Fasting Rules (2:184), (2:185)
2:184 is read as allowing a wealthy person to buy himself out of fasting by feeding the poor. 2:185 is read as allowing no such compensation. The claim is that these two verses contradict each other.
The false claim: 2:184 lets the rich pay to skip fasting while 2:185 permits no such opt-out - a contradiction.
2:184 contains no concession for the rich. The claim introduces a category - the wealthy person paying to avoid fasting - that the verse does not mention.
[2:184] Specific days (are designated for fasting); if one of you is ill or traveling, an equal number of other days may be substituted. Those who can barely fast shall make up for it by feeding a poor person for each day of breaking the fast. If one volunteers (more righteous works), it is better. But fasting is the best for you, if you only knew.
The concession of feeding a needy person is granted to those who cannot fast due to a condition that makes fasting genuinely difficult or impossible - specifically, chronic or permanent illness. Wealth is not the criterion. The ability to feed a poor person is mentioned because someone who is permanently ill and too poor to provide food for another has no obligation either way, since God does not place hardship on anyone beyond their capacity. The concession addresses inability to fast combined with sufficient means to compensate in another way. It has nothing to do with a healthy wealthy person purchasing an exemption.
[2:185] Ramadan is the month during which the Quran was revealed, providing guidance for the people, clear teachings, and the statute book. Those of you who witness this month shall fast therein. Those who are ill or traveling may substitute the same number of other days. GOD wishes for you convenience, not hardship, that you may fulfill your obligations, and to glorify GOD for guiding you, and to express your appreciation.
2:185 addresses those who are well and present - they fast. The concessions of 2:184 apply to those with genuine incapacitating conditions. The rules that emerge from both verses together are straightforward: those in good health fast Ramadan; those travelling or temporarily ill make up the missed days later; those with permanent illness who can afford it feed a needy person for each missed day; those with permanent illness who cannot afford even that are exempt entirely, because God wants ease and not hardship.
The two verses govern different categories of people under different circumstances. Together they form a complete and internally consistent set of fasting rules with no contradiction at any point.