Valid cancellation
\(\frac{6x^2y}{9xy^2}\)factorise / cancel factors\(\frac{2x}{3y}\)
The 3, \(x\) and \(y\) are factors of both numerator and denominator.
Chapter 21
Algebraic fractions
Learn to simplify algebraic fractions by factorising and cancelling common factors, then combine fractions accurately using common denominators.
By the end of this chapter, you should be able to
- identify common numerical and algebraic factors in fractions;
- factorise numerators and denominators before cancelling;
- simplify products of algebraic fractions;
- recognise why individual terms cannot be cancelled across addition or subtraction;
- find a common denominator for algebraic fractions;
- expand and simplify the numerator after combining fractions.
1
Subtopic 1
Simplifying algebraic fractions
Exam-ready method
Always factorise complete numerators and denominators. Record excluded values from the original denominator before cancelling. Cancel factors, never separate terms.
An algebraic fraction is simplified in the same way as a numerical fraction: cancel common factors from the numerator and denominator. The factors may be numbers, letters or whole brackets.
Invalid cancellation
\(\frac{x+3}{x}\)do not cancel the \(x\)not \(1+3\)
The numerator is a sum. The \(x\) is a term, not a factor of the whole numerator.
Reliable method
1 Factorise numerator and denominator fully.2 Rewrite products clearly if needed.3 Cancel factors that appear in both.4 Multiply what remains.
Worked example · Easy
Cancel numerical and letter factors
Simplify \(\frac{12a^2b}{18ab^2}\).
Cancel the common factor \(6ab\):
\[\frac{12a^2b}{18ab^2}=\frac{2a}{3b}\]Answer: \(\boxed{\frac{2a}{3b}}\).
Worked example · Medium
Simplify a product of algebraic fractions
Simplify \(\frac{8a^2b}{3c^2}\times\frac{9c^2}{4ab^2}\).
\[\frac{8a^2b\times9c^2}{3c^2\times4ab^2}\]
Cancel \(c^2\), one \(a\), one \(b\), and simplify the numbers:
\[\frac{72a}{12b}=\boxed{\frac{6a}{b}}\]Before factorising\(\frac{x^2+4x}{x^2+7x+12}\)Nothing obvious cancels.
factorise both parts →
After factorising\(\frac{x(x+4)}{(x+3)(x+4)}\)Now \((x+4)\) is a common factor.
\(\boxed{\frac{x}{x+3}}\)
Worked example · Medium
Factorise before cancelling
Simplify \(\frac{x^2+4x}{x^2+7x+12}\).
Factorise both parts:
\[x^2+4x=x(x+4)\]\[x^2+7x+12=(x+3)(x+4)\]Cancel the common factor \((x+4)\):
\[\frac{x(x+4)}{(x+3)(x+4)}=\boxed{\frac{x}{x+3}}\]Worked example · Medium
A difference of squares
Simplify \(\frac{x^2-25}{x^2+2x-15}\).
Factorise:
\[x^2-25=(x-5)(x+5)\]\[x^2+2x-15=(x+5)(x-3)\]Cancel \((x+5)\):
\[\boxed{\frac{x-5}{x-3}}\]Worked example · Exam-style
Multiple factors
Simplify \(\frac{3x^2-12x}{x^2-16}\).
Factorise numerator and denominator:
\[3x^2-12x=3x(x-4)\]\[x^2-16=(x-4)(x+4)\]Cancel \((x-4)\):
\[\boxed{\frac{3x}{x+4}}\]Common error
In \(\frac{x+6}{x}\), the \(x\) cannot be cancelled because the numerator is not a product containing \(x\) as a factor. But in \(\frac{x(x+6)}{x}\), the \(x\) is a factor and can be cancelled.
New worked example · Exam-style
Factorise, simplify and state restrictions
Simplify \(\frac{x^2-9}{x^2-2x-3}\) and state which values are not allowed.
Worked method — show your mathematical steps and check restrictions or units.
\[x^2-9=(x-3)(x+3)\]
\[x^2-2x-3=(x-3)(x+1)\]
\[\frac{(x-3)(x+3)}{(x-3)(x+1)}=\boxed{\frac{x+3}{x+1}}\]
\[x\ne3,\quad x\ne-1\]
Try it yourself — from confidence to examination level
Attempt each question before opening its step-by-step solution.
