Conference

[small_unicorn]

A conference begins between 6:00 and 7:00 p.m., and ends between 9:00 and 10:00 p.m. The minute and hour hands have exchanged places.
When does the conference start and end ?


from book "The Moscow puzzles" by Boris A. Kordemsky

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[isabella2296]

Do you mean that when it started somewhere between 6 and 7, the hour hand was on A and the minute hand was on B, and then when it ended, the hour hand was on B and minute hand on A?

[small_unicorn]

[O M G]

you know since its between 6 and 7, it should be 9:30 ending, and 6:45 since 9 corresponds to 45. so it should last 2 hours and 45 seconds

[steve123456]

O M G wrote:

[steve123456]

Here's the solution. It's not exact, but it's really close. I estimated one of the values.

Solution

Say that the conference starts at 6: x and ends at 9:y. An hour is from one number to the next, so it travels as far as the minute hand does in 5 minutes. There are 5 minutes from the number 6 to 7. On the whole clock there are 60 minutes. Therefore the hour hand is on 6*5+ 5/60*x for the little marks on the clock that mark the minutes. The minutes is on x. Similarly, at 9:y, the hour hand is on 9*5+5/60*y and the minute hand is on y. Because you're supposed to switch the hands, 6*5+5/60*x=y, and x=9*5+5/60*y. Substituting can give (6*5+5/60)(9*5+5/60*y)=y. Solving this will give y is about 34, and x would then be 48. Therefore the conference would start at 6:48 and end at 9:34. You can check this by finding where the hands are. The only flaw is that at 6:48 the minute hand would be 18 degrees past 9, and the hour hand at 9:34 would be 17 degrees past 9. This is because we estimated for y.

Hope this helps.