GCSE Maths Key Topic Exercises: 15 Practice Questions by Subject Area
15 GCSE maths practice exercises across Number, Algebra, Geometry, Statistics and Probability. Question types, solution approach, and common errors for AQA, Edexcel and OCR.
I have a confession. In nine years of teaching GCSE maths in Sheffield, the question I dread most isn't from a student. It's from a parent: "Which exercises should she actually be doing?"
It sounds simple. It isn't. Most Year 11 students waste revision time on questions they already know how to do, then walk into their AQA or Edexcel paper underprepared for the five or six question types that appear every single year without fail. This article is my attempt to fix that. Fifteen exercises, grouped by topic, with the format each question takes and the approach that earns full marks.
These exercises are drawn from past AQA 8300, Edexcel 1MA1 and OCR J560 papers. The question types below appear across Foundation and Higher tier. I'll flag when something is Higher-only.
What the 15 exercises actually cover
Practising GCSE maths key topic exercises gives you the fastest route to closing mark gaps. The five topic areas below account for roughly 95% of all marks across Papers 1, 2 and 3. Spend your time here and you'll see the difference in a mock paper within a fortnight.
Number exercises (Exercises 1–3)
Exercise 1: Fractions — adding, multiplying, and mixed numbers
Question format: Calculate $\frac{3}{4} + \frac{2}{5}$. Give your answer as a mixed number.
Solution approach: Find the lowest common denominator (20). Convert: $\frac{15}{20} + \frac{8}{20} = \frac{23}{20}$. Convert to a mixed number: $1\frac{3}{20}$.
For multiplication, the method flips: multiply numerators together, multiply denominators together, simplify. The trap on Foundation tier is forgetting to simplify. Examiners give a follow-through mark, but they expect the fully simplified answer for the final mark.
I had a student last year, Maya, who kept getting fractions questions right in class and wrong in timed papers. Turned out she was confusing the addition and multiplication methods under pressure. We spent a single session drilling just that distinction. Her next mock: full marks on every fractions question.
What examiners want: Show every step. Write the common denominator explicitly. Do not skip straight to the answer even if you can do it in your head. Method marks exist for a reason.
Exercise 2: Percentage change — increase, decrease, reverse
Question format: A jacket costs £85 after a 15% reduction. What was the original price?
Solution approach (reverse percentage): The sale price represents 85% of the original. So £85 ÷ 0.85 = £100. Avoid the classic mistake of adding 15% back onto £85 (that gives £97.75, which is wrong).
For percentage increase: multiply by (1 + rate). For decrease: multiply by (1 − rate). For reverse: divide by that multiplier.
On AQA Higher, this question often appears inside a multi-step problem (for example: a property value increases by 12% one year then decreases by 8% the next). Understand each step as a separate multiplier and you won't get tangled.
Common error: Calculating 15% of £85 and adding it back. Write "85% = £85" as your first line. It forces the correct setup.
Exercise 3: Surds — simplifying and rationalising (Higher tier)
Question format: Simplify $\sqrt{72}$.
Solution approach: Find the largest square factor of 72 (which is 36). $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.
For rationalising the denominator: multiply numerator and denominator by the surd. $\frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$.
Surds appear on every Higher paper. Paper 1 (non-calculator) uses them heavily because they let you give exact answers without a decimal approximation.
Algebra exercises (Exercises 4–7)
Exercise 4: Linear equations
Question format: Solve $4(2x - 3) = 5x + 9$.
Solution approach: Expand: $8x - 12 = 5x + 9$. Subtract 5x: $3x - 12 = 9$. Add 12: $3x = 21$. Divide: $x = 7$.
Always substitute back into the original equation to check. It takes 15 seconds and catches sign errors before they cost you a mark.
The OCR J560 paper in particular likes equations with brackets on both sides. Expanding carefully, writing out every line, is the difference between a method mark and nothing.
Exercise 5: Simultaneous equations
Question format: Solve $3x + 2y = 13$ and $x - y = 1$.
Solution approach (substitution): From the second equation, $x = y + 1$. Substitute: $3(y + 1) + 2y = 13 \Rightarrow 5y + 3 = 13 \Rightarrow y = 2$. Then $x = 3$.
