 12
i)
18 to 15
j)
7 to 10
k)
... to 70
30 %
l)
36 out of ...
80 %
m)
... to 30
130 %
n)
... out of 5
10 %
o)
... out of 64
75 %
p)
... to 10
0.15
r)
... out of 35
1.4
s)
6 to ...
0.08
t)
80 out of ...
0.64
u)
... out of 256
0.125
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50
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The Book of Fractions
Iulia & Teodoru Gugoiu
Fractions and Number Line
1. To create a number line assign points to 0 and 1:
0
4. Number lines can be used to compare numbers
(fractions). Example 3:
3 2
1
0
2. Each number corresponds to a point on the number
line.
3
Example 1:
1/4 2/4 3/4
0
0
0
1
3/5
4/5
2/5
1
4
5
0
2
1
3
1
1
5. To calibrate a ruler means to assign numbers to the
points on the ruler.
Example 4:
3. Each point on the number line corresponds to a
number.
2
1
Example 2:
0
<
1
1/5
5
5
2/3
1/3
3
4
1
1
4
1
1
1
2
1
3
4
2
1
2
F115. Use the number line to compare fractions (use > = or < symbols):
1
6
1
12
0
1
10
a) 1
1
4
1
5
6
12
2
f) 2 5
5 12
0
A
B
1
3
5
12
3
10
1
2
7
12
2
5
2
3
3
5
3
4
7
10
5
6
4
5
c) 3 7
5 12
d) 4 5
5
6
e) 11 9
12 10
g) 4 3
h) 1
6
i)
4
5
3
4
j) 1 1
12 10
4
1
5
1
C
W X
Y
F
G
11
10
b) 1 3
3 10
5
E
13
12
1
9
10
F116. Find a point on the number line corresponding to each fraction:
D
11
12
H
I
J
K
L
M N
O
P
Q
R
S
T
U
V
a)
1
2
b)
3
4
c)
1
12
d)
1
f)
3
24
g)
7
12
h) 1 5
24
i)
1
Z
a
b
c
d
e
1
4
e)
1
12
j)
f
g
h
i
j
3
8
1
1
8
F117. Find the fraction corresponding to each point on the number line:
B
D
F
H
J
0
1
A
C
E
G
I
F118. Calibrate the ruler:
0
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1
51
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The Book of Fractions
Iulia & Teodoru Gugoiu
Comparing fractions
1. Fractions are ordered numbers. That means you can
compare them and decide if they are equal (=) or which
one is greater (>) or less (<) than the other.
The main idea is that a small number is less (<) than a
big number:
small < big (1)
4. To compare two fractions in the general case:
n1
n2
?
d1
d2
use cross multiplication to convert the initial comparison to
another equivalent one:
1< 3
5>2
4=4
Example 1:
2. If you divide relation (1) by any number a, you’ll get:
n1 ´ d 2
Example 3:
1 1
<
3 2
2 2
>
5 7
3 2
>
4 5
Example 4:
small big
<
( 2)
a
a
So, if you compare two fractions having the same
denominator, the smallest one has the smallest
numerator.
2 4
7
5
3 3
<
>
=
Example 2:
5 5
11 11
4 4
3. If you cross exchange the factors in relation (2), you’ll
a
a
get:
<
(3)
big small
So, if you compare two fractions having like numerators,
the smallest fraction has the biggest denominator.
? n 2 ´ d1
because 3 ´ 5 > 4 ´ 2
5. To compare two fractions in the general case, you can
also find the LCD (Least or Lowest Common Denominator),
convert the original fractions to equivalent fractions having
like denominators and then use the relation (2).
Example 5:
5
7
5 20 21 7
<
because
=
<
=
12 16
12 48 48 16
6. You can use the LCD method when you have to order a
set of fractions:
Example 6:
3 3
=
8 8
2 3 5
< <
3 4 6
because
2 8 3 9 5 10
=
< =
< =
3 12 4 12 6 12
F119. Compare the whole numbers. Use <, =, and > operators :
a) 2 and 9
b) 10 and 10
c) 1 and 0
F120. Compare the fractions (see example 2):
a) 3 and 5
b) 5 and 1
c) 2 and 1
7
7
6
6
6
3
7
5
h) 2 and
i) 1 and
j) 1 5 and 2 1
3
5
4
4
F121. Compare the fractions (see example 3):
a) 1 and 1
b) 2 and 2
c) 1 1 and 3
2
3
7
5
2
2
5
5
7
2
h) and
i) 1 and
j) 3 and 3 2
4
6
9
7
3
F122. Compare the fractions (see example 4):
a) 2 and 3
b) 4 and 6
c) 1 1 and 1 3
3
4
5
7
2
5
5
7
2
h) 5 and 15
i)
j) 2 and 2 5
and
6
9
5
2
4
12
F123. Compare the fractions (see example 5):
a) 3 and 8
b) 3 and 11
c) 3 and 5
5
15
8
24
4
6
d) 7 and 5
e) 123 and 132
f)
d) 3 and 5
e) 13 and 23
f) 1 2 and 4
11
11
2
k) 3 and 2 4
5
5
d) 3 1 and 2 3
4
5
l)
f) 10 and 10
1
3
and
2
4
4
27
1 and
5
15
f) 7 and 8
e) 4 and 5
f) 5 and 7
l)
d) 3 and 5
e)
k)
l)
4
7
3
2
and
2
3
d) 4 and 7
15
20
12
15
g) 2 4 and 14
3
3
1
m)3 and 2 4
3
3
4
4
and
7
3
4
and 4
5
e)
1
1 and
3
k)
25
25
1
2 and 3
5
g) 12 and 21
0 and 3
5
n) 11 and 2 1
4
4
g) 2 4 and 2 4
7
11
2
m) 1 and 7
5
6
n)
7
5
13
3
and 2
7
5
3
4
and
8
9
3
4
and
5
7
g)
10
11
3
m) and 4
10
13
6
5
n)
g) 2 and 4
10
9
15
F124. Write each set of fractions in order from least to greatest:
a) ìí 2 , 1 , 3 üý
î7 7 7 þ
b) ìí 3 , 3 , 3 üý
î 5 11 7 þ
c) ìí 3 , 2 , 1 , 4 üý
î4 3 2 5þ
d) ìí 5 , 3 , 2 üý
î12 10 5 þ
e) ìí 1 , 7 , 3 , 4 üý
î 2 12 4 15 þ
f) ìí1 1 , 1 5 , 1, 11 , 1 5 üý
î 4
24
8
6þ
F125. Write each set of fractions in order from greatest to least:
a) ìí 3 , 4 , 1 üý
î5 5 5 þ
© La Citadelle
b) ìí 2 , 2 , 2 üý c) ìí 5 , 3 , 2 , 6 üý d)
î7 5 6 þ
î6 4 3 7 þ
ì1 3 1ü
í , , ý
î4 5 3þ
52
e) ìí 7 , 3 , 2 , 9 üý
î12 4 3 10 þ
f) ìí1 1 , 1 3 , 1 1 , 1 5 , 1 11 üý
î 4
8
2
16
32 þ
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The Book of Fractions
Iulia & Teodoru Gugoiu
Solving equations by working backward method
The working backward method requires to identify the
operations applied to the unknown quantity x, and do
the opposite operations in the opposite order.
