Olympiad topic · P2–P6
Clocks and time
Every year of the Hua Xia Cup has clock questions, from time passed in P1 to the angle between the hands in P6.
Try one
★★ MediumAt 3:16 pm, Ming, doing homework at home, looks up at the wall clock.
What is the smaller angle between the hour hand and the minute hand?
- Hint 1: The minute hand goes round in an hour, the hour hand in twelve hours; a full turn is 360 degrees. How far does each move per minute, per hour?
- Hint 2: The hour hand is not exactly on 3: in these 16 minutes it has moved on too, one degree every two minutes.
- 1.The minute hand moves 360 ÷ 60 = 6 degrees a minute. At 16 minutes past, it is 6 × 16 = 96 degrees round from 12.
- 2.The hour hand moves 360 ÷ 12 = 30 degrees an hour, which is 1 degree every 2 minutes. At 3:16 it is 30 × 3 + 16 ÷ 2 = 98 degrees round from 12.
- 3.The hands are 98 − 96 = 2 degrees apart.
- ⚠️Common mistake: leaving the hour hand on 3, which gives 6 degrees.
Answer: 2 degrees
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsTime passed, gaps between strikes.
- ★★MediumFree3 question types · unlimited questionsThe angle at a given time, a fast clock.
- ★★★HardFree2 question types · unlimited questionsHow often the hands make a right angle, when they make an angle.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsTwo clocks fast and slow, a digital clock's digits.
Clocks and time: what it is and how to solve it
Clock problems: time passed, a clock striking the hour, the angle between the hands, clocks that run fast or slow.
For example: At 2:30, what's the smaller angle between the hands? The minute hand is at 6 (180°), the hour hand halfway between 2 and 3 (75°): 180° − 75° = 105°.
The key idea: The minute hand turns 6° a minute, the hour hand 0.5°, so the minute hand gains 5.5° a minute. A clock striking has one gap fewer than strikes. For a fast or slow clock, ask how far it moves in one real hour.
Practise this topic free →Easy to Hard (★ to ★★★) all free, unlimited questions, no download