Summer Problem Solving Marathon Solution #21

Let a be Ann's current age, and let b be Barbara's age. Then we are told a + b = 44.

Now consider the constraint that "Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is". We will work out way backward through this.

When Barbara was half as old as Ann is, she was a/2. Since she is now b, that was b - a/2 years ago.

b - a/2 years ago, Ann was a - (b - a/2) = 3a/2 - b years old.

So Barbara was that age b - (3a/2 - b) = 2b - 3a2 years ago.

2b - 3a/2 years ago, Ann was a - (2b - 3a/2) = 5a/2 - 2b years old.

That's the age Barbara is now. So b - 5a/2 - 2b, or 3b = 5a/2, or b = 5a/6.

We now substitute this in a + b = 44. So a + 5a/6 = 44, or 11a/6 = 44, or a = 24.