1. If the operation is an additions with a number then
apply a subtraction with the same number.
Example 1:
x+
1 3
=
2 4
so
x=
5. If the operation is an inversion then apply another
inversion.
Example 5:
1 2
=
x 3
so
2
3=1
4
2
x-
1 1 5
x= + =
3 2 6
2 3
=
3 4
so
x=
1
1
3´ x 2
2
3 2 9
¸ =
4 3 8
4. If the operation is a division by a number then apply a
multiplication by the same number.
Example 4:
3 4
x¸ =
5 7
so
x=
3
2
so
x-
2 1
= ´ 4 = 2 so
3 2
x = 2+
2 8
=
3 3
Example 7:
3. If the operation is a multiplication by a number then
apply a division by the same number.
Example 3:
x´
so
6. If more than one operation are implied then identify
the operations and the order in witch they appear and
then do the opposite operations in the opposite order.
Example 6:
3 1 1
- =
4 2 4
2. If the operation is a subtraction with a number then
apply an addition with the same number.
Example 2:
1 1
x- =
2 3
x 3
=
1 2
so
+
4
3
3´ x -
so 3 ´ x -
4 3 12
x= ´ =
7 5 35
1
= 2 so
1 3
=
2 8
1
2
so 3 ´ x =
+
4
= 4 so
3
7
8
so
x=
1
3´ x -
1
2
=
8
3
7
24
F126. Solve for x working backward:
a) x + 2 = 1
b)
1 4
x- =
3 5
f) 2 ´ x + 2 = 4
g)
3´ x -
k) x + 1 = 2
l)
x 2 1
- =
5 3 2
5
2
3
3
2
5
3
2
p) x ´ 3 - 3 1
2
t)
4
+ =
3 5
1 1
x´2 ¸ 3 2´2 = 4
2 1
3 3
3 2
1 1
=
2 3
1 1
=
2 3
c)
x´
h)
x 2 3
´ =
3 3 4
d) x ¸ 4 = 2 1
7
m) ( x - 1) ´ 1 = 4
2
1
2 -1 =1
3
2 3
q)
x´2 +
u)
æ x´3 1 1 ö
¸ - ÷
ç
2 3 ÷´ 2 = 4
ç 4
1
1
ç
÷ 3 5
+
ç
÷
2 3
è
ø
r)
5
e) 1 = 5
x
3
i)
x 2 4
¸ =
3 3 5
n)
1ö 2 1
æ
çx+ ÷¸ =
5ø 3 2
è
1ö
æ
ç x´2 - ÷ 2 4
2 ÷´ =
ç
2
ç
÷ 3 5
ç
÷
è
ø
1
x¸22 ¸2=1
v)
3
3 6
7
1
2
=
x-2 3
1
3
=
o)
1 4
x+
3
j)
s)
w)
æx
ö
ç -1 2 ÷ 3 2
ç2
- ÷¸ =
3÷ 5 5
ç 3
ç
÷
è
ø
2
3
=
x -1 2 4
2
3
F127. Solve for x working backward:
a)
x´3 5 1
- ¸
8
8 2 ´ æç 1 - 1 ö÷ = 1 ´ 3 - 1
3 2
è 2 6ø 2 4 4
4 3
c)
ìé
ü
1 1 ù
ïê 1
ï
ú
ï
4 5 ú ´ 4 + 1ï ¸ 3 = 1
ê
+
í
ý
ïê10 2 ¸ æç x - 2 ö÷ ú 5 5 ï 5 2
ïîêë
ïþ
3 è
5 ø úû
© La Citadelle
1 3 1
¸ 2 2 6 ¸ æç 2 - 1 ö÷ - æç 4 - 2 ö÷ = 1 + 2 ¸ 5
1
1 4 è 3 4ø è 5 3ø 5 3 4
- ´
x´3 2 3
b)
æ
ö
ç
1
2 ÷÷ æ 2 3 2 ö
æ1 4 3 1ö ç
- -ç ¸ + ÷ = 0
ç ´ ¸ - ÷¸
5
1 3 ÷ è 5 10 3 ø
è3 5 5 9ø ç
+
ç
÷
è x´2 +3 5
ø
d)
53
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The Book of Fractions
Iulia & Teodoru Gugoiu
Final Test
Find the fractions:
1)
2)
3)
three eights
9)
3
4) five out of seven
5) four and three quarters
Express the numbers as improper fractions:
7) 2
6)
1
3
2
5
8) 3
10
7
10)
11) 2.4
five and two thirds
12) one and five thirds
Express the numbers as mixed numbers:
13)
10
4
14)
15
4
15)
16) 1
5
3
17) eight fifths
Write the fractions in lowest terms:
21)
24
36
22)
20
8
23)
18)
15
7
1.25 19)
20) eleven quarters
Check each pair of fractions for equivalence:
12
18
24)
18
32
1
6
5
6
10
15
and
12
20
25)
26)
4
5
and
12
15
33)
1
100
75
and
64
48
27)
Add the fractions:
28) 1+ 2
2
3
2 1
+
5 5
29)
31)
1 3
+
3 4
32)
5 3
7 7
38)
3 2
4 3
39) 3 - 2
1 2
2 3
45)
2 2
1 ´1
3 5
46)
3 4
´
4 9
47)
5 2
´1
3 3
48)
