When do the clock hands meet?
Tonight's idea
At 12:00 the two hands of a clock point the same way. When do they next meet? Not at 1:05. By then the hour hand has crept past the 1. The minute hand is chasing a moving target.
Measure the chase in degrees. The minute hand turns a full 360° in 60 minutes: 6° a minute. The hour hand moves from one number to the next in an hour, which is 30°: half a degree a minute.
So each minute the minute hand gains \(6 - \tfrac12 = 5\tfrac12\) degrees on the hour hand. To meet it again after 12:00 it must gain a whole lap, 360°.
Time to catch up = gap ÷ \(5\tfrac12\). From 12:00: \(360 \div 5\tfrac12 = \tfrac{720}{11} = 65\tfrac{5}{11}\) minutes.
That is about 27 seconds after 1:05. The hands go on meeting every \(65\tfrac{5}{11}\) minutes, which is \(\tfrac{12}{11}\) of an hour.
Watch one: the meeting after 3:00
At 3:00 the hour hand is on the 3, so the gap is \(3 \times 30 = 90°\). Dividing by \(5\tfrac12 = \tfrac{11}{2}\) is multiplying by \(\tfrac{2}{11}\):
\[90 \div 5\tfrac12 = 90 \times \tfrac{2}{11} = \tfrac{180}{11} = 16\tfrac{4}{11} \text{ minutes.}\]The hands meet at 3:16 and \(\tfrac{4}{11}\) of a minute.
Check: the minute hand turns \(6 \times \tfrac{180}{11} = \tfrac{1080}{11}\) degrees. The hour hand starts at 90° and adds \(\tfrac12 \times \tfrac{180}{11} = \tfrac{90}{11}\), reaching \(\tfrac{1080}{11}\) degrees too.
Your turn
- (a) How many degrees does the minute hand turn in 10 minutes? (b) How many does the hour hand turn in 10 minutes? (c) What is the angle between the hands at 4:00?
- What is the angle between the hands at 3:30? It is not 90°. Find where each hand points, in degrees clockwise from the 12.
- When do the hands first meet after 2:00? Give the minutes as a mixed number.
- (a) Use the same method to find when the hands meet after 11:00. What do you notice? (b) From 12:00 noon to 12:00 midnight, how many times do the hands meet, counting midnight but not noon? Explain why it is not 12.
Stretch: a straight line
After 12:00, when do the hands first point in exactly opposite directions, making a straight line? How many times does that happen between noon and midnight?
Answers
- (a) \(6 \times 10 = 60°\). (b) \(\tfrac12 \times 10 = 5°\). (c) \(4 \times 30 = 120°\).
- The minute hand is on the 6, at 180°. The hour hand is halfway from the 3 to the 4, at \(90 + 15 = 105°\). The angle is \(180 - 105 = 75°\).
- The gap is 60°. \(60 \div 5\tfrac12 = \tfrac{120}{11} = 10\tfrac{10}{11}\) minutes, so 2:10 and \(\tfrac{10}{11}\) of a minute. Just short of 2:11.
- (a) The gap is \(11 \times 30 = 330°\), and \(330 \div 5\tfrac12 = 60\) minutes exactly. The hands meet at 12:00, so there is no meeting during the eleven o'clock hour itself. (b) 11 times. In 12 hours the minute hand makes 12 laps and the hour hand makes 1, so the minute hand gains only 11 laps. The meetings, with minutes as mixed numbers: 1:05\(\tfrac{5}{11}\), 2:10\(\tfrac{10}{11}\), 3:16\(\tfrac{4}{11}\), 4:21\(\tfrac{9}{11}\), 5:27\(\tfrac{3}{11}\), 6:32\(\tfrac{8}{11}\), 7:38\(\tfrac{2}{11}\), 8:43\(\tfrac{7}{11}\), 9:49\(\tfrac{1}{11}\), 10:54\(\tfrac{6}{11}\), 12:00.
Stretch. The minute hand must gain half a lap, 180°. \(180 \div 5\tfrac12 = \tfrac{360}{11} = 32\tfrac{8}{11}\) minutes, so 12:32 and \(\tfrac{8}{11}\) of a minute. After that it happens every \(65\tfrac{5}{11}\) minutes, halfway between one meeting and the next: 11 times in the 12 hours. One of them is 6:00.