tdol said:
What is the difference between them?

They took their time.
their is a possessive pronoun: as a possessive marker it modifies the noun 'time' and as a pronoun it refers back to 'They'.

They took they're time.
they're is a subject pronoun (i.e. they) plus a contracted verb (i.e. are). Simple sentences have only one verb phrase, not two as in 'They took' and 'they're'. Subject pronouns (e.g. They, We, She) occur before the verb and object pronouns (i.e. them, us, her) occur after the verb. The pronoun 'they' cannot modify a noun. 'their' is used to modify a noun.

They took there time.
there can be an adverb of place (e.g. I agree with you
there) or a noun of location (e.g. They took us
there - to that place).
Cas
