How to Answer GCSE Maths Problem-Solving Questions

Problem-solving questions are the part of GCSE maths papers that separate students who know the content from students who can actually apply it under exam conditions. These are the worded, multi-step and "show that" questions that don't tell you which method to use, and they are disproportionately represented among the questions students describe as "the ones I couldn't even start." The good news is that they respond very well to a repeatable decoding method, because the difficulty is usually in structuring the problem, not in the underlying calculations, which are often no harder than questions elsewhere on the paper.

This guide sets out a four-step method you can apply to almost any worded or multi-step GCSE question, then works through several examples in prose to show the method in action. If you're building this into a wider revision plan, see our emergency one-week revision plan and our calculator skills guide for related exam technique.

Why problem-solving questions feel harder than they are

Standard GCSE questions often name the method directly: "use Pythagoras' theorem," "solve the simultaneous equations," "calculate the mean." Problem-solving questions strip this scaffolding away and instead describe a situation, leaving you to work out which mathematical tools apply. This shift from "here is a method, apply it" to "here is a situation, find a method" is what makes these questions feel disproportionately hard, even though the arithmetic and algebra involved are usually within your existing skill set.

The fix is not to learn more content — it's to learn a consistent process for extracting the mathematical structure from the words before you touch a calculation.

The four-step method

Step 1: Extract the information

Read the question once for general understanding, then read it again slowly, pulling out every numerical fact and constraint and writing it down separately from the prose, in your own shorthand. Don't try to solve anything yet. At this stage you are simply translating words into a list of known values, unknown values, and relationships between them.

For example, if a question describes a rectangular garden with a given perimeter and a length that is a stated multiple of the width, step 1 is simply writing down: perimeter = [value], length = [some expression involving width], and noting that you're looking for the width or length.

Step 2: Identify what is actually being asked

It sounds obvious, but a significant number of marks are lost because students solve for the wrong quantity — finding the width when the question asks for the area, or finding a probability when the question asks for a number of people. Underline or circle the actual question at the end of the worded problem before you begin calculating, and check it again once you have an answer.

Step 3: Choose the mathematical structure that connects what you know to what you want

This is the step that replaces "which method do I use?" with a more concrete question: what type of relationship links your known values to your unknown value? Common structures at GCSE include:

• A linear equation (when one unknown appears with a straightforward relationship to known values)
• A pair of simultaneous equations (when there are two unknowns and two independent pieces of information)
• A ratio or proportion relationship (when quantities scale together)
• A geometric formula (area, volume, Pythagoras, trigonometry) triggered by a description of shape or measurement
• A percentage change or compound interest structure (triggered by words like "increased by," "original price," "per year")

Matching the words in the question to these structures is a skill you build through repeated exposure to past papers, but the key discipline is to pause and name the structure explicitly before writing any working, rather than diving straight into calculation and hoping a method emerges.

Step 4: Solve, then check against the original question

Carry out the calculation using clear, labelled working, then go back to Step 2 and check your final answer actually answers the question asked, in the units or form requested. This catches a surprising number of otherwise-avoidable errors, particularly in multi-part questions where an early answer is a stepping stone rather than the final answer required.

Worked example 1: a multi-step percentage and ratio problem

Consider a question describing a school where the ratio of students who walk to school, cycle, and travel by car is given, along with the total number of students and a percentage increase in the number who cycle after a new bike scheme, then asking how many students now travel by car.

Applying the method: Step 1, extract that you have a ratio (say 3:2:5 for walk:cycle:car), a total number of students, and a percentage increase applying specifically to the cycling group. Step 2, identify that the final question is about the "car" group specifically, not the total or the cycling group, which is a common place a rushed student would stop early. Step 3, recognise the structure: use the ratio and total to find the original numbers in each group, since ratio parts sum to a known total; the percentage increase then only affects the cycling figure and does not directly affect the car figure unless the question states the total is redistributed. Step 4, calculate the original car number from the ratio, check whether the question implies the total stays fixed or whether other groups are affected by the cycling change, and give the final car number with correct units (a number of students, not a percentage or ratio).