You can also use elimination. Multiply the second equation by 2 to get $2x - 2y = 2$, add to the first: $5x = 15$, so $x = 3$.
Examiners accept either method. Use whichever you're faster with under timed conditions. What they do not accept: giving an answer without substituting back to verify.
Exercise 6: Quadratic equations — factorising and the formula
Question format: Solve $x^2 + 5x + 6 = 0$.
Solution approach (factorising): Find two numbers that multiply to 6 and add to 5: they're 2 and 3. Write $(x + 2)(x + 3) = 0$. Solutions: $x = -2$ or $x = -3$.
When factorising fails (usually when the discriminant isn't a perfect square), use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Completing the square is also on the Higher specification (Edexcel 1MA1 in particular). Know it: $x^2 + 5x + 6 = (x + \frac{5}{2})^2 - \frac{1}{4}$.
My opinion: too many Year 11 students treat the quadratic formula as a last resort. For anything where the discriminant looks messy, use it first. It's faster and you're less likely to make an error than with trial-and-error factorising.
Exercise 7: Sequences — nth term and geometric
Question format: A sequence is 3, 7, 11, 15… Find the nth term.
Solution approach: The common difference is 4. The nth term is $4n + c$. When $n = 1$: $4(1) + c = 3$, so $c = -1$. nth term: $4n - 1$.
For geometric sequences (Higher): if the sequence is 2, 6, 18, 54…, the common ratio is 3. The nth term is $2 \times 3^{n-1}$.
Geometric sequences appear on about 40% of Higher papers. They're free marks once you've seen the pattern. Students who haven't practised them blank completely.
Geometry exercises (Exercises 8–11)
Exercise 8: Pythagoras' theorem
Question format: A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
Solution approach: $c^2 = 6^2 + 8^2 = 36 + 64 = 100$. So $c = 10$ cm.
For the converse (proving a triangle is right-angled), calculate all three sides squared and check whether the sum of the two smaller values equals the largest.
The trap: labelling the wrong side as the hypotenuse. The hypotenuse is always opposite the right angle and always the longest side. Write this on your diagram before you start calculating.
Exercise 9: Trigonometry (SOHCAHTOA)
Question format: In a right-angled triangle, the hypotenuse is 13 cm and the opposite side to angle $\theta$ is 5 cm. Find $\theta$.
Solution approach: $\sin\theta = \frac{5}{13}$. So $\theta = \sin^{-1}(0.385) \approx 22.6°$.
SOHCAHTOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Write it on your working every time until it's automatic.
On Higher papers, the Sine Rule ($\frac{a}{\sin A} = \frac{b}{\sin B}$) and Cosine Rule ($a^2 = b^2 + c^2 - 2bc\cos A$) extend this to non-right-angled triangles. These appear on almost every Edexcel Higher paper.
Exercise 10: Circle theorems (Higher tier)
Question format: O is the centre of a circle. Angle AOB = 140°. Find the angle ACB where C is a point on the major arc.
Solution approach: The angle at the centre is twice the angle at the circumference (same arc). Angle ACB = 140° ÷ 2 = 70°.
Eight circle theorems come up on the specification. The four that appear most often on AQA and OCR past papers are: angle at centre = twice angle at circumference; angles in the same segment are equal; opposite angles in a cyclic quadrilateral sum to 180°; angle in a semicircle = 90°.
Write them out. Draw a diagram for each one from memory. If you can do that, circle theorems become four to six guaranteed marks.
Exercise 11: Area and volume
Question format: A cylinder has radius 4 cm and height 9 cm. Calculate its volume. Give your answer in terms of $\pi$.
Solution approach: Volume = $\pi r^2 h = \pi \times 16 \times 9 = 144\pi$ cm³.
Units are the most common error here. If the question gives dimensions in metres, the volume is in m³, not cm³. If it asks for litres, convert: 1 m³ = 1000 litres.
On Foundation tier, formulae for sphere volume ($\frac{4}{3}\pi r^3$) and cone volume ($\frac{1}{3}\pi r^2 h$) are provided on the formula sheet. But you still need to know when to use them.
