What's half of $25?

So I was at a nameless doctor's office and a man came in to pay his bill.  
"I would like to pay on my bill, how much do I owe?" he asked the financial secretary.

"You owe $25.00, how much would you like to pay?"

"I'll pay half," the man responded.  "How much would that be?"  

The financial secretary proceeded to get out her calculator.

I couldn't resist, I said, "It would be $12.50 right?  Half of $24 is $12, split the last dollar in half to get 50 cents more."

Both the man and financial secretary looked at me, then looked at each other,  then proceeded to take out their calculators.  

Guess they didn't trust my reasoning :)