AMC 10 · 2022 · #11

Grade 6 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

logical-deductionif-then-reasoningcontrapositiveset-partition caseworkidentify-subproblems ↑ Prerequisites: if-then-reasoninglogical-deduction

📏 Medium solution 💡 2 insights

Problem

We need to find the statement that means exactly the same thing as: "No school bigger than Euclid HS sold more T-shirts than Euclid HS." In other words, find the contrapositive — the equivalent way to say the same rule.

Pick an answer.

(A)

All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.

(B)

No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.

(C)

All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.

(D)

All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.

(E)

All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Tool #3 (Eliminate) is the textbook AMC tactic for multiple-choice equivalence: build a test situation that satisfies the original statement, then knock out any choice that disagrees. Tool #15 (Reorganize) rewrites the original as a clean If-Then: "If a school is bigger than Euclid, then it did NOT sell more than Euclid." That If-Then form makes its contrapositive easy to read: "If a school sold more than Euclid, then it is NOT bigger than Euclid" — exactly choice (B). Tool #1 (Draw) sketches a simple two-circle picture (size category, T-shirt category) so we can see which choice flips the arrow correctly.

1STEP 1

Rewrite the rule as If-Then: if a school is bigger than Euclid, then it did NOT sell more than Euclid.

If B → not M

2STEP 2

Flip to the contrapositive — negate both parts and swap them, giving if it sold more, then it is not bigger.

If M → not B

3STEP 3

Read it back as a sentence: a school that sold more than Euclid is not bigger than Euclid — that is choice (B).

(B): M → not B

4STEP 4

Eliminate the rest: (A) and (E) swap "not bigger" for "smaller," (C) forbids ties, and (D) is the converse — none is equivalent.

(A),(C),(D),(E) all fail

5STEP 5

Test a value: if Euclid sold 100, a bigger school may tie at 100 or fewer, so only (B) survives.

Big school sells 100 ≤ 100 ✓

Answer

No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.

The contrapositive of "If B then not M" is "If M then not B," which reads as "No more-seller is bigger than Euclid" — choice (B). Double-check with a borderline case: a school that sold the same number as Euclid is not a "more-seller," so the rule says nothing about its size, which matches (B). A school exactly the same size as Euclid is not "bigger," so it can sell any amount — again, (B) is silent there. Both statements agree on every possible situation, so (B) is genuinely equivalent.

💡Key takeaway

This AMC 10 problem only needs Grade 6 category logic you already know — flipping "If bigger, then not more" into its mirror form "If more, then not bigger" gives choice (B), and a quick test case knocks out the other four.

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