Nem tudom, a létra a garázsban van-e.

Because Hungarian has two main conjugation patterns: indefinite and definite.

Here, tudom is the definite form of tudni (to know), because the verb is taking a specific object: the whole clause a létra a garázsban van-e.

So Hungarian treats this a bit like:

Nem tudom (azt), ... = I don’t know (that / whether), ...

Compare:

• Tudok úszni. = I can swim.
• Tudok valamit. = I know something.
• Tudom a választ. = I know the answer.
• Tudom, hogy... / Tudom, ... = I know that...

So nem tudom is the natural form here.

-e marks a yes/no question, especially inside a larger sentence.

In this sentence, it works like English if or whether:

• a létra a garázsban van-e = whether the ladder is in the garage

So the full sentence means:

• Nem tudom, a létra a garázsban van-e.
• I don’t know whether the ladder is in the garage.

A few useful points:

• -e is written with a hyphen
• it usually attaches to the finite verb
• here that verb is van

So:

• van = is
• van-e = is ... ? / whether it is

That is a very common and very good question.

Hungarian often drops van in the 3rd person singular present tense when the sentence is describing identity or a quality:

• A létra magas. = The ladder is tall.
• Péter orvos. = Peter is a doctor.

But van is normally kept when the sentence expresses location, existence, or presence:

• A létra a garázsban van. = The ladder is in the garage.
• A könyv az asztalon van. = The book is on the table.

So in your sentence, van is needed because the sentence is about where the ladder is.

Garázsban breaks down like this:

• garázs = garage
• -ban = in

So:

• garázsban = in the garage

This is a very common Hungarian pattern: instead of using a separate word like English in, Hungarian often adds a case ending directly to the noun.

More examples:

• házban = in the house
• szobában = in the room
• iskolában = in the school

Because of vowel harmony.

Hungarian suffixes often change depending on the vowels in the word. The pair -ban / -ben both mean in, but:

• -ban is used after words with back vowels
• -ben is used after words with front vowels

garázs contains back vowels (a, á), so it takes -ban:

• garázsban

Compare:

• házban = in the house
• kertben = in the garden
• székben = in the chair

So the ending is not random; it follows Hungarian vowel harmony.

Because both nouns are being treated as specific.

• a létra = the ladder
• a garázsban = in the garage

Hungarian uses a / az much like English the.

If the meaning were less specific, you might see egy (a/an):

• egy létra = a ladder
• egy garázsban = in a garage

So this sentence is talking about a particular ladder and a particular garage, not just any ladder or any garage.

Yes, Hungarian word order is flexible, but the different orders usually change the focus or emphasis.

The version you have is a neutral, natural way to say:

• whether the ladder is in the garage

Structure:

• a létra = topic (the ladder)
• a garázsban = location
• van-e = is / whether it is

Other orders are possible, but they sound different:

• Nem tudom, a garázsban van-e a létra.
  This puts more attention on the garage as the important location.

• Nem tudom, a létra van-e a garázsban.
  This can sound contrastive, as if you are asking whether it is the ladder that is in the garage, not something else.

So the given order is a good neutral model to learn first.

Yes. That is also grammatical and natural.

Both of these work:

• Nem tudom, a létra a garázsban van-e.
• Nem tudom, hogy a létra a garázsban van-e.

The word hogy means that, and it often introduces subordinate clauses. But when you already have -e marking an embedded yes/no question, hogy is not required.

So:

• without hogy = a bit more compact
• with hogy = also very common and natural

Both mean the same thing here.

Not if you want the meaning I don’t know whether the ladder is in the garage.

The -e is what makes the embedded clause into a yes/no question.

Without -e, the sentence would no longer clearly mean whether the ladder is in the garage. Hungarian normally needs -e here to express that idea.

So for this meaning, keep:

• van-e

That is one of the most important parts of the sentence.