Brainteasers and Puzzles

Angle at a Quarter Past Three

At 3:15 the minute hand rests exactly on the 3 while the hour hand has crept past it by half a degree for each of the 15 minutes, leaving a gap of 7.5 degrees where the eye expects none.

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)}(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

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)}(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)}(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.