Two years ago Cathy was three times as old as her cousin Claire. Two years before that, Cathy was four times as old as Claire. In how many years will the ratio of their ages be 2:1?

Let's denote Cathy's current age as C and Claire's current age as c.

From the problem, we know that:

1. Two years ago, Cathy was three times as old as Claire. This can be written as: C - 2 = 3 * (c - 2).
2. Four years ago, Cathy was four times as old as Claire. This can be written as: C - 4 = 4 * (c - 4).

We now have a system of two equations that we can solve to find the current ages of Cathy and Claire.

Solving the system of equations, we find that C = 14 and c = 6.

We want to find out when the ratio of their ages will be 2:1. This can be written as: (C + x) / (c + x) = 2 / 1, where x is the number of years in the future.

Substituting C = 14 and c = 6 into the equation, we get: (14 + x) / (6 + x) = 2 / 1.

Solving this equation for x, we find that x = 8.

So, in 8 years, the ratio of Cathy's age to Claire's age will be 2:1.

Let's denote Cathy's current age as C and Claire's current age as Cl.

From the problem, we know that:

1. Two years ago, Cathy was three times as old as Claire. This can be written as: C - 2 = 3 * (Cl - 2).

2. Two years before that (which is four years ago from now), Cathy was four times as old as Claire. This can be written as: C - 4 = 4 * (Cl - 4).

We now have a system of two equations that we can solve to find the current ages of Cathy and Claire.

Solving the system of equations, we find that Cathy is 8 years old and Claire is 4 years old.

We want to find out when the ratio of their ages will be 2:1.

Let's denote the number of years in the future when this will happen as x.

So, we want to find x such that (C + x) / (Cl + x) = 2 / 1.

Substituting the current ages of Cathy and Claire into this equation, we get:

(8 + x) / (4 + x) = 2 / 1.

Solving this equation for x, we find that x = 4.

So, in 4 years, the ratio of Cathy's age to Claire's age will be 2:1.