Expert: Robi Bhattacharjee - 11/30/2008

Question
Dear Robi, do you have any formula in which you can identify how the minute hand and hour hand interchange? Sorry, but I'm not that good in english. For you to understand, here's an example:

"Ben was watching a soccer match on television. Before the start of the match, he looked at the clock. After the match, he looked at the clock again and noticed that the two hands on the clockface just interchanged positions. If the soccer match lasted more that 1 hour but less than 2 hours, how long did actually last? Give your answer correcrt to the nearest minute."

Thank you very much...

Answer
Hi David,

This is a very complicated question.

You don't need a formula to solve this question. You only need to think about it.

First, consider that it doesn't matter where you situate your problem. Theoretically, you could have times like 12:30,2:20 (just an example) or 4:30, 2:20. So let us situate ours with out first hour hand in between the 12 and the 1.

Let x be the number of minutes it is past the 12. Now, would you agree that the position of the minute hand would be 12*x. This is because we know that x/5 = y/60 where y is the position of the minute hand. This equation comes from the fact that when the hour hand goes 5 minutes, the minute hand goes 60 minutes.

So now we have two positions x, 12x. Since it has to be more than one hour and less than two hours, it has to be in between the two and the three. If it isn't, then the time will be of 1 hour.

Now when the hour hand is at 12x, the minute hand is at x. So (12x - 10)/5 = x/60. This is because the hour hand is moving from the 2 to the 3, so since the 2 is at 10 minutes we put 12x - 10. The 5 and 60 are from the same reasons they were above. Anyways, when you solve this you get that x = 120/143 minutes. So the two times are roughly 12:10 and 2:01. So the time between them is 1 hour and 51 minutes.

I hope this helped.
This was a challenging problem so feel free to ask if you have any concerns.

Robi