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The Book of Fractions
Citation preview
The Book ofFractions
Iulia & Teodoru Gugoiu
Copyright © 2006 by La Citadelle
www.la-citadelle.com
Iulia & Teodoru Gugoiu
The Book of Fractions
© 2006 by La Citadelle4950 Albina Way, Unit 160Mississauga, OntarioL4Z 4J6, [email protected]
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.
ISBN 0-9781703-0-X
Edited by Rob Couvillon
Topic
Understanding fractionsThe graphical representation of a fractionReading or writing fractions in wordsUnderstanding the fraction notationUnderstanding the mixed numbersReading and writing mixed numbers in wordsUnderstanding mixed number notationUnderstanding improper fractionsUnderstanding improper fraction notationThe link between mixed numbers and improper fractionsConversion between mixed numbers and improper fractionsWhole numbers, proper fractions, improper fractions and mixed numbersUnderstanding the addition of like fractionsUnderstanding the addition of like fractions (II)Adding proper and improper fractions with like denominatorsAdding mixed numbers with like denominatorsAdding more than two like fractionsUnderstanding equivalent fractionsFinding equivalent fractionsSimplifying fractionsChecking fractions for equivalence Equations with fractionsAdding fractions with unlike denominatorsAdding fractions with unlike denominators using the LCD methodUnderstanding the subtraction of fractions with like denominatorsSubtracting fractions with like denominatorsSubtracting mixed numbers with like denominatorsSubtracting fractions with unlike denominatorsSubtracting fractions with unlike denominators using the LCD methodOrder of operations (I)Multiplying fractionsMore about multiplying fractionsThe order of operations (II)Reciprocal of a fractionDividing fractionsDivision operatorsOrder of operations (III)Order of operations (IV)Raising fractions to a powerOrder of operations (V)Converting fractions to decimalsConverting decimals to fractionsOrder of operations (VI)Time and FractionsCanadian coins and fractionsFractions, ratio, percent, decimals, and proportionsFractions and Number LineComparing fractionsSolving equations by working backward methodFinal TestAnswers
Page
56789
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Content
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
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)
0 1
© La Citadelle www.la-citadelle.com
The Book of Fractions
Understanding fractions
5
F01. Write the fraction that represents the part of the object that has been shaded:
1. A fraction represents a part of a whole.Example 1.
The whole is divided into four equal parts.Three part are taken (considered).
2. The corresponding fraction is:
4
3 The numerator represents how many parts are taken.
Fraction line or division bar
The denominator represents the number ofequal parts into which the whole is divided.
Iulia & Teodoru Gugoiu
a) b) c) d) e)
h) i) j) l) n)
o) q) r) s) t)
v)
www.la-citadelle.com
The Book of Fractions
The graphical representation of a fraction
6
F02. Draw a diagram to show each fraction (use the images on the bottom of this page):
1. A fraction represents a part of a whole.Example 1.
The whole is divided into four equal parts.Three part are taken (considered).
2. A corresponding graphical representation (diagram) is:
4
3
f)
k)
p) u)
m)
w)
g)
x) y) z)
0 10 1
2
1
3
1
4
1
5
2
6
1
4
2
3
0
9
2
6
5
12
2
10
9
1
1
3
3
6
4
4
3
12
4
10
5
9
4
4
2
12
8
13
5
16
5
8
1
49
7
100
37
18
11
The numerator represents how many parts are taken.
Fraction line or division bar
The denominator represents the number ofequal parts into which the whole is divided.
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Reading or writing fractions in words
7
F03. Write the following fractions in words:
1. You can use words to refer to a part of a whole.So one whole has:
2 halves3 thirds4 quarters5 fifths6 sixths
7 sevenths8 eighths9 ninths10 tenths11 elevenths
12 twelfths13 thirteenths20 twentieths30 thirtieths50 fiftieths
100 hundredths1000 thousandths1000000 millionths1000000000 billionths
Example 1.
The fraction
can be written in words as:
three quarters
4
3
a) b) c)
w)
d)
g) h) i)
j) l)
m) o)
p) q)
s)
F04. Find the fraction written in words:
e)
x)
n)
r)
k)
t)
f)
u)
v)
one third
two fifths
eleven fiftieths
z)y)
eight ninths
fifteen millionths
eleven billionths
eleven twelfths
one fifth
six tenths
one half
four sevenths
seven twentieths
six tenths
eight sixths
twenty-three hundredths
three billionths
one eleventh
six twelfths
one sixth
seven eighths
five twelfths
nine thousandths
three fiftieths
seven thirteenths
thirteen thirtieths
eight ninths
3
2
2
1
6
5
5
4
7
3
8
5
50
3
20
3
12
11
9
2
10
1
1000
7
100
3
13
8
50
2
1000000
11
100
21
11
7
30
8
1000000000
9
50
11
5
2
1000
1
9
8
10
7
12
6
n)
u)
g)f)
m)
t)
e)
s)
z)
l)
d)
r)
k)
y)
c)
q)
x)
j)
b)
p)
w)
i)
a)
h)
o)
v)
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Understanding the fraction notation
8
1. A fraction also represents a quotient of two quantities:
2. The dividend (numerator) represents how many parts are taken.The divisor (denominator) represents the number ofequal parts into which the whole is divided.
divisor
divident
4
3Example 1. The dividend (numerator) is 3.
The divisor (denominator) is 4.The fraction in words is three quarters.
F05. Fill out the following table:
Fraction Numerator(Dividend)
Denominator(Divisor) The fraction written in words Graphical representation
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
3
22 3 two thirds
1 4
three fifths
3
25
25
3.......... quarters
3
five ..........
4 .......... sixths
5 three ..........
A possible graphical representation of this fraction is:
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Understanding the mixed numbers
9
a) b) c)
w)
d)
g) h) i)
j) l)
m) o)
p) q)
s)
e)
x)
n)
r)
k)
t)
f)
u)
v)
F06. Find the mixed number that corresponds to the shaded region:
F07. Find a possible graphical representation of each mixed number:
n)
u)
g)f)
m)
t)
e)
s)
z)
l)
d)
r)
k)
y)
c)
q)
x)
j)
b)
p)
w)
i)
a)
h)
o)
v)
0 1 2 3
3
21
2
13
6
52
7
33
5
43
8
51
11
54
6
13
9
22
5
22
10
22
20
31
10
66
6
31
8
53
16
71
10
72
5
42
1
15
6
52
10
73
6
11
2
15
9
73
8
32
4
36
0 1 2
1. A mixed number is an addition of wholes and a part of a whole.Example 1.
There are one complete whole andthree 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.
4
31
whole-number part(the number ofcomplete wholes)
fraction part
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Reading and writing mixed numbers in words
10
F08. Write the following mixed numbers in words:
1. You can use words to refer to a part of a whole.So one whole has:
2 halves3 thirds4 quarters5 fifths6 sixths
7 sevenths8 eighths9 ninths10 tenths11 elevenths
12 twelfths13 thirteenths20 twentieths30 thirtieths50 fiftieths
100 hundredths1000 thousandths1000000 millionths1000000000 billionths
Example 1.
The fraction
can be written in words as:two wholes and three quarters ortwo and three quarters
a) b) c)
w)
d)
g) h) i)
j) l)
m) o)
p) q)
s)
F09. Find the mixed numbers written in words:
e)
x)
n)
r)
k)
t)
f)
u)
v)
two and two thirds
two and one third
two and three quarters
z)y)
nine and one half
one and two elevenths
five and three millionths
eleven and four thirtieths
one and two fifths
five and nine tenths
three and one half
four and five sevenths
three and two ninths
eight and eleven fiftieths
eight and five sixths
twenty and three hundredths
eight and seven tenths
three and two elevenths
one and eleven twelfths
five and five sixths
seven and five fiftieths
six and seven hundredths
one and five billionths
three and two twelfths
six and four fifteenths
four and one third
eight and six ninths
2
11
5
32
15
21
9
51
6
51
20
73
50
72
7
32
15
52
17
32
4
11
30
92
3
12
11
21
60
12
16
53
1000
93
40
31
10
32
90
32
19
72
100
32
8
53
12
53
14
34
1000000
72
n)
u)
g)f)
m)
t)
e)
s)
z)
l)
d)
r)
k)
y)
c)
q)
x)
j)
b)
p)
w)
i)
a)
h)
o)
v)
4
32
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Understanding mixed number notation
11
1. A mixed number is represented by the expression: Example 1. The whole-number part is 2(the number of complete wholes).The numerator is 3.The denominator is 5.The fraction part is:
A possible graphical representation of this mixed number is:
F10. Fill out the following table:
Numberof wholes
Numerator Denominator The mixed numberin words
Graphical representation
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
2 5 two and three fifths
MixedNumber
rdenominato
numeratorwholes
5
32
This mixed number written in words is two wholes and three fifths.
5
32 3
1 43
3
12
52 3
23 5
3
2 2
three and a half
two and four ...........
.......... and three fifths
four and .......... thirds
6
3
2
2 .......
5
3
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Understanding improper fractions
12
a) b) c)
w)
d)
g) h) i)
j) l)
m) o)
p) q)
s)
e)
x)
n)
r)
k)
t)
f)
u)
v)
F11. Find the improper fraction that corresponds to the shaded region:
F12. Find a possible graphical representation of each improper fraction:
n)
u)
g)f)
m)
t)
e)
s)
z)
l)
d)
r)
k)
y)
c)
q)
x)
j)
b)
p)
w)
i)
a)
h)
o)
v)
0 1 2 3
2
3
3
4
2
5
5
7
3
7
4
11
6
8
8
10
9
20
5
12
10
22
12
25
16
30
24
40
6
21
13
28
3
11
18
40
49
55
6
25
9
17
12
30
4
15
5
17
10
24
3
13
0 1 2
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).
3
5This is a possible graphical representation of this improper fraction:Example 1.
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Understanding improper fraction notation
13
1. An improper fraction is represented by the expression:
rdenominato
numerator
F13. Fill out the following table:
Fraction Numerator Denominator The fraction in words Graphical representation
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
4
55 4 five quarters
7 4
seven fifths
16
125
75
11.......... quarters
13
seven ..........
13 .......... sixths
5 thirteen ..........
where the numerator is equal to or greater than the denominator.
Example 1. The numerator is 5The denominator is 3
The improper fraction in words is five thirds. A possible graphical representation of this improper fraction is:
3
5
........
........
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
The link between mixed numbers and improper fractions
14
a) b) c)
w)
d)
g) h) i)
j) l)
m) o)
p) q)
s)
e)
x)
n)
r)
k)
t)
f)
u)
v)
F14. Find the mixed number and the improper fraction that correspond to each picture:
0 1 2 3
0 1 2
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.
3
21
3
21
3
5+== Example 2:
6
31
6
31
6
9+==Example 1:
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Conversion between mixed numbers and improper fractions
15
F15. Write each mixed number as an improper fraction:
1. To convert a mixed number to an improper fraction,use the formula:
2
11
2
13
6
53
4
35
7
32
8
52
50
32
20
35
0
23
9
20
4
33
100
712
3
22
13
51
3
33
100
112
100
212
11
23
30
72
10
152
50
492
5
44
10
35
9
82
10
02
12
12
n)
u)
g)f)
m)
t)
e)
s)
z)
l)
d)
r)
k)
y)
c)
q)
x)
j)
b)
p)
w)
i)
a)
h)
o)
v)
F16. Write each improper fraction as a mixed number:
2
3
2
9
6
50
5
14
7
13
8
60
5
13
20
33
7
11
9
0
4
5
10
17
3
4
13
80
15
20
10
111
10
21
11
70
3
8
9
70
50
111
5
22
10
125
9
80
0
7
12
16
n)
u)
g)f)
m)
t)
e)
s)
z)
l)
d)
r)
k)
y)
c)
q)
x)
j)
b)
p)
w)
i)
a)
h)
o)
v)
d
ndw
d
nw
+×=
Example 1.5
13
5
310
5
352
5
32 =
+=
+×=
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:
Example 2.
d
rq
d
n=
4
12
4
9=2. Fractions that have a denominator of 0 are not
defined.
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Whole numbers, proper fractions, improper fractions and mixed numbers
16
F17. Convert fractions to whole numbers. Identify the expressions that are not defined.
