Precisely how many minutes and seconds past 10:20 is it when the minute and hour hands on a clock form a perfectly straight line, as shown in the diagram?

 

Solution

 

The straight line formed by the clock hands is shown in red. The other straight line is the one that can be drawn straight through the 10 and 4 for reference purposes. Angles x are opposite and therefore equal to each other. In the time that it takes the hour hand to move from exactly 10:00 to its desired position(in other words, in the time that it takes to cover angle x), the hour hand will have moved from 12 to 4(20 minutes) plus the time that it takes to cover its angle x. To create an equation, we need to figure out the speed of each hand.

 

The hour hand covers 360^o (full circle) in 12 hours or

720 minutes, so it covers only 0.5 ^o per minute.

That’s why it seems to be stationary. On the other hand,

the minute hand covers 360^o in only 60 minutes,

so it goes 12 times faster at 6 ^o per minute.

 

The equation therefore becomes:

Hour hand’s time = minute hand’s time

x^o(minute/ 0.5 ^o) = 20 minutes  + x ^o(minute/ 6 ^o)

Solving for x: (multiply through by 6)

12 x = 120 + x

x = 120/11 ^O

But on the clock every 5 minutes represents 30^o, so

120/11 ^o (5 min/30 ^o) = 20/11 minutes

 = 1.818181….minutes

or 1 minute and 49.0909…seconds