Jeg gætter på, at batterierne i fjernbetjeningen er gamle, for lyset på den blinker næsten ikke.

Because at gætte på is a very common Danish expression meaning to guess / to suppose / to think that.

So Jeg gætter på, at ... is a natural way to say I guess that ... or I’m guessing that ....

You can say jeg gætter in some contexts, but by itself it often feels more like I’m making a guess. The på is part of the idiomatic expression here.

Here at means that and introduces a subordinate clause:

• Jeg gætter på, at batterierne ... er gamle
• I guess that the batteries ... are old

In English, that is often optional, and in Danish at can also sometimes be omitted in informal language, but including it is very normal and clear.

Because Danish usually adds the definite article to the end of the noun instead of putting a separate word in front.

Here is what is happening:

• batteri = battery
• batterier = batteries

• batterierne = the batteries

• fjernbetjening = remote control

• fjernbetjeningen = the remote control

• lys = light
• lyset = the light

So those endings are the Danish way of saying the.

Because i means in, and the batteries are physically inside the remote control.

So:

• batterierne i fjernbetjeningen = the batteries in the remote control

This is a very natural Danish way to express possession or location. Danish often prefers this kind of phrasing instead of something more possessive-looking.

You could also say fjernbetjeningens batterier for the remote control’s batteries, but batterierne i fjernbetjeningen sounds very everyday and natural.

Because batterierne is plural, and the adjective has to agree with that.

The adjective gammel changes like this:

• gammel = singular, common gender
• gammelt = singular, neuter
• gamle = plural or definite

Since batterierne means the batteries and is plural, you use gamle:

• batterierne er gamle = the batteries are old

Here for means because or since.

So this part:

• for lyset på den blinker næsten ikke

means something like:

• because the light on it hardly blinks

This can confuse English speakers because Danish for does not always mean English for. As a conjunction, Danish for can introduce a reason.

Both can express a reason, but they are not used in exactly the same way.

• for links to a new main clause
• fordi introduces a subordinate clause

That affects word order.

With for:

• ... for lyset på den blinker næsten ikke

With fordi:

• ... fordi lyset på den næsten ikke blinker

Notice how næsten ikke moves before the verb with fordi.

In many situations, fordi is the more straightforward everyday word for because, but for is also very common and natural.

It refers to fjernbetjeningen.

So:

• lyset på den = the light on it
• it = the remote control

It cannot refer to batterierne, because batterierne is plural. If Danish wanted to refer to the batteries, it would use dem, not den.

Because fjernbetjening is a common-gender noun in Danish, and common-gender nouns take den.

Danish has two grammatical genders:

• en-words → referred to with den
• et-words → referred to with det

Since it is:

• en fjernbetjening

the pronoun is:

• den

So på den means on it, referring back to fjernbetjeningen.

Yes, you could say lyset på fjernbetjeningen, and it would mean the same thing.

But once fjernbetjeningen has already been mentioned, Danish often uses a pronoun to avoid repeating the noun:

• lyset på den = the light on it

That sounds natural and efficient.

You could also sometimes hear other structures, but lyset på den is very idiomatic here.

Næsten ikke literally means almost not, but in natural English it usually means:

• hardly
• barely
• almost not at all

So:

• lyset på den blinker næsten ikke

means the light hardly blinks / barely blinks.

This is a very common Danish combination.

Because Danish treats those two clauses differently.

1. After at, you have a subordinate clause:
   • at batterierne i fjernbetjeningen er gamle

In subordinate clauses, Danish does not use normal main-clause inversion. If there were an adverb like ikke, it would usually come before the finite verb:

• at batterierne ikke er gamle

2. After for, you have a new main clause:
   • for lyset på den blinker næsten ikke

Main clauses in Danish follow the verb-second pattern. In this sentence, the subject lyset på den comes first, so the verb blinker comes second. That is why the order looks straightforward here.

The comma before for is very normal because for connects two clauses.

The comma before at depends on comma style. In Danish, two systems are used:

• a style with start comma
• a style without start comma

So both of these can be seen:

• Jeg gætter på, at batterierne ...
• Jeg gætter på at batterierne ...

Both are normal depending on the punctuation system being followed.