Overview
What you’ll learn
- Simplify algebraic fractions by cancelling common factors, including factorised quadratics.
- Add and subtract algebraic fractions using a common denominator.
- Multiply and divide algebraic fractions, inverting before dividing.
- Solve equations containing algebraic fractions by clearing denominators.
- Change the subject of a formula using inverse operations.
- Evaluate a formula by substitution and rearrange formulas that need two inverse steps.
In 30 seconds
- 1Cancel factors, never terms: (6x²)/(2x) = 3x works because x and 2 divide both parts; x + x over 2 does not become x.
- 2Adding needs a common denominator: x/3 + x/4 = 4x/12 + 3x/12 = 7x/12.
- 3To divide, flip the second fraction and multiply: (x/5) ÷ 2 = x/10.
- 4Equations with fractions are cleared by multiplying every term by the common denominator.
- 5Changing the subject = undoing in reverse order: peel the letter away from the subject side one operation at a time.
- 6Always check fractional equations back in the original equation — and never let a denominator be zero.
Notes · 1 of 6
Simplifying algebraic fractions
Factor first, then cancel factors — terms are not cancelable.
An algebraic fraction has letters in its numerator or denominator. Simplifying works exactly like arithmetic: divide top and bottom by the same factor. Since 6x² = 6·x·x and 2x = 2·x, the shared 2x cancels: (6x²)/(2x) = 3x.
The cancel-safety rule: only whole factors may be cancelled. In (x + 2)/5 the numerator is a single sum, not a product — nothing cancels. But (5x + 10)/5 = 5(x + 2)/5 = x + 2, because factorising first exposed a real factor of 5.
| Expression | Verdict | Reason |
|---|---|---|
| (9a²)/(3a) | cancels to 3a | 3 and a are factors of both |
| (x + 2)/5 | stays | the numerator is a sum, not a product |
| (5x + 10)/5 | cancels to x + 2 | factorise first: 5(x + 2)/5 |
| (x² − 9)/(x + 3) | cancels to x − 3 | x² − 9 = (x + 3)(x − 3) |
Notes · 2 of 6
Adding and subtracting
One denominator to rule them all: convert, then combine the numerators.
- 11. Find a common denominator
Use the LCM of the denominators, e.g. 12 for 3 and 4.
- 22. Convert each fraction
Multiply top and bottom by whatever the denominator is missing.
- 33. Combine the numerators
Add or subtract over the single denominator, keeping brackets around minus signs.
- 44. Simplify
Collect like terms and cancel any common factor that appears.
| Sum | Common denominator | Result |
|---|---|---|
| x/3 + x/4 | 12 | 4x/12 + 3x/12 = 7x/12 |
| x/2 − x/5 | 10 | 5x/10 − 2x/10 = 3x/10 |
| 2x/3 + x/4 | 12 | 8x/12 + 3x/12 = 11x/12 |
Notes · 3 of 6
Multiplying and dividing
Straight across for times; flip the second fraction for divide.
Multiplication runs straight across: (x/5) × (10/x) = 10x/(5x) = 2, after cancelling the x and the 5. Cancelling before multiplying keeps numbers small — cross out any factor shared by a numerator and a denominator.
Division flips the second fraction: (4a/7) ÷ 2 becomes (4a/7) × (1/2) = 2a/7. A whole number is itself over 1, which is what makes the flip visible. Forgetting to flip — or flipping the wrong fraction — is the most common slip in this topic.
Notes · 4 of 6
Equations with fractions
Multiply every term by the common denominator and the fractions vanish.
To solve x/3 = 6, multiply both sides by 3: x = 18. The same idea clears harder equations: in (x + 1)/4 = 3, multiplying by 4 leaves x + 1 = 12, so x = 11. Multiply every term — including any plain numbers — or the balance breaks.
| Step | Statement |
|---|---|
| Multiply both sides by x | 2 = 8x |
| Solve the linear result | x = 2/8 = 1/4 |
| Check in the original | 2 ÷ (1/4) = 8 ✓ |
Notes · 5 of 6
Changing the subject of a formula
Treat the formula like an equation and peel operations away in reverse.
A formula’s subject is the letter standing alone. To make another letter the subject, undo the operations binding it, in reverse order — the same moves as solving an equation, performed symbol by symbol.
| Formula | Target | Undo sequence | Result |
|---|---|---|---|
| v = u + at | make t the subject | subtract u, then divide by a | t = (v − u)/a |
| C = 2 pi r | make r the subject | divide by 2 pi | r = C/(2 pi) |
| A = bh/2 | make b the subject | multiply by 2, then divide by h | b = 2A/h |
Notes · 6 of 6
Formulas in use
Substitute first or rearrange first — choose the cheaper route.
Two questions exist about every formula: evaluate it (substitute numbers and compute) and rearrange it (change the subject). When you need many values, rearrange first and substitute into the tidy result; for a single value, direct substitution is quicker.
| Task | Best route | Why |
|---|---|---|
| One value of d in C = 40 + 15d when C = 130 | substitute: 130 = 40 + 15d | one tidy linear solve |
| d for many values of C | rearrange: d = (C − 40)/15 | each later value is a one-step swap |
| r from A = pi r² | rearrange to r = √(A/pi), then substitute | the square root comes last |
Revise
Key terms
The words examiners expect you to use precisely.
