Fractions
Index
1.
2.
3.
4.
5.
6.
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


What are Fractions
Examples
Need of Fractions
Parts of Fractions
Example
Types of Fractions-
1.
Operations of Fractions




Addition
Subtraction
Multiplication
Division
6.
Comparison
 Greater than and Smaller than
Proper fractions & Improper Fractions
 How does the Denominator controls th
Mixed Fractions
 How does the Denominator controls th
Like Fractions & Unlike Fractions
Equivalent Fractions
What are Fractions?
Fractions are for counting part of something.
Loosely speaking, a fraction is a quantity that cannot be
represented by a whole number.
A fraction (from the Latin fractus, broken) is a number that
can represent part of a whole. The earliest fractions were
reciprocals of integers: ancient symbols representing one part
of two, one part of three, one part of four, and so on. A much
later development were the common or "vulgar" fractions
which are still used today (½, ⅝, ¾, etc.)
Examples
/60
/8
55
1
/12
11
1½
/12
1
1 2/10
Need Of Fractions?
Consider the following scenario.
Can you finish the whole cake?
If not, how many cakes did you eat?
1 is not the answer, neither is 0.
This suggest that we need a new
kind of number i.e. Fraction.
Parts Of Fractions?
The denominator tells us how
many congruent pieces the whole is
divided into, thus this number
cannot be 0. (b)
a
b
The numerator tells us how many
such pieces are being considered. (a)
I’m the NUMERATOR.
I tell you the number of parts
I’m the DENOMINATOR.
I tell you the name of part
Example
How much of a pizza do we have below?
The blue circle is our whole.
if we divide the whole into 8 congruent pieces,
- the denominator would be 8.
We can see that we have 7 of these pieces.
Therefore the numerator is 7, and we
have
7 of a pizza.
8
Types Of Fractions
Proper fractions
&
Improper Fractions
Like Fractions
&
Unlike Fractions
Mixed Fractions
Equivalent
Fractions
Unit Fractions
&
Whole Fractions
Proper & Improper Fractions
In Proper Fractions the
numerator is less than the
denominator.
E.g..
1/2 ,3/4 ,2/7 etc.
I’m– Bigger
7
4
I’m Smaller
1
4
In Improper Fractions the
numerator is greater than (or
equal to) the denominator.
E.g. – 4/2 ,9/5 ,6/6 etc.
Every whole no is Improper
Fraction
E.g. – 24 = 24/1
Mixed Fractions
In Mixed Fractions a whole number
and a proper fraction are together.
E.g.. –2 1/4, 16 2/5 etc.
Mixed Fractions and Improper
Fractions are same.
1¾
We can use any to show the same
amount.
=
7/8
Conversion
Improper Fraction to Mixed Fraction
8
divide the numerator by the denominator the
quotient is the leading number, the remainder as
the new numerator.
9
1
= 1
8
8
Mixed Fraction to Improper Fraction
multiply the whole number with the denominator
and add the numerator to it. The answer is the
numerator and the denominator is same
3 2 × 7 + 3 17
=
2 =
7
7
7
1
9
8
1
Like & Unlike Fractions
In Like Fractions the denominators of
the Fractions are same
In Unlike Fractions the denominators
of the Fractions are different.
Unit Fractions &
Whole Fractions
In Unit Fractions the
numerator of the
Fraction is 1.
In Whole Fractions the
denominators of the Fraction is
1.
Convert Unlike Fractions to Like
Fractions
 Simplify all the Fractions.
 Find LCM of all the Denominators.
 Multiply all the fractions with a special form of 1
to get 84 (here). Now these are Like Fractions.
2 × 2× 3× 7 = 84
3
4
,
5
3
,
4
7
= 63
113
,
84 84
,
48
84
2
2
3
7
4,3,7
2,3,7
1,3,7
1,1,7
1,1,1
Equivalent Fractions
They are the fractions that may have many
different appearances, but are same.
In the following picture we have ½ of a cake as the
cake is divided into two congruent parts and we have
only one of those parts.
But if we cut the cake into smaller congruent
pieces, we can see that
½ = 2/4 = 4/8 = 3/6
Equivalent Fractions
To know that two or more Fractions are
Equivalent we must simplify (change to its lowest
term) them.
Simplify: A fraction is in its lowest terms (or is
reduced) if we cannot find a whole number (other
than 1) that can divide into both its numerator and
denominator.
E.g.- 5 : 5 & 10 can be divided by 5. 5 1
10
10 2
Example
Are 14 & 30 equal ?
21
45
14
21
2 = 2
3 3
reduce
14 ÷ 7 = 2
21÷ 7 3
14 = 30
21
45
30
45
reduce
30 ÷15 = 2
45÷15 3
Making Equivalent Fractions
To make Equivalent Fractions we multiply the
Fraction with a special form of 1 (same numerator &
denominator- 4/4, 10/10 etc.)
E.g. : 4 = 4 × 5 = 20
5
5× 5
25
Operations Of Fractions
Addition
Subtraction
Division
Multiplication
Addition Of Fractions
Things To Know !!!!
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


