Unit3

Indice Definition Proper fractions Equivalent fractions Operations Exercises

Fractions

Matematicas 1o E.S.O.Alberto Pardo Milanes

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1 Definition

2 Proper fractions and improper fractions

3 Equivalent fractions

4 Operations

5 Exercises

Alberto Pardo Milanes Fractions


Definition



Definition

What is a fraction?

A fraction is a number that represents a part of something.

Fractions are written in the forma

b, where a and b are naturals,

and the number b is not zero. Reada

ba over b.

The number a is called numerator, and the number b is calleddenominator.Example:

3

9

3 is the numerator,9 is the denominator.



Proper fractions and improperfractions



Proper fractions and improper fractions

Proper fractionA proper fraction is a fraction such that

a

b< 1.

(The numerator is less than the denominator)

Improper fractionA fraction

a

b> 1 is called an improper fraction.

(The numerator is greater than the denominator)

Examples:2

7is a proper fraction.

9

7is an improper fraction.

2 is less than 7 9 is greater than 7



Equivalent fractions




Definition

Equivalent fractions are different fractions which represent thesame amount.Example:

2

7=

4

14because they represent the same amount.

2

7

4

14




Amplify and reduce fractions

To amplify a fraction we must multiply the numerator and thedenominator by any number.To reduce a fraction we must divide the numerator and thedenominator by any common factor.Examples:

To amplify2

7we can multiply the numerator and the denominator

by 3 =⇒ 6

21.

To reduce15

45we can divide the numerator and the denominator by

5 =⇒ 3

9.




Simplest form of a fraction

A lowest terms fraction is a fraction that can not be reducedanymore. If you reduce a fraction to the lowest terms fraction youfind the simplest form of the fraction.To reduce a fraction to the simplest form, we can use twomethods:

• Divide the numerator and the denominator by any commonfactor and keep dividing until there are no common factors(only 1).

• Divide the numerator and the denominator by their GreatestCommon Factor.




How to obtain the simplest form of a fraction

Method 1

• Divide the numerator and the denominator by any commonfactor and keep dividing until there are no common factors(only 1).

Example:

To reduce18

60to the lowest terms fractions we can divide by 2 and

keep dividing until there are no common factors (only 1)18

60=

9

30=

3

10, and GCF (3, 10) = 1.




How to obtain the simplest form of a fraction

Method 2

• Divide the numerator and the denominator by their GreatestCommon Factor.

Example:

To reduce18

60to the lowest terms fractions we can divide 18 and

60 by GCF (18, 60) = 6, so18

60=

3

10.



Operations



Operations

Add and subtract

To add or subtract fractions with the same denominator, we haveto add the numerators and keep the same denominator.

Examples:2

5+

1

5=

3

5.

4

7− 2

7=

2

7.

To add or subtract fractions with different denominators, first wehave to amplify these fractions changing the denominators to thesame number (using the LCM or any multiple of thedenominators).

Examples:3

15+

1

6=

11

30, because

LCM(15, 6) = 303

15=

6

30

1

6=

5

303

15+

1

6=

6

30+

5

30=

11

30.



Operations

Multiply and divide

To multiply fractions, multiply the numerators and multiply thedenominators.

Example:2

5· 34=

2 · 35 · 4

=6

20=

3

10.

You can get the reciprocal of a fraction by switching its numeratorand denominator.

Example: The reciprocal of4

7is

7

4.

To divide by a fraction, multiply by its reciprocal.

Example:1

8:4

7=

1

8· 74=

1 · 78 · 4

=7

32.



Exercises



Exercises

Exercise 1

Complete:

A half=

2

2=

A/one third=

2

3=

A quarter=

2

4=

Two fifths=

4

5=

An/one eighth=

1

6=

Three quarters=

3

7=

Forty over twenty-three=

35

75=



Exercises

Exercise 1

Complete:

A half=1

22

2=Two halves/ Two over two

A/one third=1

32

3=Two thirds / Two over three

A quarter=1

42

4=Two fourths / Two over four

Two fifths=2

5Alberto Pardo Milanes Fractions


Exercises

Exercise 1

Complete:

A half=1

22


A/one third=1

32


A quarter=1

42


Two fifths=2



Exercises

Exercise 1

Complete:

A half=1

22


A/one third=1

32


A quarter=1

42


Two fifths=2



Exercises

Exercise 1

Complete:

A half=1

22


A/one third=1

32


A quarter=1

42


Two fifths=2



Exercises

Exercise 1

Complete:

