A 9kg LPG cylinder holds roughly 441 megajoules of usable energy (9kg × 49 MJ/kg). How long it lasts depends entirely on how hard you burn it.
On an average two-plate home stove at medium flame (~9 MJ/h), you'll get around 45–50 days if you cook for about an hour a day. Heavy cooks (2+ hours daily) will burn through the same bottle in 3–4 weeks.
Gas geysers are a different story. A 20 L/min instant geyser at full burn pulls ~80 MJ/h — a 9kg cylinder will only give you about 5–6 total hours of hot water, or roughly 7–10 days at two showers a day.
Rule of thumb: for cooking-only households a 9kg cylinder is convenient and cheap. For hot water or heating, upgrade to a 19kg or 48kg to save money per kilogram and reduce refill trips.