1. Although written in fraction notation,some numbers are actually whole numbers.
1
1
2
4
6
00
1
11
3
9
8
162
1
0
3
24
7
7
100
3002
2
2
5
50
0
00
0
0
3
93
3
32
5
554
0
2
10
100
9
02
12
242 n)
u)
g)f)
m)
t)
e)
s)
l)
d)
r)
k)
c)
q)
j)
b)
p)
i)
a)
h)
o)
F18. Convert whole numbers to fractions (the conversion is not unique, so give at leasttwo solutions):
1 4
80
2
13
57
17
3
10025
10
11 n)
g)f)
m)
e)
l)
d)
k)
c)
j)
b)
i)
a)
h)
52
10=Example 1: 3
3
32 =Example 2:
2. A whole number can be converted into a fraction.This conversion is not unique.
Example:7
143
10
302
3
34
10
104
10
505 =====
3. A proper fraction is:
An improper fraction is:
A mixed number is:
3
2dnwhere
d
n< Example:
3
7dnwhere
d
n³ Example:
3
54Example:
d
nw
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:
2
3
9
2
16
5
0
4
7
12
8
53
3
32
20
33
4
5
10
172
1
0
10
030
11
70
3
0
50
111
5
22
10
252
9
80
2
16
n)
u)
g)f)
m)
t)
e)
s)
l)
d)
r)
k)
c)
q)
j)
b)
p)
i)
a)
h)
o)
5
32
11
10
1
7
5
22
2
16z)y)x)w)v)
A mixed numberin standard form is:
3
25Example:dnwhere
d
nw £
4. Fractions that have a denominator of 0 are not defined.
© La Citadelle
Iulia & Teodoru Gugoiu
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)
www.la-citadelle.com
The Book of Fractions
Understanding the addition of like fractions
17
F20. Add the fractions that correspond to the shaded regions:
Two fractions with the same denominators are called like fractions. When you add two fractions, you add the parts of the whole they represent.
0 1
4
3
4
2
4
1=+
So, by adding 1 quarter and 2 quarters you get 3 quarters.
Example 1.
© La Citadelle
Iulia & Teodoru Gugoiu
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)
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The Book of Fractions
Understanding the addition of like fractions (II)
18
F21. Add the fractions that correspond to the shaded regions. Express the result both as animproper fraction and as a mixed number.
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.
4
21
4
6
4
3
4
3==+
=+
Or, in mathematical symbols:
Example 1.
© La Citadelle
Iulia & Teodoru Gugoiu
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The Book of Fractions
Adding proper and improper fractions with like denominators
19
F22. Add the fractions:
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:
4
1
4
2+
11
5
11
3+
41
10
41
2+
7
3
7
2+
6
2
6
1+
13
11
13
0+
20
3
20
7+
4
0
4
0+
5
1
5
2+
3
1
3
1+
100
10
100
20+
25
11
25
7+
10
3
10
2+
17
10
17
3+
19
7
19
5+
9
2
9
5+
54
11
54
21+
35
23
35
6+
13
2
13
3+
12
4
12
5+
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
Example 1.5
3
5
2
5
1=+
2. If the result is an improper fraction, you can change it to a mixed number.
4
11
4
5
4
2
4
3==+d
nn
d
n
d
n 2121 +=+
Example 2.
30
13
30
7+
19
5
19
2+
13
8
13
3+
100
25
100
15+
10
3
10
2+ y)x)w)v)u)
F23. Add the fractions. Write the result as a mixed number in standard form.
3
2
3
2+
6
5
6
3+
12
5
12
9+
8
7
8
5+
7
5
7
3+
15
11
15
10+
40
19
40
33+
9
6
9
7+
5
4
5
3+
4
2
4
3+
10
7
10
7+
3
9
3
4+
25
20
25
10+
100
11
100
99+
9
8
9
7+
10
9
10
15+
20
19
20
4+
11
10
11
8+
10
9
10
8+
50
44
50
44+
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
50
77
50
22+
9
5
9
12+
3
12
3
13+
10
34
10
15+
10
12
10
20+
y)x)w)v)u)
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Adding mixed numbers with like denominators
20
F24. Add the mixed numbers. Write the result as a mixed number in standard form or as a whole number:
1. To add mixed fractions with like denominators, add separately the wholes and separately the numerators, and keep the denominator unchanged:
2
12
2
11 +
5
33
5
11 +
15
62
15
53 +
7
23
7
32 +
6
21
6
32 +
5
7
5
32 +
30
222
30
113 +
9
82
9
25 +
4
22
4
11 +
3
12
3
11 +
9
10
9
23 +
3
92
3
73 +
10
123
10
12 +
11
83
11
83 +
8
33
8
22 +
10
253
10
152 +
7
62
7
44 +
10
3
10
52 +
10
115
10
222 +
20
115
20
122 +
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
Example 1.
5
26
5
75
5
43)32(
5
43
5
32 ==
++=+
2. In the same way you can add whole and mixed numbers.
4
15
4
10)32(
4
13
4
02
4
132 =
++=+=+
d
nnww
d
nw
d
nw 21
212
21
1 )(+
+=+
Example 2.
23
221
23
23 +
19
121
19
212 +
35
222
35
334 +
100
952
100
151 +
30
415
30
102 + y)x)w)v)u)
F25. Add the wholes and the mixed numbers. Write the result as a mixed number in standard formor as a whole number.
12
12 + 5
5
22 +
15
020 +
7
935 +
6
525 +
25
93 +
30
11311+9
9
199 +
4
131+
3
221+
9
103+
3
1039 +
310
111 +
11
131313 +
68
57 +
410
44 +
7
815 +
1110
11 +
10
1001020 +
20
2222 +
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
23
4022 +
19
1991+
35
3711+
100
1111111 +5
30
144 + y)x)w)v)u)
© La Citadelle
Iulia & Teodoru Gugoiu
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The Book of Fractions
Adding more than two like fractions
21
F26. Add the fractions:
1. To add more than two fractions or whole numbers, start to add in order, from left to right.
2
13
2
31 ++
2
11
2
12
2
13 ++
6
74
6
53
6
32
6
11 +++
3
3
3
2
3
11 +++
7
6
7
5
7
4
7
3
7
2
7
1+++++
50
3
50
72
50
32 ++
2
55
2
32
2
11 ++
9
72
9
11 ++
24
12
4
5
4
31 +++
4
3
4
2
4
1++
2
33
2
52 ++
3
33
3
22
3
11 ++
100
313
100
2121 ++
11
9
11
7
11
5
11
3
11
1++++
10
172
10
3
10
72 ++
10
203
10
15
10
101
10
52 +++
5
41
5
31 ++
9
21
9
5
9
11 ++
10
52
10
3
10
11 ++
12
71
12
5
12
32 ++
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
2. Because the addition is a commutative operation, the order in which you add the fractions is not important.So, group them conveniently.
84225
21
5
32
3
11
3
22
5
212
3
11
5
32
3
2=++=÷
ø
öçè
æ++÷
ø
öçè
æ++=++++
4
35
4
74
4
5
4
24
4
5
4
231
4
5
4
231 ==+=+÷
ø
öçè
æ+=++
4
51
4
32
4
13 ++
9
81
9
4
9
10 ++
3
11
3
22
3
33 ++
5
43
5
32
5
21
5
1+++
10
50
10
31
10
12 ++ y)x)w)v)u)
F27. Add the whole and the mixed numbers. Write the result as a mixed number in standard formor as a whole number.
2
3
2
11 ++ 2
2
1
2
1++
26
7
6
91 +++
3
42
3
21 +++
50
113
50
71 ++
12
75
2
52
2
31 +++
9
81
9
712 ++
24
31
4
1+++
4
12
4
2++
2
52
2
7++
23
21
3
42 ++
100
2212
100
113 +++
11
812
11
432
11
21 ++++
210
71 ++
210
73
10
41 +++
25
211 ++ 3
9
31
9
22 ++
10
73
10
521 ++
212
71
12
51 ++
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
24
51
4
53 ++
9
712
9
20 ++ 2
3
11
3
42 ++
25
31
5
3+++
010
72
10
91 ++ y)x)w)v)u)
10
531
10
71
10
92 +++
Example 1.
Example 2.
© La Citadelle
Iulia & Teodoru Gugoiu
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)
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The Book of Fractions
Understanding equivalent fractions
22
F28. For each image find the equivalent fractions that correspond to the shaded part:
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.
2
1
The shaded region can be expressed as:
4
2
6
3
12
6or
So these fractions are considered equivalent (equal),and you can write:
12
6
6
3
4
2
2
1===oror
© La Citadelle
Iulia & Teodoru Gugoiu
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The Book of Fractions
Finding equivalent fractions
23
F29. Find at least three equivalent fractions by using the method 1:
Example 1.
2
1
4
1
7
1
5
1
4
3
7
2
11
5
9
2
3
2
6
1
15
2
100
1
11
2
5
3
5
4
10
1
6
5
7
3
12
5
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
50
3
7
5
200
3
1000
330
1y)
x)w)v)u)
F30. Find equivalent fractions by using the method 2:
4
2
25
5
20
15
100
10
36
6
45
30
154
66
35
20
16
4
9
3
25
20
420
56
60
18
150
60
12
4
128
32
40
25
6
4
120
75
90
60
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
121
22
81
27
625
225
1024
64
77
63y)
x)w)v)u)
ad
an
d
n
´
´= ad
an
d
n
¸
¸=
Example 2.
4
2
22
21
2
1=
´
´=
30
15
56
53
6
3
32
31
2
1=
´
´==
´
´=
2
1
36
33
6
3
212
26
12
6=
¸
¸==
¸
¸=
So:
30
15
6
3
4
2
2
1=== So:
2
1
6
3
12
6==
3
1
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:
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Simplifying fractions
24
F31. Write each fraction in lowest terms by using the method 1:
72
6
128
32
147
21
160
32
80
60
126
36
264
60
225
50
75
50
132
22
360
48
135
105
132
24
90
54
270
135
350
75
240
200
112
48
4096
512
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
600
264
252
180
300
45
480
336270
126y)
x)w)v)u)
F32. Write each fraction in lowest terms by using the method 2:
24
18
120
90
80
64
98
28
72
56
45
30
154
66
45
20
81
54
60
48
120
48
420
56
60
18
150
60
60
32
128
32
40
25
52
28
120
75
75
60
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
121
22
81
27
625
225
1024
64770
630y)
x)w)v)u)
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.
...=¸
¸=
ad
an
d
n
Example 1.
5
2
315
36
15
6
230
212
30
12
260
224
60
24=
¸
¸==
¸
¸==
¸
¸=
42
18
where a is a common divisor of n and d
Method 2. You can simplify a fraction by division of the numerator and denominator by the Greatest Common Factor (GCF).
=¸
¸=
GCFd
GCFn
d
nfraction in lowest terms
24
2 12
2 6
2 3
60
2 30
2 15
3 5
To find the GCF build the factor trees for the numerator and denominator.Example 2.
1232
53260
3224
12
112
13
=´=
´´=
´=
GCF
5
2
1260
1224
60
24=
¸
¸=
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Checking fractions for equivalence
25
F33. Check if the fractions are equivalent (use the lowest terms method):
Example 1.
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
F34. Check if the fractions are equivalent (use the cross-multiplication method):
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
Given two or more fractions, you can check whether or not they are equivalent (equal).
1. Express each fraction in lowest terms and then compare them. If you get the same fraction, the original fractions are equivalent.
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.
Example 2.
cbdaifequivalentared
cand
b
a´=´32
24
20
15
30
21and and
In lowest termsthey are: 4
3
4
3
10
7and and
So:
20
15
32
24=
30
21
32
24¹
30
21
20
15¹
6
3;
4
2
9
3;
6
2
8
6;
3
2
12
8;
6
4
18
6;
10
5
120
48;
75
30
24
20;
35
30
35
15;
56
24
40
15;
25
9
54
30;
70
40
50
15;
80
25
66
30;
220
100
96
56;
24
14
104
40;
40
15
75
40;
45
25
30
20;
24
16;
15
10
40
25;
30
18;
10
6
48
40;
6
5;
30
25
35
12;
15
6;
21
10
40
25;
24
15;
96
60
10
5;
14
7;
8
4;
2
1
14
12;
12
10;
10
8;
4
3
4
3;
20
15;
28
21;
44
33
16
8;
8
4;
4
2;
2
1
5
4;
4
3;
3
2;
2
1
3
2;
2
1
8
4;
4
2
10
4;
12
6
20
15;
16
12
12
4;
35
12
5
6;
4
5
4
3;
12
9
30
12;
10
4
5
7;
7
5
56
48;
35
30
80
72;
45
40
60
36;
40
24
81
49;
64
40
8
2;
20
5
20
9;
9
4
5
9;
4
31
24
200;
15
125
15
13;
13
11
51
40;
85
65
19
15;
133
105
20
15;
12
9;
4
3
7
6;
6
5;
5
4
16
12;
12
8;
8
4
50
30;
30
18;
20
12
56
40;
35
25;
14
10
21430330
21
4
3´¹´becauseequivalentnotareand
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Equations with fractions
26
e)
o)
e)
j)
d)
n)
i)
d)
c)
m)
c)
h)
b)
l)
b)
g)
a)
f)
k)
a)
j)i)h)g)f)
e)
o)
e)
j)
d)
n)
i)
d)
c)
m)
c)
h)
b)
l)
b)
g)
a)
f)
k)
a)
j)i)h)g)f)
4
12=
82 =
53 =
40
12
8=
75
255=
18
18
6=
10010
3=
12816
9=
12
5
11=
64
2=
7
9
3=
10
12
8=
2128
12=
10
53=
2012
9=
24
15
12=
16
42
12=
48
18
72=
56
4060=
63
1 x=
x
10
3
2=
20
123=
x x
20
5
4=
35
153=
x
x
155 =
6
34=
x 20
15
16=
x
18
1520=
x 45
2416=
x
x
40
72
45=
66
x=
48
8030=
x 3025
40 x=
2520
28 x=
32
1 x=
35
2 x=
x
5
3
4=
x
5
2
11 =
316
9 x=
87
6 x=
516
12 x=
x
3
24
20=
264
16 x=
1535
202
x=
ad
an
d
n
´
´=
ad
an
d
n
¸
¸=
Example 1:
Method 1. You can solve simple equations with fractions if you use two properties of equivalent fractions:
48
?
