Learn German: Der fünfzehnte und der sechzehnte Mai sind frei. - The fifteenth and the sixteenth of May are free.

For numbers 2 through 19, you form the ordinal number by adding the suffix -te to the regular number. So fünfzehn (fifteen) becomes fünfzehnte (fifteenth), and sechzehn (sixteen) becomes sechzehnte (sixteenth).

Why do these ordinal numbers end in -e?

Ordinal numbers take adjective endings. They describe the month Mai. All months in German are masculine, and because these dates are the subject of the sentence (nominative case) following the definite article der, the correct ending is -e.

Why isn't the word Mai written after fünfzehnte?

Just like in English, you can drop the first noun to avoid repeating it. Instead of saying der fünfzehnte Mai und der sechzehnte Mai, you just say der fünfzehnte. It still keeps its masculine ending as if Mai were right there.

Does frei here mean the days cost no money?

No, when talking about days on a calendar, frei means having time off from work or school. If you want to say a product or service is free of charge, you would usually use kostenlos or gratis.