Question

At 3:00 the hour hand and the minute hand of a clock point in directions that are 90° apart. What is the first time after 3:00 that the angle between the two hands has decreased to 26.0°?

Answer 1

Let's first try and work out how many degrees each hand moves each minute.

The hour hand does one lap of the clockface every 12 hours, or 12*60 = 720 minutes. So the hour hand travels 360 degrees every 720 minutes, and therefore travels 360/720 = 0.5 degrees per minute.

The minute hand does one lap of the clockface every 1 hour, or 60 minutes. So the minute hand travels 360 degrees every 60 minutes, and therefore travels 360/60 = 6 degrees per minute.

At 3:00, the hour hand is ahead of the minute hand, meaning the gap between the two hands is closing. But how quickly will the gap close? The amount travelled by the minute hand per minute, minus the amount travelled by the hour hand per minute, of 6 deg - 0.5 deg = 5.5 deg.

So the gap between the hands will close by 5.5 deg per minute. We want to find how long it will take for the gap to close from 90 deg to 26 deg, in other words to close by 64 deg.

If it's closing by 5.5 deg/min, then the time we are looking for is (64 deg)/(5.5 deg/min) = 11.64 minutes = 11 minutes 38 seconds

So the angle will have decreased to 26.0 degrees 11 minutes 38 seconds after 3:00, or 3:11:38.