Index
Index
1. What are Fractions2. Examples
3. Need of Fractions
4. Parts of Fractions
5. Example
6. Types of
Fractions Proper fractions & Improper Fr actions
Mixed Fractions
Like Fractions & Unlike Fracti ons
Equivalent Fractions
7. Operations of Fractions Addition
Subtraction
Multiplication
Division
8. Comparison
Greater than and Smaller than
How does the Denominator con trols the Fraction
What are Fractions?
What are Fractions?
Fractions are for counting part of something._{Fractions are for counting part of something.}
Loosely speaking, a fraction is a quantity that cannot be _{Loosely speaking, a fraction is a quantity that cannot be } represented by a whole number.
represented by a whole number.
A fraction (from the Latin fractus, broken) is a number that _{A fraction (from the Latin fractus, broken) is a number that } can represent part of a whole. The earliest fractions were
can represent part of a whole. The earliest fractions were
reciprocals of integers: ancient symbols representing one part reciprocals of integers: ancient symbols representing one part of two, one part of three, one part of four, and so on. A much of two, one part of three, one part of four, and so on. A much later development were the common or "vulgar" fractions later development were the common or "vulgar" fractions which are still used today (½, ⅝, ¾, etc.)
1
1
22/
/
1010
1
1
22/
/
1010
1
1
/
/
12
12
1
1
/
/
12
12
1
1
/
/
8
8
1 1
/
/
8 8
1
1
½
½
1
1
½
½
11
11
/
/
12
12
11 11
/
/
12 12 55
55
/
/
60
60
55
55
/
/
60
60
Examples
Need Of Fractions?
Need Of Fractions?
Consider the following scenario.
Consider the following scenario.
Can you finish the whole cake?
Can you finish the whole cake?
If not, how many cakes did you eat?
If not, how many cakes did you eat?
1 is not the answer, neither is 0.
1 is not the answer, neither is 0.
This suggest that we need a new
This suggest that we need a new
kind of number i.e.
b
a
Parts Of Fractions?
Parts Of Fractions?
I’m the NUMERATOR.
I’m the NUMERATOR.
I tell you the number of parts
I tell you the number of parts
I’m the DENOMINATOR.
I’m the DENOMINATOR.
I tell you the name of part
I tell you the name of part
The
The
denominator
denominator
tells us how
tells us how
many congruent pieces the whole is
many congruent pieces the whole is
divided into, thus this number
divided into, thus this number
cannot be 0. (b)
cannot be 0. (b)
The
The
numerator
numerator
tells us how many
tells us how many
such pieces are being considered. (a)
Example
Example
8
7
How much of a pizza do we have below?
How much of a pizza do we have below?
The blue circle is our whole.
The blue circle is our whole.
if we divide the whole into 8 congruent pieces,
if we divide the whole into 8 congruent pieces,
 the denominator would be
 the denominator would be
8
8
.
.
We can see that we have 7 of these pieces.
We can see that we have 7 of these pieces.
Therefore the numerator is
Therefore the numerator is
7
7
, and we
, and we
have
have
Proper fractions Proper fractions & & Improper Fractions Improper Fractions Proper fractions Proper fractions & & Improper Fractions
Improper Fractions Equivalent Equivalent Equivalent Equivalent _{Fractions}_{Fractions}_{Fractions}_{Fractions}
Like Fractions Like Fractions & & Unlike Fractions Unlike Fractions Like Fractions Like Fractions & & Unlike Fractions Unlike Fractions Mixed Fractions Mixed Fractions Mixed Fractions
Mixed Fractions Unit FractionsUnit Fractions_{&}_{&}
Whole Fractions Whole Fractions Unit Fractions Unit Fractions & & Whole Fractions Whole Fractions
Types Of Fractions
In Proper Fractions the
In Proper Fractions the
numerator is less than the
numerator is less than the
denominator.
denominator.
E.g.. – 1/2 ,3/4 ,2/7 etc.
E.g.. – 1/2 ,3/4 ,2/7 etc.
In Proper Fractions the
In Proper Fractions the
numerator is less than the
numerator is less than the
denominator.
denominator.
E.g.. – 1/2 ,3/4 ,2/7 etc.
E.g.. – 1/2 ,3/4 ,2/7 etc.
1
1
4
4
1
1
4
4
In Improper Fractions the
In Improper Fractions the
numerator is greater than (or
numerator is greater than (or
equal to)
equal to) the denominator.the denominator. E.g. – 4/2 ,9/5 ,6/6 etc.
E.g. – 4/2 ,9/5 ,6/6 etc.
