Ten bilet na pociąg jest najtańszy.

Polish does not have articles, so it has no direct equivalent of English the or a/an.

That means Polish often relies on:

• context
• word order
• demonstratives like ten
• other clues in the sentence

So bilet by itself could mean a ticket or the ticket, depending on context. English has to choose; Polish often does not.

Ten is a demonstrative adjective. It most basically means this or that specific.

In real Polish, ten is also often used where English might simply use the, especially when the speaker means a particular, identifiable thing. So in a sentence like this, ten bilet can feel like this ticket or that particular ticket, depending on context.

Because bilet is a masculine singular noun, and ten has to agree with it in gender, number, and case.

The nominative singular forms are:

• masculine: ten
• feminine: ta
• neuter: to

So:

• ten bilet = this ticket
• ta książka = this book
• to okno = this window

Because ten bilet na pociąg is the subject of the sentence.

The sentence structure is basically:

• Ten bilet na pociąg = subject
• jest = is
• najtańszy = cheapest

Subjects are normally in the nominative case in Polish, so bilet stays nominative.

Because Polish commonly uses the pattern bilet na + accusative for tickets for transport or for events.

So you get:

• bilet na pociąg = train ticket
• bilet na autobus = bus ticket
• bilet na koncert = ticket for a concert

By contrast, do usually points more clearly to a destination:

• bilet do Warszawy = a ticket to Warsaw
• bilet do Krakowa = a ticket to Kraków

So na pociąg is the normal expression here.

Because na in bilet na pociąg takes the accusative, but for many masculine inanimate nouns, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: pociąg
• accusative: pociąg

The case has changed, but the form happens to be identical.

Compare that with a masculine animate noun, where you often do see a change:

• nominative: kot
• accusative: kota

It is the superlative form of tani (cheap).

The pattern is:

• tani = cheap
• tańszy = cheaper
• najtańszy = cheapest

A very common way to make the superlative in Polish is: naj- + comparative form

So:

• szybszy = faster
• najszybszy = fastest

With tani, the stem changes a bit, so you need to learn:

• tani
• tańszy
• najtańszy

Because the adjective has to agree with bilet, which is masculine singular nominative.

So the adjective also appears in masculine singular nominative:

• najtańszy

Compare:

• Ten bilet jest najtańszy. — masculine
• Ta książka jest najtańsza. — feminine
• To okno jest najtańsze. — neuter

Yes, in a normal full sentence like this, jest is the standard form.

Polish often drops subject pronouns, but it usually does not drop the present-tense form of to be in ordinary sentences of this type.

So:

• Ten bilet na pociąg jest najtańszy. = normal, complete sentence

You might see omission only in very informal notes, headlines, or special styles, but for standard Polish, keep jest.

Yes. Polish word order is more flexible than English word order, because case endings show the grammatical relationships.

The most neutral version is:

• Ten bilet na pociąg jest najtańszy.

But other orders are possible for emphasis, for example:

• Najtańszy jest ten bilet na pociąg. — emphasizes the cheapest
• Ten bilet jest najtańszy. — shorter, if na pociąg is already understood from context

Even though word order can change, the original sentence is a very natural neutral way to say it.