SOLUTION: It is now past 3 o'clock in the afternoon. The minute hand 3 minutes from now will be directly opposite the hour hand 9 minutes ago. What time is it?

Let the current time under the question be 3 hours and "t" minutes, where t is the real number, so t can be fraction of minutes and seconds.


The minute hand is at angular position  of   = 6t degrees, counting from its position "vertically up" at 3:00 pm.

     *** Minute hand makes its full revolution of  360 degrees in 60 minutes. ***


The hour hand at this moment is at the angular position   degrees, counting from its position "vertically up" at noon.

     *** Hour hand has the angular velocity 12 times less than the minute hand: it is 12 times slower. ***


Three minutes from now the minute hand will be in position of    degrees.


Nine minutes ago the hour hand was in position of    degrees.


Therefore, the governing equation is 


 =  + 180  degrees.


          The major part is done: the equation/the model is set up.

          The rest is just arithmetic. 


Simplify this equation

6t + 18 =   ====>

 =   ====>   =   =  ====>

t =  = 45 minutes