Knowing a topic and knowing how to answer an exam question are different skills.
You might understand percentages in class but get stuck when a question gives you the sale price and asks for the original cost. You might know Pythagoras' theorem but struggle to spot the right-angled triangle inside a word problem.
The most useful GCSE Maths revision connects three things: how to recognise a question, which method to choose and how to check your answer.
This guide covers common GCSE Maths question types with original worked examples, mistakes to avoid and practice questions with explained answers. Use it to identify weak spots, build confidence with individual methods and practise choosing the right approach independently.
Note
These are revision priorities, not predictions or a statistical ranking of question frequency. No topic list can guarantee what will appear on your exam.
Which GCSE Maths topics should you revise first?
Start with skills that support several other topics. Accurate arithmetic makes percentages and ratio easier. Confident algebra helps with equations, graphs and geometry.
| Question type | What to look for | Useful first step |
|---|---|---|
| Fractions | Different denominators; multiplication or division | Identify the operation |
| Percentages | Original value, final value or percentage change | Decide what represents 100% |
| Ratio | A total, one share or a difference between shares | Match the amount to the correct number of parts |
| Algebra | Simplify, expand, factorise or solve | Read the instruction carefully |
| Equations | An unknown linked by an equals sign | Apply the same operation to both sides |
| Sequences | A pattern or an nth-term rule | Calculate the differences |
| Graphs | Gradient, intercept or a rate in context | Read the axis labels and scales |
| Geometry | Angle facts, triangles or measurements | Sketch and label the information |
| Probability | Repeated events or changing totals | Check whether replacement happens |
| Statistics | Values paired with frequencies | Work out what each frequency represents |
Foundation or Higher: where should you focus?
For GCSE Maths using the 9–1 grading scale, Foundation targets grades 1–5 and Higher targets grades 4–9 (see, for example, the AQA GCSE Maths specification). Check your specification if you take a different qualification.
- Foundation students: build accuracy with fractions, percentages, ratio, equations, graphs, geometry, probability and statistics. Include Pythagoras and right-angled trigonometry: both appear in Foundation content.
- Higher students: secure those foundations before moving into more demanding algebra and geometry. Topics such as circle theorems, vector proofs and the sine and cosine rules extend the geometry covered at Foundation.
The examples below are a starting point, rather than a complete syllabus checklist.
15 common GCSE Maths question types
1. Fractions: choose the operation before the method
Fraction questions become easier when you stop treating every calculation the same way:
- Addition and subtraction: use a common denominator.
- Multiplication: multiply the numerators and multiply the denominators.
- Division: multiply by the reciprocal of the second fraction.
Worked example
Subtracting fractions. Work out 5/6 − 3/8. Give your answer in its simplest form.
The lowest common denominator is 24:
5/6 = 20/24 and 3/8 = 9/24
Now subtract the numerators:
20/24 − 9/24 = 11/24
The answer is already in its simplest form.
Check your answer: 5/6 is about 0.83 and 3/8 is 0.375. A difference slightly below 0.5 makes sense.
Common mistake: subtracting the denominators as well as the numerators. You need equal-sized parts before you can add or subtract fractions.
For division, the method changes:
3/5 ÷ 9/10 = 3/5 × 10/9 = 2/3
Turn over only the second fraction, then multiply and simplify.
2. Percentages: identify what represents 100%
Before calculating, decide what the question asks you to find: a new value, an original value or a percentage change.
| Task | Method |
|---|---|
| Increase an amount by 12% | Multiply by 1.12 |
| Decrease an amount by 12% | Multiply by 0.88 |
| Find the original amount after a 12% increase | Divide the final amount by 1.12 |
| Find the original amount after a 12% decrease | Divide the final amount by 0.88 |
| Find a percentage change | Change ÷ original × 100 |
Worked example
Reverse percentage. A jacket costs £68 after a 15% discount. Find its original price.
The original price represents 100%. After a 15% discount, the customer pays 85%. Therefore:
0.85 × original price = 68
Work backwards by dividing:
original price = 68 ÷ 0.85 = £80
Check your answer: 15% of £80 is £12. Subtracting £12 gives £68.
Common mistake: adding 15% of £68. The discount was calculated from the original price, so using the sale price as your starting amount gives the wrong answer.
Worked example
Repeated percentage change. A bicycle worth £350 increases in value by 10%, then decreases by 10%. Find its final value.
Apply the multipliers in order:
350 × 1.10 × 0.90 = £346.50
The changes do not cancel out. The increase is £35, but the subsequent decrease is 10% of £385, which is £38.50.
