Originally Answered: Critical Thinking?
Have you ever noticed that it is often easier to solve a maze by starting at the end instead of the beginning?
That's also true with this problem.
You know that at the end, all 3 of them have 40 tickets. That means that there are 120 tickets total. During the course of this problem, they each pass the tickets back and forth between themselves, but never give (or get) tickets to (or from) anyone else. So keep in mind that, at each step along the way, the total number of tickets is 120.
To answer your question, we'll work through each step in reverse.
To cut back on typing, Abe = A, Jorge = J, and Rafer = R.
Altogether, there were 3 ticket "trades".
At the end.
A = 40
J = 40
R = 40
note: 40 + 40 + 40 = 120
During the final trade, Jorge doubled the amount of tickets that A & J had. So, before trade 3,
A = 20
R = 20
J = 80 (J gave away 40, and still had 40 left)
note: 20 + 20 + 80 = 120
Now we need to step back to the trade before that. During trade 2, Rafer doubled the amount of tickets that A & J had. So, before trade 2,
A = 10
J = 40
R = 70 (he gave away 50 and had 20 leftover)
note: 10 + 40 + 70 = 120
Now we step back to look at the 1st trade, when Abe doubled the amount of tickets that J & R had. So, before that first trade,
J = 20
R = 35
A = 65 (Abe gave away 55 and still had 10 left)
note: 20 + 35 + 65 = 120
These are the amounts they each started with then.
Abe had 65. Jorge had 20. Rafer had 35.
To check it, start with those amounts and step through the problem forward, as it is was given to you, to see if they all end up with 40.