There is no solution in whole numbers; the statement of the problem is faulty.
"...5 times older than..." is NOT
the same as "...5 times as old as...".
If James's age is x and his father is 5 times AS OLD AS James, then his father's age is clearly 5 times x, or 5x.
But if James's age is x and his father is 5 times OLDER THAN James, then his father's age is x, PLUS 5 TIMES MORE x, or x+5x = 6x.
So in this problem, if x is James's age, then his dad's age is 6x and his sister's age is (1/2)x. That, with the given information about the sum of their ages 2 years from now, leads to a non-integer value for x, making the problem faulty and unsolvable.
Unfortunately, in everyday language, "5 times older than" and "5 times as old as" are carelessly used to mean the same thing; but grammatically that is incorrect; they mean different things.
Bottom line:
*****************************************************************************
* Phrases like "3 times older than" or "4 times greater than" should NEVER *
* be used in the statement of a math problem. *
*****************************************************************************