angle in a clock

[r_walid]

if a clock points to 4:40, what is the angle (minor) between the hour and minute handle?

[palvarez]

Here is a little trick.

Convert time into mins. 4:40 = 4(60)+40 = 280

The angle between min n hour handle = arccos(cos(5.5*above))

5.5* 280 = 1540, get the remainder wrt 360 = 1540 - 360(4) = 1540 - 1440 = 100

Answer is 100 degrees

----------------------------------explanation---------------------------

In an hour, minutes hand travels 360 degrees; but hours hand travels 30 degrees.
The relative speed: 330 degrees per hour or 330/60 degrees per min or 5.5 degrees per min

4:40 = 280 mins.

280 * relative speed = 280*5.5 = 1440
get rid of multiples of 360.

[RoadtoIVY]

That was quite an explanation but one should stay away from this kind of complex calculation in order to find a solution of the problem.
I found an easier formula for this:

Angle = |30*H - (11/2)*M|, where H is the hours hand and M is the minutes hand

in this case: H = 4, M= 40
30*4 - (11/2)*40 = 120 - 220 = -100 = |-100| = 100

[MG368]

hey RoadtoIvy,

I didn't get your explanation.Could you elaborate?GMAT destroyer's explanation just went over my head!

[RoadtoIVY]

hey RoadtoIvy,

I didn't get your explanation.Could you elaborate?GMAT destroyer's explanation just went over my head!

Here you go:

H = represents hours hand
M = minutes hand
So when you say 4.40, it means: H = 4 and M =40
Now use this expression to find the angle:
mode of(30*H - (11/2) * M) = mode of(30*4 - (11/2) * 40) = |120 - 220| = |-100| = 100

Hope it is more clear now. [/u]

[MG368]

thnx i got it now!

[tgf]

Another less advanced solution than those posted:

The minute hand travels 360 deg i one hour. The hour hand travels (360/60)*5 =30 deg in one hour (360/60 is 6 degrees pr minute and in one hour the hour hand passes 5 minute markers).

If you assume the minute hand is at 40 min and hour hand is at exactly 4, the angle between the hands is (20/60)*360=120

But you obviously know this is to much since the hour hand has moved. All you need to do is subtract the degrees the hour hand has moved from 4 to 4:40, which is 30*(40/60)=20. (30 degrees per hour times 4/6 of an hour)

So the answer is 120-20=100.