der Betrag
amount
A specific sum of money.
Bitte überweisen Sie den offenen Betrag bis Ende des Monats.
Please transfer the outstanding amount by the end of the month.

amount / absolute value
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amount
A specific sum of money.
Bitte überweisen Sie den offenen Betrag bis Ende des Monats.
Please transfer the outstanding amount by the end of the month.

The noun 'der Betrag' (amount) is derived from the verb 'betragen', which means 'to amount to a certain sum'. A 'Betrag' is therefore what something or an invoice amounts to.
Die Reparaturkosten betragen insgesamt 250 Euro.
The repair costs amount to 250 euros in total.
In a financial context, 'Betrag' (amount) is often used with verbs such as 'überweisen' (to transfer), 'bezahlen' (to pay), 'einzahlen' (to deposit), or 'abbuchen' (to debit). Combinations with adjectives such as 'offen' (outstanding), 'fällig' (due), or 'gering' (small/low) are also common.
Der fällige Betrag wird automatisch von Ihrem Konto abgebucht.
The due amount will be automatically debited from your account.
To name an exact sum of money, the formal phrase 'ein Betrag in Höhe von' (an amount of) followed by the dative is often used. This is typical for invoices, offers, and formal letters.
Wir stellen Ihnen einen Betrag in Höhe von 500 Euro in Rechnung.
We will invoice you for an amount of 500 euros.
You can recognize the mathematical meaning from the context. When numbers, calculations, vectors, or negative values are being discussed, it refers to the absolute value without a sign.
In der Physik ist der Betrag einer Geschwindigkeit immer positiv.
In physics, the magnitude of a velocity is always positive.
value
'der Wert' and 'der Betrag' can both have to do with numbers. With money, 'Wert' means what something is worth, that is, roughly what price it has. 'Betrag' means a specific amount of money that you pay, receive, or transfer. In mathematics, 'Betrag' is also a special word for a number's distance from zero. The risk of confusion arises because both words occur with numbers and money. For the price or importance of something, you are more likely to say 'Wert'; for a concrete sum, 'Betrag' is more likely.

Bitte überweisen Sie den Betrag bis Ende der Woche.
Please transfer the amount by the end of the week.
Der Wert des alten Fahrrads ist höher, als Max gedacht hat.
The value of the old bicycle is higher than Max thought.