AMC 10 · 2007 · #5

Grade 5 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

logical-deductionif-then-reasoning if-then-reasoning ↑ Prerequisites: logical-deduction

📏 Medium solution 💡 2 insights

Problem

In a certain land there are four kinds of things: Arogs, Brafs, Crups, and Dramps. The rules are that every Arog is a Braf, every Crup is a Braf, every Dramp is an Arog, and every Crup is a Dramp. Decide which of the five statements must always be true because of these rules.

Pick an answer.

(A)

All Dramps are Brafs and are Crups.

(B)

All Brafs are Crups and are Dramps.

(C)

All Arogs are Crups and are Dramps.

(D)

All Crups are Arogs and are Brafs.

(E)

All Arogs are Dramps and some Arogs may not be Crups.

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Venn Diagram

Each rule "all X are Y" says the X group sits completely inside the Y group, so draw one circle inside another. Once the circles are nested correctly, every answer choice can be read straight off the picture, and choices that reverse a containment can be eliminated.

1STEP 1

Turn each rule into a circle inside a circle

"All X are Y" means the X circle sits entirely inside the Y circle, so draw Arog inside Braf, Dramp inside Arog, and Crup inside Dramp.

2STEP 2

Stack the circles into one chain

Chain the nestings from the inside out and the four rules collapse into one stack: Crups inside Dramps inside Arogs inside Brafs.

Crups ⊆ Dramps ⊆ Arogs ⊆ Brafs

3STEP 3

Test every choice against the stack

Choices (A), (B), (C), (E) all reverse the nesting, forcing an outer circle inside a smaller one. Only (D) runs outward, so it holds.

Answer

All Crups are Arogs and are Brafs.

Swap the names for familiar nested groups: Crups = squares, Dramps = rhombi... any concrete stack works. With Crups inside Dramps inside Arogs inside Brafs, the only safe claims run from an inner group to an outer one. "All Crups are Arogs and Brafs" runs outward from the innermost circle, so it must hold, while every other choice tries to force an outer group into an inner one and breaks.

💡Key takeaway

Turn each "all X are Y" into a circle inside a circle, then only trust claims that go from a smaller circle out to a bigger one.

Turn each rule into a circle inside a circle
Stack the circles into one chain
Test every choice against the stack

Turn each rule into a circle inside a circle

Stack the circles into one chain

Test every choice against the stack

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