Huslejen er høj, men lejligheden har en lille altan.

Because Danish usually adds the definite article to the end of the noun.

• husleje = rent

• huslejen = the rent

• lejlighed = apartment / flat
• lejligheden = the apartment / flat

So instead of a separate word like English the, Danish often uses a suffix:

• en lejlighed = an apartment
• lejligheden = the apartment

This is one of the most important differences between English and Danish noun structure.

For the same reason: Danish commonly expresses definiteness by attaching it to the noun itself.

So:

• huslejen already means the rent
• lejligheden already means the apartment

A separate the-word is not needed here.

Danish does also have separate definite words like den, det, and de, but those are mainly used when there is an adjective before the noun, for example:

• den lille altan = the small balcony

But in your sentence, huslejen and lejligheden appear without adjectives, so the ending -en is enough.

Because høj is a predicative adjective here: it comes after er and describes huslejen.

In Danish, predicative adjectives usually agree like this:

• common gender singular: base form
• neuter singular: often -t
• plural: often -e

Here, huslejen is a singular common-gender noun, so the adjective stays in its basic form:

• Huslejen er høj = The rent is high

Compare:

• Huset er højt = The house is tall/high
• Priserne er høje = The prices are high

So høj is correct because huslejen is singular and not neuter.

Because altan is a common-gender noun, so it takes en in the indefinite singular:

• en altan = a balcony

The adjective lille means small/little and is the normal everyday form used before singular nouns:

• en lille altan = a small balcony
• et lille hus = a small house

So even though the noun gender changes from en to et, the adjective often stays lille in modern Danish.

Because the two adjectives are used in different ways.

1. Attributive adjective: directly before a noun

   • en lille altan = a small balcony

2. Predicative adjective: after a verb like er

   • huslejen er høj = the rent is high

English works the same way:

• a small balcony
• the rent is high

So the placement here is very natural in Danish.

Because har is the normal verb for possession: to have.

• lejligheden har en lille altan = the apartment has a small balcony

Danish uses have/har just like English does for this kind of idea.

You would not normally use er here, because the sentence is expressing that the apartment possesses a balcony, not that it equals one.

Because men is a coordinating conjunction, like English but. After men, Danish normally starts a new main clause, and main clauses usually follow normal word order:

• subject + verb

So:

• lejligheden har = the apartment has

This is different from cases where another element comes first and triggers inversion, for example:

• I dag har lejligheden en lille altan would be wrong in meaning, but structurally it shows the pattern
• I dag er huslejen høj = Today the rent is high

After men, you usually just get a fresh main clause:

• Huslejen er høj, men lejligheden har en lille altan.

Yes. Danish uses høj very naturally for prices, costs, rent, and similar things, just as English uses high.

Examples:

• en høj pris = a high price
• høj husleje = high rent
• huslejen er høj = the rent is high

So this is a completely normal choice of adjective.

They refer to different things:

• husleje = rent, the money paid to live somewhere
• lejlighed = apartment / flat, the place itself

So the sentence contrasts the cost with a feature of the apartment:

• the rent is high
• but the apartment has a small balcony

Because the sentence is talking about:

• the rent of that apartment
• the apartment itself

Those are specific, known things in the context, so Danish uses the definite form:

• huslejen
• lejligheden

But en lille altan introduces a balcony as a feature, not as something already identified. So it stays indefinite:

• en lille altan = a small balcony

If you were referring to a specific balcony already mentioned, you could say:

• den lille altan = the small balcony

Yes, altan is a common-gender noun, so its basic article is en:

• en altan
• altanen = the balcony

That is why the sentence says en lille altan, not et lille altan.

A fairly literal breakdown is:

• Huslejen = the rent
• er = is
• høj = high
• men = but
• lejligheden = the apartment
• har = has
• en lille altan = a small balcony

So the Danish structure is actually very close to English:

The rent is high, but the apartment has a small balcony.

Not completely. Danish pronunciation is often less straightforward than spelling suggests.

A rough guide:

• huslejen: the middle part can sound smoother than an English speaker might expect
• lejligheden: this word is especially tricky, because unstressed syllables are often reduced
• høj: sounds roughly like hoy, but with a Danish vowel quality
• altan: fairly straightforward, with stress on the last syllable

A useful learner tip is to treat long words like lejligheden as chunks:

• lej-lig-he-den

But in real speech, several syllables are reduced, so listening to native audio is very important.