How many times in a day are the hands of a clock in a straight line?

The hands of a clock lie along a single straight line in exactly two situations — when they coincide (0 degrees apart) or when they point in exactly opposite directions (180 degrees apart). Since the minute hand moves at 360 degrees/hour and the hour hand at 30 degrees/hour, the minute hand gains on the hour hand at a relative speed of 360 minus 30 = 330 degrees/hour, and this relative speed alone determines how often each straight-line configuration recurs.

1. The hands return to the same relative angle every 360/330 = 12/11 hours, about 65.45 minutes — this is the fixed gap between any two consecutive coincidences.

2. In a 12-hour period the number of coincidences is 12 divided by (12/11) = 11, not 12, because the last coincidence of one 12-hour block is the same instant as the first coincidence of the next block.

3. By the identical relative-speed argument, the hands are also exactly opposite (180 degrees apart) 11 times in every 12-hour period, since that configuration also recurs every 12/11 hours.

4. So the hands are straight (coincide or opposite) 11 + 11 = 22 times in 12 hours, and in a full 24-hour day this doubles to 22 x 2 = 44 times.

Cross-check: this matches the two standard clock facts used independently in aptitude tests — the hands coincide 22 times a day and are exactly opposite 22 times a day (11 times each in every 12-hour half). Adding these two straight-line cases gives 22 + 22 = 44, confirming the result.