The Book of Fractions - La Citadelle

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Adding fractions with unlike denominators using the LCD method ... of the concepts, notations, and conventions that constitute the topic of the worksheet;.
Iulia & Teodoru Gugoiu

The Book of Fractions

Copyright © 2006 by La Citadelle www.la-citadelle.com

Iulia & Teodoru Gugoiu

The Book of Fractions ISBN 0-9781703-0-X

© 2006 by La Citadelle 4950 Albina Way, Unit 160 Mississauga, Ontario L4Z 4J6, Canada www.la-citadelle.com [email protected]

Edited by Rob Couvillon

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writting from the publisher.

Content Page Topic 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Understanding fractions The graphical representation of a fraction Reading or writing fractions in words Understanding the fraction notation Understanding the mixed numbers Reading and writing mixed numbers in words Understanding mixed number notation Understanding improper fractions Understanding improper fraction notation The link between mixed numbers and improper fractions Conversion between mixed numbers and improper fractions Whole numbers, proper fractions, improper fractions and mixed numbers Understanding the addition of like fractions Understanding the addition of like fractions (II) Adding proper and improper fractions with like denominators Adding mixed numbers with like denominators Adding more than two like fractions Understanding equivalent fractions Finding equivalent fractions Simplifying fractions Checking fractions for equivalence Equations with fractions Adding fractions with unlike denominators Adding fractions with unlike denominators using the LCD method Understanding the subtraction of fractions with like denominators Subtracting fractions with like denominators Subtracting mixed numbers with like denominators Subtracting fractions with unlike denominators Subtracting fractions with unlike denominators using the LCD method Order of operations (I) Multiplying fractions More about multiplying fractions The order of operations (II) Reciprocal of a fraction Dividing fractions Division operators Order of operations (III) Order of operations (IV) Raising fractions to a power Order of operations (V) Converting fractions to decimals Converting decimals to fractions Order of operations (VI) Time and Fractions Canadian coins and fractions Fractions, ratio, percent, decimals, and proportions Fractions and Number Line Comparing fractions Solving equations by working backward method Final Test Answers

Preface “The Book of Fractions" presents one of the primary concepts of middle and high school mathematics: the concept of fractions. This book was developed as a workbook and reference useful to students, teachers, parents, or anyone else who needs to review or improve their understanding of the mathematical concept of fractions. The structure of this book is very simple: it is organized as a collection of 50 quasi-independent worksheets and an answer key. Each worksheet contains: · a short description of the concepts, notations, and conventions that constitute the topic of the worksheet; · step-by-step examples (completely solved) demonstrating the techniques and skills the student should gain by the end of each worksheet; and · an exhaustive test to be completed independently by the students. The concept of fractions and the relations between fractions and other types of numbers, like many abstract mathematical concepts, is not always easy to understand. Bearing this in mind, the authors of this book introduce each topic gradually, starting with the basic concepts and operations and progressing to the more difficult ones. Geared specifically to help the beginners, the first part of the book contains graphical representations of the fractions. The techniques for solving both simple and complex equations implying fractions are explained. As well, complete worksheets are provided, starting with very simple and basic equations and progressing to extremely complex equations requiring the application of a full range of operations with fractions. "The Book of Fractions" also presents the link between fractions and other related mathematical concepts, such as ratios, percentages, proportions, and the application of fractions to real life concepts like time and money. The importance of the concept of fractions comes both from its link to natural numbers and its link to more complex mathematical concepts, like rational numbers. As such, the concept of fractions is a milestone in the mathematical evolution of a student, being a concept that is simultaneously concrete (as a part of a whole) and abstract (as a set of two numbers and a hidden division operation). The concept of equivalent fractions is an essential part of understanding fractions, and a full range of techniques is presented, starting with graphical representations (suitable for students in lower grades) and progressing to advanced uses, like the factor tree method of finding the LCD. The order of operations is also presented, gradually, after each main operation with fractions: addition, subtraction, multiplication, and division; using multi-term expressions; expressions containing grouping symbols of one or more levels; and more complex operations with fractions, like powers with positive and negative exponents. Single-step questions (requiring a basic knowledge and understanding of the topic presented in the worksheets) and multi-step questions (requiring a complete understanding of all of the concepts presented in the worksheets to that point) are presented throughout the entire book. Combining more than 15 years of academic studies and 30 years of teaching experience, the authors of this book wrote it with the intention of sharing their knowledge, experience and teaching strategies with all the partners involved in the educational process. Iulia & Teodoru Gugoiu, Toronto, 2006

The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding fractions 1. A fraction represents a part of a whole. Example 1.

2. The corresponding fraction is:

3 4

The whole is divided into four equal parts. Three part are taken (considered).

The numerator represents how many parts are taken. Fraction line or division bar The denominator represents the number of equal parts into which the whole is divided.

F01. Write the fraction that represents the part of the object that has been shaded:

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

0

u) © La Citadelle

v)

w)

x) 5

1

y) www.la-citadelle.com

The Book of Fractions

Iulia & Teodoru Gugoiu

The graphical representation of a fraction 1. A fraction represents a part of a whole. Example 1.

2. A corresponding graphical representation (diagram) is:

3 4

The whole is divided into four equal parts. Three part are taken (considered).

The numerator represents how many parts are taken. Fraction line or division bar The denominator represents the number of equal parts into which the whole is divided.

F02. Draw a diagram to show each fraction (use the images on the bottom of this page):

a)

1 2

b)

1 3

c)

1 4

d)

2 5

e)

1 6

f)

2 4

g)

0 3

h)

2 9

i)

5 6

j)

2 12

k)

9 10

l)

1 1

m)

3 3

n)

4 6

o)

3 4

p)

4 12

q)

5 10

r)

4 9

s)

2 4

t)

8 12

u)

5 13

v)

5 16

w)

1 8

x)

7 49

y)

37 100

z)

11 18

0

© La Citadelle

1

6

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The Book of Fractions

Iulia & Teodoru Gugoiu

Reading or writing fractions in words 1. You can use words to refer to a part of a whole. So one whole has: 2 halves 3 thirds 4 quarters 5 fifths 6 sixths

7 sevenths 8 eighths 9 ninths 10 tenths 11 elevenths

12 twelfths 13 thirteenths 20 twentieths 30 thirtieths 50 fiftieths

Example 1. 100 hundredths 1000 thousandths 1000000 millionths 1000000000 billionths

The fraction

3 4

can be written in words as: three quarters

F03. Write the following fractions in words: a)

2 3

b)

3 100

c)

1 10

d)

1 2

e)

3 7

f)

3 20

g)

1 1000

h)

4 5

i)

8 30

j)

8 13

k)

8 9

l)

5 6

m)

5 8

n)

7 1000

o)

3 50

p)

2 5

q)

21 100

r)

6 12

s)

7 11

t)

11 50

u)

11 1000000

v)

2 9

w)

7 10

x)

11 12

y)

2 50

z)

9 1000000000

F04. Find the fraction written in words: a)

one third

b)

one half

c)

one sixth

d)

two fifths

e)

four sevenths

f)

seven eighths

g)

eleven fiftieths

h)

seven twentieths

i)

five twelfths

j)

eight ninths

k)

six tenths

l)

nine thousandths

m)

fifteen millionths

n)

eight sixths

o)

three fiftieths

p)

eleven billionths

q)

twenty-three hundredths

r)

seven thirteenths

s)

eleven twelfths

t)

three billionths

u)

thirteen thirtieths

v)

one fifth

w)

one eleventh

x)

eight ninths

y)

six tenths

z)

six twelfths

© La Citadelle

7

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding the fraction notation 1. A fraction also represents a quotient of two quantities:

Example 1. The dividend (numerator) is 3. 3 The divisor (denominator) is 4. The fraction in words is three quarters. 4

divident divisor

2. The dividend (numerator) represents how many parts are taken. The divisor (denominator) represents the number of equal parts into which the whole is divided.

A possible graphical representation of this fraction is:

F05. Fill out the following table: Fraction

a)

2 3

b)

Numerator (Dividend)

Denominator (Divisor)

2

3

1

4

c)

The fraction written in words

Graphical representation

two thirds

three fifths

d)

e)

f)

3

5

2

2

g)

5 3

h)

.......... quarters

i)

five .......... 3

j)

k)

l)

© La Citadelle

4

.......... sixths

5

three ..........

8

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding the mixed numbers fraction part

1. A mixed number is an addition of wholes and a part of a whole. Example 1. whole-number part (the number of complete wholes) There are one complete whole and three quarters of the second whole

The numerator indicates how many parts are taken from the last whole. The denominator represents the number of equal parts into which the whole is divided.

3 1 4

F06. Find the mixed number that corresponds to the shaded region: a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

u)

v)

w) 0

1

2

0

x)

2

1

F07. Find a possible graphical representation of each mixed number: a)

1

2 3

b)

3

1 2

c)

2

5 6

d)

3

4 5

e)

3

3 7

f)

1

h)

3

1 6

i)

2

2 9

j)

2

2 5

k)

2

2 10

l)

1

3 20

m)

o)

3

5 8

p)

1

7 16

q)

2

7 10

r)

2

4 5

s)

5

1 1

t)

v)

1

1 6

w)

5

1 2

x)

3

7 9

y)

2

3 8

z)

6

3 4

© La Citadelle

9

5 8

g)

4

5 11

6

6 10

n)

1

3 6

2

5 6

u)

3

7 10

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3

The Book of Fractions

Iulia & Teodoru Gugoiu

Reading and writing mixed numbers in words 1. You can use words to refer to a part of a whole. So one whole has: 2 halves 3 thirds 4 quarters 5 fifths 6 sixths

7 sevenths 8 eighths 9 ninths 10 tenths 11 elevenths

12 twelfths 13 thirteenths 20 twentieths 30 thirtieths 50 fiftieths

Example 1. The fraction 2

100 hundredths 1000 thousandths 1000000 millionths 1000000000 billionths

3 4

can be written in words as: two wholes and three quarters or two and three quarters

F08. Write the following mixed numbers in words: a) 1

1 2

b)

h) 1

5 9

i)

o) 2

7 50

p) 2

3 17

w) 4 3

v) 2

1 3

c)

1

3 10

j)

1

3 100

q)

3

2

2

14

x)

1 4

d)

2

2 11

k)

3

9 1000

r) 2

2

5 15

y)

3 5

e) 1

5 6

f)

2

3 7

g)

3

5 12

l) 1

2 15

m) 3

7 20

n)

2

t)

7 19

u)

7 3 s) 1 1000000 40

2

1 60

z)

2

2

5 8

9 30

3

5 16

3 90

F09. Find the mixed numbers written in words: a)

two and two thirds

b)

three and one half

c)

five and five sixths

d)

two and one third

e)

four and five sevenths

f)

seven and five fiftieths

g)

two and three quarters

h)

three and two ninths

i)

six and seven hundredths

j)

nine and one half

k)

eight and eleven fiftieths

l)

one and five billionths

m)

one and two elevenths

n)

eight and five sixths

o)

three and two twelfths

p)

five and three millionths

q)

twenty and three hundredths

r)

six and four fifteenths

s)

eleven and four thirtieths

t)

eight and seven tenths

u)

four and one third

v)

one and two fifths

w)

three and two elevenths

x)

eight and six ninths

y)

five and nine tenths

z)

one and eleven twelfths

© La Citadelle

10

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding mixed number notation 1. A mixed number is represented by the expression:

wholes

Example 1.

numerator denominator

2

3 5

This mixed number written in words is two wholes and three fifths. A possible graphical representation of this mixed number is:

The whole-number part is 2 (the number of complete wholes). The numerator is 3. The denominator is 5. 3 The fraction part is: 5

F10. Fill out the following table: Mixed Number

a)

2

3 5

b)

2

1 3

c)

Number of wholes

Numerator

The mixed number in words

Denominator

2

3

5

1

3

4

Graphical representation

two and three fifths

d)

e)

2

f)

3

g)

3 5

2

2 3

5

2

three and a half

h) 6

i)

j)

3

k)

l)

© La Citadelle

two and four ...........

.......... and three fifths

2

four and .......... thirds

2

.......

11

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding improper fractions 1. For an improper fraction the number of parts taken (the numerator) is equal to or greater than the number of parts the whole is divided into (the denominator). Example 1.

5 3

This is a possible graphical representation of this improper fraction:

F11. Find the improper fraction that corresponds to the shaded region: a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

p)

0

1

2

0

1

2

3

o)

q)

r)

s)

t)

u)

v)

w)

x)

F12. Find a possible graphical representation of each improper fraction: a)

3 2

b)

4 3

c)

5 2

d)

7 3

e)

7 5

f)

11 4

g)

8 6

h)

10 8

i)

20 9

j)

12 5

k)

22 10

l)

25 12

m)

30 16

n)

40 24

o)

21 6

p)

28 13

q)

11 3

r)

40 18

s)

55 49

t)

25 6

u)

17 9

v)

30 12

w)

15 4

x)

17 5

y)

24 10

z)

13 3

© La Citadelle

12

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding improper fraction notation 1. An improper fraction is represented by the expression:

Example 1.

numerator denominator

5 3

The numerator is 5 The denominator is 3

The improper fraction in words is five thirds. A possible graphical representation of this improper fraction is:

where the numerator is equal to or greater than the denominator.

F13. Fill out the following table: Fraction

a)

5 4

b)

Numerator

Denominator

5

4

7

4

c)

The fraction in words

Graphical representation

five quarters

seven fifths

d)

e)

f)

16

5

........

12

7

g)

5 11

h)

.......... quarters

i)

seven ..........

........

13

j)

k)

l)

© La Citadelle

13

.......... sixths

5

thirteen ..........

13

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The Book of Fractions

Iulia & Teodoru Gugoiu

The link between mixed numbers and improper fractions 1. There is a direct link between a mixed number and an improper fraction. A mixed number is a short way to write the sum of a whole number and a fraction. 5 2 2 = 1 = 1+ 3 3 3

Example 1:

9 3 3 = 1 = 1+ 6 6 6

Example 2:

F14. Find the mixed number and the improper fraction that correspond to each picture: a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

0

m)

1

n)

0

1

2

3

o)

2

p)

q)

r)

s)

t)

u)

v)

w)

x)

© La Citadelle

14

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The Book of Fractions

Iulia & Teodoru Gugoiu

Conversion between mixed numbers and improper fractions 1. To convert a mixed number to an improper fraction, use the formula:

3. To convert an improper fraction to a mixed number, divide the numerator n into the denominator d to obtain the quotient q and the remainder r. Then write:

n w×d + n = d d 3 2 × 5 + 3 10 + 3 13 Example 1. 2 = = = 5 5 5 5 w

n r =q d d

2. Fractions that have a denominator of 0 are not defined.

9 1 =2 4 4

Example 2.

F15. Write each mixed number as an improper fraction: a) 1

1 2

b)

2 3

c)

3

h) 5

3 4

i)

7 30

j)

o) 2

3 50

p) 4

4 5

v) 0

2 9

w) 2

0 10

2

2

3 4

d)

3

1 2

e) 2

3 7

f)

5

3 20

g)

5

1

5 13

k)

2

8 9

l) 3

5 6

m)

2

5 8

n)

12

q)

2

21 100

r)

2

1 12

s) 3

2 11

t)

2

49 50

u)

x)

3

2 0

y)

3

3 3

z) 2

3 10 7 100

2

11 100

15 10

F16. Write each improper fraction as a mixed number: a)

3 2

b)

4 3

c)

5 4

d)

9 2

e)

13 7

f)

h)

14 5

i)

8 3

j)

80 13

k)

80 9

l)

50 6

m)

o)

13 5

p)

22 5

q)

21 10

r)

16 12

s)

70 11

t)

v)

0 9

w)

7 0

x)

11 7

y)

20 15

z)

70 9

© La Citadelle

15

33 20

g)

125 10

60 8

n)

17 10

111 50

u)

111 10

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The Book of Fractions

Iulia & Teodoru Gugoiu

Whole numbers, proper fractions, improper fractions and mixed numbers 1. Although written in fraction notation, some numbers are actually whole numbers. Example 1:

10 =5 2

Example 2: 2

3 =3 3

2. A whole number can be converted into a fraction. This conversion is not unique. Example:

5=

n d

where n < d

Example:

An improper fraction is: n d

where n ³ d

Example:

3. A proper fraction is:

A mixed number is:

w

A mixed number in standard form is:

w

n d

Example:

2 3 7 3 4

5 3

2 n where n £ d Example: 5 d 3 4. Fractions that have a denominator of 0 are not defined.

50 10 3 30 14 =4 =4 =2 =3 10 10 3 10 7

F17. Convert fractions to whole numbers. Identify the expressions that are not defined. 1 1

b)

h) 1

1 1

i)

2

o)

0 1

p)

2 0

a)

2 2

c)

3 3

j)

q)

2

7 7

d)

4 2

e)

9 3

f)

24 3

g)

24 12

k)

3

9 3

l) 4

55 5

m) 2

16 8

n)

0 0

r)

0

5 5

s) 2

0 9

t)

0 6

u)

0

100 10

2

300 100

0

0 0

F18. Convert whole numbers to fractions (the conversion is not unique, so give at least two solutions): a)

1

b)

3

c)

7

d)

4

e) 2

f)

5

g)

10

h)

0

i)

25

j)

100

k)

11

l)

m)

13

n)

17

8

F19. Identify each of the following expressions as a whole number, a proper fraction, an improper fraction, a mixed number, or a not defined expression: a)

3 2

b)

1

c)

5 4

d)

2 9

e) 2

h)

4 0

i)

0 3

j)

0

k)

80 9

o)

2

3 3

p)

22 5

q)

30

r)

v)

2

3 5

w)

22 5

x)

10 11

y)

© La Citadelle

1 7

f)

l)

5 16

m)

16 2

s)

70 11

t)

16 2

z)

7 1

16

33 20

g)

2

25 10

3

5 8

n)

2

17 10

1

11 50

u)

0 10

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding the addition of like fractions Two fractions with the same denominators are called like fractions. When you add two fractions, you add the parts of the whole they represent.

1 2 3 + = 4 4 4

Example 1.

So, by adding 1 quarter and 2 quarters you get 3 quarters.

F20. Add the fractions that correspond to the shaded regions:

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

0

u)

© La Citadelle

v)

w)

17

1

x)

y)

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding the addition of like fractions (II) 1. Sometimes when you add two like fractions, the number of parts you add exceeds a whole. The result is an improper fraction or a mixed number. Example 1.

=

+

Or, in mathematical symbols:

3 3 6 2 + = =1 4 4 4 4

F21. Add the fractions that correspond to the shaded regions. Express the result both as an improper fraction and as a mixed number.

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

u)

v)

w)

x)

y)

© La Citadelle

18

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The Book of Fractions

Iulia & Teodoru Gugoiu

Adding proper and improper fractions with like denominators 1. To add proper or improper fractions with like denominators (called like fractions), add the numerators and keep unchanged the denominator, according to the rule:

n1 n2 n1 + n2 + = d d d

1 2 3 + = 5 5 5

Example 1.

