2007 AMC 10B Problem 5
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?
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Key insight
The facts stack up like circles inside circles: Crups inside Dramps inside Arogs inside Brafs. So every Crup is also an Arog and a Braf.
Solution
"All Crups are Dramps" means the group of Crups sits entirely inside the group of Dramps. Draw each fact as a smaller circle inside a bigger one.
1. All Crups are Dramps: Crups circle inside the Dramps circle.
2. All Dramps are Arogs: Dramps circle inside the Arogs circle.
3. All Arogs are Brafs: Arogs circle inside the Brafs circle.
4. Put these together. The circles nest from smallest to largest:
(The fact "all Crups are Brafs" is already built in.)
5. A Crup is in the smallest circle, so it is inside every bigger circle. It is an Arog and a Braf. That is choice (D).
6. Each other choice tries to put a bigger circle inside a smaller one: (A) says every Dramp is a Crup, (B) every Braf is a Crup, (C) every Arog is a Crup, (E) every Arog is a Dramp. The facts never say any of these; there could be a Braf that is nothing else.
The answer is
Why this works
"All are " only works in one direction: it puts inside , not the other way around. Once the circles are drawn, a true conclusion always moves from an inner circle to an outer one. Any statement that moves inward is not guaranteed.
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
Translate each "all are " into a subset relation :
Chaining them gives a single nested picture:
Now test the choices. The innermost set, Crups, is contained in everything, so "all Crups are Arogs and are Brafs" is certainly true.
The others each claim a larger set sits inside a smaller one: (A) Dramps inside Crups, (B) Brafs inside Crups, (C) Arogs inside Crups, (E) Arogs inside Dramps. None of these follow; the facts allow, for instance, a Braf that is nothing else.
The answer is
Review in the AoPS books
- Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.4)
- Prealgebra, Chapter 7: Ratios, Conversions, and Rates (§7.4)
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