FRACTIONS

MATHEMATICS

Bombay Cambridge Gurukul

FRACTIONS

Standard IV

Standard III

Standard V

Choose the level

What are fractions?

Parts of a collection

RevisionMore about fractions……numerator and denominator

More about fractions…III

Back

How to read fractions?

Back

Equivalent fractions

Types of fractionsFraction as divisionMixed numbersComparison of fractionsAddition of like fractionsSubtraction of like fractions

IV

Addition of……unlike fractions, mixed numbersSubtraction of……unlike fractions, mixed numbersMultiplication of fractions

Reciprocal of a fractionDivision of fractions

V

Back

Reduced form of fractionsFactors and Multiples

Standard III

What are

fractions?

Look at the figure given below.

We can divide itinto 2 equal partsby drawing a line.

Shade only one part of the figure.

Each part is called one half of the whole.

It is a whole figure.

We write it as 12

12

12

Look at the figure given below.

We can divide itinto 2 equal partsby drawing a line.

Shade only one part of the figure.

Each part is called one half of the whole.

It is a whole figure.

We write it as 12

12

12

Back

How to read fractions?

How to read a fraction?

12

is read as 1 upon 2 or 1 by 2.

37

25

79

is read as

is read as

is read as

3 upon 7 or

2 upon 5 or

7 upon 9 or

3 by 7.

2 by 5.

7 by 9.

Back

More about fractions…

The following figures are divided into two equal parts.

When a whole is divided into two equal parts,

Two halves make a whole.

12

One half is written as

each part is called half of the whole.

12

12

12

12whole whole

Each figure is divided into two parts.Are both parts equal?

Yes Yes No Yes

No No Yes No

Which of the following figures are divided into two equal parts?

The following figures are divided into three equal parts.

When a whole is divided into three equal parts,

13

One third is written as

each part is called one third of the whole.

13

13

13

13

13

13

Each figure is divided into three parts.Are all the three parts equal ?

No Yes Yes No

Yes No No Yes

Which of the following figures are divided into three equal parts?

When a whole is divided into four equal parts,

14

One fourth is written as

each part is called one fourth of the whole.

The following figures are divided into four equal parts.

14

14

14

14

14

14

14

14

YesNo YesNo

Each figure is divided into four parts.Are all the four parts equal?

NoYes NoYes

Which of the following figures are divided into four equal parts?

Draw a line or lines to divide each of the

following shapes into:

two equal partsfour equal partsthree equal parts

Shade half (1/2) of each shapeShade one fourth (1/4) of each shapeShade one third (1/3) of each shape

12

12

13

13

131

4

14

14

14

12

12

13

13

13

14

14

14

14

Look at the figure given below:

23

The fraction for the shaded part is

It has 3 equal parts.2 parts are shaded.

It is read as two third.

34

The fraction for the shaded part is

It has 4 equal parts.3 parts are shaded.

It is read as three fourth.

Back

Match the following12

13

34

14

23

One fourth

One third

One half

Two third

Three fourth

Parts of a

collection

The box given below has 12 stars.They can be divided into 2 equal parts.

Each part has 6 stars.

6 6

To find the number of objects in one half of a collection, we

divide the total number of objects by 2.

They can be divided into 3 equal parts.

Each part has 4 stars.

4

4

4

The box given below has 12 stars.

To find the number of objects in one third of a collection, we

divide the total number of objects by 3.

They can be divided into 4 equal parts.

Each part has 3 stars.

3 33 3

The box given below has 12 stars.

To find the number of objects in one fourth of a collection, we

divide the total number of objects by 4.

Encircle one half(

43

6

One half of 12 is 6One fourth of 12 is 3One third of 12 is 4

Total number of insects shown below is 12.How many insects are there in 1

2of the collection?

=12 2 612)of each collection.

How many insects are there in 13

of the collection? How many insects are there in 14

of the collection?

Encircle one third( 13)of each collection.Encircle one fourth(14)of each collection.

=12 3 4=12 4 3

Colour one half of the collection.Colour one fourth of the collection.Colour one third of the collection.

Back

Revision

How many equal parts is each rod divided into?

2 equal parts

4 equal parts

3 equal parts

5 equal parts

What fraction do the colored portions in each of the following show?

25

34

14

23

Match the following fractions to the figures.59

28

15

67

26

46

59

28

15

67

26

46

HALF

QUARTER (ONE FOURTH)

THREE QUARTERS(THREE FOURTH)

ONE THIRD

WHOLE 1

12

13

34

14

TWO THIRD 23

Back

More about fractions…

…numerator and denominator

PARTS OF A WHOLE ARE CALLED FRACTIONS.

e.g.

12

Parts considered

Total number ofequal parts

NUMERATOR

DENOMINATOR

FRACTIONNUMERATOR

DENOMINATOR=

3838

Remember : Letter‘u’ is in the word ‘numerator’ and the word ‘ up’ .

