eso1-05_lesson_04_fractions.pdf

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LessonLesson 44

FractionsFractions

1st course of ESO Carlos Alberto Cortijo Bon

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Table of contents

1.Fractions..................................................................................................................................................12.Proper and improper fractions. Mixed numbers......................................................................................23.Equivalent fractions.................................................................................................................................3

3.1.Building equivalent fractions..................................................................................................................4

3.2.Irreducible fractions...............................................................................................................................4

4.Reduction of fractions to a common denominator...................................................................................54.1.Reduction to the product of the denominators.......................................................................................5

4.2.Reduction to the least common multiple................................................................................................5

5.Operations with fractions.........................................................................................................................65.1.Addition and subtraction of fractions.....................................................................................................6

5.2.Multiplication of fractions......................................................................................................................65.3.Division of fractions...............................................................................................................................7

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Mathematics 1st course of ESOLesson 4: Fractions

1. Fractions

A fraction represents the result of dividing a whole in a specified number of equal parts and taking a

specified number of those equal parts.

Example:3

4

3 is the numerator,of the fraction.

4 is the denominator of the fraction.

3

4represents the result of dividing a whole in 4 equal parts and taking three of those parts.

There are different ways in which a fraction can be seen.

3

4of a pizza

Fraction as a part of a unit:3

4of a pizza results from

dividing the pizza in 4 equal parts and taking 3 of those

parts. The whole pizza would be4

4, which is equivalent to

1 (one unit).

The whole pizza weighs 480g1

4of the pizza weighs

480g4=120g3

4of the pizza weigh 3120g=360g

In short: 34

of 480={48043

48034 }=360

Fraction as an operator: If the pizza weighs 480g, then:

Each portion weighs 480g

4=

120 g . Thiscorresponds to

1

4.

Thus,1

4of 480 is 120 .

3

4represents three times

1

4which is

3120g=360

3

4of 480={

multiply by 3 and divide by 4

ordivide by 4 and multiply by 3 }=360

Thus,3

4of 480g is 360g .

Carlos Alberto Cortijo Bon 1


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The whole pizza is the unit, 13

4=34=0.75

Fraction as a division: A fraction can be seen as a divisionthat is left undone. But we can actually make the division.

3

4=34=0.75

The whole pizza is 1=100%3

4of the pizza is 0.75=75%

Fraction as a percentage: The result of dividing ismultiplied by 100. In our case:

3

4=34=0.75=0.75100%=75%

2. Proper and improper fractions. Mixed numbers

A proper fraction has a numerator that is smaller than the denominator. A proper fraction represents a

quantity less than one unit.

An improper fraction has a numerator that is equal to or bigger than the denominator. An improper

fraction represents a quantity bigger than one unit.

3

4is a proper fraction.

The numerator is smaller than the denominator.

It is less than one unit

4

4is an improper fraction.

The numerator is equal to the denominator.

It is equal to one unit


3

41

4

4=1


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5


The numerator is bigger than the denominator.

It is bigger than one unit.

An improper fraction can be represented as a mixed number, consisting of a natural number and a

proper fraction.

5


It is equal to one unit plus1

4.

This is represented as 1 14

as a mixed number.

11


It is equal to two units plus3

4.

This is represented as 23

4as a mixed number.

How do we obtain the mixed number equivalent to a given improper fraction? For instance:

17

5 .

1. We divide numerator by denominator, without decimals, obtaining quotient and remainder.17 5

2 3

2. We build the mixed number as follows:

17

5

__=

2

5

__3

17 5

2 3

Denominator: Denominator of the original fraction.

Integer part: Quotient of the division.

Numerator: Remainder of the division.

3. Equivalent fractions

Two fractions are equivalent if their cross products are the same.

Are1

2and

3

6equivalent? Their cross products are 16=6 and 23=6 . The cross products

are the same. Thus,1

2and

3

6are equivalent.


5

41


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3

6

__

1

2

__ {16=623=6}equivalent fractions

Are2

3and

5

6equivalent? Their cross products are 26=12 and 35=15 . The cross

products are different. Thus,2

3and

5

6are not equivalent.

5

6

__

2

3

__ {26=1235=15} non-equivalent fractions

3.1. Building equivalent fractions

An equivalent fraction can be obtained by:

1. Expanding the fraction. A fraction is expanded by multiplying numerator and denominator by

the same natural number.