Every whole no is Improper
Every whole no is Improper
Fraction
Fraction
E.g. – 24 = 24/1
E.g. – 24 = 24/1
In Improper Fractions the
In Improper Fractions the
numerator is greater than (or
numerator is greater than (or
equal to)
equal to) the denominator.the denominator. E.g. – 4/2 ,9/5 ,6/6 etc.
E.g. – 4/2 ,9/5 ,6/6 etc.
Every whole no is Improper
Every whole no is Improper
Fraction
Fraction
E.g. – 24 = 24/1
E.g. – 24 = 24/1
7
7
4
4
7
7
4
4
Proper
Proper
&
&
Improper
Improper
Fractions
Fractions
I’m Smaller
I’m Smaller
I’m Bigger
1 ¾
1 ¾
1 ¾
1 ¾
7/8
7/8
7/8
7/8
Mixed Fractions
Mixed Fractions
In Mixed Fractions a whole number
In Mixed Fractions a whole number
and a proper fraction are together.
and a proper fraction are together.
E.g.. –2 1/4, 16 2/5 etc.
E.g.. –2 1/4, 16 2/5 etc.
Mixed Fractions and Improper
Mixed Fractions and Improper
Fractions are same.
Fractions are same.
We can use any to show the same
We can use any to show the same
amount.
amount.
=
Conversion
Conversion
8 1 9 8 19 1
= 1
8 8
7
17
7
3
7
2
7
3
2
Improper
Improper
Fraction
Fraction
to
to
Mixed
Mixed
Fraction
Fraction
divide the numerator by the denominator the
divide the numerator by the denominator the
quotient is the leading number, the remainder as
quotient is the leading number, the remainder as
the new numerator.
the new numerator.
Mixed
Mixed
Fraction to Improper Fraction
Fraction to Improper Fraction
multiply the whole number with the denominator
multiply the whole number with the denominator
and add the numerator to it. The answer is the
and add the numerator to it. The answer is the
numerator and the denominator is same
Like
Like
&
&
Unlike
Unlike
Fractions
Fractions
In Like Fractions the denominators of
In Like Fractions the denominators of
the Fractions are same
the Fractions are same
In Unlike Fractions the denominators
In Unlike Fractions the denominators
of the Fractions are different.
Unit Fractions
Unit Fractions
&
&
Whole Fractions
Whole Fractions
In Unit Fractions the
In Unit Fractions the
numerator of the
numerator of the
Fraction is 1.
Fraction is 1.
In Whole Fractions the
In Whole Fractions the
denominators of the Fraction is
denominators of the Fraction is
1.
Convert Unlike Fractions to Like
Convert Unlike Fractions to Like
Fractions
Fractions
Convert Unlike Fractions to Like
Convert Unlike Fractions to Like
Fractions
Fractions
Simplify all the Fractions.
Simplify all the Fractions.
Find LCM of all the Denominators.
Find LCM of all the Denominators.
Multiply all the fractions with a special form of 1
Multiply all the fractions with a special form of 1
to get 84 (here). Now these are Like Fractions.
to get 84 (here). Now these are Like Fractions.
3 5 4
3 5 4
4 3 7
4 3 7
,
,
4,3,7 4,3,7 2,3,7 2,3,7 1,3,7 1,3,7 1,1,7 1,1,7 1,1,1 1,1,1 2 2 2 3 7
2 2 3 7 = 84
2 2 3 7 = 84
63 113 48
63 113 48
84 84 84
84 84 84
,
,
=
Equivalent Fractions
Equivalent Fractions
They are the fractions that may have many _{They are the fractions that may have many }
different appearances, but are same.
different appearances, but are same.
In the following picture we have ½ of a cake as the _{In the following picture we have ½ of a cake as the }
cake is divided into two congruent parts and we have
cake is divided into two congruent parts and we have
only one of those parts.
only one of those parts.
But if we cut the cake into smaller congruent _{But if we cut the cake into smaller congruent }
pieces, we can see that
pieces, we can see that
½ = 2/4 = 4/8 = 3/6
Equivalent Fractions
Equivalent Fractions
To know that two or more Fractions are
To know that two or more Fractions are
Equivalent we must simplify (change to its lowest
Equivalent we must simplify (change to its lowest
term) them.
term) them.
Simplify: A fraction is in its lowest terms (or is
Simplify: A fraction is in its lowest terms (or is
reduced) if we cannot find a whole number (other
reduced) if we cannot find a whole number (other
than 1) that can divide into both its numerator and
than 1) that can divide into both its numerator and
denominator.
denominator.