For repeated growth or depreciation:
final value = initial value × (multiplier)ⁿ
Watch out
A value depreciating by 8% each year uses a multiplier of 0.92, not 0.08.
3. Ratio: work out what the given amount represents
The difficult part of a ratio question is often deciding which number of parts to divide by.
Worked example
A difference between shares. Aisha and Ben share money in the ratio 3 : 5. Ben receives £18 more than Aisha. How much money do they share altogether?
The difference between their shares is 5 − 3 = 2 parts. Those two parts represent £18:
1 part = 18 ÷ 2 = £9
There are eight parts altogether:
total = 8 × 9 = £72
Check your answer: Aisha receives £27 and Ben receives £45. Their shares are in the ratio 3 : 5, and the difference is £18.
Common mistake: dividing £18 by eight. Eight parts represent the total, but the question gives you the difference.
Decision rule
- Given the total? Divide by the sum of the ratio parts.
- Given one person's share? Divide by that person's number of parts.
- Given the difference? Divide by the difference between the parts.
This small decision prevents many ratio errors.
4. Proportion and best buys: compare equal quantities
Scaling recipes and comparing prices both involve proportional reasoning. Find the amount for one unit, or scale both options to the same quantity.
Worked example
Best value. A 750 g box of cereal costs £2.70. A 1.2 kg box costs £4.08. Which offers better value?
First, use matching units: 1.2 kg = 1,200 g. Then compare the cost per 100 g.
For the smaller box: 2.70 ÷ 7.5 = £0.36
For the larger box: 4.08 ÷ 12 = £0.34
The 1.2 kg box offers better value: 34p per 100 g, compared with 36p.
Common mistake: choosing the box with the lower total price. A cheaper packet can still cost more per gram.
Show your comparison and state your conclusion. Two unexplained calculations leave the reader to work out what you mean.
For direct proportion, doubling one quantity doubles the other. For inverse proportion, increasing one quantity reduces the other: for example, more equally productive workers can take less time to complete a fixed job. Identify the relationship before scaling.
5. Algebra: distinguish simplifying, expanding and factorising
These instructions ask for different results:
| Instruction | What it asks for |
|---|---|
| Simplify | Combine like terms or rewrite more compactly |
| Expand | Multiply out brackets |
| Factorise | Write an expression as a product of factors |
| Solve | Find values that make an equation true |
Worked example
A negative outside a bracket. Expand and simplify 4(2x − 3) − 3(x − 5).
Multiply every term inside each bracket:
8x − 12 − 3x + 15
Then collect like terms:
5x + 3
The final constant is positive because −3 × −5 = +15.
Check your answer: substitute x = 2 into the original expression and your simplified expression. Both give 13. This can reveal an error, although checking one value does not prove that two expressions are identical.
Common mistake: writing −3x − 15 when expanding −3(x − 5). The negative multiplier applies to both terms.
Worked example
Factorising with a negative constant. Factorise x² + 2x − 15.
Find two numbers that multiply to −15 and add to +2. Those numbers are +5 and −3:
x² + 2x − 15 = (x + 5)(x − 3)
Check by expanding the brackets.
Watch out
Do not give x = −5 and x = 3 unless the question asks you to solve an equation, such as x² + 2x − 15 = 0.
6. Equations and formulae: show the inverse operations
An equals sign means both sides have the same value. Whatever operation you apply to one side must also apply to the other.
Worked example
An equation with brackets. Solve 3(2x − 4) = 2x + 10.
Expand: 6x − 12 = 2x + 10
Subtract 2x from both sides: 4x − 12 = 10
Add 12: 4x = 22
Divide by four: x = 5.5
Check your answer: 3(2 × 5.5 − 4) = 21 and 2 × 5.5 + 10 = 21. Both sides agree.
Worked example
Rearranging a formula. Make a the subject of v = u + at.
Subtract u: v − u = at
Divide by t: a = (v − u) ÷ t
The whole expression v − u is divided by t. When substituting into a calculator, use brackets: (v − u) ÷ t. This rearrangement assumes t ≠ 0.
7. Sequences: connect the term number to the term value
A term-to-term rule tells you how to get the next term. An nth-term rule lets you calculate any term directly.
Worked example
Finding the nth term. Find the nth term of 8, 13, 18, 23, …, and decide whether 103 is in the sequence.
