Working back through thirteen years of our own IMC-style practice questions, one thing stands out: the mistakes repeat. The same handful of wrong instincts turn up year after year, on completely different questions, because they're not really about the maths content — they're about which habit a tense brain reaches for first. Here they are, grouped into families, each with the instinct that actually works. For the facts these mistakes attach to, see our companion list of IMC facts to know.

Scaling: length, area and volume don't scale together

This is the single biggest family of mistakes on the paper, and it always has the same shape: something is enlarged, and the wrong dimension is scaled by the wrong power.

  • Doubling every length of a shape doubles its area. It doesn't — it quadruples it, because area is two-dimensional, so the scale factor applies twice (k²).
  • Two similar triangles with sides in ratio 1 : 3 have areas in ratio 1 : 3. They're actually in ratio 1 : 9 — length ratio squared, not length ratio itself.
  • A circle's radius is doubled, so its area doubles too. Same error, same fix: area goes up by a factor of four.
  • Going the other way — from an area ratio to a length ratio — you halve the area ratio. You take its square root instead.

The underlying habit is simple once it's named: length scales by k, area by k², volume by k³. Every one of the mistakes above is the same slip in a different costume.

Counting: add or multiply?

The second-biggest family. Whenever a question involves combining choices, there's a strong pull toward addition when multiplication is needed.

  • A meal made of one starter (3 options) and one main (4 options) gives 3 + 4 = 7 possible meals. Choices made independently multiply: 3 × 4 = 12.
  • Two overlapping shapes' combined area is just the sum of the two areas. That double-counts the overlap; the fix is sum minus overlap.
  • Two shapes glued face to face lose one hidden face of surface area. They lose two — one from each shape.
  • Joining shapes edge to edge along a length k reduces the total perimeter by k. It falls by 2k, because both shapes lose that edge from their own perimeter.
  • Counting ordered arrangements (AB and BA counted separately) when the question wants unordered choices. Choosing 2 items from n is n(n − 1) ÷ 2 precisely because AB and BA are the same choice, counted twice.

Signs and negatives

Everyone can state the rule for negative numbers. Fewer apply it correctly mid-calculation, especially once several signs are stacked up.

  • Subtracting a negative number is treated the same as subtracting a positive one. 5 − (−3) is 5 + 3 = 8, not 5 − 3 = 2 — subtracting a negative behaves exactly like adding.
  • "Two negatives make a positive" is applied to addition, not just multiplication. −2 − 3 is still −5. The rule about two negatives producing a positive belongs to multiplying and dividing, never to adding two negative numbers together.
  • A carry of 2 is allowed when adding two numbers in a column. With only two numbers being added, the largest possible carry into the next column is 1 — a useful check when reconstructing a hidden digit sum.

Geometry properties that get swapped

KS3 geometry has a lot of "almost-the-same" shape properties, and the paper likes to test exactly the difference.

  • A rhombus's diagonals are equal in length, because that's easy to associate with "special quadrilateral". They're not — equal diagonals is the rectangle's property. A rhombus's diagonals are perpendicular bisectors of each other, but not equal.
  • The exterior angle of a triangle equals the interior angle right next to it. It equals the sum of the two opposite interior angles — a genuinely different, and more useful, fact.
  • An isosceles triangle's apex angle equals its two base angles. That's only true if the triangle is also equilateral; ordinarily the two base angles are the equal pair, and the apex is different.
  • Two angles inside a quadrilateral adding to 180° means they lie on a straight line. It actually tells you the two sides between them are parallel — a completely different geometric fact hiding behind the same number.
  • Any shape with two equal angles counts as a kite. A kite's one equal pair sits specifically between its two pairs of unequal sides — not any two equal angles anywhere in the shape.

Index and power slips

Powers are compact notation, and compact notation is easy to misread under time pressure.

  • (aᵐ)ⁿ is worked out by adding the indices, the way aᵐ × aⁿ is. A power of a power multiplies the indices instead: (aᵐ)ⁿ = aᵐⁿ.
  • a^(mⁿ) is read as if it were (aᵐ)ⁿ. They're different expressions — the power inside the exponent is worked out first.
  • Halving an index halves the value. Halving 2¹⁴ to "get" 2⁷ divides the actual number by 2⁷, not by 2 — a very easy trap when index questions move fast.

Fractions, decimals and percentages

  • 0.3̇ recurring is written as 3/10, or 33/100. Both stop the pattern too early; a single recurring digit is always that digit over 9, so 0.3̇ = 3/9 = 1/3.
  • 30% of one amount minus 30% of another is treated as a separate, harder calculation from 30% of the difference. They're identical — take 30% of the difference and you're done in one step.
  • A phone battery falling from three-quarters to two-thirds charged: the "amount used" is calculated as a fraction of what was there before, rather than of a full battery. The question usually wants the drop measured against the whole, which means subtracting the two fractions directly.

The misconception that isn't about the maths at all: guessing

Here's one that has nothing to do with content and everything to do with strategy — and it catches out plenty of otherwise well-prepared candidates, because it's true on a different UKMT paper.

"Guessing never hurts, so always fill in every question." On the Junior Maths Challenge, that's correct — there's no penalty for a wrong answer, so a blank question is strictly worse than a guess. The Intermediate Challenge is marked differently: a wrong answer to one of questions 16-20 costs 1 mark, and a wrong answer to 21-25 costs 2 marks, while a blank answer always scores 0. Blindly guessing among five options on those later questions is, on average, a losing move — you're more likely to turn a 0 into a −1 or −2 than into a mark. The right habit is to guess only once you've eliminated at least one or two options, which tips the arithmetic back in your favour; on questions 1-15, where there's no penalty at all, guess freely.

Where these come from, and what to do about them

Every misconception above is drawn from real IMC-style practice cards our maths tutor, Professor Pi, flags as a candidate writes or reviews them — built and refined across thirteen years of papers, from the IMC toolkit deck through each individual year's deck. Pairing a fact with the mistake that usually attaches to it is exactly why the facts list and this one are meant to be read together. If you'd rather see which of these ideas comes up most often, rather than which trips people up most often, our data-driven look at recurring IMC topics is the companion piece — and spaced repetition is the practical way we've found to make a misconception stop recurring, rather than just naming it once.

FAQ

Are these mistakes specific to the IMC, or just general maths errors?

Most of them are general KS3 misconceptions — the IMC just puts them under time pressure, which is exactly when a wrong instinct is most likely to win over the correct one. A handful, like the guessing-penalty mistake, are specific to how the IMC is marked.

Should my child avoid guessing on the Intermediate Maths Challenge?

Not entirely, but they should be selective about it. Blank answers on questions 16 to 25 score 0, but a wrong answer there loses marks — 1 for questions 16 to 20, 2 for questions 21 to 25 — so guessing among five options with no idea is, on average, a losing move on those later questions. It's a different calculation from the Junior Challenge, which carries no penalty at all.

Is this list official UKMT content?

No. These misconceptions are drawn from our own tutor's IMC-style practice cards, built and refined across thirteen years of papers. For official past papers and solutions, see the UK Mathematics Trust site.

Duke Harewood built aitutors.me's KS3 maths tutor (Professor Pi) for his Year 8 daughter. Updated 26 August 2026.