16
3=
Solution:
48
9
316
33
16
3=
´
´=
Method 2. When applying the method 1, sometimes you need first to express the fraction in the lowest terms: :
Example 2:
10
?
8
4=
Solution:
10
5
52
51
2
1
8
4=
´
´==
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.
c
bax
c
b
a
x ´=Û=
Example 3 (the result is a whole number):
103
30
3
215
3
2
15==
´=Û= x
x
Example 4 (the result is a fraction):
7
6
7
32
7
3
2=
´=Û= x
x
82
1=
9
6
3=
324
2=
16
123=
12
5
3=
40
306=
F37. Solve for x using the method 3 (see example 3 above):
F38. Solve for x using the method 3 (see example 4 above):
F36. Find the unknown factor of the fraction using the method 2:
F35. Find the unknown factor of the fraction using the method 1:
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Adding fractions with unlike denominators
27
F39. Add the fractions using the method 1:
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
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:
3
1
2
1+
4
3
3
2+
5
2
4
1+
6
1
5
1+
7
3
5
2+
8
5
7
3+
9
5
8
3+
10
3
9
5+
11
1
10
3+
11
2
10
1+
15
4
10
3+
10
3
3
10+
3
2
9
5+
15
2
10
1+
20
4
15
3+
3
2
2
11 +
5
42
3
2+
6
32
3
51 +
10
11
11
101 +
4
12
5
21 +
16
4
20
1+
25
8
40
31 +
4
31
12
5+
8
31
6
12 +
32
3
16
11 +
db
cbda
d
c
b
a
´
´+´=+
Example 1:
4
35
4
23
4
158
45
3542
4
3
5
2==
+=
´
´+´=+
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:?
15
3
4
2=+
...10
5
8
4
6
3
2
1
4
2===== ...
10
2
5
1
15
3===
So:
10
7
10
2
10
5
15
3
4
2=+=+
F40. Add the fractions using the method 2:
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
3
2
2
1+
4
1
6
2+
10
2
4
3+
6
5
5
2+
7
1
10
6+
8
3
14
8+
9
3
8
4+
10
5
9
5+
9
3
10
2+
12
2
10
5+
15
5
10
2+
12
3
3
10+
3
2
9
5+
15
2
10
1+
20
6
15
3+
18
12
2
11 +
5
32
3
21 +
6
32
3
41 +
10
8
3
21 +
4
32
5
31 +
16
4
20
4+
25
5
40
51 +
16
41
12
4+
8
61
6
22 +
48
5
16
51 +
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Adding fractions with unlike denominators using the LCD method
28
F41. Add the fractions using the LCD method (see example 1 above):
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
6
5
4
1+
16
3
6
2+
20
3
16
1+
12
1
15
4+
14
3
10
3+
4
5
14
3+
18
5
16
3+
9
2
12
5+
22
1
10
3+
33
2
6
1+
15
4
10
3+
20
3
12
7+
30
5
9
5+
25
2
10
1+
25
4
15
2+
30
7
25
11 +
50
32
30
1+
16
32
36
51 +
50
1
15
11 +
45
12
50
31 +
16
3
20
3+
25
3
40
71 +
40
31
12
7+
84
31
14
12 +
64
3
48
11 +
40
11
120
33
120
8
120
25
430
42
524
55
30
2
24
5==+=
´
´+
´
´=+
?30
2
24
5=+
120532
53230
3224
113
111
13
=´´=
´´=
´=
LCD
1. First find the Least (Lowest) Common Denominator (LCD), and then add fractions.Example 1:
24
2 12
2 6
2 3
30
2 15
3 5
430120
524120
=¸
=¸
2. You can use the LCD method to add more than two fractions.Example 2:
?40
1
45
1
12
1=++
12
2 6
2 3
40
2 20
2 10
2 5
45
3 15
3 5
360532
524053453212123
131212
=´´=
´=´=´=
LCD
360
47
940
91
845
81
3012
301
40
1
45
1
12
1=
´
´+
´
´+
´
´=++
F42. Add the fractions using the LCD method (see example 2 above):
4
3
3
2
2
1++ d)
l)
h)
p)
c)
k)
o)
g)
b)
j)
n)
f)
a)
e)
i)
m)
t)s)r)q)
10
7
8
3
6
1++
10
3
6
5
3
1++
12
1
10
3
4
3++
30
1
15
1
10
3++
14
2
9
2
12
5++
6
1
25
1
10
3++
16
3
15
4
10
3++
15
2
20
3
12
1++
36
1
30
1
9
1++
35
2
25
2
15
1++
5
1
4
1
3
1
2
1+++
6
1
5
1
4
1
3
1+++
20
3
15
7
10
3
5
1+++
16
3
12
7
6
5
4
1+++
6
1
5
1
4
1
3
1
2
1++++
25
1
20
3
15
1
10
3
5
1++++
20
3
15
2
12
5
6
1
4
1++++
50
3
32
5
20
1
4
3
15
1++++
8
3
6
5
4
1++
© La Citadelle
Iulia & Teodoru Gugoiu
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)
www.la-citadelle.com
The Book of Fractions
Understanding the subtraction of fractions with like denominators
29
F35. Subtract the fractions that correspond to the shaded regions using the pattern:all shaded parts - all light shaded parts=all dark shaded parts
1. When you subtract two fractions, you subtract the parts of the whole they represent.Example 1.
0 1
4
1
4
2
4
3=-
So, by taking 2 quarters away from 3 quarters you get 1 quarter.
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Subtracting fractions with like denominators
30
F44. Subtract the fractions (write the results in lowest terms):
1. To subtract proper or improper fractions with like denominators, subtract the numerators and keep the denominators unchanged.
4
1
4
2-
11
3
11
10-
40
3
40
7-
7
2
7
5-
6
2
6
5-
12
1
12
5-
24
7
24
13-
4
1
4
3-
5
1
5
4-
3
2
3
4-
1000
1
1000
5-
40
7
40
19-
100
3
100
13-
7
9
7
23-
19
11
19
13-
10
17
10
19-
54
1
54
7-
35
13
35
23-
16
3
16
15-
32
7
32
11-
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
Example 1.
5
1
5
2
5
3=-
d
nn
d
n
d
n 2121 -=-
Example 2.
50
7
50
17-
9
1
9
7-
33
2
33
13-
60
7
60
13-30
1
30
7-
y)x)w)v)u)
F45. Subtract the mixed numbers (write the results in lowest terms):
2. To subtract mixed numbers with like denominators, first change mixed numbers to improper fractions, and then subtract the numerators. Keep the denominators unchanged.
3
2
3
5
3
7
3
21
3
12 =-=-
3
2
3
11 -
11
32
11
13 -
40
52
40
34 -1
7
53 -
6
23
6
55 -
12
7
12
52 -
24
7
24
31 -
4
3
4
12 -
5
31
5
42 -
2
3
2
12 -
1000
31
1000
52 -
40
91
40
193 -
100
172
100
133 -
7
92
7
23 -
19
112 -
10
11
10
110 -
14
3
14
73 -
35
13
35
32 -
16
5
16
155 -
32
171
32
112 -
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
50
72
50
133 -
9
52
9
15 -
33
42
33
13 -
60
72
60
135 -30
12
30
75 -
y)x)w)v)u)
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
31
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:
Example 1:
5
22
5
13)13(
5
11
5
33 =
--=-
d
nnww
d
nw
d
nw 21
212
21
1 )(-
-=-
Subtracting mixed numbers with like denominators
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.
Example 2:
5
21
5
24)12(
3
21
3
42
3
21
3
13 =
--=-=-
3
11
3
22 -
11
55
11
1011 -
40
73
40
175 -
7
22
7
55 -
6
12
6
44 -
12
51
12
74 -
24
52
24
174 -
4
12
4
35 -
5
12
5
310 -
2
13
2
35 -
1000
172
1000
375 -
40
175
40
277 -
100
273
100
475 -
7
21
7
54 -
19
15
19
116 -
10
12
10
95 -
14
33
14
75 -
35
122
35
173 -
16
73
16
157 -
32
97
32
1710 -
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
50
5
50
115 -
9
12
9
73 -
33
5
33
112 -
60
175
60
377 -30
117
30
179 -
y)x)w)v)u)
F46. Subtract the mixed numbers using the method 1 (write the results in lowest terms).
3
21
3
12 -
11
51
11
24 -
40
52
40
34 -
6
52
6
27 -
12
7
12
52 -
24
171
24
132 -
4
315 -
5
42
5
35 -
7
32
7
15 -
1000
71
1000
173 -
40
292
40
395 -
100
172
100
136 -
7
65
7
38 -
19
112 -
10
11
10
110 -
14
21
14
96 -
35
13
35
32 -
16
17
16
75 -
32
275
32
117 -
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
50
17
50
35 -
9
55
9
27 -
33
231
33
112 -
60
151
60
74 -
30
73
30
19 - y)x)w)v)u)
F47. Subtract the mixed numbers using the method 2 (write the results in lowest terms):
9
51
9
13 -
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Subtracting fractions with unlike denominators
32
F48. Subtract the fractions using the method 1:
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
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:
3
1
2
1-
3
2
4
3-
4
1
5
2-
6
1
5
1-
5
2
7
3-
7
3
8
5-
8
3
9
5-
10
3
9
5-
11
1
10
3-
10
1
11
2-
15
4
10
3-
10
3
3
2-
9
5
3
2-
10
1
15
2-
20
4
15
3-
3
2
2
11 -
3
2
5
42 -
6
32
3
51 -
10
11
11
101 -
5
21
4
12 -
20
3
16
4-
25
8
40
31 -
12
5
4
31 -
8
31
6
12 -
32
3
16
11 -
db
cbda
d
c
b
a
´
´-´=-
Example 1:
20
7
20
815
54
2453
5
2
4
3=
-=
´
´-´=-
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: ?15
3
4
2=-
...8
4
6
3
2
1
4
2=====
10
5...
5
1
15
3===
10
2
So:
10
3
10
2
10
5
15
3
4
2=-=-
F49. Subtract the fractions using the method 2:
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
2
1
3
2-
4
1
6
2-
10
2
4
3-
5
2
6
5-
7
3
10
6-
8
3
14
8-
9
3
8
4-
10
5
9
5-
10
2
9
3-
12
5
10
5-
10
2
15
5-
12
5
3
2-
9
5
3
2-
15
2
10
3-
15
3
20
6-
18
12
2
11 -
5
32
3
23 -
6
31
3
42 -
10
8
3
21 -
4
32
5
33 -
16
4
20
7-
25
5
40
51 -
16
41
12
42 -
8
61
6
22 -
48
5
16
51 -
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Subtracting fractions with unlike denominators using the LCD method
33
F50. Subtract the fractions using the LCD method (see the example 1 above):
e)
o)
t)
j)
d)
n)
i)
s)
c)
m)
r)
h)
b)
l)
q)
g)
a)
f)
k)
p)
y)x)w)v)u)
