Working Backwards
Unwind the problem from the answer
When a problem gives you the end result and asks for the starting value, work backwards by reversing every step in reverse order.
See It in Action
The Problem
A number is doubled, then 9 is added, then the result is halved. The answer is 11. What was the starting number?
Common Mistake
Start from 11. Undo doubling: 11 ÷ 2 = 5.5. Undo +9: 5.5 − 9 = −3.5. Undo halving: −3.5 × 2 = −7.
The operations are being undone in the original (forward) order instead of the reverse order. You must undo the LAST operation first.
Correct Approach
Forward order: ×2, +9, ÷2. Reverse order to undo: ×2 (undo ÷2), −9 (undo +9), ÷2 (undo ×2). Start with 11: 11 × 2 = 22. 22 − 9 = 13. 13 ÷ 2 = 6.5.
Answer: 6.5
Forward check: 6.5 × 2 = 13. 13 + 9 = 22. 22 ÷ 2 = 11. ✓
The Core Concept
Sometimes a problem tells you where things ended up, and asks where they started. Instead of guessing and checking forward, you can work backwards: start with the final number and undo each step in reverse order.
The golden rule: undo the LAST step first. And each operation gets flipped — addition becomes subtraction, multiplication becomes division. If the sequence forward was "×3, then +7," working backwards is "−7 first, then ÷3."
This also applies to fraction problems: if 2/3 of a number is 18, then the number is 18 ÷ 2 × 3 = 27. Thinking of it as "undo the fraction": if 2/3 remains, divide by 2 to get 1/3, then multiply by 3 to get the whole.
Common Mistakes to Avoid
Undoing operations in the forward (original) order
The last thing that happened must be undone first. Think of it like unwrapping a gift — the last paper you put on is the first you remove.
Using the wrong inverse operation
The inverse of ×3 is ÷3. The inverse of +5 is −5. The inverse of squaring is square root. Double-check each flip.
Not verifying by working forwards
Always check your answer by plugging it back into the original forward sequence. If it gives the stated result, you're correct.
Struggling with fraction operations in reverse
If "3/4 of the number is 24," then: 1/4 of the number = 24 ÷ 3 = 8, so the whole = 8 × 4 = 32. Or: 24 ÷ (3/4) = 24 × 4/3 = 32.
Try It Yourself
Priya had some stickers. She gave half to her sister, then received 8 more, then gave away 3. She now has 13 stickers. How many did she start with?
Hint: Work backwards from 13. Last operation was "gave away 3" — undo it first.
Key Tips
Write out the forward steps first (in order), then reverse the list and flip each operation.
Always verify your answer by doing the problem forward with your found starting value.
For fractions: "3/4 of X is 18" → X = 18 ÷ 3 × 4 = 24. Divide by numerator, multiply by denominator.
Ready to practise Working Backwards?
Reading is the start. The Maths Gym has exercises designed around this skill — with instant feedback and progress tracking.