1. “Teach A Level Maths” Vol. 2: A2 Core Modules
35: Algebraic Fractions
© Christine Crisp

2. We need to be able to
add and subtract fractions
multiply and divide fractions
Before we look at algebraic fractions we’ll have a reminder of arithmetic fractions.

3. 21 is the lowest common denominator.
Adding and subtracting fractions.
e.g. 1.
We need the denominators to be the same.
The smallest number both denominators divide into is 21.
21 is the lowest common denominator.
So,
The 1st denominator has been multiplied by 7, so we multiply the numerator by 7.
The 2nd denominator has been multiplied by 3, so we multiply the numerator by 3.

4. The 2nd denominator is unchanged.
e.g. 2.
Since 3 is a factor of 9, the lowest common denominator is 9 not 27.
So,
The 2nd denominator is unchanged.
The 1st denominator has been multiplied by 3, so we multiply the numerator by 3.

5. e.g. 3.
3 is now a factor of both denominators.
We’ll write the factors out to see what we’ve got.
So,
The lowest common denominator is the smallest number that both denominators divide into.
So, we need
So,
The 1st denominator has been multiplied by 5, so we multiply the numerator by 5.
The 2nd denominator has been multiplied by 7 so we multiply the numerator by 7.

6. We use the same ideas when adding and subtracting algebraic fractions.
I’ll put a similar arithmetic fraction beside each example.

7. Arith.
e.g. 1.
The denominators don’t share any factors
The 1st denominator has been multiplied by (x + 2), so we multiply the numerator by (x + 2).
The 2nd denominator has been multiplied by (x - 1), so we multiply the numerator by (x - 1).
The difference here is that we don’t multiply the factors together.

8. The 2nd denominator is unchanged.
e.g. 2.
Arith.
The 1st denominator is a factor of the 2nd
The 1st denominator has been multiplied by (x - 1), so we multiply the numerator by (x - 1).
The 2nd denominator is unchanged.
We factorise the numerator if possible

9. e.g. 3.
Arith.
We have to find the factors
The denominators share a factor

10. e.g. 3.
Arith.
We have to find the factors
The denominators share a factor
The 2nd denominator has been multiplied by (x - 3), so we multiply the numerator by (x - 3).
The 1st denominator has been multiplied by (x - 2), so we multiply the numerator by (x - 2).

11. And one more . . .
e.g. 4. 1

12. SUMMARY
To simplify a sum or difference of algebraic fractions,
factorise any quadratic terms,
find the lowest common denominator, taking care not to repeat factors ( unless they are repeated in one of the fractions ),
multiply out the numerator and collect like terms,
factorise the numerator.

13. Exercise
Express each of the following as a single fraction in its simplest form. 1. 2. 3. 4.

14. Solutions: 1.

15. Solutions: 2. 1

16. Solutions: 3.

17. Solutions: 4.

18. Multiplying and dividing fractions is easier than adding or subtracting them.
Method:
For division, turn the 2nd fraction upside down and multiply.
Factorise any parts that will factorise.
Cancel factors.

19. e.g Simplify
Solution:

20. e.g Simplify
Solution:

21. Exercise
Simplify the following: 1. 2.

22. 1.
Solutions:

23. 2.

24. I’ve put the x first because it makes a later stage easier.
Solving equations containing algebraic fractions
e.g. 1 Solve the following to find the value of x
Solution:
We just multiply the whole equation by the lowest common denominator of BOTH sides of the equation.
So, multiply by :
I’ve put the x first because it makes a later stage easier.
The denominators have all cancelled so we just have a quadratic equation to solve.

25. So,

26. Exercise
Solve the following equations to find the value of x: 1. 2. 3.

27. 1.
Solution:
Multiply by :
Answer:

28. 2.
Solution:
Multiply by :
Answer:

29. 3.
Solution:
Multiply by :
Answer:

31. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

32. To simplify a sum or difference of algebraic fractions,
SUMMARY
factorise any quadratic terms,
find the lowest common denominator, taking care not to repeat factors ( unless they are repeated in one of the fractions ),
multiply out the numerator and collect like terms,
factorise the numerator.

33. e.g. 1.
The denominators don’t share any factors
The difference here is that we don’t multiply the factors together.
Arith.

34. e.g. 2.
The 1st denominator is a factor of the 2nd
We factorise the numerator if possible
Arith.

35. The denominators share a factor
e.g. 3.
We have to find the factors
Arith.

36. e.g. 4.

37. Multiplying and dividing fractions is easier than adding or subtracting them.
Method:
Factorise any parts that will factorise.
Cancel factors.
For division, turn the 2nd fraction upside down and multiply.

38. Solution:
e.g Simplify

39. e.g Simplify
Solution: