Question 631371
At exactly 4 o'clock, the big hand is on the 12 
and the little hand is on the 4. They span 20
minutes out of 60, or 120 degrees out of 360 degrees
------------------
The minute hand will move at the rate of 360 degrees / hr
The hour hand will move at the rate of {{{ 5/60)*360 = 30 }}}
degrees / hr
------------------
I want the difference between them to be 90 degrees with
the difference starting out at 120 degrees
Let {{{ a }}} = the angle between the hands
Let {{{ t }}} = the time in hours for the angle between them
to equal 90 degrees
-------------------
{{{ a = ( 120 + 30t ) - ( 360t ) }}}
{{{ 90 = ( 120 + 30t ) - ( 360t ) }}}
{{{ 90 = 120 - 330t }}}
{{{ 330t = 120 - 90 }}}
{{{ 330t = 30 }}}
{{{ t = 1/11 }}} hrs
{{{ 1/11*60 = 5 + 5/11 }}} minutes
{{{ 5/11*60 = 27.27 }}}
---------------
The hands will be 90 degrees apart at
4:05 and 27 sec
check:
{{{ a = ( 120 + 30t ) - ( 360t ) }}}
{{{ a = ( 120 + 30*(1/11) ) - ( 360*(1/11) ) }}}
{{{ a = ( 120 + 2.7272 ) -  ( 32.7272 ) }}}
{{{ a = 120 - 30 }}}
{{{ a = 90 }}}
OK