annoyance / trouble
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annoyance
A feeling of dissatisfaction, anger, or irritation about something that is not going as desired or is perceived as disruptive.
Sein Ärger über die ständige laute Musik des Nachbarn wuchs von Tag zu Tag.
His annoyance about the neighbor's constant loud music grew from day to day.
trouble
Colloquial for problems, difficulties, or inconveniences.
Ich hatte heute nur Ärger mit dem neuen Computer.
I had nothing but trouble with the new computer today.
'Der Ärger' (annoyance/trouble) belongs to a word family with the reflexive verb 'sich ärgern' (to be annoyed) and the adjective 'ärgerlich' (annoying/irritating). All three describe a feeling of anger or dissatisfaction.
Ich ärgere mich über das schlechte Wetter.
I'm annoyed about the bad weather.
The prepositions are important: you have 'Ärger mit' (trouble with) a person or thing (dative). You feel 'Ärger über' (annoyance about) a situation or action (accusative).
Sie hat oft Ärger mit ihrem alten Auto.
She often has trouble with her old car.
In the colloquial meaning 'Probleme' (problems), 'Ärger' (trouble) is often used with the verbs 'haben' (to have), 'bekommen' (to get), or 'machen' (to make/cause).
Wenn du zu spät kommst, bekommst du Ärger mit dem Chef.
If you come too late, you'll get in trouble with the boss.
The meaning 'Schwierigkeiten' (difficulties) or 'Probleme' (problems) is typical of colloquial language. In formal or written contexts, you would more likely use 'Schwierigkeiten' (difficulties) or 'Unannehmlichkeiten' (inconveniences).
Ich will keinen Ärger mit der Polizei.
I don't want any trouble with the police.
problem
In colloquial language, Problem and Ärger can both mean an unpleasant difficulty. Problem sounds more factual and focuses on the situation that needs a solution, while Ärger expresses personal annoyance more strongly. Use “Problem” in a factual description; when displeasure or troublesome consequences are emphasized, use “Ärger”.
Der Ärger mit der Lieferung kostet mich den ganzen Nachmittag.
The trouble with the delivery is costing me the entire afternoon.
Das Problem mit der Lieferung müssen wir schnell lösen.
We need to solve the problem with the delivery quickly.
