To adapt is "to adjust," "to make suitable." Note the second syllable (dapt), which resembles apt, meaning "fit" or "suited to the purpose." Adept has something of the meaning of apt ("skilled," "proficient"): one can say "Bob was apt in science" or "Bob was adept in science." Adopt means "to accept" or "to take as one's own." "You must adapt yourself to this situation." "He is adept in dancing the latest steps." "I shall adopt your proposal." "This dress designer is adept in adapting styles from abroad and finds that women here adopt them eagerly."
Dictionary of problem words and expressions. Harry Shaw. 1975.