I am stuck on this one problem:

Preferred Stock: $100 par, Credit Balance of $80,000
Common Stock: $10 par, Credit Balance of $270,000
Paid-In Capital in Excess of Par, Common Stock: $50,000

Dividends Declared, payable on December 20, to stockholders of record on December 13 as follows:
a. Quarterly cash dividends of $1.00 per share on common stock outstanding
b. Quarterly cash dividends of $3.00 per share on preferred stock outstanding

How do I calculate the cash dividends?

Actually, what you really need is the number of shares outstanding. The problem has already told you how much PER SHARE it's going to pay in dividends. But it's not giving the number of shares you have to pay it on.

The thing to look at here is that stock accounts are always listed at the par value. i.e. the common stock, even though worth a total of $320,000, has to be split between the par and the excess over the par. So the $270,000 is the par only. Since par is $10 per each share, and its total worth is $270,000, how many shares must you have?

You can do this with both preferred and common, and just ignore any excess amounts.