Intermath: Investigations: Number Concept: Ratio

A tortoise and a hare start at the same time in a 10-mile race. The hare runs for 20 minutes at 12 mph; stops to rest; falls asleep for several minutes; awakes; and finishes the race, again running at 12 mph. Meanwhile, the tortoise plugs along at the same constant rate of 2 mph throughout the race. What is the least amount of time the hare was asleep if the tortoise wins the race?

The first thing that we need to know is how long it took the tortoise to finish the race. So the tortoise went along at 2mph for some amount of time (t) and traveled 10 miles or

So the tortoise finished the race after 5 long hours of dragging along. Now we need to come up with a nap time that will have the hare finishing just after 5 hours. One thing we can figure out is how far the hare got before it took a nap. The hare ran at 12 mph for 20 minutes and got some distance or

The hare had already gotten 4 miles before it stopped to nap, and only had 6 miles left to go. We need to find out how long it will take him to go the remaining six miles at 12 mph, and then we can make him sleep for the right amount of time.

So the hare can finish the race in a half hour after he wakes up. Let's add the time running plus the time napping to come up with the amount of time he would sleep to tie the race with the tortoise. First he ran for 20 minutes (as given by the problem) then he napped (n) and then he ran for 30 minutes, which all needs to equal five hours or 300 minutes.

20 + n + 30 = 300

50 + n = 300

n = 250

Now we know that if the tortoise slept for 250 minutes he would tie with the hare. But we don't want him to tie we want him to LOSE! So the hare would lose if he slept for 251 minutes. For a photo finish - 250 minutes and 1 second!