A half=1

22


A/one third=1

32


A quarter=1

42


Two fifths=2



Exercises

Exercise 1

Complete:

A half=1

22


A/one third=1

32


A quarter=1

42


Two fifths=2



Exercises

Exercise 1

Complete:

A half=1

22


A/one third=1

32


A quarter=1

42


Two fifths=2



Exercises

Exercise 1

Complete:

A half=1

22


A/one third=1

32


A quarter=1

42


Two fifths=2



Exercises

Exercise 1

Complete:4

5=Four fifths / Four over five

An/one eighth=1

81

6=A sixth / One over six

Three quarters=3

43

7=Three sevenths / Three over seven

Forty over twenty-three=4

2335

75=Thirty five over seventy-five



Exercises

Exercise 1

Complete:4


An/one eighth=1

81


Three quarters=3

43



2335




Exercises

Exercise 1

Complete:4


An/one eighth=1

81


Three quarters=3

43



2335




Exercises

Exercise 1

Complete:4


An/one eighth=1

81


Three quarters=3

43



2335




Exercises

Exercise 1

Complete:4


An/one eighth=1

81


Three quarters=3

43



2335




Exercises

Exercise 1

Complete:4


An/one eighth=1

81


Three quarters=3

43



2335




Exercises

Exercise 1

Complete:4


An/one eighth=1

81


Three quarters=3

43



2335




Exercises

Exercise 1

Complete:4


An/one eighth=1

81


Three quarters=3

43



2335




Exercises

Exercise 2

Calculate the following and write down how we read the answer:

3

5+

1

5=

2

4+

5

6=

1

3·(1

5+

4

15

)=

3

20− 2

4· 15=(

2− 1

5

):3

10=(

1

2+

1

3

)· 35=



Exercises

Exercise 2

Calculate the following and write down how we read the answer:3

5+

1

5=4

5four over five

2

4+

5

6=

6

12+

10

12=

16

12=

4

3four over three

1

3·(1

5+

4

15

)=1

3·(

3

15+

4

15

)=

1

3·7

15=

7

45

seven over forty-five3

20− 2

4· 15=

3

20−

2

20=

1

20one over twenty(

2− 1

5

):3

10=

(10

5−

1

5

):

3

10=

9

5:

3

10=

90

15= 6 six(

1

2+

1

3

)· 35=

(3

6+

2

6

)·3

5=

5

6·3

5=

15

30=

1

2A half



Exercises

Exercise 2


5+

1

5=4

5four over five

2

4+

5

6=

6

12+

10

12=

16

12=

4

3four over three

1

3·(1

5+

4

15

)=1

3·(

3

15+

4

15

)=

1

3·7

15=

7

45


20− 2

4· 15=

3

20−

2

20=

1

20one over twenty(

2− 1

5

):3

10=

(10

5−

1

5

):

3

10=

9

5:

3

10=

90

15= 6 six(

1

2+

1

3

)· 35=

(3

6+

2

6

)·3

5=

5

6·3

5=

15

30=

1

2A half



Exercises

Exercise 2


5+

1

5=4

5four over five

2

4+

5

6=

6

12+

10

12=

16

12=

4

3four over three

1

3·(1

5+

4

15

)=1

3·(

3

15+

4

15

)=

1

3·7

15=

7

45


20− 2

4· 15=

3

20−

2

20=

1

20one over twenty(

2− 1

5

):3

10=

(10

5−

1

5

):

3

10=

9

5:

3

10=

90

15= 6 six(

1

2+

1

3

)· 35=

(3

6+

2

6

)·3

5=

5

6·3

5=

15

30=

1

2A half



Exercises

Exercise 2


5+

1

5=4

5four over five

2

4+

5

6=

6

12+

10

12=

16

12=

4

3four over three

1

3·(1

5+

4

15

)=1

3·(

3

15+

4

15

)=

1

3·7

15=

7

45


20− 2

4· 15=

3

20−

2

20=

1

20one over twenty(

2− 1

5

):3

10=

(10

5−

1

5

):

3

10=

9

5:

3

10=

90

15= 6 six(

1

2+

1

3

)· 35=

(3

6+

2

6

)·3

5=

5

6·3

5=

15

30=

1

2A half



Exercises

Exercise 2


5+

1

5=4

5four over five

2

4+

5

6=

6

12+

10

12=

16

12=

4

3four over three

1

3·(1

5+

4

15

)=1

3·(

3

15+

4

15

)=

1

3·7

15=

7

45


20− 2

4· 15=

3

20−

2

20=

1

20one over twenty(

2− 1

5

):3

10=

(10

5−

1

5

):

