Suppose the 5 numbers are a,b,c,d,e

where a ≤ b ≤ c ≤ d ≤ e 

Then: 

1. c=5 since it is the median and the median is the middle number when 
there are an odd number of numbers, and 5 is an odd number of numbers.

2. a=b=4 since the mode is 4. Reasoning: There must be more 4's than 
anything else.  4 is less than 5, and there must be more than 1 4. 
But there can only be two numbers less than 5, and that can only be 
if a=b=4. (We also know that d and e must be different and greater 
than 5, but we may not need that.)

3. e-a=7 since the range is 7, and since a=4, e-4=7, so e=11

So a=4, b=4, c=5, d=?, e=11

4. (a+b+c+d+e)/5=6 since the mean is 6. Therefore
  (4+4+5+d+11)/5=6
        (24+d)/5=6
            24+d=30
               d=6

So the numbers are 4,4,5,6,11   

Edwin