Question 1106200
As posted, I interpret "the third" to mean the third number in a list of numbers,
where the first, second, and third are in the ratio 4:5:6 respectively.
That will not yield 3 integers, but fractions are numbers too.
 
Let the numbers be {{{4x}}} , {{{5x}}} , and {{{6x}}} .
That makes sure that they are in the ratio of 4:5:6. 
{{{6x+4x=10x}}} is the sum of the largest and the smallest.
{{{6x+55}}} is the sum of the third and 55.
So, the equation to solve is
{{{10x=6x+55}}}
{{{10x-6x=55}}}
{{{4x=highlight(55)}}} is the smallest number
From {{{4x=55}}} we get
{{{x=55/4}}}
{{{5x=5(55/4)=highlight(275/4)}}} , and
{{{6x=6(55/4)=6*55/4=3*55/2=highlight(165/4)}}} .