AMC 10 · 2008 · #11

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateunit-conversionoptimization dimensional-analysisextremal-construction ↑ Prerequisites: rate

📏 Short solution 💡 2 insights

Problem

A boat is 1 mile from shore. Water leaks in at 10 gallons per minute. The boat sinks if it ever takes in more than 30 gallons. Steve rows toward shore at 4 miles per hour while LeRoy bails water out. Find the slowest steady rate, in gallons per minute, at which LeRoy can bail so the boat reaches shore without sinking.

Pick an answer.

(A)

$\ 2$

(B)

$\ 4$

(C)

$\ 6$

(D)

$\ 8$

(E)

$\ 10$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

The numbers come in different units: miles, miles per hour, gallons, and gallons per minute. Tool #8 (Analyze the Units) says line the units up first — turn the trip into a number of minutes, then everything can be measured in gallons per minute. Because the question asks for the slowest safe rate, Tool #14 (Extreme Principle) points to the boundary case where the water builds up to exactly the 30-gallon limit and not a drop more. Tool #4 (Introduce a Variable) names the bailing rate so the balance between water coming in and going out becomes one clean inequality.

1STEP 1

Find how long the trip takes

Time is distance over speed: 1 mile at 4 mph takes $\frac{1}{4}$ hour, so the trip lasts 15 minutes.

t = (1 mi)/(4 mi/hr) = 1/4 hr = 15 min

2STEP 2

Set the largest net gain that stays safe

Spread the 30-gallon budget over those 15 minutes: the water may gain at most 2 gallons per minute on net.

max net rate = (30 gal)/(15 min) = 2 gal/min

3STEP 3

Balance water in against water out

Let b be the bailing rate. Net gain $10-b$ must be at most 2, so $b\ge 8$ . The slowest safe rate is 8 gallons per minute, choice (D).

10 - b ≤ 2 → b ≥ 8 → b_min = 8 (D)

Answer

Check the boundary directly: bailing 8 gal/min against a 10 gal/min leak leaves a net gain of 2 gal/min, and over 15 minutes that is 2 × 15 = 30 gallons — exactly the limit, so 8 is safe and nothing slower is. A slower rate like 6 would let 4 gal/min build up, reaching 60 gallons and sinking the boat, so smaller choices (A), (B), (C) fail; 10 would keep the boat bone dry but is more bailing than needed, so (E) is not the slowest.

💡Key takeaway

Turn everything into gallons-per-minute over the 15-minute trip: the boat can only gain 2 of the 10 gallons leaking in each minute, so LeRoy must bail at least 8.

Find how long the trip takes
Set the largest net gain that stays safe
Balance water in against water out

Find how long the trip takes

Set the largest net gain that stays safe

Balance water in against water out

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