Fractions
A fraction represents a part of the whole. In a
fraction, the top number is called the numerator and the
bottom number is called the denominator. In ¾, 3 is
the numerator and 4 is the denominator.
Remember that the numerator tells us the number of parts taken
while the denominator tells us the total number of parts that the whole number
is divided into.
Types of Fractions
There are mainly three types of fractions:
1.
Proper fractions
2.
Improper fractions
3.
Mixed fractions
We have two other types of fractions as follows:
1.
Like fractions
2.
Unlike fractions
Proper Fractions
Fractions in which the numerator is smaller
than the denominator are called the proper fractions. A proper fraction is a
part of a whole. For example, 1⁄2 , 3⁄4 , 5⁄9 , 11⁄13
Improper Fractions
Fractions in which the numerator is greater
than the denominator are called the improper fractions. They are greater than a
whole. For example, 7⁄5 , 9⁄5
Mixed Fractions
When we combine a whole number and a proper fraction together, we get a mixed fraction. For example, 21⁄2 , 53⁄4
Converting Improper Fractions into Mixed Fractions
To convert 7⁄5 into mixed
fraction, divide the numerator by the denominator. Write the quotient as the
whole number, remainder as the numerator and divider as the denominator. Thus, 7⁄5 = 12⁄5
Converting Mixed Fractions into Improper Fractions
To convert 7⁄5 into improper
fraction, multiply the denominator with the whole number and add the product
with numerator and write the denominator as it is.
Like Fractions
All
those fractions whose denominators are the same are called like fractions. For
example, 1/7, 3/7, 4/7, 6/7, etc. are all like fractions.
Unlike Fractions
All
those fractions whose denominators are not the same are called unlike
fractions. For example, 1/2, 5/8, 3/4, 9/16, etc. are all unlike fractions.
Converting Unlike Fractions into Like Fractions
To
convert 1/2, 5/8, 3/4, 9/16, in like fractions, we first find the LCM of the
denominators of all the unlike fractions, i.e., 2, 8, 4 and 16.
LCM
of 2, 8, 4 and 16 = 16
Now,
find the equivalent fractions for all the fractions with denominator 16.
1/2
= 1 × 8 / 2 × 8 = 8/16
5/8
= 5 × 2 / 8 × 2 = 10/16
3/4
= 3 × 4 / 4 × 4 = 12/16
9/16
= 9 × 1 / 16 × 1 = 9/16
Thus,
8/16, 10/16, 12/16 and 9/16 are like fractions.
Equivalent Fractions
Two or more than two fractions are said to be equivalent if both have
the same value after simplification. Let us say, a/b and c/d are two fractions,
if after simplification they both result in equal fraction, say e/f, then they are
equivalent to each other.
For example, 1/3 and 5/15 are the equivalent fractions, because if
we simplify 5/15, its value is the same as 1/3. Similarly, 1/2 and 2/4 are also
the equivalent fractions.
The biggest question here can be, why do they
have equal values in spite of having different number?
The answer to this question is that, as the numerator and
denominator are not co-prime numbers, therefore they have a common multiple
which on division gives an exactly the same value.
Take for an example:
1/2 = 2/4 = 4/8
But, it is clearly seen that the above fractions have different numbers
as numerators and denominators.
Dividing both numerator and denominator by their common factor, we
have:
4/8 = 4÷4 / 8÷4 = 1/2
In the same way, if we simplify 2/4, we again get 1/2.
2/4 = 2÷2 / 4÷2 = 1/2
Here’s an example of equivalent fractions.
How to find equivalent fractions?
By multiplying
the numerator and the denominator of a fraction by the same non-zero whole
number, we can get an equivalent fraction. But it will not change its value.
Equivalent fractions may look different, but they have the same value. Let's
look at some more examples of equivalent fractions.
For example: to find equivalent fraction
of 2/3, we multiply both the numerator and the
denominator by 2, then we get equivalent fraction 4/6.
Again to find one more equivalent fraction of
2/3, we multiply both the numerator and the denominator by 3, then we get
equivalent fraction 6/9.
Similarly, we can multiply both the numerator
and the denominator by 2, 3, 4, 5, 6, etc. to get equivalent fractions of
a given fraction.
How to check two or more fractions are equivalent fractions?
Simplify all fractions. If they reduce to be
the same fraction, then the fractions are equivalent.
For example: Check the fractions 6/15 and 10/50 are equivalent or not.
We will simplify both the fractions-
6÷3 / 15÷3 =
2/5
10÷10 / 50÷10 =
1/5
The fractions 2/5 and 1/5 are not the same, hence fractions are not equivalent.
Use
cross-multiplication to check two fractions
are equivalent or not
The products are equal. Therefore, the fractions are
equivalent.
Lowest Form of a Fraction
A fraction is said to be in its lowest form if
the only common factor of the numerator and the denominator is 1.
For example, /3, 2/5, 3/7, 4/9, etc. are in their lowest forms.
Reducing a Fraction to its Lowest Form
A
fraction can be reduced to its lowest form by dividing both the numerator and
the denominator by a common factor.
Example:
Reduce 45/75 to its lowest form.
Solution:
(45 ÷ 3)/(75 ÷ 3) = 15/25
But 15/25 is not in its lowest form, so repeat the process till
the numerator and the denominator have no common factor except 1.
15/25 = (15 ÷ 5)/(25 ÷ 5) = 3/5
Therefore, the lowest form of 45/75 is 3/5 .