Percentage questions in the 11 plus: why 25% off and then 20% off is not 45% off

There is one question in 11 plus maths, and in grammar school entrance papers generally, that catches children who are genuinely good at percentages. It can even come back after it has been put right.

It looks like this.

A coat costs £80. In the sale it is reduced by 25%. On the last day, the sale price is reduced by a further 20%. What does the coat cost now?

Or like this.

Maya spends 30% of her money on a book. She then spends 40% of what is left on a present. What percentage of the money she started with does she have left?

Both are asking the same thing, and both catch the same children.

The two answers, and why the wrong one feels so right

On the coat, a great many children add 25 and 20, call it 45% off, and answer £44.

The right answer is £48.

The first discount takes £80 down to £60. The second discount is 20% of £60, not 20% of £80. That is £12, so the coat costs £48. The four pound gap is the whole question.

On Maya, many children add 30 and 40 to make 70, and say she has 30% left. That is wrong too. She spends 30%, so 70% is left. Then she spends 40% of that 70%, which is 28% of her money. She has 42% left.

Why it survives being taught

Nothing about it feels like a gap. Ask any of these children for 20% of £60 and they will do it quickly and correctly. The arithmetic is fine.

The trouble is a few small words: of what is left, of the rest, a further. They change what the next percentage is a percentage of. A child reading at speed takes the second number back to the starting amount.

So it can be put right one week and turn up again a few weeks later. The child has not forgotten how to do percentages. They have stopped noticing those words.

It is not only a percentages question

This is the part that is easy to miss, and it is why this one is worth real attention.

The same shape turns up all over an 11 plus maths paper:

• Fractions. He gives a third of his sweets to his sister, then a quarter of what remains to his brother.
• Money. She saves half her pocket money, then spends 60% of the rest.
• Word problems and ratio. The rest are shared between…

A child who has not spotted the shape loses marks in three or four different topics. On a score sheet it looks like three or four separate weaknesses. It is one.

In our own records the clearest case cost one child nine marks in four different topics: percentages, fractions, statistics and a mixed paper. Some of those marks were lost after the mistake had already been found and practised once.

How to test whether your child has it

Asking "do you know what 'of the rest' means?" will not tell you much, because every child says yes. Give them this instead, with a pen and no help:

A jacket costs £60. It is reduced by 10%. A week later, the new price is reduced by 10%. Is that the same as 20% off the original price? Show why.

A child who has the idea will say no. They will work out £54, then £48.60, and compare it with £48 for 20% off.

A child who has not will say yes, and be quite sure.

The 60p gap is small on purpose. You are not testing whether they can take away. You are testing whether they notice that the starting amount changed.

The fix: one question before the second step

"Read the question carefully" is good advice, but it is hard to act on halfway through a timed paper. This is something a child can actually do, and it takes seconds:

Before the second step, ask: percent of what?

Then write that number down first. In the coat question, before touching the 20%, write £60. Once the second percentage has a number to hang on, the mistake is very hard to make.

And then check it again in a month

This is the one to be patient with. For the child above, it came back weeks after it seemed settled. That is not a failure. There was never a gap in their knowledge to fill, only a habit of reading past a few small words, and a habit needs meeting more than once.

So the routine that works is simple: find it, fix it, and then ask again a month later when nobody is expecting it. If it holds then, it has gone.