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 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).
Best Answer
$62$ 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