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Simplifying expressions Collecting like terms Expanding brackets Factorising Factorising quadratics Difference of two squares Adding and subtracting fractions Simplifying fractionsThis topic is relevant for:
Here we will learn about adding and subtracting algebraic fractions, including algebraic fractions with single and binomial denominators. We will also look at some problems involving quadratics and the difference of two squares.
There are also adding and subtracting algebraic fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.
Adding and subtracting algebraic fractions is the skill of adding and subtracting two or more fractions that contain algebraic terms.
For example, or
In order to add or subtract fractions we must ensure that the fractions have a common denominator.
For example,
Finding the LCM of the algebraic denominators and using a Venn diagram.
LCM
So the first denominator, , needs multiplying by in order to make .
The second denominator, , needs multiplying by in order to make .
Remember to multiply the numerator and denominator of a fraction by the same thing in order to write an equivalent fraction.
Step-by-step guide: Adding and subtracting fractions
Step-by-step guide: Algebraic terms
If the fractions already have a common denominator then the numerators can be easily added/subtracted and the final answer will also have the same denominator. For algebraic fractions you may need to use additional skills such as simplifying expressions by collecting like terms.
For example,
If the fractions have different denominators, we then need to find the lowest common multiple (LCM) of those denominators. Then we can use equivalent fractions to write all the fractions in the question with a common denominator.
Finding the LCM of numerical denominators will be a familiar skill. Additional skills such as expanding brackets and collecting like terms may be required for the algebraic numerators.
For example,
The LCM of and is
Finding the lowest common multiple of algebraic denominators can involve many different skills including identifying unique algebraic factors and factorising algebraic expressions.
Here are some examples of finding common algebraic denominators.
Step-by-step guide: Factorising
Step-by-step guide: Factorising quadratics
Step-by-step guide: Difference of two squares
In order to add and subtract algebraic fractions:
Get your free adding and subtracting algebraic fractions worksheet of 20+ algebraic fractions questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEGet your free adding and subtracting algebraic fractions worksheet of 20+ algebraic fractions questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEWrite as a single fraction in the simplest form,
Sometimes the lowest common multiple can be found by simply multiplying the two denominators together. This is the case here.
The lowest common multiple of and is which we can write as
2For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
3For each fraction, simplify the numerator expression and simplify the denominator expression.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
4Add/subtract the numerators together and write over the common denominator.
The numerator is
This expression cannot be simplified because and are not like terms.
The final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
Sometimes the lowest common multiple can be found by simply multiplying the two denominators together. This is the case here.
The lowest common multiple of and is which we can write as
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator.
The numerator is
This expression cannot be simplified because and are not like terms.
Then final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
The lowest common multiple of and is
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
As we need to multiply the numerator and the denominator of the first fraction by
As we need to multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator.
The numerator is These are like terms so we can simplify this expression.
The final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
The lowest common multiple of and is
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator.
The numerator is .
These are like terms so we can simplify this expression.
The final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
The lowest common multiple of and is
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
We have to think carefully at this point because the numerators have more than one term. We must remember to multiply all terms in the numerator by the identified value.
It can help to write each numerator in a single bracket before multiplying, like this,
By expanding the brackets, the numerator of the first fraction becomes
By expanding the brackets, the numerator of the second fraction becomes
This gives us the updated calculation
Add/subtract the numerators together and write over the common denominator. Simplify the numerator expression by collecting like terms.
Imagine that each numerator is within a single set of brackets.
When we add the numerators together (because the denominators are now the same), we get
Expanding each bracket gives us the expression which can be simplified to by collecting like terms.
This step is important when subtracting fractions (see example 6).
The final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
The lowest common multiple of and is
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
We have to think carefully at this point because the numerators have more than one term. We must remember to multiply all terms in the numerator by the identified value.
Writing each numerator in a single bracket before multiplying makes our calculation look like this.
By expanding the brackets, the numerator of the first fraction becomes
By expanding the brackets, the numerator of the second fraction becomes
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator. Simplify the numerator expression by collecting like terms.
Imagine that each numerator is within a single set of brackets.
When we subtract the numerators, we get
Expanding each bracket gives us the expression
Notice how the last term has changed from to .
