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# The Bicycles and the Fly

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Two boys on bicycles 20 miles apart, began racing directly toward each other.

The instance they started, a fly on the handle of one bicycle started flying straight toward the other cyclist.

As soon as it reached the other handle bar it turned and started back.

The fly went back and forth this way, from handle bar to handle bar, until the two bicycles met.

If each bicycle had a constant speed of 10 miles per hour, and the fly flew at a constant speed of 15 miles per hour, how far did the fly fly?

Guest May 4, 2015

#1
+81007
+5

Let's see.....

The bicyclists will take one hour to reach each other at the midway point, 10 miles distant from where each started. Since the fly travels continuously in this time....he must travel 15 miles in the one hour.

Seen another way....

The total distance the fly travels in meeting the first distant bike = 12 miles. The successive distances traveled between the converging bikes is just (1/5) of the preceding distance. To see this.....note that the distant bike and fly are 20 miles apart when they begin. They close 2.5 miles of distance between them every 6 minutes. So...they meet  in (20/2.5) x 6 = 48 min. And in this time, the fly has traveled 4/5 hr x 15 mph = 12 miles.

So. the bikes have traveled 16 miles total in this time and are 4 miles apart. It takes the fly and the first bike  (4/2.5) x 6 = 9.6 minutes to close this distance. And in this time, the fly travels (9.6/60) x 15 = 2.4 miles. Note that this is 1/5 of the previous distance it has traveled  = 12 miles.  And this pattern continues.

So....summing the series, we have :

12 + 12/5 + 12/25 + 12/125 + ....   = 12 /(1 - 1/5)  = 15 miles

CPhill  May 4, 2015
Sort:

#1
+81007
+5

Let's see.....

The bicyclists will take one hour to reach each other at the midway point, 10 miles distant from where each started. Since the fly travels continuously in this time....he must travel 15 miles in the one hour.

Seen another way....

The total distance the fly travels in meeting the first distant bike = 12 miles. The successive distances traveled between the converging bikes is just (1/5) of the preceding distance. To see this.....note that the distant bike and fly are 20 miles apart when they begin. They close 2.5 miles of distance between them every 6 minutes. So...they meet  in (20/2.5) x 6 = 48 min. And in this time, the fly has traveled 4/5 hr x 15 mph = 12 miles.

So. the bikes have traveled 16 miles total in this time and are 4 miles apart. It takes the fly and the first bike  (4/2.5) x 6 = 9.6 minutes to close this distance. And in this time, the fly travels (9.6/60) x 15 = 2.4 miles. Note that this is 1/5 of the previous distance it has traveled  = 12 miles.  And this pattern continues.

So....summing the series, we have :

12 + 12/5 + 12/25 + 12/125 + ....   = 12 /(1 - 1/5)  = 15 miles

CPhill  May 4, 2015
#2
+91451
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A better question is

Where was the fly when the cyclists met ?

Melody  May 4, 2015
#3
+81007
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Looking at my series answer.......the distances narrow.....thus....he was squashed between the handlebars....

Alas.....all that flying was for naught.....!!!!!

CPhill  May 4, 2015
#4
+1068
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How many times did the fly change the direction on its 15 mile fly?

civonamzuk  May 4, 2015
#5
+91451
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I think Chris has answered this question.  If I am interpreting correctly it is an infinite number.

Looks like that fly gets awfully beaten up.

Someone should give it a more humane death.