Word problem

Question

A canoe takes one hour to make a  up river and 45mins to make the return trip down river. If the river flows at 1mile per hour, what is the speed of t canoe in still water? How long is trip? 

Answer 1

When speeds and times are involved, the most useful equation "d = rt," which means "distance = (rate)(time)."

 

First, make an equation for the upstream scenario. I'll use subscripts of "u" to mean "upstream."

 

D_u = r_ut_u

 

Imagine walking or biking uphill. It's a lot harder than going downhill. Likewise, it's more difficult to go upstream, and therefore, the person in the canoe will go slower. The current is going against the direction of the canoe. This is like trying to walk in the opposite direction of a moving escalator, and then getting in trouble with mall security.

 

When traveling against the direction of the current, because the canoe person's moving slower, the speed will be subtracted. Let's call this person Mr. Canoe. Let's call Mr. Canoe's boat speed "c." He can usually row at a speed of c, but with the current pushing against him, his speed going upstream would be "c - 1," because the river flows at 1 mile per hour. Therefore r_u = c - 1. The upstream trip takes an hour, so t_u = 1.

 

D_u = r_ut_u

D_u = (c - 1)(1)

 

Now it's time to make an equation for the downstream case. Biking downhill is a lot easier. When going in the same speed of the stream, Mr. Canoe's speed will add up with the stream to make him super fast. His speed for the downstream case will be r_d = c + 1. It says his downstream trip takes 45 minutes. Wait, the rate unit is in miles per hour, so time must also be in hours.

 

There are 60 minutes in an hour, so 45 minutes is 45/60 of an hour. That fraction reduces to 3/4 of an hour, so t_d = 3/4.

 

D_d = r_dt_d

D_d = (c + 1)(3/4)

 

Find the relationship between the two equations. Because he's taking a round trip, the path that he takes upstream is the exact same path he takes downstream. The upstream distance and downstream distances are the same. It's safe to set the two equations equal to each other.

 

D_u = (c - 1)(1)

D_d = (c + 1)(3/4)

D_u = D_d

(c - 1)(1) = (c + 1)(3/4)

c - 1 = 3c/4 + 3/4

c/4 = 7/4

c = 7

 

Because c = 7, this mean's Mr. Canoe's speed is 7 miles per hour in still water. Find out how long the total trip was by adding together the upstream time, which was 1 hour, and the downstream time, which was 45 minutes. The whole trip was 1 hour, and 45 minutes.