Clock hands: relative speed in a costume
The minute hand gains 5.5° a minute on the hour hand. Hold that number and every clock question is one subtraction or one division.
- Find the angle between the hands at any time, remembering the hour hand moves
- Find the next time the hands overlap, or make a right angle, by dividing a gap by 5.5
- Explain why the hands meet 11 times in 12 hours, not 12
- Handle the fold: angles are reported between 0° and 180°
The intuition
At 3:20 the minute hand is at 120°. The hour hand is not at 90°: in twenty minutes it has crept a third of the way to the 4, so it is at 100°. The angle is 20°, not 30°. Forgetting that the hour hand moves is the classic mistake, and it is wrong at every time except the top of the hour.
Everything else is relative speed. The minute hand turns 6° a minute, the hour hand 0.5°, so the gap between them changes at 5.5° a minute. At 3:00 the hour hand is 90° ahead; the minute hand closes that at 5.5° a minute and catches it 90 ÷ 5.5 = 16.36 minutes later. In twelve hours the minute hand makes twelve laps and the hour hand one, so it overtakes eleven times, 65 and 5/11 minutes apart, and the eleven o'clock hour is the one with no meeting: its meeting lands on 12:00.
Hour hand = 30h + 0.5t degrees; minute hand = 6t. Angle = |30h − 5.5t|, folded into 0°–180°. The hands meet at t = 60h ÷ 11 minutes past h. They are at right angles when the signed gap is ±90° (mod 360°). 11 meetings per 12 hours, 22 per day; 22 right angles per 12 hours, 44 per day.
Why it works
- The conventions here: a 12-hour dial with continuously moving hands. Angles are the smaller of the two between the hands. 'Next' means strictly after the stated time. Hours run 1 to 11; 12 has its meeting at the start.
- Two speeds, one difference. 6° per minute against 0.5° per minute. The signed gap, hour hand minus minute hand, starts at 30h at the top of the hour and falls 5.5° every minute. Every question is about when that gap hits a value.
- Overlap: divide the gap by 5.5. 30h ÷ 5.5 = 60h ÷ 11 minutes. It is a whole number only when 11 divides h, at h = 11, which gives 60 minutes, i.e. 12:00.
- Right angles and opposite hands. Find the first t > 0 where the gap is +90 or −90 (or 180) modulo 360. Each sweep of the gap through a full turn passes 90° twice, so 22 right angles per 12 hours.
- The fold. The raw gap always changes by 5.5° a minute, but the reported angle bounces back at 0° and at 180°. Ten minutes always move the raw gap by 55°; the reported angle changes by 55° only if no fold happened in between.
- Why eleven, not twelve. The minute hand gains one lap for every twelve it makes, so it overtakes the hour hand once for each lap it gains: 12 − 1 = 11 times in 12 hours.
| Hour hand: 30 × 3 + 0.5 × 20 | 100° |
| Minute hand: 6 × 20 | 120° |
| Angle: |100 − 120| | 20° (30° if the hour hand were parked on the 3) |
| Next overlap after 3:00: 90 ÷ 5.5 | 16.36 minutes, 16 min 22 s |
| Next right angle after 3:00: gap 90 − 5.5t = −90 | 32.73 minutes |
| Ten minutes later, at 3:30: hour 105°, minute 180° | 75°, a change of 55° |
Meetings per day: 2 × 11 = 22, spaced 720 ÷ 11 = 65.45 minutes apart. The 11 o'clock hour has none; its meeting is 12:00.
The formulas
Degrees, t minutes past h o'clock. The hour hand moves too.
The signed gap, then the smaller angle.
The gap 30h closed at 5.5° a minute.
One overtake per lap gained; each sweep passes 90° twice.
Worked example
Place each hand, subtract, and fold into 0°–180°. The follow-up is the parked-hour-hand mistake.
See it move
Same clock. Change the hour and the minutes past it.
- Move the hour on. The hour hand's position rises by 30° a step and the overlap comes later, 60 ÷ 11 minutes per hour.
- Move the minutes on. The minute hand advances 30° per five minutes and the hour hand creeps 2.5°; watch the angle, which falls, folds at 0°, rises, and folds again at 180°.
- Set the minutes to zero. The hour hand sits exactly on its number, and the parked-hand shortcut is right for once.
Run it backwards
The hands are 30h degrees apart at the top of the hour. When do they next meet?
The minute hand closes the gap at 5.5° a minute, so t = 30h ÷ 5.5. It is a fraction of a minute at every hour but 11, when it is 60 minutes exactly: 12:00.
The same division answers 'next right angle' with a gap of 30h − 90 or 30h + 90, taken modulo 360 and choosing the first positive time.
Traps
Say it in the interview
“What is the angle between the hands at 3:20, and when do they next overlap?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Hour hand 30h + 0.5t, minute hand 6t; the gap changes 5.5° a minute.
- Overlap at 60h ÷ 11 minutes past h; 11 meetings per 12 hours, 22 per day.
- Fold the answer into 0°–180°, and remember the fold when time passes.
- Hold the mechanism, not the answers.