Cancelling Fractions

This article explores how cancelling fractions work in mathematics. Cancelling fractions is an important skill in mat...

This article explores how cancelling fractions work in mathematics. Cancelling fractions is an important skill in mathematics. In this article we shall be looking at how to simplify fractions by cancelling the common factors and factorising to identify common factors.

You must have some basic knowledge of indices and factorising quadratics because this won’t be explained in detail here.

Simplifying a Fraction

In this section we shall look at simplifying fractions by cancelling out similar factors in the denominator. The following examples deal with simplifying fractions.

Example: Simplify the following fraction.
30m3n2/5m2n
30m×m×m×n×n/5×m×m×n
30m×m×m×n×n/5×m×m×n
30mn/5
6mn
Explanation:

To see exactly how this is simplified we’re going to work out the entire expression in parts by expanding it, for example we shall write m3 as

m3 = m×m×m

Here is the expression expanded.

30m×m×m×n×n/5×m×m×n

Since this is division we’re going to cancel the values at the top with the ones at the bottom as follows as shown below;

30m×m×m×n×n/5×m×m×n

The ms and ns in the denominator have all disappeared. so we rewrite the expression as;

30mn/5

The expression can be simplified further. 30 divided by 5 gives 6.

6mn

Factorising and simplifying

In this article we shall look at factorising and simplifying fractions.

Example: Simplify the following fraction;
9a + 72/9a2
Answer:
= 9(a + 8)/9a2
= 9(a + 8)/9a2
= a + 8/a2
Explanation:

You might be attempted here to cancel out as we did before. That would be wrong. The 8 in the numerator is not a multiple of a It is an addition. Therefore we can’t cancel.

What we will need to do is factorise the numerator first.

= 9(a + 8)/9a2

Now we can do some cancelling in the fraction. If you look at the expression you will see that both the numerator and the denominator have a factor of 9, so we can cancel them out.

= 9(a + 8)/9a2

Therefore the expression becomes;

= a + 8/a2
This is the simplified version of the original expression. Each term in the original expression was a multiple of 8=9. We divided each term in the expression by 9 to get a simplified version of the fraction.
Example: Work out the following division
2ab + 4a2/6a
Answer:
= 2a(b + 2a)/6a
= 2a(b + 2a)/6a
= 12(b + 2a)/63
= b + 2a/3
Explanation:

First we factorise the numerator as we did before

= 2a(b + 2a)/6a

The numerator and the denominator both have a factor of a, so we can cancel out this in the fraction.

= 2a(b + 2a)/6a

Both the numerator and denominator also have a common factor of 2 so we divide the numerator and denominator by 2 to cancel this out as follows;

= 12(b + 2a)/63

This calculaton resuts in a simplified version;

= b + 2a/3

Each term in the original fraction expression had a multiple of 2 and a, therefore we have divided each term by 2a to get a simplified fraction.

Quadratic Fractions

This is example we shall explore fractions involving quadratic expressions.

Example: Simplify the following fraction as much as possible;
x + 3/x2 + 5x + 6
Answer:
x + 3/x2 + 5x + 6
= x + 3/(x + 3)(x + 2)
= x + 3/(x + 3)(x + 2)
= 1/x + 2
Explanation:

The same principle and methods applies with quadratic expressions. Expect you have to watch out for factorisation before moving forward with the factorisation.

x + 3/x2 + 5x + 6

The denominator needs to be factorised. Maybe there will be factors in the numerator that match with the denominator.

= x + 3/(x + 3)(x + 2)

The common factor in both the numerator and denominator is a (x+3), we can cancel these out.

= x + 3/(x + 3)(x + 2)

This should provide a simplified fraction of the original fraction;

= 1/x + 2
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