Initial Amount Sabrina & Samantha had

But what was that? 

This could be solved using simultaneous equations but going in reverse makes it easy.

At each stage, one's money is doubled. At last stage, Samantha doubled money to 80 by receiving 40 from Sabrina. So both had 40 before that.

How Sabrina had 40? Since Samantha gave her 20 at previous transaction. So Samantha had 40+20=60 & Sabrina had 20 before this transfer. 

And how Samantha had 60? Since she received 30 from Sabrina at previous transfer. So Sabrina had 30+20=50 & Samantha had 30 before the transfer.

Hence at start , Sabrina had 50 & Samantha had 30!