E.g. 5 : 5 & 10 can be divided by 5. 5 1
E.g. 5 : 5 & 10 can be divided by 5. 5 1
Example
Example
Example
Example
Are 14 & 30 equal ?
21 45
2 = 2 14 = 30
3 3 21 45
Making Equivalent Fractions
Making Equivalent Fractions
To make Equivalent Fractions we multiply the _{To make Equivalent Fractions we multiply the }
Fraction with a special form of 1 (same numerator &
Fraction with a special form of 1 (same numerator &
denominator 4/4, 10/10 etc.)
denominator 4/4, 10/10 etc.)
E.g. : 4 = 4 5 = 20
E.g. : 4 = 4 5 = 20
Operations Of Fractions
Operations Of Fractions
Subtraction
Subtraction
Subtraction
Subtraction
Multiplication
Multiplication
Multiplication
Multiplication
Addition
Addition
Addition
Addition Of Fractions
Addition Of Fractions
Things To Know !!!!
Things To Know !!!!
Simplifying
Simplifying
Like and Unlike Fractions
Like and Unlike Fractions
Like fractions are Compulsory to add.
Like fractions are Compulsory to add.
If there are Unlike Fractions then convert them to like
If there are Unlike Fractions then convert them to like
fractions.
fractions.
The Denominator should not be added.
The Denominator should not be added.
Always change Improper fraction to a mixed
Always change Improper fraction to a mixed
fraction.
Adding Fractions
Adding Fractions
Addition means combining objects in two or more _{Addition means combining objects in two or more }
sets
sets
The objects must be of the same type, i.e. we _{The objects must be of the same type, i.e. we }
combine bundles with bundles and sticks with sticks.
combine bundles with bundles and sticks with sticks.
In fractions, we can only combine pieces of the _{In fractions, we can only combine pieces of the }
same size. In other words, the denominators must be
same size. In other words, the denominators must be
the same.
Adding Fractions With
Adding Fractions With
Same Denominators
Same Denominators
+
=
Add the numerator
and
leave the denominator as it is.
1 2 3 1
1 2 3 1
Adding Fractions with
Adding Fractions with
Different Denominators
Different Denominators
If there are different denominators in the
If there are different denominators in the
fractions, then we change them to like
fractions, then we change them to like
fractions.
fractions.
15
5
3
1
5
2
15
6
5
2
3
1
15
11
15
6
15
5
5
2
3
Adding Mixed Fractions
Adding Mixed Fractions
Change the Mixed fractions to
Change the Mixed fractions to
Improper Fractions and then to
Improper Fractions and then to
Like Fractions.
Like Fractions.
At last add the Improper Like
At last add the Improper Like
Fractions.
Fractions.
Don’t forget to change the
Don’t forget to change the
answer to Mixed Fraction again.
Subtraction Of Fractions
Subtraction Of Fractions
Things To Know !!!!
Things To Know !!!!
Simplifying
Simplifying
Like and Unlike Fractions
Like and Unlike Fractions
Like fractions are Compulsory to subtract.
Like fractions are Compulsory to subtract.
If there are Unlike Fractions then convert them to like
If there are Unlike Fractions then convert them to like
fractions.
fractions.
The Denominator should not be subtracted.
The Denominator should not be subtracted.
Always change Improper fraction to a mixed
Always change Improper fraction to a mixed
fraction.
fraction.
Subtracting Fractions
Subtracting Fractions
Subtraction means taking objects away.
The objects must be of the same type, i.e. we
can only take away apples from a group of
apples.
In fractions, we can only take away pieces of
the same size.
In other words, the denominators must be
Subtracting Fractions With
Subtracting Fractions With
Same Denominators
Same Denominators

=
Subtract the numerator
and
leave the denominator as it is.
4 2 2 1
4 2 2 1
= =
= =
6 6 6 3
Subtracting Fractions with
Subtracting Fractions with
Different Denominators
Different Denominators
If there are different denominators in the
If there are different denominators in the
fractions, then we change them to like
fractions, then we change them to like
fractions.
fractions.
15
10
3
2
5
2
15
6
5
2
3
2
15
4
15
6
15
10
5
2
3
Subtracting Mixed Fractions
Subtracting Mixed Fractions
Change the Mixed fractions to
Change the Mixed fractions to
Improper Fractions and then to
Improper Fractions and then to
Like Fractions.
Like Fractions.
At last subtract the Improper
At last subtract the Improper
Like Fractions.
Like Fractions.