1

2=

12

22=

2

4 We can multiply numerator and denominator by by other numbers to obtain more fractions

equivalent to1

2.1

2=2

4=

3

6=4

8=

5

10=

6

12=

7

14=

8

16=...

2. Simplifying the fraction. A fraction is simplified by dividing numerator and denominator by a

common divisor.

Let's try to find equivalent fractions to12

18. The common divisors of 12 and 18 are: 2 , 3

and 6 . Thus, equivalent fractions to12

18are

122

182=

6

9,

123

183=

4

6and

126

186=

2

3. In

other words:12

18=6

9=4

6=2

3 .

3.2. Irreducible fractions

An irreducible fraction is a fraction that cannot be reduced. In other words, a fraction in which the

numerator and denominator have no common divisors other than 1.

There are two ways of obtaining the irreducible fraction equivalent to a given fraction:

1. Simplify repeatedly a fraction until we obtain an irreducible fraction.



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For instance:12

18=

122

182=

6

9=

63

93=

2

3.

2

3is irreducible because 2 and 3 have no

common divisors other than 1.

2. Divide both numerator and denominator by their gcd (greatest common divisor).

For instance:12

18. 12=2

23 , 18=23

2 . gcd 12,18=23=6. The irreducible fraction

equivalent to12

18is

126

186=

2

3.

4. Reduction of fractions to a common denominator

It is very easy to compare, sum or subtract fractions with the same denominator.

For instance:2

5

,1

5

and6

5

.

We can order them from smallest to largest by comparing their numerators:1

5

2

5

6

5 We can sum and subtract them and the result is another fraction with the same denominator:

1

5

2

5=

12

5=

3

5,1

5

2

56

5=126

5=9

5,

6

5

2

5=

62

5=

4

5,

6

5

1

5

2

5=

612

5=

7

5

If two fractions have not the same denominator, we cannot directly sum, subtract or compare them:

2

34

5=? Which is larger,

2

3or

4

5?

We first have to reduce them to common denominator. This consists in obtaining two equivalent fractionswith the same denominator.

There are two ways of reducing two fractions to common denominator.

4.1. Reduction to the product of the denominators

We multiply numerator and denominator of each fraction by the denominator of the other fraction.

Example: reduce5

6and

3

4to common denominator.

We multiply numerator and denominator of5

6by 4 :

5

6=

54

64=

20

24

We multiply numerator and denominator of3

4by 6 :

3

4=

36

46=

18

24

4.2. Reduction to the least common multiple



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The common denominator is the least common multiple.

The new denominator is the old numerator multiplied by the result of dividing the least common multiple

by the old denominator.

Example: reduce 56

and 34

to common denominator.

The common denominator is lcm 6,4=12

5

6=5126

12=10

12

3

4=3124

12=

9

12

In general, it is better to reduce to the least common multiple, because we obtain simpler fractions

5. Operations with fractionsThe fractions are numbers, so that they can be added, subtracted, multiplied and divided.

A natural number can be expressed as a fraction with denominator 1 . For example: 5=5

1.

5.1. Addition and subtraction of fractions

If the fractions have the same denominator:

The denominator of the result is the same.

The numerator is the addition or subtraction of the numerators.

3

72

7=32

7=5

7

3

7

2

7=

32

7=

1

7

If the fractions have different denominators, we first reduce them to common denominator and

then add or subtract them.

8

15

3

10

=16

30

9

30

=25

30

8

15

3

10

=16

30

9

30

=7

30

lcm 15,10=30 ,8

15=83015

30=82

30=16

30,

3

10=33010

30=33

30=

9

30

5.2. Multiplication of fractions

a

bc

d=

ac

bd

The numerator is the product of the numerators.

The denominator is the product of the denominators.



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Example:1

3

2

5=

12

35=

2

15

5.3. Division of fractions

The inverse of a fraction results from swapping numerator and denominator.

The inverse of2

3is

3

2. The inverse of

1

3is

3

1=3 . The inverse of 3=

3

1is

1

3.

The division of two fractions is the product of the first and the inverse of the second.

a

b

c

d=

a

bd

c=

ad

bcThe numerator is the product of the numerators.

The denominator is the product of the denominators.

Example: 1325

=1352=1

532=56



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Glossary

Fill in the blanks to complete the glossary

English Spanish

Numerador

Denominador

Proper fraction

Improper fraction

Nmero mixto

Fraccin irreducible

Fraccin inversa


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eso1-05_lesson_04_fractions.pdf