If perferred shareholders had $300,000 in stock outstanding and they were paid $340,000 for one year because of arrears in prior years that accumulated what would the perferred dividend per share be?

$340,000 has been paid in one year and not for one year, since their dividends remain unpaid for several years. There must be some agreed rate of preferred dividend, say 10%, that makes dividends payment as $30,000 per year. If the par value of shares is say $1, that means there are 300,000 shares outstanding. Therefore the per share dividend is $0.10 per share per year. Based on such information from your problem just do the required calculations.

Anything per something is always the first thing divided by the second. It can be used for anything in any subject.

Hence dividends per share. Divide the first by the second.