If your rental situation is governed by Chapter 59.18 RCW (Residential landlord-tenant act), you should review RCW 59.18.200 (Tenancy from month to month or for rental period – Termination – Armed Forces exception – Exclusion of children – Conversion to condominium – Notice). The statutes are at http://apps.leg.wa.gov/RCW/default.aspx?cite=59.18 .
RCW 59.18.200 provides: "(1)(a) When premises are rented for an indefinite time, with monthly or other periodic rent reserved, such tenancy shall be construed to be a tenancy from month to month, or from period to period on which rent is payable, and shall be terminated by written notice of twenty days or more, preceding the end of any of the months or periods of tenancy, given by either party to the other."
The tenants need to give 20-day notice before the end of a month. If they are staying for September, their notice of vacating in the middle of the month is not good. They may be able to move out at the end of August.
The statutes do not require a landlord to prorate the rent if the tenant moves out early. The statutes do not prohibit the landlord from prorating the rent if the landlord wants to.