AMC 10 · 2017 · #3
Grade 6 number-theoryPick an answer.
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
The promise runs one way only.
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
Flipping and negating gives the contrapositive.
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 that did the thing and missed the reward would break the promise outright.
▸ Why?
With that case ruled out, the only reading left is that the thing was not done.
Negate "all right" the careful way
Negating all correct gives at least one wrong.
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
Only one option is that statement, choice (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