Why does the adjective “wenig” require declension in one instance, but not in another?

Question

1^st: Das wenige Geld muss ausreichen.
2^nd: Ich hab genauso wenig Ahnung wie du.
3^rd: Hier haben wir wenig Abwechslung.

I’m not sure why in the second and third sentences, you do not need to say wenige, as opposed to how it works in the first sentence.

Answer 1

The difference is, that in (1) the noun phrase is definite (it contains a definite article), but in (2) and (3) the noun phrase is indefinite (contains an indefinite article or no article). Definiteness is one of the four grammatical properties, that influence declension of adjectives.

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More General:

There are four grammatical properties, that influence the declension of an adjective (when it is used attributively):

1. (grammatical) gender
   The gender is derived from the noun to which the attributive adjective refers:

   • masculine:
     der Löffel – ein kleiner Löffel (a small spoon)
   • feminine:
     die Gabel – eine kleine Gabel (a small fork)
   • neuter:
     das Messer – ein kleines Messer (a small knife)

2. (grammatical) case
   The case is given by the grammatical function of the part of speech, to which the attributive adjective belongs:

   • nominative:
     In diesem Glas ist das frische Wasser. (The fresh water is in this glas.)
   • genitive:
     Achte auf das Glitzern des frischen Wassers. (Watch the sparkles of the fresh water.)
   • dative:
     Ich habe dem frischen Wasser nichts hinzugefügt. (I added nothing to the fresh water.)
   • accusative:
     Ich trinke das frische Wasser. (I’m drinking the fresh water.)

3. (grammatical) number
   The number of an attributive adjective matches with the number of the noun to which it refers:

   • singular:
     ein roter Apfel (a red apple)
   • plural:
     viele rote Äpfel (many red apples)

4. (grammatical) definiteness
   Only articles and attributive adjectives are influenced by definiteness, nouns are not. In a noun phrase, definiteness of article and adjective must match:

   • definite:
     der rote Apfel, die roten Äpfel (the red apple, the red apples)
   • indefinite:
     ein roter Apfel, rote Äpfel (a red apple, red apples)

_{Note 1: Definitenes is a relatively new concept in German grammar. The classification of German (and English) grammar is strongly influenced by the classifications for Latin language. But because Latin has no definiteness, this classification for long times was not used in German grammar. Instead you will find descriptions of strong, weak and mixed declension in many grammar books. But in fact strong/weak/mixed is based on definiteness.}

_{Note 2: Please consult your grammar book for more details.}

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Also important:

There are three ways to use adjectives:

1. attributvely
   Used as an attribute of a noun.

   Das ist ein lustiger Clown. (This is a funny clown.)

2. predicatively
   Used after a form of sein (to be), bleiben (to stay) and werden (to become). Here the adjective is not an attribute of the clown, but via the verb sein/bleiben/werden still describes the noun.

   Dieser Clown ist lustig. (This clown is funny.)

3. adverbially
   User after any other verb. Here the adjective no longer describes the noun. It describes the action (i.e. the verb).

   Dieser Clown lacht lustig. (This clown laughs funny.)

This is important for declension, because only attributive adjectives will be declined. Neither predicative nor adverbial adjectives will be declined:

• attributive:
  Das ist ein junger Mann. Das ist eine junge Frau.
• predicative:
  Der Mann ist jung. Die Frau ist jung.
• adverbial:
  Der Mann sieht jung aus. Die Frau sieht jung aus.