River banks and boats...

8 viewed last edited 5 years ago

Two boats on opposite banks of a river start moving towards each other. They first pass each other 140 meters from one bank. They each continue to the opposite bank, immediately turn around and start back to the other bank. When they pass each other a second time, they are 60 meters from the other bank. We assume that each boat travels at a constant speed all along the journey. Find the width of the river.

Vivekanand Vellanki

First, lets draw a figure.

There are 3 variables: speed of boat 1, speed of boat 2, and width of the river.

Looks like we can write 2 equations. Given this, we cannot find all 3 variables - since we dont have 3 equations.

If we can reduce to 2 variables, we can solve for the 2 variables. Lets keep this in mind as we write down the equations.

Equation 1: Boat 1 covers a distance of (d - 140) while boat 2 covers a distance of 140. The time taken by both the boats is the same. This gives us the first equation.


Equation 2: Before the boats meet again, boat 1 covers 140 + (d - 60); and

boat 2 covers (d - 140) + 60

\frac{140\ +\ \left(d-60\right)}{s_1}=\frac{\left(d-140\right)\ +\ 60}{s_2}

That's it. We have 2 equations in 3 variables. We cannot solve for all 3. But, can we find d?

How about multiplying both equations by s1? We get

d-140=140\cdot\frac{s_1}{s_2} and


Replacing s1/s2 with r (relative speed) gives us 2 equations in 2 variables. Now, we can solve for d and r.

We can figure out the width of the river; and the relative speeds of the boats. But, we cannot figure out the actual speed of the individual boats.