SOLUTION: How many minutes after 5 o�clock, will the minute hand be 16 minute spaces behind the hour hand?

Question 1012022: How many minutes after 5 o�clock, will the minute hand be 16 minute spaces behind the hour hand?
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
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How many minutes after 5 o�clock, will the minute hand be 16 minute spaces behind the hour hand?
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Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
at 5 o'clock, the hour hand is at the 25 minute mark and the minute hand is at the 0 minute mark.

at 6 o'clock, the hour hand is at the 30 minute mark and the minute hand is at the 0 minute mark, or at the 60 minute mark, depending on how you want to look at it.

based on this, we can develop a formula for the hour hand versus the minute hand.

the formula is:

hour hand in terms of minutes = 25 + x/12.

minute hand in terms of minutes = x.

based on this formula, we can determine the position of the hour hand in terms of minutes.

when x = 0, the hour hand is at the 25 + 0/12 = 25 minute mark.

when x = 60, the hour hand is at the 25 + 60/12 = 30 minute mark.

this tracks with what we know, because at 5 o'clock, the hour hand is at the 25 minute mark and at 6 o'clock, the hour hand is at the 30 minute mark.

so now we want to find out at how many minutes will the hour hand be 16 minutes behind the hour hand.

our formula becomes:

25 + x/12 - x = 16

that says that the difference of the position of the hour hand in minutes minus the position of the minute hand must be equal to 16.

if we solve for x, we will find that x = 9.81818...

when x = 9.81818..., 25 + x/12 = 25 + 9.81818.../12 = 25.81818....

the difference between the hour hand in terms of minutes a and the minute hand is 25.81818... - 9.81818... = 16.

in 9.81818... minutes after 5:00 o'clock, the hour hand and the minute hand will be exactly 16 minutes apart, with the minute hand being behind the hour hand.

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