The Book of Fractions - La Citadelle, Ontario, Canadala-citadelle.com/mathematics/the_book_of_fractions.pdf · Each worksheet contains: ... The Book of Fractions Understanding the

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The Book ofFractions

Iulia & Teodoru Gugoiu

Copyright © 2006 by La Citadelle

www.la-citadelle.com


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

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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


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.


a) b) c) d) e)

h) i) j) l) n)

o) q) r) s) t)

v)



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




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




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




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




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




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:


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




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




Understanding improper fraction notation

13

1. An improper fraction is represented by the expression:

rdenominato

numerator


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




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




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




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


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)



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


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)



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




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




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




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


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)



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




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




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




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




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




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




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


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)



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




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




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




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




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




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




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




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


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




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)


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




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




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




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)


3

1:

2

1

4

1:

2

11

10

12:

5

11 4:

5

27:3

o)n)m)l)k)


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)


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




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:



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




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

æ¸´÷

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




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




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

æ¸´+

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´-+

---



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

æ¸´

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

æ¸´÷

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




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




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




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)


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)


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

æ¸´

¸´

© La Citadelle




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




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




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




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




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):



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


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




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




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

æ¸´÷

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

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The Book of Fractions - La Citadelle, Ontario, Canadala-citadelle.com/mathematics/the_book_of_fractions.pdf · Each worksheet contains: ... The Book of Fractions Understanding the