The difference is +5, so start with 5n. At n = 1, this gives 5. The actual first term is 8, so add three:
nth term = 5n + 3
To test whether 103 is in the sequence, solve:
5n + 3 = 103
5n = 100
n = 20
103 is in the sequence: it is the 20th term.
Common mistake: stopping at the difference and giving 5n. Always check your rule against the first few terms.
If solving produces a value of n that is not a positive integer, the number is not a term of the sequence.
Higher students should also practise quadratic sequences, where the second differences are constant.
8. Straight-line graphs: find the gradient, then the intercept
In y = mx + c, m is the gradient and c is the y-intercept.
The y-intercept is the value of y when x = 0. It is not simply the y-coordinate of any point given in the question.
Worked example
Finding a line's equation. A straight line passes through (−1, 7) and (3, −5). Find its equation.
Calculate the gradient:
m = change in y ÷ change in x = (−5 − 7) ÷ (3 − (−1)) = −12 ÷ 4 = −3
So y = −3x + c. Substitute (−1, 7):
7 = −3(−1) + c
7 = 3 + c
c = 4
Therefore y = −3x + 4
Check your answer: substituting x = 3 gives y = −5. The negative gradient also makes sense: the line falls as x increases.
For graphs in context, explain what the numbers mean:
| Graph | What it tells you |
|---|---|
| Gradient of a distance–time graph | Speed |
| Gradient of a speed–time graph | Acceleration |
| Area under a speed–time graph | Distance travelled |
Read the axis units before calculating.
9. Angles: give the reason as well as the calculation
Useful angle facts include:
- Angles on a straight line total 180°.
- Angles around a point total 360°.
- Angles in a triangle total 180°.
- Vertically opposite angles are equal.
- Corresponding and alternate angles are equal when the relevant lines are parallel.
- Co-interior angles total 180° when the relevant lines are parallel.
Worked example
A regular polygon. Each interior angle of a regular polygon is 156°. How many sides does it have?
An interior angle and its adjacent exterior angle total 180°:
exterior angle = 180 − 156 = 24°
The exterior angles total 360°. Because the polygon is regular, they are equal:
number of sides = 360 ÷ 24 = 15
Common mistake: calculating 360 ÷ 156. This method requires the exterior angle.
Watch out
If a question asks for reasons, write the angle fact. "Z angles" may help you remember the diagram, but "alternate angles are equal because the lines are parallel" explains the mathematics.
10. Pythagoras and trigonometry: label the triangle first
For a right-angled triangle, use this starting rule:
| Information given | What you need | Method |
|---|---|---|
| Two sides | Third side | Pythagoras |
| An acute angle and a side | Another side | Trigonometry |
| Two sides | An acute angle | Inverse trigonometry |
Worked example
Finding a shorter side. A 6.5 m ladder rests against a wall. Its foot is 2.5 m from the wall. Assuming a right angle between the wall and ground, how high does it reach?
The ladder is the hypotenuse because it is opposite the right angle. Let the height be h:
h² + 2.5² = 6.5²
h² = 42.25 − 6.25 = 36
h = √36 = 6 m
Check your answer: the height is shorter than the ladder.
Common mistake: always adding the squares. When finding a shorter side, subtract its known partner's square from the hypotenuse's square.
Worked example
Finding an angle. In a right-angled triangle, the side opposite angle θ is 7 cm and the adjacent side is 10 cm. Find θ to one decimal place.
Opposite and adjacent identify tangent:
tan θ = 7/10
Use inverse tangent:
θ = tan⁻¹(0.7) = 35.0°
Check your calculator
Make sure it is in degree mode. Opposite and adjacent depend on the angle you are using; the hypotenuse is always opposite the right angle.
11. Area, volume and units: identify what is being measured
Perimeter measures a boundary. Area measures a surface. Volume measures space inside a solid.
| Measurement | Example unit |
|---|---|
| Length or perimeter | cm |
| Area | cm² |
| Volume | cm³ |
Worked example
Volume of a prism. A triangular prism has a triangular cross-section with base 8 cm and perpendicular height 5 cm. The prism is 12 cm long. Find its volume.
Calculate the cross-sectional area:
A = ½ × 8 × 5 = 20 cm²
Multiply by the prism's length:
V = 20 × 12 = 240 cm³
Common mistake: multiplying all three dimensions without halving. The cross-section is a triangle, not a rectangle.
Watch out
Be particularly careful with area and volume conversions: 1 m = 100 cm, but 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.
For a composite shape, split it into familiar shapes and add their areas. Alternatively, calculate a larger shape's area and subtract the missing section.
