Ratio questions in the 11 plus: why the answer is almost never the number you divide by

Ratio is the topic where a child can understand every word of the question, do real arithmetic, and still hand in a wrong answer. It is also one of the few topics in 11 plus maths where the wrong answer looks completely sensible written down, which is why it survives so much practice.

Here is the question, in the form it usually arrives.

Anna and Dev share £40 in the ratio 5 : 3. How much does Dev get?

The answer is £15.

The answer most often given is £8, and sometimes £13.33.

Where £8 comes from, and why it is not carelessness

There are two numbers in that ratio and one amount of money. A child who has been taught to share looks at £40, looks at 5 and 3, and divides by the number nearest to hand. Divide by 5 and you get £8. Divide by 3 and you get £13.33.

Both are real arithmetic, done correctly. Neither has anything to do with the question.

The step that goes missing is the one nobody says out loud: the number you divide by is not in the question. You have to make it. Five parts and three parts are eight parts, and eight is the number that matters. Once a child has 8, the rest is easy: £40 divided by 8 is £5 a part, so Dev gets three of them, £15.

That single hidden number is the whole topic. Almost every ratio question a grammar school sets is a test of whether a child invents it.

The second version, which catches the children who have learned the first

A box holds red, blue and green counters in the ratio 6 : 4 : 1. There are 10 more red counters than blue ones. How many counters are in the box altogether?

A child who has learned to add the parts adds 6, 4 and 1 to get 11, and then stops, because there is no total to divide.

There is no total in this question. What there is instead is a difference. Six parts and four parts are two parts apart, and that gap is the 10 counters we were given. So two parts are 10 counters, one part is 5 counters, and eleven parts are 55 counters.

The reason this one is worth practising is not that it is harder. It is that it asks the same question from the other end, and a child who has only ever met "share this total" will read it for a total that is not there.

How to tell which of the two problems your child has

Ask them to talk you through this one out loud, without writing anything.

Ben and Sara share sweets in the ratio 4 : 1. Sara gets 6 sweets. How many sweets do they share altogether?

The answer is 30.

Listen for the first thing they say.

If they say "6 divided by 4" or "6 times 4", they are still looking for the number to divide by rather than making it. That is the first problem, and the fix is the sentence above: count the parts first, before you touch any of the numbers.

If they say "one part is 6", and then get lost, they have understood the parts perfectly and are losing the thread of a multi-step question. That is a different problem with a different fix, and more ratio practice will not touch it.

Two children can get the same question wrong for those two completely different reasons. A score out of forty cannot tell you which one you have. That is why we ask every skill more than once, in a different form each time.

The thing to say, and it is short

When a ratio question appears, the useful instruction is not "remember your ratios". It is:

"Before you divide anything, tell me how many parts there are altogether."

If the answer comes quickly, the rest of the question is ordinary arithmetic. If it does not, you have found the gap, and you have found it in about four seconds.

Where ratio quietly turns up when it is not called ratio

Worth knowing, because a child can be solid in the ratio chapter and still lose these:

• Recipes scaled up or down for a different number of people
• Maps and scale drawings, where 1 cm stands for 2 km
• Best value questions, where two packet sizes are compared
• Mixing paint, squash or cement in a given proportion
• Pie charts, where a sector is a share of a whole

None of those pages says "ratio" at the top. All of them fail in exactly the same place, on the number that has to be made rather than read.

Try it on one paper

Our free paper takes about forty five minutes and marks itself. You will see which skills are solid, which are not, and the named mistake behind each one that went wrong. It costs nothing and there is no card.