AMC 8 · 2022 · #9
Easy mode Grade 4Problem
Picture a cup of boiling water sitting in a room. The water is very hot, at 212∘F. The room stays at 68∘F the whole time.
The water cools down in a special way. Every 5 minutes, the gap between the water's temperature and the room's temperature shrinks to half of what it was.
What is the water's temperature after 15 minutes have passed?
Pick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A cup of water starts at $212^{\circ}\text{F}$ in a room that stays at $68^{\circ}\text{F}$. The gap between the water and the room cuts in half every $5$ minutes. What is the water's temperature after $15$ minutes?
Givens: Starting water temperature $= 212^{\circ}\text{F}$; Constant room temperature $= 68^{\circ}\text{F}$; The water-to-room temperature gap is halved every $5$ minutes; Elapsed time $= 15$ minutes; Answer choices: (A) $77$, (B) $86$, (C) $92$, (D) $98$, (E) $104$
Unknowns: The water temperature (in $^{\circ}\text{F}$) after $15$ minutes
Understand
Restated: A cup of water starts at $212^{\circ}\text{F}$ in a room that stays at $68^{\circ}\text{F}$. The gap between the water and the room cuts in half every $5$ minutes. What is the water's temperature after $15$ minutes?
Givens: Starting water temperature $= 212^{\circ}\text{F}$; Constant room temperature $= 68^{\circ}\text{F}$; The water-to-room temperature gap is halved every $5$ minutes; Elapsed time $= 15$ minutes; Answer choices: (A) $77$, (B) $86$, (C) $92$, (D) $98$, (E) $104$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #7 Identify Subproblems, #6 Guess and Check
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 "$D_n = D_0 \cdot (\tfrac{1}{2})^n$" 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.
Execute — Answer: B
4.NBT.B.4 Step 1 - Find the starting gap between the water and the room.
- This is the quantity that will get halved over time, so we want it on the table before anything else.
💡 Subtracting two multi-digit whole numbers ($212 - 68$) is a Grade 4 fluency skill.
3.OA.A.3 Step 2 - Count how many times the gap is halved.
- The gap halves every $5$ minutes, and $15$ minutes is three $5$-minute chunks, so the gap is halved exactly $3$ times.
💡 "How many groups of $5$ fit in $15$?" is a Grade 3 division word-problem move.
4.OA.C.5 Step 3 - Apply the halving rule three times.
- Make a tiny table — this is the pattern: at each row the gap is the previous gap divided by $2$.
💡 Generating a sequence from the explicit rule "divide by $2$" is exactly the Grade 4 "shape or number pattern from a given rule" standard.
4.NBT.B.4 Step 4 - Translate the final gap back into a water temperature.
- After $15$ minutes the water is $18^{\circ}\text{F}$ warmer than the room, and the room is still at $68^{\circ}\text{F}$.
💡 Adding the room temperature back to the gap is the final subproblem and a Grade 4 multi-digit addition.
4.NBT.B.4 Find the starting gap between the water and the room. This is the quantity that 3.OA.A.3 Count how many times the gap is halved. The gap halves every $5$ minutes, and $1 4.OA.C.5 Apply the halving rule three times. Make a tiny table — this is the pattern: at 4.NBT.B.4 Translate the final gap back into a water temperature. After $15$ minutes the wa Review
Reasonableness: Common-sense check: the water cools from $212^{\circ}\text{F}$ toward $68^{\circ}\text{F}$, 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 $86^{\circ}\text{F}$ 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.
Alternative: Tool #6 (Guess and Check) on the choices works cleanly. For each candidate water temperature $T$, the implied final gap is $T - 68$. After three halvings starting from $144$, the gap must be $18$, so we need $T - 68 = 18$, i.e. $T = 86$. The other choices give gaps of $9, 24, 30, 36$ — none of which is $144$ halved three times.
CCSS standards used (min grade 4)
4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Computing the initial gap $212 - 68 = 144$ and the final water temperature $68 + 18 = 86$.)3.OA.A.3Solve multiplication and division word problems within 100 (Dividing $15 \div 5 = 3$ to count how many five-minute halving intervals fit in the elapsed $15$ minutes.)4.OA.C.5Generate a number or shape pattern following a given rule (Generating the gap sequence $144, 72, 36, 18$ by applying the rule "divide by $2$" three times.)
⭐ This AMC 8 problem only needs Grade 4 pattern-making — halving a number a few times — that you already know!
⭐ This AMC 8 problem only needs Grade 4 pattern-making — halving a number a few times — that you already know!
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