# How to deduce the simplified equation for angle between clock hands?

I'm trying to understand how wikipedia simplifies the equation for the angle between clock hands.

https://en.wikipedia.org/wiki/Clock_angle_problem

The angle between clock hands can be found by

H = hour M = minute

step 1:

| 0.5 degrees x (60 X H + M) - 6 degrees x M |


which is simplified to:

step 2:

| 0.5 degrees x (60 x H x M) - 0.5 degrees X 12 x M |


I understand the above step because 0.5 x 12 = 6, but I do not under stand the following step where the further simplify the equation:

step 3:

| 0.5 degrees x (60 x H - 11 x M) |


Where is the 11 coming from? It seems to me that they pulled this out of thin air. Could someone please explain to me how this final equation is derived? Specifically how do you get from step 2 to step 3? Thanks for the help.

• 0.5 degrees x (60 x H + M) - 0.5 degrees x 12 x M = (0.5)x[60H+M - 12M]=(0.5)x[60H +(M - 12M)] = (0.5)x[60H - 11M]. Commented Jul 17, 2016 at 19:57

You have a mistake ( maybe a typo) in step 1. The correct formula is: $$|0.5°(60\cdot H+M)-0.5°\cdot 12 \cdot M|$$ that becomes: $$|0.5°(60\cdot H+M-12 \cdot M)|=|0.5°(60\cdot H-11 \cdot M)|$$
• $0.5$ is a common factor, so you can use distributivity of the multiplication ..... Commented Jul 17, 2016 at 20:38
The problem is with your equation in step 1. You have "60 X H X M". It should be "60 X H + M" Then you can write $0.5 \times (60 \times H + M)-6\times M=0.5 \times (60 \times H + M)-0.5\times 12\times M=0.5 \times (60 \times H + M -12\times M)=0.5 \times (60 \times H -11 \times M)$