4
1
6
5-
16
3
6
2-
16
1
20
3-
12
1
15
4-
14
3
10
3-
14
3
4
5-
16
3
18
5-
9
2
12
5-
22
1
10
3-
33
2
6
1-
15
4
10
3-
20
3
12
5-
30
5
9
4-
25
2
10
1-
15
2
25
4-
30
7
25
11 -
50
31
30
12 -
16
1
36
51 -
50
12
15
12 -
45
21
50
31 -
16
3
20
5-
25
3
40
31 -
40
3
12
1-
84
31
14
12 -
64
3
48
11 -
120
17
120
8
120
25
430
42
524
55
30
2
24
5=-=
´
´-
´
´=-
?30
2
24
5=-
120532
53230
3224
113
111
13
=´´=
´´=
´=
LCD
1. First find the Least (Lowest) Common Denominator (LCD), then subtract fractions.Example 1.
24
2 12
2 6
2 3
30
2 15
3 5
430120
524120
=¸
=¸
2. You can use the LCD method to add or subtract more than two fractions.Example 2.
?40
1
45
1
12
1=-+
12
2 6
2 3
40
2 20
2 10
2 5
45
3 15
3 5
360532
524053453212123
131212
=´´=
´=´=´=
LCD
360
29
940
91
845
81
3012
301
40
1
45
1
12
1=
´
´-
´
´+
´
´=-+
F51. Add and subtract the fractions using the LCD method (see the example 2 above):
4
1
3
2
2
1-+ d)
l)
h)
p)
c)
k)
o)
g)
b)
j)
n)
f)
a)
e)
i)
m)
t)s)r)q)
10
3
8
3
6
5--
10
3
6
1
3
2--
12
1
10
3
4
3+-
30
1
15
1
10
3--
14
1
9
1
12
1-+
6
1
25
2
10
3--
16
3
15
2
10
3+-
15
2
20
3
12
5--
36
1
30
1
9
1-+
35
1
25
3
15
2+-
5
4
4
11
3
1
2
12 ---
6
1
5
1
4
1
3
1--+
20
3
15
7
10
3
5
2-+-
16
3
12
7
6
1
4
3-+-
6
1
5
1
4
1
3
1
2
1+-+-
25
2
20
3
15
1
10
3
5
2-+--
20
3
15
2
12
5
6
1
4
1+-+-
480
7
32
1
20
1
4
3
15
13----
8
3
6
1
4
3+-
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Order of operations (I)
34
F52. Solve each exercise by following the proper order of operations:
a)5
1
4
1
3
1
2
1+++
12
51
4
1
6
7
4
1
3
1
2
3
4
1
3
1
2
3=+=+÷
ø
öçè
æ-=+-
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:
2. To change the order of operations, use: parentheses ( ), brackets [ ] or braces { }. An inner bracket has a greater priority.Example 2:
12
11
12
7
2
3
4
1
3
1
2
3=-=÷
ø
öçè
æ+-
12
111
12
5
2
3
12
11
3
4
2
3
12
11
3
4
2
3
4
1
3
11
3
4
2
3
=+=þýü
îíì
-+=
=þýü
îíì
úû
ùêë
é--+=
ïþ
ïýü
ïî
ïíì
úû
ùêë
é÷ø
öçè
æ---+
4. One single type of brackets is enough to change the order of operations.Example 4:
12
11
12
7
2
1
12
51
2
1
12
1
2
11
2
1
4
1
3
1
2
11
2
1
=+=÷ø
öçè
æ-+=
=÷÷ø
öççè
æ÷ø
öçè
æ--+=÷
÷ø
öççè
æ÷÷ø
öççè
æ÷ø
öçè
æ---+
b)5
1
4
1
3
1
2
1-+- c) 5
5
13
4
14
3
11
2
121 +-+-+
d)6
1
5
1
4
1
3
1
2
1+-+- e) f)
÷ø
öçè
æ--÷
ø
öçè
æ--
6
1
5
1
4
1
3
1
2
1÷ø
öçè
æ-+-÷
ø
öçè
æ-+
6
1
5
1
4
1
3
1
2
11g) ÷
ø
öçè
æ+-
3
11
3
1
3
22 h) i)
÷ø
öçè
æ--
5
31
5
7
÷ø
öçè
æ--÷
ø
öçè
æ-
4
3
4
21
4
32
j) ÷ø
öçè
æ--÷
ø
öçè
æ+-
2
1
6
5
2
1
3
2
2
11 k) l)
úû
ùêë
é÷ø
öçè
æ---÷
ø
öçè
æ-
7
51
4
335
7
42
7
38 ú
û
ùêë
é÷ø
öçè
æ+-÷
ø
öçè
æ+-+ 1
5
11
4
1
3
11
2
121m)
þýü
îíì
úû
ùêë
é÷ø
öçè
æ----úû
ùêë
é+-+
4
3
5
4
10
1
2
1)
5
1
2
1(
10
7
5
3
n)
þýü
îíì
úû
ùêë
é÷ø
öçè
æ--÷
ø
öçè
æ--÷
ø
öçè
æ+-
6
1
12
5
9
5
3
2
6
5
3
2
9
22o)
4
3
10
1
5
3
10
7
2
1
8
3123 +
þýü
îíì
+úû
ùêë
é÷ø
öçè
æ--+÷
ø
öçè
æ---
p)
÷ø
öçè
æ--
þýü
îíì
÷ø
öçè
æ-+ú
û
ùêë
é÷ø
öçè
æ--
50
7
4
1
5
31
20
3
25
8
5
3q) r)
s)
þýü
îíì
-úû
ùêë
é÷ø
öçè
æ---ú
û
ùêë
é÷ø
öçè
æ----
þýü
îíì
÷ø
öçè
æ--ú
û
ùêë
é÷ø
öçè
æ---
10
3
5
211
2
1
5
3
5
21
3
11
2
1
3
21
2
12t)
þýü
îíì
úû
ùêë
é÷ø
öçè
æ--÷
ø
öçè
æ--ú
û
ùêë
é÷ø
öçè
æ--÷
ø
öçè
æ--
þýü
îíì
úû
ùêë
é÷ø
öçè
æ--÷
ø
öçè
æ--ú
û
ùêë
é÷ø
öçè
æ--÷
ø
öçè
æ-
6
1
5
1
5
1
4
1
4
1
3
1
3
1
2
1
5
1
4
1
4
1
3
1
3
1
2
1
2
11u)
÷÷ø
öççè
æ÷ø
öçè
æ---+
÷÷
ø
ö
çç
è
æ-÷
÷ø
öççè
æ-÷÷
ø
öççè
æ-÷
ø
öçè
æ-----
4
1
3
2
2
11
5
1
2
1
3
2
4
3
5
4
4
3
3
4
2
11
úû
ùêë
é÷ø
öçè
æ---
5
1
4
1
3
1
2
1÷÷ø
öççè
æ÷÷ø
öççè
æ÷ø
öçè
æ----
4
1
3
1
2
112
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
35
1. To multiply two proper or improper fractions, you multiply the numerators first and then the denominators, according to the rule:
Example 1:
Multiplying fractions
2. To multiply mixed numbers, first convert them to improper fractions.Example 2:
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F53. Multiply the proper or improper fractions (write the results in lowest terms):
21
21
2
2
1
1
dd
nn
d
n
d
n
´
´=´
15
8
53
42
5
4
3
2=
´
´=´
8
71
8
15
42
53
4
5
2
3
4
5
2
11 ==
´
´=´=´
3. To multiply a whole and a fraction, rewrite the whole as a fraction.Example 3:
5
31
5
8
51
42
5
4
1
2
5
42 ==
´
´=´=´
4
3
2
1´
3
2
2
1´
5
3
4
1´
4
15
5
2´
4
1
2
3´
2
3
3
4´
4
2
2
1´
4
1
2
1´
6
5
4
3´
20
5
10
1´
9
8
2
1´
15
12
6
3´
5
22
11
5´
5
21
7
4´
6
2
4
3´
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F54. Multiply the mixed numbers or fractions (write the results in lowest terms):
2
1
2
11 ´
3
12
2
1´
3
5
5
11 ´
3
1
5
22 ´
4
31
3
2´
12
31
5
11 ´
2
12
2
11 ´
5
22
3
21 ´
2
32
4
11 ´
12
11
11
22 ´
3
22
2
11 ´
5
22
6
51 ´
11
5
10
11 ´
5
14
7
31 ´
6
21
4
12 ´
o)n)m)l)k)
F55. Multiply the wholes and fractions (write the results in lowest terms):
32
11 ´
15
225´
6
523´ 14
7
41 ´
6
112´
t)s)r)q)p)9
22
2
11 ´
13
3
6
12 ´
5
24
11
41 ´
5
14
7
31 ´
3
51
4
12 ´
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
4
13´ 2
2
1´
5
22´ 5
5
2´
4
53´
15
35´ 6
2
1´
9
23´ 6
4
3´
20
415´
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
36
1. Although the most common multiplication operator is “x”, there are two other acceptable operators: “of” and “ ”
Example 1:
More about multiplying fractions
2. When multiplying fractions, first try to cancel out the common factors between numerators and denominators. Example 3:
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F56. Multiply the fractions (write the results in lowest terms):
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F58. Cancel out the common factors and then multiply the fractions:
o)n)m)l)k)3
2
2
11 ´
15
61
3
21 ´
5
42
7
15´ 21
7
11 ´
3
22
2
11 ´
t)s)r)q)p)9
22
2
11 ´
13
3
6
12 ´
8
5
10
41 ´
23
25
20
31 ´
18
42
40
12 ´
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
4
15
5
1´
2
3
3
2´
4
10
5
2´
9
14
7
3´
10
6
3
5´
15
35´ 8
6
2´
9
12
4
3´
44
40
4
33´
20
121
22
4´
10
3
52
31
5
3
2
1
5
3
2
1=
´
´=´=of
Example 2:
10
3
52
31
5
3
2
1=
×
×=×
5
2
51
21
5
4
2
1=
´
´=´
2
1
3. This process of cancellation can be applied more than one time.Example 4:
3
2
13
12
10
15
9
4=
´
´=´
51
1
2
3 2
2
1
3
2of
4
3
3
2of
5
1
5
1of
3
1
5
1of
4
3
2
5of
2
3
2
3of
4
5
5
4of
4
2
2
1of
6
9
3
2of
903
2of
2
110 of 32 of
4
1
3
11 of
2
11
5
12 of
5
4
3
2of
F57. Multiply the fractions (write the results in lowest terms):
3
2
2
1×
4
3
3
2×
3
5
5
3×
5
2
2
11 ×
3
12
2
11 ×
6
14
7
3×
2
5
10
1×
20
15
5
4×
9
12
5
3×
5
1
4
1×
1
2
2
1×
2
1
2
1×
15
18
12
5×
30
22
12
10× 30
21
6×
×
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
37
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:
The order of operations (II)
3. If the expression contains addition, subtraction and multiplication, do the multiplication first. Example 3:
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F60. Find the value of each expression (write the results in lowest terms):
4
115
4
2
1
15
8
2
1
5
4
3
2
2
1
5
4
3
2=´=´÷
ø
öçè
æ´=´´
2. Do not forget to cancel out the common factors, before multiplying the fractions .Example 2:
21
1
3
432
1
8
6
9
15
5
2=´´
1
1 1
2
8
11
8
9
8
5
2
1
6
5
4
3
2
1
6
5
4
3
2
1==+=÷
ø
öçè
æ´+=´+
4. If the expression contains brackets, replace the brackets with the result of the operation(s) inside the brackets.Example 4:
24
11
24
25
6
5
4
5
6
5
4
3
2
1==´=´÷
ø
öçè
æ+
e)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F59. Multiply the fractions (write the results in lowest terms):
F61. Find the value of each expression (write the results in lowest terms):
4
3
3
2
2
1´´
2
3
5
2
3
1´´
4
3
8
3
9
4´´
4
312
3
1´´
4
2
3
1
5
2´´
4
32
3
2
2
11 ´´
4
31
5
2
2
12 ´´
5
3
3
21
2
12 ´´
9
22
4
12 ´´
8
15
3
2
5
11 ´´
6
5
5
4
4
3
3
2
2
1´´´´
8
15
16
10
5
4
2
1´´´
16
14
12
10
10
8
6
4
2
1´´´´
10
9
8
7
6
5
4
3
2
1´´´´
5
4
2
1
2
1´+
6
5
3
2
4
1+´
3
1
2
1
3
2´-
6
1
3
2
2
1-´
10
1
4
2
3
1
5
4-´´
2
1
3
2
2
1
3
1-´+
3
1
3
2
2
1
2
1+´-
3
2
2
1
2
1
3
2´-+
3
2
6
5
9
15
5
3-+´
6
5
5
3
4
3
3
2´+´
8
3
3
2
3
2
2
1´-´
3
2
3
2
4
3
2
1+´´
4
3
3
1
2
1
3
2´´-
3
21
5
6
3
2
2
11 ´+´
8
15
5
31
4
31
7
21 ´+´-
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
5
4
3
1
2
1´÷
ø
öçè
æ+ ÷
ø
öçè
æ+´
6
1
3
2
5
1
5
3
2
1
3
2´÷
ø
öçè
æ- ÷
ø
öçè
æ-´
3
1
2
1
2
3÷ø
öçè
æ-´÷
ø
öçè
æ+
3
2
4
3
3
2
4
3
2
1
5
3
2
1
3
1-´÷
ø
öçè
æ+ ÷
ø
öçè
æ+´-
2
1
3
2
7
1
2
1
3
2
6
1
2
11
3
2´÷
ø
öçè
æ-+
5
1
4
1
3
1
7
4-÷
ø
öçè
æ+´
7
6
3
1
2
1
5
2´÷
ø
öçè
æ+´
÷ø
öçè
æ´-´
5
2
2
1
3
1
4
11 ÷
ø
öçè
æ+´´
3
1
3
2
4
3
5
3
4
3
3
2
2
1
3
2´÷
ø
öçè
æ´- ÷
ø
öçè
æ´+´
3
2
5
6
3
2
11
5
9
11
3
3
2
1
5
21 ´÷
ø
öçè
æ+´÷
ø
öçè
æ-
q)p)3
2
4
1
2
11
4
3
3
2
2
11
4
1
3
11 ´ú
û
ùêë
é´+÷
ø
öçè
æ-´-´
7
2
4
1
2
1
5
3
11
5
3
13
5
1
4
1
3
11
2
21
2
3´
ïþ
ïýü
ïî
ïíì
úû
ùêë
é-÷
ø
öçè
æ-´-ú
û
ùêë
é´÷
ø
öçè
æ+-+´
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
38
1. The reciprocal of a fraction is the fraction obtained by interchanging its numerator and denominator.So, the reciprocal of
Reciprocal of a fraction
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
F62. Find the reciprocal of each fraction:
2
1
3
2
4
5
3
7
10
0
6
7
3
13
4
2
5
3
0
3
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
F63. Find the reciprocal of each mixed number:
2
11
3
22
5
22
7
11
3
23
3
33
11
22
7
52
4
11
4
33
F64. Find the reciprocal (multiplication inverse) of each whole number:
e)d)c)b)a) 1 2 5 100 0
d
n
n
dis
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).
