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
Ming studies in the library from 8:00 am to 4:00 pm, with a clock on the wall.
During this time (counting both 8:00 am and 4:00 pm themselves), how many times do the hour and minute hands overlap?
Show the hints (3)
💡Hint 1: In 12 hours, how many more times round does the minute hand go than the hour hand? Each extra lap, how often do they overlap?
💡Hint 2: The hands overlap at about 1:05, 2:11, 3:16, … roughly once an hour. Which hour misses out?
💡Hint 3: Between 11 and 12 they never overlap; the next time is 12 exactly. Count how many such hours this stretch passes through.
Worked solution
Have a go first, then read the solution step by step.
1.In 12 hours the minute hand goes round 12 times and the hour hand once: 11 extra laps, so the hands overlap 11 times.
2.They overlap at 12 exactly, then once each in the 1 o'clock hour, the 2 o'clock hour, … up to the 10 o'clock hour. The 11 o'clock hour has none: the next is 12 exactly.
3.From 8:00 am to 4:00 pm is 8 hours. Counting each hour from its start, there is usually 1 per hour: 8.
4.Of these, 1 is an 11-to-12 hour with none: take away 1.
5.The final moment, 4:00 pm, belongs to no hour, so check it alone: the hands don't overlap then.
6.In all 8 − 1 + 0 = 7.
⚠️Common mistake: thinking they overlap once every hour, and answering 8.
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