3 1
1 ´1
4 3
52)
4 6
1 ¸
5 10
53)
1
1
1 ¸3
6
2
54)
22 5
¸1
3
6
55)
1
2
2 ¸2
4
5
15
12
60)
62)
5
16
7
8
64)
30) 1 + 2
10 5
+1
3
6
2
3
+2
15
20
34)
5
2
+1
4 3
41)
9 1
-1
5 4
Subtract the fractions:
35)
1-
2
3
10
-2
3
36)
37)
4
5
7
1
2
40) 5 - 2
10
4
3
Multiply the fractions:
42)
2 3
´
3 5
43) 2´
3
5
44) 1 ´
50) 2 ¸
2
5
51)
8
¸4
9
58)
4
10
Divide the fractions:
49)
2 3
¸
3 4
Convert each fraction to a decimal:
56)
2
5
57) 2
2
5
59)
2
3
61) 1
2
7
63)
100
32
Convert each decimal to a fraction:
65) 0.12
66) 1.5
67) 2.25
68) 0.0125
69) 0.4
70) 1.63
71)
0.725
72) 1.65
73) 0.875
Do the required conversions (use a fraction to express the result if possible):
74) 0.45 min = ? s
75) 0.1 h = ? min
79) 1.5 dimes = ? nickels
83)
2
= ? twelfths
3
80)
84)
76)
100 s = ? min
7.5 quarters = ? $
1
= ? tenths
5
85) 1
81)
1
= ? sixths
2
77) 2000 s = ? h
78) 20 min 20 s = ? h
0.35 $ = ? twonies
86)
82) 1 $ 1 quarter = ? dimes
3
= ? eighths
16
87)
2
= ? fifteenths
3
Do the required operations:
1 1 1
- +
2 3 4
æ2 1ö 2
92) ç - ÷ ´
è 3 4ø 3
1 4 1
´ 2 3 6
2 æ 1 5ö
¸ ç1 - ÷
93)
3 è 3 6ø
88)
89)
90)
94)
2 5 2
1 ¸ +
3 6 3
4ö
æ 2ö æ
ç1 - ÷ ´ ç 2 ¸ ÷
3ø
è 3ø è
1
3 1
2 ¸3 ´
2
4 2
91)
1ö æ2 3 ö
æ1
¸2 ÷-ç - ÷
2 ø è 5 10 ø
è2
95) ç
Solve for x:
96)
1 1
x- =
3 4
© La Citadelle
97)
1 2
x¸ =
3 5
98)
x 5
=
3 6
99)
54
2
3
=
x -3/ 2 4
100)
1
2
=
2 + 3/ x 5
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The Book of Fractions
Iulia & Teodoru Gugoiu
Answers
1
2
1
5
58
b)
c)
d)
e)
2
3
4
9
100
6
2
9
1
3
n)
o)
p)
q)
r)
10
4
13
4
8
F01.
6
5
4
2
1
6
1
2
g)
h)
i)
j)
k)
l)
m)
10
6
6
6
3
18
1
5
7
21
6
0
12
7
10
s)
t)
u)
v)
w)
x)
y)
9
49
12
12
12
10
16
a)
f)
F02. The answers may vary.
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)
z)
F03.
a ) two thirds
b ) three hundredths
h ) four fifths
i ) eight thirtieths
o ) three fiftieths
p ) two fifths
u ) eleven millionths
F04.
c ) one tenth
d ) one half
j ) eight thirteenths
k ) eight ninths
q ) twenty - one hundredths
v ) two ninths
w) seven tenths
e) three sevenths
l ) five sixths
r ) six twelfths
x ) eleven twelfths
f ) three twentieths
m) five eighths
s ) seven elevenths
y ) two fiftieths
1
1
1
2
4
7
11
7
5
8
6
9
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
3
2
6
5
7
8
50
20
12
9
10
1000
8
3
11
23
7
11
3
13
1
n)
o)
p)
q)
r)
s)
t)
u)
v)
6
50
1000000000
100
13
12
1000000000
30
5
a)
2
3
F05.
a)
F06.
a) 1
b)
1
4
c)
3
5
d)
3
6
e)
3
12
f)
2
5
g)
2
7
h)
3
4
i)
5
12
j)
3
10
k)
4
6
l)
55
n ) seven thousandths
t ) eleven fiftieths
z ) nine billionths
15
1000000
1
8
w)
x)
11
9
m)
y)
6
10
z)
6
12
3
5
1
1
2
1
3
1
5
7
33
7
1
b) 2
c) 1
d) 2
e) 1
f)2
g) 3
h) 6 i ) 1
j) 1
k) 4
l) 2
2
4
3
3
6
4
6
9
100
10
6
8
6
3
9
13
5
6
6
10
3
7
n) 1
o) 5
p) 6
q) 2
r) 2
s) 3
t) 2
u) 1
v) 5
w) 1
x) 2
18
10
4
13
49
9
8
12
16
10
10
© La Citadelle
g ) one thousandth
m) 3
2
5
www.la-citadelle.com
The Book of Fractions
Iulia & Teodoru Gugoiu
F07. The answers may vary.
