The Hardest GCSE Maths Questions and How to Solve Them

What Are the Hardest GCSE Maths Question Types?

The hardest GCSE Maths questions typically combine multiple skills, hide the method, and reward clear reasoning. The three question types students most consistently struggle with are: long algebra problems with an intermediate step, quadratic sequence nth term questions, and multi-step ratio and percentage word problems.

Introduction

You can probably recognise the feeling. Most of a paper feels manageable, then you turn to the last page and the questions suddenly look longer, wordier, or strangely drawn. There are more marks on offer, the working space is huge, and it is tempting to skip them altogether.

Those problems are not magic. They are usually testing a familiar skill, but in a way that combines several ideas or asks you to plan your own method. This guide looks at three Higher Tier question types that students consistently find difficult, with a full worked solution for each. For a broader look at how to approach the paper as a whole, see our guide on how to get a Grade 9 in GCSE Maths.

For each type you will see a typical exam-style question, why students find it hard, a clear strategy you can follow, and a worked solution written in the way examiners like to see it.

What Makes a GCSE Maths Question Hard?

The hardest GCSE Maths questions usually share three features:

• They combine ideas - you might have to use ratio and percentages together, or algebra and geometry, rather than practising one skill in isolation
• They hide the method - you are not told to 'solve', 'factorise', or 'expand'. You need to decide which technique to use and when
• They reward clear reasoning - there are marks for correct steps and for clear statements, not just for the final answer

Here is a quick overview of the three question types covered in this guide:

                                                                                                               

Question Type	Why Students Find It Hard	Key Strategy
Long algebra with intermediate step	Three brackets means many chances for small errors; two parts seem unrelated	Expand two brackets first, then use that result directly in the second part
Quadratic sequences	Differences are changing, not constant; students know to find the second difference but not what to do with it	Second difference ÷ 2 gives the n² coefficient; subtract that to reveal a simple linear sequence
Multi-step ratio and percentage word problems	Language packs several steps into one paragraph; easy to lose track of which amount belongs to which person	Draw a bar model first; find each person's share before applying the percentage

Worked Example 1: Long Algebra with an Intermediate Step

Question

Expand and simplify:

(x − 3)(x + 1)(2x − 1)

Hence solve the equation:

2x³ − 5x² − 4x + 3 = 0

Why This Feels Hard

The expression contains three brackets, so there are many chances for small mistakes. The two parts also seem unrelated at first. The key idea is that the first part is there to help you. Once you have expanded the brackets, you are supposed to use that result in the second part rather than starting again from scratch.

Strategy

Treat it as two smaller tasks:

• First, expand carefully by multiplying two brackets at a time
• Then, solve the cubic equation using the factorised form you already have

You do not have to factorise the cubic from scratch. In a GCSE setting, the equation is designed so that it comes from a product of simpler brackets, and you can use that structure directly to find the solutions.

Solution

Step 1: Expand the first two brackets.

(x − 3)(x + 1) = x² + x − 3x − 3 = x² − 2x − 3

Step 2: Multiply the result by the third bracket.

(x² − 2x − 3)(2x − 1)

Multiply each term in turn:

• x² × 2x = 2x³
• x² × (−1) = −x²
• (−2x) × 2x = −4x²
• (−2x) × (−1) = 2x
• (−3) × 2x = −6x
• (−3) × (−1) = 3

Step 3: Collect like terms.

2x³ + (−x² − 4x²) + (2x − 6x) + 3 = 2x³ − 5x² − 4x + 3

This confirms that:

(x − 3)(x + 1)(2x − 1) = 2x³ − 5x² − 4x + 3

Step 4: Solve the equation using the factorised form.

(x − 3)(x + 1)(2x − 1) = 0

A product equals zero if at least one factor equals zero:

• x − 3 = 0 gives x = 3
• x + 1 = 0 gives x = −1
• 2x − 1 = 0 gives x = ½

The solutions are x = 3, x = −1 and x = ½.

The important lesson from this example is not the exact numbers, but the structured approach: expand two brackets at a time, use the expanded line as a checkpoint, and then apply the factorised form directly. If quadratic expressions come up as part of this, our post on how to quickly find the vertex of a quadratic covers that technique in detail.

