A low mock score does not automatically mean a student is “bad at maths”. More often, it shows a small number of gaps that keep reappearing: a negative-sign error in algebra, uncertainty with fractions, or a formula remembered without knowing when to use it. A worked GCSE Maths diagnostic with explanations helps turn that vague concern into a clear starting point for revision.

The purpose is not to test every GCSE topic in one sitting. It is to identify how a student thinks, where marks are being lost, and which skill should be practised first. Complete the questions without a calculator where possible, then read the explanations carefully. The method matters as much as the final answer.

How to use this GCSE Maths diagnostic

Set aside 35 to 45 minutes and work in exam conditions: quiet space, no notes, and full workings shown. If a question feels difficult, make a genuine attempt rather than skipping immediately. A half-completed method can reveal more than a blank answer.

Afterwards, mark each question as secure, nearly there, or revisit. “Nearly there” might mean the approach was right but arithmetic went wrong. “Revisit” means the method itself was unclear. This distinction makes revision more efficient and prevents students spending hours on topics they already understand.

Number: fractions, percentages and ratio

Question 1: Calculate \(\frac{3}{4} + \frac{2}{5}\)

The lowest common denominator for 4 and 5 is 20. Convert both fractions:

\[ \frac{3}{4}=\frac{15}{20}, \qquad \frac{2}{5}=\frac{8}{20} \]

So:

\[ \frac{15}{20}+\frac{8}{20}=\frac{23}{20}=1\frac{3}{20} \]

Answer: \(1\frac{3}{20}\)

A common error is adding the denominators to make \(\frac{5}{9}\). Fractions can only be added directly when their denominators are the same. This is a foundational skill because it also appears in algebraic fractions and probability.

Question 2: A jacket costs £68 after a 15% discount. What was its original price?

After a 15% discount, 85% of the original price remains. Therefore, £68 represents 85%.

\[ \text{Original price}=68 \div 0.85=80 \]

Answer: £80

Students often calculate 15% of £68 and add it back. That does not work because £68 is the discounted amount, not the original amount. The multiplier method is usually the most reliable approach for reverse percentages.

Question 3: Share £126 in the ratio 2:5.

First find the total number of parts:

\[ 2+5=7 \]

Each part is:

\[ 126 \div 7=18 \]

The two shares are:

\[ 2 \times 18=36, \qquad 5 \times 18=90 \]

Answer: £36 and £90

Check that the two amounts add to £126. This quick check catches many ratio mistakes. If a student can set up the ratio but cannot find one part, ratio should become a priority revision topic.

Algebra: forming, solving and simplifying

Question 4: Solve \(3(2x-5)=21\)

Expand the bracket first:

\[ 6x-15=21 \]

Add 15 to both sides:

\[ 6x=36 \]

Divide by 6:

\[ x=6 \]

Answer: \(x=6\)

An alternative is to divide both sides by 3 first, giving \(2x-5=7\). Either route is valid. What matters is doing the same operation to both sides of the equation. A student who writes \(6x-15=21\) but then subtracts 15 has made a sign error, not necessarily an algebra misunderstanding.

Question 5: Simplify \(4a-3b+7a+5b\)

Collect like terms. Terms containing \(a\) go together, and terms containing \(b\) go together:

\[ 4a+7a=11a \]

\[ -3b+5b=2b \]

Answer: \(11a+2b\)

You cannot combine \(a\) and \(b\) into \(13ab\). They represent different terms. This question checks whether a student understands algebraic notation rather than simply follows a memorised rule.

Question 6: The perimeter of a rectangle is 34 cm. Its length is \(3x+2\) cm and its width is \(x-1\) cm. Find \(x\).

A rectangle has two lengths and two widths, so:

\[ 2(3x+2)+2(x-1)=34 \]

Expand and simplify:

\[ 6x+4+2x-2=34 \]

\[ 8x+2=34 \]

\[ 8x=32 \]

\[ x=4 \]

Answer: \(x=4\)

The key diagnostic point is whether the student uses all four sides. If they write \((3x+2)+(x-1)=34\), they understand the expressions but have not applied the perimeter rule correctly.

Geometry and measures

Question 7: A triangle has angles \(2x\), \(3x\) and \(4x\). Find the largest angle.

Angles in a triangle total 180°:

\[ 2x+3x+4x=180 \]

\[ 9x=180 \]

\[ x=20 \]

The largest angle is \(4x\):

\[ 4 \times 20=80 \]

Answer: 80°

This checks both angle knowledge and equation solving. If a student reaches \(x=20\) but answers 20°, they need to slow down and return to what the question actually asks.

Question 8: Find the area of a circle with radius 6 cm. Give your answer in terms of \(\pi\).

Use \(A=\pi r^2\):

\[ A=\pi \times 6^2=36\pi \]

Answer: \(36\pi\text{ cm}^2\)

The most frequent mistake is using the diameter as the radius. Here the radius is already given, so there is no need to halve anything. Giving the answer in terms of \(\pi\) means do not convert it to a decimal.

Interpreting the result

This worked GCSE Maths diagnostic is deliberately short, so it should guide the next step rather than label a student by a score. A student who gets five or six questions correct may still need targeted support if the errors cluster around one topic. Equally, a lower score may reflect rushed reading, weak exam habits, or anxiety rather than an absence of mathematical potential.

Look for patterns. Difficulty with Questions 1 to 3 suggests a number skills gap that can affect percentages, proportion and later financial maths. Errors in Questions 4 to 6 point towards algebra, a major GCSE topic that needs regular, structured practice. Questions 7 and 8 reveal whether formulae and geometry facts are being selected accurately.

For revision, choose one weak area and practise it across several short sessions. Begin with fully worked examples, move to similar questions with prompts, then attempt mixed exam-style questions independently. Repeating only familiar questions may build speed, but it will not repair the misunderstanding behind recurring mistakes.

When a student is unsure why an answer is wrong, personalised feedback can make a substantial difference. A qualified GCSE Maths tutor can identify the exact step where confidence drops, explain the method in a way that suits the learner, and build a realistic plan around their target grade. TutorsValley’s free assessment and tailored tutor matching can provide that clearer next step.

The most useful diagnostic is one that leads to action. Keep the workings, revisit the questions in two weeks, and notice not only the improved score but the growing ability to explain why each method works.