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# How many degrees are in the smaller angle formed by the minute and hour hands on a clock at 12:30?

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How many degrees are in the smaller angle formed by the minute and hour hands on a clock at 12:30?

Nov 15, 2018

#1
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165 degrees

12:30 (without considering hour hand movement) is 180 degrees. However, the hour hand moves half of 30 degrees, which is 15 degrees. 180 - 15 = 165.

Nov 15, 2018
#2
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165 degrees.

Nov 15, 2018
edited by pepitio  Nov 15, 2018
#3
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The Clock is divided into 12 hours. But, the Clock is also a "circle" and has 360 degrees.
So: 360 / 12 =30 degrees between 12 and 1(or every 5 minutes).
Since the Minute hand has moved 30 minutes, or 180 degrees, the Hour hand has moved half the distance between 12 and 1. Or: 30 degees/2 =15 degrees.
Then: 180 - 15 = 165 degrees between the two hands at 12:30.

Nov 15, 2018
#4
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We can also use the clock formula, which is $$|30H-5.5M|$$, so $$|30(12)-5.5(30)|=360-165=195^\circ$$. Since the problem is asking us for the smaller angle, the answer is $$\boxed{165^\circ}$$.

Nov 15, 2018
#5
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How many degrees are in the smaller angle formed by the minute and hour hands on a clock at 12:30?

$$\boxed{\large{\Delta\varphi=330\cdot t}} \\~ \text{t is the time in hours }\\~ \text{\Delta\varphi is the angle in degrees between minute and hour hands. }$$

$$\text{time at 12:30 }$$

$$\begin{array}{|rcll|} \hline t &=& 12+\dfrac{30}{60} = 12.5 \ h \\ \Delta\varphi &=& 330 \cdot 12.5 \\ &=& 4125^{\circ} \\ \Delta\varphi &=& 4125^{\circ} - 11\cdot 360^{\circ} \\ && \text{A multiple of 360^{\circ} degrees must be deducted from the angle.}\\ \Delta\varphi &=& 4125^{\circ} - 3960^{\circ} \\ \mathbf{\Delta\varphi} &\mathbf{=}& \mathbf{165^{\circ}} \\ \hline \end{array}$$

The smaller angle formed by the minute and hour hands on a clock at 12:30 are $$\mathbf{165^{\circ} }$$

Nov 15, 2018