My son had this same exact problem on his 5th grade math homework which uses the Singapore math method. This kind of two-variable problem is easy to solve with algebra, but they have not learned algebra in 5th grade, so the expectation (and the challenge) is to solve this using bar models (bar models are a fundamental concept in the Singapore math method).
I think we figured it out!
1. DRAW KATHERINE'S BAR
The first thing to do is to draw a bar representing Katherine's money, and divide it into thirds. We know that 2/3 is what she has now, and 1/3 is what she gave to Sally. I actually drew the last 1/3 with a dotted line to show that it is the "ghost" of the money that Katherine used to have.
2. DRAW SALLY'S BAR
We know that after this transaction (which includes the father's donation), that Sally has the same amount of money that Katherine now has, so we can draw a second bar under Katherine's, labeled "Sally" that is exactly the same length as 2/3 of Katherine's bar. We can also divide Sally's bar by the amount that we know came from Katherine (extend the line that you used to make Katherine's first 1/3rd). We can see visually that half of the money Sally has now came from the 1/3 that Sally gave her.
What about the money that Sally has that Katherine didn't give her? We know that $68 of that came from her dad, the rest - the mystery to be solved - is what she had at first.
Sally's bar should have three labelled sections now, "from Katherine," "from dad," and "had."
3. FIND OUT HOW MUCH THEY HAVE NOW
This should be the easiest one. If they didn't buy anything and just moved around money, together, with the money Sally's dad provided should have $288+$68, or $356.
And if they have the same amount of money now, we know that they each have $178.
4. START FILLING-IN THE MYSTERIES IN THE BAR MODEL WITH YOUR FACTS
For example we know that Katherine's two remaining boxes are $178. They're equal, so we know that each box is worth $89. Now we're cooking! This is also the same amount as Sally's box that represents what Katherine gave her.
5. BRING IT HOME WITH SOME SIMPLE ADDITION AND SUBTRACTION
This is where the visualization of the bar models really help. If we've labelled the parts in Sally's bar correctly, we should see a "whole-part" problem with $89 (from Katherine) +$68 (from Dad) + mystery = $178 (what each girl has).
Or $178 - ($89+$68) = $21
Sally had $21 at first.
To check our work and logic we went back through the problem using all of the values now labeled on our bar model.
(If this isn't clear let me know and I'm happy to share a picture of our bar model).
