 # Simplify Algebraic Fractions

In this worksheet, students will simplify and solve algebraic fractions using factorisation. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Number, Algebra

Curriculum subtopic:   Structure and Calculation, Algebraic Expressions

Difficulty level:   ### QUESTION 1 of 10

You may know how to simplify fractions by dividing by the same number, but do you understand why this works?

Let's look at an example.

If you were asked to cancel 6/8, you would divide by 2 and get 3/4.

Awesome and perfectly fine, but let's dig a bit deeper.

We can rewrite 6/8 like this...

 6 8
=
 3 x 2 4 x 2

We also know that when we are multiplying fractions, we multiply the denominators and multiply the numerators so we can also say that:

 3 x 2 4 x 2
=
 3 4
x
 2 2

We should also be able to say that 2/2 = 1 and multiplying by 1 can be ignored. so now we can say that:

 6 8
=
 3 4

Why is this important?

You would never be asked to show all this working if you were cancelling down a fraction like 6/8 so why is this important?

It's because the technique shown, which is called cancelling by factorisation, is the only way to cancel fractions that involve algebra.

How to cancel by factorisation

Whenever you see a fraction involving algebra, remember that you have to factorise it first.

e.g. Simplify the following fraction:

 6x + 3 10x + 5

If we look at each line individually:

6x + 3 can be factorised to 3 (2x + 1)

10x + 5 can be factorised to 5 (2x + 1)

So we can rewrite our fraction as:

 6x + 3 10x + 5
=
 3 (2x + 1) 5 (2x + 1)

Anything that is the same on the top and bottom can now cancel each other out:

 3 (2x + 1) 5 (2x + 1)
=
 3 5

e.g. Simplify the following fraction:

 15x2 + 5x 30x + 10

If we look at each line individually:

15x2 + 5x can be factorised to 5x (3x + 1)

30x + 10 can be factorised to 10 (3x + 1)

So we can rewrite our fraction as:

 15x2 + 5x 30x + 10
=
 5x (3x + 1) 10 (3x + 1)

Anything that is the same on the top and bottom can now cancel each other out:

 5x (3x + 1) 210 (3x + 1)
=
 x 2

In this activity, you will simplify and solve algebraic fractions using a deeper understanding of dividing fractions to simplify as shown in the examples above.

Complete the sentence below to summarise how to simplify algebraic fractions.

Which of the options in the list is the correct factorisation of the expression below?

45x2 + 30x

15x (3x + 2)

5x (9x + 6)

15x (3 + 2x)

Consider these two expressions:

6x2 + 2x

10x3 + 5x2

For each of the elements shown in the table, select if it is a common factor of both expressions.

What is the highest common factor both the expressions below?

6x2 + 2x

15x + 5

Hint: Try factorising them both first.

Simplify the expression:

 10x + 5 12x + 6

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

Consider the expression:

 6x + 15 8x + 20

Then complete the sentence below to describe how to factorise it.

Simplify the expression:

 3x + 12 2x2 + 8x

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

Simplify the expression:

 6a2 + 15a 2ab + 5b

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

Simplify the expression:

 6x2 + 2x 15x2 + 5x

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

Which of the expressions in the list is a fully simplified version of the expression below?

3x3 + 6x2

3(x3 + 2x2)

3x2(x + 2)

3x(x2 + 2x)

• Question 1

Complete the sentence below to summarise how to simplify algebraic fractions.

EDDIE SAYS
Yep, it's always this step first. You can only simplify an algebraic fraction by factorising first. Remember that 'factorising' means like splitting an expression into a multiplication of simpler expressions. Let's try doing this now...
• Question 2

Which of the options in the list is the correct factorisation of the expression below?

45x2 + 30x

15x (3x + 2)
EDDIE SAYS
When we are factorising, we need to find the highest common factor. For the two elements in this expression, both can be divided by 15 and have an x present. We need to be really careful though and not jump on the first answer we see that has 15x. We need to look to find the x outside the bracket. This means that it is affecting the entire rest of the expression. The correct answer is 15x (3x +2) as we need to see an x beside the first number in the bracket to create the x2. 15x (3 + 2x) was close, but had the x beside the 2 inside the bracket rather than the 3. Hopefully that didn't catch you out!
• Question 3

Consider these two expressions:

6x2 + 2x

10x3 + 5x2

For each of the elements shown in the table, select if it is a common factor of both expressions.

EDDIE SAYS
These two expressions look like they should have lots of common, but they actually only have 2 things which are the same. For an option to be a common factor, it must appear in all four elements of the expressions. For example, x2 is not a common factor because you can't divide 2x by x2. Both the expressions have an x in them, and can be divided by 1.
• Question 4

What is the highest common factor both the expressions below?

6x2 + 2x

15x + 5

Hint: Try factorising them both first.

3x + 1
3x +1
3x+ 1
EDDIE SAYS
This is quite difficult to spot, unless you factorise both expressions first. If you did this, you should have reached: 2x(3x+1) 5(3x +1) Remember that a common factor is anything that you can divide both expressions by. In this case, the answer is 3x + 1.
• Question 5

Simplify the expression:

 10x + 5 12x + 6

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

5/6
EDDIE SAYS
Remember the golden rule: If you want to simplify an algebraic fraction, you have to factorise first. If you do this correctly you should have...
 5(2x + 1) 6(2x + 1)
Can you see the factor to cancel from this? That's right, there is (2x + 1) on both the top and bottom rows, so these cancel each other out. So we are left with 5/6 as our final answer.
• Question 6

Consider the expression:

 6x + 15 8x + 20

Then complete the sentence below to describe how to factorise it.

EDDIE SAYS
We looked at this in the Introduction. When you are simplifying fractions, you need to factorise and cancel out any common factors. The top row becomes: 3(2x + 5) The bottom row becomes: 4(2x + 5) As (2x + 5) appears in both the top and bottom rows, they cancel each other out. So the expression can be simplified to 3/4.
• Question 7

Simplify the expression:

 3x + 12 2x2 + 8x

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

3/2x
EDDIE SAYS
Remember the golden rule: If you want to simplify an algebraic fraction, you have to factorise first. If you do this correctly you should have:
 3(x + 4) 2x(x + 4)
Can you see the factor to cancel from this? (x + 4) appears in both the top and bottom rows, so they can be crossed out, leaving us with an answer of 3/2x
• Question 8

Simplify the expression:

 6a2 + 15a 2ab + 5b

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

3a/b
EDDIE SAYS
If you want to simplify an algebraic fraction, you have to factorise first. If you do this correctly you should have:
 3a(2a + 5) b(2a + 5)
Can you see the factor to cancel from this? (2a + 5) appears in both the top and bottom rows, so they cancel each other out. This leaves us with an answer of 3a/b
• Question 9

Simplify the expression:

 6x2 + 2x 15x2 + 5x

Give your answer in the form a/b, without any spaces and using the / key to create your fraction bar.

2/5
EDDIE SAYS
If you factorise this expression, you will reach:
 2x(3x + 1) 5x(3x + 1)
Can you see the factors to cancel from this? Yep, there's two of them! Firstly, we can cancel out (3x + 1) to leave us with 2x/5x. We can then cancel this further by removing the x's, leaving us with 2/5.
• Question 10

Which of the expressions in the list is a fully simplified version of the expression below?

3x3 + 6x2

3x2(x + 2)
EDDIE SAYS
When you're looking to fully factorise an expression, you need to find the highest common factor. Both of these expressions can be divided by 3, x and x2. This makes the highest common factor 3x2. The second option is correct as it shows 3x2 outside the bracket, and a correctly factorised sum inside the bracket. Great work!
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