3

10=

9

5:

3

10=

90

15= 6 six(

1

2+

1

3

)· 35=

(3

6+

2

6

)·3

5=

5

6·3

5=

15

30=

1

2A half



Exercises

Exercise 2


5+

1

5=4

5four over five

2

4+

5

6=

6

12+

10

12=

16

12=

4

3four over three

1

3·(1

5+

4

15

)=1

3·(

3

15+

4

15

)=

1

3·7

15=

7

45


20− 2

4· 15=

3

20−

2

20=

1

20one over twenty(

2− 1

5

):3

10=

(10

5−

1

5

):

3

10=

9

5:

3

10=

90

15= 6 six(

1

2+

1

3

)· 35=

(3

6+

2

6

)·3

5=

5

6·3

5=

15

30=

1

2A half



Exercises

Exercise 2


5+

1

5=4

5four over five

2

4+

5

6=

6

12+

10

12=

16

12=

4

3four over three

1

3·(1

5+

4

15

)=1

3·(

3

15+

4

15

)=

1

3·7

15=

7

45


20− 2

4· 15=

3

20−

2

20=

1

20one over twenty(

2− 1

5

):3

10=

(10

5−

1

5

):

3

10=

9

5:

3

10=

90

15= 6 six(

1

2+

1

3

)· 35=

(3

6+

2

6

)·3

5=

5

6·3

5=

15

30=

1

2A half



Exercises

Exercise 3

Fill in the blank with a number so the fractions are equivalent:

7=

16

14

27=

11

9

1

5=

25

11

22=

6

144

60=

12

2

5=

22



Exercises

Exercise 3

Fill in the blank with a number so the fractions are equivalent:8

7=

16

14

33

27=

11

9

1

5=

5

25

11

22=

3

6

144

60=

12

5

2

5=

22



Exercises

Exercise 4

Order these fractions from least to greatest:4

6,3

2,

9

15,7

5,1

3,13

6.



Exercises

Exercise 4


6,3

2,

9

15,7

5,1

3,13

6.

LCM(2, 3, 5, 6, 15) = 2 · 3 · 5 = 60

4

6=

20

30,3

2=

45

30,

9

15=

18

30,7

5=

42

30,1

3=

10

30,13

6=

65

30

1

3<

9

15<

4

6<

7

5<

3

2<

13

6.



Exercises

Exercise 4


6,3

2,

9

15,7

5,1

3,13

6.

LCM(2, 3, 5, 6, 15) = 2 · 3 · 5 = 60

4

6=

20

30,3

2=

45

30,

9

15=

18

30,7

5=

42

30,1

3=

10

30,13

6=

65

30

1

3<

9

15<

4

6<

7

5<

3

2<

13

6.



Exercises

Exercise 4


6,3

2,

9

15,7

5,1

3,13

6.

LCM(2, 3, 5, 6, 15) = 2 · 3 · 5 = 60

4

6=

20

30,3

2=

45

30,

9

15=

18

30,7

5=

42

30,1

3=

10

30,13

6=

65

30

1

3<

9

15<

4

6<

7

5<

3

2<

13

6.



Exercises

Exercise 5

Find the lowest terms fraction dividing the numerator anddenominator by common factors until the only common factor is 1:14

6=

18

45=

60

80=

22

44=



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 5


6=7

6

dividing by2

18

45=

6

15=

2

5

dividing by2 and 3

60

80=30

40=

3

4

dividing by2 and 10

22

44=11

22=

1

2

dividing by2 and 11



Exercises

Exercise 6

Find the lowest terms fraction using prime decomposition of thenumerator and denominator:

18

30=

63

42=

14

21=

22

33=



Exercises

Exercise 6


18

30=

2© · 3© · 32© · 3© · 5

=3

5

63

42=

3© · 3 · 7©2 · 3© · 7©

=3

2

14

21=

2 · 7©3 · 7©

=2

3

22

33=

2 · 11©3 · 11©

=2

3



Exercises

Exercise 6


18

30=

2© · 3© · 32© · 3© · 5

=3

5

63

42=

3© · 3 · 7©2 · 3© · 7©

=3

2

14

21=

2 · 7©3 · 7©

=2

3

22

33=

2 · 11©3 · 11©

=2

3



Exercises

Exercise 6


18

30=

2© · 3© · 32© · 3© · 5

=3

5

63

42=

3© · 3 · 7©2 · 3© · 7©

=3

2

14

21=

2 · 7©3 · 7©

=2

3

22

33=

2 · 11©3 · 11©

=2

3



Exercises

Exercise 6


18

30=

2© · 3© · 32© · 3© · 5

=3

5

63

42=

3© · 3 · 7©2 · 3© · 7©

=3

2

14

21=

2 · 7©3 · 7©

=2

3

22

33=

2 · 11©3 · 11©

=2

3



Exercises

Exercise 6


18

30=

2© · 3© · 32© · 3© · 5

=3

5

63

42=

3© · 3 · 7©2 · 3© · 7©

=3

2

14

21=

2 · 7©3 · 7©

=2

3

22

33=

2 · 11©3 · 11©

=2

3



Exercises

Exercise 7

Find the lowest terms fraction dividing the numerator anddenominator by the GCF:

10

6=

30

60=

18

24=

23

46=



Exercises

Exercise 7


10

6=5

3dividing by GCF(10,6)=2

30

60=1


18

24=3


23

46=1




Exercises

Exercise 7


10

6=5


30

60=1


18

24=3


23

46=1




Exercises

Exercise 7


10

6=5


30

60=1


18

24=3


23

46=1




Exercises

Exercise 7


10

6=5


30

60=1


18

24=3


23

46=1




Exercises

Exercise 7


10

6=5


30

60=1


18

24=3


23

46=1




Exercises

Exercise 8

Alice runs12

5miles on Monday. On Wednesday, she runs

11

6miles.

How many miles does Alice run on both days?



Exercises

Exercise 8

Alice runs12


11

6miles.


Data: She runs:12

5m on Monday.

11

6m on Wednesday.

12

5+

11

6=

72

30+

55

30=

=122

30=

61

15Using: GCF (5, 6) = 30 and

GCF (30, 122) = 2 Answer: Alice

runs61

15miles.



Exercises

Exercise 8

Alice runs12


11

6miles.


Data: She runs:12

5m on Monday.

11

6m on Wednesday.

12

5+

11

6=

72

30+

55

30=

=122

30=

61

15Using: GCF (5, 6) = 30 and


runs61

15miles.



Exercises

Exercise 8

Alice runs12


11

6miles.


Data: She runs:12

5m on Monday.

11

6m on Wednesday.

12

5+

11

6=

72

30+

55

30=

=122

30=

61

15Using: GCF (5, 6) = 30 and


runs61

15miles.



Exercises

Exercise 9

Find:

1 My soccer team wins 3 games and lost 5.Write down a fraction of the games they win:

2 Tim sells3

5of his cookies, and now he has 14 left.

How many did he have originally?

3 Of Tim’s stone collection1

5are white stones. He has total of

75 stones.How many of them are not white?



Exercises

Exercise 9

Find:

1 My soccer team wins 3 games and lost 5.

Write down a fraction of the games they win:3

8

2 Tim sells3


How many did he have originally?2

5of them all is 14 , so

1

5of them all is 7, so He had

originally 35.



75 stones. How many of them are not white?1

5of 75 =

1

5· 75 =

75

5= 15.

75 − 15 = 60 60 of the stones are not white.



Exercises

Exercise 9

Find:



8

2 Tim sells3




1


originally 35.




5of 75 =

1

5· 75 =

75

5= 15.




Exercises

Exercise 9

Find:



8

2 Tim sells3




1


originally 35.




5of 75 =

1

5· 75 =

75

5= 15.




Exercises

Exercise 9

Find:



8

2 Tim sells3




1


originally 35.




5of 75 =

1

5· 75 =

75

5= 15.




Exercises

Exercise 10

When you write music in the stave you must write signs callednotes, and you need to say how long a note is. The note value is acode which determines the note‘s duration. The length of a wholenote is equal to four beats (in 4/4 time), half notes are playedfor one half the duration of the whole note, crotchet are playedfor one quarter the duration of the whole note, quavers are eachplayed for one eighth the duration of the whole note, semiquaver

are each played for one sixteenth the duration of the wholenote,...In this piece from Tchaikovsky‘s The Nutcracker draw a verticalline every two beats.



Exercises

Exercise 10

When you write music in the stave you must write signs callednotes, and you need to say how long a note is. The note value is acode which determines the note‘s duration. The length of a wholenote is equal to four beats (in 4/4 time), half notes are playedfor one half the duration of the whole note, crotchet are playedfor one quarter the duration of the whole note, quavers are eachplayed for one eighth the duration of the whole note, semiquaver

are each played for one sixteenth the duration of the wholenote,...In this piece from Tchaikovsky‘s The Nutcracker draw a verticalline every two beats.


Education

Unit3