Your second model should be Betty 5 units and Amy 1 unit.

Working backwards , Amy gives back 2 units to Betty so she no longer has 3 units now. She has only 1 unit now.

So the ratio at this stage is :

Amy : Betty = 1 : 5

Next, recall that Amy gave ⅓ of her share to Betty at first. That's 1 unit of 3. So after that, Amy would have 2 units left.

But our ratio now is 1 : 5. This tells us that the 2 units actually make up 1 bigger unit.

So change the ratio.

Amy : Betty = 1 : 5 = 2 : 10

Now work backwards again.

Amy takes 1 unit back from Betty.
So Ratio at first = 3 : 9 = 1 : 3

Total bigger units at first = 1 + 3 = 4

Since they shared $360 at first,
$360 = 4 bigger units
1 bigger unit = $360 ÷ 4 = $90

Amy at first = $90
Betty at first = 3 x $90 = $270

Alternatively,

After working backwards the first time,

Ratio of Amy : Betty = 1 : 5
Total units here = 6

This is a 'total unchanged, internal transfer' question so the total is always $360.

So $360 = 6 units
1 unit = $360 ÷ 6 = $60

So at this stage, Amy has $60.
Since Amy gave ⅓ of her $ to Betty at first, she has ⅔ of her original amount of $ after that.

Therefore ,⅔ of her original amount $ = $60
⅓ of her original amount of $ = $60 ÷ 2 = $30
Her original amount of $ = $30 x 3 = $90

Betty's $ at first = $360 - $90 = $270