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, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: Four kinds of things follow nested rules: 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.

Givens: All Arogs are Brafs.; All Crups are Brafs.; All Dramps are Arogs.; All Crups are Dramps.

Unknowns: Which answer choice is forced to be true by the four rules

Understand

Givens: All Arogs are Brafs.; All Crups are Brafs.; All Dramps are Arogs.; All Crups are Dramps.

Plan

Primary tool: #12 Draw a Venn Diagram

Secondary: #3 Eliminate Possibilities

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.

Execute — Answer: D

#12 Draw a Venn Diagram 5.G.B.4 Step 1

Turn each rule into a circle inside a circle

"All X are Y" means the whole X circle fits inside the Y circle.
So: the Arog circle sits inside the Braf circle; the Crup circle sits inside the Braf circle; the Dramp circle sits inside the Arog circle; and the Crup circle sits inside the Dramp circle.

💡 A containment rule is just one circle swallowed by a bigger one.

#12 Draw a Venn Diagram 5.G.B.3 Step 2

Stack the circles into one chain

Follow the nesting from the smallest group out.
Crups are inside Dramps, Dramps are inside Arogs, and Arogs are inside Brafs.
That gives one clean stack: Crups inside Dramps inside Arogs inside Brafs.
(The rule "all Crups are Brafs" is already guaranteed by this stack.)

$$\text{Crups} \subseteq \text{Dramps} \subseteq \text{Arogs} \subseteq \text{Brafs}$$

💡 If each group hides inside the next, everything a small group belongs to, a bigger surrounding group belongs to too.

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

Test every choice against the stack

Read each claim off the stack Crups inside Dramps inside Arogs inside Brafs.
(A) says all Dramps are Crups, but Crups is the smaller inner circle, so this reverses the containment and fails.
(B) says all Brafs are Crups and Dramps, reversing the whole stack, so it fails.
(C) says all Arogs are Crups and Dramps, again pulling a big circle into small ones, so it fails.
(E) says all Arogs are Dramps, but Arogs is the larger circle, so it fails.
(D) says all Crups are Arogs and are Brafs: Crups sits inside Dramps inside Arogs inside Brafs, so every Crup really is an Arog and a Braf.
Only (D) matches the picture, so the answer is (D).

💡 A true claim only ever moves outward in the stack, from a smaller circle to a larger one.

[1] #12 5.G.B.4 "All X are Y" means the whole X circle fits inside the Y circle. So: the Arog ci

[2] #12 5.G.B.3 Follow the nesting from the smallest group out. Crups are inside Dramps, Dramps

[3] #3 5.G.B.4 Read each claim off the stack Crups inside Dramps inside Arogs inside Brafs. (A)

Review

Reasonableness: 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.

Alternative: Instead of a diagram, chain the rules symbolically: all Crups are Dramps, all Dramps are Arogs, so all Crups are Arogs; and all Arogs are Brafs, so all Crups are Brafs. That transitive chain lands directly on choice (D) without drawing anything.

CCSS standards used (min grade 5)

5.G.B.3 Understand that attributes belonging to a category apply to all subcategories (Recognizing that once a small group sits inside a larger one, membership in the larger group is inherited automatically.)
5.G.B.4 Classify two-dimensional figures in a hierarchy based on properties (Arranging Crups, Dramps, Arogs, and Brafs into a single nested hierarchy and reading valid claims off it.)

⭐ 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.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Turn each rule into a circle inside a circle

Stack the circles into one chain

Test every choice against the stack

Review

CCSS standards used (min grade 5)

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