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FRACTIONS
Definitions
1. Fraction : If any unit be divided into any number of equal parts, one or more of
these parts is called a fraction of the unit. The fractions one-fifth, two-fifths, threefifths are written as , and respectively.
I.
II.
III.
Numerator : The upper number, which indicates the number of parts
taken to form the fraction, is called the numerator.
Denominator : The lower number, which indicates the number of equal
parts into which the unit is divided, is called the denominator.
Terms : The numerator and the denominator of a fraction are called its
terms.
Notes :
1. A fraction is also called a rational number.
2. A fraction is equal to zero when its numerator alone is zero. The denominator of
a fraction is always assumed to be non-zero.
3. A fraction is unit when its numerator and denominator are equal.
4. The value of a fraction is not altered by multiplying or dividing the numerator and
the denominator by the same number.
Example :
=
=
5. When the numerator and denominator of a fraction have no common factor other
than 1, the fraction is said to be in its lowest terms.
Example :
=
=
6. When the fraction is reduced to its lowest terms, its numerator and denominator
are prime to each other i.e., they have no common factor.
7. An integer can be expressed as a fraction with any denominator we want.
Example : If we want to express 19 as a fraction whose denominator is 17, the
process will be as follows :
19
=
2. Kinds of Fractions :
I.
Proper Fraction : A proper fraction is one whose numerator is less than the
denominator.
Example :
II.
Improper Fraction : A fraction whose numerator is equal to or greater than the
denominator is called an improper fractions.
Example :
III.
, , , , etc. are proper fractions.
,
, , , etc. are improper fractions.
Mixed Number : We see that an improper fraction is made up of a whole number
and a proper fraction. When an improper fraction is changed to consist of a
whole number and a proper fraction it is called a mixed number.
Example : 2 , 3 , 4 , 8 etc. are mixed numbers.
IV.
Equal Fractions : The fractions which have equal value are called equal
fractions.
Example :
V.
, ,
,
etc.
Reciprocal Fractions : Interchange of the numerator and denominator of a
fraction to find the new fraction is called the reciprocal fraction of that fraction.
Example : Reciprocal fraction of
VI.
,
,
,
,
etc.
Unlike Fractions : Fractions in which the denominators are different are called
unlike fractions.
Example :
VIII.
.
Like Fractions : Fractions in which the denominators are the same are called
like fractions.
Example :
VII.
=
,
,
,
etc.
Complex Fractions : A complex fraction is one in which the numerator or
denominator or both are fractions.
,
Example :
IX.
Decimal Fractions : Fractions in which the denominators are powers of 10 are
called decimal fractions.
,
Example :
X.
,
,
etc.
Compound Fractions : A fraction of a fraction is called a compound fraction.
Example :
XI.
etc. are complex fractions.
of
is a compound fraction.
Continued Fractions : Fractions that contain an additional fraction in the
numerator or the denominator are called continued fractions.
Example : Fractions of the form
(a) 3 +
(b)
Are called continued fractions.
3. Arrangement of fractions in Ascending and Descending order : When two
or more than two fractions are given and are to be arranged in ascending or
descending order, first we bring change in the fraction for having the same
denominator and according to numerator we arrange the fractions in ascending
or descending order.
Example : Arrange the fractions
We have LCM of 2, 3, 4 =12.
=
×
=
× =
=
× =
=
, and in ascending or descending order.
6 < 8 < 9.
Ascending order of fractions :
Descending order of fractions :
< <
>
>
and
.
4. Changed of mixed fraction into improper fraction : Denominator of a fraction
is multiplied by whole integral and added to the numerator.
Example : 3 =
=
( improper fraction )
5. Change of improper fraction into mixed fraction : Numerator is divided by
denominator and changed into mixed fraction.
Example :
= 9 (mixed fraction)
1. Standard form of Fraction : Numerator and denominator of fraction is divided
by the HCF of a fraction and converted into standard form.
