Every problem has the answer built in. No-one is asking you to make up a process.
Use the data given and simple equations, manipulate the equations, and the answer pops out.
The sum of the ages of A and his father is 100.
When A is as old as his father is now, he will be five times as old as his son B is now.
B will be eight years older than A is now, when A is as old as his father is now.
Start with the first statement.
1) A + F = 100
The solution revolves around A being as old as his father is now, but x years from now
2) F = A + x
Substitute the value of F from equation 2 into equation 1
3) A + (A + x) = 100
At that time in the future, A will be 5 times as old as his son is now
4) A + x = 5B
At that future date, the son will be 8 years older than A is right now
5) B + x = A + 8
Solving that equation for B gives us the son's age right now
6) B = A - x + 8
That takes care of the preliminaries.
Substitute the value of B from equation 6 into equation 4
7) A + x = 5 (A - x + 8)
Multiply through on the right side
8) A + x = 5A - 5x + 40
Add 5x to both sides, subtract A from both sides
9) 6x = 4A + 40
Divide both sides by 6
10) x = 2/3 A + 6 2/3
Substitute the value of x from equation 10 into equation 3
11) A + A + (2/3 A + 6 2/3) = 100
Combine the A terms, then subtract 6 2/3 from both sides
12) 2 2/3 A = 93 1/3
Multiply both sides by 3
13) 8A = 280
Divide both sides by 8
14) A = 35
We now know that A is 35 years old.
From equation 1, we find the age of A's father
A + F = 100; 35 + F = 100; F = 100 - 35; F = 65
Putting that into equation 2, we find how far we are looking into the future
F = A + x; 65 = 35 + x; 30 = x
Substituting the appropriate values into equation 6, we find B's age
B = A - x + 8; B = 35 - 30 + 8; B = 13
Check: 30 years in the future, B will be
B + 30 = ??; 13 + 30 = 43
That should be 8 years older than A is now
A + 8 = ??; 35 + 8 = 43
Done
