# Math Help - angles between clock hands of the Time = 11:08

1. ## angles between clock hands of the Time = 11:08

The questions asks what is the small angle between clock hands where it shows 11:08

This is not a difficult questions, what I need is the fastest possible solution.

My Solution: We first consider that two hands are on 00:00
I would say an hour is 60 minutes so 360/60 = 6 degree for each minute. Therefore we have 8*6=48 degree for the 08 minutes.
The hour is 11 and is 5*6=30 degree away from 00:00 and we have 30+48=78 degree
BUT, we should also consider that if it's 11:08 then hour hand is moved slightly closer to 00:00 ... but how much? it's that an hour has 60 minutes and (8/60)*6 = 0.8

Therefore the final answer is 78 - 0.8 = 77.2

Is this correct ??? Any faster way???

2. Originally Posted by Narek
The questions asks what is the small angle between clock hands where it shows 11:08

This is not a difficult questions, what I need is the fastest possible solution.

My Solution: We first consider that two hands are on 00:00
I would say an hour is 60 minutes so 360/60 = 6 degree for each minute. Therefore we have 8*6=48 degree for the 08 minutes.
The hour is 11 and is 5*6=30 degree away from 00:00 and we have 30+48=78 degree
BUT, we should also consider that if it's 11:08 then hour hand is moved slightly closer to 00:00 ... but how much? it's that an hour has 60 minutes and (8/60)*6 = 0.8

Therefore the final answer is 78 - 0.8 = 77.2

Is this correct ??? Any faster way???
1. The hour-hand describes an angle of 30° per hour, that means the hour hand moves $\dfrac{30^\circ}{60}=\dfrac{1}{2} \dfrac{ ^\circ}{min}$

2. Therefore the hour hand has made during the time 11 h 8 min an angle of

$(11\ h\cdot 60\ \frac{min}h+8\min)\cdot 0.5\ \frac{ ^\circ}{min}=334^\circ$

3. Therefore the included angle is (360°-334°)+48° = 74°

3. You are correct until this.

BUT, we should also consider that if it's 11:08 then hour hand is moved slightly closer to 00:00 ... but how much? it's that an hour has 60 minutes and (8/60)*6 = 0.8
8/60 should be multiplied by 30 degrees, which is the angle between 11 and 12.

Another way is no notice that the minute hand gains 330 degrees over the hour hand each hour. Since there is 52 minutes between 11:08 and 12:00, the gain is 330 * 52 / 60 = 286 degrees. So the small angle is 360 - 286 = 74 degrees. (I guess it's easier to imagine here that the hands move counterclockwise, but it does not matter.)

4. Hello, Narek!

What is the small angle between clock hands where it shows 11:08?

The minute hand moves: . $\dfrac{360\text{ degrees}}{60\text{ min}} \:=\:6\text{ deg/min.}$

The hour hand moves: . $\dfrac{30\text{ degrees}}{60\text{ min}} \:=\: \frac{1}{2}\text{ deg/min}$

Using $12\!:\!00 = 0^o$, label the dial from $\text{-}180^o$ to $+180^o.$

At exactly 11 o'clock, the minute hand is at $0^o$
. . and the hour hand is at $\text{-}30^o$

In the next eight minutes:

. . The minute hand has advanced: . $8 \times 6^o \:=\:48^o$
. . The minute hand is at: $0^o + 48^o\:=\:+48^o$

. . The hour hand has advanced: . $8 \times \frac{1}{2}^o \:=\:4^o$
. . The hour hand is at: . $\text{-}30^o + 4^o \:=\:\text{-}26^o$

The angle between the hands is: . $48^o - (\text{-}26^o) \;=\;74^o$

5. Alternatively (yet again),

$\frac{360}{12}=30^o\Rightarrow$ there is $30^o$ between the minute and hour hands at $11:00$.

In 8 minutes, the hour hand moves through $30^o\frac{8}{60}=4^o$ clockwise.

in 8 minutes, the minute hand moves through $360^o\frac{8}{60}=48^o$ clockwise.

Therefore the angle between the hands increases by $(48-4)^o$