Statistics exercises (Exercises 12–13)
Exercise 12: Averages — mean from a frequency table
Question format: The table shows the number of books read by 20 students. Calculate the mean.
| Books | Frequency |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 6 |
| 4 | 3 |
Solution approach: Multiply each value by its frequency: $(1×4) + (2×7) + (3×6) + (4×3) = 4 + 14 + 18 + 12 = 48$. Divide by the total frequency: $48 ÷ 20 = 2.4$ books.
The median from a frequency table: add a cumulative frequency column. The median is the $\frac{n+1}{2}$th value. With 20 values, it falls between the 10th and 11th. Both are in the "2 books" group, so the median is 2.
Do not confuse the mean calculation with the median. I see this error in roughly one in three mock papers.
Exercise 13: Scatter graphs and correlation
Question format: A scatter graph shows hours of revision and exam scores for 12 students. Describe the correlation and draw a line of best fit.
Solution approach: Identify the trend: positive (both increase), negative (one increases as the other decreases), or no correlation. A line of best fit passes through the middle of the data points with roughly equal numbers above and below. Use it to read off estimated values.
On AQA papers, you're often asked to interpret a line of best fit: "Predict the score for a student who revised 6 hours." Read up from the x-axis to the line, then across to the y-axis.
Extrapolation (reading beyond the plotted data) is not reliable, and some mark schemes dock a mark if you make a strong claim from it.
Probability exercises (Exercises 14–15)
Exercise 14: Tree diagrams
Question format: A bag contains 3 red and 5 blue counters. A counter is drawn and not replaced. A second counter is then drawn. Find the probability that both counters are the same colour.
Solution approach: Draw the tree.
First draw: Red $\frac{3}{8}$, Blue $\frac{5}{8}$.
Second draw if first was Red: Red $\frac{2}{7}$, Blue $\frac{5}{7}$. Second draw if first was Blue: Red $\frac{3}{7}$, Blue $\frac{4}{7}$.
P(both red) = $\frac{3}{8} \times \frac{2}{7} = \frac{6}{56}$. P(both blue) = $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56}$. P(same colour) = $\frac{6}{56} + \frac{20}{56} = \frac{26}{56} = \frac{13}{28}$.
Without replacement changes the denominator for the second branch. This is the mistake that costs the most marks on probability questions.
Exercise 15: Venn diagrams and conditional probability (Higher tier)
Question format: In a class of 30 students, 18 study French, 12 study Spanish, and 5 study both. One student is chosen at random. Find the probability that they study French, given they study Spanish.
Solution approach: Fill in the Venn diagram. Both = 5. French only = 13. Spanish only = 7. Neither = 5.
Conditional probability: P(French | Spanish) = P(French and Spanish) ÷ P(Spanish) = $\frac{5}{30} \div \frac{12}{30} = \frac{5}{12}$.
The formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$ is not given on the formula sheet. Memorise it.
On OCR J560 Higher papers, Venn diagram questions often include a written description rather than a ready-drawn diagram. You must build it yourself from the information in the problem. Practise that.
How to use these exercises in practice
These fifteen question types cover the territory. But doing them once isn't revision. It's reading. Revision means closing this article, opening a blank page, and attempting each question type from memory. Then checking your working against the approach above. Then trying again three days later.
If you're building a structured timetable, our 12-week GCSE maths revision plan maps exactly when to work on each topic area above. If your exam is closer, the 8-week GCSE maths revision plan covers the same ground at a faster pace.
For the topics where you keep dropping marks (quadratics, circle theorems, conditional probability), a few sessions with targeted feedback make a bigger difference than working through another ten practice sheets alone. EduBoost's GCSE maths tutoring walks you through problems with guided hints rather than just answers, which is how you actually fix a method rather than just see the right number.
One last thing. Every Year 11 student I've taught who improved significantly did the same thing: they stopped reading about revision and started doing it. The exercises above are the start. The work is yours.
Olivia Hartwell has taught GCSE maths in Sheffield for nine years, working across Foundation and Higher tier students on AQA 8300. She joined EduBoost as a content contributor in 2025.
