# Hat in the River

##### Jan 27, 2018

RAY: Imagine this: It's the first day of summer. You decide to rent a little rowboat at the dock and go for a ride. You begin to paddle upstream. Did I tell you you've got two oars in the water?

TOM: People say I don't have two oars in the water. What do they mean by that?

RAY: You're rowing upstream. You get a mile from where you put your boat in the water, and your hat flies off. You say, "Ah, forget it!" You keep rowing.

Suddenly, you realize that your two tickets to that night's Sleepy LaBeef concert are inside the sweatband of the hat. At that moment, when you realize you've also lost the tickets, you've rowed away from the hat for 10 minutes.

You turn around and you start rowing again, trying to get to the hat. You're rowing with the same intensity that you were when you were going upstream.

You catch up with the hat right at the point that you first put the boat in the water.

How fast is the current moving?

RAY: So, let me go over the salient points here.

TOM: Wow, yeah.

RAY: You go upstream for a mile.

TOM: A mile.

RAY: Against the current.

TOM: The current is working against you.

RAY: The hat falls off into the water. You say, forget it. You keep rowing for ten minutes. At which point, you remember the tickets.

TOM: You turn around.

RAY: You turn around, during which time, that hat is --

TOM: All this time, the hat's been going the other way.

RAY: Going the other way, baby.

TOM: Because the current is pushing it.

RAY: And you catch up with the hat right at the dock where you in fact rented the boat. So, the hat has actually traveled a mile.

TOM: Traveled one mile.

RAY: That's the only thing you really know.

TOM: Yeah, because you dropped it when you were a mile away from the dock, and you caught it at the dock. So the hat, I got that, the hat has traveled; that's a key point.

RAY: That's key, well, and good thing we gave you that information. So, the hat traveled a mile. So, we want to find out how fast the river's going. If we only knew how long the hat was traveling that mile, we could use the famous formula, distance = rate x time --

TOM: And we do know.

RAY: And we do know.

TOM: We do know.

RAY: Because imagine that there is no current, and your hat falls off into the water. If you row away from the hat for ten minutes, how long would it take you to get back to the hat? It would take you ten minutes.

TOM: Exactly.

RAY: Well, it turns out the current doesn't make any difference, because the same current that's pushing the hat downstream, once you've turned around, is pushing you downstream at the same rate. So, in fact, if you row away from the hat for ten minutes and then turn around when you've decided to go back and retrieve it, it takes you ten minutes to get back to the hat.

TOM: What was the question?

RAY: What row were those Sleepy La Beef tickets in?

TOM: The question was, how fast is the river flowing?

RAY: Right. You row away from the hat for ten minutes. It takes you ten minutes to get back to the hat. So, in fact, you've been rowing for 20 minutes, and the hat, been drifting for 20 minutes downstream, during which time it's gone a mile. So, in a third of an hour, it went a mile. The current must be going at three miles per hour.

TOM: Yeah, and that's all you really had to know, because you know that the hat has gone one mile in 20 minutes. You know that. That's it.

RAY: There you go.

TOM: And that's how fast, the answer is, the river is flowing at one mile in 20 minutes. No one says it's got to be in miles per hour. You didn't tell us that.