AMC 8 · 2012 · #2

Easy mode Grade 4

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Problem

In the town of East Westmore, a baby is born every $8$ hours. One person dies every day.

A day has $24$ hours, and a year has $365$ days. Figure out how many people are added to the town in one year.

Then round your answer to the nearest hundred.

Pick an answer.

(A)

600

(B)

700

(C)

800

(D)

900

(E)

1000

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: In East Westmore, a baby is born every $8$ hours and one person dies every day. Estimate the yearly population increase, rounded to the nearest hundred.

Givens: Birth rate $= 1$ birth every $8$ hours; Death rate $= 1$ death every day ($24$ hours); A standard year has $365$ days; Answer choices: (A) $600$, (B) $700$, (C) $800$, (D) $900$, (E) $1000$

Unknowns: The net yearly population increase, rounded to the nearest hundred

Understand

Restated: In East Westmore, a baby is born every $8$ hours and one person dies every day. Estimate the yearly population increase, rounded to the nearest hundred.

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

The rates are in mixed units — births per $8$ hours, deaths per day — and the answer wants people per year. Tool #8 (Analyze the Units) lines them up: convert the birth rate into births per day, then per year. Tool #7 (Identify Subproblems) splits the task into three clean pieces — births per day, net change per day, and net change per year — so each step is just one arithmetic move.

Execute — Answer: B

#8 Analyze the Units 4.MD.A.2 Step 1

Find births per day

Convert the birth rate to births per day.
Since there are $24$ hours in a day and one birth every $8$ hours, divide.

$$\dfrac{24 \text{ hr/day}}{8 \text{ hr/birth}} = 3 \text{ births/day}$$

💡 Dividing hours per day by hours per birth cancels "hours" and leaves "births per day" — a Grade 4 distance/time-style word-problem move.

#7 Identify Subproblems 4.OA.A.3 Step 2

Subtract deaths from births

Find the net daily population change by subtracting deaths from births.

$$3 \text{ births/day} - 1 \text{ death/day} = 2 \text{ people/day}$$

💡 Once both rates share the same time unit (per day), the change is a simple subtraction — Tool #7's subproblem move.

#8 Analyze the Units 4.NBT.B.5 Step 3

Scale up to one year

Scale the daily net change up to a year using $365$ days per year.

$$2 \text{ people/day} \times 365 \text{ days/year} = 730 \text{ people/year}$$

💡 Multiplying (people/day) by (days/year) cancels "day" and leaves "people/year" — multi-digit multiplication is Grade 4.

#8 Analyze the Units 4.NBT.A.3 Step 4

Round to the nearest hundred

Round $730$ to the nearest hundred.
The tens digit is $3 < 5$, so round down.

$$730 \approx 700 \;\Rightarrow\; \textbf{(B)}$$

💡 Rounding multi-digit whole numbers to any place is a Grade 4 number-and-operations standard.

[1] #8 4.MD.A.2 Convert the birth rate to births per day. Since there are $24$ hours in a day an

[2] #7 4.OA.A.3 Find the net daily population change by subtracting deaths from births.

[3] #8 4.NBT.B.5 Scale the daily net change up to a year using $365$ days per year.

[4] #8 4.NBT.A.3 Round $730$ to the nearest hundred. The tens digit is $3 < 5$, so round down.

Review

Reasonableness: Three births a day minus one death a day is a net gain of $2$ people a day. Two a day for roughly $365$ days is about $730$, which sits squarely between $700$ and $800$ but closer to $700$. Choice (B) $700$ matches; the other choices ($600$, $800$, $900$, $1000$) are too far from $730$ to round there.

Alternative: Tool #6 (Guess and Check) on the choices: if the answer were (C) $800$, the net daily gain would be about $\frac{800}{365} \approx 2.19$, which would require more than $3$ births per day. With exactly $3$ births and $1$ death per day, the gain is exactly $2$/day, so the yearly total $2 \times 365 = 730$ rounds to $700$ — only choice (B) fits.

CCSS standards used (min grade 4)

4.MD.A.2 Solve word problems involving distances, time, liquid volumes, and money (Converting the birth rate from "$1$ birth per $8$ hours" into "$3$ births per day" using the $24$-hour day.)
4.OA.A.3 Solve multistep word problems with whole numbers using the four operations (Combining births and deaths into the net daily change ($3 - 1 = 2$ people/day) inside a multi-step word problem.)
4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number (Computing the yearly total $2 \times 365 = 730$ to scale the daily change up to one year.)
4.NBT.A.3 Use place value understanding to round multi-digit whole numbers to any place (Rounding $730$ to the nearest hundred to match the form of the answer choices.)

⭐ This AMC 8 problem only needs Grade 4 skills: convert a rate, subtract, multiply, and round!

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Find births per day

Subtract deaths from births

Scale up to one year

Round to the nearest hundred

Review

CCSS standards used (min grade 4)

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