Sometimes a tenant owes the landlord for more than just the current month rent.
If a tenant offers money short of the total amount owed, it is important that the landlord apply the payment to the oldest month first. Otherwise, in an eviction or civil action the court might hold that the landlord waived the prior months. For example, if the current month is February and the tenant has not paid any rent this year, then offers the amount of one month rent, the landlord should apply the payment to January, not February.
If the tenant notes on the payment instrument (an old-school hard copy check, for instance) that the payment is for the current month, the landlord may want to refuse or return the payment and insist that the tenant pay the prior rent before paying the current rent.
The landlord should give a receipt or a written statement showing that the payment was applied to the oldest periods first.
If a formal notice was served, such as a 3-day notice or a 10-day notice, the landlord should seek legal advice before accepting any payments after the applicable cure period has expired. Accepting payments after the notice cure period may require the landlord to start the eviction process over.
In Seattle for residential tenancies the landlord is required to apply all payments to rent before applying them to other charges such as deposits or unpaid utilities. This does not apply to commercial tenancies.
Outside Seattle the landlord may prefer to apply tenant payments other obligations first before applying the payment to rent. Ideally this would be supported by good language in a written lease.
This is a short summary of some of the legal issues that may be present when a tenant offers a payment but owes more than the current month rent. For advice about your circumstances always consult with an attorney