Brainteasers & Puzzles

Angle at a Quarter Past Three

At 3:15 the minute hand points straight at the 3. The hour hand looks like it does too. How wide is the angle that is actually between them?

solvedeasy1 min

How many degrees, if any, are there between the hour and minute hands of a clock at a quarter past three?

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#Place each hand

The minute hand turns a full 360∘360^\circ each hour, so 15 minutes is a quarter turn,

15⋅6=90∘,(1)15 \cdot 6 = 90^\circ, \tag{1}

landing it exactly on the 3. The hour hand turns 30∘30^\circ per hour, which is 0.5∘0.5^\circ per minute, so by 3:15 it has moved past the 3 by a quarter of an hour,

3⋅30+15⋅0.5=97.5∘.(2)3 \cdot 30 + 15 \cdot 0.5 = 97.5^\circ. \tag{2}

#Take the difference

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The minute hand sits exactly on 3, but the hour hand has already drifted a quarter of the way toward 4, half a degree for each of the 15 minutes. That leftover quarter-step is the whole of the angle, a slender 7.5 degrees.

The gap is the difference of the two positions,

∣97.5−90∣=7.5∘.(3)\lvert 97.5 - 90 \rvert = 7.5^\circ. \tag{3}

The whole angle is just the small drift of the hour hand off the 3, half a degree for each of the 15 minutes. The tidy formula ∣30H−5.5M∣=∣90−82.5∣=7.5∘\lvert 30H - 5.5M \rvert = \lvert 90 - 82.5 \rvert = 7.5^\circ gives the same answer in one step.