2. If the result is an improper fraction, you can change it to a mixed number. Example 2.

3 2 5 1 + = =1 4 4 4 4

F22. Add the fractions: a)

2 1 + 4 4

b)

1 1 + 3 3

c)

2 1 + 5 5

d)

3 5 + 11 11

e)

1 2 + 6 6

f)

2 3 + 7 7

g)

5 7 + 19 19

h)

20 10 + 100 100

i)

6 23 + 35 35

j)

2 10 + 41 41

k)

0 11 + 13 13

l)

21 11 + 54 54

m)

2 3 + 10 10

n)

5 4 + 12 12

o)

3 10 + 17 17

p)

0 0 + 4 4

q)

3 2 + 13 13

r)

7 3 + 20 20

s)

7 11 + 25 25

t)

5 2 + 9 9

u)

2 5 + 19 19

v)

2 3 + 10 10

w)

7 13 + 30 30

x)

3 8 + 13 13

y)

15 25 + 100 100

F23. Add the fractions. Write the result as a mixed number in standard form. a)

2 2 + 3 3

b)

3 2 + 4 4

c)

3 4 + 5 5

d)

3 5 + 6 6

e)

3 5 + 7 7

f)

5 7 + 8 8

g)

7 8 + 9 9

h)

7 7 + 10 10

i)

8 10 + 11 11

j)

9 5 + 12 12

k)

10 11 + 15 15

l)

4 19 + 20 20

m)

10 20 + 25 25

n)

44 44 + 50 50

o)

99 11 + 100 100

p)

7 6 + 9 9

q)

8 9 + 10 10

r)

33 19 + 40 40

s)

4 9 + 3 3

t)

15 9 + 10 10

u)

12 5 + 9 9

v)

20 12 + 10 10

w)

22 77 + 50 50

x)

13 12 + 3 3

15 34 + y) 10 10

© La Citadelle

19

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The Book of Fractions

Iulia & Teodoru Gugoiu

Adding mixed numbers with like denominators Example 1. 2 3 + 3 4 = ( 2 + 3) 3 + 4 = 5 7 = 6 2

1. To add mixed fractions with like denominators, add separately the wholes and separately the numerators, and keep the denominator unchanged:

5

5

5

5

5

2. In the same way you can add whole and mixed numbers.

n n n +n w1 1 + w2 2 = ( w1 + w2 ) 1 2 d d d

Example 2. 2 + 3

1 0 1 0 +1 1 = 2 + 3 = (2 + 3) =5 4 4 4 4 4

F24. Add the mixed numbers. Write the result as a mixed number in standard form or as a whole number: 1 1 1 1 1 2 1 3 3 2 a) 1 + 2 b) 1 + 2 c) 1 + 2 d) 1 + 3 e) 2 + 1 2 2 3 3 4 4 5 5 6 6 3 2 f) 2 + 3 7 7

2 3 g) 2 + 3 8 8

2 1 h) 3 + 0 9 9

3 7 k) 2 + 5 5

4 6 l) 4 + 2 7 7

m) 2

2 8 p) 5 + 2 9 9

q) 2

22 11 +5 10 10

v) 2

10 41 +5 30 30

u) 2

21 12 +1 19 19

i) 2

5 3 + 10 10

j) 3

5 6 +2 15 15

1 12 +3 10 10

n) 2

12 11 +5 20 20

o) 3

8 8 +3 11 11

r) 3

11 22 +2 30 30

7 9 s) 3 + 2 3 3

t) 2

15 25 +3 10 10

w) 3

2 22 +1 23 23

x) 4

33 22 +2 35 35

y) 1

15 95 +2 100 100

F25. Add the wholes and the mixed numbers. Write the result as a mixed number in standard form or as a whole number. 1 2 1 2 5 a) 2 + 1 b) 1+ 2 c) 1+ 3 d) 2 + 5 e) 5 + 2 2 3 4 5 6 f)

5+3

9 7

g)

5 7 +6 8

k)

9 3 +2 5

l)

5 +1

p)

9

19 +9 9

q)

20 + 10

u)

1+ 9

19 19

v)

4

© La Citadelle

h)

8 7

m) 1

100 10

14 +5 30

10 9

i)

1

11 +3 10

n)

11 30

40 23

3+

r)

11 + 3

w)

2+2

20

1 + 11 10

j)

0+2

0 15

2+2

22 20

o)

13 + 13

s)

9+3

10 3

t)

4

x)

1+ 1

37 35

y)

111 + 1

13 11

4 +4 10 111 100

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The Book of Fractions

Iulia & Teodoru Gugoiu

Adding more than two like fractions 1. To add more than two fractions or whole numbers, start to add in order, from left to right. 2 5 æ 2ö 5 2 5 7 3 1 + 3 + = ç1 + 3 ÷ + = 4 + = 4 = 5 4 4 è 4ø 4 4 4 4 4

Example 1.

2. Because the addition is a commutative operation, the order in which you add the fractions is not important. So, group them conveniently. 2 3 1 2 2ö æ 2 1ö æ 3 + 2 +1 + 2 +1 = 2 + ç +1 ÷ + ç 2 +1 ÷ = 2 + 2 + 4 = 8 3 5 3 5 3 3 5 5 è ø è ø

Example 2.

F26. Add the fractions: a) 1 + 3 + 3 1 2

f)

2

1 2 3 1+ + + 3 3 3

1 2 3 + + 4 4 4

g)

2

7 3 17 + +2 10 10 10

3 5

4 5

l)

1 5 2 1 + +1 9 9 9

1 9

7 9

q)

1

v)

1 3 5 3 + 2 +1 4 4 4

k) 1 + + 1

p) 1 + + 2

1 9

b)

4 9

u) 0 + + 1

8 9

1 3 5 + +2 10 10 10

e)

1 2 3 4 +1 + 2 + 3 5 5 5 5

3 7 3 +2 + 50 50 50

j)

1 3 5 7 1 +2 +3 +4 6 6 6 6

3 5 7 + +1 12 12 12

o)

1 3 5 7 9 + + + + 11 11 11 11 11

21 31 +3 100 100

t)

2

1 3 5 +1 + 0 10 10 10

y)

1 2 3 4 5 6 + + + + + 7 7 7 7 7 7

c)

3 5 1 1 + +2 +2 4 4 4

d)

1 1 1 3 + 2 +1 2 2 2

h)

5 3 2 +3+ 2 2

i)

2

3 3

n)

2

1 3 5 1 +2 +5 2 2 2

s)

1+ 2

x)

2

1 3

2 3

m) 1 + 2 + 3

r)

3 3

2 3

w) 3 + 2 + 1

1 3

5 10 15 20 +1 + + 3 10 10 10 10

F27. Add the whole and the mixed numbers. Write the result as a mixed number in standard form or as a whole number. a)

1+

1 3 + 2 2

b)

f)

1+

2 4 +2+ 3 3

g) 1 +

2 5

k) 1 + 1 + 2

l)

2 1 +2+ 4 4

7 +2 10

2 3 2 +1 + 3 9 9

p)

7 8 2 +1 +1 9 9

q) 1 + 2

u)

2 7 0 + 2 +1 9 9

v) 3 5 + 1 5 + 2

© La Citadelle

4

5 7 +3 10 10

4

c) 1 + 1 + 3 + 2

d)

1 1 + +2 2 2

h) 7 + 2 + 5

i)

1

4

4

2

2

4 3

2 3

7 11 +3+ 50 50

n) 1 5 + 1 7 + 2

m) 2 + 1 + 2

12

12

11 22 + 2 + +1 100 100

r) 1 3 + 2 5 + 5 7 + 1

s)

w) 2 4 + 1 1 + 2

x) 1 9 + 2 7 + 0

2

3

2

2

3

21

3

10

10

e)

3 3 +1+ + 2 5 5

9 6

7 6

j) 1 + + + 2

o) 1 2 + 2 + 3 4 + 2 + 1 8 11

t) 1

11

11

4 7 +3+ + 2 10 10

y) 2 9 + 1 7 + 1 + 3 5 10

10

10

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding equivalent fractions 1. Two fractions are considered equivalent if they represent the same part of the whole. We’ll see later that equivalent fractions are equal in value and correspond to the same decimal number. Example 1. The shaded region can be expressed as: So these fractions are considered equivalent (equal), and you can write: 1 or 2 or 3 6 1 2 3 6 or = = =

2

4

6

12

2

4

6

12

F28. For each image find the equivalent fractions that correspond to the shaded part:

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

u)

v)

w)

x)

y)

© La Citadelle

22

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The Book of Fractions

Iulia & Teodoru Gugoiu

Finding equivalent fractions There are two methods to find equivalent fractions. Method 1. Multiply both the numerator and the denominator by the same number, according to the formula:

Method 2. Divide both the numerator and the denominator by a common factor, according to the formula: n n¸a

Example 1.

Example 2.

n n´a = d d ´a

d

1 1´ 2 2 1 1´ 3 3 3 ´ 5 15 = = = = = = 2 2´ 2 4 2 2 ´ 3 6 6 ´ 5 30 So: 1 = 2 = 3 = 15 2 4 6 30

=

d ¸a

6 6¸2 3 3¸3 1 = = = = 12 12 ¸ 2 6 6 ¸ 3 2 6 3 1 So: = = 12 6 2

F29. Find at least three equivalent fractions by using the method 1: a)

1 2

b)

1 3

c)

2 3

d)

1 4

e)

3 4

f)

1 5

g)

3 5

h)

1 6

i)

5 6

j)

1 7

k)

2 7

l)

1 10

m)

1 100

n)

5 12

o)

2 11

p)

2 9

q)

3 7

r)

5 11

s)

2 15

t)

4 5

u)

5 7

v)

1 30

w)

3 50

x)

3 200

y)

3 1000

F30. Find equivalent fractions by using the method 2: a)

2 4

b)

3 9

c)

4 16

d)

5 25

e)

6 36

f)

10 100

g)

4 12

h)

20 25

i)

4 6

j)

15 20

k)

30 45

l)

25 40

m)

n)

60 90

o)

60 150

p)

20 35

q)

75 120

r)

66 154

s)

56 420

t)

u)

27 81

v)

63 77

w)

22 121

x)

225 625

64 y) 1024

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18 60

23

32 128

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Iulia & Teodoru Gugoiu

The Book of Fractions

Simplifying fractions To simplify (or reduce) a fraction means to find the equivalent fraction having the simplest form (in lowest terms). Method 1. You can simplify a fraction by repetitive division of the numerator and the denominator by a common factor.

n n¸a = = ... where a is a common divisor of n and d d d ¸a

Method 2. You can simplify a fraction by division of the numerator and denominator by the Greatest Common Factor (GCF). n n ¸ GCF = = fraction in lowest terms

d

d ¸ GCF

To find the GCF build the factor trees for the numerator and denominator. 24 = 23 ´ 31 Example 2. 24

Example 1.

12

2

2

15

2 3

GCF = 2 2 ´ 31 = 12

30

2 6

2

24 24 ¸ 2 12 12 ¸ 2 6 6¸3 2 = = = = = = 60 60 ¸ 2 30 30 ¸ 2 15 15 ¸ 3 5

60 = 2 2 ´ 31 ´ 51

60

3

5

24 24 ¸ 12 2 = = 60 60 ¸ 12 5

F31. Write each fraction in lowest terms by using the method 1: a)

6 72

b)

18 42

c)

50 75

d)

32 128

e)

60 80

f)

32 160

g)

54 90

h)

22 132

i)

200 240

j)

21 147

k)

36 126

l)

75 350

m)

105 135

n)

512 4096

o)

24 132

p)

50 225

q)

48 112

r)

60 264

s)

48 360

t)

u)

180 252

v)

126 270

w)

264 600

x)

45 300

y)

336 480

135 270

F32. Write each fraction in lowest terms by using the method 2: a)

18 24

b)

48 60

c)

54 81

d)

90 120

e)

56 72

f)

28 98

g)

32 60

h)

48 120

i)

28 52

j)

64 80

k)

30 45

l)

25 40

m)

18 60

n)

60 75

o)

60 150

p)

20 45

q)

75 120

r)

66 154

s)

56 420

t)

u)

27 81

v)

630 770

w)

22 121

x)

225 625

y)

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24

32 128 64 1024

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The Book of Fractions

Iulia & Teodoru Gugoiu

Checking fractions for equivalence Given two or more fractions, you can check whether or not they are equivalent (equal). 2. For two fractions you can use the cross-multiplication method. If the cross-products you get are equal then the two fractions are equivalent.

1. Express each fraction in lowest terms and then compare them. If you get the same fraction, the original fractions are equivalent. Example 1.

24 32

In lowest terms they are: So:

24 15 = 32 20

3 4

and 15

and

20 and

24 21 ¹ 32 30

3 4

and

21 30

a b

7 10

and

c d

are equivalent if

a´d = b´c

Example 2.

3 4

15 21 ¹ 20 30

and

21 are not equivalent because 3 ´ 30 ¹ 4 ´ 21 30

F33. Check if the fractions are equivalent (use the lowest terms method): a)

2 3 ; 4 6

b)

2 3 ; 6 9

c)

2 6 ; 3 8

d)

4 8 ; 6 12

e)

5 6 ; 10 18

f)

30 48 ; 75 120

g)

30 20 ; 35 24

h)

24 15 ; 56 35

i)

9 15 ; 25 40

j)

40 30 ; 70 54

k)

25 15 ; 80 50

l)

100 30 ; 220 66

m) 14 ; 56 24 96

n)

15 40 ; 40 104

o)

25 40 ; 45 75

p)

10 16 20 ; ; 15 24 30

q)

6 18 25 ; ; 10 30 40

r)

25 5 40 ; ; 30 6 48

s)

10 6 12 ; ; 21 15 35

t)

60 15 25 ; ; 96 24 40

u)

1 4 7 5 ; ; ; 2 8 14 10

v)

3 8 10 12 ; ; ; 4 10 12 14

w) 33 ; 21 ; 15 ; 3 44 28 20 4

x)

1 2 4 8 ; ; ; 2 4 8 16

y)

1 2 3 4 ; ; ; 2 3 4 5

F34. Check if the fractions are equivalent (use the cross-multiplication method): a) 1 ; 2 2 3

b)

2 4 ; 4 8

c)

6 4 ; 12 10

d)

12 15 ; 16 20

e)

12 4 ; 35 12

5 6 ; 4 5

g)

9 3 ; 12 4

h)

4 12 ; 10 30

i)

5 7 ; 7 5

j)

30 48 ; 35 56

k) 40 ; 72 45 80

l)

24 36 ; 40 60

m) 40 ; 49 64 81

n)

5 2 ; 20 8

o)

4 9 ; 9 20

p) 1 3 ; 9 4 5

q)

125 200 ; 15 24

r)

11 13 ; 13 15

s)

65 40 ; 85 51

t)

105 15 ; 133 19

u) 3 ; 9 ; 15 4 12 20

v)

4 5 6 ; ; 5 6 7

w) 4 ; 8 ; 12 8 12 16

x)

12 18 30 ; ; 20 30 50

y)

10 25 40 ; ; 14 35 56

f)

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25

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The Book of Fractions

Iulia & Teodoru Gugoiu

Equations with fractions Method 1. You can solve simple equations with fractions if you use two properties of equivalent fractions:

n n¸a = d d ¸a

Method 3. This is the best method for solving simple equations with fractions: To get a term of a fraction, multiply the adjacent terms and divide by the opposite term.

n n´a = d d ´a

Example 1:

x b = a c

Solution:

3 ? = 16 48

3 3´ 3 9 = = 16 16 ´ 3 48

a´b c

Example 3 (the result is a whole number):

Method 2. When applying the method 1, sometimes you need first to express the fraction in the lowest terms: : Example 2: Solution:

4 ? = 8 10

x=

Û

x 2 = 15 3

Û

x=

15 ´ 2 30 = = 10 3 3

Example 4 (the result is a fraction):

x 3 = 2 7

4 1 1´ 5 5 = = = 8 2 2 ´ 5 10

Û

x=

2´3 6 = 7 7

F35. Find the unknown factor of the fraction using the method 1:

a) f) k)

1 = 2 8 6 30 = 40

5

=

25 75

b) 3

g) l)

=

6 9

12 4 6 18 = 18 =

c) h) m)

2 = 4 32 8 2=

d)

3 = 10 100

n)

i)

3

e)

3 12 = 5

j)

8 40 = 12

9 = 16 128

o)

11 12 = 5

=

3=

12 16 5

F36. Find the unknown factor of the fraction using the method 2:

a)

2 = 4 6

b)

3 7 = 9

c)

8 10 = 12

d)

12 = 28 21

e)

f)

9 = 12 20

g)

12 24 = 15

h)

12 16 = 42

i)

=

18 48

j)

72

3

=

60

5 10

=

40 56

F37. Solve for x using the method 3 (see example 3 above):

a)

1 x = 3 6

b)

2 10 = 3 x

c)

3 12 = x 20

d)

4 20 = 5 x

e)

3 15 = x 35

f)

5=

15 x

g)

4 3 = x 6

h)

x 15 = 16 20

i)

20 15 = x 18

j)

16 24 = x 45

k)

45 40 = 72 x

l)

6=

x 6

m)

30 80 = x 48

n)

40 x = 25 30

o) 28 = x 20 25

F38. Solve for x using the method 3 (see example 4 above):

a)

1 x = 2 3

b)

2 x = 5 3

c)

4 5 = 3 x

d)

1 5 1 = 2 x

e)

9 x = 16 3

f)

6 x = 7 8

g)

12 x = 16 5

h)

20 3 = 24 x

i)

16 x = 64 2

j)

2

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26

20 x = 35 15

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The Book of Fractions

Iulia & Teodoru Gugoiu

Adding fractions with unlike denominators To add fractions with different denominators, you must first replace them with equivalent fractions having the same denominators. Method 1. This is a general method that works for two fractions and uses cross-multiplication according to the formula:

Method 2. First, express the fractions in lowest terms and then write down equivalent fractions by multiplication until you get the lowest common denominator: Example 2: 2 3

a c a´d + b´c + = b d b´d

+ =? 4 15

2 1 3 4 5 = = = = = ... 4 2 6 8 10

Example 1:

2 3 2 ´ 4 + 5 ´ 3 8 + 15 23 3 + = = = =5 5 4 5´ 4 4 4 4

So:

3 1 2 = = = ... 15 5 10

2 3 5 2 7 + = + = 4 15 10 10 10

F39. Add the fractions using the method 1: a)

1 1 + 2 3

b) 2 + 3 3 4

c)

1 2 + 4 5

d) 1 + 1 5 6

e) 2 + 3 5 7

f)

3 5 + 7 8

g) 3 + 5 8 9

h)

5 3 + 9 10

i)

3 1 + 10 11

j)

1 2 + 10 11

k)

3 4 + 10 15

l) 10 + 3 3 10

m)

5 2 + 9 3

n)

1 2 + 10 15

o)

3 4 + 15 20

p)

1 2 1 + 2 3

q) 2 + 2 4 3 5

r)

5 3 1 +2 3 6

s) 110 + 11 11 10

t) 1 2 + 2 1 5 4

u)

1 4 + 20 16

v) 1 3 + 8 40 25

w)

5 3 +1 12 4

x) 2 1 + 1 3 6 8

y) 1 1 + 3 16 32

e)

6 1 + 10 7

F40. Add the fractions using the method 2: a)

1 2 + 2 3

b) 2 + 1 6 4

c)

3 2 + 4 10

d) 2 + 5 5 6

f)

8 3 + 14 8

g) 4 + 3 8 9

h)

5 5 + 9 10

i)

2 3 + 10 9

j)

5 2 + 10 12

k)

2 5 + 10 15

l) 10 + 3 3 12

m)

5 2 + 9 3

n)

1 2 + 10 15

o)

3 6 + 15 20

p)

1 12 1 + 2 18

q) 1 2 + 2 3 3 5

r)

4 3 1 +2 3 6

s) 1 2 + 8 3 10

t) 1 3 + 2 3 5 4

u)

4 4 + 20 16

v) 1 5 + 5 40 25

w)

4 4 +1 12 16

x) 2 2 + 1 6 6 8

y) 1 5 + 5 16 48

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27

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The Book of Fractions

Iulia & Teodoru Gugoiu

Adding fractions with unlike denominators using the LCD method 1. First find the Least (Lowest) Common Denominator (LCD), and then add fractions. Example 1:

2. You can use the LCD method to add more than two fractions. Example 2:

5 2 + =? 24 30 24 = 23 ´ 31

1 1 1 + + =? 12 45 40

1

1

24

30

12

2

15

2

2

3

6 2

5

3

1

30 = 2 ´ 3 ´ 5

12 = 2 2 ´ 31

120 ¸ 24 = 5 120 ¸ 30 = 4

LCD = 23 ´ 31 ´ 51 = 120

12

2

45

2

15

3

6

3

3

45 = 32 ´ 51

40

20

2

10

2

5

2

40 = 23 ´ 51

LCD = 23 ´ 32 ´ 51 = 360

5 2 5´ 5 2´ 4 25 8 33 11 + = + = + = = 24 30 24 ´ 5 30 ´ 4 120 120 120 40

1 1 1 1´ 30 1´ 8 1´ 9 47 + + = + + = 12 45 40 12 ´ 30 45 ´ 8 40 ´ 9 360

F41. Add the fractions using the LCD method (see example 1 above): 1 3 a) 1 + 5 b) 2 + 3 c) d) 4 + 1 + 4 6 6 16 16 20 15 12

e)

3 3 + 10 14

f)

3 5 + 14 4

g)

3 5 + 16 18

h)

5 2 + 12 9

i)

3 1 + 10 22

j)

1 2 + 6 33

k)

3 4 + 10 15

l)

7 3 + 12 20

m)

5 5 + 9 30

n)

1 2 + 10 25

o)

2 4 + 15 25

p)

1

q)

1 3 +2 30 50

r)

1

s) 1 1 + 1 15 50

t)

1

u)

3 3 + 20 16

v)

1

7 3 + 40 25

w)

7 3 +1 12 40

x) 2 1 + 1 3 14 84

1 3 1 + y) 48 64

1 7 + 25 30

5

5 3 +2 36 16

3 1 +2 50 45

F42. Add the fractions using the LCD method (see example 2 above): a)

1 2 3 + + 2 3 4

b)

1 5 3 + + 4 6 8

c)

1 3 7 + + 6 8 10

d)

1 5 7 3 + + + 4 6 12 16

e)

3 3 1 + + 4 10 12

f)

3 1 1 + + 10 15 30

g)

5 2 2 + + 12 9 14

h)

1 1 1 1 1 + + + + 2 3 4 5 6

i)

3 4 3 + + 10 15 16

j)

1 3 2 + + 12 20 15

k)

1 1 1 + + 9 30 36

l)

1 1 5 2 3 + + + + 4 6 12 15 20

m)

1 5 3 + + 3 6 10

n)

3 1 1 + + 10 25 6

o)

1 2 2 + + 15 25 35

p)

1 3 1 5 3 + + + + 15 4 20 32 50

q)

1 1 1 1 + + + 2 3 4 5

r)

1 1 1 1 + + + 3 4 5 6

s)

1 3 7 3 + + + 5 10 15 20

t)

1 3 1 3 1 + + + + 5 10 15 20 25

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The Book of Fractions

Iulia & Teodoru Gugoiu

Understanding the subtraction of fractions with like denominators 1. When you subtract two fractions, you subtract the parts of the whole they represent. Example 1.

