AMC 8 · 2012 · #2

Grade 4 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateunit-conversionestimation identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticrate

📏 Medium solution 💡 3 insights

Problem

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.

Pick an answer.

(A)

$hspace{.05in}600$

(B)

$hspace{.05in}700$

(C)

$hspace{.05in}800$

(D)

$hspace{.05in}900$

(E)

$hspace{.05in}1000$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

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.

1STEP 1

A birth every 8 hours means 24 ÷ 8 = 3 births per day.

(24 hr/day)/(8 hr/birth) = 3 births/day

2STEP 2

Now both rates are per day, so subtract: 3 - 1 = 2 people per day.

3 births/day - 1 death/day = 2 people/day

3STEP 3

Scale the daily gain to a year: 2 × 365 = 730 people per year.

2 people/day × 365 days/year = 730 people/year

4STEP 4

The tens digit of 730 is 3, below 5, so 730 rounds down to 700.

730 ≈ 700 → (B)

Answer

hspace{.05in}700

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.

💡Key takeaway

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

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