beams_unit 2 fractions

8/9/2019 BEAMS_Unit 2 Fractions

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Unit 1:Negative Numbers

UNIT 2

FRACTIONS

B a s i c E s s e n t i a l

A d d i t i o n a l M a t h e m a t i c s S k i l l s

Curriculum Development Division

Ministry of Education Malaysia


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TABLE OF CONTENTS

Module Overview 1

Part A: Addition and Subtraction of Fractions 2

1.0 Addition and Subtraction of Fractions with the Same Denominator 5

1.1 Addition of Fractions with the Same Denominators 5

1.2 Subtraction of Fractions with The Same Denominators 6

1.3 Addition and Subtraction Involving Whole Numbers and Fractions 7

1.4 Addition or Subtraction Involving Mixed Numbers and Fractions 9

2.0 Addition and Subtraction of Fractions with Different Denominator 10

2.1 Addition and Subtraction of Fractions When the Denominator

of One Fraction is A Multiple of That of the Other Fraction 11

2.2 Addition and Subtraction of Fractions When the Denominators

Are Not Multiple of One Another 13

2.3 Addition or Subtraction of Mixed Numbers with Different

Denominators 16

2.4 Addition or Subtraction of Algebraic Expression with Different

Denominators 17

Part B: Multiplication and Division of Fractions 22

1.0 Multiplication of Fractions 24

1.1 Multiplication of Simple Fractions 28

1.2 Multiplication of Fractions with Common Factors 29

1.3 Multiplication of a Whole Number and a Fraction 29

1.4 Multiplication of Algebraic Fractions 312.0 Division of Fractions 33

2.1 Division of Simple Fractions 36

2.2 Division of Fractions with Common Factors 37

Answers 42


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Basic Essential Additional Mathematics Skills (BEAMS) Module

UNIT 2: Fractions

1Curriculum Development DivisionMinistry of Education Malaysia

PART 1

MODULE OVERVIEW

1. The aim of this module is to reinforce pupils understanding of the conceptof fractions.

2. It serves as a guide for teachers in helping pupils to master the basiccomputation skills (addition, subtraction, multiplication and division)

involving integers and fractions.

3. This module consists of two parts, and each part consists of learningobjectives which can be taught separately. Teachers may use any parts of the

module as and when it is required.


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UNIT 2: Fractions


PART A:ADDITION AND SUBTRACTION

OF FRACTIONS

LEARNING OBJECTIVES

Upon completion ofPart A, pupils will be able to:

1. perform computations involving combination of two or more operationson integers and fractions;

2. pose and solve problems involving integers and fractions;3. add or subtract two algebraic fractions with the same denominators;4. add or subtract two algebraic fractions with one denominator as a

multiple of the other denominator; and

5. add or subtract two algebraic fractions with denominators:(i) not having any common factor;(ii) having a common factor.


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UNIT 2: Fractions


TEACHING AND LEARNING STRATEGIES

Pupils have difficulties in adding and subtracting fractions with different

denominators.

Strategy:

Teachers should emphasise that pupils have to find the equivalent form of

the fractions with common denominators by finding the lowest common

multiple (LCM) of the denominators.


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UNIT 2: Fractions


numerator

denominator

Fraction is written in the form of:

b

a

Examples:

3

4,

3

2

Proper Fraction Improper Fraction Mixed Numbers

The numerator is smaller

than the denominator.

Examples:

20

9,

3

2

The numerator is larger

than or equal to the denominator.

Examples:

12

108,

4

15

A whole number and

a fraction combined.

Examples:

65

71 8,2

Rules for Adding or Subtracting Fractions

1. When the denominators are the same, add or subtract only the numerators andkeep the denominator the same in the answer.

2. When the denominators are different, find the equivalent fractions that have thesame denominator.

Note: Emphasise that mixed numbers and whole numbers must be converted to improper

fractions before adding or subtracting fractions.

LESSON NOTES


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UNIT 2: Fractions


1.0 Addition And Subtraction of Fractions with the Same Denominator

1.1 Addition of Fractions with the Same Denominators

8

5

8

4

8

1i)

2

1

8

4

8

3

8

1ii)

fff

651

iii)

EXAMPLES

Add only the numerators and keep the

denominator same.

Write the fraction in its simplest form.


denominator the same.



