Clock Hands

How many minutes past 4 o'clock are the hands of a standard 12-hour clock first perpendicular to each other?

It should be five minutes and some fraction of a minute

angle from 12 for the minute hand is 360 * M/60 = 6 M
angle from 12 for the hour hand is 360 deg/12 for each hour + 1/12 of minute hand angle
The angle between them is absolute value of the difference

6 * M = Am
30 *H + M/12 = Ah
Ah - Am = 90
30 H + M/12 - 6M = 90
M(1/12 - 6) = 90-30H
M = (90 - 30 H)/(1/12-6)
M = 5.07042253521127 minutes

 

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