A bank transfers sum of money to Mr. Cathrall, who spends $20,000 of

The key here is that Mr. Cathrall receives money twice (once from the bank, and once from Mr. Leeds). Let x denote the lump sum amount from the bank.

Bank to Cathrall: x
Cathrall to Carlyle: (x - 20,000)*(3/4)
Carlyle to Leeds: (((x - 20,000)*(3/4))-15,000)*(2/3)
Leeds to Cathrall: (((((x - 20,000)*(3/4))-15,000)*(2/3))-5,000)*(4/5)

Now we know that the total amount received by Cathrall would be the money he received from the bank PLUS the money he eventually receives from Leeds. This amounts to $260,000; re-expressed as:
x + (((((x - 20,000)*(3/4))-15,000)*(2/3))-5,000)*(4/5) = 260,000

Solving for x, we get $200,000 (the money received from the bank). We can then plug x into the expression for Carlyle to Leeds, obtaining $80,000.

Answer (IMO): A