Technique Encyclopedia

Working Backwards

What it is

Working backwards means beginning at the end result a problem describes and undoing each stated operation in reverse order, until you arrive at the unknown starting value. It turns a problem that looks like it needs an unknown-then-solve setup into a sequence of simple, direct computations.

Signals that suggest using it

A problem describes a sequence of operations applied in order, ending in a stated final value, and asks for the starting value.
Each operation (add, double, remove a fraction, etc.) has an easy, unambiguous inverse.
Setting up a single forward equation would require tracking an unknown through several steps at once.

When it's effective

This is fastest when every operation is cleanly invertible — reversing addition with subtraction, doubling with halving, and so on — turning a multi-step algebra problem into simple arithmetic done in reverse order.

When it's not effective

If a step isn't cleanly invertible — flooring, rounding, taking a remainder, or any operation that could have come from multiple different inputs — working backwards needs extra case analysis to handle the ambiguity, or breaks down entirely.

Simple example

Problem

A number is doubled, then 6 is added, giving 20. What was the original number?

Solution

Reverse the operations in reverse order: undo "add 6" first (20-6=14), then undo "double" (14/2=7). The original number was 7.

AMC-style example

Problem

A number is tripled, decreased by 4, and then the result is halved, giving 10. What was the original number?

Solution

Reverse each step in reverse order. Undo "halved": 10 × 2 = 20. Undo "decreased by 4": 20+4=24. Undo "tripled": 24/3=8. The original number was 8.

Strategic Substitution Testing Small Cases