# Thread: Critical Thinking System of 3 Equations Problem

1. ## Critical Thinking System of 3 Equations Problem

Abe gives Rafer as many raffle tickets as Rafer already had and Jorge as many as Jorge already had. Then Rafer gives Abe and Jorge as many tickets as each of them has. Similarly, Jorge gives Abe and Rafer as many tickets as each of them has. They each end up with 40 tickets. How many tickets did Rafer have at the beginning?

I know Abe, Jorge, and Rafer all have 40 which gives me 120. But now I'm trying to work backwards or create equations. Please help and explain.

2. ## Re: Critical Thinking System of 3 Equations Problem

Originally Posted by petey615
Abe gives Rafer as many raffle tickets as Rafer already had and Jorge as many as Jorge already had. Then Rafer gives Abe and Jorge as many tickets as each of them has. Similarly, Jorge gives Abe and Rafer as many tickets as each of them has. They each end up with 40 tickets. How many tickets did Rafer have at the beginning?

I know Abe, Jorge, and Rafer all have 40 which gives me 120. But now I'm trying to work backwards or create equations. Please help and explain.
I would use a table:

1. Let a, r, j denote the number of tickets of Abe, Rafer and Jorge.

2. Then they have:
$\displaystyle \begin{array}{ccccccc}\text{step}&|&\text{Abe} &|& \text{Rafer} &|& \text{Jorge} \\ 0&|& a &|& r &|& j \\ 1 &|& a-r-j &|& 2r &|& 2j \\ 2 &|& 2(a-r-j) &|& 2r-(a-r-j)-2j &|& 4j \\ 3 &|& 4(a-r-j) &|& 2(3r-a-j) &|& 4j-2(a-r-j)-(3r-a-j) \\ \end{array}$

3. The terms in the last line have the value 40. That means you have a system of 3 simultaneous equations.
Spoiler:
I've got (a,r,j)=(65, 35, 20)