8

1

8

4

8

5


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UNIT 2: Fractions


1.2 Subtraction of Fractions with The Same Denominators

2

1

8

4

8

1

8

5i)

7

4

7

5

7

1ii)

nnn

213iii)


Subtract only the numerators and keep

the denominator the same.





8

5

8

1

2

1

8

4


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UNIT 2: Fractions


1.3 Addition and Subtraction Involving Whole Numbers and Fractions

.

8

11Calculatei)

7

29

7

1

7

28

7

14

7

14

5

18

5

2

5

20

5

24

5

33

3

12

3

1

3

12

3

14

y

yy

First, convert the whole number to an improper fraction with thesame denominator as that of the other fraction.

Then, add or subtract only the numerators and keep the denominatorthe same.

18

1

8

11

8

9

+

8

8

+

8

1


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UNIT 2: Fractions


n

n

nn

n

n

52

5252

k

k

k

k

kk

32

323

2

First, convert the whole number to an improper fraction withthe same denominator as that of the other fraction.

Then, add or subtract only the numerators and keep thedenominator the same.


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UNIT 2: Fractions


1.4 Addition or Subtraction Involving Mixed Numbers and Fractions

.8

4

8

11Calculatei)

7

5

7

15

7

5

7

12

= 7

20

= 7

6

2

9

4

9

29

9

4

9

23

= 9

25

= 9

7

2

88

11

88

31

xx

= 8

11 x

First, convert the mixed number to improper fraction. Then, add or subtract only the numerators and keep the


811

84

8

51

8

13

+

8

9

+

8

4


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UNIT 2: Fractions


2.0 Addition and Subtraction of Fractions with Different Denominators

.2

1

8

1Calculatei)

To make the denominators the same, multiply both the numerator and the denominator of

the second fraction by 4:

Now, the question can be visualized like this:

?

The denominators are not the same.

See how the slices are different insizes? Before we can add the

fractions, we need to make them the

same, because we can't add them

to ether like this!

8

1

8

4

+

8

5

8

4

2

1

4

4

Now, the denominators

are the same. Therefore,we can add the fractions

together!

8

1

2

1

+

?


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UNIT 2: Fractions


Hint: Before adding or subtracting fractions with different denominators, we must

convert each fraction to an equivalent fraction with the same denominator.2.1 Addition and Subtraction of Fractions When the Denominator of One Fraction is

A Multiple of That of the Other Fraction

Multiply both the numerator and the denominator with an integer that makes the

denominators the same.

(i)6

5

3

1

6

5

6

2

6

7

=6

11

(ii)4

3

12

7

12

9

12

7

12

2

6

1

Change the first fraction to an equivalent

fraction with denominator 6.

(Multiply both the numerator and thedenominator of the first fraction by 2):

6

2

3

1 2

2



Change the second fraction to an equivalentfraction with denominator 12.

(Multiply both the numerator and the

denominator of the second fraction by 3):

12

9

4

3 3

3

Subtract only the numerators and keep the



Convert the fraction to a mixed number.


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UNIT 2: Fractions


(iii) vv 5

91

vv 5

9

5

5

v5

14

Change the first fraction to an equivalent

fraction with denominator 5v.(Multiply both the numerator and thedenominator of the first fraction by 5):

vv 5

51 5

5




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UNIT 2: Fractions


2.2 Addition and Subtraction of Fractions When the Denominators Are Not Multiple of

One Another

Method I

4

3

6

1

(i) Find the Least Common Multiple (LCM)

of the denominators.

2) 4 , 62) 2 , 3

3) 1 , 3- , 1

LCM = 2 2 3 = 12

The LCM of 4 and 6 is 12.

(ii) Change each fraction to an equivalent

fraction using the LCM as thedenominator.

( Multiply both the numerator and thedenominator of each fraction by a whole

number that will make their

denominators the same as the LCM

value).

=

4

3

6

1

=12

9

12

2

=12

11

Method II

4

3

6

1

(i) Multiply the numerator and thedenominator of the first fraction withthe denominator of the second fraction

and vice versa.

=4

3

6

1

=24

18

24

4

=24

22

=12

11

Write the fraction in its

simplest form.

This method is preferred but youmust remember to give the

answer in its simplest form.3

3

22

44 6

6


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UNIT 2: Fractions


Multiply the first fraction with the second denominator

and

multiply the second fraction with the first denominator.

