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An odd mathematician and engineer decides to redesign an analog clock. It now has 2 periods of 10 hours, each hour is divided into 100 minutes and each minute into 100 seconds.

He starts the clock at midnight. What would the time of 12:30 PM be on the on the new clock?

If the new clock shows 5:75:75 PM, What would be the time on a normal clock?

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closed as off-topic by LogicianWithAHat, JMP, Glorfindel, Ankoganit, Mike Earnest Oct 17 '17 at 14:36

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    $\begingroup$ Do they redefine what time is a second (200 000 new sec vs 86 400 old sec) ? Or are the new days lasting more time ? $\endgroup$ – Damien Bannerot Oct 17 '17 at 13:38
  • $\begingroup$ @DamienBannerot: Doesn't the information in the puzzle mean: 200 000 new sec = 86 400 old sec? $\endgroup$ – Mea Culpa Nay Oct 17 '17 at 14:03
  • $\begingroup$ No, it's confusing as you presented it, you should have create labels for the new units. $\endgroup$ – Damien Bannerot Oct 17 '17 at 14:18
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First:

12:30PM means 12 hour 30 min passed, which is 12*3600+30*60=45000 second. On the new clock it is 04:50AM, which is 4*10000+50*100=45000 second.

Second:

05:75:75PM means 15*10000+75*100+75=157575 second. On the original clock it is 07:46:15PM(more than a day passed), because 157575 = (24+12+7)*3600+46*60+15

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I assume that the new clock runs at a speed such that its 20 hour period is equivalent to a full 24 hour day. So its minutes and seconds are not the same length as the minutes and seconds of a regular clock.

Part 1:

12:30pm is half an hour after midday, and a regular half hour is $1/24$ of a half day. A half day is 10 hours on the decimal clock. $1/24 = 0.041666...$ so on the decimal clock it will be 0h:41m:67s into the afternoon period, and could be displayed as 10:41:67PM.

Part 2:

It is $0.57575 * 12$ hours after midday. $$0.57575 * 12 \text{ hours} = 6.909 \text{ hours}\\ 0.909*60\text{ minutes} = 54.54 \text{ minutes}\\ 0.54*60\text{ seconds} = 32.4 \text{ seconds}\\$$

This means it is 6:54:32PM.

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For the first part (12:30 PM standard = ? New)

See the explanation below for the relationship between "seconds" and standard seconds. 12:30 PM standard clock is 45,000 out of 86,400 standard seconds through the day. This is a ratio of 0.5208'3 of a day. Multiply this by the 200,000 "seconds" of a day, and we find that we are 104,166.'6 "seconds" into the day; we can convert this to the clock reading by inserting colons: 10:41:66.'6, or 10:41:66.'6 PM (rounded to the nearest "second", 10:41:67)

For the second part (5:75:75 PM = ? standard)

Your modified clock contains 200,000 "seconds" in a 24-standard-hour day. PM in a printed time indicates that it’s in the second ten-"hour" period, so to convert it to non-AM/PM time, just stick a 1 in front. We can then just delete the colons (because they represent divisions at 100), meaning that the time is 157,575/200,000 of the way through the day. The relationship between "seconds" and standard seconds is 86,400 standard seconds = 200,000 "seconds". We thus multiply 157,575 by 86,400, then divide by 200,000, to get the number of standard seconds into the day we are: 68,072.4. Two successive divisions by 60 (to convert from seconds to minutes, then from minutes to hours) gives us 18.909 hours out of 24. Taking the fractional part (0.909) and multiplying back by 60 to get minutes yields 54.54, and taking the fractional part of that (0.54) and multiplying by 60 to get seconds yields 32.4. Thus, the time as shown on a normal clock would be 18:54:32.4, or 6:54:32 PM (rounded to the nearest "second").

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