Competition · AMC preparation · step 4 of 4
AMC 8 · 2022 · #9
Grade 4 arithmeticalgebraPick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 · (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.
Find the starting gap
Subtract to get the starting gap: the water is 144^°F above the room.
Subtracting two multi-digit whole numbers (212 - 68) is a Grade 4 fluency skill.
4.NBT.B.4Identify SubproblemsCount the halvings
15 minutes is three 5-minute chunks, so the gap gets halved 3 times.
"How many groups of 5 fit in 15?" is a Grade 3 division word-problem move.
3.OA.A.3Identify SubproblemsHalve the gap three times
Follow the pattern, dividing by 2 each step: 144 → 72 → 36 → 18^°F.
Generating a sequence from the explicit rule "divide by 2" is exactly the Grade 4 "shape or number pattern from a given rule" standard.
After 15 minutes the gap between the water and the room has shrunk from 144^°F to 18^°F, because that starting gap is halved once for each of the three 5-minute intervals.
▸ Why?
The gap is halved once every 5 minutes, and 15 minutes is made of three 5-minute intervals, so the starting gap is halved exactly three times.
▸ Why?
Counting how many 5-minute intervals fill 15 minutes means seeing 15 as three equal groups of 5.
▸ Why?
Halving the 144^°F gap three times steps it down 144 → 72 → 36 → 18, each new gap being half of the one before it.
▸ Why?
To halve a gap you split it into two equal parts and keep one, so 144 splits into two 72s, 72 into two 36s, and 36 into two 18s.
Turn the gap back into a temperature
The room is still 68^°F, so add it back: 68 + 18 = 86^°F → (B).
Adding the room temperature back to the gap is the final subproblem and a Grade 4 multi-digit addition.
4.NBT.B.4Identify SubproblemsThis AMC 8 problem only needs Grade 4 pattern-making — halving a number a few times — that you already know!
- Find the starting gap
- Count the halvings
- Halve the gap three times
- Turn the gap back into a temperature
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