Is it grammatically correct to use present perfect and past simple tense in the following sentence?
She has been mad at John for five days, as she couldn't go to Paris because of him.
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Yes. Being mad is a condition which continues to the present, so the perfect is appropriate. (Not) going to Paris is an event which is over, so the past fits.
... as she hasn't been able to go to Paris (perfect) would be fine, but would have a slightly different meaning: it would treat the possibility of her going to Paris as something which might have happened throughout the five days.