FRACTION
Fraction – A fraction is a part of
a whole. If a whole thing is divided into equal parts, each part is a fraction.
Or, numbers of the form
where
are called fractions.
e.g. –
A fraction is made up
of two parts.
Numerator – It tells us how many of the equal parts
Denominator – It tells us how many
equal parts are
you have taken. there
in the whole.
Fraction as
Division – A
fraction can be expressed as a division sum and vice-versa.
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Here, 1 part is divided into 5
equal parts.
So, each part is
one-fifth or
of the whole.
Here,
1 is the numerator and 5 is the denominator.
e.g.
– the fraction
means
Here,
is a fraction and one-fifth is a fractional
number.
Fraction on
number line –
The fraction
on the number line.
O A
Divide
OA into 7 equal parts and take 4 parts out of it to reach the point P.
Then,
the point P represents the number
.
Fractions are
of different kinds –
(A)Proper
fraction – A
fraction whose number is less than the denominator is called a proper fraction.
e.g. –
(B)Improper
fraction – A
fraction whose numerator is more than the denominator is called an improper
Fraction.
e.g. –
ð
All
improper fractions represents numbers greater than or equal to 1.
(C) Mixed Numbers or Mixed Fractions – When an improper fraction is written
as a combination of a whole
number and a proper fraction it is
called a mixed number or mixed fraction.
Or, A number which has a whole
number mixed with a fraction, is called a mixed number.
e.g. –
(D)
Decimal
fraction – A
fraction whose denominator is a multiple of 10 is called a decimal fraction.
e.g. -
(E)Vulgar
fraction – Fractions
having denominators as whole numbers other than multiples of 10 are called
Vulgar fractions.
e.g. –
(F) Equivalent fraction – An equivalent fraction of a given fraction can be
obtained by multiplying or dividing
its numerator and denominator by the
same non-zero number.
e.g. –
Thus,
=
(G)Like
fractions – The
fractions which have the same denominator are called like fractions.
e.g. –
(H)
Unlike
fractions – The
fractions which have different denominators are called unlike fractions.
Or, fractions having different
denominators are called unlike fractions.
e.g. –
(I) Unit fractions – A fraction which has 1 as its numerator is called a
unit fraction.
Or, A fraction having 1 as the
numerator is called a unit fraction.
e.g. –
(J) Simple fractions – A fraction of the form
are whole numbers is called a simple
Fraction.
e.g. –
(K)Complex
fraction – A
fraction of the form
and q (say
= c/d) are fractions is called a
complex fraction.
e.g. –
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is
a complex fraction.
Fractions in the Lowest Terms
Or, Simplest
Form Reducing a fraction to its lowest terms –
A fraction is said to be
in its lowest terms when its numerator and denominator are co-primes i.e. , they have only 1 as the common
factor.
Or, a fraction
is said to be in its lowest form, if the HCF
of
and
= 1.
Or, a fraction is in its
simplest form or in the lowest terms when its numerator and denominator have no
common factors other than 1.
In other words, a
fraction is said to be in the lowest terms or in the simplest form if the HCF
of the numerator and the denominator is 1.
e.g. -
soln. here,
numerator = 7 and denominator = 9.
Factors of 7 are : 1 and 7
Factors of 9 are : 1, 3 and 9.
Common factors of 7 and 9 is 1 only.
So, the HCF of 7 and 9 is 1.
Hence,
is in its lowest term.
Comparison of
fractions –
ⓐ When two fractions have the same
numerator, the fraction having a smaller denominator is the greater fraction.
e.g. -
ⓑ When two fractions have the same
denominator, the fraction having a smaller numerator is the smaller fraction.
e.g. -
ⓒ When the given fractions are unlike i.e.
they have different numerators and denominators, then we can compare
them
by two methods.
M- I – By Cross-multiplication –
In this method,
we cross-multiply the two fractions and find two products.
N1D2
> D1N2, then ⇒ first fraction > second fraction
N1D2
< D1N2, then ⇒ first fraction < second fraction
N1D2
= D1N2, then ⇒ first fraction = second fraction
(equivalent)
M-II – By Converting Unlike Fractions Using
LCM –
In this method,
first we convert the given unlike fractions into like fractions and then
compare them.
ⓐ
First,
we find out the LCM of the denominators of the given fractions i.e. 8 and 10.
ⓑ Then, we divide this LCM by both the
denominators.
ⓒ Then, we multiply the numerators and the
denominators of both the fractions by the respective quotients found in the
previous step.
We get two like fractions equivalent to
the given unlike fractions.
ⓓ
Finally,
we compare these like fractions by comparing their numerators.
e.g. –
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2
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8, 10
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4, 5
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LCM
=
As 25 < 28 thus,
or,
Division of
fractions –
Reciprocals – The product of such numbers is always 1
and they are called reciprocals.
Thus,
reciprocals are number pairs that have the product 1.
e.g.
–
Division of a
Fraction by a Fraction –
Rule
– In
order to divide a fraction by another fraction, we multiply the dividend by the
reciprocal of the divisor.
e.g. –
Reciprocal
of
∴
Division of a
Fraction by a Whole Number –
Rule – (A fraction)
(A whole number) = (A fraction)
(Reciprocal of the whole number)
e.g. –
Reciprocal
of 12 = Reciprocal of
.
∴
=
Division of a
Whole Number by a Fraction –
Rule – (A whole number)
(A fraction)
(Reciprocal of the fraction)
e.g. –
Reciprocal of
.
∴
.
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