- Algebraic fraction
- A fraction whose numerator or denominator contains letters.
- Common factor
- A factor shared by every term, cancelable from the top and bottom of a fraction.
- Common denominator
- A denominator shared by all fractions in a sum, usually the LCM of the originals.
- LCM
- Lowest common multiple: the smallest number both denominators divide.
- Reciprocal
- The flipped fraction; dividing by a number equals multiplying by its reciprocal.
- Clearing denominators
- Multiplying every term of an equation by the common denominator to remove fractions.
Show 6 more key termsShow fewer key terms
- Subject of a formula
- The single letter left alone on one side of a formula.
- Inverse operation
- The operation that undoes another: add/subtract, multiply/divide, square/square root.
- Substitution
- Replacing letters by their numerical values to evaluate a formula.
- Factorise
- Write an expression as a product, exposing factors that can cancel.
- Extraneous value
- A value that appears when solving but breaks the original equation, such as a zero denominator.
- Bracket discipline
- Keeping numerators in brackets when subtracting fractions or substituting negatives.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Cancelling terms instead of factors, e.g. (x + 2)/2 = x + 1.
Only multiplied factors can be cancelled; a sum is not a product.
Instead: (x + 2)/2 stays whole unless the numerator factorises — 2 does not divide x + 2.
Mistake: Adding denominators: x/3 + x/4 = x/7.
Denominators never add; the fractions need a common denominator.
Instead: x/3 + x/4 = 4x/12 + 3x/12 = 7x/12.
Show 8 more mistakesShow fewer mistakes
Mistake: Forgetting to flip when dividing: (x/5) ÷ 2 = (x/5) × 2.
Division by 2 means multiplying by its reciprocal, 1/2.
Instead: (x/5) ÷ 2 = x/10.
Mistake: Multiplying only the fractional terms by the common denominator.
Every term must be multiplied or the equation is unbalanced.
Instead: In x/2 + 1 = 5, multiplying by 2 gives x + 2 = 10 — the 1 and the 5 were multiplied too.
Mistake: Writing t = v − u/a instead of t = (v − u)/a.
Without brackets only u is divided by a.
Instead: The whole (v − u) is divided by a; brackets carry that meaning.
Mistake: Keeping a negative root for a length or radius, e.g. r = −7 from A = pi r².
The algebra offers two roots, but the context rejects one.
Instead: State that the positive root is the only sensible measurement and discard the negative.
Mistake: Expanding 2(2x + 1) as 2x + 1 after clearing denominators.
The multiplier multiplies every term inside the bracket.
Instead: 2(2x + 1) = 4x + 2; expand before collecting.
Mistake: Cancelling the x in (x + 5)/x to leave 5.
The top is a sum, so x is not a factor of it — the same trap as cancelling in a sum of numbers.
Instead: (x + 5)/x = x/x + 5/x = 1 + 5/x; split the fraction instead of cancelling.
Mistake: Flipping the wrong fraction when dividing, e.g. (x/3) ÷ (2/5) = (x/3) × (2/5).
The divisor is the fraction that flips: 2/5 becomes 5/2.
Instead: (x/3) ÷ (2/5) = (x/3) × (5/2) = 5x/6.
Mistake: Solving x + 2 = 3y as x = 3y + 2.
Moving the 2 across undoes it by subtraction, so the sign changes.
Instead: x = 3y − 2. Substitute a number to check: y = 1 gives x = 1, and indeed 1 + 2 = 3.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Simplify algebraic fractions
Simplify (6x²)/(2x).
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 5Foundation
- 6Standard
- 4Challenging
15 questions worth 57 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 4 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 4.
Mark scheme
PROChapter 4 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 2Coming soonWeighted Assessment 2 (WA2)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 4 about?
Chapter 4 covers algebraic fractions and formulae: simplifying fractions by cancelling factors, adding and subtracting with common denominators, multiplying and dividing, solving equations with fractions, and changing the subject of a formula.
Why can I cancel x in (6x²)/(2x) but not in (x + 2)/2?
Cancelling divides a factor from the whole top and bottom. In (6x²)/(2x) the top is 6·x·x, so 2x is a genuine factor. In (x + 2)/2 the top is a sum — 2 is not a factor of it — so nothing cancels unless you factorise first.
How do I solve an equation like (2x + 1)/3 = (x − 1)/2?
Multiply every term by the common denominator (here 6): 2(2x + 1) = 3(x − 1). Expand the brackets, collect terms, and check the answer in the original fractional equation.
How do I change the subject of a formula?
Treat it like an equation: undo the operations attached to your target letter in reverse order, doing the same thing to both sides. For v = u + at making t the subject: subtract u, then divide by a, giving t = (v − u)/a.
Up next
Chapter 5: Direct and Inverse Proportion