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)
z)
F08.
a ) one and one half
b ) two and one third
f ) two and three sevenths
c ) one and one quarter
g ) three and five eighths
d ) two and three fifths
h ) one and five ninths
e) one and five sixths
i ) two and three tenths
j ) one and two elevenths
k ) three and five twelfths l ) one and two fifteenths m) three and seven twentieths n ) two and nine thirtieths
o) two and seven fiftieths p ) two and three hundredths q ) three and nine thousandths r ) two and seven millionths
s ) one and three fourtieths t ) two and seven nineteenths u ) three and five sixteenths v) two and three seventeenths
w) four and three fourteenths
x) two and five fifteenths
z ) two and three ninetieths
F09. a) 2 2 b) 3 1 c) 5 5 d ) 2 1
3
2
6
3
5
2
3
n) 8
o) 3
p) 5
6
12
1000000
F10. a) 2
3
1
b) 2
5
3
c) 1
3
4
d)1
4
6
e) 4
q ) 20
e) 2
F11. a) 5 b) 3 c) 11 d ) 15 e) 10
3
17
q)
10
1
15
r)
6
4
6
31
48
s)
t)
9
13
5
7
6
u)
32
12
f)7
3
100
3
5
f)3
5
50
y ) two and one sixtieth
g) 2
3
4
h) 3
r) 6
4
15
s ) 11
2
5
g) 2
2
3
4
30
h) 3
2
9
i) 6
t) 8
7
10
1
2
25
5
14
10
g)
h)
i)
9
2
4
3
133
74
62
v)
w)
x)
100
16
49
f)
7
100
j) 9
u) 4
1
3
i) 2
4
6
j) 3
j)
6
5
k)
3
5
1
2
v) 1
k) 8
2
5
k) 4
33
11
l)
10
8
11
50
w) 3
2
3
2
11
l) 2
m)
5
2
m) 1
1000000000
11
6
9
11
x) 8
y) 5
z) 1
9
10
12
l) 1
3
4
19
4
n)
21
42
o)
10
18
p)
13
10
F12. The answers may vary.
a)
g)
© La Citadelle
b)
h)
c)
i)
d)
j)
56
e)
f)
k)
l)
www.la-citadelle.com
The Book of Fractions
Iulia & Teodoru Gugoiu
m)
n)
o)
p)
q)
s)
t)
u)
v)
w)
y)
z)
F13.
a)
F14.
a)
F15.
a)
F16.
F17.
5
4
b)
7
4
c)
7
5
d)
4
3
e)
16
12
f)
12
5
7
1
3
1
9
1
=2
b) = 1
c) = 2
d)
3
3
2
2
4
4
12
2
25
5
20
4
j)
=2
k)
=2
l)
=2
5
5
10
10
8
8
16
4
24
6
32
6
r)
=2
s)
=2
t)
=2
6
6
9
9
13
13
g)
7
5
h)
11
7
i)
4
3
j)
13
10
k)
r)
x)
13
13
l)
6
5
20
2
13
1
23
5
7
1
11
3
8
2
=3
e)
=2
f)
=2
g) = 2
h) = 2
i) = 2
6
6
6
6
9
9
3
3
4
4
3
3
15
3
21
1
24
6
13
3
34
4
m)
=3
n)
=2
o)
=1
p)
=1
q)
=3
4
4
10
10
18
18
10
10
10
10
24
136
36
54
6
111
13
u)
= 2 v)
=1
w)
=3
x)
=2
12
100
100
16
16
49
49
3
8
15
7
17
103
53
23
67
18
26
23
21
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
2
3
4
2
7
20
10
4
30
13
9
6
8
1207
103
24
221
25
35
149
211
2
20
n)
o)
p)
q)
r)
s)
t)
u)
v)
w)
x) not defined
100
50
5
100
12
11
50
100
9
10
y)
12
3
z)
1
1
1
1
6
13
5
4
2
2
8
2
6
b) 1
c) 1
d) 4
e) 1
f )1
g ) 12
h) 2
i) 2
j) 6
k) 8
l) 8
m) 8
2
3
4
2
7
20
10
5
3
3
9
6
8
7
3
2
1
4
4
11
1
0
4
5
n) 1
o) 2
p) 4
q) 2
r) 1
s) 6
t) 2
u ) 11
v) 0
w) not defined x) 1
y) 1
10
5
5
10
12
11
50
10
9
7
15
35
10
a) 1
a ) 1 b ) 1 c ) 1 d ) 2 e) 3
q ) not defined
f ) 8 g ) 10 h) 2 i ) 3
j ) 4 k ) 6 l ) 15 m) 4 n) 5 o) 0
z) 7
7
9
p ) not defined
r ) 1 s ) 2 t ) 0 u ) not defined
F18. The answers may vary.
2 5
6 9
14 21
8 12
2 12
10 45
100 30
0 0
50 125
100 700
;
b) ;
c) ;
d) ;
e) ;
f) ;
g)
;
h) ;
i) ;
j)
;
2 5
2 3
2 3
2 3
1 6
2 9
10 3
2 3
2 5
1
7
22 55
16 24
26 39
34 51
k)
;
l) ;
m)
;
n) ;
2 5
2 3
2 3
2 3
a ) improper b) whole c) improper d ) proper e) mixed f ) improper g ) mixed h) not defined i ) proper
j ) whole k ) improper l ) proper m) mixed n) mixed o) mixed p ) improper q ) whole r ) improper s ) improper
t ) mixed u ) proper v) mixed w) improper x) proper y ) improper z ) improper
a)
F19.
F20.
r)
F21.
