Tjeneren kommer tilbage med byttepengene, selvom jeg troede, at jeg havde betalt præcist.

Tjener means waiter. The ending -en makes it definite, so tjeneren means the waiter.

This is very common in Danish: instead of putting a separate word like the in front, Danish usually adds the definite ending to the noun.

• en tjener = a waiter
• tjeneren = the waiter

Kommer tilbage means comes back / returns.

It is built from:

• kommer = comes
• tilbage = back

So Tjeneren kommer tilbage literally means The waiter comes back.

In Danish, verbs are often combined with words like tilbage, ud, op, ned, etc., much like English phrasal verbs.

Byttepengene means the change, as in the money you get back after paying.

It comes from byttepenge, which refers to change in cash transactions. In this sentence, -ne makes it definite:

• byttepenge = change
• byttepengene = the change

Even though English uses singular change, Danish treats byttepenge like a plural-type noun.

Med means with.

So kommer tilbage med byttepengene means comes back with the change.

This is a very natural way in Danish to describe someone returning while carrying or bringing something.

• Han kommer tilbage med kaffen = He comes back with the coffee
• Tjeneren kommer tilbage med byttepengene = The waiter comes back with the change

Selvom means although / even though.

Yes, you may also see selv om written as two words. In modern Danish, selvom as one word is very common.

It introduces a subordinate clause:

• selvom jeg troede ... = even though I thought ...

Because the sentence is talking about a past thought:

• jeg tror = I think / I believe
• jeg troede = I thought

The waiter is coming back with change, and the speaker’s belief existed before that moment, so Danish uses the past tense troede.

At here means that.

So:

• jeg troede, at ... = I thought that ...

It introduces the content of the thought. In English, that is often omitted, but in Danish at is very common in this kind of sentence.

• Jeg troede, at jeg havde betalt præcist = I thought that I had paid the exact amount

Havde betalt is the past perfect, meaning had paid.

It is used because the paying happened before the thinking:

1. I paid.
2. I thought I had paid exactly.
3. The waiter came back with change.

So the sentence uses:

• troede = thought
• havde betalt = had paid

If you said jeg troede, at jeg betalte præcist, it would sound wrong or at least very unnatural in this context.

Because after at, you are in a subordinate clause, and Danish word order changes.

In a main clause, Danish usually has verb-second word order:

• Jeg troede det = I thought that

But in a subordinate clause, the subject usually comes before the finite verb:

• at jeg havde betalt præcist

So this is normal subordinate clause order.

The same thing happens after selvom:

• selvom jeg troede ...

Here præcist means exactly, in the sense of paying the exact amount.

So jeg havde betalt præcist means:

• I had paid exactly
• in natural English: I had paid the exact amount

It is the adverb form of præcis:

• et præcist beløb = an exact amount
• jeg betalte præcist = I paid exactly

Yes, it is understandable and natural. It means the speaker thought no change would be needed.

Danish can also express this idea in other ways, for example:

• jeg havde betalt det præcise beløb = I had paid the exact amount
• jeg havde betalt lige = I had paid the exact amount / paid evenly

But betalt præcist works well and is clear.

They mark subordinate clauses.

In this sentence:

• selvom jeg troede, at jeg havde betalt præcist

both selvom and at introduce subordinate clauses, so commas are used to separate the parts of the sentence.

Danish comma rules can be a bit tricky, and you may see some variation depending on comma style, but the punctuation here is perfectly normal and standard for learners to follow.

Yes. Tjener can also mean servant in some contexts, especially older or more formal language. But in a restaurant context like this one, tjeneren clearly means the waiter.

So the context tells you which meaning is intended.