3 2 1 - = 4 4 4

So, by taking 2 quarters away from 3 quarters you get 1 quarter.

F35. Subtract the fractions that correspond to the shaded regions using the pattern: all shaded parts - all light shaded parts=all dark shaded parts

a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

0

u) © La Citadelle

v)

w) 29

1

x)

y) www.la-citadelle.com

The Book of Fractions

Iulia & Teodoru Gugoiu

Subtracting fractions with like denominators 1. To subtract proper or improper fractions with like denominators, subtract the numerators and keep the denominators unchanged.

Example 1.

2. To subtract mixed numbers with like denominators, first change mixed numbers to improper fractions, and then subtract the numerators. Keep the denominators unchanged.

n1 n2 n1 - n2 - = d d d 3 2 1 - = 5 5 5

1 2 7 5 2 2 -1 = - = 3 3 3 3 3

Example 2.

F44. Subtract the fractions (write the results in lowest terms): a)

2 1 4 4

b)

4 2 3 3

c)

4 1 5 5

d)

10 3 11 11

e)

5 2 6 6

f)

5 2 7 7

g)

13 11 19 19

h)

19 17 10 10

i)

23 13 35 35

j)

7 3 40 40

k)

5 1 12 12

l)

7 1 54 54

m)

7 1 9 9

n)

11 7 32 32

o)

23 9 7 7

p)

3 1 4 4

q)

15 3 16 16

r)

13 7 24 24

s)

19 7 40 40

t)

5 1 1000 1000

v)

7 1 30 30

w)

17 7 50 50

x)

13 2 33 33

y)

13 7 60 60

u)

13 3 100 100

F45. Subtract the mixed numbers (write the results in lowest terms): 1 2 a) 1 3 3

b)

1 3 2 2 2

5 f) 3 - 1 7

g)

2-

l)

3

7 3 14 14

q)

5

15 5 16 16

v)

5

7 1 -2 30 30

k) 2

5 7 12 12

1 3 p) 2 4 4

u) 3

13 17 -2 100 100

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11 19

d)

3

1 3 -2 11 11

e)

5 2 5 -3 6 6

i)

2

3 13 35 35

j)

4

1 5 m) 5 - 2 9 9

n)

2

11 17 -1 32 32

o)

2 9 3 -2 7 7

r) 1

3 7 24 24

s)

3

19 9 -1 40 40

t)

2

w) 3

13 7 -2 50 50

x)

3

1 4 -2 33 33

c) 2 4 - 1 3 5 5 h) 10

1 11 10 10

30

3 5 -2 40 40

5 3 -1 1000 1000

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y)

5

The Book of Fractions

Iulia & Teodoru Gugoiu

Subtracting mixed numbers with like denominators Method 1. To subtract mixed numbers with like denominators, subtract separately the wholes, and separately the numerators. Keep the denominator unchanged, according to the rule:

Method 2. If the numerator of the first fraction is less than the numerator of the second fraction, you must rewrite the first fraction as an improper fraction by exchanging one whole.

n1 n n -n - w2 2 = ( w1 - w2 ) 1 2 d d d

w1

Example 2:

1 2 4 2 4-2 2 3 - 1 = 2 - 1 = (2 - 1) =1 3 3 3 3 5 5

Example 1:

3 1 3 -1 2 3 - 1 = (3 - 1) =2 5 5 5 5

F46. Subtract the mixed numbers using the method 1 (write the results in lowest terms). 2 1 a) 2 - 1 3 3

b)

3 1 5 -3 2 2

5 2 f) 5 - 2 7 7

g)

6

l)

k) 4

7 5 -1 12 12

3 1 p) 5 - 2 4 4

u) 5

47 27 -3 100 100

c)

3 1 10 - 2 5 5

11 1 -5 19 19

h)

5

5

7 3 -3 14 14

7 1 m) 3 - 2 9 9

q)

7

15 7 -3 16 16

r)

4

v)

9

17 11 -7 30 30

w)

5

10 5 -5 11 11

e)

4 1 4 -2 6 6

17 12 -2 35 35

j)

5

o)

5 2 4 -1 7 7

27 17 -5 40 40

t)

5

37 17 -2 1000 1000

11 5 33 33

y)

7

37 17 -5 60 60

d)

11

i)

3

n)

10

17 5 -2 24 24

s)

7

11 5 50 50

x)

2

9 1 -2 10 10

17 9 -7 32 32

17 7 -3 40 40

F47. Subtract the mixed numbers using the method 2 (write the results in lowest terms): a)

1 2 2 -1 3 3

b)

1 3 5 -2 7 7

c)

3 4 5 -2 5 5

f)

1 5 3 -1 9 9

g)

2-

11 19

h)

10

k)

2

5 7 12 12

l)

6

9 2 -1 14 14

2 5 m) 7 - 5 9 9

p)

5 -1

3 4

q)

5

7 17 16 16

r)

2

u)

6

v)

9

1 7 -3 30 30

w)

5

13 17 -2 100 100

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d)

4

2 5 -1 11 11

e)

2 5 7 -2 6 6

i)

2

3 13 35 35

j)

4

n)

7

11 27 -5 32 32

o)

3 6 8 -5 7 7

13 17 -1 24 24

s)

5

39 29 -2 40 40

t)

3

17 7 -1 1000 1000

3 17 50 50

x)

2

11 23 -1 33 33

y)

4

7 15 -1 60 60

1 11 10 10

31

3 5 -2 40 40

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The Book of Fractions

Iulia & Teodoru Gugoiu

Subtracting fractions with unlike denominators To subtract fractions with unlike denominators, first you must replace the fractions with equivalent fractions having like denominators. Method 1. This is a general method that works for two fractions and uses cross-multiplication according to the formula:

Method 2. First, express the fractions in lowest terms and then write down equivalent fractions by multiplication until you get the lowest common denominator:

2 3 - =? 4 15

Example 2:

a c a´d -b´c - = b d b´d

2 1 3 4 5 = = = = = ... 4 2 6 8 10

Example 1:

3 2 3 ´ 5 - 4 ´ 2 15 - 8 7 - = = = 4 5 4´5 20 20

So:

3 1 2 = = = ... 15 5 10

2 3 5 2 3 - = - = 4 15 10 10 10

F48. Subtract the fractions using the method 1: a)

1 1 2 3

b) 3 - 2 4 3

c)

2 1 5 4

d) 1 - 1 5 6

f)

5 3 8 7

g) 5 - 3 9 8

h)

5 3 9 10

i)

3 1 10 11

j)

2 1 11 10

k)

3 4 10 15

l)

m)

2 5 3 9

n)

2 1 15 10

o)

3 4 15 20

p)

1 2 1 2 3

q) 2 4 - 2 5 3

r)

5 3 1 -2 3 6

s) 110 - 11 11 10

t) 2 1 - 1 2 4 5

u)

4 3 16 20

v) 1 3 - 8 40 25

w)

3 5 1 4 12

x) 2 1 - 1 3 6 8

1 3 y) 116 - 32

e)

6 3 10 7

2 3 3 10

e) 3 - 2 7 5

F49. Subtract the fractions using the method 2: a)

2 1 3 2

b) 2 - 1 6 4

c)

3 2 4 10

d) 5 - 2 6 5

f)

8 3 14 8

g) 4 - 3 8 9

h)

5 5 9 10

i)

3 2 9 10

j)

5 5 10 12

k)

5 2 15 10

l)

m)

2 5 3 9

n)

3 2 10 15

o)

6 3 20 15

p)

1 12 1 2 18

q) 3 2 - 2 3 3 5

r)

4 3 2 -1 3 6

s) 1 2 - 8 3 10

t) 3 3 - 2 3 5 4

u)

7 4 20 16

v) 1 5 - 5 40 25

w)

2

x) 2 2 - 1 6 6 8

5 5 1 y) 16 - 48

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2 5 3 12

4 4 -1 12 16

32

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The Book of Fractions

Iulia & Teodoru Gugoiu

Subtracting fractions with unlike denominators using the LCD method 1. First find the Least (Lowest) Common Denominator (LCD), then subtract fractions. Example 1. 24

5 2 =? 24 30 3

1

1

1

30

12

2

2. You can use the LCD method to add or subtract more than two fractions. Example 2.

2

3

6

24 = 2 ´ 3

2

1 1 1 + =? 12 45 40

15

2

12

5

2

45

15

3

6 2

40

3

3

20

2

2

3

1

30 = 2 ´ 3 ´ 5

2

1

12 = 2 ´ 3

120 ¸ 24 = 5 120 ¸ 30 = 4

LCD = 23 ´ 31 ´ 51 = 120

10

2

5

2

1

3

45 = 3 ´ 5

5

1

40 = 2 ´ 5

LCD = 23 ´ 32 ´ 51 = 360

5 2 5´ 5 2´ 4 25 8 17 = = = 24 30 24 ´ 5 30 ´ 4 120 120 120

1 1 1 1´ 30 1´ 8 1´ 9 29 + = + = 12 45 40 12 ´ 30 45 ´ 8 40 ´ 9 360

F50. Subtract the fractions using the LCD method (see the example 1 above): 3 1 a) 5 - 1 b) 2 - 3 c) d) 4 - 1 e) 3 - 3 6 4 6 16 20 16 15 12 10 14 f)

5 3 4 14

g)

5 3 18 16

h)

5 2 12 9

i)

3 1 10 22

j)

1 2 6 33

k)

3 4 10 15

l)

5 3 12 20

m)

4 5 9 30

n)

1 2 10 25

o)

4 2 25 15

p)

1

q)

2

1 3 -1 30 50

r)

1

s)

2

t)

1

u)

5 3 20 16

v)

1

3 3 40 25

w)

1 3 12 40

1 7 25 30

5 1 36 16

1 1 -2 15 50

x) 2 1 - 1 3 14 84

3 2 -1 50 45

y) 1 1 - 3 48 64

F51. Add and subtract the fractions using the LCD method (see the example 2 above): a)

1 2 1 + 2 3 4

b)

3 1 3 - + 4 6 8

c)

5 3 3 - 6 8 10

d)

3 1 7 3 - + 4 6 12 16

e)

3 3 1 - + 4 10 12

f)

3 1 1 - 10 15 30

g)

1 1 1 + 12 9 14

h)

1 1 1 1 1 - + - + 2 3 4 5 6

i)

3 2 3 - + 10 15 16

j)

5 3 2 - 12 20 15

k)

1 1 1 + 9 30 36

l)

1 1 5 2 3 - + - + 4 6 12 15 20

m)

2 1 3 - 3 6 10

n)

3 2 1 - 10 25 6

o)

2 3 1 - + 15 25 35

p)

13 3 1 1 7 - - - 15 4 20 32 480

q)

1 1 1 4 2 - -1 2 3 4 5

r)

1 1 1 1 + - 3 4 5 6

s)

2 3 7 3 - + 5 10 15 20

t)

2 3 1 3 2 - - + 5 10 15 20 25

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The Book of Fractions

Iulia & Teodoru Gugoiu

Order of operations (I) 1. Additions and subtractions are operations of the same priority, so the order in which they are done is not important. By convention, these operations are done one by one from left to right. Example 1.

3. By convention the order in which the brackets appear is { [ ( ) ] }. Example 3: 3 ìï 4 é æ 1 1 öù üï 3 ì 4 é 1 ùü + í - ê1 - ç - ÷ú ý = + í - 1 ý= 2 ïî 3 ë è 3 4 øû ïþ 2 î 3 êë 12 úû þ =

3 1 1 æ 3 1ö 1 7 1 5 - + = ç - ÷ + = + =1 2 3 4 è 2 3 ø 4 6 4 12 2. To change the order of operations, use: parentheses ( ), brackets [ ] or braces { }. An inner bracket has a greater priority. Example 2:

3 ì 4 11 ü 3 5 11 +í - ý= + =1 2 î 3 12 þ 2 12 12

4. One single type of brackets is enough to change the order of operations. Example 4:

1 æ æ 1 æ 1 1 ööö 1 æ æ 1 1 öö + ç1 - ç - ç - ÷ ÷ ÷ = + ç1 - ç - ÷ ÷ = 2 çè çè 2 è 3 4 ø ÷ø ÷ø 2 çè è 2 12 ø ÷ø

3 æ 1 1 ö 3 7 11 -ç + ÷ = - = 2 è 3 4 ø 2 12 12

=

1 æ 5ö 1 7 1 + ç1 - ÷ = + = 1 2 è 12 ø 2 12 12

F52. Solve each exercise by following the proper order of operations: a) 1 + 1 + 1 + 1 2 3 4 5 d) 1 - 1 + 1 - 1 + 1 2 3 4 5 6 2 1 1 g) 2 - æç + 1 ö÷ 3 è 3 3ø 1 2 1 5 1 j) 1 - æç + ö÷ - æç - ö÷ 2 è 3 2ø è6 2ø

b) 1 - 1 + 1 - 1 2 3 4 5

c) 1 + 2 1 - 1 1 + 4 1 - 3 1 + 5 2 3 4 5

7 æ 3ö e) 5 - çè1 - 5 ÷ø 1 æ1 1ö æ1 1ö h) 2 - çè 3 - 4 ÷ø - çè 5 - 6 ÷ø

k)

æ æ 1 1 1 öö l) 2 - ç1 - çç - æç - ö÷ ÷÷ ÷ ç ÷ è è 2 è 3 4 øøø

1 é 1 æ 1 1 öù - -ç - ÷ 2 êë 3 è 4 5 øúû

3 4 é 3 5 ù m) æç 8 - 2 ö÷ - ê5 - æç 3 - 1 ö÷ú 7 ø ë è 4 7 øû è 7

3 2 3 f) æç 2 - ö÷ - æç1 - ö÷ 4ø è 4 4ø è æ 1 1ö æ 1 1 1ö i) 1 + çè 2 - 3 ÷ø - çè 4 + 5 - 6 ÷ø

n)

1 éæ 1 1 ö æ 1 öù 1 + 2 - êç1 + ÷ - ç1 + 1÷ú 2 ëè 3 4 ø è 5 øû

3 é7 1 1 ù ì 1 é 1 æ 4 3 öù ü + ê - ( + )ú - í - ê - ç - ÷ú ý 5 ë10 2 5 û î 2 ë10 è 5 4 øû þ

p)

2 ìæ 2 5 ö éæ 2 5 ö æ 5 1 öù ü 2 - íç + ÷ - êç - ÷ - ç - ÷ú ý 9 îè 3 6 ø ëè 3 9 ø è 12 6 øû þ

ì é 3 1 7 3 ù 1ü 3 q) í3 - ê2 - æç1 - ö÷ + - æç - ö÷ú + ý + î ë è 8 ø 2 è 10 5 øû 10 þ 4

r)

ìé 3 æ 8 3 öù æ 3 öü æ 1 7 ö í ê - ç - ÷ ú + ç1 - ÷ ý - ç - ÷ îë 5 è 25 20 øû è 5 øþ è 4 50 ø

o)

æ1 æ4 æ3 4 3 2ö 1ö 1ö æ 1 2 1 ö s) 1 - ç - ç - çç - æç - ö÷ - ÷÷ - ÷ - ÷ + çç1 - - æç - ö÷ ÷÷ ç 2 ç 3 è 4 è 5 4 ø 3 ø 2 ÷ 5 ÷ è 2 è 3 4 øø è ø è ø ì 1 é 2 1 ù 1 ü ì é2 3 1 ù é 2 ù 3ü t) í2 - ê1 - æç - ö÷ú - æç1 - ö÷ý - í1 - ê - æç - ö÷ú - ê1 - æç1 - ö÷ú - ý î 2 ë è 3 2 øû è 3 øþ î ë 5 è 5 2 øû ë è 5 øû 10 þ ìé 1 1 1 ù é 1 1 1 1 ù ü ìé 1 1 1 1 ù é 1 1 1 1 ùü u) íêæç1 - ö÷ - æç - ö÷ú - êæç - ö÷ - æç - ö÷ú ý - íêæç - ö÷ - æç - ö÷ú - êæç - ö÷ - æç - ö÷ú ý îëè 2 ø è 2 3 øû ëè 3 4 ø è 4 5 øû þ îëè 2 3 ø è 3 4 øû ëè 4 5 ø è 5 6 øû þ

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The Book of Fractions

Iulia & Teodoru Gugoiu

Multiplying fractions 1. To multiply two proper or improper fractions, you multiply the numerators first and then the denominators, according to the rule:

2. To multiply mixed numbers, first convert them to improper fractions. Example 2:

1 5 3 5 3 ´ 5 15 7 1 ´ = ´ = = =1 2 4 2 4 2´ 4 8 8

n1 n2 n1 ´ n2 ´ = d1 d 2 d1 ´ d 2 Example 1:

3. To multiply a whole and a fraction, rewrite the whole as a fraction. Example 3: 4 2 4 2´ 4 8 3

2 4 2´ 4 8 ´ = = 3 5 3 ´ 5 15



5

= ´ = = =1 1 5 1´ 5 5 5

F53. Multiply the proper or improper fractions (write the results in lowest terms): a) 1 ´ 3 2 4

b)

1 2 ´ 2 3

c) 1 ´ 3 4 5

d) 2 ´ 15 5 4

e) 3 ´ 1 2 4

4 3 ´ 3 2

g)

1 2 ´ 2 4

h) 1 ´ 1 2 4

i) 3 ´ 5 4 6

j)

k) 1 ´ 8 2 9

l)

3 12 ´ 6 15

m) 5 ´ 22 11 5

n) 4 ´ 21 7 5

o) 3 ´ 2 4 6

f)

1 5 ´ 10 20

F54. Multiply the mixed numbers or fractions (write the results in lowest terms): a) 1 1 ´ 1 2 2

b)

f) 1 1 ´1 3 5 12 k) 1 1 ´ 2 2 2 3 p) 1 1 ´ 2 2 2 9

1 1 ´2 2 3

c) 1 1 ´ 5 5 3

d) 2 2 ´ 1 5 3

e) 2 ´1 3 3 4

g) 1 1 ´ 2 1 2 2

h) 1 2 ´ 2 2 3 5

i) 1 1 ´ 2 3 4 2

j)

l)

5 2 1 ´2 6 5

m) 1 1 ´ 5 10 11

n) 1 3 ´ 14 7 5

o) 2 1 ´1 2 4 6

q)

1 3 2 ´ 6 13

r) 1 4 ´ 4 2 11 5

s) 1 3 ´ 14 7 5

t)

2

2 1 ´1 11 12

1 5 2 ´1 4 3

F55. Multiply the wholes and fractions (write the results in lowest terms): a) 3´ 1 4

b)

1 ´2 2

c) 2´ 2 5

d) 2 ´ 5 5

e) 3´ 5 4

f) 5´ 3 15

g)

1 ´6 2

h) 3´ 2 9

i) 3 ´ 6 4

j) 15´ 4 20

k) 1 1 ´ 3 2

l)

5´ 2

m) 3´ 2 5 6

n) 1 4 ´14 7

o) 2´1 1 6

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2 15

35

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The Book of Fractions

Iulia & Teodoru Gugoiu

More about multiplying fractions 1. Although the most common multiplication operator is “x”, there are two other acceptable operators: “of” and “×” Example 1:

2. When multiplying fractions, first try to cancel out the common factors between numerators and denominators. Example 3: 2