Remember : Letter ‘d’ starts the word denominator’ and the word ‘down’ .

So, in the fraction , 3 is the numerator.38

So, in the fraction , 8 is the denominator.3 8

Write the numerator and denominator for each of the following fractions.

Fraction Numerator Denominator23

34

15

57

23

34

15

57

Write the fraction for the numerator and denominator given below.

Numerator Denominator Fraction

15

47

34

58

15

47

34

58

15

47

34

58

15

47

34

58

15

47

34

58

Write the fraction for the shaded part.

Numerator

Denominator

Fraction

Numerator

Denominator

Fraction

5

58

510

5

810

(shaded parts) (shaded parts)

(total parts)

(total parts)

The EndThe End

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Standard IV

Equivalent fractions

Is the shaded part in each pair of figures same?

Yes YesYes

Is the shaded part in both the figures same?

What is the fraction for the shaded part?

So, we see that =

Yes

Is the shaded part in both the figures same?

Yes

What is the fraction for the shaded part?

So, we see that =

Is the shaded part in both the figures same?

Yes

What is the fraction for the shaded part?

So, we see that =

Fractions which are equal in value to each other are calledequivalent fractions.

is equivalent to12

24

is equivalent to34

68

e.g.

Match the following equivalent fractions.

33

28

24

26

13

14

12

1

Back

Types of fractions

27

35

14

49

etc.

Fractions where the numerator is smaller than the denominatorare called

proper fractions.

e.g.

43

72

94

87

etc.

Fractions where the numerator is greater than the denominatorare calledimproper fractions.

e.g.

Fractions which have same denominator are calledlike fractions.

29

39

59

49

e.g. etc.

Fractions which have different denominators are calledunlike fractions.

23

34

59

45

e.g. etc.

Fractions which have numeral 1 as numerator are calledunit fractions.

13

14

19

15

e.g. etc.

Back

Fraction as

division

We can write each division sum as a fraction. 4

12=4 12

36

=3 6

15

=1 5

710

=7 10

We can write each fraction as a division sum.18 = 1 8

69 = 6 9

412 = 4 12

29 = 2 9

Back

Mixed numbers

Mixed numbers include a whole number and a fraction.

=+

(whole number) (fraction) (mixed number)+ =

Converting mixed numbers to improper fractions.

Convert 12

4 to a improper fraction.

Step 1 : Multiply the denominator 2 with whole number 4. =2 4 8

Step 2 : Add numerator 1 to 8 =1 8 9

Step 3 : Write 9 as the numerator of the improper fraction.

9

Step 4 : Write denominator 2 as the denominator of the improper fraction.

92

12

4 = 92(improper fraction)(mixed number)

Converting improper fractions to mixed numbers.

Convert 73

to a mixed number.

Step 1 : Divide 7 by 3.

Step 2 : Write the mixed number.

The quotient becomes the whole number.

Divisor : 3

Quotient: 2

Remainder : 1

The divisor becomes the denominator.

The remainder becomes the numerator.

73

13

2=(improper fraction) (mixed number)

**

2

*3

2

13

2

Improper fractions

Mixed numbers

Improper fractions

Mixed numbers

can be changed to

Converting improper fractions to mixed numbers.

can be changed to

Back

Comparison of fractions

like fractionsHow to compare like fractions ?

Look at the figures shown below.

Each figure is divided into 4 equal parts.

Which figure has more shaded parts?

The first figure (A) has more shaded parts.

(A) (B)

like fractionsHow to compare like fractions ?

Look at the figures shown below.Write the fraction for both figures.

26

46

Which fractions has more shaded area?

So, we can say that 46

26

>

46

like fractionsHow to compare like fractions ?

Look at the figures shown below.Write the fraction for both figures.

27

67

Which fraction has less shaded area?

So, we can say that 27

67

<

27

like fractions

If there are two like fractions, then the fraction with greater numerator is greater in value.

If there are two like fractions, then the fraction with smaller numerator is lesser in value.

How to compare like fractions ?

e.g. 47

37

e.g. 89

29

>

<

Compare the following using < , > or = .

45

37

29

46

15

67

29

36

>

<

=

>Back

Addition oflike fractions

Addition of like fractions

Two more parts of the circle are shaded.

In the circle given below only one part out of five is shaded.

The circle has three shaded parts.

Addition of like fractions

14

24

34

14

24

34

=+

=+

Addition of like fractions

26

36

56

=+

=+

26

36

56

Addition of like fractions

13

13

23

=+

+

13

13

23

When two or more like fractions are added, then only the numerators are added together.

The denominators are not added together.

Addition of like fractions

Addition of like fractions

The answer should be written in the reduced form of fractions.