1Cancel monomial factors · Build confidence[2 marks]
Simplify \(\frac{18a^3b^2}{24a^2b^4}\).
Show Answer
Step-by-step solution
\[\frac{18}{24}=\frac34,\quad a^{3-2}=a,\quad b^{2-4}=\frac1{b^2}\]
\[\boxed{\frac{3a}{4b^2}}\]
2Factorise fully · Develop skill[3 marks]
Simplify \(\frac{x^2+6x+8}{x^2+5x+4}\).
Show Answer
Step-by-step solution
\[x^2+6x+8=(x+2)(x+4)\]
\[x^2+5x+4=(x+1)(x+4)\]
\[\boxed{\frac{x+2}{x+1}},\quad x\ne-4,-1\]
3Factorise a difference of squares · Exam-style[4 marks]
Simplify \(\frac{2x^2-18}{x^2+4x+3}\).
Show Answer
Step-by-step solution
\[2x^2-18=2(x-3)(x+3)\]
\[x^2+4x+3=(x+1)(x+3)\]
\[\boxed{\frac{2(x-3)}{x+1}},\quad x\ne-1,-3\]
4Multiple cancellation · Exam challenge[5 marks]
Simplify \(\frac{x^2-5x}{x^2-25}\times\frac{x+5}{3x}\).
Show Answer
Step-by-step solution
\[\frac{x(x-5)}{(x-5)(x+5)}\times\frac{x+5}{3x}\]
\[\boxed{\frac13},\quad x\ne0,5,-5\]
2
Subtopic 2
Adding and subtracting algebraic fractions
Exam-ready method
Find the least common denominator first. Preserve brackets around an entire numerator, especially after a subtraction; factorise the final numerator only if useful.
As with numerical fractions, algebraic fractions must have a common denominator before you add or subtract them. Multiply each numerator by exactly the factor used to change its denominator.
\(\frac{x+4}{2}\)×3 top and bottom\(\frac{3(x+4)}{6}\)
−
\(\frac{x-1}{3}\)×2 top and bottom\(\frac{2(x-1)}{6}\)
→ same denominator
Worked example · Easy
Numerical common denominator
Simplify \(\frac{x+4}{2}-\frac{x-1}{3}\).
The common denominator is 6.
\[\frac{3(x+4)-2(x-1)}{6}\]Expand the brackets carefully:
\[\frac{3x+12-2x+2}{6}=\boxed{\frac{x+14}{6}}\]Worked example · Medium
Common denominator containing \(x\)
Simplify \(\frac{5}{x}-\frac{2}{3x}\).
The common denominator is \(3x\).
\[\frac{15}{3x}-\frac{2}{3x}=\boxed{\frac{13}{3x}}\]\(x\)\(x+2\)
combine the factors
common denominator: \(x(x+2)\)Worked example · Medium
Two algebraic denominators
Simplify \(\frac{3}{x+2}+\frac{1}{x}\).
The common denominator is \(x(x+2)\).
\[\frac{3x}{x(x+2)}+\frac{x+2}{x(x+2)}\]\[=\frac{3x+x+2}{x(x+2)}=\boxed{\frac{4x+2}{x(x+2)}}\]Worked example · Medium
Subtract with brackets
Simplify \(\frac{4}{x+1}-\frac{3}{x}\).
Use common denominator \(x(x+1)\):
\[\frac{4x-3(x+1)}{x(x+1)}\]\[=\frac{4x-3x-3}{x(x+1)}=\boxed{\frac{x-3}{x(x+1)}}\]Worked example · Exam-style
Factor denominators first
Simplify \(\frac{2}{x-3}+\frac{1}{x+3}\).
Common denominator:
\[(x-3)(x+3)\]Combine:
\[\frac{2(x+3)+(x-3)}{(x-3)(x+3)}\]\[=\frac{2x+6+x-3}{(x-3)(x+3)}=\boxed{\frac{3x+3}{(x-3)(x+3)}}\]Worked example · Exam-style
Combine, then simplify
Simplify \(\frac{x}{x+1}-\frac{1}{x-1}\).