1.5

1

3

2

=5

5

3

2

+

3

3

5

1

15

3

15

10

=15

13

2.8

3

6

5

=

8

8

6

5

6

6

8

3

=48

18

48

40

=48

22

=24

11


EXAMPLES

Multiply the first fraction by the

denominator of the second fraction andmultiply the second fraction by the

denominator of the first fraction.


denominator of the second fraction and

multiply the second fraction by thedenominator of the first fraction.






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UNIT 2: Fractions


3.7

1

3

2g

=3

3

7

7

7

1

3

2

g

=21

3

21

14

g

=21

314 g

4.53

2 hg

3

3

55

5

3

2

hg

15

3

15

10 hg

15

310 hg

5.dc

46

=

c

c

d

d

dc

46

cd

c

cd

d 46

=cd

cd 46

Multiply the first fraction by thedenominator of the second fraction and

multiply the second fraction by the

denominator of the first fraction.

Write as a single fraction.




denominator of the second fraction and


Multiply the first fraction by thedenominator of the second fraction and



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UNIT 2: Fractions


Convert the mixed numbers to improper fractions.


2.3 Addition or Subtraction of Mixed Numbers with Different Denominators

1.4

32

2

12

=4

11

2

5

=4

11

2

5

2

2

=4

11

4

10

=4

21

4

15

2.4

31

6

53

=4

7

6

23

=6

6

4

4

4

7

6

23

=24

42

24

92

=24

50

=12

25

=12

12

Change the first fraction to an equivalent fraction

with denominator 4.(Multiply both the numerator and the denominator

of the first fraction by 2)

The denominators are not multiples of one another:

Multiply the first fraction by the denominatorof the second fraction.

Multiply the second fraction by thedenominator of the first fraction.





Change the fraction back to a mixed number.






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UNIT 2: Fractions


The denominators are not multiples of one anotherMultiply the first fraction with the second denominator

Multiply the second fraction with the first denominator

The denominators are not multiples of one anotherMultiply the first fraction with the second denominatorMultiply the second fraction with the first denominator

2.4Addition or Subtraction of Algebraic Expression with Different Denominators

1.22

m

m

m

=)2(

)2(

2

2

22

m

mm

m

m

=

22

2

22

2

m

mm

m

m

=)2(2

)2(2

m

mmm

=)2(2

222

m

mmm

=)2(2

2

m

m

2.y

y

y

y 1

1

=

)1(

)1(1

1

y

y

y

y

y

y

y

y

=

)1(

)1)(1(2

yy

yyy

=)1(

)1(22

yy

yy

=)1(

122

yy

yy

=)1(

1

yy

Remember to usebrackets

Write the above fractions as a singlefraction.




Expand:

m (m2) = m22m

Expand:

(y1) (y + 1) =y2 +yy12

=y21

Expand:

(y21) =y2 + 1

Write the fractions as a singlefraction.





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UNIT 2: Fractions


The denominators are not multiples of one anotherMultiply the first fraction with the second denominator

Multiply the second fraction with the first denominator

3.

24

5

8

3

n

n

n

=n

n

n

n

n

n

n 8

8

24

4

4

5

8

3

2

2

=)4(8

)5(8

)4(8

1222

2

nn

nn

nn

n

= )4(8

)5(812

2

2

nn

nnn

=)4(8

84012

2

22

nn

nnn

=)4(8

404

2

2

nn

nn

=

)8(4

)10(4

2

nn

nn

=2

8

10

n

n

Factorise and simplify the fraction by canceling

out the common factors.

Expand:

8n (5 + n) =40n8n2

Subtract the like terms.

Write as a singlefraction.





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UNIT 2: Fractions


Calculate each of the following.

1. 7

1

7

2

2.

12

5

12

11

3. 14

1

7

2

4.

12

5

3

2

5. 5

4

7

2

6. 7

5

2

1

7. 313

22

8. 9

72

5

24

9. ss

12

10.

ww

511

TEST YOURSELF A


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UNIT 2: Fractions


11. aa 2

12

12. ff 3

52

13. ba

42

14. qp

51

15. nmnm5

3

7

2

5

2

7

5

16.

)2(2

1p

p

17. 5

3

2

32 yxyx

18. xx

x 5

2

412

19.

x

x

x

x 1

1 20.

2

4

2 x

x

x

x


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UNIT 2: Fractions


21.

4

84

2

36 yxyx

22.

2

9

4

3

2

n

n

n

23.

r

rr

15

25

5

2

24.

p

p

p

p

2

232

25.

n

n

n

n

10

34

5

322

26.

n

n

mn

nm 33

27.

mn

nm

m

m

5

5

28.

mn

mn

m

m

3

3

29.

24

5

8

3

n

n

n 30.

m

p

m

p 1

3


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UNIT 2: Fractions


PART B:

MULTIPLICATION AND DIVISION

OF FRACTIONS

LEARNING OBJECTIVES

Upon completion ofPart B, pupils will be able to:

1. multiply:

(i) a whole number by a fraction or mixed number;

(ii) a fraction by a whole number (include mixed numbers); and

(iii) a fraction by a fraction.

2. divide:

(i) a fraction by a whole number;

(ii) a fraction by a fraction;

(iii) a whole number by a fraction; and

(iv) a mixed number by a mixed number.

3. solve problems involving combined operations of addition, subtraction,

multiplication and division of fractions, including the use of brackets.


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UNIT 2: Fractions


TEACHING AND LEARNING STRATEGIES

Pupils face problems in multiplication and division of fractions.

Strategy:

Teacher should emphasise on how to divide fractions correctly. Teacher should

also highlight the changes in the positive (+) and negative () signs as follows:

Multiplication Division

(+) (+) = + (+) (+) = +

(+) () = (+) () =

() (+) = () (+) =

() () = + () () = +


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UNIT 2: Fractions


1.0Multiplication of Fractions

Recall that multiplication is just repeated addition.

Consider the following:

32

First,lets assume this box as 1 whole unit.

Therefore, the above multiplication 32 can be represented visually as follows:

This means that 3 units are being repeated twice, or mathematically can be written as:

6

3332

Now, lets calculate 2 x 2. This multiplication can be represented visually as:

This means that 2 units are being repeated twice, or mathematically can be written as:

4

2222

LESSON NOTES

3 + 3 = 6

2 + 2 = 4

2 groups of 3 units

2 groups of 2 units


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UNIT 2: Fractions


Now, lets calculate 2 x 1. This multiplication can be represented visually as:

This means that 1 unit is being repeated twice, or mathematically can be written as:

21112

It looks simple when we multiply a whole number by a whole number. What if we

have a multiplication ofa fraction by a whole number? Can we represent it visually?

Lets consider .2

12

Since represents 1 whole unit, therefore2

1unit can be represented by the

following shaded area:

Then, we can represent visually the multiplication of2

12 as follows:

This means that2

1unit is being repeated twice, or mathematically can be written as:

1

2

2

2

1

2

1

2

12

1 + 1 = 2

2

1+

2

1= 1

2

2

2 groups of 1 unit

2 groups of2

1unit


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UNIT 2: Fractions


Lets consider again .22

1 What does it mean? It means

2

1out of 2 units and the

visualization will be like this:

Notice that the multiplications2

12 and 2

2

1 will give the same answer, that is, 1.

How about ?2

3

1

Since represents 1 whole unit, therefore3

1unit can be represented by the

following shaded area:

Then, we can represent visually the multiplication 23

1 as follows:

This means that3

1unit is being repeated twice, or mathematically can be written as:

3

2

3

1

3

12

3

1

3

1+

3

1=

3

2

The shaded area is 3

1unit.

2

1out of 2 units 12

2

1


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UNIT 2: Fractions


Lets consider 23

1 . What does it mean? It means

3

1out of 2 units and the visualization

will be like this:

Notice that the multiplications3

12 and 2

3

1 will give the same answer, that is,

3

2.

Consider now themultiplication of a fraction by a fraction, like this:

21

31

This means 3

1out of

2

1units and the visualization will be like this:

Consider now thismultiplication:

2

1

3

2

This means

3

2out of

2

1units and the visualization will be like this:

2

1unit

3

1out of 2 units

3

22

3

1

3

1out of

2

1units

6

1

2

1

3

1

2

1unit

3

2out of

2

1units

6

2

2

1

3

2


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UNIT 2: Fractions


What do you notice so far?

The answer to the above multiplication of a fraction by a fraction can be obtained by

just multiplying both the numerator together and the denominator together:

6

1

2

1

3

1

9

2

3

1

3

2

So, what do you think the answer for3

1

4

1 ? Do you get

12

1as the answer?

The steps to multiply a fraction by a fraction can therefore be summarized as follows:

1.1 Multiplication ofSimple Fractions

Examples:

a)35

6

7

3

5

2 b)

35

6

5

3

7

2

c)35

12

5

2

7

6 d)

35

12

5

2

7

6

Steps to Multiply Fractions:

1) Multiply the numerators together andmultiply the denominators together.

