Expert: Bobby Soltani - 4/3/2006

Question
I have this problem that I am stuck trying to figure out the answer:

Hal gives Tom as many raffle tickets as Tom has and Gary as many as Gary
has. In like manner, Tom then gives Hal and Gary as many tickets as each
then has. Similarly, Gary gives HAl and Tom as many tickets as each then
has. If each finally has 40 tickets, with how many does Tom begin?  

Answer
Hi Anthony,

This is a confusing problem.  It is best to work backwards from the last transaction.

If they each end up with 40 tickets, that means that before the last transaction, Gary must have had 80 tickets, and Hal and Tom had 20 each.  This is so because he gave them each as many as they had (20) and they all ended up with 40.  In the second transaction, Tom gave Hal and Gary as many as they had.  So, Gary must have had 40, and Hal must have had 10, since they ended up with 80 and 20, respectively.  Since Tom gave away 40+10 = 50, he must have had 50+20 = 70.  Finally, in the first transaction, Hal gave Gary and Tom half of what they had.  So, Tom must have had 35, and Gary must have had 20.  Since Gary gave away 35+20 = 55, and he already had 10, then he started out with 65. Here are the results of each step.

          Hal   Tom   Gary
4          40   40   40
3           20   20   80
2           10   70   40
1           65   35   20

The answer is that Tom started out with 35 tickets.  Try and understand the steps and let me know if you have any questions.

Bobby