AMC 10 · 2011 · #8

Grade 6 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination
if-then-reasoningcontrapositivelogical-deduction logical-deduction ↑ Prerequisites: if-then-reasoning
📏 Medium solution 💡 2 insights
Problem
A beach rule says: if it is at least 80°F and sunny, then the beach is crowded. On June 10, the beach was not crowded. Which statement about that day's weather must be true?

Pick an answer.

(A)
The temperature was cooler than } 80^{\circ} \text{F and it was not sunny.
(B)
The temperature was cooler than } 80^{\circ} \text{F or it was not sunny.
(C)
If the temperature was at least } 80^{\circ} \text{F, then it was sunny.
(D)
If the temperature was cooler than } 80^{\circ} \text{F, then it was sunny.
(E)
If the temperature was cooler than } 80^{\circ} \text{F, then it was not sunny.

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Change Focus / Count the Complement

The rule is a one-way if-then whose condition is a compound "hot AND sunny." Tool #11 (Work Backwards) turns the not-crowded fact into the only statement that must travel with the rule — its contrapositive: not crowded → not (hot and sunny). Tool #16 (Change Focus / Count the Complement) then does the load-bearing move: the complement of "hot AND sunny" is not "cool AND cloudy" but "cool OR cloudy" — negating an AND flips it to an OR (De Morgan). With that disjunction in hand, Tool #3 (Eliminate Possibilities) tests the five choices; only the correct negation survives while the AND-trap and the converse/inverse traps fall away.

1STEP 1

Write the rule in symbols

"At least 80°F" is T ≥ 80, and the rule fires only when BOTH parts hold: [(T ≥ 80) and sunny] → crowded — one way only.

[(T ≥ 80) and sunny] → crowded
2STEP 2

Reverse it: the contrapositive

Flip the rule to its contrapositive: no crowd → not (hot and sunny). That is the only new fact the rule forces.

¬ crowded → ¬[(T ≥ 80) and sunny]
3STEP 3

Negate the AND — it becomes an OR

A two-part AND breaks if just one part fails, so its negation is an OR: cooler than 80°F or not sunny.

¬[(T ≥ 80) and sunny] = (T ＜ 80) or (not sunny)
4STEP 4

Test the five choices

(A) demands AND, (C)(D)(E) invent links the rule never states; only the plain OR statement survives — (B).

(A) AND ✗, (C) ✗, (D) ✗, (E) ✗, (B) OR ✓ → (B)
Answer
The temperature was cooler than 80°F or it was not sunny.
Stress-test with concrete uncrowded days. A cool AND sunny day is not hot-and-sunny, so it can be uncrowded — it satisfies (B) via "cooler than 80°," but breaks (A) and (E), which demand "not sunny." A hot AND cloudy day can be uncrowded too — it satisfies (B) via "not sunny," but breaks (C). A cool AND cloudy day breaks (D). In every uncrowded case at least one part of (B) holds, while each other choice fails some case. This confirms (B).
💡Key takeaway

To break a "both A and B" rule you only need ONE part to fail, so "not (hot and sunny)" means cool OR cloudy — joined by OR, never AND.

  • Write the rule in symbols
  • Reverse it: the contrapositive
  • Negate the AND — it becomes an OR
  • Test the five choices

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