AMC 10 · 2022 · #11

Grade 6 arithmetic

Archetype Small-Domain Constrained Search

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

📏 Medium solution 💡 2 insights

Problem

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?

$\textbf{(A) }$ All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.

$\textbf{(B) }$ No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.

$\textbf{(C) }$ All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.

$\textbf{(D) }$ All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.

$\textbf{(E) }$ All schools smaller than Euclid HS sold more T-shirts than Euclid HS.

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.

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Every high school in the district sold T-shirts; Original claim: "No school bigger than Euclid HS sold more T-shirts than Euclid HS"; Five answer choices, each a different statement about size vs. T-shirt count

Unknowns: Which choice (A)-(E) is logically equivalent to the original

Understand

Plan

Primary tool: #3 Eliminate Possibilities

Secondary: #15 Reorganize Information, #1 Draw a Diagram

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.

Execute — Answer: B

#15 Reorganize Information 5.G.B.3 Step 1

Rewrite the original sentence in If-Then form.
"No school bigger than Euclid sold more than Euclid" means: every bigger school failed to outsell Euclid.
So If (a school is bigger than Euclid) Then (it did NOT sell more than Euclid).

$$\text{If } B \Rightarrow \text{not } M$$

💡 Grade 5: rewriting a category rule as If-Then is like saying "every member of the bigger group belongs to the not-more group."

#15 Reorganize Information 5.G.B.3 Step 2

Flip to the contrapositive.
The contrapositive of "If $B$ then not $M$" is "If $M$ then not $B$." Negate both parts and swap the order.
The contrapositive is always equivalent to the original.

$$\text{If } M \Rightarrow \text{not } B$$

💡 Grade 5 hierarchy logic: if every $B$ lives in not-$M$, then every $M$ must live in not-$B$ — same rule, viewed from the other side.

#1 Draw a Diagram 5.G.B.3 Step 3

Translate the contrapositive back to English: "If a school sold more T-shirts than Euclid, then it is NOT bigger than Euclid." That is exactly choice (B): "No school that sold more T-shirts than Euclid HS is bigger than Euclid HS."

$$\text{(B): } M \Rightarrow \text{not } B$$

💡 Draw two boxes (bigger schools, more-sellers); the rule forces them to never overlap. Choice (B) says exactly that.

#3 Eliminate Possibilities 5.G.B.3 Step 4

Eliminate the other choices by spotting a hidden flip.
(A) talks about "smaller" schools, but "not bigger" includes same-size — not the same.
(C) says bigger schools sold fewer, but the rule allows them to tie Euclid — not just fewer.
(D) reverses size and sales (converse, not contrapositive — not equivalent).
(E) is the opposite of the rule.

$$\text{(A),(C),(D),(E) all fail}$$

💡 Grade 5: testing each choice against "could a same-size school exist?" knocks four of them out.

#3 Eliminate Possibilities 6.EE.B.5 Step 5

Confirm with a tiny example.
Suppose Euclid sold 100 shirts.
The rule allows: a bigger school sold 100 (tie) or fewer; any school that sold 101 must be same size or smaller.
Choice (B) matches this perfectly; the other choices break it.

$$\text{Big school sells } 100 \le 100 \;\checkmark$$

💡 Grade 6: pick a value and check which statement still holds — only (B) survives.

[1] #15 5.G.B.3 Rewrite the original sentence in If-Then form. "No school bigger than Euclid sol

[2] #15 5.G.B.3 Flip to the contrapositive. The contrapositive of "If $B$ then not $M$" is "If $

[3] #1 5.G.B.3 Translate the contrapositive back to English: "If a school sold more T-shirts th

[4] #3 5.G.B.3 Eliminate the other choices by spotting a hidden flip. (A) talks about "smaller"

[5] #3 6.EE.B.5 Confirm with a tiny example. Suppose Euclid sold 100 shirts. The rule allows: a

Review

Reasonableness: 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.

Alternative: Tool #4 (Matrix Logic): build a $2 \times 2$ table with rows {bigger, not bigger} and columns {more, not more}. The original rule forbids the (bigger, more) cell. Choice (B) forbids the (more, bigger) cell — the same cell, just renamed. (A), (C), (D), (E) each forbid different cells, so they say different things.

CCSS standards used (min grade 6)

5.G.B.3 Understand that attributes belonging to a category apply to all subcategories (Reading the original rule as a category claim (the bigger-than-Euclid group is contained in the not-more-than-Euclid group) and flipping to its contrapositive form.)
6.EE.B.5 Understand solving an equation or inequality as a process of finding values (Testing each candidate statement against a sample situation (Euclid sells 100) to confirm only choice (B) survives.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

More like this