1 3 1 3 1´ 3 3 of = ´ = = 2 5 2 5 2 ´ 5 10 Example 2:

1 4 1´ 2 2 ´ = = 2 5 1´ 5 5 1

3. This process of cancellation can be applied more than one time. Example 4:

1 3 1× 3 3 × = = 2 5 2 × 5 10

1 2 5 4 15 2 ´1 2 ´ = = 9 10 3 ´1 3 3 2 1

F56. Multiply the fractions (write the results in lowest terms): a) 2 of 1 3 2

b)

2 3 of 3 4

c)

1 1 of 5 5

d)

2 4 of 3 5

e)

1 1 of 5 3

f) 5 of 3 2 4

g)

3 3 of 2 2

h)

4 5 of 5 4

i)

1 2 of 2 4

j)

2 9 of 3 6

k) 2 of 90 3

l)

10 of

m) 2 of 3

n)

1 1 1 of 3 4

o) 2 1 of 1 1 5 2

1 2

F57. Multiply the fractions (write the results in lowest terms): a) 1 × 2 2 3

b)

2 3 × 3 4

c)

3 5 × 5 3

d)

1 2 1 × 2 5

e) 1 1 × 2 1 2 3

f) 3 × 14 7 6

g)

1 5 × 10 2

h)

4 15 × 5 20

i)

3 12 × 5 9

j)

1 1 × 4 5

k) 1 × 2 2 1

l)

1 1 × 2 2

m)

5 18 × 12 15

n)

10 22 × 12 30

o)

6 × 30 21

F58. Cancel out the common factors and then multiply the fractions: a) 1 ´ 15 5 4

b)

2 3 ´ 3 2

c)

2 10 ´ 5 4

d)

3 14 ´ 7 9

e)

5 6 ´ 3 10

f) 5´ 3 15

g)

2 ´8 6

h)

3 12 ´ 4 9

i)

33 40 ´ 4 44

j)

4 121 ´ 22 20

k) 1 1 ´ 2 2 3

l)

2 6 1 ´1 3 15

m) 15 ´ 2 4 7 5

n)

1 1 ´ 21 7

o) 1 1 ´ 2 2 2 3

p) 1 1 ´ 2 2 2 9

q)

1 3 2 ´ 6 13

r)

4 5 ´ 10 8

s)

1

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1

36

3 25 ´ 20 23

t)

2

1 4 ´2 40 18

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The Book of Fractions

Iulia & Teodoru Gugoiu

The order of operations (II) 1. Because multiplication is a commutative operation, the order in which you multiply them is not important. By convention, the order is from left to right. Example 1: 4 2 4 1 æ 2 4ö 1 8 1 4 ´ ´ = ç ´ ÷´ = ´ = 3 5 2 è 3 5 ø 2 15 2 15 1 2. Do not forget to cancel out the common factors, before multiplying the fractions . Example 2:

3. If the expression contains addition, subtraction and multiplication, do the multiplication first. Example 3: 1 3 5 1 æ 3 5ö 1 5 9 1 + ´ = +ç ´ ÷ = + = =1 2 4 6 2 è 4 6ø 2 8 8 8

4. If the expression contains brackets, replace the brackets with the result of the operation(s) inside the brackets. Example 4:

1 1 1 3 2 2 15 6 1 ´ ´ = 5 9 8 2 1 3 4 1 2

1 æ 1 3 ö 5 5 5 25 =1 ç + ÷´ = ´ = 24 è 2 4 ø 6 4 6 24

F59. Multiply the fractions (write the results in lowest terms): a) 1 ´ 2 ´ 3 b) 1 ´ 2 ´ 3 c) 4 ´ 3 ´ 3 d) 1 ´12 ´ 3 2 3 4 3 5 2 9 8 4 3 4

e) 2 ´ 1 ´ 2 5 3 4

f) 1 1 ´ 2 ´ 2 3 2 3 4

j) 1 1 ´ 2 ´ 15 5 3 8

k) 1 ´ 4 ´ 10 ´ 15 2 5 16 8

g) 2 1 ´ 2 ´1 3 2 5 4 l)

h) 2 1 ´1 2 ´ 3 2 3 5

1 2 3 4 5 ´ ´ ´ ´ 2 3 4 5 6

i) 2 1 ´ 2 ´ 2 4 9

m) 1 ´ 4 ´ 8 ´ 10 ´ 14 2 6 10 12 16

n)

1 3 5 7 9 ´ ´ ´ ´ 2 4 6 8 10

F60. Find the value of each expression (write the results in lowest terms): a) 1 + 1 ´ 4 b) 1 ´ 2 + 5 c) 2 - 1 ´ 1 d) 1 ´ 2 - 1 e) 4 ´ 1 ´ 2 - 1 2 2 5 4 3 6 3 2 3 2 3 6 5 3 4 10 f) 1 + 1 ´ 2 - 1 3 2 3 2

g) 1 - 1 ´ 2 + 1 2 2 3 3

h) 2 + 1 - 1 ´ 2 3 2 2 3

i) 3 ´ 15 + 5 - 2 5 9 6 3

j)

2 3 3 5 ´ + ´ 3 4 5 6

k) 1 ´ 2 - 2 ´ 3 2 3 3 8

l)

1 3 2 2 ´ ´ + 2 4 3 3

m) 2 - 1 ´ 1 ´ 3 3 2 3 4

n) 1 1 ´ 2 + 6 ´1 2 2 3 5 3

o) 1 - 2 ´1 3 + 1 3 ´ 15 7 4 5 8

F61. Find the value of each expression (write the results in lowest terms): a) æç 1 + 1 ö÷ ´ 4 è2

f)

3ø 5

æ1 1ö 3 1 ç + ÷´ è3 2ø 5 2

k) 1 1 ´ æç 1 - 1 ´ 2 ö÷ 4 è3

p)

2 5ø

b)

1 æ2 1ö ´ç + ÷ 5 è 3 6ø

c)

æ2 1ö 3 ç - ÷´ è 3 2ø 5

d) 3 ´ æç 1 - 1 ö÷

g)

1 1 æ2 1ö - ´ç + ÷ 2 7 è 3 2ø

h)

1 1ö 2 æ2 ç +1 - ÷´ 3 2 6ø 3 è

i)

l)

3 æ 3 2 1ö ´ç ´ + ÷ 5 è 4 3 3ø

m) æç 2 - 1 ´ 2 ö÷ ´ 3

n)

1 1 é 1 æ 2 3ö 1 1ù 2 1 ´ - ê1 ´ ç - ÷ + 1 ´ ú ´ 3 4 ë 2 è 3 4 ø 2 4û 3

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è3

q)

2 3ø 4

e)

æ 3 2ö æ 3 2ö ç + ÷´ç - ÷ è 4 3ø è 4 3ø

4 æ1 1ö 1 ´ç + ÷ 7 è3 4ø 5

j)

2 æ 1 1ö 6 ´ç + ÷´ 5 è 2 3ø 7

5 æ 2 6 2ö ´ç + ´ ÷ 11 è 3 5 3 ø

o)

æ 2ö æ 1 3ö 1 ç1 - ÷ ´ ç + ÷ ´ 1 è 5ø è 2 3ø 9

2 è2



ìï 3 é 2 1 æ 1 1 ö 1 ù é 5 æ 3 1 ö 1 ù üï 2 í ´ ê1 + 1 - ç + ÷ ´ 3 ú - ê ´ ç - ÷ - ú ý ´ 3 è 4 5 ø 3 û ë11 è 5 2 ø 4 û ïþ 7 ïî 2 ë 2

37

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The Book of Fractions

Iulia & Teodoru Gugoiu

Reciprocal of a fraction 1. The reciprocal of a fraction is the fraction obtained by interchanging its numerator and denominator. So, the reciprocal of

n d Example 1: The reciprocal of

3 4

d n

is

2. To find the reciprocal of a mixed number, express it as an improper fraction and then interchange the numerator and the denominator (invert the fraction). Example 2. The reciprocal of

3=

Example 3. The reciprocal of

4 3

is

3. To find the reciprocal of a whole number, express it as an improper fraction and then interchange (invert) the numerator and the denominator.

3 7 1 = 4 4

is

3 1

is

1 3

4. If you multiply a fraction by its reciprocal, the product is always 1. Thus, the reciprocal of a fraction is called also the multiplication inverse of that fraction. Example 4:

3 4 ´ =1 4 3

4 7

F62. Find the reciprocal of each fraction: a)

1 2

b)

2 3

c)

5 4

d)

7 3

e)

0 10

f)

7 6

g)

13 3

h)

2 4

i)

3 5

j)

3 0

F63. Find the reciprocal of each mixed number: a) 1

1 2

b) 2

2 3

c)

2

2 5

d) 1

1 7

e) 3

2 3

f) 3

3 3

g) 2

2 11

h) 2

5 7

i) 1

1 4

j) 3

3 4

F64. Find the reciprocal (multiplication inverse) of each whole number: a) 1

b) 2

c) 5

d) 100

e) 0

F65. Check if the pair of fractions are reciprocal: a)

1 2 and 2 1

1 3 f) 3 and 3 11

1 2 b) 1 and 2 4

c)

7 1 and 2 3 7

d)

1 and 3 3

e)

1 10

h)

3 5 and 5 2

i)

3 6 and 9 2

3 15 j) 1 and 5 24

g) 10 and

F66. Find the unknown fraction (f): 1 1 a) f ´ = 1 b) c) ´ f =1 4 2 f)



2 =1 7

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g) 7 ´ f = 1

h)



2 =1 5

2 f ´1 = 1 9

38

3 2 and 2 5 5

d) f ´ 5 = 1

e)

5 f ´1 = 1 4

3 i) 1 ´ f = 1 4

j)

f ´2

4 =1 5

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The Book of Fractions

Iulia & Teodoru Gugoiu

Dividing fractions 1. To divide two proper or improper fractions, change the divisor to its reciprocal and then multiply, according to the rule:

2. To divide mixed numbers, first change them to improper fractions. Example 2:

1 2 5 5 5 3 3 1 2 ¸1 = ¸ = ´ = = 1 2 3 2 3 2 5 2 2

n1 n2 n1 d 2 ¸ = ´ d1 d 2 d1 n2

3. To divide a whole number and a fraction, rewrite the whole as a fraction. Example 3:

In short, invert the divisor and multiply. Example 1:

2 4 2 5 5 ¸ = ´ = 3 5 3 4 6



4 2 4 2 5 5 1 = ¸ = ´ = =2 5 1 5 1 4 2 2

F67. Divide the proper or improper fractions (write the results in lowest terms): a) 1 ¸ 3 2 4 f)

4 12 ¸ 3 5

k) 1 ¸ 8 3 9

b)

1 1 ¸ 2 3

c)

2 4 ¸ 3 9

d)

2 8 ¸ 5 15

e) 3 ¸ 6 2 5

g)

5 3 ¸ 2 4

h)

1 1 ¸ 2 4

i)

10 5 ¸ 4 6

j)

l)

3 12 ¸ 6 15

m)

5 25 ¸ 11 22

n)

4 16 ¸ 7 21

o) 3 ¸ 6 4 16

1 3 ¸ 10 20

F68. Divide the mixed numbers or fractions (write the results in lowest terms): a) 2 1 ¸ 1 2 2

b) 1 1 ¸ 2 1 2 4

c)

1 1 1 ¸1 5 15

d)

2 3 2 ¸ 5 5

e) 2 ¸ 1 1 3 6

f) 1 1 ¸ 1 3 5 15

g) 1 1 ¸ 2 1 2 4

h)

2 1 1 ¸1 3 9

i)

1 3 1 ¸6 4 2

j) 2 3 ¸ 1 3 11 22

k) 1 1 ¸ 2 2 2 8

l)

5 2 1 ¸3 6 3

m) 1 2 ¸ 3 10 25

n)

3 10 1 ¸ 7 21

o) 2 1 ¸ 1 5 4 16

p) 1 1 ¸ 2 2 3 9

q)

1 1 2 ¸3 6 4

r)

2 5 ¸ 13 26

s)

3 8 1 ¸ 7 35

t) 2 1 ¸ 2 1 4 3

1

F69. Divide the wholes and fractions (write the results in lowest terms): a) 1¸ 1 2

b)

1 ¸2 3

c)



f) 5 ¸ 2 1 7

g)

1 ¸2 3

h)

3¸2

k) 1 1 ¸ 3 2

l)

5¸2

© La Citadelle

3 11

4 5 2 5

m) 13 ¸ 3 5 7 39

d)

5 ¸ 15 2

e) 32 ¸ 5 1 3

i)

3 ¸6 4

j) 15 ¸ 6 2 3

n)

4 1 ¸ 44 7

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The Book of Fractions

Iulia & Teodoru Gugoiu

Division operators There are four operators used to express the division operation between two numbers or fractions. 1. The operator is ¸ Example 1.

2. The operator is : Example 2.

3. The operator is (the division line)

3 9 3 10 2 ¸ = ´ = 5 10 5 9 3

_

1 2 = 1´4 = 2 3 2 3 3 4

Example 3. 4. The operator is

3 6 3 21 3 : = ´ = 7 21 7 6 2

/

3 9 3 25 5 2 / = ´ = =1 5 25 5 9 3 3

Example 4.

F70. Divide the fractions (write the results in lowest terms): a) 2 ¸ 4 3 5

b) 1 1 ¸ 2 2 3

c)

2 1 1 ¸2 3 2

2 7

d)



e) 4 ¸ 5

d)

2 :4 5

e) 3 : 7

d)

3 5 = 1 2 10

1 2 = e) 3 2 4

F71. Divide the fractions (write the results in lowest terms): a) 1 : 1 2 3

b) 1 1 : 1 2 4

c)

1 1 1 :2 5 10

F72. Divide the fractions (write the results in lowest terms):

a)

f)

2 3= 4 6 2 = 4 6

2 k) 3 = 4

b)

g)

l)

4 5 = 8 15 5 = 10 3

5 2= 4

2 9 = 4 6

1

c)

h)

3 = 1 2 4

4 5= 10

i)

j)

n)

1 3= 3

o)

1

m)

3 = 1 3

1

5 = 2 1 3

2 5= 4

2

F73. Divide the fractions (write the results in lowest terms): a) 1 / 2 2 3

b)

2 3 / 3 5

c)

5 3 / 3 5

d)

1 3 / 4 8

e) 5 / 3 1 3

f) 3 / 2 1 4

g)

1 /2 3

h)

1 2 /3 4

i)

2/5

j) 1 2 / 3 3 3 4

k) 5 / 15 7 14

l)

2 1 4 /1 3 6

m)

3 /1

1 3

n)

2 2 /4 7

o) 7 / 2

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40

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Iulia & Teodoru Gugoiu

Order of operations (III) 1. Division is not a commutative operation, so the order in which you divide numbers (or fractions) is important. If an expression contains more than one division operation, then by convention the divisions must be done, one by one, from left to right. Example 1:

2. Division and multiplication are considered operations of equal priority. If an expression contains both division and multiplication, then by convention the operations must be done, one by one, from left to right. Example 2:

1 2 5 æ1 2ö 5 3 5 3 ¸ ¸ =ç ¸ ÷¸ = ¸ = 2 3 4 è 2 3ø 4 4 4 5

4 1 2 æ4 1ö 2 2 2 3 ´ ¸ =ç ´ ÷¸ = ¸ = 5 2 3 è5 2ø 3 5 3 5

3. Division and multiplication are considered operations of greater priority than addition and subtraction. If an expression contains all these kinds of operations, do the multiplication and division first, then the addition and subtraction. Example 3: 3 2 6 3 6 3 æ 2 6 ö æ3 6ö 3 2 1 1 1 - ´ + ¸ =1 -ç ´ ÷+ç ¸ ÷ =1 - + =1 5 3 10 5 1 5 è 3 10 ø è 5 1 ø 5 5 10 10

F74. Use the correct order of operations to find the value of each expression (see example 1): a) 1 ¸ 2 ¸ 4 2 3 5

b) 2 ¸ 3 ¸ 5

c) 1 ¸ 2 ¸ 3 ¸ 4 ¸ 5

d)

3 1 1 2 ¸ ¸4 5 15 3

e) 1 ¸ 3 ¸ 5 ¸ 7 2 4 6 8

f) 1 ¸ 1 ¸ 3 ¸ 2 ¸ 4 2 3

g) 1 ¸ 2 ¸ 3 ¸ 4 ¸ 5 2 3 4 5 6

h)

1 2 3 4 1 ¸2 ¸3 ¸4 2 3 4 5

j) 1 2 ¸ 2 1 ¸ 3 3 ¸ 4 1 3 4 5 6

k) 1 1 ¸ 6 ¸ 4 ¸ 6 11 22 9

l)

3 6 5 3 ¸1 ¸ ¸1 7 21 6 5

i)

2 4 2 15 ¸ ¸ ¸ 3 3 5 4

F75. Use the correct order of operations to find the value of each expression (see example 2): a) 1 ¸ 3 ´ 5 2 4 6

b) 2 3 ´ 2 ¸ 4 4 5 15

c)

e) 1 ´ 2 ¸ 3 ¸ 4 5 5 5 5

f) 1 1 ´ 2 3 ¸ 3 3 ¸ 4 4 2 3 4 5

g) 3 ¸ 2 ´ 3 ¸ 9 2 3 16 8

i)

1 3 5 1 3 6 ¸ ´ ´ ¸ ´ 2 4 6 2 4 5

j)

1 3 5 7 9 ´ ¸ ´ ¸ 2 4 6 8 10

2 3 12 3 3 3 ¸ ´ ¸ ¸ ´ 5 5 5 10 4 20

d)

1´ 2 ¸ 3 ´ 4 ¸ 5

h)

1 1 1 1 1 1¸ ´ ¸ ´ ¸ 2 4 8 16 32

k) 3 ¸ 6 ´ 9 ¸ 3 ´ 15 ¸ 12 ´ 3 2 4 6 2 10 8 2

F76. Use the correct order of operations to find the value of each expression (see example 3): a) 1 + 3 - 5 ´ 2 ¸ 1 2 4 6 15 9

b) 1 + 2 ´ 6 - 3 ¸ 6 3 8 5 10

c) 1 2 - 3 ´ 10 ¸ 3 + 2 ¸ 1 - 1 15 5 12 2 5 3 2

d) 5 + 1´ 2 ¸ 3 - 4 + 5 ´ 6 ¸ 7

e) 1 1 ´ 2 ¸ 2 + 1 ¸ 1 ´1 1 2 3 2 6 4

f) 1 ¸ 1 - 2 ´ 3 + 1 ¸ 1 2 3 4 4 2

g) 1 1 ´ 1 - 1 ¸ 2 - 1 ´ 1 + 1 ¸ 1 + 1 ¸ 4 2 2 3 4 2 2 6 © La Citadelle

h)

41

1 1 3 3 1 3 1 1 1 1 6 1 +1 ´ ¸ + ¸ ´ - ´ - 2 ¸ + 12 2 4 4 2 4 2 2 2 2 2 12

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The Book of Fractions

Iulia & Teodoru Gugoiu

Order of operations (IV) 2. When an expression contains fraction (division) lines, the numerators and the denominators must be calculated independently. Example 2:

1. You can change the order of operations using brackets. Example 1:

1 æ1 2ö æ1 1ö 7 7 1 7 1 ç + ÷´ç - ÷ ¸ = ´ ¸ = è 2 3ø è 2 4ø 3 6 4 3 8

-

2 2¸

1

1

1 2 2 4 1 8 25 7 = 6 + 1 = ´ + ´ = + = =2 1 1 3 1 1 6 3 1 3 9 3 9 9 + ´ + 2 2 2 2 2 4

3+ 4 1 3

2

2

F77. Solve each exercise by following the proper order of operations (see example 1): a)

15 æ 1 2 ö æ 3 2 ö 5 ´ç + ÷ -ç - ÷ ¸ 3 è 2 5 ø è 4 3 ø 12

d)

11 2 æ 1 2 ö é 1 æ 2 3 ö 6 ù + ´ç + ÷ ¸ ê + ç - ÷ ¸ ú 25 17 è 6 5 ø ë 3 è 5 10 ø 5 û

e)

f)

é1 æ 3 1 ö 8ù æ 2 1 ö ê + ç - ÷´ ú ¸ç + ÷ ë2 è 4 6 ø 7û è 3 2 ø

b)

e)

3 æ 10 5 ö 2 5 ´ ç ¸ ÷ - ´ ¸ (1 ¸ 3) 5 è 9 27 ø 25 6

c)