2) Simplify the fraction (if needed).

Remember!!!

(+) (+) = +

(+) () =

() (+) =

() () = +

Multiply the two numerators together and the two denominators together.


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UNIT 2: Fractions


1.2 Multiplication of Fractions with Common Factors

6

5

7

12 or

6

5

7

12

1.3 Multiplication of a Whole Number and a Fraction

6

152

=

6

31

1

2

=

6

31

1

2

=3

31

=3

110

Second Method:

(i) Simplify the fraction by cancelingout the common factors.

6

5

7

12

(i) Then, multiply the twonumerators together and the twodenominators together, andconvert to a mixed number, if

needed.

6

5

7

12

7

31

7

10

2

1

Convert the mixed number to improper

fraction.

Simplify by canceling out the common

factors.

Remember

2 =1

2

First Method:

(ii)Multiply the two numeratorstogether and the twodenominators together:

6

5

7

12 =

42

60

(ii) Then, simplify.

7

31

7

10

42

60

10

7

3Multiply the two numerators together and

the two denominators together.

Remember: (+) () = ()


1

1

2


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UNIT 2: Fractions


1. Find10

15

12

5

Solution:10

15

12

5

=8

5

2. Find5

2

6

21

Solution :5

2

6

21

=5

2

6

21

5

7

=5

21

Simplify by canceling out the common

factors.

Note that3

21can be further simplified.

Simplify further by canceling out the

common factors.

3

1

Simplify by canceling out the common factors.

EXAMPLES

Multiply the two numerators together and the

two denominators together.

Remember: (+) () = ()

Multiply the two numerators together and


Remember: (+) () = ()

3

1

1

7

Change the fraction back to a mixed

number.

2

1

4

5


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UNIT 2: Fractions


1.4 Multiplication of Algebraic Fractions

1. Simplify 4

52 x

x

Solution :4

52 x

x

=2

5

=212

2. Simplify

m

n

n4

9

2

Solution:

mn

n4

9

2

=

1

4

2

9

2

mn

n

n

=1

)2(

2

9 mn

= nm22

9

1 2

1 1Simplify the fraction by canceling out the xs.



Simplify the fraction by canceling the

common factor and the n.

Multiply the two numerators together

and the two denominators together.


Change the fraction back to a mixednumber.

2

1

1

1


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UNIT 2: Fractions


1. Calculate 27

25

5

92. Calculate

20

14

7

3

12

45

3. Calculate

4

112 4. Calculate

5

14

3

1

5. Simplify

k

m3 6. Simplify )5(

2m

n

7. Simplify

14

3

6

1

1

x

8. Simplify )32(2 dan

9. Simplify

yx10

95

3

2 10. Simplify

x

x 120

4

TEST YOURSELF B1


35/45


UNIT 2: Fractions


2.0 Division of Fractions

Consider the following:

36

First,lets assume this circle as 1 whole unit.

Therefore, the above division can be represented visually as follows:

This means that 6 units are being divided into a group of 3 units, or mathematically

can be written as:

236

The above division can also be interpreted as how many 3s can fit into 6. The answer is

2 groups of 3 units can fit into 6 units.

Consider now a division of a fraction by a fraction like this:

.8

1

2

1

LESSON NOTES

How many8

1is in

6 units are being divided into a group of 3

units:

236


36/45


UNIT 2: Fractions


This means How many is in ?

8

1

2

1

The answer is 4:

Consider now this division:

.4

1

4

3

This means How many is in ?

4

1

4

3

The answer is 3:But, how do you

calculate the answer?

How many4

1is in ?

4

3


37/45


UNIT 2: Fractions


Consider again .236

Actually, the above division can be written as follows:

3

16

3

636

Notice that we can write the division in the multiplication form. But here, we have to

change the second number to its reciprocal.

Therefore, if we have a division of fraction by a fraction, we can do the same, that is,

we have to change the second fraction to its reciprocal and then multiply the

fractions.

Therefore, in our earlier examples, we can have:

4

2

8

1

8

2

1

8

1

2

1(i)

The reciprocal of a

fraction is found by

inverting the

fraction.

Change the second fraction to its

reciprocal and change the sign to .

The reciprocal

of8

1is .

1

8

These operations are the same!