4
7
4
31 =
7
4Example 2. The reciprocal of is
1
33 =
3
1Example 3. The reciprocal of 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.
Example 1:The reciprocal of is
4
3
3
44. 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:
13
4
4
3=´
F65. Check if the pair of fractions are reciprocal:
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
F66. Find the unknown fraction (f):
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
14
1=´f 1
2
1=´ f 1
5
2=´f 15 =´f 1
4
51 =´f
17
2=´f 17 =´ f 1
9
21 =´f 1
4
31 =´ f 1
5
42 =´f
1
2
2
1and
4
2
2
11 and
7
12
3
7and 3
3
1and
5
22
5
3and
11
3
3
13 and
10
110 and
2
5
5
3and
2
6
9
3and
24
15
5
31 and
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
39
1. To divide two proper or improper fractions, change the divisor to its reciprocal and then multiply, according to the rule:
Example 1:
Dividing fractions
2. To divide mixed numbers, first change them to improper fractions.Example 2:
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F67. Divide the proper or improper fractions (write the results in lowest terms):
3. To divide a whole number and a fraction, rewrite the whole as a fraction.Example 3:
4
3
2
1¸
3
1
2
1¸
9
4
3
2¸
15
8
5
2¸
5
6
2
3¸
5
12
3
4¸
4
3
2
5¸
4
1
2
1¸
6
5
4
10¸
20
3
10
1¸
9
8
3
1¸
15
12
6
3¸
22
25
11
5¸
21
16
7
4¸
16
6
4
3¸
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F68. Divide the mixed numbers or fractions (write the results in lowest terms):
2
1
2
12 ¸
4
12
2
11 ¸
15
11
5
11 ¸
5
3
5
22 ¸
6
11
3
2¸
15
31
5
11 ¸
4
12
2
11 ¸
9
11
3
21 ¸
2
36
4
11 ¸
22
31
11
32 ¸
8
22
2
11 ¸
3
23
6
51 ¸
25
3
10
21 ¸
21
10
7
31 ¸
16
51
4
12 ¸
o)n)m)l)k)
F69. Divide the wholes and fractions (write the results in lowest terms):
32
11 ¸
11
325 ¸
7
5313¸ 44
7
41 ¸ 35 ¸
t)s)r)q)p)9
22
3
11 ¸
4
13
6
12 ¸
26
5
13
21 ¸
35
8
7
31 ¸
3
12
4
12 ¸
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
2
11¸ 2
3
1¸
5
44 ¸ 15
2
5¸
3
1532 ¸
7
125 ¸ 2
3
1¸
5
223 ¸ 6
4
3¸
3
2615 ¸
2
2
1
1
2
2
1
1
n
d
d
n
d
n
d
n´=¸
6
5
4
5
3
2
5
4
3
2=´=¸
2
11
2
3
5
3
2
5
3
5
2
5
3
21
2
12 ==´=¸=¸
2
12
2
5
4
5
1
2
5
4
1
2
5
42 ==´=¸=¸
In short, invert the divisor and multiply.
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
40
There are four operators used to express the division operation between two numbers or fractions.
1. The operator is
Example 1.
Division operators
e)d)c)b)a)
F70. Divide the fractions (write the results in lowest terms):
5
4
3
2¸
3
2
2
11 ¸
2
12
3
21 ¸
7
22 ¸ 54 ¸
e)d)c)b)a)
F71. Divide the fractions (write the results in lowest terms):
3
1:
2
1
4
1:
2
11
10
12:
5
11 4:
5
27:3
o)n)m)l)k)
F72. Divide the fractions (write the results in lowest terms):
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
¸
2. The operator is :
Example 2.
3. The operator is(the division line)
Example 3.
_
4. The operator is /
Example 4.
3
2
9
10
5
3
10
9
5
3=´=¸
2
3
6
21
7
3
21
6:
7
3=´=
3
2
3
4
2
1
4
32
1
=´=
3
21
3
5
9
25
5
3
25
9/
5
3==´=
o)n)m)l)k)
F73. Divide the fractions (write the results in lowest terms):
14
15/
7
5
6
11/
3
24
3
11/3 4/
7
22 2/7
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
3
2/
2
1
5
3/
3
2
5
3/
3
5
8
3/
4
1
3
13/5
4
12/3 2/
3
13/
4
12 5/2
4
33/
3
21
=
6
43
2
=
6
4
2
=43
2
=
15
85
4
=
6
49
21
=
10
12
5
3
=
4
32
2
11
=
3
10
5=
4
12
3=
3
1
3=
3
21
5
=42
5
=10
5
41
=33
1
=45
22
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
41
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:
Order of operations (III)
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:
d)
l)
h)
c)
k)
g)
b)
j)
f)
a)
e)
i)
F74. Use the correct order of operations to find the value of each expression (see example 1):
5
4
3
2
2
1¸¸ 532 ¸¸ 54321 ¸¸¸¸
3
14
15
1
5
32 ¸¸
8
7
6
5
4
3
2
1¸¸¸ 4
3
23
2
11 ¸¸¸¸
6
5
5
4
4
3
3
2
2
1¸¸¸¸
5
44
4
33
3
22
2
11 ¸¸¸
4
15
5
2
3
4
3
2¸¸¸
6
14
5
33
4
12
3
21 ¸¸¸ 6
9
4
22
6
11
11 ¸¸¸
5
31
6
5
21
61
7
3¸¸¸
5
3
4
5
4
3
4
5
3
2
2
1
4
5
3
2
2
1=¸=¸÷
ø
öçè
æ¸=¸¸
5
3
3
2
5
2
3
2
2
1
5
4
3
2
2
1
5
4=¸=¸÷
ø
öçè
æ´=¸´
10
11
10
1
5
2
5
31
1
6
5
3
10
6
3
2
5
31
1
6
5
3
10
6
3
2
5
31 =+-=÷
ø
öçè
æ¸+÷
ø
öçè
æ´-=¸+´-
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:
F75. Use the correct order of operations to find the value of each expression (see example 2):
F76. Use the correct order of operations to find the value of each expression (see example 3):
d)
h)
c)
k)
g)
b)
j)
f)
a)
e)
i)
6
5
4
3
2
1´¸
15
4
5
2
4
32 ¸´
10
9
8
7
6
5
4
3
2
1¸´¸´ 54321 ¸´¸´
32
1
16
1
8
1
4
1
2
11 ¸´¸´¸
5
44
4
33
3
32
2
11 ¸¸´
8
9
16
3
3
2
2
3¸´¸
5
4
5
3
5
2
5
1¸¸´
5
6
4
3
2
1
6
5
4
3
2
1´¸´´¸
20
3
4
3
10
3
5
12
5
3
5
2´¸¸´¸
2
3
8
12
10
15
2
3
6
9
4
6
2
3´¸´¸´¸
c)
f)
b)
h)
e)
a)
d)
g)
9
1
15
2
6
5
4
3
2
1¸´-+
10
6
5
3
8
6
3
21 ¸-´+
2
1
3
1
5
2
2
3
12
10
5
3
15
21 -¸+¸´-
4
11
6
1
2
12
3
2
2
11 ´¸+¸´
2
1
4
1
4
3
3
2
2
11 ¸+´-¸76543215 ¸´+-¸´+
46
1
2
11
2
1
4
12
3
1
2
1
2
11 ¸+¸+´-¸-´
12
1
2
6
2
12
2
1
2
1
2
1
4
3
2
1
4
3
4
3
2
11
12
1+¸-´-´¸+¸´+
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
42
1. You can change the order of operations using brackets.Example 1:
Order of operations (IV)
2. When an expression contains fraction (division) lines, the numerators and the denominators must be calculated independently.Example 2:
F77. Solve each exercise by following the proper order of operations (see example 1):
8
1
3
7
4
1
6
7
3
7
4
1
2
1
3
2
2
1=¸´=¸÷
ø
öçè
æ-´÷
ø
öçè
æ+
F78. Solve each exercise by following the proper order of operations (see example 2):
f)
a) e)=
÷ø
öçè
æ
÷ø
öçè
æ
6
5
4
3
2
1
=
÷ø
öçè
æ
+¸
´÷ø
öçè
æ¸
÷ø
öçè
æ
÷ø
öçè
æ
2/13
28
3
4
3
3
2
5
3
3
5
4
33
9
2
4
3
3
2
-¸
´+
9
72
9
25
3
8
9
1
3
4
1
2
3
2
6
1
4
1
2
11
2
2
36
1
2
1
2
1
2
1
2
3
42
3
1
2
1
==+=´+´=
+
+=
´+
+
¸
-
c)b)
e)
a)
d)
12
5
3
2
4
3
5
2
2
1
3
15¸÷
ø
öçè
æ--÷
ø
öçè
æ+´ ( )31
6
5
25
2
27
5
9
10
5
3¸¸´-÷
ø
öçè
渴÷
ø
öçè
æ+¸ú
û
ùêë
é´÷
ø
öçè
æ-+
2
1
3
2
7
8
6
1
4
3
2
1
úû
ùêë
é¸÷
ø
öçè
æ-+¸÷
ø
öçè
æ+´+
5
6
10
3
5
2
3
1
5
2
6
1
17
2
25
11
12
1
5
6
3
1
2
1
6
2
2
1
4
3
2
1
4
3
4
5
4
3
2
11
2
1+¸-´-´¸+¸´+¸+ ÷
ø
öçè
æúû
ùêë
é÷ø
öçè
æúû
ùêë
é÷ø
öçè
æ÷ø
öçè
æ
f)
e)
úû
ùêë
é÷ø
öçè
æ
þýü
îíì
úû
ùêë
é÷ø
öçè
æ¸-¸+¸-´-¸-¸- 6
3
2
4
3
4
3
2
1
6
1
3
1
4
1
2
1
3
1
2
1
2
1
4
12
÷ø
öçè
æ¸-´ú
û
ùêë
é´÷
ø
öçè
æ-¸+-ú
û
ùêë
é+¸÷
ø
öçè
æ¸-´+´
9
8
3
2
6
5
5
2
5
3
3
2
10
3
5
1
3
2
2
1
10
3
5
1
2
3
2
1
5
2
2
1
b) =
÷ø
öçè
æ
5
25
4
3
c) =
÷ø
öçè
æ
÷ø
öçè
æ
6
5
3
3
2
d) =
÷ø
öçè
æ
÷ø
öçè
æ
÷ø
öçè
æ
3
2
3
1
2
1
3
g)2321
3432
+´¸
¸-´ i)
3
1
4
3
4
9
2
3
2
13
2
3
2
2
1
8
15
5
2
-´¸+
+¸-´h)
2
1
3
2
6
15
3
53
1
8
9
2
3
3
2
´-¸
-´¸
k)
13
2
4
9
2
13
1
10
3
5
1
13
4
3
1
2
13
2
6
10
5
2
5
1
4
5
2
14
5
2
1
5
3
-´+
-¸¸
-´¸
+´+
+¸
¸-j)
15
9
15
13
9
2
5
3
154
12
5
2
13
8
2
1
4
31
´
+´
+¸¸
-
´-
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
43
1. If you multiply a fraction by itself k times, you can use exponential notation to make the expression more compact (simpler):
Raising fractions to a power
2. To calculate the value an expression written in exponential notation, rewrite it in expanded notation and do the multiplication.
d)
h)
c)
g)
b)
f)
a)
e)
F79. Write each expression in exponential notation:
d)
h)
c)
g)
b)
f)
a)
e)
k
timesk
d
n
d
n
d
n
d
n÷ø
öçè
æ=´´´
44 344 21L
3
2
1
2
1
2
1
2
1÷ø
öçè
æ=´´
16
9
44
33
4
3
4
32
22
=´
´===÷
ø
öçè
æ
5. By convention, any number (including a fraction) raised to the power of 0 equals 1.Example 6:
15
40
=÷ø
öçè
æ
k
timesk
nnnn =´´´ 43421 L
433333 =´´´
The left side is the expanded notation and the right side is the exponential notation.
