To solve this problem, we first need to convert the mixed numbers to improper fractions.

4 1/6 = (6*4 + 1)/6 = 25/6

2 7/8 = (8*2 + 7)/8 = 23/8

Now, the expression becomes:

(25/6 - 23/8) * 2.4 : (6/1 + 5/1)

Next, we need to subtract the fractions and divide:

(25/6 - 23/8) = (100/24 - 69/24) = 31/24

Now the expression becomes:

(31/24) * 2.4 : (6/1 + 5/1)

Multiplying 31/24 by 2.4 gives us:

(31/24) * 2.4 = 74.4/24 = 3.1

So now the expression becomes:

3.1 : (6/1 + 5/1)

Adding 6/1 and 5/1 gives us:

6/1 + 5/1 = 11/1 = 11

Finally, dividing 3.1 by 11 gives us the result:

3.1 / 11 = 0.2818181818 or approximately 0.282

Therefore, the final answer is approximately 0.282.