2
3
5
52
c)
d)
e)
3
4
9
100
7
25
10
5
s)
t)
u)
v)
9
49
12
9
a ) 1 b)
6
8
7
10
6
w)
12
f)
4
5
4
h)
i)
6
6
6
7
12
x)
y)
10
16
g)
j)
2
3
k)
10
18
l)
4
6
m)
3
5
n)
7
10
o)
3
4
p)
10
13
q)
3
4
2
4
1
9
1
12
3
120
20
24
4
8
2
20
2
7
1
= 1 b) = 1
c) = 2
d)
=1
e)
=1
f)
=2
g) = 1
h)
=3
i) = 1
2
3
3
4
4
9
9
100
100
10
10
6
6
6
6
6
6
4
1
19
1
19
1
6
1
13
3
5
1
15
2
10
2
j) = 1
k)
=1
l)
=3
m) = 1
n)
=1
o) = 1
p)
=1
q)
=2
3
3
18
18
6
6
5
5
10
10
4
4
13
13
4
4
a)
© La Citadelle
57
www.la-citadelle.com
The Book of Fractions
r)
10
2
=1
8
8
s)
13
4
=1
9
9
F22. a) 3 b) 2 c) 3 d ) 8
4
10
r)
20
3
18
s)
25
5
63
14
=1
49
49
t)
11
7
u)
19
7
t)
9
Iulia & Teodoru Gugoiu
3
6
e)
v)
f)
5
10
u)
5
7
3
4
p) 1
9
4
7
q) 1
10
5
r) 1
12
40
3
4
3
q ) 10
10
s) 4
1
4
t) 2
3
10
2
o) 27
11
3
5
f)9
6
11
q ) 40 r ) 14
30
s ) 15
o) 2
3
11
p) 3
4
8
9
q) 3
2
7
1
2
F27. a) 3 b) 2 3 c) 4 d ) 3 e) 4 1
4
o) 10
3
11
p) 5
q) 7
2
10
r ) 16
52
100
f ) 5 g) 3
5
6
9
s) 6
1
2
s) 6
h) 1
33
100
2
10
t) 7
49
50
3
9
i) 2
21
30
w) 5
h) 4
1
9
18
6
=1
12
12
x)
28
12
=1
16
16
12
11
32
k)
l)
41
13
54
7
11
j) 1
x) 8
8
10
1
23
u ) 11 v) 9
2
12
1
3
k) 1
y) 4
m)
6
15
4
9
h) 8 i ) 4
11
3
k ) 4 l) 7
15
7
20
10
x) 7
y) 4
35
100
1
10
14
30
w) 5
j) 2 k ) 6
17
23
1
10
u ) 4 v) 8
j) 5
4
6
2
4
w) 6
k) 4
x) 3
2
3
2
5
4
5
m) 1
6
10
5
9
m) 6
l) 7
2
35
l) 6
x) 4
1 3
1 2
f) =
g) =
3 9
5 10
2 6
2 4
4
1 2 4 8 16
1 5
o) = = =
=
p) =
24
2 4 8 16 32
4 20
1 2 4
2 4 8 16
w) = =
x) = =
=
y)
4 8 16
3 6 12 24
9
12
o)
5
25
13
17
n) 1
p)
0
4
38
50
q)
o) 1
5
13
10
100
9
10
j) 5
i ) 12
8
2
=1
6
6
n)
3
20
l) 1
1
7
y ) 113
13
4
2
8
j ) 12
k) 3
l) 2
50
6
5
9
1
9
v) 8
w) 8 x) 3
y) 3
4
10
18
50
y)
5
10
h) 9 i ) 4
F28. a) 1 = 2 b) 1 = 2 c) 1 = 2 = 4 d ) 2 = 6 e) 1 = 2
2 4
3 6
2 4 8
1 2 4
1 2
1 2
l) = =
m) =
n) =
=
4 8 16
2 4
6 12
1 3
1 2
4
1 3
t) =
u) =
=
v) =
4 12
9 18 36
4 12
j)
w) 1
v) 8
t ) 11 u ) 2
7
10
29
35
i) 1
h) 3
5
8
w)
4
10
5
8
g ) 13
10
r ) 12
40
400
1
4
t) 8
3
10
2
9
10
y)
6
9
g) 5
F26. a) 6 b) 1 2 c) 7 1 d ) 7 1 e) 8 f ) 3 g ) 6 7
4
i)
v) 3
6
1
14
s ) 10
t ) 9 u) 4
3
19
4
1
p ) 20
9
8
9
5
7
30
100
11
13
g) 1
u) 1
f)5
F25. a) 3 1 b) 3 2 c) 4 1 d ) 7 2 e) 7 5
2
4
8
13
4
=1
9
9
v)
h)
x)
f )1
7
5
3
r) 6
30
12
19
20
30
6
F24. a) 4 b) 3 2 c) 3 3 d ) 4 4 e) 3 5
1
p) 8
9
g)
w)
F23. a) 1 1 b) 1 1 c) 1 2 d ) 1 2 e) 1 1
19
7
=1
12
12
3
10
m) 5
n) 8
1
10
3
20
n) 5
o) 7
5
11
2
20
11
100
m) 8 n) 4
3
12
m) 7 n) 5
y) 9
1
10
1 2
2 6
1 2
1 3
=
i) =
j) =
k) =
3 6
3 9
4 8
6 18
1 2
1 2
1 2
4
8
q) =
r) =
s) =
=
=
6 12
4 8
5 10 20 40
3 6 12 24 48
= =
=
=
4 8 16 32 64
h)
F29. The answers may vary.
