Problem-solving questions at GCSE maths — the multi-step, contextualised, less-structured problems worth 4–6 marks each — are where the difference between a Grade 4 and a Grade 7 is most often decided. Unlike routine practice questions, problem-solving questions require you to identify which mathematics to apply before applying it, and this selection skill is teachable with deliberate practice.
What makes a GCSE maths question a "problem-solving" question?
Problem-solving questions at GCSE share several features that distinguish them from straightforward calculation questions:
| Feature | Standard question | Problem-solving question |
|---|---|---|
| Structure | Tells you the method to use | Requires you to decide the method |
| Context | Abstract numbers only | Set in a real-world context |
| Steps | Typically one or two steps | Typically three to five steps |
| Information | Exactly what you need is given | May include redundant or missing information |
| Answer form | Usually a number | Sometimes a decision, comparison, or justification |
An example of a standard question: "Solve 3x + 7 = 22."
An example of a problem-solving question: "Tia wants to tile a rectangular floor that is 3.6 m by 2.4 m using square tiles with side length 30 cm. Tiles are sold in packs of 10. How many packs does she need to buy?"
The second question requires you to: convert units, calculate areas, divide to find number of tiles, round up (partial packs cannot be bought), and divide by 10 — a five-step process that a student must sequence without being told how.
How do I approach an unfamiliar problem-solving question?
Use this four-step reading and planning approach before writing any working:
- Read the question twice. On the first read, understand the context. On the second read, identify the specific mathematical question being asked.
- Extract the numbers and units. Write them on your working space, noting any unit differences (metres vs centimetres, litres vs millilitres, percentages vs decimals).
- Identify the end goal. What exactly does the question ask for? A number of packs? A profit? A percentage increase? Whether a claim is correct? Keep this in view throughout.
- Plan the steps. In rough, jot down the sequence of operations you will need: "find area → convert units → divide → round up → divide by 10."
Students who plan before calculating make far fewer errors than those who start calculating immediately. The plan also means that if your final answer is wrong, you can still pick up method marks for each correct step.
A worked example: a typical GCSE problem-solving question
Question: A car travels from city A to city B, a distance of 270 km, in 2 hours 30 minutes. On the return journey, the car travels at 80 km/h. How much longer does the return journey take than the outward journey? Give your answer in minutes.
Step 1: Convert the outward journey time to hours: 2 h 30 min = 2.5 hours.
Step 2: Calculate the outward average speed: 270 ÷ 2.5 = 108 km/h.
Step 3: Calculate the return journey time: 270 ÷ 80 = 3.375 hours.
Step 4: Find the difference in time: 3.375 − 2.5 = 0.875 hours.
Step 5: Convert to minutes: 0.875 × 60 = 52.5 minutes.
Answer: The return journey takes 52.5 minutes (or 52 minutes 30 seconds) longer.
Check for sense: The return speed is slower (80 km/h vs 108 km/h) so a longer return journey makes sense. The difference of just under an hour for a 270 km journey is plausible.
Always check that your answer makes sense in the context of the question — this catches errors before you move on.
What mathematical topics appear most in problem-solving questions?
Certain topics appear in multi-step problem-solving questions more frequently than others across all three major GCSE specifications:
| Topic | Common problem-solving applications |
|---|---|
| Ratio and proportion | Sharing in a ratio, recipe scaling, best-value comparisons |
| Percentages | Percentage increase/decrease, reverse percentages, compound interest |
| Area and volume | Composite shapes, unit conversion, painting/tiling/filling problems |
| Speed, distance, time | Multi-leg journeys, different units, average speed for whole journey |
| Algebra | Setting up and solving equations from a word description |
| Probability | Combined probability, relative frequency, experimental vs theoretical |
| Sequences | Finding the nth term, deciding if a number is in a sequence |
Practice problems from each of these areas specifically. When practising, focus on problems where you do not know the method in advance — the point is to develop the selection skill.
How do I improve my problem-solving skills between now and the exam?
Five practical strategies that develop the selection skill:
- Practise sorting problems before solving them. Look at 10 questions and classify each one: "This needs ratio / This needs percentage / This needs forming an equation." Practise the sorting without solving to build method-recognition skill.
- Study mark schemes alongside your working. When you attempt a multi-step problem and get it wrong, read the mark scheme step by step and identify exactly which step you missed or misapplied.
- Practise "worst-case" problems. Seek out the five- and six-mark problems on past papers specifically — these are the hardest and build the most resilience.
- Write out your reasoning in words. When you practise, write a sentence before each calculation explaining what you are doing and why: "I need to find the total surface area of the cylinder, so I will calculate 2πr² + 2πrh using the given radius and height." This habit slows impulsive errors.
- Work in mixed topics. Rather than practising all ratio problems together, mix topics in each session — one ratio, one percentage, one area question. This replicates the exam, where you cannot predict which topic comes next.
Frequently asked questions
What should I do if I am stuck on a multi-step problem in the exam?
Write down what you know. Transfer numbers and units onto the page with their labels. Identify what you are trying to find. Then look at the gap between what you have and what you need — even if you cannot bridge it fully, a partial method (correctly set up but arithmetically incomplete) can earn method marks. Never leave a multi-step problem completely blank: examiners can award marks for correct steps even when the final answer is wrong.
Does it help to memorise problem-solving strategies?
Knowing named strategies (e.g., "draw a diagram", "use a table", "work backwards from the answer", "try a simpler case") is genuinely useful — these are recognised mathematical problem-solving heuristics. For example, "work backwards" is the correct strategy for reverse percentage problems (finding the original amount after a percentage change), and recognising this quickly saves time. Build a short list of strategies and practise applying each one to an example problem.
Why do I understand the topic but still get problem-solving questions wrong?
Understanding a topic in isolation (knowing how to calculate percentage increase) is a different skill from recognising when to apply it in an unfamiliar context. The gap between "I know percentages" and "I can spot when a question needs percentages and apply the right type" is where most problem-solving marks are lost. The solution is mixed, contextualised practice — not more practice of percentage calculations in isolation.
Is there a way to get partial marks on a question I cannot fully complete?
Yes — GCSE maths awards method marks (M marks) for correct mathematical procedures, even when the final answer is wrong. For example, setting up the correct equation (even if you solve it incorrectly) earns an M mark. Writing a correct first step earns marks. Show every step of your working clearly, even if you doubt the answer — partial marks for a correct method are better than zero marks for no working shown.
For a Socratic AI maths tutor that develops your problem-solving reasoning through guided questioning — helping you find the method, not just giving you the answer — visit aitutors.me.