SOLUTION: At exactly 12 o'clock noon the hour hand of a clock begins to move at six times its normal speed, and the minute hand begins to move backwards at five sixths its normal speed. When

As everybody knows, the hour hand makes one full rotation in 12 hours.

Hence, the regular angular speed of the hour hand is  of 360 degs, or 30 degs per hour, or  0.5 degs per minute.


Again, as everybody knows, the minute hand makes one full rotation in 1 hour = 60 minutes. 

Hence, the regular angular speed of the minute hand is 360 degs per hour,  or 6 degs per minute.


The equation for the angular position of the "crazy" hour hand is

    H(t) = 90 - (6*0.5)*t                     (1)

where 90 = 90 degs is its initial position in degrees at noon ("vertically up");  (6*0.5) is its "crazy" angular velocity 
and "t" is time (! the CORRECT time !).


The equation for the angular position of the "crazy" minute hand is

    M(t) =        (2)

where 90 = 90 degs is its initial position in degrees at noon ("vertically up");  ((5/6)*6) is its "crazy" angular velocity and 
"t" is time in minutes (! the CORRECT time !). ( ! Notice that the sign "+" in formula (2) reflects anti-clockwise rotation). 


The condition that the hour and the minute hands coincide is

    M(t) = H(t) + 360 degs,   or    = 90 - (6*0.5)*t + 360   (3)


Simplify equation (3) step by step.

    5*t = -3*t + 360  ====>  5t + 3t = 360  ====>  8t = 360  ====>  t =  = 45 minutes.

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