Questions: Working Backwards

Question 1

A student starts with a number, multiplies by 3, subtracts 4, and gets 20. Working backwards, what is the original number?

• A. 5
• B. 7
• C. 8
• D. 24

Answer: (C)

Working backwards: start from 20. The last forward step was 'subtract 4,' so reverse it: 20 + 4 = 24. The step before that was 'multiply by 3,' so reverse it: 24 ÷ 3 = 8. Check forward: 8 × 3 = 24, 24 - 4 = 20. Confirmed.

Question 2

When using the working backwards strategy, the reverse of 'divide by 2 then add 7' is 'subtract 7 then multiply by 2.'

• T. True
• F. False

Answer: True

When reversing a sequence of operations, you reverse both the operations themselves AND their order. The last operation applied forward ('add 7') is the first to be undone ('subtract 7'). The first operation applied forward ('divide by 2') is the last to be undone ('multiply by 2'). This is the same principle as unwinding a stack of function calls.

Question 3

In a tournament, the final winner beat the player who beat the player who beat Player A. If the tournament is single-elimination and you know the final winner, explain how working backwards helps identify who Player A lost to.

Model answer: Start from the final winner. The winner beat someone in the final — identify that opponent. That opponent beat someone in the semifinal — identify who. That person beat Player A. Working backwards through the bracket from the known winner traces the chain of results to find exactly who eliminated Player A.

The bracket structure makes working backwards natural: each match has a known winner, and you can trace connections backward through the results. The end state (final winner) is specific and known, while the forward question (who will Player A lose to?) depends on many unknowns. This asymmetry — specific end, uncertain start — is exactly when working backwards excels.