Common denominator: \((x+1)(x-1)\).
\[\frac{x(x-1)-(x+1)}{(x+1)(x-1)}\]\[=\frac{x^2-x-x-1}{(x+1)(x-1)}\]\[=\boxed{\frac{x^2-2x-1}{(x+1)(x-1)}}\]High-grade habit
When subtracting, put the whole second numerator in brackets before expanding. The minus sign must apply to every term in that numerator.
New worked example · Exam-style
Combine three fractions
Simplify \(\frac{2}{x}+\frac{1}{x+2}-\frac{1}{2x}\).
Worked method — show your mathematical steps and check restrictions or units.
\[\text{LCD}=2x(x+2)\]
\[\frac{4(x+2)+2x-(x+2)}{2x(x+2)}=\frac{5x+6}{2x(x+2)}\]
\[\boxed{\frac{5x+6}{2x(x+2)}},\quad x\ne0,-2\]
Try it yourself — from confidence to examination level
Attempt each question before opening its step-by-step solution.
1Numerical common denominator · Build confidence[2 marks]
Simplify \(\frac{x+1}{4}+\frac{2x-3}{6}\).
Show Answer
Step-by-step solution
\[\text{LCD}=12\]
\[\frac{3(x+1)+2(2x-3)}{12}=\boxed{\frac{7x-3}{12}}\]
2Subtract two rational expressions · Develop skill[3 marks]
Simplify \(\frac{4}{x}-\frac{1}{x+3}\).
Show Answer
Step-by-step solution
\[\frac{4(x+3)-x}{x(x+3)}\]
\[\boxed{\frac{3x+12}{x(x+3)}}\]
3Factorable denominator · Exam-style[4 marks]
Simplify \(\frac{2}{x-2}+\frac{3}{x+2}\).
Show Answer
Step-by-step solution
\[\frac{2(x+2)+3(x-2)}{(x-2)(x+2)}\]
\[\boxed{\frac{5x-2}{x^2-4}},\quad x\ne\pm2\]
4Combine and simplify · Exam challenge[5 marks]
Simplify \(\frac{x}{x-1}-\frac{1}{x^2-x}\).
Show Answer
Step-by-step solution
\[x^2-x=x(x-1)\]
\[\frac{x^2-1}{x(x-1)}=\frac{(x-1)(x+1)}{x(x-1)}\]
\[\boxed{\frac{x+1}{x}},\quad x\ne0,1\]
Chapter 21 practice
Start with direct cancellation, then move to factorising and common-denominator questions. Keep answers hidden until you have attempted each problem.
A. Build confidence
1Cancel factors[2 marks]
Simplify \(\frac{15a^2b}{20ab^2}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
Cancel the common factor \(5ab\):
\[\frac{15a^2b}{20ab^2}=\boxed{\frac{3a}{4b}}\]2Product of fractions[3 marks]
Simplify \(\frac{6a^2}{5b}\times\frac{10b^2}{9a}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[\frac{60a^2b^2}{45ab}=\frac{4ab}{3}\]\(\boxed{\frac{4ab}{3}}\)
3Factorise first[3 marks]
Simplify \(\frac{x^2+5x}{x^2+8x+15}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[\frac{x(x+5)}{(x+3)(x+5)}=\boxed{\frac{x}{x+3}}\]4Difference of squares[3 marks]
Simplify \(\frac{x^2-9}{x^2+x-6}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[x^2-9=(x-3)(x+3)\]\[x^2+x-6=(x+3)(x-2)\]\[\boxed{\frac{x-3}{x-2}}\]B. Common denominators
5Numerical denominator[3 marks]
Simplify \(\frac{x+2}{3}+\frac{x-1}{4}\).
Show Answer
Common denominator = 12.
\[\frac{4(x+2)+3(x-1)}{12}=\frac{4x+8+3x-3}{12}=\boxed{\frac{7x+5}{12}}\]6Subtract carefully[3 marks]
Simplify \(\frac{2x+1}{5}-\frac{x-3}{2}\).
