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AMC 8 · 2012 · #2

Grade 4 arithmetic
Archetype Arithmetic Expression Decomposition
rateunit-conversionestimation identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticrate
📏 Medium solution 💡 3 insights
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Problem

In the country of East Westmore, statisticians estimate there is a baby born every 88 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?

Pick an answer.

(A)
$hspace{.05in}600$
(B)
$hspace{.05in}700$
(C)
$hspace{.05in}800$
(D)
$hspace{.05in}900$
(E)
$hspace{.05in}1000$
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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.

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) $1000$

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
  • 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

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 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 $730$ 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 $800/365 \approx 2.19$, 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!

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

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