Question 1091689
<br>I find it easiest to work problems involving the angle between the hands of a clock by measuring all angles relative to 12 o'clock.<br>
The arithmetic required to solve this kind of problem is based on the fact that the hour hand moves 1/12 of a revolution, or 30 degrees, in 1 hour (60 minutes), so its rate of rotation is 0.5 degrees per minute; and the minute hand moves one full rotation (360 degrees) in an hour (60 minutes), so its rate of rotation is 6 degrees per minute.<br>
You want to find the time after 4 o'clock when the minute hand is "9 minutes" behind the hour hand.  9 minutes is 3/20 of an hour; so you want the minute hand to be 3/20 of 360 degrees, or 54 degrees, behind the hour hand.<br>
At x minutes after 4, the angle the hour hand makes with 12 o'clock, in degrees, is
{{{120+0.5x)}}}  (120 degrees to get to 4; then 0.5 degrees for each minute after 2)<br>
And x minutes after 4, the angle the minute hand makes with 12 o'clock, in degrees,  is
{{{6x}}}<br>
You want the minute hand to be "9 minutes", or 54 degrees, behind the hour hand, so you want
{{{(120+0.5x)-6x = 54}}}
{{{-5.5x = -66}}}
{{{x = 12}}}<br>
The minute hand will be "9 minutes" behind the hour hand at 4:12.