12. HCF and LCM: let the context choose the method
Use the highest common factor for the largest equal grouping that divides quantities exactly. Use the lowest common multiple for the earliest repeat or smallest common quantity.
Worked example
Events happening together. One light flashes every 18 seconds and another every 24 seconds. They flash together now. How long until they next flash together?
You need the LCM because the time must be a multiple of both intervals. Write the prime factorisations:
18 = 2 × 3²
24 = 2³ × 3
Take the highest power of each prime:
LCM = 2³ × 3² = 72 seconds
Check your answer: 72 ÷ 18 = 4 and 72 ÷ 24 = 3. Both lights complete a whole number of intervals.
For the HCF, take only the shared primes at their lowest powers. Here, HCF = 2 × 3 = 6.
Common trap
Choosing HCF just because the question contains two numbers. Decide what the context requires before calculating.
13. Probability: check whether the totals change
Before calculating repeated selections, ask: is the item replaced? Without replacement, the number of available items and the total may change.
Worked example
Two selections without replacement. A bag contains five red counters and three blue counters. Two counters are chosen without replacement. Find the probability that both are blue.
The probability that the first counter is blue is 3/8. After selecting a blue counter, two blue counters remain out of seven:
P(both blue) = 3/8 × 2/7 = 3/28
Common mistake: using 3/8 for the second selection as well. That would describe a situation where the first counter was replaced.
For probability trees:
- Multiply along a path to find the probability of that sequence.
- Add the probabilities of distinct successful paths to find the total probability of an event.
For "one of each colour", include both red–blue and blue–red.
14. Statistics: use the frequencies
A frequency tells you how many times a value occurs. Treating every row as one observation gives the wrong mean.
Worked example
Mean from a frequency table. Find the mean number of pets.
| Number of pets | Frequency |
|---|---|
| 0 | 3 |
| 1 | 5 |
| 2 | 4 |
| 3 | 2 |
The number of people is 3 + 5 + 4 + 2 = 14.
The total number of pets is (0 × 3) + (1 × 5) + (2 × 4) + (3 × 2) = 19.
Therefore mean = 19 ÷ 14 ≈ 1.36 pets
The mean need not be a whole number, even though each individual's pet count is a whole number.
Common mistake: calculating (0 + 1 + 2 + 3) ÷ 4. That ignores how many people have each number of pets.
For grouped data, use class midpoints to estimate the mean. Call your answer an estimate because the exact values within each interval are unknown.
15. Higher extension: simultaneous equations
When two equations describe the same pair of unknowns, your answer must satisfy both.
Worked example
Elimination. Solve 2x + 3y = 19 and 3x − 2y = 9.
Multiply the first equation by two and the second by three:
4x + 6y = 38
9x − 6y = 27
Add the equations to eliminate y:
13x = 65, so x = 5
Substitute into the first original equation:
2(5) + 3y = 19
3y = 9, so y = 3
Therefore x = 5, y = 3
Check your answer: the second equation gives 15 − 6 = 9.
Common mistake: multiplying the terms on the left but forgetting to multiply the number after the equals sign. Scale every term.
How to approach unfamiliar GCSE Maths questions
A longer question may combine familiar skills without naming them. Calculating the cost of flooring, for example, could require area, unit conversion, percentages and rounding up to whole packs.
Use this routine:
- Read the final instruction. Identify exactly what you must find, compare or prove.
- Organise the information. Draw a sketch, label units or make a small table.
- Find a useful intermediate value. You may need an area before a cost, or one ratio part before a total.
- Write the mathematical relationship. Use an equation, formula or proportion rather than an unexplained calculation.
- Check the result in context. Does it fit the units, size and practical constraints?
Worked example
Interpreting the answer. One flooring pack covers 2 m² and you need to cover 9 m².
9 ÷ 2 = 4.5
That does not mean you can buy 4.5 packs. You need five whole packs.
Getting the calculation right is only part of the task. You must also interpret it correctly.
Common GCSE Maths mistakes and how to prevent them
| Mistake | Better habit |
|---|---|
| Treating the final value as 100% in reverse percentages | Label the original 100% |
| Dividing by total ratio parts when given a difference | Write what the given amount represents |
| Dropping a minus sign outside a bracket | Multiply each term separately |
| Rounding during a multi-step calculation | Keep full precision until the final answer |
| Using a diameter as a radius | Label r and d before substituting |
| Dividing percentage change by the new value | Divide by the original value |
| Giving an angle without a requested reason | Write the relevant angle fact |
| Assuming a diagram is accurately drawn | Use stated information unless measurement is requested |
Show enough working for someone to follow your method. Depending on the mark scheme, a correct method may earn marks even if a later calculation is wrong.
