At a certain point of time someone observes a clock and find out that the hour hand is exactly at the minute mark and the minute hand is six minutes ahead it. The clock is observed again to find out that hour hand is exactly on a different minute mark but the minute hand is seven minutes ahead of it this time.

Can you calculate the time that has elapsed between the two observations?

two hours and twelve minutes.

If you know about clocks, you must be knowing the fact that the hour hand is exactly on the minute mark five times each hour i.e. on the hour, twelve minutes past the hour, twenty four minutes past the hour, thirty six minutes past the hour and forty eight minutes past the hour.

Now let us suppose that X are the number of hours and Y are the number of minutes past the hour.
When the hour hand is on minute mark,
Position of hour hand:
5X + Y/12
Position of the minute hand:
Y

Now when the first observation is taken,
Y = Y = 5X + Y/12 + 6
This is also equivalent to 60X = 11Y - 72.

But from what we know, Y can only be equal to 0, 12, 24, 36 or 48. Thus keeping that in mind, the only value possible for X and Y are 1 and 12 respectively. This implies that the time is 1:12 on the first observation.

In the second observation, the equation will be:
60X = 11Y - 84
The possible values for X and Y here are 3 and 24 respectively. This implies that the time is 3:24.

Now time that has elapsed between 1:12 and 3:24 = two hours and twelve minutes.

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