Royal Academy to copy the frantic pace of city life (3-4)

I believe the answer is:

rat-race

I'm a little stuck... Click here to teach me more about this clue!

'frantic pace of city life' is the definition.
I can't tell whether this definition defines the answer.

'royal academy to copy' is the wordplay.
'royal academy' becomes 'RA'.
'to copy' becomes 'trace' (I've seen this in another clue).
'ra'+'trace'='RAT-RACE'

'the' is the link.

Can you help me to learn more?

(Another definition for rat-race that I've seen is " futile competition".)

I've seen this clue in the Sydney Morning Herald.

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