This is because we must subtract each term that is in the second bracket. When you subtract negative four it is equivalent to adding four.
Simplifying this expression by collecting like terms gives us
The final answer is
We could also write as our solution if we factorised the numerator.
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
Sometimes the lowest common multiple can be found by simply multiplying the two denominators together. This is the case here.
The lowest common multiple of and is which we can write as or
Top tip: Leave the denominator in its factorised form as this may help you simplify your solution at the end of the question.
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator. Simplify the numerator expression by collecting like terms.
by collecting like terms.
Then final answer is
Note: By expanding the denominator, is also a valid solution.
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
Sometimes the lowest common multiple can be found by simply multiplying the two denominators together. This is the case here.
The lowest common multiple of and is which we can write as or
Remember it is often helpful to leave the denominator in its factorised form.
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator. Simplify the numerator expression by collecting like terms.
Then, by collecting like terms, the numerator is equal to
The numerator can factorise to be
This gives us the fraction
The numerator and the denominator share a common factor of and so we can cancel this factor.
This gives the final answer
By expanding brackets, another form of the solution is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
When working with binomial denominators it is helpful to first see if they factorise. Here factorises to
We can now see that the denominators share a common factor of and the lowest common multiple is
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
To write the fractions with a common denominator, we only need to multiply the numerator and the denominator of the first fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
The numerator of the first fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator. Simplify the numerator expression by collecting like terms.
The final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
Sometimes the lowest common multiple can be found by simply multiplying the two denominators together. This is the case here.
Remember you can leave the denominator in its factorised form.
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator.
and so by collecting like terms, the numerator is equal to which we can factorise to be
The numerator can factorise to be
The final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
Sometimes the lowest common multiple can be found by simply multiplying the two denominators together. This is the case here.
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
Multiply the numerator and the denominator of the first fraction by
Multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
The numerator of the first fraction becomes
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/subtract the numerators together and write over the common denominator. Simplify the numerator expression by collecting like terms.
then by collecting like terms, the numerator is equal to
The numerator can factorise to be
The final answer is
Write as a single fraction in the simplest form,
Determine the lowest common multiple of the denominators.
The denominator of the first fraction is in the difference of two squares form. Factoring this expression gives us a pair of double brackets.
Now we can see that the denominators share the common binomial factor of
Therefore the lowest common multiple of and is
For each fraction, multiply the numerator and the denominator by the same value to obtain the common denominator.
To write the fractions with a common denominator we only need to multiply the numerator and the denominator of the second fraction by
For each fraction, simplify the numerator expression and simplify the denominator expression.
Note: This will use skills such as expanding brackets, multiplying terms and collecting like terms. However, do not simplify the fraction by cancelling or dividing.
The numerator of the second fraction becomes
The common denominator is
This gives us the equivalent calculation
Add/Subtract the numerators together and write over the common denominator. Simplify the numerator expression by collecting like terms.
. This gives us the fraction
There is a chance that this fraction will simplify if we can factorise the quadratic numerator into a pair of double brackets and show that the numerator and denominator share a common binomial factor.
does not factorise.
The final answer is therefore
1. Write as a single fraction in the simplest form,
2. Write as a single fraction in the simplest form,
3. Write as a single fraction in the simplest form,
4. Write as a single fraction in the simplest form,
5. Write as a single fraction in the simplest form,
6. Write as a single fraction in the simplest form,
7. Write as a single fraction in the simplest form,
8. Write as a single fraction in the simplest form,
9. Write as a single fraction in the simplest form,
10. Write as a single fraction in the simplest form,
11. Write as a single fraction in the simplest form,
12. Write as a single fraction in the simplest form,
1. Write as a single fraction,
(2 marks)
(1)
(1)
Alternative method 1
(1)
or
(1)
Alternative method 2
(1)
or
(1)
Alternative method 3
(1)
or
(1)
2. Lucia is answering the question,
(a) Describe the mistake that Lucia has made in her working out.
(b) Determine the correct solution to
(4 marks)
(a) She has added to the numerator and denominator of the second fraction in an attempt to make a common denominator of
However is not equivalent to
(1)
(b)
(1)
(1)
(1)
3. Write as a single fraction of the form where and are integers.
(4 marks)
(1)
(1)
(1)
or
(1)
You have now learned how to:
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