Maths olympiad question · P5–P6★★★★ Expert
Clocks and time: olympiad question 9
An expert question on clocks and time for P5–P6, with a step-by-step solution, diagrams and the answer. Every year of the Hua Xia Cup has clock questions, from time passed in P1 to the angle between the hands in P6.
The question
A clock runs slow: for every 60 minutes of real time it moves only 55.
One day at exactly 11 o'clock it is set right and left to run.
After it is set, when its hands are 33° apart (the smaller angle) for the third time, how many minutes of real time have passed?
Show the hints (3)
- Hint 1: Forget the slowness first: on the clock's own face from 11 o'clock, how much does the minute hand gain on the hour hand for each minute it shows?
- Hint 2: At 11 o'clock, how far is the minute hand ahead, clockwise? 33° apart means a lead of 33° or 327° (plus whole turns). Find the third.
- Hint 3: The clock must show 66 minutes passing. Each 60 real minutes, it shows only 55.
Worked solution
Have a go first, then read the solution step by step.
- 1.On the clock's face: the minute hand turns 6° a minute, the hour hand 1/2°, so the minute hand gains 11/2° a minute.
- 2.At 11 o'clock the minute hand is at 12 and the hour hand at 330°, so the minute hand leads by 360 − 330 = 30°.
- 3.33° apart means a lead of 33° or 327°, plus whole turns. Above 30° these come at 33, 327, 393, 687, …; the third is 393°.
- 4.The lead must grow 393 − 30 = 363°, so the clock shows 363 ÷ 11/2 = 66 minutes.
- 5.Every 55 minutes it shows, 60 real minutes pass: 66 ÷ 55 × 60 = 72 real minutes.
- 6.Check: in 72 real minutes the clock shows 72 × 55 ÷ 60 = 66; the lead is 30 + 66 × 11 ÷ 2 = 393°, an angle of 33° ✓
- ⚠️Common mistake: giving the clock's 66 minutes, forgetting it is slow; or using the ratio upside down (66 × 55 ÷ 60). A slow clock shows less, so real time is longer. And 33° apart happens at leads of 33° and 327°: sort them before counting.
Answer: 72 minutes
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