Dances runs around a track in sixty seconds. Wolves, running in the opposite direction around the track, meets Dances every twenty seconds. How long does it take for Wolves to run around the track?

-Problem modified from UNL Math Day

Let the speed of Dances be ‘u’ units/sec. and that of Wolves be ‘v’ units/sec.

Now, the circumference of the track is ‘2(pi)r’ units, where ‘r’ is the radius of the track and ‘(pi)’ as we know is the ratio of the circumference to the diameter of the circular track.

So, according to the problem, we see that Dances runs around the track in 60 seconds.
Therefore, we have:

2(pi)r = 60u --------------(1)

Now, since both Dances and Wolves are running in opposite directions, their relative speed is (u+v) units/sec.

Therefore, once again, from the problem we have:

[2(pi)r]/(u+v) = 20 ----------------(2)
Substituting the value of 2(pi)r = 60u from equation (1) in equation (2), we have:

u + v = 3u
or, v = 2u

That is, the speed of Wolves is twice the speed of Dances, therefore the Wolves will run around the rack in half the time in which Dances runs around the track.

Thus, it takes 30 seconds for Wolves to run around the track.

Posted by Ravi Raja on 2002-12-24 20:03:51