AMC 8 · 2022 · #9

Grade 4 arithmeticalgebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

sequences-geometricexponentsfraction-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: fraction-arithmeticexponents

📏 Short solution 💡 2 insights

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Problem

A cup of water starts at 212^°F in a room that stays at 68^°F. The gap between the water and the room cuts in half every 5 minutes. What is the water's temperature after 15 minutes?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

104

AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The rule "halve the gap every 5 minutes" is a perfect Tool #5 (Pattern) setup: list the gap at 0, 5, 10, 15 minutes and a clean geometric pattern 144, 72, 36, 18 appears. Tool #7 (Identify Subproblems) splits the problem into three clean pieces — (a) find the starting gap, (b) halve it three times, (c) add the room temperature back to recover the water temperature. We avoid Tool #13 (Algebra) and any "D_n = D₀ · $(\frac{1}{2})$ ⁿ" formula on purpose: a bright elementary student can just halve 144 three times. Tool #6 (Guess and Check) is held in reserve as a verification pass against the multiple choices.

1STEP 1

Subtract to get the starting gap: the water is 144^°F above the room.

212 - 68 = 144 ^°F

2STEP 2

15 minutes is three 5-minute chunks, so the gap gets halved 3 times.

(15 min)/(5 min) = 3 halvings

3STEP 3

Follow the pattern, dividing by 2 each step: 144 → 72 → 36 → 18^°F.

144 → 72 → 36 → 18

4STEP 4

The room is still 68^°F, so add it back: 68 + 18 = 86^°F → (B).

68 + 18 = 86 ^°F → (B)

Answer

Common-sense check: the water cools from 212^°F toward 68^°F, so the answer must be strictly between those two values — and closer to 68 than to 212 after 15 minutes of fast cooling. Our answer 86^°F sits inside the interval (68, 212) and is much nearer the room temperature, which is exactly what "the gap halves three times" should produce. Choices (D) 98 and (E) 104 would only happen if the gap halved fewer times; (A) 77 would need an extra halving. Only (B) 86 matches three halvings.

💡Key takeaway

This AMC 8 problem only needs Grade 4 pattern-making — halving a number a few times — that you already know!

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