Ohio Rev. Code § 5155.35
The expense of operating a district home shall be apportioned among the counties of the district in proportion to the amount of care given to former residents of the respective counties. The superintendent or administrator of such home shall keep exact records showing each day's enrollment and attendance of residents from each county, and shall report such enrollment and attendance monthly to the county auditor of each county of the district. The auditor and the county treasurer of the county in which the district home is located shall be the auditor and treasurer of the home. Preceding the occupancy of the district home, the board of trustees of the home as provided by section 5155.34 of the Revised Code shall prepare an estimate of each county's share of operating expense for the next six months. When such estimate has been filed with the auditor of each county such auditor shall draw a warrant on the treasurer for the payment of such estimated amount to the treasurer of the home, who shall credit such amount to the home fund. Semiannually thereafter, or more often if advisable, the auditors of the respective counties shall audit the accounts for the preceding period, balance the accounts of their respective counties according to the service rendered, review estimates of expenses for the next six months as submitted by the board of trustees, and agree on amounts to be transferred from the respective counties for operating expenses for the ensuing period of six months.
R.C. §5155.35