The reciprocal

of3 is .3

1


38/45


UNIT 2: Fractions


3

1

4

4

3

4

1

4

3(ii)

The steps to divide fractions can therefore be summarized as follows:

2.1 Division of Simple Fractions

Example:

7

3

5

2

=3

7

5

2

=15

14

Change the second fraction to its reciprocal

and change the sign to .



Steps to Divide Fractions:

1. Change the second fraction to itsreciprocal and change the sign to .

2. Multiply the numerators together andmultiply the denominators together.

3. Simplify the fraction (if needed).

Tips:

(+) (+) = +

(+) () =

() (+) =

() () = +

Change the second fraction to its

reciprocal and change the sign to .

The reciprocal

of4

1is .

1

4


39/45


UNIT 2: Fractions


2.2 Division of Fractions With Common Factors

Examples:

9

2

21

10

=2

9

21

10

=2

9

21

10

=7

15

=7

12

7

6

5

3

6

7

5

3

10

7

7

65

3

1

5 3

7

1

2

Express the fraction in division form.

Change the second fraction to its reciprocal and

change the sign to .

Simplify by canceling out the common factors.




Then, simplify by canceling out the common

factors.



Remember: (+) () = ()

Multiply the two numerators together and thetwo denominators together.


40/45


UNIT 2: Fractions


1. Find6

25

12

35

Solution :6

25

12

35

=25

6

12

35

=

10

7

2. Simplify 4

52 x

x

Solution : xx 5

42

= 2

5

8

x

3. Simplify2

x

y

Solution :

2x

y

2

1x

y

x

y

2

5

7


and change the sign to .Then, simplify by canceling out the common

factors.

Method I

EXAMPLES



Multiply the two numerators together and the twodenominators together.

Express the fraction in division form.


and change to .

Multiply the two numerators together and the two

denominators together.

Remember: (+) () = ()



2

1


41/45


UNIT 2: Fractions


Multiply the numerator and the denominator of

the given fraction with x

2

x

y

=

2x

y

x

x

=x

xx

y

2

=

x

y

2

4. Simplify5

)1( 1r

Solution:

5

)1( 1r

=

5

)1

1(r

r

r

=

r

r

5

1

The given fraction.

ris the denominator ofr

1.

Multiply the given fraction withr

r.

Note that:

1)1

1( rrr

Method II

The numerator is also

a fraction with

denominatorx

Multiply the numerator and the denominator of the

given fraction byx.


42/45


UNIT 2: Fractions


1. Calculate2

21

7

3 2. Calculate165

87

95

3. Simplify3

48 y

y

4. Simplify

k

2

16

5. Simplify

3

5

2

x 6. Simplify

n

m

n

m

3

242

7. Simplify8

1

4

y

8. Simplify

x

x

11

TEST YOURSELF B2


43/45


UNIT 2: Fractions


9. Calculate5

)1(341

10. Simplifyy

x15

11. Simplify

3

2

9

41 x

12. Simplify

15

1

1

p


44/45


UNIT 2: Fractions


TEST YOURSELF A:

1.7

3

2.

2

1

3.

14

5

4.4

1

5.

35

38or

35

31

6.

14

3

7.13

67or

13

25

8.45

73or

45

281

9.s

3

10.w

6

11.

a2

5

12.

f3

1

13.ab

ab 42

14.

pq

pq 5

15. nm

16.2

33 p

17.10

1716 yx

18.x

x 12

19.)1(

1

xx

20. 2 21.2

8 yx

22.2

9

47

n

n

23.

r

r

3

12

24.2

2

2

6

p

p

25.2

2

10

647

n

nn

26.m

m1

27.

n

n

5

5

28.n

n

3

3

29.

28

10

n

n

30.

m

p

3

34

ANSWERS


45/45


UNIT 2: Fractions

TEST YOURSELF B1:

1.3

21

3

5or 2.

8

11

8

9 or 3.

2

15

2

11or

4.5

21

5

7 or 5.

k

m36.

2

5mn

7.4

x8. ndna

2

3 9. yx

5

3

3

10

10.

4

15 x

TEST YOURSELF B2:

1.49

22.

9

51

9

14 or 3.

2

6

y

4. 8k5.

x5

66.

m

6

7.)1(2

1

y 8.

1

2

x

x

9.20

9

10.xy

x 15 11.

6

13x 12.

p4

5

beams_unit 2 fractions

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