"This is a difficult problem for a fifteen-year-old boy."
My guess is that "fifteen-year-old" forms an adjective, but I'm not really sure.
Thank you in advance for your help.
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In your sentence
is used as an adjective for boy
In the case of
He is a fifteen-year-old.
it is used as a compound noun.
in contrast to
He is fifteen years old
to say how many years he has been alive.
http://www.grammar.cl/english/compound-adjectives.htm Compound adjectives are written with hypen, and nouns after numbers don't get a plural-s.