Tumult Over Tadir

Shimon and Boruch had just closed their Gemaras after finishing the daf. Boruch leaned back with a grin.
“Wow,” he said. “Today’s daf was incredible. We just learned tadir v’she’eino tadir, tadir kodem in Zevachim 89. I’m officially a posek now—ready for any shailah that comes my way.”
Shimon chuckled. “I’m happy to discuss, but just for the record—this is only for informational purposes. Any actual shailah should be asked to your rabbi.”
Boruch frowned. “Why are you telling me that?”
Shimon shrugged. “I don’t know… Somehow, every private conversation we have ends up published in a newspaper! Anyway, let me give you a scenario. At the end of davening on the second day of Rosh Chodesh Elul, the custom is to say Borchi Nafshi because it’s Rosh Chodesh. But we also begin saying L’Dovid Hashem Ori because it’s the first day of Elul. Which should be said first?”
Boruch waved dismissively. “Easy! Borchi Nafshi is only said once or twice a month. L’Dovid is recited for over a month and a half straight, all the way to Shemini Atzeres. So L’Dovid is much more common—you say the more common one first.”