På tåget hade barnen varsitt äpple och varsin liten kaka i väskan.

For ordinary travel, Swedish normally says på tåget = on the train.

This is the usual pattern with several forms of transport:

• på tåget = on the train
• på bussen = on the bus
• på planet = on the plane

By contrast, i tåget is not the normal way to say you are riding a train. It can even mean in the procession/parade, because tåg can also mean procession in another context.

So here på tåget is simply the standard expression.

Because Swedish main clauses follow the V2 rule: the finite verb must come in second position.

Here, På tåget has been moved to the front to set the scene. Once that happens, the verb must come next:

• På tåget hade barnen ...

If the subject came first, the order would be:

• Barnen hade ...

So the word order changes because Swedish wants the verb in position 2, not because the meaning changes much.

Barnen is the definite plural form of barn.

The forms are:

• ett barn = a child
• barnet = the child
• barn = children
• barnen = the children

A thing that often surprises learners is that barn is used for both:

• singular indefinite: ett barn
• plural indefinite: barn

So barnen specifically means the children.

Hade is the past tense of ha = to have.

So:

• har = has/have
• hade = had

In this sentence it means the children had those items with them / in their possession.

They mean one each or one apiece.

So:

• barnen hade varsitt äpple = the children had an apple each
• och varsin liten kaka = and a small cake each

These words are called distributive determiners: they distribute one item to each member of a group.

So the sentence is not saying there was just one apple and one cake total. It says each child had:

• one apple
• one small cake

Because they agree with the gender of the noun that follows.

• varsitt is used with neuter nouns
• varsin is used with common gender nouns

Here:

• äpple is a neuter noun: ett äpple
  → therefore varsitt äpple
• kaka is a common-gender noun: en kaka
  → therefore varsin kaka

A useful summary:

• varsin
  • en-word
• varsitt
  • ett-word
• var sina
  • plural noun

Because varsin/varsitt already function as the determiner.

You do not say:

• varsitt ett äpple
• varsin en kaka

That would be like saying one-each an apple.

So:

• varsitt äpple is correct
• varsin liten kaka is correct

The varsin/varsitt already carries the idea of a/an each.

Because kaka is:

• common gender (en kaka)
• singular
• indefinite

For an indefinite singular common-gender noun, the adjective form is liten:

• en liten kaka
• varsin liten kaka

Compare:

• ett litet äpple = a small apple
• den lilla kakan = the small cake

So:

• liten = common gender singular indefinite
• litet = neuter singular indefinite
• lilla = definite form

Since varsin works like an indefinite determiner, liten is the correct form here.

Yes.

Because both nouns are introduced with distributive forms:

• varsitt äpple
• varsin liten kaka

That means each child had:

• one apple
• and one small cake

If Swedish only said ett äpple och en liten kaka, that could be more ambiguous in some contexts. But varsitt/varsin makes the one-each meaning very clear.

Literally, i väskan means in the bag.

By itself, it does not fully specify whose bag it is. Depending on context, it could mean:

• in one known bag
• in a shared bag
• possibly in each child’s bag, if the context makes that clear

So this part can be a little context-dependent.

If you wanted to be more explicit, Swedish could say for example:

• i en väska = in a bag
• i väskorna = in the bags
• i sina väskor = in their own bags

So i väskan is grammatically straightforward, but the exact ownership is not spelled out in the sentence itself.

In normal reading, it is understood as part of the whole situation: the children had the apple and the cake in the bag.

In other words, most readers will naturally understand that the items mentioned were in the bag, not just the second item.

That said, Swedish—like English—can sometimes leave attachment slightly open if the context is thin. If a speaker wanted to remove all doubt, they could rephrase the sentence.

But in ordinary reading, the most natural understanding is:

• each child had an apple and a small cake,
• and these were in the bag.

Normally, no.

These words are used when there is a plural subject or otherwise more than one participant, and each participant gets one of something.

So this works:

• Barnen hade varsitt äpple. = The children had an apple each.

But with a singular subject, varsin/varsitt does not make sense, because there is no group to distribute across.

So you would not normally say:

• Barnet hade varsitt äpple.

Instead:

• Barnet hade ett äpple. = The child had an apple.