You have the misfortune to own an unreliable clock. This one loses exactly 20 minutes every hour. It is now showing 4:00am and you know that is was correct at midnight, when you set it. The clock stopped 4 hours ago, what is the correct time now?
OK.. I did really solve it. However, my solution was simple:
to literally write the time while an hour is 40min opposite to time when an hour is just an hour, 60 mins:
12:40 – 1 AM
1:20 – 2
2:00 – 3
2:40 – 4
3:20 – 5
4:00 AM- 6:00AM
and then add 4 hours so the answer is 10:00 AM.
Now another solution goes like this:
Since the clock is losing 20 minutes every hour, for every real hour that has passed, the clock will only show 40 minutes. Since the clock shows 4:00am, we know that 240 clock minutes have passed. This therefore equals 360 real minutes and hence 6 hours. The clock stopped 4 hour ago and the time must now be 10:00am.
Assumably, you could do 240:40 = 6 and there you go.. 6 additional hours to 12 AM.
What I don't understand is why divide 240 by 40?
240 mins reflect the "real" time when an hour is 60 minutes, why divide it by "40 minutes hour"?
Best Answer
The quantity "$240$ minutes" describes the amount of time shown on the clock. Since we know the clock is inaccurate, these "$240$ minutes" do not reflect "real" time. They reflect "clock" time.
$40$ minutes of "clock" time elapse in one hour of real time. So $40n$ minutes of "clock" time is $n$ hours of real time. Letting $240 = 40n$, we want to solve for $n$. What is the first step in solving that equation?
An alternative approach is: since the clock records only $\frac23$ hour for each hour of real time that passes, it takes $\frac32$ hour of real time for the clock to advance one hour. For the clock to advance $m$ hours takes $\frac32 m$ real hours. Since the clock advanced four hours, set $m = 4.$ But as noted in a comment, this is really doing the same thing, just in a slightly different order of computation and using different units.