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2008 AMC 10A Problem 11

2008 AMC 10A Problem 11AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 2 min · position P11-15

While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing toward the shore at a constant rate of 4 miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?

A)
$2$
B)
$4$
C)
$6$
D)
$8$
E)
$10$

Answer

Key insight

The trip takes 15 minutes, so the net inflow may be at most 30/15 = 2 gallons per minute, meaning LeRoy bails at least 10 - 2 = 8.

Solution

First find how long they are in the water. One mile at $4$ miles per hour takes $\frac{1}{4}$ hour, which is $15$ minutes.

During those $15$ minutes the boat may accumulate at most $30$ gallons, so the net rate of water coming in can be at most
$\frac{30}{15} = 2\ \text{gallons per minute}.$

Water enters at $10$ gallons per minute, so LeRoy must remove at least $10 - 2 = 8$ gallons per minute.

The answer is $\boxed{\textbf{(D)}\ 8}$ .

Why this works

Two rates compete: water in and water out. Only their difference matters, and the constraint is on the total accumulated over a fixed time, so convert the total into a maximum net rate. The one real step is computing that time in the same units (minutes) as the rates.

The trap

Mixing units: treating 4 miles per hour as 4 minutes, or 1 mile at 4 mph as 4 minutes, instead of the correct 15 minutes.

Common mistakes

Mixing units: treating 4 miles per hour as 4 minutes, or 1 mile at 4 mph as 4 minutes, instead of the correct 15 minutes.
Reporting the allowed net inflow $2$ (choice A) instead of the bailing rate $10 - 2$ .

Techniques

Set up the equation/formula and compute; no special trick needed