ì 1 1 éæ 1 1 ö 1 1 ù 1 1 ü é 1 3 æ 3 2 ö ù í2 - ¸ êç - ÷ ¸ - ú ´ - ý ¸ ê + ¸ ç - ÷ ¸ 6ú î 4 2 ëè 2 3 ø 2 4 û 3 6 þ ë 2 4 è 4 3 ø û

éæ 1 ö æ 1 3 öù 5 éæ 3 1 ö 3 1 2 ù æ 1 1 ö 6 1 êç 2 + 1÷ ¸ ç 2 + 4 ÷ú ´ 4 ¸ êç 4 + 2 ÷ ¸ 4 ´ 2 - 6 ú ´ ç 2 - 3 ÷ ¸ 5 + 12 ø è øû ø ø ëè ëè û è 1 é 2 1 æ 3 1 3 ö 1 2 ù é1 3 æ 2 3 ö 2 ù æ 5 2 8 ö ´ ê + ´ç - ¸ ÷ ¸ + ú - ê + ¸ç - ÷´ ú´ç - ¸ ÷ 2 ë 5 2 è 2 5 10 ø 2 3 û ë 5 10 è 3 5 ø 5 û è 6 3 9 ø

F78. Solve each exercise by following the proper order of operations (see example 2):

a)

æ1ö ç ÷ è 2ø æ 3ö ç ÷ è 4ø = 5 6

b)

æ 3ö ç ÷ è 4ø 5 = 2 5

f)

2 3 2 + ´ 3 4 9 3 5 3¸ 4 3

j)

3 1 8 1 - ´ 4 2 3 ¸ 4 ¸ 5 +1 ´ 9 1 5 3 2 13 15 ´ + 2 12 5 9 15

© La Citadelle

c)

g) 2 ´ 3 - 4 ¸ 3 1¸ 2´ 3 + 2

æ 2ö ç ÷ è 3ø æ 3ö ç ÷ è5ø = 6

d)

3 æ1ö ç ÷ è 2ø = æ1ö ç ÷ è 3ø æ 2ö ç ÷ è 3ø

2 3 9 1 ¸ ´ h) 3 2 8 3 5 15 2 1 ¸ - ´ 3 6 3 2

k)

42

e)

æ 3ö ç ÷ 3 è 5 ø ¸æ 3 ö´ = ç ÷ 2 4 æ ö è ø æ 2 ö 8¸ç ç ÷ ÷ è 3ø è 3 + 1/ 2 ø

i)

2 15 1 2 2 ´ - ¸ + 5 8 2 3 3 1 3 9 3 1 + ¸ ´ 2 2 4 4 3

3 1 5 2 10 2 - ¸ ´ + 5 2 4+ 5 6 3 ¸ 1 5 1 1 1 4 ¸ + ¸ ´ -1 2 4 5 2 3 3

1 3 1 ¸ 5 10 3 1 9 2 + ´ -1 2 4 3

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The Book of Fractions

Iulia & Teodoru Gugoiu

Raising fractions to a power 1. If you multiply a fraction by itself k times, you can use exponential notation to make the expression more compact (simpler):

n n n ænö ´ ´ L´ = ç ÷ d4 d 244 d èdø 1 4 3

n1´42 n ´L ´ n = nk 43 k times

3. You can use also the following rule to calculate the value of a fraction raised to a power:

n1´42 n ´L ´n 43 k n n n ænö nk k times ´ ´ L´ = ç ÷ = k = d d 244 d èdø d d1´42 d ´L ´d 14 4 3 4 43 4

k

k times

The left side is the expanded notation and the right side is the exponential notation. Example 1:

3 ´ 3 ´ 3 ´ 3 = 34

Example 2:

1 1 1 æ1ö ´ ´ =ç ÷ 2 2 2 è 2ø

Example 4:

3

4

3 3´ 3 9 æ 3ö = ç ÷ == 2 = 4 4 ´ 4 16 è 4ø

2 2 2 2 16 æ 2ö ç ÷ = ´ ´ ´ = 3 3 3 3 81 è 3ø

ænö ædö ç ÷ =ç ÷ èdø ènø -3 3 2 5 53 125 5 Example 5: æç ö÷ = æç ö÷ = = = 15 3 8 8 è5ø è 2ø 2 5. By convention, any number (including a fraction) raised to the power of 0 equals 1. Example 6:

F79. Write each expression in exponential notation: a) 1 ´ 1 ´ 1 ´ 1 b) 3 ´ 3 c) 3 ´ 3 ´ 3 ´ 3 ´ 3 2 2 2 2 4 4 2 2 2 2 2 g) 4 ´ 4 f) 1 2 ´1 2 ´1 2 3 3 3 5 5 F80. Write each expression in expanded notation:

e) 2 ´ 2 ´ 2 ´ 2

a) æç 3 ö÷ è 4ø e) æç 3 ö÷ è7ø

2

2

b) æç 5 ö÷ è 3ø

2

4. If the exponent is negative, replace the fraction with its reciprocal raised to a positive exponent according to the rule: -k k

2. To calculate the value an expression written in exponential notation, rewrite it in expanded notation and do the multiplication. Example 3.

k times 2

k times

4

f) æç 3 ö÷ è 2ø

c) æç 1 ö÷ è5ø

3

d)

3 3 3 3 ´ ´ ´ 2 2 2 2

h)

1 1 1 1 1 ´ ´ ´ ´ 4 4 4 4 4

3

g) æç 5 ö÷ è 4ø

0

æ 4ö ç ÷ =1 è5ø

d)

æ 2ö ç ÷ è 3ø

h)

æ 4ö ç ÷ è5ø

d)

æ 3ö ç ÷ è5ø

h)

æ 3ö ç ÷ è 2ø

d)

æ 2ö ç ÷ è 3ø

h)

æ 2ö ç ÷ è5ø

-1

-2

3

F81. Calculate the value of each expression: a) æç 3 ö÷ è7ø

0

b) æç 3 ö÷ è 2ø f) æç 1 ö÷ è 3ø

e) 23

3

c) æç 2 ö÷ è5ø

4

g) æç 2 ö÷ è1ø

2

4

3

4

F82. Calculate the value of each expression: b) æç 5 ö÷ è1ø

a) 3-2 e) æç 3 ö÷ è 4ø

-2

© La Citadelle

-2

f) æç 5 ö÷ è7ø

c) æç 1 ö÷ è 2ø

-1

g) æç 4 ö÷ è 3ø 43

-4

-2

-3

-3

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The Book of Fractions

Iulia & Teodoru Gugoiu

Order of operations (V) 1. If the base of a power contains other operations, replace the base with the result of that operations, then raise the new base to the exponent. 2

2. If the exponent of a power contains other operations, replace the exponent with the result of that operations, then raise the base to the new exponent.

2

25 æ 1 1ö æ 5ö ç + ÷ =ç ÷ = 36 è 2 3ø è 6ø

Example 1.

1+ 2´2

æ 2ö ç ÷ è 3ø

Example 2.

5

32 æ 2ö =ç ÷ = 243 è 3ø

3. If an expression contains addition, subtraction, multiplication, division, and powers, do the powers first. 2

1 16 6 æ 2 ö 1 16 6 4 1 16 6 9 1 1 + ´ ¸ ç ÷ -1 = + ´ ¸ -1 = + ´ ´ -1 = + 3 -1 = 2 2 9 8 è 3ø 2 9 8 9 2 9 8 4 2 2

Example 3.

4. The order of operations can be changed by brackets. -1

2 -1 -1 1 æç 1 1 æ 1 9ö 1 æ1ö 1 4 1 1 æ 3 ö ö÷ + 2 - 4´ç ÷ ¸8 = + ç2 - 4´ ÷ ¸8 = + ç ÷ ¸8 = + ¸8 = + = 1 2 çè 2 2 è 2 16 ø 2 è 4ø 2 1 2 2 è 4 ø ÷ø

Example 4.

F75. Calculate the value of each expression (see example 1): 2

a)

æ1 1ö ç + ÷ è3 4ø

e)

æ1 3ö æ1 1ö ç ´ ÷ ´ç ¸ ÷ è3 2ø è3 2ø

3

b)

æ 1 15 8 ö ç ´ ´ ÷ è5 4 7ø

f)

æ 2 1 4ö ç - ´ ÷ è5 4 5ø

2

2

2

c)

æ1 6 4ö ç ´ ¸ ÷ è3 5 5ø

g)

æ2 5 1ö ç + - ÷ è 3 6 2ø

-1

2

2

1

2

d) æç 1 - 1 ö÷ + æç 1 + 1 ö÷ è 2 3ø è 2 3ø

-2

h) æç 1 - 1 ö÷ + æç 2 - 1 ö÷ è 2 3ø è 3 4ø

F76. Calculate the value of each expression (see example 2): 3´1- 4´

1 2

a)

æ 2ö ç ÷ è 3ø

e)

æ 2 2 ö2 ç + ÷ è 3 3ø

1 1 2

1 1 ¸ -1 6

b)

æ 3 ö2 ç ÷ è5ø

f)

æ 1 3 ö3 ç + ÷ è 2 4ø

1

1 1 ¸ 6

2 1 ´6 + ¸ 3 3

c)

æ 1 ö3 ç ÷ è 2ø

g)

æ 1 3 ö3 ç ´ ÷ è3 2ø

d) æç 2 ö÷ è5ø

2 1 ¸ 6

1 3 3 ( ´3+ ) ¸ 2 2 2

3

h) æç 3 - 2 ö÷ 2 è 4 3ø

2 2 ´( ´ 2 - ) 3 3

F77. Calculate the value of each expression (see example 3): -1

0

1

-2

0

2

1

a) æç 1 ö÷ + æç 1 ö÷ + æç 1 ö÷ è 2ø è 2ø è 2ø

b) æç 2 ö÷ ´ æç 2 ö÷ ´ æç 2 ö÷ è 3ø è 3ø è 3ø

2 2 3 ö é 4 3 æ 3 öù æ ç ÷ú d) ç ÷ ´ ê + ¸ ç ÷ 4 3 2 2 è ø ëê è øûú

e) 1 + æç 2 ö÷ ´ 25 - 3 + æç 1 ö÷ ´ 2 5 è 2ø è5ø

2

2

3

æ1ö æ 3ö æ 3ö ç ÷ +ç ÷ ¸ç ÷ è 2ø è 4ø è 2ø = g) é 3 æ 3 ö -1 ù 2´ ê - ç ÷ ú êë 4 è 2 ø úû

3

2

2

æ 2ö ç ÷ 33 j) è 3 2ø ´ = 2 2 æ 3ö ç ÷ è 4ø © La Citadelle

1

3

5

2

f) 1 + æç 3 ö÷ ¸ æç 1 ö÷ ´ æç 1 ö÷ - 1 è 4ø è 2ø è 3ø 2

2

æ1 1 3ö æ1ö 2 ç + ´ ÷ ç ÷ + 4 2 2 3 9 è ø h) ´ è ø = 2 2 æ1ö 3 æ1 2 4ö ç ÷ + ç + ¸ ÷ è 2ø 4 è6 9 3ø 3

2

2

c) æç 1 ö÷ + æç 2 ö÷ ´ æç 3 ö÷ ¸ æç 1 ö÷ è 2ø è 3ø è 4ø è 3ø

2

æ4 1 ö -1 ç + ÷ i) ç 5 10 ÷ ¸ 3 - æç 1 ö÷ = ç 3 ÷ 4 è 2ø ç ÷ è 5 ø -2

2 2 é ù é ù k) ê12 ´ æç 1 ö÷ ´ æç1 - 1 ö÷ - 2 - 2 ú ¸ ê 3 ¸ 1 - æç 1 ö÷ ú ¸ 52 = è 2ø è 4ø ëê ûú êë 4 2 è 2 ø ûú 2 3 ìïæ 3 2 ö é 7 æ 1 1 öù -1 æ 1 ö 2 üï éæ 1 ö 2 æ 1 ö -3 æ 1 ö 2 ù æ 3 ö 2 l) íçè 4 - 3 ÷ø ´ ê 6 - çè 2 + 3 ÷øú + çè 2 ÷ø ý ¸ êçè 2 ÷ø ´ çè1 - 2 ÷ø - çè 2 ÷ø - 1ú ´ çè 2 ÷ø = ïî ïþ ëê ë û ûú

44

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The Book of Fractions

Iulia & Teodoru Gugoiu

Converting fractions to decimals 1.212... 3. If the denominator of the fraction written in lowest terms has factors other than 2 or 5, 33 40.000 the fraction can be written as a non33 terminating repeating decimal. Example 2: 70

1. The fraction line is a division operation, so to write a fraction as a decimal, divide the numerator (dividend) by the denominator (divisor). 0.375 2. Usually we are dealing with the 8 3.000 decimal system (the base is 10). The single prime factors of 10 are 2 and 5. 24 If the denominator of the fraction written 60 in lowest terms has only 2 or 5 as prime 56 factors, then the fraction can be written 40 as a terminating decimal. 40 Example 1:

8 = 2´ 2´ 2

3 = 0.375 8

40 = 1.2121... = 1.21 33

33 = 3 ´ 11

66 40 33 70 66 4...

4. The part of the decimal that repeats is called the period. The period is 21 in example 2. The number of repeating digits is the length of the period. The length of the period is 2 digits in example 2.

00

F86. Build the factor tree for the denominator and classify each fraction as a terminating or a non-terminating decimal: a)

1 2

b)

2 3

c)

3 4

d)

4 5

e)

5 6

f)

2 7

g)

7 9

h)

3 10

i)

8 11

j)

2 15

k)

11 20

l)

3 6

m)

3 40

n)

13 100

o)

25 120

d)

1 5

e)

3 20

i)

3 500

j)

1 625

F87. Write each fraction as a terminating decimal: 1 3 5 a) b) c) 2 4 8 f)

11 80

g)

13 100

h)

1 125

F88. Write each fraction as a non-terminating repeating decimal. Use a bar over the repeating decimals: a)

1 3

b)

1 6

c)

11 36

d)

7 9

e)

2 11

f)

1 15

g)

7 30

h)

2 7

i)

2 13

j)

1 27

F89. Write each fraction as either a terminating or non-termonating repeating decimal. Use a bar over the repeating decimals: a)

1

1 2

b)

6 15

c)

1

f)

1

5 4

g)

5 64

h)

k)

3 250

l)

1 7

m)

© La Citadelle

7 21

d)

1 75

e)

7 3

1 32

i)

3 64

j)

1 128

20 14

n)

5 12

o)

2

45

3 13

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The Book of Fractions

Iulia & Teodoru Gugoiu

Converting decimals to fractions 1. To convert a terminating decimal to a fraction, use the place value of each digit of the decimal part. Example 1:

1.25 = 1 +

2 5 1 1 5 1 + = 1+ + =1 =1 10 100 5 20 20 4

1000

3. To convert a non-terminating repeating decimal to a fraction, you can use algebra. Example 3:

x = 0.666... = 0.6 10 x = 6.666... 10 x - x = 6.666... - 0.666... 9x = 6 x=

2. A faster method to convert a terminating decimal to a fractions is: a) the numerator is the number without the decimal point b) the denominator is 1 followed by a 0 for each digit of the decimal part 1625 13 5 1.625 = = =1 Example 2:

6 2 = 9 3

8

8

4. A faster method to convert a non-terminating repeating decimal to a fraction is the following: 1) the numerator is a mixed number: a) the whole is the decimal written without the decimal point and the period b) the numerator is the period c) the denominator is one 9 for each digit of the period 2) the denominator is 1 followed by one 0 for each digit between the decimal point and the period 45 Example 4: 6123

6.12345 =

99 = 6 679 1000 5500

F90. Write each terminating decimal as a fraction in lowest terms (see the example 1): a)

0 .1

b)

0 .5

c)

1 .4

d) 1.25

e)

0.75

f)

0.035

g)

2.125

h)

10.125

i)

j)

100.725

5.075

F91. Write each terminating decimal as a fraction in lowest terms (see the example 2): a)

0.625

b)

1 .5

c)

0.125

d) 0.4

e)

2.16

f)

0.275

g)

0.24

h)

0.35

i)

j)

0.640625

2.45

F92. Write each non-terminating decimal as a fraction in lowest terms (see the example 3): a)

0.3

b)

1.2

c)

0.12

d) 1.21

e)

4.025

f)

0.23

g)

1.25

h)

2.012

i)

j)

1.23456

0.123

F93. Write each non-terminating decimal as a fraction in lowest terms (see the example 4): a)

0.7

b)

1.3

c)

2.53

d) 1.32

e)

1.129

f)

0.12

g)

3.01

h)

1.312

i)

j)

1.0123

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46

6.12345

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The Book of Fractions

Iulia & Teodoru Gugoiu

Order of operations (VI) 2. If an expression contains operations with non-terminating repeating decimal numbers, convert decimal numbers to fractions and do all the operations with fractions. Example 2: 1

1. If an expression contains only decimal numbers, you can do all the operations using decimal numbers. Example 1:

3 1 25 533 1.3 ´1.25 - 0.31 = 1 ´1 - 9 = 1 @ 1.359 3 99 10 1485

1 .2 + 0 .5 ´ 1 .6 - 2 .4 ¸ 2 = 1 .2 + 0 .8 - 1 .2 = 0 .8

3. If an expression contains both fractions and decimal numbers, it is recommended that you first convert the decimal numbers to fractions, and do all the operations with fractions. Example 3:

1 .2 +

2 4 3 6 2 1 4 3 3 7 - 0 .5 ´ + ¸ 1 .5 = + - ´ + ¸ = = 1 .4 5 5 10 5 5 2 5 10 2 5

F94. Find the value of each expression (only use operations with decimals):

a) 0.25 + 0.15 =

b)

0.50 - 0.15 =

c)

d)

2 .4 ´ 0 .5 =

0.10 ¸ 0.05 =

e) 0.5 2 + 0.25 =

1.2 ¸ 0.75 =

e) 1 - 0.4 2 =

F95. Find the value of each expression by converting decimals to fractions:

a) 0.75 + 1.25 =

b) 0.60 - 0.25 =

c)

d)

0 .5 ´ 1 .4 =

F96. Find the value of each expression by converting repeating decimals to fractions:

a) 1.3 + 0.6 =

b) 2.4 - 0.45 =

c)

d)

0.5 ´ 2.2 =

0.28 ¸ 0.01 =

e) 0.6 2 - 0.32 =

F97. Find the value of each expression by converting fractions to decimals:

a) 2 + 0.5 ´ 1 ¸ 0.2 - 0.2 5 2

b)

2 1 1 .5 ´ ¸ - 0 .5 2 5 2

c)

4 - 0.3 ¸ 1.5 + 0.82 d) 5

1ö 3 æ 2 ç 0 . 1 + ÷ ´ - 0 .2 5 ø 10 è

F98. Find the value of each expression by converting decimals to fractions:

a) 1 + 0.20 ´ 3 4 2

b)

5 2 0.25 ¸ - 0.15 ´ 6 3

c)

2 1 ¸ 0.25 ´ - 0.3 3 4

3

d)

æ1ö 0.52 - ç ÷ ´ 0.8 è 2ø

F99. Find the value of each expression by converting decimals to fractions:

a) 1 + 0.5 ´ 3 - 3 ¸ 1.5 ´ 2 2 5 4 5 æ 2 5 3ö ç1.5 ´ - ÷ ¸ 24 6 8ø d) è 3 0.4 ´ - 0.25 4

b) 1.25 ¸ 5 ´ 0.5 + 0.5 ´ 0.752 ´ 2 4 3 2

e)

1ö 3 æ ç 0.25 + ÷ + 2 ø 16 è ´ 0.5 æ4 ö ç - 0.3 ÷ ´ 3 è5 ø

2 5 é 2 ù 3 2 g) êæç + 1.3 ö÷ ¸ 5 - 0.4ú ´ æç1.6 - ö÷ + 0.2 ¸ 5ø 3 ø êëè 3 úû è

i)

© La Citadelle

æ3 ö æ3 ö ç - 0.1÷ ¸ ç - 0.5 ÷ ø è4 ø ´3 f) è 5 1ö æ1 æ ö 4 ç1.25 - ÷ ´ ç + 1.25 ÷ 2ø è 4 è ø 3

2

h) é2.5 ¸ æç 4 + 0.7 ö÷ + 1 ù ¸ éæç1.8 - 3 ö÷ ´ 5 ù + 1 ê ú ê 5 ø 2 úû 9 è5 ø 3 û ëè ë 2

3

4ù é 2 1 êë2.4 - 9 úû ´ 0.5 0.6 ´ 0.3 + 9 ¸ 52 ¸ 2 2ö æ 1ö æ 2.4 - ç1.2 + ÷ 1.5 + ÷ ç 5ø è 9ø è

c) 4 ´ 0.5 ´ 5 - 0.3 ´ 1 + 0.25 ´ 1 ´ 4 5 3 2 3

j)

