AMC 10 · 2017 · #6
Grade 6 number-theoryPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The promise is a one-way if-then: "all right → A." Tool #11 (Work Backwards) builds the one statement that must travel with it — the contrapositive, which flips the arrow and negates both sides. Tool #16 (Count the Complement) handles the tricky negation: the opposite of "all right" is "at least one wrong," not "all wrong." With the contrapositive in hand, Tool #3 (Eliminate Possibilities) tests all five choices against it; only the true contrapositive survives, and the converse/inverse traps fall away.
Write the promise as one-way if-then
One way only: all right leads to an A. It never says an A comes only that way, so "A means all right" reads in too much.
A promise "do X and you get Y" guarantees Y only when X happens; it never claims X is the only road to Y.
6.EE.B.5Change Focus Count The ComplementFlip and negate: the contrapositive
Every if-then has one guaranteed partner — its contrapositive — made by reversing the arrow and negating both sides: not A → not all right.
If doing X always earns Y, then missing Y proves you didn't do X — same fact, read in reverse.
If doing something always earns a reward, then missing the reward proves it was not done.
▸ Why?
A single case where the promised reward fails is enough to rule out the condition holding.
▸ Why?
The promise runs one way only, so it must be read backwards, never forwards.
Negate "all right" the careful way
"Not all right" means at least one wrong, not all wrong — a single missed question already breaks "all right."
To break "every one is right" you only need a single wrong answer, not a whole sheet of them.
6.EE.B.5Change Focus Count The ComplementTest all five, keep the contrapositive
Test all five: (A) too strong, (C) inverse, (D)/(E) backward — only the true contrapositive survives, so the answer is (B).
Only the contrapositive is logically forced; the converse and inverse are the classic traps that look right but aren't.
6.EE.B.5Eliminate PossibilitiesAn "if you do X you get Y" promise only forces one extra fact — its contrapositive: "no Y means you didn't do X" — and remember the opposite of "all right" is just "at least one wrong," not "all wrong."
- Write the promise as one-way if-then
- Flip and negate: the contrapositive
- Negate "all right" the careful way
- Test all five, keep the contrapositive