Kassan hade ingen växel kvar, så jag betalade med två mynt.

Kassan usually means the cash register, the checkout, or the till, depending on context.

In a shop sentence like this, it does not usually mean a literal box full of money only. It often refers to the whole checkout point or register area.

It is the definite form of kassa:

• en kassa = a cash register / a checkout
• kassan = the cash register / the checkout

So the sentence is talking about the register/checkout not having any change left.

Because Swedish uses ha (to have) when talking about possession or availability.

So:

• Kassan hade ingen växel kvar = The register had no change left

Using var (was) would not sound natural here, because the idea is not that the register was change, but that it had change available.

This is very similar to English:

• natural: The register had no change left
• unnatural: The register was no change left

Because ingen is used to mean no / not any before a noun.

So:

• ingen växel = no change

By contrast, inte usually negates a verb, adjective, or whole clause:

• Jag betalade inte = I did not pay
• Det var inte dyrt = It was not expensive

Since växel is a noun, Swedish uses ingen here.

The form depends on the gender and number of the noun.

• ingen
  • common gender singular nouns
• inget
  • neuter singular nouns
• inga
  • plural nouns

Växel is a common gender singular noun (en växel), so:

• ingen växel = no change

Compare:

• ingen butik = no shop
• inget mynt = no coin
• inga mynt = no coins

Here växel means change, as in money returned after paying.

So ingen växel kvar means:

• no change left
• no small change remaining

This word can also have other meanings in other contexts, such as gear or switch/exchange, so context matters.

In a shopping/payment sentence, växel almost always means change.

Kvar means left, remaining, or still there.

So:

• ingen växel kvar = no change left

It often appears after the noun or object it describes:

• Det finns kaffe kvar = There is coffee left
• Har du pengar kvar? = Do you have money left?

In this sentence, kvar tells you that the change has been used up or is no longer available.

Here så means so in the sense of therefore / as a result.

The sentence has two main clauses:

• Kassan hade ingen växel kvar
• så jag betalade med två mynt

Together:

• The register had no change left, so I paid with two coins.

This is a very common Swedish use of så to connect cause and result.

Because this så is acting as a coordinating conjunction, similar to English so.

After a coordinating conjunction, Swedish usually keeps normal main-clause word order:

• så jag betalade = so I paid

That means:

• conjunction
• subject
• verb

Compare:

• Jag var trött, så jag gick hem.
• I was tired, so I went home.

If something else were placed first in the clause, then inversion could happen:

• Så med två mynt betalade jag
  But that sounds marked or unnatural here.

So så jag betalade is the normal wording.

Because med means with, and Swedish uses it to show the means or instrument used for paying.

So:

• betala med kort = pay with a card
• betala med kontanter = pay with cash
• betala med två mynt = pay with two coins

This works very much like English.

Because mynt is one of the Swedish nouns whose indefinite singular and indefinite plural look the same.

• ett mynt = a coin
• två mynt = two coins

So the word does not change in that form.

The definite plural is different:

• mynten = the coins

This is common with many neuter nouns.

It is in the preterite (simple past).

The verbs are:

• hade = had
• betalade = paid

So the whole sentence describes a completed situation in the past:

• the register had no change left
• therefore I paid with two coins

Sometimes in everyday speech kassan can loosely refer to the checkout area and, by extension, what is happening at the checkout. But literally it means the register / the till / the checkout, not the cashier as a person.

If you specifically mean the cashier, Swedish would more naturally use:

• kassören = the male cashier
• kassörskan = the female cashier, though this can sound old-fashioned
• often just personen i kassan = the person at the checkout

So in this sentence, kassan hade ingen växel kvar is best understood as the register/checkout had no change left.

In this sentence, växel behaves more like an uncountable mass noun, similar to English change.

That is why Swedish naturally says:

• ingen växel = no change

You are not counting individual units of change here. You are talking about change as a general money resource.

If you wanted to talk about specific items, you would use words like:

• mynt = coins
• sedlar = banknotes

So the sentence moves from the general idea:

• ingen växel kvar = no change left

to the specific solution:

• två mynt = two coins