Worked Example 2: Quadratic Sequence nth Term

Question

A sequence has these terms:

−2, 1, 8, 19, ...

Find an expression for the nth term of the sequence.

Why This Feels Hard

Linear sequences are straightforward once you know the difference is constant. Quadratic sequences are a step up because the differences themselves are changing. Students often know they should find the second difference, but are not sure how that turns into a formula.

Strategy

• Use the second difference to find the coefficient of n²
• Subtract that quadratic part from each term to see what is left
• Turn what is left into a simple linear expression

Solution

Step 1: Find the first and second differences.

                                                                                                                                       

n	1	2	3
Term	−2	1	8
1st difference		+3	+7
2nd difference			+4

The second difference is constant and equal to 4. This tells you the sequence is quadratic, and that the coefficient of n² is half of this number. So the nth term contains 2n².

Step 2: Subtract 2n² from each term to find the linear part.

                                                                                                                                                               

n	1	2	3	4
Term	−2	1	8	19
2n²	2	8	18	32
Term − 2n²	−4	−7	−10	−13

The remaining sequence −4, −7, −10, −13 goes down by 3 each time. A linear sequence with difference −3 has the form −3n + k.

Step 3: Find k using the first term. When n = 1, −3n + k = −4:

−3(1) + k = −4

k = −1

So the linear part is −3n − 1.

Step 4: Combine.

nth term = 2n² − 3n − 1

Check by substituting n = 1, 2, 3, 4 to confirm you get −2, 1, 8, 19.

The key habits here are: always set up a clear differences table, use the second difference to fix the n² coefficient, and reduce the remaining part to a linear sequence. For more on how retrieval practice can help lock in these methods, see our post on strategies to enhance retrieval in the Maths classroom.

Worked Example 3: Multi-Step Ratio and Percentage Word Problem

Question

Anna and Ben share £480 in the ratio 3 : 5. Ben then gives 20% of his money to Carla. Work out the amount of money that Carla receives.

Why This Feels Hard

The language packs several steps into a short paragraph and it is easy to lose track of which amount belongs to which person. Students sometimes try to jump straight to equations without organising the information first.

Strategy

Translate the story into a bar model or a simple table before doing any arithmetic:

• Use the ratio to find Anna and Ben's starting amounts
• Find 20% of Ben's amount
• That amount becomes Carla's money

Solution

Step 1: Use the ratio to find each share.

The ratio is 3 : 5, so there are 3 + 5 = 8 equal parts in total.

£480 ÷ 8 = £60 per part

Anna: 3 × £60 = £180

Ben: 5 × £60 = £300

Step 2: Find 20% of Ben's share.

20% means one fifth:

£300 ÷ 5 = £60

Carla receives £60.

On a whiteboard, this is best drawn as two bars side by side: Anna's bar split into three equal blocks of £60, and Ben's bar split into five. Shading one of Ben's blocks and labelling it '20% to Carla' makes the structure obvious before any arithmetic.

This kind of visual approach to ratio and percentage problems is also directly relevant beyond school - see our post on GCSE Maths you will use every day as an adult for more real-world examples.

How to Use These Examples

You make the most progress when you do not just read the solutions, but attempt each question type yourself first:

• Pick one question type at a time and attempt similar problems without looking at notes
• Afterwards, compare your work line by line with a fully written solution and identify where the method marks and reasoning marks are coming from
• Practise drawing the same diagrams and tables: difference tables for sequences, bar models for ratios

For revision techniques that help this kind of method practice actually stick, see the science of effective revision. And for a structured plan that gives each topic enough time, our 10 GCSE revision tips covers how to approach the full paper systematically.

If you often reach the last page of a paper and freeze, remember that those questions are built from skills you already have. What you are really practising is how to combine them and explain your thinking clearly.

If you are finding that difficult to do consistently on your own, a GCSE Maths tutor can slow each step down, build diagrams together on an online whiteboard, and turn each question type into a familiar pattern.

Once you are thinking about what comes after GCSEs, it is also worth reading navigating the transition from GCSE to A-level Mathematics and should I take Further Maths to understand where these skills lead.