# Difference between revisions of "2011 AMC 12A Problems/Problem 12"

## Problem

A power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock $A.$ How many hours did it take the power boat to go from $A$ to $B$?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 3.5 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 4.5 \qquad \textbf{(E)}\ 5$

## Solution

### Solution 1

Since the speed of the river is not specified, the outcome of the problem must be independent of this speed. We may thus trivially assume that the river has a speed of $0$. In this case, when the powerboat travels from $A$ to $B$, the raft remains at $A$. Thus the trip from $A$ to $B$ takes the same time as the trip from $B$ to the raft. Since these times are equal and sum to $9$ hours, the trip from $A$ to $B$ must take half this time, or $4.5$ hours. The answer is thus $\boxed{\textbf{D}}$.

### Solution 2

What's important in this problem is to consider everything in terms of the power boat and the raft, since that is how the problem is given to us. Let us denote the speed of the boat as $b$ and the speed of the raft at $r$.

$[asy] size(8cm,8cm); pair A, B; A=(-3,4); B=(3,4); draw(A--B); label("A",A,S); label("B",B,S); pair A, B; dot((-3,3),blue); A=(-3,5); B=(3,5); draw(A--B); label("A",A,S); label("B",B,S); pair A, B; A=(-3,6); B=(3,6); draw(A--B); label("A",A,S); label("B",B,S);[/asy]$