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Would Rather:
When the person himself expresses a preference on something referring to the present or future:
Subject + would rather ('d rather) + base form of verb
example:
I would rather (I'd rather) stay at home than go out right now.
When referring to the past:
Subject + would rather ('d rather) + have + past participle form of verb
I
would rather (I'd rather) have stayed at home than (have) gone out last night.
When the person expresses a preference on someone else's actions referring to the present or future:
Subject + would rather ('d rather) + object + past tense
I would rather (I'd rather) you gave up smoking.
When referring to the past:
Subject + would rather ('d rather) + object + past perfect
I would rather (I'd rather) you hadn't lied to me about your school marks yesterday.
Would Prefer:
When the person himself expresses a preference on something referring to the present or future:
Subject + would prefer + to infinitive + verb
I would prefer ('d prefer) to watch TV rather than (to) do shopping.
When referring to the past:
Subject + would prefer ('d prefer) to + have + past participle form of a verb
I would prefer (I'd prefer) to have been born somewhere else.
When the person expresses a preference on someone else's actions referring to the present or future:
Subject + would prefer ('d prefer) + object + to infinitive + verb
I would prefer ('d prefer) my son to live with me rather than (to) live abroad.
When referring to the past:
Subject + would prefer ('d prefer) + object + to + have + past participle form of a verb
I would prefer your daughter to have accepted my apology rather than (to have) ignored me last
week.