AMC 8 · 2012 · #3

Easy mode Grade 4

Problem

On February $13$ , the sun came up at $6:57\textsc{am}$ . That day, the sun was up for $10$ hours and $24$ minutes.

The newspaper printed a sunset time of $8:15\textsc{pm}$ , but that was wrong. The sunrise time and the length of daylight were both correct.

What time did the sun actually set?

Pick an answer.

(A)

5:10 PM

(B)

5:21 PM

(C)

5:41 PM

(D)

5:57 PM

(E)

6:03 PM

AMC 8 2012 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.

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Toolkit + CCSS Solution

Understand

Restated: On February 13, the newspaper reports daylight as $10$ hours $24$ minutes, sunrise at $6{:}57$ AM, and sunset at $8{:}15$ PM. The daylight length and sunrise are correct, but the printed sunset is wrong. Find the actual sunset time.

Givens: Sunrise time: $6{:}57$ AM (correct); Length of daylight: $10$ hours $24$ minutes (correct); Printed sunset: $8{:}15$ PM (wrong — ignore); Answer choices: (A) $5{:}10$ PM, (B) $5{:}21$ PM, (C) $5{:}41$ PM, (D) $5{:}57$ PM, (E) $6{:}03$ PM

Unknowns: The correct sunset time, written in $12$-hour PM format

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

Sunset = sunrise + daylight is a single sum, but it is cleaner to split it into two subproblems (Tool #7): add the hours, then add the minutes, then carry. After computing, Tool #3 (Eliminate Possibilities) lets us check the answer against the five choices — a free verification on any AMC multiple-choice problem. Tool #8 (Units) is implicitly used, but the bookkeeping here is hours vs. minutes, not a unit conversion across systems, so #7 captures the structure best.

Execute — Answer: B

#7 Identify Subproblems 4.MD.A.2 Step 1

Set up the time sum

Set up the relationship.
Daylight is the time from sunrise to sunset, so the correct sunset equals the correct sunrise plus the correct daylight length.
Ignore the printed (wrong) sunset.

$$\text{sunset} = 6{:}57 \text{ AM} + 10 \text{ h } 24 \text{ min}$$

💡 Picking the right relationship before computing is the first move of Tool #7 — name the subproblems before doing them.

#7 Identify Subproblems 4.NBT.B.4 Step 2

Add the hours

Subproblem 1: add the hours.
$6$ hours (from $6{:}57$) plus $10$ hours of daylight gives $16$ hours.

$$6 + 10 = 16 \text{ h}$$

💡 Treating hours as their own column is exactly the "add place by place" Grade 4 algorithm.

#7 Identify Subproblems 4.NBT.B.4 Step 3

Add the minutes

Subproblem 2: add the minutes.
$57$ minutes plus $24$ minutes is $81$ minutes.

$$57 + 24 = 81 \text{ min}$$

💡 Same place-value addition, this time in the minutes column.

#7 Identify Subproblems 4.MD.A.1 Step 4

Carry 60 minutes into an hour

Carry.
$81$ minutes is more than $60$, so peel off $60$ minutes as $1$ hour.
That leaves $21$ minutes and bumps the hour count from $16$ to $17$.

$81 \text{ min} = 60 \text{ min} + 21 \text{ min} = 1 \text{ h } 21 \text{ min}$, so total $= 17 \text{ h } 21 \text{ min}$

💡 Trading $60$ minutes for $1$ hour is the same idea as carrying $10$ ones into a ten — Grade 4 unit conversion within the time system.

#3 Eliminate Possibilities 4.MD.A.2 Step 5

Convert to PM time

Convert $17{:}21$ in $24$-hour format to PM time by subtracting $12$ from the hours.
Then check the answer choices: $5{:}21$ PM is choice (B).

$$17 - 12 = 5 \;\Rightarrow\; 17{:}21 = 5{:}21 \text{ PM} \;\Rightarrow\; \textbf{(B)}$$

💡 Matching the computed result against the five choices is Tool #3 (Eliminate Possibilities) — only (B) survives, the rest are off by $11$, $20$, $36$, or $42$ minutes.

[1] #7 4.MD.A.2 Set up the relationship. Daylight is the time from sunrise to sunset, so the cor

[2] #7 4.NBT.B.4 Subproblem 1: add the hours. $6$ hours (from $6{:}57$) plus $10$ hours of daylig

[3] #7 4.NBT.B.4 Subproblem 2: add the minutes. $57$ minutes plus $24$ minutes is $81$ minutes.

[4] #7 4.MD.A.1 Carry. $81$ minutes is more than $60$, so peel off $60$ minutes as $1$ hour. Tha

[5] #3 4.MD.A.2 Convert $17{:}21$ in $24$-hour format to PM time by subtracting $12$ from the ho

Review

Reasonableness: Quick sanity check: $10$ hours $24$ minutes after $6{:}57$ AM should land in the late afternoon. Round to $11$ hours after $7$ AM, that is roughly $6$ PM, so a sunset near $5$ to $6$ PM is plausible. The exact value $5{:}21$ PM sits in that window. Reverse-check: from $5{:}21$ PM back to $6{:}57$ AM, count from $6{:}57$ AM to $5{:}21$ PM as $5$ hours $3$ minutes to noon, plus $5$ hours $21$ minutes after noon, totaling $10$ hours $24$ minutes. Matches.

Alternative: Tool #11 (Work Backwards): start from each PM choice, subtract $10$ h $24$ min, and see which one gives $6{:}57$ AM. $5{:}21$ PM $-$ $10$ h $24$ min $=$ $6{:}57$ AM exactly, picking out (B). The other choices land at $6{:}46$, $7{:}17$, $7{:}33$, or $7{:}39$ AM — none match the given sunrise.

CCSS standards used (min grade 4)

4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money (Recognizing that sunset = sunrise + daylight and computing the elapsed-time word problem in hours and minutes.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm (Adding the hours ($6 + 10 = 16$) and the minutes ($57 + 24 = 81$) column by column before regrouping.)
4.MD.A.1 Know relative sizes of measurement units within one system; convert larger units to smaller (Trading $60$ minutes for $1$ hour to rewrite $16$ h $81$ min as $17$ h $21$ min, and converting $17{:}21$ from $24$-hour to $12$-hour PM format.)

⭐ If you can add hours, add minutes, and carry $60$ minutes as $1$ hour, you have all the Grade 4 skills this AMC 8 problem needs.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Set up the time sum

Add the hours

Add the minutes

Carry 60 minutes into an hour

Convert to PM time

Review

CCSS standards used (min grade 4)

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