Key point
An unexplained answer gives the examiner less evidence to assess. Useful working includes the formula you chose, the values you substituted and the steps linking them to your answer.
Mixed GCSE Maths practice questions
Try these without looking back at the examples. Choosing the method is part of the practice.
- Work out 7/10 − 1/4. Give your answer in its simplest form.
- A coat costs £91 after a 30% reduction. Find its original price.
- Two lengths are in the ratio 4 : 7. The longer is 15 cm longer than the shorter. Find their total length.
- Expand and simplify 2(3x − 4) − 5(x − 2).
- Solve 4(2x + 1) = 3x + 19.
- Find the nth term of 6, 10, 14, 18, … Is 82 in the sequence?
- Find the equation of the line through (0, −2) and (4, 10).
- Each exterior angle of a regular polygon is 18°. How many sides does it have?
- A right-angled triangle has hypotenuse 17 cm and one shorter side 8 cm. Find the other side.
- A bag contains four green and six yellow counters. Two are chosen without replacement. Find the probability that both are green.
AnswersShow answersHide answers
| Question | Answer | Key method |
|---|---|---|
| 1 | 9/20 | 14/20 − 5/20 |
| 2 | £130 | £91 represents 70%: 91 ÷ 0.70 |
| 3 | 55 cm | Three parts represent 15 cm; eleven parts represent 11 × 5 |
| 4 | x + 2 | Expand to 6x − 8 − 5x + 10 |
| 5 | x = 3 | 8x + 4 = 3x + 19, so 5x = 15 |
| 6 | 4n + 2; yes, the 20th term | Solve 4n + 2 = 82 |
| 7 | y = 3x − 2 | Gradient = 12 ÷ 4 = 3; intercept = −2 |
| 8 | 20 sides | 360 ÷ 18 |
| 9 | 15 cm | √(17² − 8²) = √225 |
| 10 | 2/15 | 4/10 × 3/9, then simplify |
How to use this guide in your revision
Reading a worked solution can make a topic feel easier than it is. Check that you can reproduce the method without seeing the steps.
Use a 30-minute revision session like this:
| Time | What to do |
|---|---|
| Five minutes | Attempt a few questions without notes to identify a weakness |
| Ten minutes | Study the relevant method, then answer fresh questions on that skill |
| Ten minutes | Attempt a mixed set where you must choose the methods yourself |
| Five minutes | Mark your answers and record why each mistake happened |
Make your error notes specific. "I'm bad at ratio" is difficult to act on. "I divided by the total parts when the question gave the difference" tells you exactly what to change.
Return to the missed skill in a later session using a different question. Once you can solve it independently, practise it in a timed past paper for your exam board and tier: Starley has AQA, Edexcel and OCR GCSE Maths past papers. For a full weekly routine, see our GCSE Maths revision timetable.
Frequently asked questions
What are the most common GCSE Maths questions?
Useful recurring question types include fractions, percentages, ratio, algebra, equations, sequences, graphs, angles, measurement, probability and statistics.
Their value comes partly from how these skills connect to other problems. This guide does not claim a measured frequency ranking; for the topics that turn up in almost every series, see GCSE Maths topics that come up every year.
Can any topic appear on the non-calculator paper?
Do not assume non-calculator means only arithmetic. AQA's GCSE Maths specification states that content from any part of the specification may be assessed on each paper. Practise exact fractions, algebra and suitable geometry questions without a calculator, and check your own board's assessment guidance.
Is trigonometry only on Higher GCSE Maths?
No. Right-angled trigonometry and Pythagoras are included in Foundation content. More advanced work, such as the sine and cosine rules, is Higher content.
What should I revise to get a grade 4 or 5?
Start with questions you currently lose marks on. Secure arithmetic, fractions, percentages, ratio, basic algebra and equations, then develop geometry, graphs, probability and statistics.
Use whole papers to identify remaining gaps. No single topic list guarantees a grade, and being able to combine skills matters as well as knowing them individually.
Should I revise only predicted topics?
Use prediction lists as extra practice prompts. Revise the full content for your tier and use past papers to practise combining skills.
A topic appearing in an earlier paper does not establish that it will be absent next time.
What should I do if I get stuck in the exam?
Write a relevant first step: a labelled diagram, formula, substitution or equation.
If you cannot make progress, move on and return later. Leave your working legible so your method can be assessed.