Can you solve this math problem without algebra?

This is something of a rant.

I came across this problem in a book I am using for tutoring. The solution given is really ugly. I can see teaching the use of algebra for these types of problems, but the intuitive solution should be included as well.

A person runs two laps on a track. The first lap is done at 5 mph and the second at 4 mph. If the total time for the two laps is 54 minutes, how long did it take to run each of the two laps?

Solution given by book.
Let t = time for five minute lap. Note that 54 minutes is .9 hours.
The person covers 5t for the first lap and 4(t – .9) for the second. Since the two laps have the same distance, we get
5t = 4(.9 – t)
9t = 3.6
t = .4 hour = .4 x 60 = 24 minutes.
I don’t have the book with me. I don’t recall if they evaluated (.9 – t) for the second lap (I sure hope not). The second lap is obviously 54 – 24 = 30 minutes.

YUK!!

This can be done without algebra and without conversion between minutes and hours. The numbers work out simply enough that it can be done in your head. Do you see how? While you are at it, figure out the length of the track.