2 3 4
2 3 4
4 6 8
2 3
4
6 9 12
2
3
4
6
9 12
= =
b) = =
c) = =
d) =
=
e) =
=
f)
=
=
g)
=
=
4 6 8
6 9 12
6 9 12
8 12 16
8 12 16
10 15 20
10 15 20
2
3
4
10 15 20
2
3
4
4
6
8
2
3
4
2
3
4
h)
=
=
i)
=
=
j)
=
=
k)
=
=
l)
=
=
m)
=
=
12 18 24
12 18 24
14 21 28
14 21 28
20 30 40
200 300 400
10 15 20
4
6
8
4
6
8
6
9 12
10 15 20
4
6
8
n)
=
=
o)
=
=
p)
=
=
q)
=
=
r)
=
=
s)
=
=
24 36 48
22 33 44
18 27 36
14 21 28
22 33 44
30 45 60
8 12 16
10 15 20
2
3
4
6
9
12
6
9
12
6
9
12
t)
=
=
u)
=
=
v)
=
=
w)
=
=
x)
=
=
y)
=
=
10 15 20
14 21 28
60 90 120
100 150 200
400 600 800
2000 3000 4000
1
1
2 1
1
3
2 1
5
2
1
2 1
4
2
3
10 6 2
5
a)
b)
c) =
d)
e)
=
=
f)
=
=
g) =
h)
i)
j)
k)
= =
l)
2
3
8 4
5
18 12 6
50 20 10
6 3
5
3
4
15 9 3
8
9
6
3
30 20 12 10 6 4 2
30 20 12 10 6
4 2
4
25 15 5
m)
=
=
n)
=
=
=
= = =
o)
=
=
=
=
=
=
p)
q)
=
=
30 20 10
45 30 18 15 9 6 3
75 50 30 25 15 10 5
7
40 24 8
33 6 3
28
14
8
4
2
16
8
4 2 1
9 3 1
9
2
45
9
r)
=
=
s)
=
=
=
=
t)
=
=
= =
u)
= =
v)
w)
x)
=
77 14 7
210 105 60 30 15
64 32 16 8 4
27 9 3
11
11
125 25
32
16
8
4
2
1
y)
=
=
=
=
=
512 256 128 64 32 16
a)
F30.
© La Citadelle
58
www.la-citadelle.com
The Book of Fractions
F31. a) 1
12
5
r)
22
Iulia & Teodoru Gugoiu
3
2
1
3
1
3
1
5
c)
d)
e)
f)
g)
h)
i)
7
3
4
4
5
5
6
6
2
1
5
7
11
3
7
s)
t)
u)
v)
w)
x)
y)
15
2
7
15
25
20
10
b)
F32. a) 3 b) 4 c) 2 d ) 3 e) 7
4
5
3
2
r)
s)
7
15
3
1
t)
4
j)
2
8
2
7
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h)
i)
7
15
5
13
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9
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11
25
16
f)
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9
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9
u)
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3
11
1
7
j)
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4
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2
7
k)
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3
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3
14
m)
7
9
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8
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8
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4
5
2
11
p)
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9
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3
7
2
5
p)
4
9
q)
5
8
o)
F33. a ) yes b) yes c) no d ) yes e) no f ) yes g ) no h) yes i ) no j ) no k ) no l ) yes m) yes
n) no o) no p ) yes q ) no r ) yes s ) no t ) yes u ) yes v) no w) yes x) yes y ) no
F34. a ) no b) yes c) no d ) yes e) no f ) no g ) yes h) yes i ) no j ) yes k ) no l ) yes m) no
n) yes o) no p ) no q ) yes r ) no s ) no t ) yes u ) yes v) no w) no x) yes y ) yes
F35. a) 4 b) 2 c) 16 d ) 4 e) 20 f ) 8 g ) 3 h) 4 i) 15 j ) 60 k ) 15 l ) 54 m) 30 n) 72 o) 5
F36. a ) 3 b) 21 c) 15 d ) 9 e) 6
f ) 15 g ) 30 h) 56 i ) 27
F37. a ) 2 b) 15 c) 5 d ) 25 e) 7
f ) 3 g ) 8 h) 12 i ) 24
F38. a) 1
1
2
b) 1
1
5
c) 3
F39. a) 5 b) 1 5
6
q) 3
7
15
F40. a) 1 1 b) 7
6
1
p) 2
6
F41. a) 1 1
q) 4
12
41
p) 1
150
b)
4
15
25
48
q) 2
d)
r) 5
19
20
c)
12
d) 3
13
20
c)
12
1
p) 2
6
3
4
7
75
5
6
24
199
p) 1
2400
e)
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1
110
7
30
s) 2
17
80
d)
r) 3
47
144
F42. a) 1 11 b) 1 11 c) 1 29
12
107
o)
525
e) 1
11
30
d)1
r) 4
c)
1
6
1
3
120
17
q) 1
60
f)6
29
35
e)
7
15
7
20
11
16
f )1
t) 3
26
35
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13
20
f)
7
20
6
7
g) 3
3
56
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u)
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20
18
13
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35
28
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37
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150
450
e)
h) 3
67
72
h)
v) 1
79
200
3
10
53
56
3
4
5
6
h) 1
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13
40
3
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5
2
77
90
1
18
5
11
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4
7
43
31
17
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110
110
30
1
13
5
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6
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32
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19
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8
2
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15
3
15
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23
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11
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144
36
55
22
30
15
18
27
59
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80
200
120
28
192
g)
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197
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15
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252
19
7
227
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60
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9
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181
240
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11
30
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31
180
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9
50
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7
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75
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75
F43. a) 2 - 1 = 1 b) 2 - 1 = 1 c) 3 - 2 = 1 d ) 5 - 3 = 2 e) 50 - 20 = 3
2 2 2
3 3 3
4 4 4
9 9 9
3 1 1
4 2 1
2 1 1
16 8 4
h) - =
i) - =
j) - =
k)
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6 6 3
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18 18 9
7 5 1
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100 100 10
10 10 5
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6 6 6
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- =
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r) - =
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=
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13 13 13
4 4 4
8 8 2
9 9 9 3
49 49 49
8 4 1
5 2 1
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7 3 2
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10 10 5
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© La Citadelle
5
1
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The Book of Fractions
F45. a) 2 b) 1 c) 1 1 d ) 9
3
p) 1
5
1
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5
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Iulia & Teodoru Gugoiu
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F46. a) 1 1 b) 3 c) 8 2 d ) 6 5 e) 2 1
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F48. a) 1 b) 1
6
5
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F50. a) 7
12
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11
13
23
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5
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F52. a) 1 17 b) 13 c) 8 13 d ) 23 e) 1 f ) 1 g ) 1 h) 23 i) 53
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F53. a)
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© La Citadelle
1
6
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F57.
