Unlike a free-response exam, the AMC hands you five candidate answers. Ignoring that information and solving purely “forward” from the problem statement is sometimes the slow way in.
Backsolving
If the problem gives you an equation or condition and the choices are numeric, plug each choice back into the condition and see which one works. Start from the middle choice — if it’s too big or too small, you often know immediately which direction to move, letting you binary-search to the answer in at most two more tries.
Bounding
Estimate a lower and upper bound for the answer using quick reasoning, then eliminate any choices outside that range — sometimes narrowing it to a single remaining option without ever finding the exact value.
Monotonicity comparisons
When comparing two large or complex quantities, ask whether the underlying function is increasing or decreasing — this can settle “which is bigger” questions without computing either value exactly.
Pattern recognition in the choices
Sometimes the structure of the five choices itself is a clue — if four choices share a common factor and one doesn’t, or if the choices are clearly built from a specific formula, that structure can guide which approach to try first.
Applying it
Scenario: A problem states satisfies and asks for the value of among choices 1, 2, 3, 4, 5.
Response: Rather than factoring the cubic, test the middle choice : ✓ — done, with no factoring required. (In fact all of 1, 2, 3 are roots here, but backsolving found a valid answer immediately.)
Practice problems
1.A problem asks to solve 2x + 5 = 17 and gives choices 4, 6, 8, 10, 12. Using backsolving from the middle choice, what do you try first, and does it work?
2.You know a quantity must be between 10 and 20. The choices are 5, 15, 25, 35, 45. How many can you eliminate by bounding alone?
3.Without computing either value exactly, determine which is larger: 2^40 or 3^25, given both grow but you know 2^40 = (2^8)^5 ≈ 256^5 and 3^25 = (3^5)^5 = 243^5.
4.Why is starting backsolving from the middle answer choice usually better than starting from the first choice listed?