Let's go forward and then backward on this problem.

Ann gave Betty as many cents as Betty had.

Ann gave Betty what she already had, which means that you have:

What Betty has now = What Betty already had + What Ann gave Betty

But this is:

What Betty has now = What Betty already had + What Betty already had

What Betty has now = 2*What Betty already had

Now, we look at the next statement:

Betty then gave Ann as many cents as Ann already had

Now, we need to find out what she has:

What Ann has now = What Ann already had + What Betty gave Ann

What Ann has now = What Ann already had + What Ann already had

What Ann has now = 2*(What Ann already had)

But now Betty has less cents than before because she gave Ann what Ann already had:

What Betty has now = 2*(What Betty already had) - What Ann already had

So, at this point they both have 12 cents, which means:

What Ann has now = 2*(What Ann already had) = 12

This give you:

(Ann already had) = 6 cents

Now, you know:

What Betty has now = 2*(What Betty already had) - What Ann already had = 12

2*(What Betty already had) - 6 = 12

2*(What Betty already had) = 18

What Betty already had = 9 cents

And there you go, you can work through it to make sure.

Ann starts with 6 cents, Betty with 9

Ann gives betty what betty already has, 9 + 9 = 18

Betty gives ann what ann already has, 6 + 6 = 12, 18 - 6 = 12

And there you go. This is a slightly harder problem than I expected though.