Die Kosten betragen nur drei Euro.

Kosten is a plural noun in this sentence, so it takes the plural article die.

• die Kosten = the costs
• Singular forms do exist in some contexts, such as die Kosten being used as a plural-only idea in everyday language, while der Preis is often used when talking about the price of one item.

So die Kosten betragen ... is a normal way to say the costs amount to ... or the cost is ...

Because the subject is die Kosten, which is plural.

The verb betragen is conjugated like this:

• ich betrage
• du beträgst
• er/sie/es beträgt
• wir betragen
• ihr betragt
• sie/Sie betragen

Since die Kosten = they in grammatical terms, German uses betragen.

Here, betragen means something like:

• to amount to
• to come to
• to total

So:

• Die Kosten betragen nur drei Euro. = The cost comes to only three euros.

This verb is common in more formal or neutral statements about amounts, sums, distances, times, and percentages.

Examples:

• Die Miete beträgt 800 Euro.
• Die Wartezeit beträgt zehn Minuten.

Yes. That is often more natural in everyday speech.

Compare:

• Die Kosten betragen nur drei Euro.
  More formal or neutral.
• Es kostet nur drei Euro.
  More conversational and very common.

Both are correct. The sentence with betragen sounds a bit more written, official, or business-like.

Because drei Euro is simply the amount, and after verbs like betragen, German usually gives the number directly.

• drei Euro = three euros

You would not normally say die drei Euro here, because that would mean the three euros, referring to specific euros already known.

So:

• Die Kosten betragen drei Euro. = correct
• Die Kosten betragen die drei Euro. = not correct in this meaning

In German, the word Euro usually stays the same after numbers.

• ein Euro
• zwei Euro
• drei Euro

So unlike English, German does not normally add -s here.

The same is true for Cent:

• ein Cent
• fünf Cent

Nur means only.

It emphasizes that the amount is small:

• Die Kosten betragen drei Euro.
  The costs are three euros.
• Die Kosten betragen nur drei Euro.
  The costs are only three euros.

So nur adds the idea of just, only, or no more than.

This is standard German main-clause word order:

1. Subject: Die Kosten
2. Verb: betragen
3. Other information: nur drei Euro

In a normal main clause, the conjugated verb goes in position 2.

So this pattern is very typical:

• Die Kosten | betragen | nur drei Euro

You can change the emphasis by moving something else to the front, but the verb still stays in second position:

• Nur drei Euro betragen die Kosten.

That version is correct, but much less neutral and more stylistically marked.

Not exactly.

• der Preis = the price
• die Kosten = the costs / the expense / the cost

In many situations, they can overlap, but they are not always identical.

For example:

• Der Preis des Produkts beträgt drei Euro.
  The price of the product is three euros.
• Die Kosten betragen drei Euro.
  The cost(s) amount to three euros.

Der Preis often refers to the listed or selling price of something.
Die Kosten can sound broader and can refer to expenses more generally.

It is in the accusative case, because betragen takes a direct object.

Structure:

• Die Kosten = subject, nominative
• betragen = verb
• drei Euro = direct object, accusative

In this sentence, though, drei Euro looks the same in nominative and accusative, so you do not see a visible case ending.

It is not extremely formal, but it does sound more formal or written than everyday spoken German.

• Die Kosten betragen nur drei Euro.
  Neutral, written, business-like
• Es kostet nur drei Euro.
  Everyday spoken German

So if you see betragen in advertisements, forms, websites, bills, or official writing, that is very normal.