Must someone who grew up in the West and did not learn the rulings of fasting until after reaching the age of twenty-five make up the fasts they missed? And how is the make-up done?

Whoever intentionally broke their fasts in multiple Ramadans must calculate all the Ramadans they missed since puberty. Then, they should make up all the missed days according to their capacity, as Allah says: "Allah does not burden a soul beyond its capacity." They are only required to make up the fasts and are not obligated to offer expiation (kaffarah). This is the view of the Shafi'i and Ahmad schools of thought.

As for whoever intentionally vomited, they must also make up the fast, based on the Prophet's (peace be upon him) saying: "Whoever intentionally vomits, let him make up the fast." The same applies to whoever engaged in sexual intercourse during the day in Ramadan, based on the Prophet's (peace be upon him) saying: "The one who engaged in sexual intercourse during the day in Ramadan was commanded to make up a day in its place."

As for the hadith "Whoever broke his fast for one day in Ramadan without an excuse, the fasting of a lifetime will not compensate for it," it is weak by the agreement of the Hadith scholars and should not be acted upon.