3 æ1ö 0 .2 - ç ÷ + 0 .6 ´ -1 2 0.1´ 0.25 ¸ 0.4 5 æ1ö è5ø -5 + 7 ¸ ç ÷ ´2 -1 2 æ ö è 3ø é 1 1 æ 1 öù 1 .6 ´ ç 0 .4 - ´ 0 .5 ÷ ê 2 ´ 5 ¸ ç 2 ÷ú 5 è ø è øû ë

47

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The Book of Fractions

Iulia & Teodoru Gugoiu

Time and Fractions 1. One hour has 60 minutes. This relation can be written in two ways:

1 h = 60 min

1 min =

4. To make conversions between minutes and seconds, use the formulas above. Example 3:

1h 1 = h 60 60

0.25 min =

2. To make conversions between hours and minutes, use the formulas above. Example 1:

Example 4:

1 4 4 1 h = h = ´ 60 min = 80 min 3 3 3

100 s = 100 ´

1 25 5 h= h= h 60 60 12

4500 s = 4500 ´

3. One minutes has 60 seconds. This relation can be written in two ways:

1 min = 60 s

1s =

1 100 5 min = min = min 60 60 3

5. The conversion between seconds and hours is a twostep task.

Example 2:

25 min = 25 ´

3 3 min = ´ 60 s = 45 s 4 4

1 1 1 min = 75 min = 75 ´ h =1 h 60 60 4

1 min 1 = min 60 60

F100. Convert hours to minutes. Write the results as mixed numbers in lowest terms: a) 0.5 h b) 1 h c) 1.75 h d) 2 3 h e) 0.1 h 3 4 5 7 h) 3 h g) 2.25 h i) j) 1.23 h h h 6 8 15 F101. Convert minutes to hours. Write the results as mixed numbers in lowest terms:

f)

a)

b)

5 min

c)

10 min

d)

15 min

25 min

e)

50 min

3 j) 2.5 min min 4 F102. Convert minutes to seconds. Write the results as mixed numbers in lowest terms: a) 1.5 min b) 1 2 min c) 0.25 min d) 1 2 min e) 1.15 min 3 5

f)

70 min

g)

36 min

h)

250 min

i)

f)

7 min 12

g)

1.75 min

h)

5 min 12

i)

11 min 45

j)

7.3 min

F103. Convert seconds to minutes. Write the results as mixed numbers in lowest terms: a)

b)

12 s

10 s

1 g) 0.75 s s 2 F104. Do the required conversions:

f)

10

a) 500 s = ? h

b) 1500 s = ? h

g) 2 h 10 min = ? s © La Citadelle

c)

45 s

h)

15

3 s 4

c) 9000 s = ? h

h) 15 min 5 s = ? s

d)

d)

90 s

e)

200 s

i)

100 s 3

j)

20.6 s

0 .2 h = ? s

i) 0.5 h 15 s = ? min 48

e) 1.75 h = ? s

f) 0.15 h = ? s

j) 2 h 10.5 min 10 s = ? min 3 www.la-citadelle.com

The Book of Fractions

Iulia & Teodoru Gugoiu

Canadian coins and fractions 1. One dollar (loonie or $) has 100 cents (pennies): Example 1:

1 $ = 100 cents

1 1 125 cents = 125 ´ $ =1 $ 100 4

2. One twonie (or toonie) has 200 cents: Example 2:

1.75 twonies =

Example 4:

25 cents = 25 ´

5. One quarter has 25 cents: Example 5:

1 twonie 1 = twonie 200 200

1 nickel = 5 cents

1 cent =

1 nickel 1 = nickel 5 5

1 dime = 10 cents

1 cent =

1 dime 1 = dime 10 10

1 quarter = 25 cents

1 cent =

1 quarter 1 = quarter 25 25

1 6 1 nickels = ´ 5 cents = 6 cents 5 5

4. One dime has 10 cents:

1 1 dimes = 2 dimes 10 2

1$ 1 = $ 100 100

1 twonie = 200 cents 1 cent =

175 ´ 200 cents = 350 cents 100

3. One nickel has 5 cents: Example 3:

1 cent =

2 2 quarters = ´ 25 cents = 10 cents 5 5

F105. Convert dollars to cents. Write the results as mixed numbers in lowest terms:

a) 0.12 $

b)

1 $ 125

d) 1 2 $ 75

c) 1.02 $

e) 0.07 $

f)

7 $ 150

F106. Convert cents to dollars. Write the results as mixed numbers in lowest terms:

a) 25 cents

b) 20 cents

c) 45 cents

d) 160 cents

e) 450 cents

f) 5.5 cents

F107. Convert twonies to cents. Write the results as mixed numbers in lowest terms:

a) 0.02 twonies

b) 0.95 twonies

c)

3 twonies 100

d)

11 twonies 150

e) 1 3 twonies 10

f) 1.2 twonies 25

F108. Convert cents to twonies. Write the results as mixed numbers in lowest terms:

a) 125 cents

b) 250 cents

c) 40 cents

d) 120 cents

e) 500 cents

f) 10.2 cents

F109. Convert nickels to cents. Write the results as mixed numbers in lowest terms:

a) 0.2 nickels

b) 1.2 nickels

c) 7 nickels 20

d) 7 nickels 5

e) 2 3 nickels 20

f) 10.5 nickels 2

F110. Convert cents to nickels. Write the results as mixed numbers in lowest terms:

a) 15 cents

b) 25 cents

c) 75 cents

d) 4 cents

e) 1.5 cents

f) 0.5 cents

F111. Convert quarters to cents. Write the results as mixed numbers in lowest terms:

a) 0.2 quarters

b) 1.6 quarters

c) 3 quarters 25

d)

9 quarters 125

e) 1 3 quarters 5

f) 1.2 quarters 5

F112. Convert cents to quarters. Write the results as mixed numbers in lowest terms:

a) 50 cents

b) 125 cents

c) 100 cents

d) 20 cents

e) 0.5 cents

f) 55 cents

F113. Do the required conversions:

a) 3 quarters = ? nickels b) 1.5 dimes = ? nickels c) 2.5 dimes = ? quarters d) e) 8 nickels = ? dimes © La Citadelle

f) 15 nickels = ? dimes

0.2 quarters = ? dimes

g) 1 quarters 2 dimes = ? nickels h) 5 quarters = ? dimes 49

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The Book of Fractions

Iulia & Teodoru Gugoiu

Fractions, ratio, percent, decimals, and proportions 1. A fraction is a comparison between a part and the whole. For example 40/100 2. A ratio is a comparison between two numbers. For example 40:100 3. A percent is a comparison between a number and 100. For example 40% 4. A decimal is a comparison between a number and 1. For example 0.40 5. The same number can be written as a fraction, ratio, percent, or decimal. To convert the number from one expression to another, use this two-step algorithm: 1) Express the number as a fraction Ex. 1

2 7 1 = 5 5

Ex. 2

7 to 5 = 7 : 5 =

7 5

Ex. 3

140 % =

140 7 = 100 5

Ex. 4

1 .4 =

14 7 = 10 5

2) Convert the fraction using a proportion and the cross-multiplication rule Ex. 5

7 2 =1 5 5

Ex. 6

7 x 7 ´ 35 = x to 35 = ; x = = 49; so 5 35 5

Ex. 7

7 = 1 .4 5

Ex. 8

7 x 7 ´100 = ; x= = 120; so 5 100 5

7 = 49 to 35 5

7 = 120% 5

F114. Fill out the table: Fraction

Ratio

a)

1/ 4

... to 20

b)

1/ 3

... out of 21

c)

3/ 2

12 to ...

d)

5/6

15 out of ...

e)

3/ 5

42 out of ...

Percent

Decimal

f)

2 out of 5

g)

10 to 45

h)

3 out of 12

i)

18 to 15

j)

7 to 10

k)

... to 70

30 %

l)

36 out of ...

80 %

m)

... to 30

130 %

n)

... out of 5

10 %

o)

... out of 64

75 %

p)

... to 10

0.15

r)

... out of 35

1.4

s)

6 to ...

0.08

t)

80 out of ...

0.64

u)

... out of 256

0.125

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50

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The Book of Fractions

Iulia & Teodoru Gugoiu

Fractions and Number Line 1. To create a number line assign points to 0 and 1: 0

4. Number lines can be used to compare numbers (fractions). Example 3: 3 2

1

0

2. Each number corresponds to a point on the number line. 3 Example 1:

1/4 2/4 3/4

0 0

0

1

3/5 4/5

2/5

1 4

5

0

2

1

3

1

1

5. To calibrate a ruler means to assign numbers to the points on the ruler. Example 4:

3. Each point on the number line corresponds to a number. 2 1 Example 2: 0


= or < symbols): 1 6

1 12

0

1 10

a) 1

1 4

1 5

6 12

2

f) 2 5 5 12 0

A

B

1 3

5 12

3 10

1 2

7 12

2 5

2 3

3 5

3 4

7 10

5 6

4 5

c) 3 7 5 12

d) 4 5 5

6

e) 11 9 12 10

g) 4 3

h) 1 6

i)

4 5

3 4

j) 1 1 12 10

4

1 5

1

C

W X

Y

F

G

11 10

b) 1 3 3 10 5

E

13 12

1

9 10

F116. Find a point on the number line corresponding to each fraction:

D

11 12

H

I

J

K

L

M N

O

P

Q

R

S

T

U

V

a)

1 2

b)

3 4

c)

1 12

d)

1

f)

3 24

g)

7 12

h) 1 5 24

i)

1

Z

a

b

c

d

e

1 4

e)

1 12

j)

f

g

h

i

j

3 8 1

1 8

F117. Find the fraction corresponding to each point on the number line:

B

D

F

H

J

0

1 A

C

E

G

I

F118. Calibrate the ruler:

0 © La Citadelle

1 51

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The Book of Fractions

Iulia & Teodoru Gugoiu

Comparing fractions 1. Fractions are ordered numbers. That means you can compare them and decide if they are equal (=) or which one is greater (>) or less ( 5 7

3 2 > 4 5

Example 4:

small big < ( 2) a a So, if you compare two fractions having the same denominator, the smallest one has the smallest numerator. 2 4 7 5 3 3 < > = Example 2: 5 5 11 11 4 4 3. If you cross exchange the factors in relation (2), you’ll a a get: < (3) big small So, if you compare two fractions having like numerators, the smallest fraction has the biggest denominator.

? n 2 ´ d1

because 3 ´ 5 > 4 ´ 2

5. To compare two fractions in the general case, you can also find the LCD (Least or Lowest Common Denominator), convert the original fractions to equivalent fractions having like denominators and then use the relation (2). Example 5: 5 7 5 20 21 7 < because = < = 12 16 12 48 48 16 6. You can use the LCD method when you have to order a set of fractions: Example 6:

3 3 = 8 8

2 3 5 < < 3 4 6

because

2 8 3 9 5 10 = < = < = 3 12 4 12 6 12

F119. Compare the whole numbers. Use operators :

a) 2 and 9

b) 10 and 10

c) 1 and 0

F120. Compare the fractions (see example 2): a) 3 and 5 b) 5 and 1 c) 2 and 1 7 7 6 6 6 3 7 5 h) 2 and i) 1 and j) 1 5 and 2 1 3 5 4 4 F121. Compare the fractions (see example 3): a) 1 and 1 b) 2 and 2 c) 1 1 and 3 2 3 7 5 2 2 5 5 7 2 h) and i) 1 and j) 3 and 3 2 4 6 9 7 3 F122. Compare the fractions (see example 4): a) 2 and 3 b) 4 and 6 c) 1 1 and 1 3 3 4 5 7 2 5 5 7 2 h) 5 and 15 i) j) 2 and 2 5 and 6 9 5 2 4 12 F123. Compare the fractions (see example 5): a) 3 and 8 b) 3 and 11 c) 3 and 5 5 15 8 24 4 6

d) 7 and 5

e) 123 and 132

f)

d) 3 and 5

e) 13 and 23

f) 1 2 and 4

11 11 2 k) 3 and 2 4 5 5

d) 3 1 and 2 3 4

5

l)

f) 10 and 10

1 3 and 2 4 4 27 1 and 5 15

f) 7 and 8

e) 4 and 5

f) 5 and 7

l)

d) 3 and 5

e)

k)

l)

4 7 3 2 and 2 3

d) 4 and 7 15

20

12

15

g) 2 4 and 14

3 3 1 m)3 and 2 4 3 3

4 4 and 7 3 4 and 4 5

e)

1 1 and 3

k)

25 25 1 2 and 3 5

g) 12 and 21

0 and 3

5

n) 11 and 2 1 4

4

g) 2 4 and 2 4

7 11 2 m) 1 and 7 5 6

n)

7 5 13 3 and 2 7 5 3 4 and 8 9 3 4 and 5 7

g)

10 11 3 m) and 4 10 13

6

5

n)

g) 2 and 4

10

9

15

F124. Write each set of fractions in order from least to greatest:

a) ìí 2 , 1 , 3 üý î7 7 7 þ

b) ìí 3 , 3 , 3 üý î 5 11 7 þ

c) ìí 3 , 2 , 1 , 4 üý î4 3 2 5þ

d) ìí 5 , 3 , 2 üý î12 10 5 þ

e) ìí 1 , 7 , 3 , 4 üý î 2 12 4 15 þ

f) ìí1 1 , 1 5 , 1, 11 , 1 5 üý î 4

24

8



F125. Write each set of fractions in order from greatest to least:

a) ìí 3 , 4 , 1 üý î5 5 5 þ

© La Citadelle

b) ìí 2 , 2 , 2 üý c) ìí 5 , 3 , 2 , 6 üý d) î7 5 6 þ

î6 4 3 7 þ

ì1 3 1ü í , , ý î4 5 3þ

52

e) ìí 7 , 3 , 2 , 9 üý î12 4 3 10 þ

f) ìí1 1 , 1 3 , 1 1 , 1 5 , 1 11 üý î 4

8

2

16

32 þ

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The Book of Fractions

Iulia & Teodoru Gugoiu

Solving equations by working backward method The working backward method requires to identify the operations applied to the unknown quantity x, and do the opposite operations in the opposite order. 1. If the operation is an additions with a number then apply a subtraction with the same number. Example 1:

x+

1 3 = 2 4

so

x=

5. If the operation is an inversion then apply another inversion. Example 5:

1 2 = x 3

so

2 3=1 4 2

x-

1 1 5 x= + = 3 2 6

2 3 = 3 4

so

x=

1 1 3´ x 2 2

3 2 9 ¸ = 4 3 8

4. If the operation is a division by a number then apply a multiplication by the same number. Example 4:

3 4 x¸ = 5 7

so

x=

3 2

so

x-

2 1 = ´ 4 = 2 so 3 2

x = 2+

2 8 = 3 3

Example 7:

3. If the operation is a multiplication by a number then apply a division by the same number. Example 3:



so

6. If more than one operation are implied then identify the operations and the order in witch they appear and then do the opposite operations in the opposite order. Example 6:

3 1 1 - = 4 2 4

2. If the operation is a subtraction with a number then apply an addition with the same number. Example 2:

1 1 x- = 2 3

x 3 = 1 2

so

+

4 3 3´ x -

so 3 ´ x -

4 3 12 x= ´ = 7 5 35

1

= 2 so 1 3 = 2 8

1 2

so 3 ´ x =

+

4 = 4 so 3

7 8

so

x=

1 3´ x -

1 2

=

8 3

7 24

F126. Solve for x working backward:

a) x + 2 = 1

b)

1 4 x- = 3 5

f) 2 ´ x + 2 = 4

g)

3´ x -

k) x + 1 = 2

l)

x 2 1 - = 5 3 2

5

2

3

3

2

5

3

2

p) x ´ 3 - 3 1 2

t)

4 + = 3 5

1 1 x´2 ¸ 3 2´2 = 4 2 1 3 3 3 2

1 1 = 2 3

1 1 = 2 3

c)



h)

x 2 3 ´ = 3 3 4

d) x ¸ 4 = 2 1 7

m) ( x - 1) ´ 1 = 4 2

1 2 -1 =1 3 2 3

q)

x´2 +

u)

æ x´3 1 1 ö ¸ - ÷ ç 2 3 ÷´ 2 = 4 ç 4 1 1 ç ÷ 3 5 + ç ÷ 2 3 è ø

r)

5

e) 1 = 5 x

3

i)

x 2 4 ¸ = 3 3 5

n)

1ö 2 1 æ çx+ ÷¸ = 5ø 3 2 è

1ö æ ç x´2 - ÷ 2 4 2 ÷´ = ç 2 ç ÷ 3 5 ç ÷ è ø 1 x¸22 ¸2=1 v) 3 3 6

7

1 2 = x-2 3 1 3 = o) 1 4 x+ 3

j)

s)

w)

æx ö ç -1 2 ÷ 3 2 ç2 - ÷¸ = 3÷ 5 5 ç 3 ç ÷ è ø 2 3 = x -1 2 4 2 3

F127. Solve for x working backward:

a)

x´3 5 1 - ¸ 8 8 2 ´ æç 1 - 1 ö÷ = 1 ´ 3 - 1 3 2 è 2 6ø 2 4 4 4 3

c)

ìé ü 1 1 ù ïê 1 ï ú ï 4 5 ú ´ 4 + 1ï ¸ 3 = 1 ê + í ý ïê10 2 ¸ æç x - 2 ö÷ ú 5 5 ï 5 2 ïîêë ïþ 3 è 5 ø úû

© La Citadelle

1 3 1 ¸ 2 2 6 ¸ æç 2 - 1 ö÷ - æç 4 - 2 ö÷ = 1 + 2 ¸ 5 1 1 4 è 3 4ø è 5 3ø 5 3 4 - ´ x´3 2 3

b)

æ ö ç 1 2 ÷÷ æ 2 3 2 ö æ1 4 3 1ö ç - -ç ¸ + ÷ = 0 ç ´ ¸ - ÷¸ 5 1 3 ÷ è 5 10 3 ø è3 5 5 9ø ç + ç ÷ è x´2 +3 5 ø

d)

53

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The Book of Fractions

Iulia & Teodoru Gugoiu

Final Test Find the fractions:

1)

2)

3)

three eights

9)

3

4) five out of seven

5) four and three quarters

Express the numbers as improper fractions:

7) 2

6)

1 3

2 5

8) 3

10 7

10)

11) 2.4

five and two thirds

12) one and five thirds

Express the numbers as mixed numbers:

13)

10 4

14)

15 4

15)

16) 1

5 3

17) eight fifths

Write the fractions in lowest terms:

21)

24 36

22)

20 8

23)

18)

15 7

1.25 19)

20) eleven quarters

Check each pair of fractions for equivalence:

12 18

24)

18 32

1 6

5 6

10 15 and 12 20

25)

26)

4 5 and 12 15

33)

1

100 75 and 64 48

27)

Add the fractions:

28) 1+ 2

2 3

2 1 + 5 5

29)

31)

1 3 + 3 4

32)

5 3 7 7

38)

3 2 4 3

39) 3 - 2

1 2 2 3

45)

2 2 1 ´1 3 5

46)

3 4 ´ 4 9

47)

5 2 ´1 3 3

48)

3 1 1 ´1 4 3

52)

4 6 1 ¸ 5 10

53)

1 1 1 ¸3 6 2

54)

22 5 ¸1 3 6

55)

1 2 2 ¸2 4 5

15 12

60)

62)

5 16

7 8

64)

30) 1 + 2

10 5 +1 3 6

2 3 +2 15 20

34)

5 2 +1 4 3

41)

9 1 -1 5 4

Subtract the fractions:

35)

1-

2 3

10 -2 3

36)

37)

4 5

7 1 2 40) 5 - 2 10 4 3

Multiply the fractions:

42)

2 3 ´ 3 5

43) 2´

3 5

44) 1 ´

50) 2 ¸

2 5

51)

8 ¸4 9

58)

4 10

Divide the fractions:

49)

2 3 ¸ 3 4

Convert each fraction to a decimal:

56)

2 5

57) 2

2 5

59)

2 3

61) 1

2 7

63)

100 32

Convert each decimal to a fraction:

65) 0.12

66) 1.5

67) 2.25

68) 0.0125

69) 0.4

70) 1.63

71)

0.725

72) 1.65

73) 0.875

Do the required conversions (use a fraction to express the result if possible):

74) 0.45 min = ? s

75) 0.1 h = ? min

79) 1.5 dimes = ? nickels 83)

2 = ? twelfths 3

80)

84)

76)

100 s = ? min

7.5 quarters = ? $

1 = ? tenths 5

85) 1

81)

1 = ? sixths 2

77) 2000 s = ? h

78) 20 min 20 s = ? h

0.35 $ = ? twonies 86)

82) 1 $ 1 quarter = ? dimes

3 = ? eighths 16

87)

2 = ? fifteenths 3

Do the required operations:

1 1 1 - + 2 3 4 æ2 1ö 2 92) ç - ÷ ´ è 3 4ø 3

1 4 1 ´ 2 3 6 2 æ 1 5ö ¸ ç1 - ÷ 93) 3 è 3 6ø

88)

89)

90) 94)

2 5 2 1 ¸ + 3 6 3 4ö æ 2ö æ ç1 - ÷ ´ ç 2 ¸ ÷ 3ø è 3ø è

1 3 1 2 ¸3 ´ 2 4 2

91)

1ö æ2 3 ö æ1 ¸2 ÷-ç - ÷ 2 ø è 5 10 ø è2

95) ç

Solve for x:

96)

1 1 x- = 3 4

© La Citadelle

97)

1 2 x¸ = 3 5

98)

x 5 = 3 6

99)

54

2 3 = x -3/ 2 4

100)

1 2 = 2 + 3/ x 5

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The Book of Fractions

Iulia & Teodoru Gugoiu

Answers 1 2 1 5 58 b) c) d) e) 2 3 4 9 100 6 2 9 1 3 n) o) p) q) r) 10 4 13 4 8

F01.

