11 plus Inverse / Think of a Number questions

What an Inverse / Think of a Number question looks like

An inverse question hands your child the finishing number and asks for the starting one. Someone thinks of a number, does one or two things to it, and says where they ended up. The job is to run the whole thing backwards. It is one of the reasoning formats on an 11 plus maths paper, and it is where a child meets algebraic thinking without a letter in sight.

The range is narrow. Most are a person thinking of a number and then adding, subtracting, multiplying or dividing: one operation in the easiest, two in the middle, three in the hardest, plus one that squares the number so the way back begins with a square root. The rest swap the number for people and things. Sam is twice as old as Maya, and Priya swam twice as many lengths in her second week as in her first. Same backwards work, dressed as a short story.

That narrowness marks the type out from its neighbours in the algebra part of a paper. It only ever runs backwards from a result to the number someone started with, so there is no machine to send a number forwards through, no hidden rule to find in a table, and no letter left standing in the answer.

Example one: one operation to undo

Marcus thinks of a number, divides it by 4, and the result is 7. What is the number? The options are 28, 1.75, 3, 32 and 11.

Start at the end, at 7. The last thing Marcus did was divide by 4, and dividing is undone by multiplying, so multiply the 7 by 4: 7 × 4 = 28.

The answer is 28, and running it forwards confirms it: 28 ÷ 4 = 7.

The option that tempts children is 1.75, which comes from doing what the question describes rather than the opposite of it: 7 ÷ 4 = 1.75. A child who reads “divides it by 4” and reaches for a division has done the arithmetic perfectly and answered a question nobody asked.

Example two: two operations, and the order they come undone

Jasmine thinks of a number. She divides it by 4 and then adds 5. The result is 14. What is the number? The options are 18, 45, 9, 76 and 36.

Two things were done, so two must be undone, and the last one done is the first undone. The last thing Jasmine did was add 5, so take 5 off the 14: 14 - 5 = 9. Now undo the dividing by multiplying: 9 × 4 = 36.

The answer is 36. The check takes seconds: 36 ÷ 4 = 9, and 9 + 5 = 14.

Two wrong options are worth a look. The 9 is the halfway number, and a child who undoes one step and hands in what is in front of them will pick it. The 76 comes from going the wrong way on that first step, adding the 5 instead of taking it off: (14 + 5) × 4 = 76.

Example three: people instead of a chain of sums

Aisha has twice as many marbles as Ben. Ben has 4 more marbles than Cleo. Aisha has 20 marbles. How many marbles does Cleo have? The options are 6, 14, 10, 4 and 16.

Only one person’s count is known, so start there and step along the chain. Aisha has twice what Ben has, so Ben has half of Aisha’s: 20 ÷ 2 = 10. Ben has 4 more than Cleo, so Cleo has 4 fewer than Ben: 10 - 4 = 6.

The answer is 6, and it checks back up: Cleo has 6, Ben has 6 + 4 = 10, Aisha has 2 × 10 = 20.

The 10 is there for the child who reaches Ben and stops, easy to do when Ben was never the person asked about. The 14 is for the child who sees “4 more” and adds the 4 instead of taking it away. Here the words carry the operations, so they are the part to slow down on.

How to answer it

1. Find the finishing number and mark it. It is nearly always the last number in the question, after “the result is” or “the answer is”.
2. Write the operations out in the order they were done. “Multiply by 3, then subtract 7” is two steps, and steps on paper are easier to trust than steps held in your head.
3. Start at the bottom of that list and undo each step going upwards. Adding undoes subtracting, subtracting undoes adding, multiplying undoes dividing, dividing undoes multiplying, and a square root undoes squaring.
4. Do one step at a time and write down the number you have after each. Undoing two operations in a single move is where most wrong answers come from.
5. Run your answer forwards through the question. If it lands on the number the question finished with, you are right. If not, look for the step you did the same way round as the question.

Common mistakes

• Doing what the question says rather than the opposite. “Multiply it by 5 and the result is 20” pulls a child towards 20 × 5 = 100. Every set of options carries one of these, and it is the most popular wrong answer.
• Undoing the steps in the order they were written. If a number was multiplied by 3 and then had 7 taken off, the 7 goes back on first. Starting with the multiplying gives an answer that looks reasonable and is not.
• Stopping halfway. The halfway number is always one of the options, because handing that in is such a common slip.
• Going the wrong way on a single step. Adding 6 back rather than taking it off, or adding 4 to Ben rather than subtracting it. Everything either side is correct, so it is hard to spot on a second read.
• Answering about the wrong person. In the word questions the person asked about is often two links along the chain from the one whose number you were given, and the middle person’s count is in the options.

How to practise 11 plus Inverse / Think of a Number questions

The habit worth building is the check, because it is quick and catches nearly everything: take the answer, run it forwards through the question as written, and see whether it lands where the question ended. A child who does that finds their own direction errors instead of being told about them, which is what makes the method stick. Ten focused minutes on one format beats an hour of mixed work, the pattern we set out in how much 11 plus practice a child needs each day. 11+ Daily has an Inverse / Think of a Number project alongside the other eleven plus maths topics, on laptop, tablet or phone, and the full list sits on our 11 plus maths practice page. Every question is original to 11+ Daily, nothing is a past paper and nothing is licensed, and the bank is re-checked in rounds.

Questions parents ask