to work through / work off
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to work through
To deal with a series of tasks or a large amount of work systematically and one after another.
Ich muss heute noch eine lange Liste von E-Mails abarbeiten.
I still have to work through a long list of emails today.
'Abarbeiten' is a separable verb. The prefix 'ab-' goes to the end in main clauses. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Sie arbeitet die ganze To-do-Liste ab.
She works through the whole to-do list.
In the meaning 'to deal with systematically,' 'abarbeiten' is often combined with things that have a structure. Typical objects are: a list, a plan, a stack of files, emails, or agenda items.
Zuerst müssen wir die dringendsten Kundenanfragen abarbeiten.
First we have to work through the most urgent customer inquiries.
The reflexive form 'sich abarbeiten' means to exert oneself to the point of exhaustion. The preposition 'an' with the dative is often used to name the object of the effort: 'sich an etwas abarbeiten' (to wear oneself out on something).
Sie hat sich an diesem schwierigen Problem völlig abgearbeitet.
She completely wore herself out on this difficult problem.
The meaning 'to pay a debt through work' is rarer today and often historical. You can recognize it from the object of the sentence, which is usually 'Schulden' (debts) or 'eine Schuld' (a debt).
Er musste seine Schulden auf dem Hof des Großbauern abarbeiten.
He had to work off his debts on the large farmer's estate.
to take care of / to finish
'erledigen' and 'abarbeiten' both have to do with completing tasks. 'erledigen' can refer to a single task or several tasks. 'abarbeiten' is used especially when there is a list, a pile of work, or many items that you do one after another. So it emphasizes the order and the amount. If only one task is meant, 'erledigen' usually sounds more natural.
Vor der Reise arbeitet Paul die lange Liste mit Aufgaben ab.
Before the trip, Paul works through the long list of tasks.
Sie erledigt vor dem Termin noch schnell die Überweisung.
Before the appointment, she quickly takes care of the bank transfer.