Show Answer
Common denominator = 10.
\[\frac{2(2x+1)-5(x-3)}{10}\]\[=\frac{4x+2-5x+15}{10}=\boxed{\frac{17-x}{10}}\]7Denominator with \(x\)[3 marks]
Simplify \(\frac{7}{x}+\frac{2}{3x}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[\frac{21}{3x}+\frac{2}{3x}=\boxed{\frac{23}{3x}}\]8Two algebraic denominators[4 marks]
Simplify \(\frac{2}{x+1}+\frac{3}{x}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[\frac{2x+3(x+1)}{x(x+1)}=\frac{5x+3}{x(x+1)}\]\(\boxed{\frac{5x+3}{x(x+1)}}\)
C. Develop exam technique
9Cancel a bracket factor[4 marks]
Simplify \(\frac{x^2-2x-8}{x^2-x-12}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[x^2-2x-8=(x-4)(x+2)\]\[x^2-x-12=(x-4)(x+3)\]\[\boxed{\frac{x+2}{x+3}}\]10Common factor first[4 marks]
Simplify \(\frac{4x^2-16x}{2x^2-32}\).
Show Answer
\[4x^2-16x=4x(x-4)\]\[2x^2-32=2(x^2-16)=2(x-4)(x+4)\]\[\frac{4x(x-4)}{2(x-4)(x+4)}=\boxed{\frac{2x}{x+4}}\]
11Subtract algebraic fractions[4 marks]
Simplify \(\frac{5}{x+2}-\frac{2}{x}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[\frac{5x-2(x+2)}{x(x+2)}=\frac{5x-2x-4}{x(x+2)}\]\[\boxed{\frac{3x-4}{x(x+2)}}\]12Bracket denominators[4 marks]
Simplify \(\frac{1}{x-2}+\frac{2}{x+2}\).
Show Answer
\[\frac{x+2+2(x-2)}{(x-2)(x+2)}\]\[=\frac{3x-2}{(x-2)(x+2)}\]
\(\boxed{\frac{3x-2}{(x-2)(x+2)}}\)
D. Exam-style challenge
13Factorise and cancel[5 marks]
Simplify fully \(\frac{2x^2+6x}{x^2+5x+6}\).
Show Answer
Method: Identify factors or the least common denominator, apply the same operation to the complete numerator and denominator as needed, then simplify the expression shown below.
\[2x^2+6x=2x(x+3)\]\[x^2+5x+6=(x+2)(x+3)\]\[\boxed{\frac{2x}{x+2}}\]14Subtraction with expansion[5 marks]
Simplify \(\frac{3}{x-1}-\frac{2}{x+1}\).
Show Answer
\[\frac{3(x+1)-2(x-1)}{(x-1)(x+1)}\]\[=\frac{3x+3-2x+2}{(x-1)(x+1)}\]\[\boxed{\frac{x+5}{(x-1)(x+1)}}\]
15Combine and factorise numerator[6 marks]
Simplify \(\frac{x}{x-2}-\frac{4}{x+2}\).
Show Answer
\[\frac{x(x+2)-4(x-2)}{(x-2)(x+2)}\]\[=\frac{x^2+2x-4x+8}{(x-2)(x+2)}\]\[=\boxed{\frac{x^2-2x+8}{(x-2)(x+2)}}\]
16Mixed simplification[6 marks]
Simplify \(\frac{x^2-4}{x^2+3x+2}\times\frac{x+1}{x-2}\).
Show Answer
Factorise:
\[x^2-4=(x-2)(x+2)\]\[x^2+3x+2=(x+1)(x+2)\]Then:
\[\frac{(x-2)(x+2)}{(x+1)(x+2)}\times\frac{x+1}{x-2}=\boxed{1}\]Chapter 21 high-grade checklist
Cancel factors only
I never cancel individual terms across a plus or minus sign.
Factorise first
I factorise numerator and denominator completely before looking for common factors.
Common denominator
I rewrite every fraction over the same denominator before adding or subtracting.
Brackets matter
When subtracting, I keep the second numerator in brackets until I have distributed the minus sign correctly.
Simplify at the end
After combining fractions, I check whether the final numerator or denominator can be factorised and simplified.
Check by structure
I can explain why a cancellation is valid by identifying the common factor explicitly.