2011 AMC 10B Problem 8

P1-10~2 minPrint

Logic Puzzles Set up the equation/formula and compute; no special trick needed

At a certain beach if it is at least $80^{\circ} F$ and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?

Show solution without answering

Key insight

Hot and sunny always means crowded. So if the beach was not crowded, the day was not both hot and sunny: at least one of the two was missing.

Solution

The rule says: if it is at least $80^\circ$ AND sunny, the beach is crowded.
Suppose June 10 had been at least $80^\circ$ and sunny. Then the rule says the beach would be crowded. But it was not crowded. So June 10 was NOT both hot and sunny.
"Not both hot and sunny" means at least one of them was missing: it was cooler than $80^\circ$ , or it was not sunny, or both.
That is exactly choice (B).
Why (A) is wrong: a cool but sunny day is not hot and sunny, so the rule says nothing about it, and the beach could be empty. (A) would say that day was not sunny, which is false.
Choices (C), (D) and (E) are new "if-then" rules that the given rule never says.

The answer is $\boxed{\textbf{(B)}\ \text{cooler than } 80^{\circ}\text{ F or not sunny}}$ .

Why this works

If a rule says "when A and B both happen, C happens," and C did not happen, then A and B did not both happen. You only learn that at least one of them failed, not which one. The word "and" turns into "or" when you say what went wrong.

Alternative approach

Test real days where the beach was empty. A day at $70^\circ$ and sunny could have an empty beach; it breaks (A) (it was sunny) and (E) (cool but sunny). A day at $70^\circ$ and cloudy could also have an empty beach; it breaks (D). A day at $85^\circ$ and cloudy could have an empty beach; it breaks (C). Choice (B) is true on all of these days, so (B) is the answer.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The rule is "(at least $80^\circ$ AND sunny) $\Rightarrow$ crowded." An implication is equivalent to its contrapositive: "not crowded $\Rightarrow$ NOT (at least $80^\circ$ AND sunny)."

Negating an "and" statement gives an "or" statement: at least one of the two conditions failed. "Not at least $80^\circ$ " means cooler than $80^\circ$ . So on June 10, the temperature was cooler than $80^\circ$ F or it was not sunny (possibly both).

Choice (A) is too strong: a cool sunny day is consistent with an empty beach. Choices (C), (D), (E) are conditionals not implied by the rule.

The answer is $\boxed{\textbf{(B)}\ \text{cooler than } 80^{\circ}\text{ F or not sunny}}$ .

Review in the AoPS books

Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.4)