Example : Convert
into standard form.
Soln. HCF of 425 and 315 = 5
Standard form of fraction =
=
Some Important Laws
Comparison of Fractions : If two fractions
=
are given, then
I.
If ad > bc then
>
II.
If ad < bc then
>
III.
When fractions have the same numerator but different denominator, then that
fraction is the smallest which has the largest denominator.
Example :
IV.
and
is the smaller.
When fractions have the same denominator but different numerators then that
fraction is the largest which has the largest numerator.
and
Example :
V.
. The fraction
. The fraction
is the larger.
If the difference between numerator and denominator of two or more than two
fractions is the same then that fraction is the largest which has the largest
numerator and denominator.
Example :
,
. The fraction
,
fractions is given as
>
is the largest, i.e., descending order of
>
Some Special Tricks
1. Addition and Subtraction of Fractions :
Trick (i) : Fractions of same base :
Rule : (a) First add or subtract integers and also its numerators.
(b) Then divide by denominator.
Example 1. 2 + 2 = ?
Soln. Quick :
(2+3)+
= 6 Ans.
Example 2. 3 - 1 = ?
Soln. Quick :
(3- 1)+
= 2 Ans.
Trick (ii) : Fractions of common base :
Rule : (a) First larger denominator is divided by smaller denominator of given
fractions.
(b) Quotient is multiplied by numerator of smaller denominator.
(c ) Now add or subtract to obtain value of numerator of larger fraction and write
the larger denominator in the form of denominator.
Example 1.
+
=?
Soln. Quick :
=
Example 2. 3 - 1
Soln. Quick :
= 1 Ans.
=?
(3- 1)+
=2
Ans.
Trick (iii) : Fractions of common base :
Rule : (a) First, add the integers of two given fractions.
(b) Second, find the value of proper fraction with simple form and add.
(c ) Now find the value of the fraction.
Example 1. 3 + 6 = ?
(3+6)+(
Soln. Quick :
) = 9+
= 10
=9+1
Example 2.
Soln. Quick :
-
Ans.
=?
=
Ans.
2. Multiplication of Fractions :
Trick (i) : Multiplication with integer and fraction
Rule : x × y = ( x × y ) +
×y
Example 1. 1 × 9 = ?
Soln. Quick :
(1
9)+
× 9 = 9 + 6 = 15 Ans.
Example 2. 8 × 16 = ?
Soln. Quick :
(8
16 ) +
× 16 = 128 + 4 = 132 Ans.
Trick (ii) : If the sum of the fractional parts is 1.
Rule : (Integer + 1) × Integer
Fraction × Fraction
Step II
Step I
Example : 4 × 4 = ?
Soln. Quick :
(4+1)×4[
×
] = 20 Ans.
Trick (iii) : If the sum of the fractional parts are
Rule : (Integer) × (Integer) +
× Integer
Fraction × Fraction
Step II
Step I
Example : 4 × 4 = ?
Soln. Quick :
(4×4)+ ×4[
×
] = 18 Ans.
3.
Division of Fractions :
Trick (i) : Division in fraction with integer :
Rule : Fraction × Reciprocal of the integer
Example :
÷ 16 = ?
×
Soln. Quick :
=
Ans.
Trick (ii) : Division in fraction with fraction :
Rule : Fraction × Reciprocal of the fraction
Example :
Soln. Quick :
÷
=?
× =
Ans.
Memorable Facts
1.
There is no change in fraction if
the same number is multiplied of divided by the numerator and denominator of a
fraction.
2.
If numerator and denominator
of any fraction is the same, the value of fraction is 1.
3.
If numerator of any fraction is
zero, then the value of fraction is also zero.
4.
In any fraction, denominator
never assumes zero, but if denominator of fraction is zero then the value of
fraction is infinite ( ).
5.
If numerator and denominator
of any fraction is 1, then the value of fraction is always 1.