Problem 20 · 2003 AMC 8
Stretch
Geometry & Measurement
proportion
What is the measure of the acute angle formed by the hands of the clock at 4:20 PM?
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Answer: D — 10°.
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Hint 1 of 2
The trap is assuming the hour hand sits right on the 4 — it has crept toward 5. Pin down each hand's exact spot before measuring the gap.
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Hint 2 of 2
At :20 the minute hand lands exactly on the 4, so the whole gap is just how far the hour hand has drifted past the 4.
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Approach: pin each hand's exact position, then take the gap
- Minute hand first: 20 minutes is 20/60 = 1/3 of the way around, which lands it exactly on the 4 (since the 4 marks 20 minutes). So both hands are near the 4 — convenient.
- Hour hand next, and here's the catch: by 4:20 it has slid 1/3 of the way from 4 toward 5. Each number-to-number gap is 360° ÷ 12 = 30°, so the hour hand sits 30° ÷ 3 = 10° past the 4.
- The minute hand is right on the 4, the hour hand 10° beyond it, so they're 10° apart.
- Worth keeping: the hour hand moves 30° per hour = 0.5° per minute; the minute hand moves 6° per minute. Knowing both speeds lets you place either hand at any time — and remembering the hour hand keeps drifting is what saves you from the 0° trap.
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