Working Backwards with Inverse Operations
Working backwards means solving a problem by starting from the final result given and undoing each step in reverse order, using the inverse operation at every stage: addition undoes subtraction, and multiplication undoes division, and the other way round. It is the standard technique for 'think of a number' puzzles and for function machine questions where the output is known but the input is not.
Method
- List the operations described in the question in the order they happen to the unknown starting number.
- Reverse the order of the list, so the last operation performed becomes the first one you undo.
- Replace each operation with its inverse: undo 'add' with subtract, undo 'subtract' with add, undo 'multiply' with divide, and undo 'divide' with multiply.
- Apply each inverse operation in turn to the given final result, working one step at a time and writing down the value after each step.
- Once you reach the start, check your answer by working forwards through the original operations to see if you arrive back at the given final result.
Worked example
Jamie thinks of a number. He adds 15, then divides the result by 3, then subtracts 4. His final answer is 6. What number did Jamie start with?
- List the operations in order: add 15, then divide by 3, then subtract 4.
- Reverse the last step first: undo 'subtract 4' by adding 4 to the final answer, 6 + 4 = 10.
- Undo 'divide by 3' by multiplying by 3: 10 x 3 = 30.
- Undo 'add 15' by subtracting 15: 30 - 15 = 15.
- Jamie's starting number was 15. Check: 15 + 15 = 30, 30 / 3 = 10, and 10 - 4 = 6, which matches.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1I think of a number, add 9, and the result is 23. What was my number?Show answer
Answer: 14 (23 - 9)
Q2I think of a number, multiply it by 5, and the result is 60. What was my number?Show answer
Answer: 12 (60 / 5)
Q3A number is divided by 4, then 3 is added, giving a result of 10. What was the original number?Show answer
Answer: 28 (undo 'add 3': 10 - 3 = 7, then undo 'divide by 4': 7 x 4 = 28)
Q4I think of a number, subtract 8, then double the result, to get 24. What was my number?Show answer
Answer: 20 (undo 'double': 24 / 2 = 12, then undo 'subtract 8': 12 + 8 = 20)
Q5A function machine does 'add 6' then 'multiply by 2' to give an output of 26. Find the input.Show answer
Answer: 7 (undo 'multiply by 2': 26 / 2 = 13, then undo 'add 6': 13 - 6 = 7)
Q6I think of a number, multiply by 3, then subtract 5, and the result is 40. What was my number?Show answer
Answer: 15 (undo 'subtract 5': 40 + 5 = 45, then undo 'multiply by 3': 45 / 3 = 15)
Q7A number has 12 added to it, then the result is divided by 4, giving 9. What was the original number?Show answer
Answer: 24 (undo 'divide by 4': 9 x 4 = 36, then undo 'add 12': 36 - 12 = 24)
Q8I think of a number, halve it, then add 11, to get 20. What was my number?Show answer
Answer: 18 (undo 'add 11': 20 - 11 = 9, then undo 'halve': 9 x 2 = 18)
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
Jamie's sister Nina also thinks of a number. She subtracts 6, then multiplies the result by 4, then adds 10. Her final answer is 42. What number did Nina start with?
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 4 available
A number is put into a function machine: first it is halved, then 9 is added, then the result is multiplied by 3. The output is 45. Find the input number, showing each step of your reversal.
Show mark scheme
Tick each line you got. Your score builds from the marks on the scheme.
Nothing ticked yet - 5 available
Free printable worksheet
Want more practice on paper? Download the working backwards with inverse operations worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 26 of 11+ Maths Workbook 4, the whole course as one free printable PDF.
Next topics
Not quite what you needed?
Tell us what is missing on working backwards with inverse operations, or which topic to write up next. Every request is read, and we reply to every one.
Build a full practice pack.
This topic is one of hundreds in the library - pick the ones a student needs and generate a printable PDF in minutes.