Rent proration question

[Nathaniel C.]

Hypothetical: You purchase a house with a tenant in place and close in the middle of the month. There were no adjustments for prepaid rent at closing. Let's say the tenant's lease is for $1,000 per month, beginning February 10, 2019, and you closed on the house on March 15, 2019. Property manager required the tenant to pay a full month's rent upfront along with the security deposit at the time of lease on February 10, so $1,000 security deposit plus $1,000 in rent. In March, property manager only charges a partial month's rent ($1,000 * 21/31 days = $677.42), since the tenant was prepaid through March 10. PM will charge tenant the full month's rent on the first of every month going forward. 

PM issues you a payment to you for the first month based on a gross rent of $349.64. They based this on the idea that $677.42 is the rent for March, and you owned the property for 16 out of 31 days ($677.42 * 16/31). I believe this is incorrect because the seller is essentially getting credited double on the proration - they owned the property for 33 days and received $1,327.78, while you owned the property for 16 days and received only $349.64. I think the fair thing to do would be to treat March's rent as being $1,000 (although 10 days of that was prepaid), and divide it pro rata between the buyer and seller. Am I correct that the new owner got the short end of the stick, or is this an industry standard?