A person is fitted with one in old age (7)

I believe the answer is:

permian

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'old age' is the definition.
The definition and answer can be both related to time as well as being singular nouns.
Perhaps they are linked in a way I don't understand?

'a person is fitted with one' is the wordplay.
'a' becomes 'per' (eg 'one a day' means 'one per day').
'person' becomes 'man' (man is a kind of person).
'is fitted with' indicates putting letters inside.
'one' becomes 'i' (Roman numeral).
'man' placed around 'i' is 'mian'.
'per'+'mian'='PERMIAN'

'in' is the link.

Can you help me to learn more?

I've seen this clue in The Times.

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