6 5 4 2 1 6 1 2 g) h) i) j) k) l) m) 10 6 6 6 3 18 1 5 7 21 6 0 12 7 10 s) t) u) v) w) x) y) 9 49 12 12 12 10 16

a)

f)

F02. The answers may vary. a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

u)

v)

w)

x)

y)

z)

F03. a ) two thirds

b ) three hundredths

h ) four fifths

i ) eight thirtieths

o ) three fiftieths

p ) two fifths

u ) eleven millionths

F04.

c ) one tenth

d ) one half

j ) eight thirteenths

k ) eight ninths

q ) twenty - one hundredths

v ) two ninths

w) seven tenths

e) three sevenths

l ) five sixths

r ) six twelfths

x ) eleven twelfths

f ) three twentieths m) five eighths

s ) seven elevenths

y ) two fiftieths

1 1 1 2 4 7 11 7 5 8 6 9 b) c) d) e) f) g) h) i) j) k) l) 3 2 6 5 7 8 50 20 12 9 10 1000 8 3 11 23 7 11 3 13 1 n) o) p) q) r) s) t) u) v) 6 50 1000000000 100 13 12 1000000000 30 5 a)

2 3

F05.

a)

F06.

a) 1

b)

1 4

c)

3 5

d)

3 6

e)

3 12

f)

2 5

g)

2 7

h)

3 4

i)

5 12

j)

3 10

k)

4 6

l)

55

n ) seven thousandths

t ) eleven fiftieths

z ) nine billionths

15 1000000 1 8 w) x) 11 9 m)

y)

6 10

z)

6 12

3 5

1 1 2 1 3 1 5 7 33 7 1 b) 2 c) 1 d) 2 e) 1 f)2 g) 3 h) 6 i ) 1 j) 1 k) 4 l) 2 2 4 3 3 6 4 6 9 100 10 6 8 6 3 9 13 5 6 6 10 3 7 n) 1 o) 5 p) 6 q) 2 r) 2 s) 3 t) 2 u) 1 v) 5 w) 1 x) 2 18 10 4 13 49 9 8 12 16 10 10

© La Citadelle

g ) one thousandth

m) 3

2 5

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The Book of Fractions

Iulia & Teodoru Gugoiu

F07. The answers may vary. a)

b)

c)

d)

e)

f)

g)

h)

i)

j)

k)

l)

m)

n)

o)

p)

q)

r)

s)

t)

u)

v)

w)

x)

y)

z)

F08. a ) one and one half

b ) two and one third

f ) two and three sevenths

c ) one and one quarter

g ) three and five eighths

d ) two and three fifths

h ) one and five ninths

e) one and five sixths

i ) two and three tenths

j ) one and two elevenths

k ) three and five twelfths l ) one and two fifteenths m) three and seven twentieths n ) two and nine thirtieths o) two and seven fiftieths p ) two and three hundredths q ) three and nine thousandths r ) two and seven millionths s ) one and three fourtieths t ) two and seven nineteenths u ) three and five sixteenths v) two and three seventeenths w) four and three fourteenths

x) two and five fifteenths

z ) two and three ninetieths F09. a) 2 2 b) 3 1 c) 5 5 d ) 2 1 3 2 6 3 5 2 3 n) 8 o) 3 p) 5 6 12 1000000

F10. a) 2

3 1 b) 2 5 3

c) 1

3 4

d)1

4 6

e) 4

q ) 20

e) 2

F11. a) 5 b) 3 c) 11 d ) 15 e) 10 3 17 q) 10

1 15 r) 6

4

6 31 48 s) t) 9 13

5 7

6

u)

32 12

f)7 3 100

3 5

f)3

5 50

y ) two and one sixtieth g) 2

3 4

h) 3

r) 6

4 15

s ) 11

2 5

g) 2

2 3

4 30

h) 3

2 9

i) 6

t) 8

7 10

1 2

25 5 14 10 g) h) i) 9 2 4 3 133 74 62 v) w) x) 100 16 49

f)

7 100

j) 9

u) 4

1 3

i) 2

4 6

j) 3

j)

6 5

k)

3 5

1 2

v) 1

k) 8 2 5

k) 4

33 11 l) 10 8

11 50

w) 3 2 3

2 11

l) 2

m)

5 2 m) 1 1000000000 11 6 9 11 x) 8 y) 5 z) 1 9 10 12

l) 1

3 4

19 4

n)

21 42 o) 10 18

p)

13 10

F12. The answers may vary. a)

g)

© La Citadelle

b)

h)

c)

i)

d)

j)

56

e)

f)

k)

l)

www.la-citadelle.com

The Book of Fractions

Iulia & Teodoru Gugoiu

m)

n)

o)

p)

q)

s)

t)

u)

v)

w)

y)

z)

F13.

a)

F14.

a)

F15.

a)

F16.

F17.

5 4

b)

7 4

c)

7 5

d)

4 3

e)

16 12

f)

12 5

7 1 3 1 9 1 =2 b) = 1 c) = 2 d) 3 3 2 2 4 4 12 2 25 5 20 4 j) =2 k) =2 l) =2 5 5 10 10 8 8 16 4 24 6 32 6 r) =2 s) =2 t) =2 6 6 9 9 13 13

g)

7 5

h)

11 7 i) 4 3

j)

13 10

k)

r)

x)

13 13 l) 6 5

20 2 13 1 23 5 7 1 11 3 8 2 =3 e) =2 f) =2 g) = 2 h) = 2 i) = 2 6 6 6 6 9 9 3 3 4 4 3 3 15 3 21 1 24 6 13 3 34 4 m) =3 n) =2 o) =1 p) =1 q) =3 4 4 10 10 18 18 10 10 10 10 24 136 36 54 6 111 13 u) = 2 v) =1 w) =3 x) =2 12 100 100 16 16 49 49

3 8 15 7 17 103 53 23 67 18 26 23 21 b) c) d) e) f) g) h) i) j) k) l) m) 2 3 4 2 7 20 10 4 30 13 9 6 8 1207 103 24 221 25 35 149 211 2 20 n) o) p) q) r) s) t) u) v) w) x) not defined 100 50 5 100 12 11 50 100 9 10

y)

12 3

z)

1 1 1 1 6 13 5 4 2 2 8 2 6 b) 1 c) 1 d) 4 e) 1 f )1 g ) 12 h) 2 i) 2 j) 6 k) 8 l) 8 m) 8 2 3 4 2 7 20 10 5 3 3 9 6 8 7 3 2 1 4 4 11 1 0 4 5 n) 1 o) 2 p) 4 q) 2 r) 1 s) 6 t) 2 u ) 11 v) 0 w) not defined x) 1 y) 1 10 5 5 10 12 11 50 10 9 7 15

35 10

a) 1

a ) 1 b ) 1 c ) 1 d ) 2 e) 3 q ) not defined

f ) 8 g ) 10 h) 2 i ) 3

j ) 4 k ) 6 l ) 15 m) 4 n) 5 o) 0

z) 7

7 9

p ) not defined

r ) 1 s ) 2 t ) 0 u ) not defined

F18. The answers may vary. 2 5 6 9 14 21 8 12 2 12 10 45 100 30 0 0 50 125 100 700 ; b) ; c) ; d) ; e) ; f) ; g) ; h) ; i) ; j) ; 2 5 2 3 2 3 2 3 1 6 2 9 10 3 2 3 2 5 1 7 22 55 16 24 26 39 34 51 k) ; l) ; m) ; n) ; 2 5 2 3 2 3 2 3 a ) improper b) whole c) improper d ) proper e) mixed f ) improper g ) mixed h) not defined i ) proper j ) whole k ) improper l ) proper m) mixed n) mixed o) mixed p ) improper q ) whole r ) improper s ) improper t ) mixed u ) proper v) mixed w) improper x) proper y ) improper z ) improper a)

F19.

F20.

r)

F21.

2 3 5 52 c) d) e) 3 4 9 100 7 25 10 5 s) t) u) v) 9 49 12 9

a ) 1 b) 6 8

7 10 6 w) 12 f)

4 5 4 h) i) 6 6 6 7 12 x) y) 10 16 g)

j)

2 3

k)

10 18

l)

4 6

m)

3 5

n)

7 10

o)

3 4

p)

10 13

q)

3 4

2 4 1 9 1 12 3 120 20 24 4 8 2 20 2 7 1 = 1 b) = 1 c) = 2 d) =1 e) =1 f) =2 g) = 1 h) =3 i) = 1 2 3 3 4 4 9 9 100 100 10 10 6 6 6 6 6 6 4 1 19 1 19 1 6 1 13 3 5 1 15 2 10 2 j) = 1 k) =1 l) =3 m) = 1 n) =1 o) = 1 p) =1 q) =2 3 3 18 18 6 6 5 5 10 10 4 4 13 13 4 4

a)

© La Citadelle

57

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The Book of Fractions r)

10 2 =1 8 8

s)

13 4 =1 9 9

F22. a) 3 b) 2 c) 3 d ) 8 4 10 r) 20

3 18 s) 25

5

63 14 =1 49 49

t)

11 7 u) 19

7 t) 9

Iulia & Teodoru Gugoiu

3 6

e)

v)

f) 5 10

u)

5 7

3 4 p) 1 9

4 7 q) 1 10

5

r) 1

12 40

3

4

3 q ) 10 10

s) 4

1 4 t) 2 3 10

2 o) 27 11

3

5

f)9

6

11 q ) 40 r ) 14 30

s ) 15

o) 2

3 11

p) 3

4

8 9

q) 3

2 7

1 2

F27. a) 3 b) 2 3 c) 4 d ) 3 e) 4 1 4

o) 10

3 11

p) 5

q) 7

2 10

r ) 16

52 100

f ) 5 g) 3

5

6 9

s) 6

1 2

s) 6

h) 1

33 100

2 10

t) 7

49 50

3 9

i) 2

21 30

w) 5 h) 4

1 9

18 6 =1 12 12

x)

28 12 =1 16 16

12 11 32 k) l) 41 13 54

7 11

j) 1

x) 8 8 10

1 23

u ) 11 v) 9

2 12

1 3

k) 1

y) 4

m)

6 15

4 9

h) 8 i ) 4

11 3 k ) 4 l) 7 15 7 20 10 x) 7 y) 4 35 100 1 10

14 30

w) 5

j) 2 k ) 6 17 23

1 10

u ) 4 v) 8

j) 5

4 6

2 4

w) 6

k) 4

x) 3

2 3

2 5

4 5

m) 1

6 10

5 9

m) 6

l) 7

2 35

l) 6

x) 4

1 3 1 2 f) = g) = 3 9 5 10 2 6 2 4 4 1 2 4 8 16 1 5 o) = = = = p) = 24 2 4 8 16 32 4 20 1 2 4 2 4 8 16 w) = = x) = = = y) 4 8 16 3 6 12 24

9 12

o)

5 25

13 17

n) 1

p)

0 4

38 50

q)

o) 1

5 13

10 100

9 10

j) 5

i ) 12

8 2 =1 6 6

n)

3 20

l) 1

1 7

y ) 113

13 4 2 8 j ) 12 k) 3 l) 2 50 6 5 9 1 9 v) 8 w) 8 x) 3 y) 3 4 10

18 50

y)

5 10

h) 9 i ) 4

F28. a) 1 = 2 b) 1 = 2 c) 1 = 2 = 4 d ) 2 = 6 e) 1 = 2 2 4 3 6 2 4 8 1 2 4 1 2 1 2 l) = = m) = n) = = 4 8 16 2 4 6 12 1 3 1 2 4 1 3 t) = u) = = v) = 4 12 9 18 36 4 12

j)

w) 1

v) 8

t ) 11 u ) 2 7 10

29 35

i) 1

h) 3

5 8

w)

4 10

5 8

g ) 13

10

r ) 12

40 400

1 4 t) 8 3 10

2

9 10

y) 6 9

g) 5

F26. a) 6 b) 1 2 c) 7 1 d ) 7 1 e) 8 f ) 3 g ) 6 7 4

i)

v) 3

6 1 14 s ) 10 t ) 9 u) 4 3 19

4

1 p ) 20 9

8 9

5 7

30 100

11 13

g) 1

u) 1

f)5

F25. a) 3 1 b) 3 2 c) 4 1 d ) 7 2 e) 7 5 2

4 8

13 4 =1 9 9

v)

h)

x)

f )1

7

5

3 r) 6 30

12 19

20 30

6

F24. a) 4 b) 3 2 c) 3 3 d ) 4 4 e) 3 5 1 p) 8 9

g)

w)

F23. a) 1 1 b) 1 1 c) 1 2 d ) 1 2 e) 1 1

19 7 =1 12 12

3 10

m) 5

n) 8

1 10

3 20

n) 5

o) 7

5 11

2 20

11 100

m) 8 n) 4

3 12

m) 7 n) 5 y) 9

1 10

1 2 2 6 1 2 1 3 = i) = j) = k) = 3 6 3 9 4 8 6 18 1 2 1 2 1 2 4 8 q) = r) = s) = = = 6 12 4 8 5 10 20 40 3 6 12 24 48 = = = = 4 8 16 32 64 h)

F29. The answers may vary. 2 3 4 2 3 4 4 6 8 2 3 4 6 9 12 2 3 4 6 9 12 = = b) = = c) = = d) = = e) = = f) = = g) = = 4 6 8 6 9 12 6 9 12 8 12 16 8 12 16 10 15 20 10 15 20 2 3 4 10 15 20 2 3 4 4 6 8 2 3 4 2 3 4 h) = = i) = = j) = = k) = = l) = = m) = = 12 18 24 12 18 24 14 21 28 14 21 28 20 30 40 200 300 400 10 15 20 4 6 8 4 6 8 6 9 12 10 15 20 4 6 8 n) = = o) = = p) = = q) = = r) = = s) = = 24 36 48 22 33 44 18 27 36 14 21 28 22 33 44 30 45 60 8 12 16 10 15 20 2 3 4 6 9 12 6 9 12 6 9 12 t) = = u) = = v) = = w) = = x) = = y) = = 10 15 20 14 21 28 60 90 120 100 150 200 400 600 800 2000 3000 4000 1 1 2 1 1 3 2 1 5 2 1 2 1 4 2 3 10 6 2 5 a) b) c) = d) e) = = f) = = g) = h) i) j) k) = = l) 2 3 8 4 5 18 12 6 50 20 10 6 3 5 3 4 15 9 3 8 9 6 3 30 20 12 10 6 4 2 30 20 12 10 6 4 2 4 25 15 5 m) = = n) = = = = = = o) = = = = = = p) q) = = 30 20 10 45 30 18 15 9 6 3 75 50 30 25 15 10 5 7 40 24 8 33 6 3 28 14 8 4 2 16 8 4 2 1 9 3 1 9 2 45 9 r) = = s) = = = = t) = = = = u) = = v) w) x) = 77 14 7 210 105 60 30 15 64 32 16 8 4 27 9 3 11 11 125 25 32 16 8 4 2 1 y) = = = = = 512 256 128 64 32 16 a)

F30.

© La Citadelle

58

www.la-citadelle.com

The Book of Fractions F31. a) 1

12 5 r) 22

Iulia & Teodoru Gugoiu

3 2 1 3 1 3 1 5 c) d) e) f) g) h) i) 7 3 4 4 5 5 6 6 2 1 5 7 11 3 7 s) t) u) v) w) x) y) 15 2 7 15 25 20 10

b)

F32. a) 3 b) 4 c) 2 d ) 3 e) 7 4 5 3 2 r) s) 7 15

3 1 t) 4

j)

2 8 2 7 g) h) i) 7 15 5 13 2 9 1 w) x) y) 11 25 16

f)

4 9 1 9 u) v) 3 11

1 7

j)

k)

4 5

2 7

k)

2 3

l)

3 14

m)

7 9

5 8

m)

3 10

l)

n)

1 8

n)

o)

4 5

2 11

p)

2 9

q)

3 7

2 5

p)

4 9

q)

5 8

o)

F33. a ) yes b) yes c) no d ) yes e) no f ) yes g ) no h) yes i ) no j ) no k ) no l ) yes m) yes n) no o) no p ) yes q ) no r ) yes s ) no t ) yes u ) yes v) no w) yes x) yes y ) no F34. a ) no b) yes c) no d ) yes e) no f ) no g ) yes h) yes i ) no j ) yes k ) no l ) yes m) no n) yes o) no p ) no q ) yes r ) no s ) no t ) yes u ) yes v) no w) no x) yes y ) yes F35. a) 4 b) 2 c) 16 d ) 4 e) 20 f ) 8 g ) 3 h) 4 i) 15 j ) 60 k ) 15 l ) 54 m) 30 n) 72 o) 5 F36. a ) 3 b) 21 c) 15 d ) 9 e) 6

f ) 15 g ) 30 h) 56 i ) 27

F37. a ) 2 b) 15 c) 5 d ) 25 e) 7

f ) 3 g ) 8 h) 12 i ) 24

F38. a) 1

1 2

b) 1

1 5

c) 3

F39. a) 5 b) 1 5 6

q) 3

7 15

F40. a) 1 1 b) 7 6 1 p) 2 6

F41. a) 1 1

q) 4

12 41 p) 1 150

b)

4 15

25 48

q) 2

d)

r) 5 19 20

c)

12

d) 3

13 20

c)

12

1 p) 2 6

3 4

7 75

5 6

24 199 p) 1 2400

e)

s) 3

1 110

7 30

s) 2

17 80

d)

r) 3

47 144

F42. a) 1 11 b) 1 11 c) 1 29 12 107 o) 525

e) 1

11 30

d)1

r) 4 c)

1 6

1 3

120 17 q) 1 60

f)6

29 35

e) 7 15

7 20

11 16

f )1

t) 3 26 35 t) 4

13 20 f)

7 20

6 7

g) 3

3 56 u)

g)

u)

g) 9 20

18 13 f )1 35 28 13 37 s) 1 t) 3 150 450 e)

h) 3

67 72

h)

v) 1

79 200

3 10

53 56

3 4

5 6

h) 1

v) 1

13 40

3 1 i) 5 2

77 90

1 18

5 11

j ) 84 j ) 30 k ) 64 l ) 36 m) 18 n) 48 o) 35 j ) 38

4 7

43 31 17 j) k) 110 110 30 1 13 5 w) 2 x) 3 y) 1 6 24 32 i)

l) 3

19 30

8 2 8 7 j) k) l) 3 15 3 15 12 7 1 5 w) 1 x) 4 y) 1 12 12 12 i)

m) 1

m) 1

2 9

2 9

n)

67 23 19 5 17 11 13 h) i) j) k) l) m) 144 36 55 22 30 15 18 27 59 79 3 13 u) v) 1 w) 1 x) 3 y) 1 80 200 120 28 192 g)

41 2 2 197 e) 1 f) g) 48 15 5 252 19 7 227 r) s) 1 t) 20 60 300

d)1

h) 1

9 20

i)

181 240

j)

11 30

k)

31 180

l) 1

7 60

n)

7 30

n)

7 30

o)

9 50

m) 1

7 15

o)

2 5

1 2

o)

22 75

n)