The core insight this example illustrates is that a percentage change described in the question sometimes applies to only part of a scenario, and careless students often apply it to the wrong part or to the whole total. Careful Step 1 extraction, listing exactly what changes and what stays fixed, prevents this.

Worked example 2: a "show that" question

"Show that" questions are a distinct category worth addressing separately, because they require a different mindset from "calculate" questions. Rather than finding an unknown value, you are asked to demonstrate that a given statement or value is correct, using a valid method.

Consider a question giving the dimensions of a cuboid in terms of an unknown x, stating its volume, and asking you to show that x satisfies a particular cubic equation.

Applying the method: Step 1, extract the expressions for length, width and height in terms of x, and the stated volume. Step 2, note precisely what you must "show" — the exact equation given in the question, not just any correct equation. Step 3, recognise the structure: volume equals length × width × height, so multiply the three expressions together and set the result equal to the given volume. Step 4, expand and rearrange your resulting expression until it matches the exact form stated in the question, showing every algebraic step, since "show that" questions are marked almost entirely on visible method rather than a final answer, and reaching the required form is itself the goal.

The key lesson from "show that" questions is that you must show full working even if you can see the answer is obviously correct by inspection, because the marks are awarded specifically for the demonstration, not for stating that the given result is true. Skipping steps, even obvious-looking ones, is the most common way marks are lost on this question type.

Worked example 3: a geometry problem embedded in a real-world context

Consider a question describing a ladder leaning against a wall, giving the length of the ladder and the distance from the foot of the ladder to the wall, and asking for the angle the ladder makes with the ground, then a follow-up part asking how much further up the wall the ladder would reach if moved to a new distance from the wall.

Applying the method: Step 1, extract the ladder length (hypotenuse), the base distance (adjacent), and note that the angle is the unknown quantity in the first part. Step 2, note that the first part wants an angle, so the answer should be in degrees, while the second part wants a length, and is really asking for a new height compared with the original height, meaning you need two Pythagoras or trigonometry calculations and a subtraction. Step 3, recognise the structure: a right-angled triangle with two known sides means SOHCAHTOA (for the angle) or Pythagoras (for the missing side); since here you know the hypotenuse and the adjacent side and want the angle, use cosine. Step 4, calculate the angle using inverse cosine, then for the second part calculate the new height using Pythagoras with the new base distance and the same hypotenuse, and subtract the original height to answer the actual question asked, which is the difference in height, not the new height alone.

This example illustrates a very common trap in multi-part problem-solving questions: the final part often asks for a difference, a total, or a comparison built from two separate calculations, rather than either calculation on its own. Re-reading the exact wording of the question at Step 2, before calculating, is what prevents this error.

Building this method into your practice

The four-step method works best when it becomes automatic, which means practising it deliberately rather than only on exam day. When working through past papers, try writing out your Step 1 extraction and Step 2 target explicitly, even for questions you find straightforward, until the habit becomes second nature. Structured resources and topic-based question sets can help you build fluency with the underlying methods (ratio, percentages, trigonometry, algebra) that Step 3 depends on, since the decoding method only works if the mathematical toolkit behind it is solid.

It's also worth timing yourself on problem-solving questions specifically, since they typically take longer per mark than routine questions; if you are consistently running out of time, the revision timetable generator can help you build practice sessions that specifically target speed on this question type rather than treating all revision time as equal.

If problem-solving questions remain a persistent weak point despite solid content knowledge, this is often best addressed with guided practice and feedback, since it's difficult to self-assess whether your extraction and structuring process is efficient; A-Level Maths Tutoring works with GCSE and A-Level students specifically on this kind of exam technique, alongside content revision.

Takeaway

Problem-solving questions are hard because they hide the method inside a worded description, not because the underlying mathematics is unusually difficult. A consistent four-step process — extract the information, identify exactly what's being asked, choose the mathematical structure that connects the two, then solve and check against the original question — turns an intimidating block of text into a standard calculation. Practise this process deliberately on past paper questions, including "show that" questions where full visible working is the entire point, and it will become close to automatic by the time you sit the real exam.