Let's start by assigning variables to their current ages.
Let Jim's age be J.
Then, according to the problem, Beth's age is two years older than Jim's age:
Beth's age = J + 2
In three years, their ages will be:
Jim's age in three years = J + 3
Beth's age in three years = (J + 2) + 3 = J + 5
According to the problem, the sum of their ages in three years will be twice the sum of their ages three years ago:
(J + 3) + (J + 5) = 2[(J - 3) + (J + 2 - 3)]
Simplifying this equation:
2J + 8 = 2J + 2
Subtracting 2J from both sides:
8 = 2
This is a contradiction, which means our initial assumption of Jim's age being J is incorrect.
Let's try a different approach.
This time, let Beth's age be B.
Then, according to the problem, Jim's age is two years younger than Beth's age:
Jim's age = B - 2
In three years, their ages will be:
Jim's age in three years = (B - 2) + 3 = B + 1
Beth's age in three years = B + 3
According to the problem, the sum of their ages in three years will be twice the sum of their ages three years ago:
(B + 1) + (B + 3) = 2[(B - 3) + (B + 2 - 3)]
Simplifying this equation:
2B + 4 = 2B - 2
Subtracting 2B from both sides:
4 = -2
Another contradiction.
This means that there is no solution that satisfies the conditions of the problem. There must be a mistake in the problem statement.