38 75

F43. a) 2 - 1 = 1 b) 2 - 1 = 1 c) 3 - 2 = 1 d ) 5 - 3 = 2 e) 50 - 20 = 3 2 2 2 3 3 3 4 4 4 9 9 9 3 1 1 4 2 1 2 1 1 16 8 4 h) - = i) - = j) - = k) - = 6 6 3 6 6 3 3 3 3 18 18 9

7 5 1 5 3 1 f) = g) - = 100 100 10 10 10 5 6 6 3 3 2 1 3 2 1 6 2 2 l) - = m) - = n) = 6 6 6 5 5 5 10 10 5

3 1 1 9 4 5 3 2 1 6 2 1 5 2 3 1 23 8 15 - = p) - = q) - = r) - = s) - = = t) = 4 4 2 13 13 13 4 4 4 8 8 2 9 9 9 3 49 49 49 8 4 1 5 2 1 8 2 1 7 3 2 11 4 7 u) = v) - = w) = x) = y) = 12 12 3 9 9 3 12 12 2 10 10 5 16 16 16 o)

F44. a) 1 b) 2 c) 3 d ) 7 4 1 p) 2

3 3 q) 4

© La Citadelle

5 1 r) 4

11 3 s) 10

1 3 2 1 2 1 1 1 f) g) h) i) j) k) l) 2 7 19 5 7 10 3 9 1 1 1 1 1 6 1 t) u) v) w) x) y) = 250 10 5 5 3 60 10

e)

59

m)

6 2 = 9 3

n)

1 8

o) 2

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The Book of Fractions F45. a) 2 b) 1 c) 1 1 d ) 9 3

p) 1

5

1 2

q) 5

5 8

e) 2

11

r)

5 6

Iulia & Teodoru Gugoiu

s) 2

1 4

1 2

t) 1

1 500

F46. a) 1 1 b) 3 c) 8 2 d ) 6 5 e) 2 1 3 1 p) 3 2

5

1 q) 4 2

11

1 r) 2 2

2

1 s) 2 4

5 7

f)2

t) 3

g) 1

u)

24 25

f)3

3 7

1 50

u) 2

8 19

h) 9 i ) 1 1 5

w) 1

3 25

10 19

h) 3

4 5

v) 3 g) 1 1 5

5 7

v) 2

1 5

w) 5

j) 1 x)

10 11

k) 1 y) 3

5 6

l) 3

2 7

j) 2

1 4

1 6

l) 2

x) 2

2 11

y) 2

k) 3

7

1 4

p) 3

q) 4

3 8

F48. a) 1 b) 1 6 5 p) 6

F50. a) 7

12 121 p) 150

3 1 1 11 13 23 23 9 1 d) e) f) g) h) i) j) k) 20 30 35 56 72 90 110 110 30 1 89 17 1 151 1 19 31 r) s) t) u) v) w) 1 x) y) 6 110 20 10 200 3 24 32

2 15

F49. a) 1 b) 1 6 5 p) 6

12 1 q) 1 15

c)

11 20

r) 1

13 6 e) 30 35 13 17 s) t) 15 20

d) 5 6

11 1 1 2 1 2 g) h) i) j) k) 56 6 18 15 12 15 1 37 1 7 5 u) v) w) 1 x) y) 1 10 40 12 12 24 f)

47 8 e) 120 48 15 13 5 31 r) s) t) 60 12 300

24 7 q) 60

f)

1 5

g)

31 252

h)

23 17 i) 60 48

F52. a) 1 17 b) 13 c) 8 13 d ) 23 e) 1 f ) 1 g ) 1 h) 23 i) 53 60 60 7 3 p) q) 2 12 40

F53. a)

3 1 b) 8 3

c)

60 60 2 18 7 7 r) s) 1 t ) 1 u) 25 12 30

3 20

d)

3 2

e)

3 8

f ) 2 g)

F54. a) 3 b) 1 1 c) 2 d ) 4 e) 1 1 4

p) 3

1 3

l)

1 4

6 1 q) 2

5

6

f )1

1 2

60

1 4

h)

g) 3

3 4

1 8

i)

60

5 8

j)

h) 4 i ) 4

1 40

3 8

j)

2 15

j) 0 k )

k)

j) 2

4 9

4 11

l)

2 7

m) 1

2 3

n) 3

1 2

m) 1

2 3

n) 1

k)

13 60

2 5

7 60

l) 1

l)

4 15

5 12

2 5

F58. a) 3 b) 1 c) 1 d ) 2 e) 1 f ) 1 g ) 2 2 h) 1 i) 7 1 4

3

© La Citadelle

1 6

o)

1 10

m)

5 18

n)

1 50

1 5

o) 3

o) 2

3 7

4 7

o) 0

o)

2 75

n)

4 75

o)

22 525

n) 4

7 60

o)

3 20

o)

m)

1 2

n) 4 o) 3

1 2

n) 22 o) 2

m) 6 n)

1 2

1 2

1 30

2 5

F56.

q)

n)

25 28

m) 8

1 3

1 4

1 4

r ) 6 s) 4 t ) 6

3 4 3 2 1 1 2 b) 1 c ) d ) 2 e) 3 f ) 1 g ) 3 h) i) 4 j) 3 k ) 4 l ) 10 4 5 4 3 2 2 3 1 1 1 8 1 7 1 1 a) b) c) d) e) f )1 g) 2 h) 1 i ) j ) 1 k ) 60 l ) 5 3 2 25 15 15 8 4 4 1 1 3 1 1 3 4 1 1 a) b) c) 1 d ) e) 3 f ) 1 g) h) i) j) k ) 1 l) m) 3 2 5 2 4 5 5 20 4

p) 3

o) 0

n)

m)

F55. a)

F57.

13 16

m) 2

m) 2 n) 2

k ) 4 l) 4

1 9

31 60

l)

1 9

m)

m)

7 7 11 3 1 13 7 14 7 1 c) d) e) f )1 g) h) i) j) k) 48 80 60 35 28 144 36 55 66 30 73 11 7 7 1 191 1 1 187 q) r) 1 s) t) u) v) w) x) 1 y) 75 144 150 450 16 200 120 28 192 d)

n)

11 30

l)

b)

F51. a) 11 b) 23 c) 19 12 1 p) 48

l) 5

c)

12

q) 2

1 5 8 5 19 5 e) 4 f )1 g) 1 h) 9 i ) 1 j) 1 k) 1 11 2 9 19 7 20 6 1 1 24 4 18 7 13 s) 3 t) 2 u) 3 v) 5 w) 4 x) y) 2 4 100 25 5 25 11 15

5 5 r) 6

5 9

1 3

F47. a) 2 b) 2 5 c) 2 4 d ) 2 8 3

m) 2

1 10

1 7

i) 1 3 25

19 20

r)

7 8

s) 1

3

1 4

t) 4

2

j) 1

1 10

k ) 1 l) 2

1 3

1 2

n)

11 18

1 3

o) 3

o) 8

1 3

3 10

4 7

m) 6 n) 24 o) 4

1 2

60

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The Book of Fractions

Iulia & Teodoru Gugoiu

1 1 1 1 3 3 1 1 15 1 7 63 b) c) d ) 3 e) f)2 g) 1 h) 2 i) 1 j ) 1 k) l) m) n) 4 5 8 15 4 4 2 2 32 6 36 256 9 1 1 1 1 1 5 1 1 11 13 1 a) b) 1 c ) d) e) f) g) h) i) 1 j) 1 k ) l) m) n) 3 o) 3 10 2 6 30 6 2 6 6 12 12 24 2

F59. a) F60.

F61. a) 2 b) 1 c) 1 3

6 65 p) 0 q) 77

2 1 2 a) 3

F64.

1 4

17 144

e)

f ) 0 g)

1 3

h) 1

1 2 i) 3 15

j)

2 7

k)

1 4 3 6 3 2 c) d) e) not defined f ) g) h) 2 i ) 1 2 5 7 7 13 3 3 5 7 3 1 11 7 4 4 b) c) d) e) f) g) h) i) j) 8 12 8 11 4 24 19 5 15 1 1 1 a ) 1 b) c) d) e) not defined 2 5 100

F62. a) F63.

10

d)

b) 1

1 6

l)

1 2

m)

1 4

n)

2 3

o) 1

j ) not defined

F65. a) yes b) no c) no d ) yes e) no f ) no g ) yes h) no i) yes j ) yes F66. a) 4 b) 2 c) 2 F67. a)

2 3

b) 1

1 2

1 2

c) 1

d) 1 2

1 5

d)

e) 3 4

4 9

e) 1

F68. a) 5 b) 2 c) 1 1 d ) 4 e) 4 3 p) 5

3 2 q) 3

F69. a) 2 b) 5 6 3 a) 2

F72. F73. F74. F75.

F78.

1 4

1 4

b) 2

b)

1 F79. a) æç ö÷

4

è 2ø

F80.

c) 5 d )

1 6

e) 6

a)

3 3 ´ 4 4

1 2

2 3

d ) 7 e)

1 2

d) 5

f)

g) 5 9

1 3

h)

g) 3

f ) 1 g)

f)2

1 7

g)

2 3

1 3

h) 1

1 6

9 4 i) 11 7

5 14 2 3 5 j) k) l) 3 8 8

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20 1 1 11 e) 4 f ) 2 g) 2 h) 21 4 2 12 3 2 3 1 1 47 a) 4 b) 1 c ) 3 d) e) f) g) 10 5 5 24 2 60 4 3 5 9 5 1 1 a) b) c) d ) 12 e) f) g) 1 h) i) 1 5 8 27 35 14 3 2

F76. a) F77.

7 27 t) 28

1 r ) 6 s) 6 4

1 4

1 2

1 4

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n)

3 4

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m) 10 n) 3 o) 1

l) 2

1 5

m) 3

1 2

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5 7

1 28

o) 1

2 3

4 5 4 1 3 b) 6 c ) d) e) 7 10 7 1 5 2 6 1 1 1 5 9 1 3 a ) 1 b) 1 c) 1 d) e) f ) 3 g) 1 h) 1 i) 9 j ) 3 k ) l) m) n) o) 2 6 7 11 2 3 6 8 50 9 5 3 1 7 2 1 1 1 3 2 4 2 1 4 7 a) b) 1 c) 2 d) e) 1 f )1 g) h) i) j) k) l ) 4 m) 2 n) o) 4 9 9 3 2 3 6 4 5 9 3 4 7 2 15 2 1 32 1 1 1 1 4 3 1 a) b) c) d ) 9 e) f) g) 1 h) i) j) k) l) 16 15 120 35 4 2 32 3 81 2 4 5 1 7 8 1 1 3 4 1 1 a) b) 4 c) d) e) f) g) h) 8 i ) j) 1 k) 1 9 8 16 15 6 4 8 9 15 2

F70. a) F71.

1 6

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2

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5 5 5 5 ´ ´ ´ 3 3 3 3

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The Book of Fractions F81. F82. F83. F84. F85. F86.

Iulia & Teodoru Gugoiu

3 4 27 1 c) d) e) 8 f ) 8 25 125 81 1 1 3 7 2 a) b) c) 16 d ) 3 e) 1 f )1 9 25 8 9 5 49 36 1 13 1 a) b) c) d) e) f)5 144 49 4 18 18 2 9 1 4 9 a) b) c) d) e) 1 f ) 1 3 25 16 25 16 1 1 15 1 a) 3 b) 1 c ) 1 d) e) 2 f)2 2 2 16 2 a ) 1 b) 3

g ) 16 h) 5 g)

9 16

m) t

n) t

h) 15

5 8

49 144 1 h) 12

g ) 1 h)

1 16 1 g) 2 h) 3 i ) 1 2 g)

Legend : t = terminating ; nt = non - terminating k ) t l ) nt

1 16

a ) t b) nt

j) 5

c) t

1 3

d) t

k)

1 2

e) nt

l) 1

1 3

f ) nt

g ) nt

h) t i ) nt

j ) nt

o) nt

F87.

a ) 0.5 b) 0.75 c) 0.625 d ) 0.2 e) 0.15

F88.

a ) 0.3 b) 0.16 c) 0.305 d ) 0.7 e) 0.18

F89.

a ) 1.5 b) 0.4 c) 1.3 d ) 0.013 e) 2.3

f ) 0.1375 g ) 0.13 h) 0.008 i ) 0.006

j ) 0.0016

f ) 0.06 g ) 0.23 h) 0.285714 i ) 0.153846

j ) 0.037

f ) 2.25 g ) 0.078125 h) 0.03125 i ) 0.046875

j ) 0.0078125 k ) 0.012

l ) 0.142857 m) 1.428571 n) 0.416 o) 2.230769

F90. F91. F92. F93.

1 1 b) 10 2 5 1 a) b) 1 8 2 1 2 a) b) 1 3 9 7 1 a) b) 1 9 3 a)

2 1 3 7 1 1 3 29 d)1 e) f) g) 2 h) 10 i) 5 j ) 100 5 4 4 200 8 8 40 40 1 2 4 11 6 7 9 41 c) d) e) 2 f) g) h) i) 2 j) 8 5 25 40 25 20 20 64 11 19 23 23 25 2 41 7811 c) d)1 e) 4 f) g) 1 h) 2 i) j) 1 90 90 900 99 99 165 333 33300 8 29 13 4 1 103 679 41 c) 2 d)1 e) 1 f) g) 3 h) 1 i) 6 j) 1 15 90 100 33 99 330 5500 3330 c) 1

F94.

a ) 0.4 b) 0.35 c) 1.2 d ) 2 e) 0.5

F95.

a ) 2 b)

F96.

7 7 3 21 c) d)1 e) 20 10 5 25 98 19 1 a ) 2 b) 1 c) 1 d ) 26 e) 99 81 3

F97.

a ) 1.45 b) 0.95 c) 1.24 d ) 0.05

F98.

a)

F99.

11 1 1 3 b) c) d) 20 5 3 20 3 11 5 1 1 a) b) c) d)1 e) 5 16 6 4 4

f )1

1 3

g)

7 10

h) 1 i )

5 6

j) 1

1 2

F100. a) 30 min b) 20 min c) 105 min d ) 165 min e) 6 min f ) 50 min g ) 135 min h) 22 min i) 28 min j ) 74 min 3 1 1 1 h h) 4 h i ) h j) h 5 6 80 24 2 a ) 90 s b) 100 s c) 15 s d ) 84 s e) 69 s f ) 35 s g ) 105 s h) 25 s i ) 14 s j ) 440 s 3 1 1 3 1 1 7 1 21 5 31 a ) min b) min c) min d ) 1 min e) 3 min f ) min g ) min h) min i ) min j ) min 5 6 4 2 3 40 80 80 9 90 5 5 1 1 2 a) h b) h c) 2 h d ) 720 s e) 6300 s f ) 540 s g ) 7800 s h) 905 s i ) 30 min j ) 50 min 36 12 2 4 3

F101. a) F102. F103. F104.

1 1 1 5 5 h b) h c ) h d ) h e) h 12 6 4 12 6

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1 f )1 h 6

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62

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The Book of Fractions

Iulia & Teodoru Gugoiu

4 5

2 3

2 3

F105. a) 12 cents b) cents c) 102 cents d ) 102 cents e) 7 cents f ) 4 cents 1 4

1 5

9 3 1 11 $ d ) 1 $ e) 4 $ f ) $ 20 5 2 200 2 3 a ) 4 cents b) 190 cents c) 6 cents d ) 14 cents e) 260 cents f ) 9 cents 3 5 5 1 1 3 1 51 a ) twonies b) 1 twonies c) twonies d ) twonies e) 2 twonies f ) twonies 8 4 5 5 2 1000 3 3 1 a ) 1 cent b ) 6 cents c ) 1 cents d ) 7 cents e) 10 cents f ) 26 cents 4 4 4 4 3 1 a ) 3 nickels b) 5 nickels c) 15 nickels d ) nickels e) nickels f ) nickels 5 10 10 4 a ) 5 cents b) 40 cents c) 3 cents d ) 1 cents e) 40 cents f ) 6 cents 5 4 1 1 a ) 2 quarters b) 5 quarters c) 4 quarters d ) quarters e) quarters f ) 2 quarters 5 50 5

F106. a) $ b) $ c) F107. F108. F109. F110. F111. F112.

F113. a) 15 nickels

b) 3 nickels

c) 1 quarter

d)

1 dimes 2

e) 4 dimes

f)7

1 dimes 2

g ) 9 nickels

h) 12

1 dimes 2

F114. 1 1 3 5 = 5 to 20 = 25 % = 0.25 b) = 7 out of 21 = 33 .3 % = 0.3 c) = 12 to 8 = 150 % = 1.5 d ) = 15 out of 18 = 83 .3 % = 0.83 4 3 2 6 3 2 2 1 e) = 42 out of 70 = 60 % = 0.6 f ) = 2 out of 5 = 40 % = 0.4 g ) = 10 to 45 = 22 .2 % = 0.2 h) = 3 out of 12 = 25 % = 0.25 5 5 9 4 6 7 3 4 i ) = 18 to 15 = 120 % = 1.2 j ) = 7 to 10 = 70 % = 0.7 k ) = 21 to 70 = 30 % = 0.3 l ) = 36 out of 45 = 80 % = 0.8 5 10 10 5 13 1 3 3 m) = 39 to 30 = 130 % = 1.3 n) = 0.5 out of 5 = 10 % = 0.1 o) = 48 out of 64 = 75 % = 0.75 p ) = 1.5 to 10 = 15 % = 0.15 10 10 4 20 7 2 16 r ) = 49 out of 35 = 140 % = 1.4 s ) = 6 to 75 = 8 % = 0.08 t ) = 80 out of 125 = 64 % = 0.64 5 25 25 1 u ) = 32 out of 256 = 12 .5 % = 0.125 8 a)

F115.

a ) = b ) > c ) > d ) < e) >

F116.

a) M

F117.

1 A => 12

b) S

c) C

1 B => 6

f)
15

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g ) O h) d 1 E => 2

j)
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G =>

7 10

4 5

41 60

H =>

47 60

I =>

11 12

J => 1

1 12

F118. 0

1 10

1 5

3 10

2 5

3 5

1 2

9 10

1

1

1 10

1

1 5

1

3 10

1

2 5

F119. a) < b) = c) > d ) > e) < f ) < g ) < F120.

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j ) = k ) > l ) < m) = n) >

F121.

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g ) < h) > i ) >

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F123.

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g)
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F124. a) < < F125.

F126. a) 1

b) 1

2 15

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5 12

b)

13 44

c)

2 3

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2 3

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10 8 p) 15

F127. a) 3

2 1 2 1 5 3 3 d)1 e) 1 f) g) h) 3 i) 1 3 3 5 15 18 8 5 9 11 5 8 2 2 q) 1 r ) 1 s) 7 t) u) v) 1 w) 7 20 25 36 9 3 3 11 15

1 2

k)

1 2

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5 6

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2 15

o) 1

d)1

5 3 3 7 17 31 17 12 8 2 1 3 5) 4 6) 7) 8) 9) 10) 11) 12) 13) 1 14) 2 15) 3 7 4 2 3 5 7 3 5 3 3 2 4 2 3 1 1 3 2 1 2 9 2 3 16) 2 17) 1 18) 1 19) 2 20) 2 21) 22) 2 23) 24) 25) no 26) yes 27) yes 28) 3 29) 30) 4 3 5 4 7 4 3 2 3 16 3 5 1 1 17 11 1 1 2 1 1 7 11 2 1 31) 1 32) 5 33) 3 34) 2 35) 36) 1 37) 38) 39) 1 40) 2 41) 42) 43) 1 44) 1 12 6 60 12 3 3 7 12 10 12 20 5 5 1 1 7 1 8 2 1 15 45) 2 46) 47) 2 48) 2 49) 50) 5 51) 52) 3 53) 54) 4 55) 56) 0.4 57) 2.4 58) 0.4 59) 1.25 3 3 9 3 9 9 3 16 3 1 1 1 4 7 29 13 60) 0.6 61) 1.285714 62) 0.3125 63) 0.875 64) 3.125 65) 66) 1 67) 2 68) 69) 70) 1 71) 72) 1 25 2 4 80 9 11 40 20 7 2 5 61 7 7 1 1 5 73) 74) 27 75) 6 76) 1 77) 78) 79) 3 80) 1 81) 82) 12 83) 8 84) 2 85) 9 86) 87) 10 88) 8 3 9 180 8 40 2 2 12 1 2 1 5 1 1 1 7 2 1 1 89) 90) 2 91) 92) 93) 1 94) 95) 96) 97) 98) 2 99) 4 100) 6 2 3 3 18 3 2 10 12 15 2 6

FT.

1)

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3 8

j) 3

4)

64

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