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This was the question I faced:

The minute-hand of a clock overtakes the hour-hand at intervals of $62$ minutes of a correct time. How much in a day does the clock gain or lose?

And since I wasn't getting the right result as per the answers and the solution didn't fit well with me so I looked up similar questions on internet and the world seemed divided on the following $2$ variations:

To make things more generalised, let the minute-hand of a clock overtakes the hour-hand at intervals of $x$ minutes of a correct time.

Variation 1:

In $65 \tfrac{5}{11}$ mins, the clock gains $\bigl(65 \tfrac{5}{11} - x\bigr)$ mins

Thus, in one day clock will gain $2(720 - 11x)$ mins.

Variation 2:

In $x$ mins, the clock gains $\bigl(65 \tfrac{5}{11} - x\bigr)$ mins

Thus, in one day clock will gain $\bigl(\tfrac{720}{11} - x\bigr) \cdot \bigl(60 \cdot \tfrac{24}{x}\bigr)$ mins.

Now, for given $x = 62$ mins, we get $76$ mins and $80 \tfrac{80}{341}$ mins or $80.23$ mins respectively for the $2$ variations which is a significant difference.

I personally attempted the way variation $1$ does and still feel comfortable with it and not the other one.

So, which among the 2 should be followed. Please help and I hope we can come at a consensus (even though this should not be opinion based at all)

Please Note:

The minute hand of a clock overtakes the hour hand at intervals of 65 minutes of correct time. How much a day does the clock gain?

The above mentioned post doesn't points out the difference between the $2$ variants and thus, the discussion doesn't addresses it either (despite the question in consideration being practically the same).

Sammy Black
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InanimateBeing
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  • I don't chat, so I am putting (hopefully last) comment here. Just are error and error % are computed with respect to the true value, error in fast clock also has to be computed with respect to the correct clock. I really don't understand why you don't get this. – true blue anil Jun 03 '22 at 10:24

1 Answers1

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$62$ minutes of correct time is equivalent to $65 \frac5{11}$ minutes on the fast clock, so variation $2$ is correct, and in $24\times60$ minutes in a day, you can work out how much the fast clock will gain.

PS
As requested, I am confirming the answer
In $62$ actual minutes, gain of fast clock is $\frac{38}{11}$ minutes on the dial
In $24$ actual hours, gain is $80 \frac{80}{341}$ minutes on the dial

true blue anil
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  • So then how would the language be so that the variation 1 is correct? – InanimateBeing Jun 03 '22 at 04:55
  • In the time from noon the minute hand of my reliable clock takes for it to overtake the hour hand, the new clock shows a time of $1:10$. How many minutes will it gain in a day ? – true blue anil Jun 03 '22 at 06:12
  • Well the wording is alright as per your explanation but not in a way question would be framed though, in my opinion. But anyways, here's the confusion: "How much in a day does the clock gain or lose?", this is the main issue here.

    Variation 1 gives answer wrt 1 day of correct watch's time while Variation 2 gives answer wrt 1 day of incorrect watch's time, i.e. Actual 24 hrs VS Wrong 24 hours respectively.

    Shouldn't we by default give our answer based on the actual 24 hours?

     – InanimateBeing Jun 03 '22 at 06:31
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    "in a day" implies "correct" day, unless the question has the wording "in a day by the new clock" – true blue anil Jun 03 '22 at 06:45
  • "in a day" implies "correct" day, unless the question has the wording "in a day by the new clock": As per this statement shouldn't variation 1 give the correct result? Here's my interpretation, 62 min ≡ 65 5/11 mins so, 65 5/11 mins ≡ 62 mins then "correct" day will finally give us the result as obtained from variation 1. – InanimateBeing Jun 03 '22 at 07:25
  • You are going round in circles. Let's stick to the original question. What I wrote in the answer is very clear and explicit. – true blue anil Jun 03 '22 at 07:45
  • I meant that you put out 2 contradicting statements. Per your answer, V2 is correct but per you comment, V1 should be correct. Could you please solve this problem? – InanimateBeing Jun 03 '22 at 14:35
  • PS added to confirm analysis – true blue anil Jun 03 '22 at 15:34
  • thanks a lot for clearing all my doubts, that final confirm analysis really helped! – InanimateBeing Jun 04 '22 at 04:20
  • You're welcome ! – true blue anil Jun 04 '22 at 04:42