81
16
3
2
3
2
3
2
3
2
3
24
=´´´=÷ø
öçè
æ
3. You can use also the following rule to calculate the value of a fraction raised to a power:
4434421 L
43421 L
44 344 21L
timesk
timesk
k
kk
timesk
ddd
nnn
d
n
d
n
d
n
d
n
d
n
´´´
´´´
==÷ø
öçè
æ=´´´
4. If the exponent is negative, replace the fraction with its reciprocal raised to a positive exponent according to the rule: kk
n
d
d
n÷ø
öçè
æ=÷
ø
öçè
æ-
8
515
8
125
2
5
2
5
5
23
333
===÷ø
öçè
æ=÷
ø
öçè
æ-
Example 1:
Example 2:
Example 4:
Example 5:
F80. Write each expression in expanded notation:
d)
h)
c)
g)
b)
f)
a)
e)
F81. Calculate the value of each expression:
d)
h)
c)
g)
b)
f)
a)
e)
F82. Calculate the value of each expression:
2
1
2
1
2
1
2
1´´´
4
3
4
3´
2
3
2
3
2
3
2
3
2
3´´´´
2
3
2
3
2
3
2
3´´´
2222 ´´´3
21
3
21
3
21 ´´
5
4
5
4´
4
1
4
1
4
1
4
1
4
1´´´´
2
4
3÷ø
öçè
æ4
3
5÷ø
öçè
æ3
5
1÷ø
öçè
æ2
3
2-
÷ø
öçè
æ
2
7
3÷ø
öçè
æ3
2
3÷ø
öçè
æ1
4
5-
÷ø
öçè
æ3
5
4÷ø
öçè
æ
0
7
3÷ø
öçè
æ3
2
3÷ø
öçè
æ2
5
2÷ø
öçè
æ3
5
3÷ø
öçè
æ
32
4
3
1÷ø
öçè
æ4
1
2÷ø
öçè
æ4
2
3÷ø
öçè
æ
23-
2
1
5-
÷ø
öçè
æ4
2
1-
÷ø
öçè
æ3
3
2-
÷ø
öçè
æ
2
4
3-
÷ø
öçè
æ1
7
5-
÷ø
öçè
æ2
3
4-
÷ø
öçè
æ3
5
2-
÷ø
öçè
æ
Example 3.
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
44
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.
Order of operations (V)
d)
h)
c)
g)
b)
f)
a)
e)
F75. Calculate the value of each expression (see example 1):
d)
h)
c)
g)
b)
f)
a)
e)
c)
f)
b)
e)
a)
d)
36
25
6
5
3
1
2
122
=÷ø
öçè
æ=÷
ø
öçè
æ+
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.
243
32
3
2
3
25221
=÷ø
öçè
æ=÷
ø
öçè
æ´+
3. If an expression contains addition, subtraction, multiplication, division, and powers, do the powers first.
2
1213
2
11
4
9
8
6
9
16
2
11
9
4
8
6
9
16
2
11
3
2
8
6
9
16
2
12
=-+=-´´+=-¸´+=-÷ø
öçè
渴+
4. The order of operations can be changed by brackets.
12
1
2
18
1
4
2
18
4
1
2
18
16
94
2
12
2
18
4
34
2
12
2
11112
=+=¸+=¸÷ø
öçè
æ+=¸÷
ø
öçè
æ´-+=¸
÷÷
ø
ö
çç
è
æ÷ø
öçè
æ´-+
---
F76. Calculate the value of each expression (see example 2):
F77. Calculate the value of each expression (see example 3):
2
4
1
3
1÷ø
öçè
æ+
1
5
4
4
1
5
2-
÷ø
öçè
æ´-
2
5
4
5
6
3
1÷ø
öçè
渴
22
3
1
2
1
3
1
2
1÷ø
öçè
æ++÷
ø
öçè
æ-
23
2
1
3
1
2
3
3
1÷ø
öçè
渴÷
ø
öçè
æ´
21
4
1
3
2
3
1
2
1÷ø
öçè
æ-+÷
ø
öçè
æ-
2
7
8
4
15
5
1÷ø
öçè
æ´´
2
2
1
6
5
3
2-
÷ø
öçè
æ-+
2
1413
3
2´-´
÷ø
öçè
æ1
6
1
2
1
5
3-¸
÷ø
öçè
æ 3
1
3
26
3
1
2
1¸+´
÷ø
öçè
æ 2
3)
2
33
2
1(
5
2¸+´
÷ø
öçè
æ
2
1
2
1
3
2
3
2-
÷ø
öçè
æ+
6
1
3
1
4
3
2
1¸
÷ø
öçè
æ+
6
1
3
2
2
3
3
1¸
÷ø
öçè
æ´
)3
22
3
2(
2
3
3
2
4
3-´´
÷ø
öçè
æ-
101
2
1
2
1
2
1÷ø
öçè
æ+÷
ø
öçè
æ+÷
ø
öçè
æ- 202
3
2
3
2
3
2÷ø
öçè
æ´÷
ø
öçè
æ´÷
ø
öçè
æ-
÷ø
öçè
æ¸÷
ø
öçè
æ´÷
ø
öçè
æ+÷
ø
öçè
æ
3
1
4
3
3
2
2
1121
úúû
ù
êêë
é
÷÷ø
öççè
æ¸+´÷
ø
öçè
æ22
2
3
2
3
3
4
4
32
2
1
5
325
5
21
23
´÷ø
öçè
æ+-´÷
ø
öçè
æ+
2
1
3
1
2
1
4
31
253
-÷ø
öçè
æ´÷
ø
öçè
æ¸÷
ø
öçè
æ+
i)
l)
h)
j)
g)
k)
=
úúû
ù
êêë
é÷ø
öçè
æ-´
÷ø
öçè
æ¸÷
ø
öçè
æ+÷
ø
öçè
æ
-1
322
2
3
4
32
2
3
4
3
2
1
=
÷ø
öçè
æ´
÷ø
öçè
æ
2
3
2
2
4
3
3
2
3
2
=÷ø
öçè
æ-¸
÷÷÷÷
ø
ö
çççç
è
æ+ -1
2
2
1
4
3
5
310
1
5
4
=
÷ø
öçè
æ¸+
+÷ø
öçè
æ
´
+÷ø
öçè
æ
÷ø
öçè
æ´+
2
2
2
2
3
4
9
2
6
1
9
2
3
1
4
3
2
1
2
3
2
1
4
1
=¸úúû
ù
êêë
é÷ø
öçè
æ-¸¸
úúû
ù
êêë
é-÷
ø
öçè
æ-´÷
ø
öçè
æ´
-
- 2
223
2
2
52
1
2
1
4
32
4
11
2
112
=÷ø
öçè
æ´
úúû
ù
êêë
é-÷
ø
öçè
æ-÷
ø
öçè
æ-´÷
ø
öçè
æ¸
ïþ
ïýü
ïî
ïíì
÷ø
öçè
æ+ú
û
ùêë
é÷ø
öçè
æ+-´÷
ø
öçè
æ-
-- 232322
21
2
31
2
1
2
11
2
1
2
1
3
1
2
1
6
7
3
2
4
3
Example 1. Example 2.
Example 3.
Example 4.
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Converting fractions to decimals
45
F86. Build the factor tree for the denominator and classify each fraction asa terminating or a non-terminating decimal:
e)
o)
j)
d)
n)
i)
c)
m)
h)
b)
l)
g)
a)
f)
k)
F87. Write each fraction as a terminating decimal:
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
1. The fraction line is a division operation, so to write a fraction as a decimal, divide the numerator (dividend) by the denominator (divisor).
2
1
3
2
4
3
5
4
6
5
7
2
9
7
10
3
11
8
15
2
20
11
6
3
40
3
100
13
120
25
2
1
4
3
8
5
5
1
20
3
80
11
100
13
125
1
500
3
625
1
00
4040
5660
24
375.0000.38
375.08
3=
2. Usually we are dealing with the decimal system (the base is 10). The single prime factors of 10 are 2 and 5.If the denominator of the fraction written in lowest terms has only 2 or 5 as prime factors, then the fraction can be written as a terminating decimal.Example 1:
3. If the denominator of the fraction written in lowest terms has factors other than 2 or 5, the fraction can be written as a non-terminating repeating decimal.Example 2:
...4
6670
3340
6670
33
...212.1000.4033
21.1...2121.133
40==
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.
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
3
1
6
1
7
2
9
7
11
2
15
1
30
7
36
11
13
227
1
2228 ´´=
11333 ´=
F88. Write each fraction as a non-terminating repeating decimal. Use a bar over the repeating decimals:
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
2
11
15
6
21
71
75
1
3
7
4
51
64
5
32
1
64
3
128
1
F89. Write each fraction as either a terminating or non-termonating repeating decimal. Use a bar over the repeating decimals:
o)n)m)l)k)250
3
7
1
14
20
12
5
13
32
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Converting decimals to fractions
46
F90. Write each terminating decimal as a fraction in lowest terms (see the example 1):
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
1.0 5.0 4.1 25.1 75.0
035.0 125.2 125.10 075.5 725.100
1. To convert a terminating decimal to a fraction, use the place value of each digit of the decimal part.Example 1:
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 periodc) 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 Example 4:
4
11
20
51
20
1
5
11
100
5
10
2125.1 ==++=++=
2. A faster method to convert a terminating decimal to a fractions is:a) the numerator is the number without the decimal pointb) the denominator is 1 followed by a 0 for each digit of the decimal partExample 2:
8
51
8
13
1000
1625625.1 ===
5500
6796
100099
456123
45123.6 ==
3. To convert a non-terminating repeating decimal to a fraction, you can use algebra.Example 3:
3
2
9
6
69
...666.0...666.610
...666.610
6.0...666.0
==
=
-=-
=
==
x
x
xx
x
x
F91. Write each terminating decimal as a fraction in lowest terms (see the example 2):
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
625.0 5.1 125.0 4.0 16.2
275.0 24.0 35.0 45.2 640625.0
F92. Write each non-terminating decimal as a fraction in lowest terms (see the example 3):
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
F93. Write each non-terminating decimal as a fraction in lowest terms (see the example 4):
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
3.0 2.1 21.0 12.1 502.4
23.0 25.1 120.2 123.0 45623.1
7.0 3.1 35.2 23.1 912.1
12.0 01.3 1230.145123.6123.1
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Order of operations (VI)
47
F94. Find the value of each expression (only use operations with decimals):
e)d)c)b)a) =+ 15.025.0 =- 15.050.0
1. If an expression contains only decimal numbers, you can do all the operations using decimal numbers.Example 1:
8.02.18.02.124.26.15.02.1 =-+=¸-´+
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:
4.15
7
2
3
10
3
5
4
2
1
5
2
5
65.1
10
3
5
45.0
5
22.1 ==¸+´-+=¸+´-+
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:
359.11485
5331
109
13
99
251
3
1113.025.13.1 @=-´=-´
=´ 5.04.2 =¸ 05.010.0 =+ 25.05.0 2
F95. Find the value of each expression by converting decimals to fractions:
e)d)c)b)a) =+ 25.175.0 =- 25.060.0 =´ 4.15.0 =¸ 75.02.1 =- 24.01
F96. Find the value of each expression by converting repeating decimals to fractions:
e)d)c)b)a) =+ 6.03.1 =- 45.04.2 =´ 2.25.0 =¸ 10.082.0 =-22
3.06.0
F97. Find the value of each expression by converting fractions to decimals:
d)c)b)a)
F98. Find the value of each expression by converting decimals to fractions:
d)c)b)a)2
320.0
4
1´+
3
215.0
6
525.0 ´-¸ 3.0
4
125.0
3
2-´¸ 8.0
2
15.0
3
2 ´÷ø
öçè
æ-
2.02.02
15.0
5
2-¸´+ 25.0
2
1
5
25.1 -¸´ 28.05.13.0
5
4+¸- 22.0
10
3
5
11.0 -´÷
ø
öçè
æ+
c)b)a)
F99. Find the value of each expression by converting decimals to fractions:
5
25.1
4
3
5
35.0
2
1´¸-´+
3
275.05.05.0
4
525.1 2 ´´+´¸ 4
3
125.0
2
13.0
3
55.0
5
4´´+´-´´
f)e)d)
25.04
34.0
248
3
6
55.1 2
-´
¸÷ø
öçè
æ-´
5.0
33.05
4
16
3
2
125.0
2
´
´÷ø
öçè
æ-
+÷ø
öçè
æ+
4
3
25.14
1
2
125.1
5.04
31.0
5
3
´
÷ø
öçè
æ+´÷
ø
öçè
æ-
÷ø
öçè
æ-¸÷
ø
öçè
æ-
h)g)
j)i)
3
22.0
5
36.14.053.1
3
252
¸+÷ø
öçè
æ-´
úúû
ù
êêë
é-¸÷
ø
öçè
æ+
9
1
2
5
5
38.1
3
17.0
5
45.2
23
+úû
ùêë
é´÷
ø
öçè
æ-¸ú
û
ùêë
é+÷
ø
öçè
æ+¸
2
2
2
3
5
9
15.1
9
13.06.0
5
22.14.2
5.09
44.2
¸
÷ø
öçè
æ+
+´¸
÷ø
öçè
æ+-
´úû
ùêë
é-
5
1
2
1
2
23
17
5.05
24.06.1
5
36.0
5
12.0
2
1
5
1
2
1
4.025.01.0 -
-
-´÷
ø
öçè
æ¸-
÷ø
öçè
æ´-´
´+÷ø
öçè
æ-
+
úû
ùêë
é÷ø
öçè
渴
¸´
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Time and Fractions
48
F100. Convert hours to minutes. Write the results as mixed numbers in lowest terms:
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
1. One hour has 60 minutes. This relation can be written in two ways:
2. To make conversions between hours and minutes, use the formulas above.Example 1:
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
hh
minminh60
1
60
11601 ===
minminhh 80603
4
3
4
3
11 =´==
Example 2:
hhhmin12
5
60
25
60
12525 ==´=
3. One minutes has 60 seconds. This relation can be written in two ways:
4. To make conversions between minutes and seconds, use the formulas above.Example 3:
minmin
ssmin60
1
60
11601 ===
ssminmin 45604
3
4
325.0 =´==
Example 4:
minminmins3
5
60
100
60
1100100 ==´=
F101. Convert minutes to hours. Write the results as mixed numbers in lowest terms:
F102. Convert minutes to seconds. Write the results as mixed numbers in lowest terms:
F103. Convert seconds to minutes. Write the results as mixed numbers in lowest terms:
F104. Do the required conversions:
5. The conversion between seconds and hours is a two-step task.
hhminmins4
11
60
17575
60
145004500 =´==´=
h5.0 h3
1h75.1 h
4
32 h1.0
h6
5h25.2 h
8
3h
15
7h32.1
min5 min10 min15 min25 min50
min70 min36 min250 min4
3min5.2
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
min5.1 min3
21 min25.0 min
5
21 min15.1
min12
7min75.1 min
12
5min
45
11min3.7
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
s12 s10 s45 s90 s200
s2
110 s75.0 s
4
315 s
3
100s6.20
e)
j)
d)
i)
c)
h)
b)
g)
a) f)hs ?500 = hs ?1500 = hs ?9000 = sh ?2.0 = sh ?75.1 = sh ?15.0 =
sminh ?102 = ssmin ?515 = minsh ?155.0 = minsminh ?105.103
2=
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Canadian coins and fractions
49
F105. Convert dollars to cents. Write the results as mixed numbers in lowest terms:
e)d)c)b)a)
1. One dollar (loonie or $) has 100 cents (pennies):
4. One dime has 10 cents:
cents100$1 =
F113. Do the required conversions:
$12.0 $125
1$02.1 $
75
21 $07.0
h)g)f)e)
quartersdimes ?5.2 =
dimesnickels ?15 = nickelsdimesquarters ?21 =
dimesquarters ?2.0 =
centsdime 101 =
3. One nickel has 5 cents: centsnickel 51 =
5. One quarter has 25 cents: centsquarter 251 =
2. One twonie (or toonie) has 200 cents: centstwonie 2001 =
Example 1: $4
11$
100
1125125 =´=cents
Example 2: centscentstwonies 350200100
17575.1 =´=
Example 3: centscentsnickels 655
6
5
11 =´=
Example 4: dimesdimescents2
12
10
12525 =´=
Example 5: centscentsquarters 10255
2
5
2=´=
f) $150
7
F106. Convert cents to dollars. Write the results as mixed numbers in lowest terms:
e)d)c)b)a) cents25 cents20 cents45 cents160 cents450 f) cents5.5
F107. Convert twonies to cents. Write the results as mixed numbers in lowest terms:
e)d)c)b)a) twonies02.0 twonies95.0 twonies100
3twonies
150
11twonies
10
31 f) twonies
25
2.1
F108. Convert cents to twonies. Write the results as mixed numbers in lowest terms:
e)d)c)b)a) cents125 cents250 cents40 cents120 cents500 f) cents2.10
F109. Convert nickels to cents. Write the results as mixed numbers in lowest terms:
e)d)c)b)a) nickels2.0 nickels2.1 nickels20
7nickels
5
7nickels
20
32 f) nickels
2
5.10
F110. Convert cents to nickels. Write the results as mixed numbers in lowest terms:
e)d)c)b)a) cents15 cents25 cents75 cents4 cents5.1 f) cents5.0
F111. Convert quarters to cents. Write the results as mixed numbers in lowest terms:
e)d)c)b)a) quarters2.0 quarters6.1 quarters25
3quarters
125
9quarters
5
31 f) quarters
5
2.1
F112. Convert cents to quarters. Write the results as mixed numbers in lowest terms:
e)d)c)b)a) cents50 cents125 cents100 cents20 cents5.0 f) cents55
d)c)b)a) nickelsquarters ?3 = nickelsdimes ?5.1 =
dimesnickels ?8 = dimesquarters ?5 =
$100
1
100
$11 ==cent
dimedime
cent10
1
10
11 ==
nickelnickel
cent5
1
5
11 ==
quarterquarter
cent25
1
25
11 ==
twonietwonie
cent200
1
200
11 ==
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Fractions, ratio, percent, decimals, and proportions
50
1. A fraction is a comparison between a part and the whole. For example 40/1002. A ratio is a comparison between two numbers. For example 40:1003. 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 0.405. 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
2) Convert the fraction using a proportion and the cross-multiplication rule5
7
5
21 =
5
75:757 ==to
5
7
100
140%140 ==
5
7
10
144.1 ==
5
21
5
7= 3549
5
7;49
5
357;
3535
5
7tosox
xtox ==
´===
4.15
7= %120
5
7;120
5
1007;
1005
7==
´== sox
x
Ex. 1 Ex. 2 Ex. 3 Ex. 4
Ex. 5 Ex. 6
Ex. 7 Ex. 8
Fraction Ratio Percent Decimal
4/1
3/1
2/3
6/5
... to 20
... out of 21
12 to ...
15 out of ...
2 out of 5
10 to 45
3 out of 12
18 to 15
30 %
80 %
130 %
10 %
... to 70
36 out of ...
... to 30
... out of 5
0.15
1.4
0.08
0.64
... to 10
... out of 35
6 to ...
80 out of ...
5/3 42 out of ...
7 to 10
75 %... out of 64
0.125... out of 256
F114. Fill out the table:
a)
b)
c)
d)
e)
g)
h)
i)
j)
l)
m)
n)
o)
r)
s)
t)
u)
f)
k)
p)
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Fractions and Number Line
51
F115. Use the number line to compare fractions (use > = or < symbols):
1. To create a number line assign points to 0 and 1:
0 1
2. Each number corresponds to a point on the number line.Example 1:
5
3
0 1
3. Each point on the number line corresponds to a number.Example 2:
5
21
0 1 2
4. Number lines can be used to compare numbers (fractions). Example 3:
0 1
0 1
0 1
1/3 2/31/3
1/4 2/4 3/4
1/5
2/5
3/5
4/5
3
2
5
3<
5. To calibrate a ruler means to assign numbers to the points on the ruler.Example 4:
0 1
4
1
2
1
4
3
2
4
11
2
11
4
31
F116. Find a point on the number line corresponding to each fraction:0 1
e)d)c)b)a)2
1
4
3
12
1
4
11
8
3
F117. Find the fraction corresponding to each point on the number line:
0 1A C E G I
B D F H J
F118. Calibrate the ruler:
0 1
j)i)h)g)f)24
3
12
7
24
51
12
11
8
11
0 12
1
4
1
4
3
12
1
6
1
3
1
12
5
12
7
3
2
6
5
12
11
12
13
10
1
5
1
10
3
5
2
5
3
10
7
5
4
10
9
10
11
e)d)c)b)a)12
6
2
1
10
3
3
1
12
7
5
36
5
5
4
10
9
12
11
j)i)h)g)f)12
5
5
24
3
5
4
5
1
6
1
4
3
5
4
10
1
12
1
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 a b c d e f g h i j
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Comparing fractions
52
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.
e)d)c)b)a)
F119. Compare the whole numbers. Use <, =, and > operators :
The main idea is that a small number is less (<) than a big number:
So, if you compare two fractions having the same denominator, the smallest one has the smallest numerator.
3. If you cross exchange the factors in relation (2), you’ll get:
92 and 1010 and 01 and 57 and 132123 and
)1(bigsmall <
Example 1: 442531 =><
2. If you divide relation (1) by any number a, you’ll get:
)2(a
big
a
small<
Example 2: 4
3
4
3
11
5
11
7
5
4
5
2=><
)3(small
a
big
a<
So, if you compare two fractions having like numerators, the smallest fraction has the biggest denominator.
8
3
8
3
7
2
5
2
2
1
3
1=><
4. To compare two fractions in the general case:
use cross multiplication to convert the initial comparison to another equivalent one:
1221 ? dndn ´´
Example 4: 24535
2
4
3´>´> because
2
2
1
1 ?d
n
d
n
Example 3:
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:
16
7
48
21
48
20
12
5
16
7
12
5=<=< because
6. You can use the LCD method when you have to order a set of fractions:Example 6:
12
10
6
5
12
9
4
3
12
8
3
2
6
5
4
3
3
2=<=<=<< because
F120. Compare the fractions (see example 2):
F121. Compare the fractions (see example 3):
F122. Compare the fractions (see example 4):
F124. Write each set of fractions in order from least to greatest:
F125. Write each set of fractions in order from greatest to least:
g)f) 30 and 2112 and
e)d)c)b)a)7
5
7
3and
6
1
6
5and
3
1
6
2and
11
5
11
3and
25
23
25
13and g)f)
3
4
3
21 and
5
14
5
42 and
l)k)j)i)h)3
72 and
5
51 and
4
12
4
51 and
5
42
5
23 and 3
5
12 and n)m)
3
42
3
13 and
4
12
4
11and
e)d)c)b)a)3
1
2
1and
5
2
7
2and
2
3
2
11 and
5
32
4
13 and
3
4
7
4and g)f)
11
10
7
10and
5
42
7
42 and
l)k)j)i)h)6
5
4
5and
9
71 and
3
23
7
23 and
3
11 and 4
5
4and n)m)
6
7
5
21 and
5
32
7
13and
e)d)c)b)a)4
3
3
2and
7
6
5
4and
5
31
2
11 and
7
5
4
3and
4
3
2
1and g)f)
11
8
10
7and
9
4
8
3and
l)k)j)i)h)12
15
4
5and
9
7
6
5and
2
52
5
22 and
3
2
2
3and
15
27
5
41 and n)m)
13
4
10
3and
7
4
5
3and
e)d)c)b)a)þýü
îíì
7
3,
7
1,
7
2
þýü
îíì
7
3,
11
3,
5
3
þýü
îíì
5
4,
2
1,
3
2,
4
3
þýü
îíì
5
2,
10
3,
12
5
þýü
îíì
15
4,
4
3,
12
7,
2
1 f)þýü
îíì
6
51,
8
11,1,
24
51,
4
11
e)d)c)b)a)þýü
îíì
5
1,
5
4,
5
3
þýü
îíì
6
2,
5
2,
7
2
þýü
îíì
7
6,
3
2,
4
3,
6
5
þýü
îíì
3
1,
5
3,
4
1
þýü
îíì
10
9,
3
2,
4
3,
12
7 f)þýü
îíì
32
111,
16
51,
2
11,
8
31,
4
11
F123. Compare the fractions (see example 5):
e)d)c)b)a)15
8
5
3and
24
11
8
3and
6
5
4
3and
20
7
15
4and
15
5
12
4and g)f)
10
7
6
5and
15
4
9
2and
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Solving equations by working backward method
53
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:
e)
j)
d)
i)
c)
h)
b)
g)
a)
f)
F126. Solve for x working backward:
4
1
2
1
4
3
4
3
2
1=-==+ xsox
2. If the operation is a subtraction with a number then apply an addition with the same number.Example 2:
6
5
2
1
3
1
3
1
2
1=+==- xsox
3. If the operation is a multiplication by a number then apply a division by the same number.Example 3:
8
9
3
2
4
3
4
3
3
2=¸==´ xsox
4. If the operation is a division by a number then apply a multiplication by the same number.Example 4:
35
12
5
3
7
4
7
4
5
3=´==¸ xsox
5. If the operation is an inversion then apply another inversion.Example 5:
2
3
2
3
13
21=== xso
xso
x6. 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
8
3
2224
2
1
3
2
2
1
43
2
=+==´=-=-
xsoxsox
Example 7:
24
7
8
73
8
3
2
13
3
8
2
13
14
3
4
2
13
12
2
3
4
2
13
1
==´=-´
=
-´
=+
-´
=
+-´
xsoxsoxso
x
so
x
sox
2
1
5
2=+x
5
4
3
1=-x
3
1
2
1=´x
3
12
7
4=¸x
7
51=
x
5
4
3
22 =+´ x
3
1
2
13 =-´ x
4
3
3
2
3=´
x
5
4
3
2
3=¸
x
3
2
2
1=
-x
o)n)m)l)k)3
2
2
1
3=+
x
2
1
3
2
5=-
x
5
4
2
1)1( =´-x 4
3
3
1
1=
+x
s)r)q)p)
5
4
3
1
23
23
=+-´x
3
1
2
1
32
12
=-+´x
5
4
3
2
22
12
=´
÷÷÷÷
ø
ö
çççç
è
æ-´x
5
2
5
3
3
2
3
12 =¸
÷÷÷÷
ø
ö
çççç
è
æ
--
x
2
1
3
2
5
1=¸÷
ø
öçè
æ+x
w)v)u)t)6
1
3
2
32
12
=¸-¸x
3
4
3
2
2
1
3
22
1
3
12
=´
-
-¸´x
5
4
3
2
3
1
2
13
1
2
1
4
3
=´
÷÷÷÷
ø
ö
çççç
è
æ
+
-¸´x
4
3
3
2
2
1
2=
--x
F127. Solve for x working backward:
b)
d)
a)
c)
4
1
4
3
2
1
6
1
2
1
3
2
4
32
1
8
5
8
3
-´=÷ø
öçè
æ-´
-
¸-´x
4
5
3
2
5
1
3
2
5
4
4
1
3
2
3
4
2
1
3
16
1
2
3
2
1
¸+=÷ø
öçè
æ--÷
ø
öçè
æ-¸
´-´
-¸
x
2
1
5
3
5
1
5
4
5
2
3
25
1
4
1
10
1=¸
ïï
þ
ïï
ý
ü
ïï
î
ïï
í
ì
+´
úúúú
û
ù
êêêê
ë
é
÷ø
öçè
æ-¸
-+
x
03
2
10
3
5
2
3
2
5
1
32
5
1
9
1
5
3
5
4
3
1=÷
ø
öçè
æ+¸-
÷÷÷÷
ø
ö
çççç
è
æ
-
++´
¸÷ø
öçè
æ-¸´
x
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Final Test
54
Find the fractions:
1) 2) three eights five out of seven four and three quartersExpress the numbers as improper fractions:
6) 7) 8) 9) five and two thirds3
12
7
103
Express the numbers as mixed numbers:
13) 14) 15) 16) eight fifths4
10
3
51
Write the fractions in lowest terms:
21) 22) 23)8
20
36
24
18
12
Check each pair of fractions for equivalence:
25) 26) 27)20
15
12
10and
15
5
12
4and
48
75
64
100and
Add the fractions:
28) 29) 30)5
1
5
2+
4
3
3
1+
6
52
6
11 + 31) 32) 33)
6
51
3
10+
20
32
15
21 +
3
221+
Subtract the fractions:
35) 36) 37)23
10-
3
21-
7
3
7
5- 38) 39) 40)
3
2
4
3-
10
72
5
43 -
3
22
4
15 -
Multiply the fractions:
42) 43) 44)5
32´
5
3
3
2´
3
2
2
11 ´ 45) 46) 47)
5
21
3
21 ´
9
4
4
3´
3
21
3
5´
Divide the fractions:
49) 50) 51)5
22 ¸
4
3
3
2¸ 4
9
8¸ 52) 53) 54)
10
6
5
41 ¸
2
13
6
11 ¸
6
51
3
22¸
Convert each fraction to a decimal:
56) 57) 58)5
22
5
2
10
459) 60) 61)
12
15
3
2
7
21
Convert each decimal to a fraction:
65) 66) 67)5.112.0 25.2 68) 69) 70)4.0 63.10125.0
Do the required conversions (use a fraction to express the result if possible):
74) 75) 76) 77) 78)mins ?100 =minh ?1.0 =smin ?45.0 = hs ?2000 = hsmin ?2020 =
79) 80) 81) 82)nickelsdimes ?5.1 = $?5.7 =quarters twonies?$35.0 = dimesquarter ?1$1 =
83) 84) 85) 86)twelfths?3
2= tenths?