Simplifying
Like and Unlike Fractions
Like fractions are Compulsory to add.
If there are Unlike Fractions then convert them to like
fractions.
 The Denominator should not be added.
 Always change Improper fraction to a mixed
fraction.
Adding Fractions
Addition means combining objects in two or more
sets
The objects must be of the same type, i.e. we
combine bundles with bundles and sticks with sticks.
In fractions, we can only combine pieces of the
same size. In other words, the denominators must be
the same.
Adding Fractions With
Same Denominators
=
+
1
2
+
3
=
1
=
6
6
6
2
Add the numerator
and
leave the denominator as it is.
Adding Fractions with
Different Denominators
If there are different denominators in the
fractions, then we change them to like
fractions.
1
+
2
3 5
2
6
5
1
=
=
5 15
3 15
1 2 5
6 11
+ =
+
=
3 5 15 15 15
Adding Mixed Fractions
Change the Mixed fractions to
Improper Fractions and then to
Like Fractions.
At last add the Improper Like
Fractions.
Don’t forget to change the
answer to Mixed Fraction again.
Subtraction Of Fractions
Things To Know !!!!




Simplifying
Like and Unlike Fractions
Like fractions are Compulsory to subtract.
If there are Unlike Fractions then convert them to like
fractions.
 The Denominator should not be subtracted.
 Always change Improper fraction to a mixed
fraction.
 The numerator can be negative.
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
the same.
Subtracting Fractions With
Same Denominators
=
4
2
2
=
1
=
6
6
6
3
Subtract the numerator
and
leave the denominator as it is.
Subtracting Fractions with
Different Denominators
If there are different denominators in the
fractions, then we change them to like
fractions.
2
2 10
=
3 15
3
−
2
5
2 2 10 6 4
− =
−
=
3 5 15 15 15
2
6
=
5 15
Subtracting Mixed Fractions
Change the Mixed fractions to
Improper Fractions and then to
Like Fractions.
At last subtract the Improper
Like Fractions.
Don’t forget to change the
answer to Mixed Fraction again.
Multiplication Of Fractions
Things To Know !!!!
Simplifying
The Denominator are always multiplied.
“ Of ” : Of means multiply. E.g. : ½ of 2 = ½ × 2 = 1
Product of two Proper Fractions is always less than both of them.
Product of two Improper Fractions is always greater than both of
them.
 Product of one Improper Fraction and one Proper
Fraction is always less than the Improper Fraction
and greater than the Proper Fraction.
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


Multiplying Fractions
To Multiply Fractions we Multiply both - The
numerators and the Denominators separately.
2
3
×
4
2×3
=
2
6
=
4×2
3
=
8
4
Multiplying Mixed Fractions
Change the Mixed fractions to
Improper Fractions.
Then multiply the Improper
Fraction.
Don’t forget to change the
answer to Mixed Fraction again.
Addition Of Fractions
Things To Know !!!!




Simplifying
Multiplication
Always change Improper fraction to a mixed fraction.
Reciprocal: The inverse of fraction
E.g. – 2/3 = 3/2 = 1 ½ , 2 ½ = 5/2 = 2/5
Dividing Fractions
To Divide Fractions we change the Second Fraction
with its Reciprocal.
Then Multiply the Reciprocal with the First
Fraction.
2
4
÷
4
2×5
=
5
10
=
4×4
5
=
16
8
Dividing Mixed Fractions
Change the Mixed fractions to
Improper Fractions.
Then Multiply the Improper
Fraction.
Don’t forget to change the
answer to Mixed Fraction again.
Comparison
Comparison
•
•
•
Greater than and Smaller than
How does the Denominator controls the Fraction
How does the Numerator controls the Fraction
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
&
3
10
=
5
15
>
9
15
How does the Denominator
controls the Fraction
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 4 people, we get ¼ of pizza.
If we share a Pizza with 8 people, we get 1/8 of pizza.
Conclusion:
The larger the denominator the smaller the pieces, and
if the numerator is kept fixed, the larger the
denominator the smaller the fraction,
How does the Numerator controls
the Fraction
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 13 people, we get 3/16 of pizza.
If we share a Pizza with 5 people, we get 5/16 of pizza.
Conclusion :
When the numerator gets larger we have more pieces.
And if the denominator is kept fixed, the larger numerator
makes a bigger fraction.