A hard question on clocks and time for P4–P5, 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
May starts piano practice at 2 pm sharp, watching the wall clock.
The first time the hour and minute hands make an angle of 38 degrees (the smaller angle between them), she stops for a rest.
How many minutes past 2 does she stop?
Show the hints (3)
💡Hint 1: At 2 o'clock, how far apart are the hands? As the minute hand catches up, how does the angle change?
💡Hint 2: Every two minutes the minute hand moves 12 degrees and the hour hand one, so the minute hand gains 11 degrees.
💡Hint 3: The angle starts at 60 degrees and shrinks. While the minute hand is still behind, it must gain 60 − 38 = 22 degrees.
Worked solution
Have a go first, then read the solution step by step.
1.At 2 o'clock the minute hand is on 12 and the hour hand on 2: the hour hand is 30 × 2 = 60 degrees ahead.
2.Every 2 minutes the minute hand moves 12 degrees and the hour hand 1, so the minute hand gains 12 − 1 = 11 degrees.
3.38 is less than 60: the first time, the minute hand is still 38 degrees behind, having gained 60 − 38 = 22 degrees.
4.22 ÷ 11 = 2, and 2 × 2 = 4: it takes 4 minutes, so 4 minutes past 2.
⚠️Common mistake: using 60 + 38 degrees. That happens after the minute hand has passed the hour hand — later, not the first time.
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