Competition · AMC preparation · step 4 of 4
AMC 10 · 2021A · #7
Grade 5 logicPick an answer.
AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Five answer choices, all multiple-choice — Tool #3 (Eliminate Possibilities) tests each in turn against the three given facts. Tool #1 (Draw a Diagram) sketches the facts as arrows between properties so the chain is visible. Tool #7 (Identify Subproblems) splits the work: first chain facts 2 and 3, then bring in the contrapositive of fact 1, then read off which arrow lands on the answer.
Turn each fact into an arrow
Turn each 'if-then' fact into an arrow between property boxes: happy → add, purple → not-subtract, not-subtract → not-add.
An if-then statement is just an arrow from one property box to another.
5.G.B.3Draw A DiagramChain the purple facts
Chain facts 2 and 3: purple → not-subtract → not-add, so every purple snake cannot add.
Arrow A→B and arrow B→C give a combined arrow A→C — like a transitive chain.
An arrow from one property to a second and on to a third gives a combined arrow straight through.
▸ Why?
Everything in the first group inherits every property the second group is stated to have.
▸ Why?
Those containments chain together, so a claim travels the whole way along the stack.
Reverse the first arrow
Read fact 1 (happy → add) backwards — its contrapositive is not-add → not-happy.
"All A are B" and "anything that is not B is not A" say the exact same thing.
5.G.B.3Identify SubproblemsChain the two results
Chain Steps 2 and 3: purple → not-happy; its contrapositive is happy → not-purple.
Combine two arrows then flip with the contrapositive to land on the answer's phrasing.
5.G.B.3Identify SubproblemsMatch against the choices
Test the choices against the chain: only (D) — happy → not-purple — is forced; (A) is the opposite, and (B), (C), (E) aren't forced.
Walk down the five options and keep only the one that the chain forces.
5.G.B.3Eliminate PossibilitiesThis AMC 10 problem only needs Grade 5 category reasoning you already know — turn each "all A are B" into an arrow, chain purple → not-subtract → not-add and flip happy → can-add to its contrapositive, and the chain gives happy → not-purple, which is choice (D).
- Turn each fact into an arrow
- Chain the purple facts
- Reverse the first arrow
- Chain the two results
- Match against the choices
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