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Tie someone down to something vs Tie someone up with something

tie someone down to somethingrestrict a person to the terms of an agreement, etc.:

  • If he tries to cheat, we must tie him down to his contract.

tie someone up with something

1. connect or associate a person with smth.:

  • Isn’t this company closely tied up with the continental industry?

2. fully involve a person with smth.:

  • I’m afraid I can’t help you now—I’m too tied up with other things.

Note: The expression is not equivalent in meaning to the phrase rope someone into something—persuade or induce a person to do smth.:

  • I was roped into the school dancing troupe and we performed in front of an audience.