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150
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3
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60
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The Book of Fractions
Iulia & Teodoru Gugoiu
1
1
1
1
3
3
1
1
15
1
7
63
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3
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77
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F64.
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7
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100
F62. a)
F63.
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F65. a) yes b) no c) no d ) yes e) no f ) no g ) yes h) no i) yes j ) yes
F66. a) 4 b) 2 c) 2
F67. a)
2
3
b) 1
1
2
1
2
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1
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5
3
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F69. a) 2 b)
5
6
3
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2
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F73.
F74.
F75.
F78.
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F77.
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The Book of Fractions
F81.
F82.
F83.
F84.
F85.
F86.
Iulia & Teodoru Gugoiu
3
4
27
1
c)
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8
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k ) t l ) nt
1
16
a ) t b) nt
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c) t
1
3
d) t
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2
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1
3
f ) nt
g ) nt
h) t i ) nt
j ) nt
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F87.
a ) 0.5 b) 0.75 c) 0.625 d ) 0.2 e) 0.15
F88.
a ) 0.3 b) 0.16 c) 0.305 d ) 0.7 e) 0.18
F89.
a ) 1.5 b) 0.4 c) 1.3 d ) 0.013 e) 2.3
f ) 0.1375 g ) 0.13 h) 0.008 i ) 0.006
j ) 0.0016
f ) 0.06 g ) 0.23 h) 0.285714 i ) 0.153846
j ) 0.037
f ) 2.25 g ) 0.078125 h) 0.03125 i ) 0.046875
j ) 0.0078125 k ) 0.012
l ) 0.142857 m) 1.428571 n) 0.416 o) 2.230769
F90.
F91.
F92.
F93.
1
1
b)
10
2
5
1
a)
b) 1
8
2
1
2
a)
b) 1
3
9
7
1
a)
b) 1
9
3
a)
2
1
3
7
1
1
3
29
d)1
e)
f)
g) 2
h) 10
i) 5
j ) 100
5
4
4
200
8
8
40
40
1
2
4
11
6
7
9
41
c)
d)
e) 2
f)
g)
h)
i) 2
j)
8
5
25
40
25
20
20
64
11
19
23
23
25
2
41
7811
c)
d)1
e) 4
f)
g) 1
h) 2
i)
j) 1
90
90
900
99
99
165
333
33300
8
29
13
4
1
103
679
41
c) 2
d)1
e) 1
f)
g) 3
h) 1
i) 6
j) 1
15
90
100
33
99
330
5500
3330
c) 1
F94.
a ) 0.4 b) 0.35 c) 1.2 d ) 2 e) 0.5
F95.
a ) 2 b)
F96.
7
7
3
21
c)
d)1
e)
20
10
5
25
98
19
1
a ) 2 b) 1
c) 1
d ) 26 e)
99
81
3
F97.
a ) 1.45 b) 0.95 c) 1.24 d ) 0.05
F98.
a)
F99.
11
1
1
3
b)
c)
d)
20
5
3
20
3
11
5
1
1
a)
b)
c)
d)1
e)
5
16
6
4
4
f )1
1
3
g)
7
10
h) 1 i )
5
6
j) 1
1
2
F100. a) 30 min b) 20 min c) 105 min d ) 165 min e) 6 min f ) 50 min g ) 135 min h) 22 min i) 28 min j ) 74 min
3
1
1
1
h h) 4 h i )
h j)
h
5
6
80
24
2
a ) 90 s b) 100 s c) 15 s d ) 84 s e) 69 s f ) 35 s g ) 105 s h) 25 s i ) 14 s j ) 440 s
3
1
1
3
1
1
7
1
21
5
31
a ) min b) min c) min d ) 1 min e) 3 min f )
min g )
min h)
min i ) min j )
min
5
6
4
2
3
40
80
80
9
90
5
5
1
1
2
a)
h b)
h c) 2 h d ) 720 s e) 6300 s f ) 540 s g ) 7800 s h) 905 s i ) 30 min j ) 50 min
36
12
2
4
3
F101. a)
F102.
F103.
F104.
1
1
1
5
5
h b) h c ) h d )
h e) h
12
6
4
12
6
© La Citadelle
1
f )1 h
6
g)
62
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The Book of Fractions
Iulia & Teodoru Gugoiu
4
5
2
3
2
3
F105. a) 12 cents b) cents c) 102 cents d ) 102 cents e) 7 cents f ) 4 cents
1
4
1
5
9
3
1
11
$ d ) 1 $ e) 4 $ f )
$
20
5
2
200
2
3
a ) 4 cents b) 190 cents c) 6 cents d ) 14 cents e) 260 cents f ) 9 cents
3
5
5
1
1
3
1
51
a ) twonies b) 1 twonies c) twonies d ) twonies e) 2 twonies f )
twonies
8
4
5
5
2
1000
3
3
1
a ) 1 cent b ) 6 cents c ) 1 cents d ) 7 cents e) 10 cents f ) 26 cents
4
4
4
4
3
1
a ) 3 nickels b) 5 nickels c) 15 nickels d ) nickels e)
nickels f )
nickels
5
10
10
4
a ) 5 cents b) 40 cents c) 3 cents d ) 1 cents e) 40 cents f ) 6 cents
5
4
1
1
a ) 2 quarters b) 5 quarters c) 4 quarters d ) quarters e)
quarters f ) 2 quarters
5
50
5
F106. a) $ b) $ c)
F107.
F108.
F109.
F110.
F111.
F112.