+

+

4

4

Addition of like fractions

39

39

69=+

58

28

78=+

17

27

37=+

25

25

45=+

Back

=

=

=

=2 + 25

1 + 27

5 + 28

3 + 39

Subtraction oflike fractions

Subtraction of like fractions

In the figure given below, three parts out of five parts are shaded.

Two parts are taken away.

One part out of five is left.

Subtraction of like fractions

34

24

14

=-

Two parts are taken away.

In the figure given below, three parts out of four are shaded.

One part out of four is left.

Subtraction of like fractions

When two like fractions are subtracted, then the smaller numerator is subtracted from the bigger numerator.

The denominators are not

subtracted.

Subtraction of like fractions

The answer should be written in the reduced form of fractions.

612

412

212

=- 2

2 = 16

1416

216

1216

=-2

2 = 68

2

2 = 34

79

59

29

=-

68

18

58

=-

57

27

37

=-

36

26

16

=-

Subtraction of like fractions

=

=

=

= 3 - 26

5 - 27

6 - 18

7 - 59

The EndThe End

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Standard V

Reduced form of fractions

A fraction is said to be in the reduced form if its numerator and denominatorcannot be divided by a common number.

Reduced form of fractions

Look at the fraction given below.

We can divide the numerator and denominator both by 2.

68

22

68

=So,

Now we can not divide 3 and 4 both by any number.

So, we can say that 34is the reduced form of 6

8

2

234

=68

6 divided by 2 is 3.8 divided by 2 is 4.

=

Reduce the given fraction to its lowest form.

39

3

313

We can divide both, the numeratorand the denominator by 3.

The reduced form of 39

13

is

=1012

2

256

We can divide both, the numeratorand the denominator by 2.

The reduced form of 1012

56

is

1012

39

28

14

Circle the fractions which are in the reduced form.

39

56

57

412

35

918

38

614

49

812

Back

3

3

2

2

9

9

2

2

2

2

4

4

Factors and Multiples

A number that divides a given number completely (without leaving a remainder) is called its

factor.

e.g. 5 divides 20 exactly.

So, 5 is a factor of 20.

And 20 is a multiple of 5.

Is 20 exactly divisible by 3? No

Is 3 a factor of 20? NoIs 20 a multiple of 3? No

List the numbers that divide 15 exactly.

1 53 15

So, we can say that factors of 15 are 1, 3, 5 and 15.

List the numbers that divide 12 exactly.

1 2 3 4 6 12

So, we can say that factors of 12 are 1, 2, 3, 4, 6 and 12.

Every number has at least 2 factors :the number itself.

1

1

15

12

1 and

Which of the following are factors of 16?

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16

1

2 4 8 16

Is 4 a factor of 14 ? No

Is 6 a factor of 24 ?

Is 4 a factor of 32 ?

Is 3 a factor of 17 ?

Yes

Yes

No

Try the following….

Which of the following are multiples of 4 ?8, 10, 12, 14, 16, 18,

20, 22, 24, 26, 28, 30

8

12 16 20 24

Is 15 a multiple of 6 ? No

Yes

Yes

No

28

Is 28 a multiple of 7 ?

Is 24 a multiple of 8 ?

Is 21 a multiple of 9 ?

Try the following….

Common factors

The factors of 24 are :

1, 2, 3, 4, 6, 8, 12 and 24.

The factors of 30 are :

1, 2, 3, 5, 6, 10, 15 and 30.

Common factors of 24 and 30 are :

1,

Highest common factor (H.C.F.) of 24 and 30 is : 6

2,

3,

6

Common multiples

The multiples of 3 are :

3, 6, 9, 12, 15, 18, 21, 24 …

The multiples of 4 are :

4, 8, 12, 16, 20, 24, 28 ...

Common multiples of 3 and 4 are :

12 ,

Least common multiple (L.C.M.)of 3 and 4 is :

12

24 …

Back

Addition of……unlike fractions,

mixed numbers

When we add two unlike fractions(with different denominators),

we need to find the least common multiple ( L.C.M.)

of the two denominators.

Addition of unlike fractions

To change both fractions to like fractions, we find the L.C.M. of 2 and 6.

Multiples of 2 are :

Multiples of 6 are :

2, 4, 6, 8, 10, 12…

6, 12, 18, 24, 30…

Common multiples of 2 and 6 are : 6, 12…

Least common multiple (L.C.M.)of 2 and 6 is : 6

16

1 2

+

Now we can addThe denominator of both the fractions is the same as the L.C.M.

12

16

+ L.C.M. of 2 and 6 is 6.

________6

=

Divide the common denominator with the denominator of the first fraction.

Step 1 :

Step 2 :

Step 3 :

Multiply 3 with the numerator of the first fraction.Write 3 in place of the first numerator.

