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The employees of corporate "Clocks" have a fixed rule for working hours. All the employees start working at 16:00 and finish their work by 21:00. They take exactly one break during that time.

All the employees stop working when their work is half done and at the same time both the minute and the hour hand of the clock coincides. They all start back working again when the minute hand makes a 30° angle with the hour hand.

Assuming the rate of work of employees is constant throughout the work period, calculate the amount of time they spend on break.

Bonus Question: What would be the length of break if they worked from 15:00 to 21:00?

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I came up with:

4 hours, 16.36363636363636 minutes during their 5 hour shift. Sounds like a great job!

Rationale:

You can get the angle between the hands with this formula (ugly, I know). Absolute value of 5.5 x minutes minus 30 x the hour. enter image description here

There are only two possible stop times where the hour and minute hand coincide and the restart time is later than the break start time: 4:21.8181818181 and 5:27.27272727272. The other three would require you to go back in time are restart work before you start your break (e.g., start break at 6:32PM and end at 6:27PM).
If the pre-break duration has to match the post-break duration, that only gives two possible restart times: 20:38.1818181818 and 19:32.727272727272. Using that same formula, the first one is 390 degrees, minus 360 for one time around the circle leaves 30 degrees.enter image description here

Double checked myself in Excel, and discovered that I should have finished my handwritten version.

The second possible breaktime works too, starting break at 5:27:16PM and getting a 2 hour, 5.4545454545 minute break. Also a solid break on a 5 hour shift.

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Side note: Excel is a pain in the rear end for working with times.

And for the bonus question:

no breaktime is acceptable. Every acceptable break time results in a restart time with an angle of 0. enter image description here

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  • $\begingroup$ So the answer is either 4.2727 or 2.0909? $\endgroup$
    – justhalf
    May 26, 2022 at 9:38
  • $\begingroup$ 2.09, I added the lateral thinking tag because it's way more far fetched to think a corporate allows a 4 hour break in a 5 hour job than 2 hours in a 5 hour job. Although both are very much correct, i think it doesn't require a lateral thinking tag anyway. $\endgroup$
    – I'm Nobody
    Aug 18, 2022 at 8:15

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