Look for a chain that ends with a known result.
Working backwards is useful when a question describes steps applied to an unknown starting value, but tells you the final value. Read the operations in order and write them as a short chain. Then undo the last operation first.
The inverse pairs are familiar: add/ subtract, multiply/divide, square/square root where appropriate. The important habit is the order. Undoing the right operation at the wrong time changes the problem.
Original worked example: reverse the full chain.
A student thinks of a number, adds 5, multiplies the result by 3, then subtracts 6. The final answer is 30. What was the number?
Start at 30. Undo “subtract 6” by adding 6: 36. Undo “multiply by 3” by dividing by 3: 12. Undo “add 5” by subtracting 5: 7.
Check forward: 7 + 5 = 12; 12 × 3 = 36; 36 − 6 = 30. The starting number is 7.
Self-check: make forward checking part of the solution.
Try this original problem: a number is multiplied by 4, then 9 is added. The result is 41. What was the starting number?
Write each reverse operation before calculating. Then substitute your answer into the forward steps.
Check your reasoning
Undo +9 first: 41 − 9 = 32. Undo ×4: 32 ÷ 4 = 8. Forward check: 8 × 4 + 9 = 41. The starting number is 8.
If your forward check fails, revisit the order before changing the arithmetic. A correct calculation can still belong to the wrong reverse step.
Choose the method that fits the information.
Working backwards is not a universal trick. If the question gives several growing cases, a table may reveal a pattern more clearly. Read how to find patterns for that approach. When you have a method, explain each choice using the solution guide.
This article uses original examples. It is independent preparation advice, not official Australian Mathematics Competition material.