F113. a) 15 nickels
b) 3 nickels
c) 1 quarter
d)
1
dimes
2
e) 4 dimes
f)7
1
dimes
2
g ) 9 nickels
h) 12
1
dimes
2
F114.
1
1
3
5
= 5 to 20 = 25 % = 0.25 b) = 7 out of 21 = 33 .3 % = 0.3 c) = 12 to 8 = 150 % = 1.5 d ) = 15 out of 18 = 83 .3 % = 0.83
4
3
2
6
3
2
2
1
e) = 42 out of 70 = 60 % = 0.6 f ) = 2 out of 5 = 40 % = 0.4 g ) = 10 to 45 = 22 .2 % = 0.2 h) = 3 out of 12 = 25 % = 0.25
5
5
9
4
6
7
3
4
i ) = 18 to 15 = 120 % = 1.2 j )
= 7 to 10 = 70 % = 0.7 k )
= 21 to 70 = 30 % = 0.3 l ) = 36 out of 45 = 80 % = 0.8
5
10
10
5
13
1
3
3
m)
= 39 to 30 = 130 % = 1.3 n)
= 0.5 out of 5 = 10 % = 0.1 o) = 48 out of 64 = 75 % = 0.75 p )
= 1.5 to 10 = 15 % = 0.15
10
10
4
20
7
2
16
r ) = 49 out of 35 = 140 % = 1.4 s )
= 6 to 75 = 8 % = 0.08 t )
= 80 out of 125 = 64 % = 0.64
5
25
25
1
u ) = 32 out of 256 = 12 .5 % = 0.125
8
a)
F115.
a ) = b ) > c ) > d ) < e) >
F116.
a) M
F117.
1
A =>
12
b) S
c) C
1
B =>
6
f)<
d ) e e) J
4
C =>
15
g ) > h) < i ) >
f)D
11
D =>
30
g ) O h) d
1
E =>
2
j) <
i) a
j) b
7
F =>
12
G =>
7
10
4
5
41
60
H =>
47
60
I =>
11
12
J => 1
1
12
F118.
0
1
10
1
5
3
10
2
5
3
5
1
2
9
10
1
1
1
10
1
1
5
1
3
10
1
2
5
F119. a) < b) = c) > d ) > e) < f ) < g ) <
F120.
a ) < b ) > c ) = d ) < e) <
f)>
g ) = h) < i ) =
j ) = k ) > l ) < m) = n) >
F121.
a ) > b ) < c ) = d ) > e) <
f)>
g ) < h) > i ) >
j ) < k ) > l ) < m) > n) <
F122.
a ) < b ) < c ) < d ) > e) <
f)<
g ) < h) = i ) >
j ) < k ) > l ) = m) < n) >
F123.
a ) > b ) < c ) < d ) < e) =
© La Citadelle
f)>
g) <
63
www.la-citadelle.com
The Book of Fractions
Iulia & Teodoru Gugoiu
1 2 3
3 3 3
1 2 3 4
3 2 5
4 1 7 3
5
1 11
5
b) < <
c) < < <
d)
< <
e)
< <
<
f )1<1 <1 < <1
7 7 7
11 7 5
2 3 4 5
10 5 12
15 2 12 4
24
4 8
6
4 3 1
2 2 2
6 5 3 2
3 1 1
9 3 2 7
1
3
11
5
1
a) > >
b) > >
c) > > >
d) > >
e)
> > >
f )1 >1 >1 >1 >1
5 5 5
5 6 7
7 6 4 3
5 3 4
10 4 3 12
2
8
32
16
4
F124. a) < <
F125.
F126. a) 1
b) 1
2
15
c)
5
12
b)
13
44
c)
2
3
2) 1
2
3
3)
10
8
p)
15
F127. a) 3
2
1
2
1
5
3
3
d)1
e) 1
f)
g)
h) 3
i) 1
3
3
5
15
18
8
5
9
11
5
8
2
2
q) 1 r ) 1
s) 7
t)
u)
v) 1
w) 7
20
25
36
9
3
3
11
15
1
2
k)
1
2
l) 5
5
6
m) 2
3
5
n)
2
15
o) 1
d)1
5
3
3
7
17
31
17
12
8
2
1
3
5) 4
6)
7)
8)
9)
10)
11)
12)
13) 1
14) 2
15) 3
7
4
2
3
5
7
3
5
3
3
2
4
2
3
1
1
3
2
1
2
9
2
3
16) 2
17) 1
18) 1
19) 2
20) 2
21)
22) 2
23)
24)
25) no 26) yes 27) yes 28) 3
29)
30) 4
3
5
4
7
4
3
2
3
16
3
5
1
1
17
11
1
1
2
1
1
7
11
2
1
31) 1
32) 5
33) 3
34) 2
35)
36) 1
37)
38)
39) 1
40) 2
41)
42)
43) 1
44) 1
12
6
60
12
3
3
7
12
10
12
20
5
5
1
1
7
1
8
2
1
15
45) 2
46)
47) 2
48) 2
49)
50) 5 51)
52) 3 53)
54) 4 55)
56) 0.4 57) 2.4 58) 0.4 59) 1.25
3
3
9
3
9
9
3
16
3
1
1
1
4
7
29
13
60) 0.6 61) 1.285714 62) 0.3125 63) 0.875 64) 3.125 65)
66) 1
67) 2
68)
69)
70) 1
71)
72) 1
25
2
4
80
9
11
40
20
7
2
5
61
7
7
1
1
5
73)
74) 27 75) 6 76) 1
77)
78)
79) 3 80) 1
81)
82) 12
83) 8 84) 2 85) 9 86)
87) 10 88)
8
3
9
180
8
40
2
2
12
1
2
1
5
1
1
1
7
2
1
1
89)
90) 2
91)
92)
93) 1
94)
95)
96)
97)
98) 2
99) 4
100) 6
2
3
3
18
3
2
10
12
15
2
6
FT.
1)
© La Citadelle
3
8
j) 3
4)
64
www.la-citadelle.com