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Die Cafeteria ist in der dritten Etage.

Etage is a feminine noun, in is dative. Der Dritte is the third (masculine noun).

Why is it dritten here?

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The phrase der Dritte (or der dritte Stock) has no -n because it is nominative singular. Im dritten Stock/Stockwerk or in der dritten Etage have -en because they are dative singular.

masculine:
Das ist der dritte Stock. (nominative)
vs.
Wir suchen den dritten Stock. (accusative)
Wir sind im dritten Stock. (dative)
die Bewohner des dritten Stocks (genitive)

neuter:
Das ist das dritte Stockwerk. (nominative)
Wir suchen das dritte Stockwerk. (accusative)
vs.
Wir sind im dritten Stockwerk. (dative)
die Bewohner des dritten Stockwerks. (genitive)

feminine:
Das ist die dritte Etage. (nominative)
Wir suchen die dritte Etage. (accusative)
vs.
Wir sind in der dritten Etage. (dative)
die Bewohner der dritten Etage. (genitive)

Weak adjectives, i.e. those following an inflected article such as der, den, im, die, der, have -e in the nominative singular and -en in forms distinct from the nominative singular. Dative and genitive are always distinct from the nominative.

Note that in the masculine singular, nominative der and accusative den are distinct and therefore the adjective has -en in the accusative: den dritten Stock. The dative and genitive also have -en.

In the neuter and feminine singular, nominative and accusative are identical and the adjective takes -e in both cases: das erste Stockwerk, die erste Etage. Only the dative and genitive have -en.

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  • Thanks a lot for your examples, I know that, in dative case, all weak adjectives (e.g: ordinal numbers) for all genders should +en. That is a rule which I couldn't figure out before. – Bằng Rikimaru Jun 19 '19 at 8:47
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It answers the question "Wo ist die Cafeteria?" which takes the Dativ case.

https://www.das-akademie.de/en/akkusativ-and-dativ-wo-and-wohin/

Nominativ: Die dritte Etage ist die höchste Etage im Haus.
Genitiv: Die Beschilderung der dritten Etage fehlt.
Dativ: "Die Cafeteria ist in der dritten Etage.
Akkusativ: "Ich gehe jetzt in die dritte Etage.

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  • Thanks, but I still don't understand why dritte in dative is dritten? Is there any reason for it? – Bằng Rikimaru Jun 18 '19 at 11:51
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    It's the adjective declension after the definite article. The rules for that seem somewhat arbitrary but after a while you find they are mostly about avoiding tongue-twisters. – Janka Jun 18 '19 at 13:10
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    @BằngRikimaru Declension generally has no "reason". To see why declension tables cannot be predicted from any underlying principle, just consider that German declension is slowly deteriorating from "many different endings" (Old High German) to "almost nothing" (now) and that the language has remained active and vibrant throughout it all. – Kilian Foth Jun 19 '19 at 6:56
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The question wo? requires the dative case. Der Etage is the Dative case of "die Etage":

die Etage (Nominative)
der Etage (Genitive)
der Etage (Dative)
die Etage (Accusative)

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  • But I still don't understand why it is "dritten" but not "dritte". – Bằng Rikimaru Jun 18 '19 at 11:55
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    It's the adjective's declension: die dritte Etage, der dritten Etage, der dritten Etage, die dritte Etage. The masculine declension would be der dritte Mann, des dritten Mannes, dem dritten Mann, den dritten Mann, the neuter declension das dritte Kind, des dritten Kindes, dem dritten Kind, das dritte Kind. – Thorsten Dittmar Jun 19 '19 at 14:06

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