5
1= sixths?
2
11 = eighths?
16
3=
Do the required operations:
Solve for x:
88) 89) 90) 91)
92) 93) 94) 95)
96) 97) 98) 99)
4
1
3
1
2
1+-
6
1
3
4
2
1-´
3
2
6
5
3
21 +¸
2
1
4
33
2
12 ´¸
3
2
4
1
3
2´÷
ø
öçè
æ- ÷
ø
öçè
æ-¸
6
5
3
11
3
2÷ø
öçè
渴÷
ø
öçè
æ-
3
42
3
21 ÷
ø
öçè
æ--÷
ø
öçè
æ¸
10
3
5
2
2
12
2
1
4
1
3
1=-x
5
2
3
1=¸x
6
5
3=
x
4
3
2/3
2=
-x
11) one and five thirds4.210)
5)4)3)
12)
18) 19) eleven quarters25.17
1517) 20)
24)32
18
34)3
21
4
5+
41)4
11
5
9-
48)3
11
4
31 ´
55)5
22
4
12 ¸
62) 63) 64)16
5
8
7
32
100
71) 72) 73)65.1 875.0725.0
100)5
2
/32
1=
+ x
87) fifteenths?3
2=
5
23
4
15
© La Citadelle
Iulia & Teodoru Gugoiu
www.la-citadelle.com
The Book of Fractions
Answers
55
16
10)
10
7)
12
12)
12
0)
12
6)
49
21)
9
7)
8
3)
4
1)
13
9)
4
2)
10
6)
5
2)
1
1)
18
6)
3
1)
6
2)
6
4)
6
5)
10
6)
100
58)
9
5)
4
1)
3
2)
2
1)
yxwvutsrqpon
mlkjihgfedcbaF01.
)a
F02. The answers may vary.
)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
billionthsninezfiftiethstwoytwelfthselevenxtenthssevenwninthstwovmillionthselevenu
fiftiethseleventeleventhssevenstwelfthssixrhundredthsonetwentyqfifthstwopfiftiethsthreeo
sthousandthsevenneighthsfivemsixthsfivelninthseightksthirteentheightjthirtiethseightififthsfourh
thousandthonegtwentiethsthreefseventhsthreeehalfonedtenthonechundredthsthreebthirdstwoa
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-
F03.
12
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10
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9
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11
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5
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12
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13
7)
100
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11)
50
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6
8)
1000000
15)
1000
9)
10
6)
9
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12
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20
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50
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8
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7
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5
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6
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3
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zyxwvutsrqpon
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5
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6
4)
10
3)
12
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4
3)
7
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5
2)
12
3)
6
3)
5
3)
4
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3
2) lkjihgfedcbaF05.
10
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10
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16
105)
12
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8
62)
9
53)
49
132)
13
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4
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10
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18
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6
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100
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9
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4
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2
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xwvutsrqpon
mlkjihgfedcbaF06.
© La Citadelle
Iulia & Teodoru Gugoiu
)a
F07. The answers may vary.
)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
The Book of Fractions
www.la-citadelle.com56
thirtiethsnineandtwontwentiethssevenandthreemfifteenthstwoandoneltwelfthsfiveandthreek
eleventhstwoandonejtenthsthreeandtwoininthsfiveandoneheighthsfiveandthreegseventhsthreeandtwof
sixthsfiveandoneefifthsthreeandtwodquarteroneandonecthirdoneandtwobhalfoneandonea
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ninetiethsthreeandtwoz
sixtiethoneandtwoyfifteenthsfiveandtwoxsfourteenththreeandfourw
hsseventeentthreeandtwovsixteenthsfiveandthreeusnineteenthsevenandtwotfourtiethsthreeandones
millionthssevenandtworsthousandthnineandthreeqhundredthsthreeandtwopfiftiethssevenandtwoo
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12
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10
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30
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100
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100
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4
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32) lkjihgfedcbaF10.
49
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100
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6
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10
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8
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5
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9
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6
10)
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15)
4
11)
1
3)
3
5)
xwvutsrq
ponmlkjihgfedcbaF11.
F12. The answers may vary.
)a )b )c )d )e )f
)g )h )i )j )k )l
© La Citadelle
Iulia & Teodoru Gugoiu
)m )n )o )p )q )r
)s )t )u )v )w )x
The Book of Fractions
www.la-citadelle.com57
)y )z
5
13)
6
13)
10
13)
3
7)
4
11)
5
7)
5
12)
12
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3
4)
5
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4
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4
5) lkjihgfedcbaF13.
49
132
49
111)
16
63
16
54)
100
361
100
136)2
12
24)
13
62
13
32)
9
62
9
24)
6
42
6
16)
10
43
10
34)
10
31
10
13)
18
61
18
24)
10
12
10
21)
4
33
4
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8
42
8
20)
10
52
10
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5
22
5
12)
3
22
3
8)
4
32
4
11)
3
12
3
7)
9
52
9
23)
6
12
6
13)
6
23
6
20)
4
12
4
9)
2
11
2
3)
3
12
3
7)
=======
========
=========
xwvutsr
qponmlkj
ihgfedcbaF14.
10
35)
3
12))
10
20)
9
2)
100
211)
50
149)
11
35)
12
25)
100
221)
5
24)
50
103)
100
1207)
8
21)
6
23)
9
26)
13
18)
30
67)
4
23)
10
53)
20
103)
7
17)
2
7)
4
15)
3
8)
2
3)
zydefinednotxwvutsrqpon
mlkjihgfedcbaF15.
9
77)
15
51)
7
41))
9
00)
10
111)
50
112)
11
46)
12
41)
10
12)
5
24)
5
32)
10
71)
8
68)
6
28)
9
88)
3
26)
3
22)
5
42)
10
512)
20
131)
7
61)
2
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4
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3
11)
2
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zyxdefinednotwvutsrqpon
mlkjihgfedcbaF16.
definednotutsrdefinednotq
definednotponmlkjihgfedcba
)0)2)1))
)0)5)4)15)6)4)3)2)10)8)3)2)1)1)1)F17.
F18. The answers may vary.
3
51;
2
34)
3
39;
2
26)
3
24;
2
16)
5
55;
2
22)
7
700;
1
100)
5
125;
2
50)
3
0;
2
0)
3
30;
10
100)
9
45;
2
10)
6
12;
1
2)
3
12;
2
8)
3
21;
2
14)
3
9;
2
6)
5
5;
2
2)
nmlk
jihgfedcba
improperzimproperyproperximproperwmixedvproperumixedt
impropersimproperrwholeqimproperpmixedomixednmixedmproperlimproperkwholej
properidefinednothmixedgimproperfmixedeproperdimpropercwholebimpropera
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16
12)
10
7)
12
6)
9
5)
12
10)
49
25)
9
7)
8
6)
4
3)
13
10)
4
3)
10
7)
5
3)
6
4)
18
10)
3
2)
6
4)
6
5)
6
4)
10
7)
100
52)
9
5)
4
3)
3
2)1)
yxwvutsr
qponmlkjihgfedcbaF20.
F21.
4
22
4
10)
13
21
13
15)
4
11
4
5)
10
31
10
13)
5
11
5
6)
6
13
6
19)
18
11
18
19)
3
11
3
4)
6
11
6
7)
6
23
6
20)
6
21
6
8)
10
42
10
24)
100
201
100
120)
9
31
9
12)
4
12
4
9)
3
11
3
4)1
2
2)
========
=========
qponmlkj
ihgfedcba
© La Citadelle
Iulia & Teodoru Gugoiu
The Book of Fractions
www.la-citadelle.com58
6
21
6
8)
16
121
16
28)
12
61
12
18)
9
41
9
13)
12
71
12
19)
49
141
49
63)
9
41
9
13)
8
21
8
10) ======== yxwvutsr
400
40)
13
11)
30
20)
10
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9
7)
25
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20
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13
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4
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17
13)
12
9)
10
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54
32)
13
11)
41
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35
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100
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19
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7
5)
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5
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3
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4
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yxwvutsr
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10
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3
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50
491)
10
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3
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40
121)
10
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9
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100
101)
50
381)
25
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20
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15
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12
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8
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6
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5
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3
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yxwvutsrqp
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100
104)
35
207)
23
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30
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19
144)9)
3
110)
30
36)
10
310)
9
18)
11
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20
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10
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7
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15
115)
10
82)
9
33)
8
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7
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6
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5
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4
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3
23)4)
yxwvutsrqp
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100
11113)
35
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23
175)
30
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10
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115)
30
1114)40)
9
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20
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yxwvutsrqpo
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3)10
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9
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100
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yxwvutsrqpo
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10
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100
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yxwvutsrqpo
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64
48
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16
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40
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4
2
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=============
===============
============
yxwvut
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4000
12
3000
9
2000
6)
800
12
600
9
400
6)
200
12
150
9
100
6)
120
4
90
3
60
2)
28
20
21
15
14
10)
20
16
15
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10
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60
8
45
6
30
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44
20
33
15
22
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28
12
21
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36
8
27
6
18
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8
33
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22
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48
20
36
15
24
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400
4
300
3
200
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40
4
30
3
20
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28
8
21
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28
4
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24
20
18
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12
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24
4
18
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20
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4
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12
12
9
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8
9
6
6
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4
9
3
6
2)
8
4
6
3
4
2)
============
============
============
==============
yxwvut
srqpon
mlkjih
gfedcba
F29. The answers may vary.
16
1
32
2
64
4
128
8
256
16
512
32)
25
9
125
45)
11
2)
11
9)
3
1
9
3
27
9)
4
1
8
2
16
4
32
8
64
16)
15
2
30
4
60
8
105
14
210
28)
7
3
14
6
77
33)
8
5
24
15
40
25)
7
4)
5
2
10
4
15
6
25
10
30
12
50
20
75
30)
3
2
6
4
9
6
15
10
18
12
30
20
45
30)
10
3
20
6
30
9)
8
5)
3
2
9
6
15
10)
4
3)
3
2)
5
4)
3
1
6
2)
10
1
20
2
50
5)
6
1
12
2
18
3)
5
1)
4
1
8
2)
3
1)
2
1)
=====
=============
================
========
y
xwvutsr
qponm
lkjihgfedcbaF30.
© La Citadelle
Iulia & Teodoru Gugoiu
The Book of Fractions
www.la-citadelle.com59
10
7)
20
3)
25
11)
15
7)
7
5)
2
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© La Citadelle
Iulia & Teodoru Gugoiu
The Book of Fractions
www.la-citadelle.com64
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Iulia & Teodoru Gugoiu