Problem 10 · 1989 AJHSME
Medium
Geometry & Measurement
clock-angles
What is the number of degrees in the smaller angle between the hour hand and the minute hand on a clock that reads seven o'clock?
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Answer: D — 150°.
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Hint 1 of 3
A full clock face is one full turn — 360° — divided into 12 equal hour gaps. What's one gap worth?
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Hint 2 of 3
Convert 'hour marks apart' into degrees by giving every hour gap a fixed value, then just count gaps.
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Hint 3 of 3
At 7:00 the minute hand points at 12 and the hour hand at 7. Count the hour gaps the SHORT way around between them.
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Approach: one hour gap = 30°, then count gaps
- The 12 hour marks split the full 360° circle into 12 equal gaps, so each gap is 360 ÷ 12 = 30°.
- At 7:00 the minute hand sits at 12 and the hour hand at 7. The short way between them spans 5 hour gaps (12→11→10→9→8→7), giving 5 × 30° = 150°.
- Why this transfers: turning a clock into '30° per hour gap' converts every clock-angle question into simple counting; the long way around would be 12 − 5 = 7 gaps = 210°, and the two always add to 360°.
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