Technique Encyclopedia
Bounding
What it is
Bounding means deriving inequalities that squeeze an unknown quantity into a narrow range, rather than solving for it directly. On its own, a bound only narrows down the possibilities — but combined with a discreteness constraint (the answer must be an integer, a digit, or one of a small number of choices), a tight enough bound can pin down a single exact value.
Signals that suggest using it
- An unknown satisfies an inequality (or two) rather than an equation.
- The problem states or implies the answer is an integer, but doesn't hand you an equation to solve for it directly.
- Multiple-choice answer options that a rough bound could immediately narrow down.
When it's effective
Extremely effective when a bound plus an integrality constraint together force a unique value — the bound does the narrowing, and "must be an integer" does the pinning.
When it's not effective
If the bound obtained isn't tight enough, multiple candidate values remain and the technique alone won't finish the problem — a sharper argument or an additional constraint is needed.
Simple example
Problem
A positive integer n satisfies 10
Solution
√10 ≈ 3.16 and √20 ≈ 4.47, so 3.16
AMC-style example
Problem
How many positive integers n satisfy 1/3
Solution
Multiplying through by 7: 7/3
Related techniques
Estimation Exploiting Answer Choices