Step 4 : = 3 +

6

6 2 = 3

3 1 = 3

Divide the common denominator with the denominator of the second fraction.

Step 5 :

Step 6 :

Step 7:

Add the numeratorsStep 8 :

= 3 + 1 6

Multiply 1 with the numerator of the second fraction.

Write 1 in place of the second numerator.

= 46

12

16

+So,

= 46

6 6 = 1

1 = 11

Addition of unlike fractions

=

The denominators are different,so, we find the L.C.M. of 2 and 4.

L.C.M. of 2 and 4 is 4.

Then, numerators are added.

Addition of mixed numbers

Change the mixed number to an improper fraction.

Add both the fractions.

15

4 25

=

+

15

4

Back

4 5 + 1

5215

=

Step1:

Step 2:

25

+215

= 21 + 25

= 235

So, 15

4 25

+ = 235

(21 + 2) 5 5

Subtraction of……unlike fractions,

mixed numbers

When we subtract two unlike fractions (with different denominators),

we need to find the least common multiple ( L.C.M.)

of the two denominators.

Subtraction of unlike fractions

To change both fractions to like fractions, we find the L.C.M. of 3 and 6.

Multiples of 3 are :

Multiples of 6 are :

3, 6, 9, 12, 15, 18…

6, 12, 18, 24, 30…

Common multiples of 8 and 4 are : 6, 12…

Least common multiple (L.C.M.)of 8 and 4 is : 6

16

2 3

_

Now we can subtract

The denominator of both the fractions is the same as the L.C.M.

23

16

- L.C.M. of 3 and 6 is 6.

________6

=

Divide the common denominator with the denominator of the first fraction.

Step 1 :

Step 2 :

Step 3: Multiply 2 with the numerator of the first fraction.Write 4 in place of the first numerator.

Step 4 :

= 4 - 6

6 3 = 2

2 2 = 4

Divide the common denominator with the denominator of the second fraction.

Step 5 :

Step 6 :

Step 7:

Subtract the numeratorsStep 8 :

= 4 - 1 6

Multiply 1 with the numerator of the second fraction.

Write 1 in place of the second numerator.

= 36

23

16

-So,

= 36

1 1 = 1

6 6 = 1

Subtraction of unlike fractions

Then, numerators are subtracted.

The denominators are different,so, we find the L.C.M. of 2 and 4.

L.C.M. of 2 and 4 is 4.

Subtraction of mixed numbers

Change the mixed number to an improper fraction.

Subtract both the fractions.

=

27

3 27

-

27

3

Back

3 7 + 2

7237

=

Step 1:

Step 2:

27

-237

= 23 - 27

= 217

So, 27

3 27

- = 217

(23 - 2) 7 7

Multiplicationof fractions

How to multiply a fraction by a whole number ?58

4

We multiply only the numerator of the fraction with the whole number.

The denominator remains the same.

= 20 8

45 8

We should write the answer in the reduced form of fractions.20 8

22

= 10 4

22

= 52

So, 58

4 = 52

How to multiply a whole number by a fraction ?

69

We multiply the whole numberonly with the numerator of the fraction.

The denominator remains the same.

= 30 9

65 9

We should write the answer in the reduced form of fractions.

30 9

33

= 10 3

So, 69 =

10 3

5

5

How to multiply a fraction by a fraction ?

67

We multiply both the numerators.

= 1221

We should write the answer in the reduced form of fractions.

1221

33

= 47

So, =47

23

And we multiply both the denominators. 62 73

672

3 Back

Reciprocalof a fraction

How to write a reciprocal fraction ?

The numerator becomes the denominator.

And the denominator becomes the numerator.

79

Fraction Reciprocal fraction

97

The reciprocal of 17

is 71

The reciprocal of 7 or is 17

or 7

71

The reciprocal of a unit fraction is a whole number.

The reciprocal of a whole number is a unit fraction.

REMEMBER

Back

Divisionof fractions

How to divide a whole number by a fraction ?68

4

=

We change the division sign to multiplication.

4 Then we write the reciprocal of the second fraction.

4 86

Multiply the numerators.32 6Reduce the fraction to its lowest form. 3

2 6

16 3

2

2

So, 68

4 = 16 3

How to divide a fraction by a whole number ?45

4

=

We change the division sign to multiplication.

Then we write the reciprocal of the whole number.4

5Multiply the numerators.

Reduce the fraction to its lowest form. 420

2

2

So, =

45

14

420 2

10

2

2

= 15

45

4 15

How to divide a fraction by a fraction ?48

=

We change the division sign to multiplication.

Then we write the reciprocal of the second fraction.4

8Multiply the numerators and the denominators.

Reduce the fraction to its lowest form. 1216

2

2

So, =

48

32

12 16 6

8

2

2

= 34

48

34

23

23

The EndThe End

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