The five most expensive fraction mistakes in the 11+

Ask any tutor which topic costs able children the most marks in a selective paper and you will hear the same answer: fractions. Not because the fractions themselves are hard — most 11+ candidates can simplify, compare and add fractions on a good day — but because fraction questions are unusually good at hiding a wrong turn inside working that looks perfectly tidy.

What follows are the five most expensive of those wrong turns. Expensive in the plain sense: each one typically appears several times in a single paper, so a child who carries it doesn’t lose one mark to it. They lose four or five.

For each, you will find the mistake, a worked example of the wrong answer, what was going through the child’s head, and one question to ask at the kitchen table that springs the trap in the open.

1. Taking the fraction of the wrong number

We call it working with what’s left, and it is the most reliably expensive mistake in the topic.

A bookshop has 36 copies of a novel. It sells a third of them on Saturday, then half of the rest on Sunday. How many copies are sold on Sunday?

Leo writes 18. Half of 36.

What he was thinking: nothing, in the best sense — the machinery ran on its own. “Half” arrived, 36 was the only number on the page, and half of 36 is a fact he has known for years. The 24 copies actually left after Saturday never existed in his working, because he never paused to build them.

The tell is that the wrong answer comes back fast. A child working the problem properly has to stop and compute the remainder first; a child in the trap answers like it is a times-table question.

The question to ask: “Half of what, exactly?” Or, if you want to make him build the missing number: “How many copies were on the shelf when Sunday started?”

2. Solving the reverse question forwards

Two thirds of a number is 24. What is the number?

Amelia writes 16.

She has taken two thirds of 24 — done the only fraction move she has rehearsed, on the only number available. The question runs backwards (here is the part, find the whole) and she has answered it forwards (here is the whole, find the part). We call it finding the whole from a part.

What makes this one so costly is that her answer is the product of correct arithmetic. Two thirds of 24 really is 16. Every step she performed was right; she just performed them on the wrong question. That is why it survives ordinary practice — nothing in the working looks broken.

The check that catches it takes two seconds: 16 is smaller than 24, but the question says 24 is only two thirds of the answer. The whole cannot be smaller than its part.

The question to ask: “Is 24 the whole thing, or a piece of it?”

3. Adding the tops and adding the bottoms

What is 1/4 + 2/5?

Noah writes 3/9, then simplifies it — carefully, correctly — to 1/3.

Tops added, bottoms added — the mistake we call adding tops and bottoms. He is treating a fraction as two independent whole numbers that happen to live in the same house, rather than as a single quantity. The cruel detail is that his answer simplified nicely, and to a ten-year-old, an answer that simplifies feels like an answer that is right.

The most dangerous wrong answers are the ones that feel finished.

The estimate exposes it instantly. He started with 2/5, which is nearly a half, and then added more — so the answer must be bigger than 2/5. His answer, 1/3, is smaller than what he started with. Adding made it shrink.

The question to ask: “Should your answer be bigger or smaller than the biggest fraction you started with?”

4. Comparing fractions by their numerators

Which is larger, 3/5 or 4/9?

Freya picks 4/9, because 4 beats 3. Sometimes the same child picks by denominator instead — ninths beat fifths because 9 beats 5 — but it is the same underlying move: comparing fractions by comparing their visible digits. We call it comparing by the top number.

The trap really bites in ordering questions, where four or five fractions have to be ranked and one digit-led swap quietly wrecks the whole line. One wrong turn, one mark, no partial credit.

The rescue skill is the benchmark check, and it is worth teaching explicitly: is each fraction more or less than a half? 3/5 is more than a half (3 is more than half of 5). 4/9 is less than a half (4 is less than half of 9). Comparison over, no common denominators required.

The question to ask: “Is each one more than a half, or less?”

5. The borrow that flips the subtraction

Work out 5 2/9 − 1 7/9.

Isaac writes 4 5/9.

Look closely at the fraction part and you can see the exact moment it happened: he needed 2/9 − 7/9, couldn’t do it, and so — without ever deciding to — did 7/9 − 2/9 instead. The subtraction quietly flipped. Wholes: 5 − 1 = 4. Fractions: 5/9. Everything on the page looks tidy, and everything on the page is wrong. We call it borrowing with mixed numbers.

The correct route needs a borrow: break one whole into ninths, turn 5 2/9 into 4 11/9, and then subtract to get 3 4/9. Children flip instead of borrowing because flipping produces an answer and borrowing produces work. Under exam pressure, the path that produces an answer always wins.

The question to ask: “Can you take seven ninths away from two ninths? So what has to happen first?”

What these five have in common

None of them makes the child stall. That is the property that matters. Each one produces a confident, tidy, plausible answer — arithmetic done correctly on the wrong quantity, the wrong question, or the wrong direction. Nothing feels wrong at the desk, which is exactly why general practice doesn’t clear them: the child completes another thirty questions, falls into the same traps, and files the losses under bad luck.

What clears them is meeting each trap deliberately, several times, in different costumes, with someone at the kitchen table asking the one question that makes the wrong turn visible. Five questions, one per trap, is a better Tuesday evening than another full past paper.