12 x - x = 11 x
11 x = 2 right angles = half an hour
Since 12 x is equal to the time passed,
12/11 * 30 minutes = 32 8/11 minutes.
It will be 32 8/11 minutes before they are at right angles again.
Thank you Chin Siang (Tao Nan School, Singapore) for this splendid solution in HTML and the excellent diagram.
James sent a good solution too and he makes the comment "I had to keep reminding myself that every time the big hand moves so does the little one."
Here is the solution from the Strabane Grammar School Key Stage 3 Maths Club:
We got a clock and set the hands to 3 o'clock i.e. at right angles.
We moved time forward and noticed that the hands got closer, were on top of each other, and then separated, the minute hand moving in front, until they were at right angles again between 3.30 and 3.35.
We tried to calculate the time more precisely. Every minute the hour hand moves forward 0.5 degrees and the minute hand moves forward 6 degrees so the angle separating the hands will change by 5.5 degrees.
Since they have to close up and then separate again this will take 180/5.5 minutes i.e. 32 8/11 minutes. After another 32 8/11 minutes the hands will be perpendicular again.