Simplifying Fractions with Variables

FractionIn Mathematics, fraction is nothing but two numbers written up and down separated by ahorizontal line. The number above the horizontal line is the numerator and the number below thehorizontal line is the denominator. Examples of simple fractions are

1     3

-- , --

2    4


There are fractions with very high values in numerator and denominator which are very complexto use. So, for ease of use, these fractions are simplified.

Simplifying Fractions

 Simplifying fractions is nothing but dividing both the numerator and the denominator by thehighest common factor to reduce the fraction into a smaller value which can be easily used forcalculations. The fraction obtained is checked for common factor again. If any common factoravailable, then the fraction is again simplified. It goes on until the fraction has become very muchsimpler and there are no common factors anymore.

Simplifying Fractions with Variables

numerical fractions contain numbers. There are rational expressions which contain fractionswith variables. We saw above that in case of numerical fractions, the fractions are simplifiedusing highest common factor. In case of fractions with variables, the common variable factorsand numerical are cancelled. An important point to note here is that the expression cannot becancelled and only the factors in the expression can be cancelled to simplify the fraction.

To Simplify Fractions with Variables, we have to follow simple steps. Let us see the Steps toSimplify Fractions

Steps to Simplify Fractions
The first step to simplify fractions is to write the given rational expression in terms ofnumerical and variable factors.Let us consider Example 1





This can be written in terms of factors as 4.x and 2.x.x. We know that the result ofanything divided by itself gives 1. In the above example, the numerator and thedenominator have one x in common. As x divided by x is 1, we can cancel one x innumerator and one x in the denominator. Also, if we see, the numerical factors inthe given fraction are 4 and 2 and 4 is divisible by 2. Thus, after simplification, thenumerator will have the numerical factor 2 and the denominator will have the variable x.




Consider the Example 2


4+ x


2+ x^2


In example 2 you cannot cancel 4 by 2 as 4 and 2 are part of expressions and are notfactors as in example 1. But if the expressions are represented as factors, then theexpression can be cancelled as in example 3.


example 3:

2(4+ x)


(2+ x^2) (4+ x)


Here the expression (4+x) is common in both the numerator and the denominator. As theexpression is a factor, it can be cancelled in both the numerator and the denominator. Weget



(2+ x^2)


But, we cannot cancel the 2 in the numerator and the 2 in the denominator, as 2 in thenumerator is a factor but 2 in the denominator is a part of the expression and not a factor.

Simplification of the fractions favors easy computations.