Don’t forget to change the
Don’t forget to change the
answer to Mixed Fraction again.
Multiplication Of Fractions
Multiplication Of Fractions
Things To Know !!!!
Things To Know !!!!
Simplifying_{Simplifying}
The Denominator are always multiplied._{The Denominator are always multiplied.}
“ _{“ }Of ” : Of means multiply. E.g. : ½ of 2 = ½ Of ” : Of means multiply. E.g. : ½ of 2 = ½ × 2 = 1× 2 = 1
Product of two Proper Fractions is always less than both of them._{Product of two Proper Fractions is always less than both of them.}
Product of two Improper Fractions is always _{Product of two Improper Fractions is always }greatergreater than both of than both of them.
them.
Product of one Improper Fraction and one Proper _{Product of one Improper Fraction and one Proper }
Fraction is always less than the Improper Fraction
Fraction is always less than the Improper Fraction
and greater than the Proper Fraction.
Multiplying Fractions
Multiplying Fractions
To Multiply Fractions we Multiply both  The _{To Multiply Fractions we Multiply both  The }
numerators and the Denominators separately.
numerators and the Denominators separately.
2 3 2
2 3 2
×
×
3 6 3
3 6 3
×
×
= =
= =
=
=
4 2 4 × 2 8 4
Multiplying Mixed Fractions
Multiplying Mixed Fractions
Change the Mixed fractions to
Change the Mixed fractions to
Improper Fractions.
Improper Fractions.
Then multiply the Improper
Then multiply the Improper
Fraction.
Fraction.
Don’t forget to change the
Don’t forget to change the
answer to Mixed Fraction again.
Addition Of Fractions
Addition Of Fractions
Things To Know !!!!
Things To Know !!!!
Simplifying
Simplifying
Multiplication
Multiplication
Always change Improper fraction to a mixed fraction.
Always change Improper fraction to a mixed fraction.
Reciprocal: The inverse of fraction
Reciprocal: The inverse of fraction
E.g. – 2/3 = 3/2 = 1 ½ , 2 ½ = 5/2 = 2/5
Dividing Fractions
Dividing Fractions
To Divide Fractions we change the Second Fraction
with its Reciprocal.
Then Multiply the Reciprocal with the First
Fraction.
Dividing Mixed Fractions
Dividing Mixed Fractions
Change the Mixed fractions to
Change the Mixed fractions to
Improper Fractions.
Improper Fractions.
Then Multiply the Improper
Then Multiply the Improper
Fraction.
Fraction.
Don’t forget to change the
Don’t forget to change the
answer to Mixed Fraction again.
Comparison
Comparison
Comparison
Comparison
• Greater than and Smaller than_{Greater than and Smaller than}
Greater than and Smaller than
Greater than and Smaller than
Change the Fractions to Like Fractions.
The fraction with greater numerator is bigger
than the other one.
If the Numerators are same, then the Fraction
with smallest Denominator is the Biggest.
>
2 3 10 9
& =
How does the Denominator
How does the Denominator
controls the Fraction
controls the Fraction
Denominator represents the total number of pieces._{Denominator represents the total number of pieces.} If we share a Pizza with 2 people, we get ½ of pizza._{If we share a Pizza with 2 people, we get ½ of pizza.} If we share a Pizza with 4 people, we get ¼ of pizza._{If we share a Pizza with 4 people, we get ¼ of pizza.} If we share a Pizza with 8 people, we get 1/8 of pizza._{If we share a Pizza with 8 people, we get 1/8 of pizza.}
Conclusion:
Conclusion:
The larger the denominator the smaller the pieces, and
The larger the denominator the smaller the pieces, and
if the numerator is kept fixed, the larger the
if the numerator is kept fixed, the larger the
denominator the smaller the fraction,
How does the Numerator controls
How does the Numerator controls
the Fraction
the Fraction
Numerator represents the number of pieces._{Numerator represents the number of pieces.}
_{If we share a Pizza with 16 people, we get 1/16 of pizza.}_{If we share a Pizza with 16 people, we get 1/16 of pizza.} If we share a Pizza with 13 people, we get 3/16 of pizza._{If we share a Pizza with 13 people, we get 3/16 of pizza.} If we share a Pizza with 5 people, we get 5/16 of pizza._{If we share a Pizza with 5 people, we get 5/16 of pizza.}
Conclusion :
Conclusion :
When the numerator gets larger we have more pieces.
When the numerator gets larger we have more pieces.
And if the denominator is kept fixed, the larger numerator
And if the denominator is kept fixed, the larger numerator
makes a bigger fraction.