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# Simplifying fractions

## Introduction

Fractions involving symbols occur very frequently in engineering mathematics. It is necessary to be able to simplify these and rewrite them in different but equivalent forms. On this leaflet we revise how these processes are carried out.

## 1. Expressing a fraction in its simplest form

An algebraic fraction can always be expressed in different, yet equivalent forms. A fraction is expressed in its simplest form by cancelling any factors which are common to both the numerator and the denominator. You need to remember that factors are multiplied together. For example, the two fractions and are equivalent. Note that there is a common factor of a in the numerator and the denominator of which can be cancelled to give .

To express a fraction in its simplest form,an y factors which are common to both the numerator and the denominator are cancelled.

Notice that cancelling is equivalent to dividing the top and the bottom by the common factor. It is also important to note that can be converted back to the equivalent fraction by multiplying both the numerator and denominator of by a.

A fraction is expressed in an equivalent form by multiplying both top and bottom by the same quantity,or dividing top and bottom by the same quantity

Example

The two fractions

are equivalent. Note that

and so there are common factors of 5 and y Ã— y . These can be cancelled to leave .

Example

The fractions

are equivalent. In the first fraction,the common factor ( x + 3) can be cancelled.

Example

The fractions

are equivalent. In the first fraction,the common factor a can be cancelled. Nothing else can be cancelled.

Example

In the fraction

there are no common factors which can be cancelled. Neither a nor b is a factor of the numerator. Neither a nor b is a factor of the denominator.

Example

Express as an equivalent fraction with denominator (2 x + 1)( x - 7).

Solution

To achieve the required denominator we must multiply both top and bottom by ( x - 7). That is

If we wished,the brackets could now be removed to write the fraction as

Exercises

1. Express each of the following fractions in its simplest form: