По возвращении домой я ещё раз посчитаю деньги, которые остались после обмена валюты.

По возвращении домой means upon returning home or after returning home.

This is a fairly compact, written-style way to express time. It is similar to:

• Когда я вернусь домой — when I return home
• После возвращения домой — after returning home

So the beginning of the sentence could be understood as:

• По возвращении домой я... = When I get back home, I...

It sounds a little more formal or bookish than a simple clause with когда.

Because the preposition по here requires the prepositional case in this expression.

The noun is:

• возвращение — return, returning

Its prepositional singular form is:

• (о) возвращении

So:

• по возвращении = upon return / after returning

This is a fixed and common pattern:

• по приезде — upon arrival
• по окончании — upon completion
• по возвращении — upon returning

For learners, it is often easiest to remember по + prepositional in these time expressions as a set phrase.

Домой means homeward / to home and is used when there is movement toward home.

Compare:

• дома — at home
• домой — to home / homeward

So:

• вернуться домой = to return home
• быть дома = to be at home

Using в дом would usually mean into the house/building, emphasizing entry into a physical structure, not the general idea of going home.

Посчитаю is the 1st person singular future of the perfective verb посчитать.

• я посчитаю = I will count
• literally, I will count it through / complete the counting

Russian often uses a perfective verb for a single completed future action. Here the speaker means they will count the money once, as a complete act, after getting home.

If you used the imperfective считать, the simple future would be:

• буду считать — I will be counting / I will count (with more emphasis on process or ongoing action)

So посчитаю is natural because the sentence refers to one completed counting event.

The main difference is aspect:

• считать — imperfective: counting as a process, repeated action, or general activity
• посчитать — perfective: to count and finish counting

In this sentence, the speaker intends to do the action once and complete it, so посчитать fits better.

Compare:

• Я считаю деньги. — I am counting the money / I count money.
• Я посчитаю деньги. — I will count the money.
  (as one completed action)

The prefix по- here helps form a perfective verb.

Ещё раз means once again, one more time, or again.

So:

• я ещё раз посчитаю деньги = I’ll count the money once again

It suggests the speaker has probably already counted the money before, but wants to check it again.

Которые is a relative pronoun meaning which / that.

It refers back to деньги and introduces the relative clause:

• деньги, которые остались...
• the money that remained...

Here которые is in the nominative plural because it is the subject of the verb остались.

You can test it like this:

• Что осталось? — What remained?
• Деньги остались. — The money remained.

Since деньги is plural, the pronoun is also plural:

• singular masculine: который
• singular feminine: которая
• singular neuter: которое
• plural: которые

Because the relative clause describes money that was left over after the exchange already happened.

The timeline is:

1. the currency exchange happened
2. some money remained
3. later, when the speaker gets home, they will count it again

So:

• которые остались после обмена валюты = which remained after the currency exchange

The remaining happened earlier, so Russian uses the past tense остались.

Also, остались is plural because it agrees with деньги, which is grammatically plural.

The preposition после requires the genitive case.

So:

• обмен → обмена after после

Then валюты is also in the genitive singular, because it depends on обмен:

• обмен валюты = exchange of currency / currency exchange

So the structure is:

• после
  • обмена
• обмена
  • валюты

In other words:

• после обмена валюты = after the currency exchange

Because валюта here is being used as a general noun meaning currency.

• обмен валюты = exchange of currency
• обмен валют would mean exchange of currencies

Both are possible in some contexts, but обмен валюты is natural when speaking more generally about exchanging money from one currency into another. Russian often uses the singular for this kind of abstract or general meaning.

Yes, usually.

Compare:

• По возвращении домой я ещё раз посчитаю деньги.
  More compact, slightly formal, somewhat written-style.

• Когда я вернусь домой, я ещё раз посчитаю деньги.
  Very natural, neutral, conversational.

So по возвращении is not wrong or unnatural, but it sounds more polished or formal than the everyday clause with когда.

Yes. Russian word order is flexible, though different orders change the emphasis slightly.

Original:

• По возвращении домой я ещё раз посчитаю деньги, которые остались после обмена валюты.

Possible variations:

• Я ещё раз посчитаю деньги по возвращении домой.
• Деньги, которые остались после обмена валюты, я ещё раз посчитаю по возвращении домой.

The original sentence puts the time frame first: upon returning home. That sets the scene before the main action.

Russian often uses word order to manage emphasis rather than basic grammatical relationships, since the cases already show those relationships clearly.