zyaguti wo aketa mama de ha, suidoudai ga takaku natte simaimasu.

〜たまま is the standard pattern for saying left in a state or while still having done something.

So:

• 開けたまま = left open
• つけたまま = left on
• 座ったまま = remaining seated

The past form 開けた here does not mean simple past in English. It combines with まま to show a continuing state that results from the action.

So 蛇口を開けたまま means:

• with the faucet left open
• if you leave the faucet open

In this sentence, まま means in that state, as it is, or left that way.

Pattern:

• Verbた + まま = leave something in the state resulting from that action
• Noun + のまま = as it is / in the state of
• Adjective + まま is less common, depending on the structure

Examples:

• 電気をつけたまま寝た。 = I went to sleep with the lights left on.
• ドアを閉めたままにする。 = Keep the door closed.
• 靴のまま入る。 = Enter with shoes on.

So here, 開けたまま describes the faucet being left open.

では here marks a condition, similar to:

• if...
• when...
• in the case that...

So:

• 蛇口を開けたままでは = if you leave the faucet open
• more literally: in the state of having the faucet left open, ...

This では is made of:

• で = in/with/by means of a state
• は = adds topical or contrastive emphasis

You will often see this after state expressions:

• そんなに遅くでは困る。 = If it’s that late, that’s a problem.
• このままでは危ない。 = At this rate / if things stay like this, it’s dangerous.

In your sentence, 〜たままでは is a common structure meaning:

• if something is left that way
• if you keep it in that state

Not naturally in the same way.

Here, 〜たままでは is a fixed and very common way to say:

• if left like that
• if you keep it that way

For example:

• このままではまずい。
• 窓を開けたままでは寒い。

If you changed it, the nuance would shift:

• 開けたままなら = if it is left open / if that's the case
  This is possible in some contexts, but less natural here.
• 開けたままだと = if it is left open, then...
  Also possible and natural in some situations.
• 開けたままで = left open, and... / while left open...
  This connects clauses, but does not by itself express the same conditional force as clearly.

So 〜たままでは is the most natural form for this sentence.

Because 蛇口 is the direct object of 開ける.

• 蛇口を開ける = to open the faucet/tap

Even though English says leave the faucet open, Japanese still builds it from the verb open:

• 蛇口を開けたまま = having opened the faucet and left it that way

So the faucet is the thing being opened, which is why it takes を.

Because 高くなる means to become expensive/high, and the thing that becomes high is 水道代.

So:

• 水道代が高くなる = the water bill gets high / the water bill goes up

Here:

• 水道代 = water bill / water charges
• が marks it as the thing undergoing the change
• 高くなってしまいます = ends up becoming high

This is very natural in Japanese: the bill itself is described as becoming high.

水道代 means water bill or water charges.

It is made up of:

• 水道 = water supply / running water / water service
• 代 = cost / charge / fee

Similar words:

• 電気代 = electricity bill
• ガス代 = gas bill
• 家賃 = rent
• 通信費 = communication expenses

So 水道代が高くなる means the amount you pay for water will increase.

Because 高い is an い-adjective, and before なる it changes to the adverbial form:

• 高い → 高く

Pattern:

• い-adjective + なる = become ...
• 高くなる = become expensive/high
• 安くなる = become cheap
• 早くなる = become early/faster

For な-adjectives, you use に:

• 静かになる = become quiet
• きれいになる = become clean/pretty

So:

• 水道代が高くなる = the water bill becomes high

〜てしまう often adds one of these nuances:

1. completion: completely do something
2. regret / undesired result: end up doing / unfortunately happens

In this sentence, it mainly gives the nuance of an unwanted consequence:

• 高くなります = it becomes expensive / goes up
• 高くなってしまいます = it ends up becoming high, unfortunately

So the speaker is not just stating a neutral fact. They are warning that leaving the faucet open will lead to an undesirable result.

しまいます is polite, so the whole sentence sounds polite or neutral-formal.

Casual versions include:

• 蛇口を開けたままだと、水道代が高くなってしまう。
• 蛇口を開けたままでは、水道代が高くなっちゃう。
• 蛇口を開けっぱなしだと、水道代が高くなるよ。

Notes:

• しまう → ちゃう is a casual contraction
• 〜よ can add a conversational warning tone

So yes, the original is polite, and there are many more casual alternatives.

Both can mean left open, but the nuance is different.

• 開けたまま = simply left open / in the open state
• 開けっぱなし = left open and not properly closed, often with a stronger sense of carelessness or negligence

Examples:

• 窓を開けたまま寝た。 = I slept with the window open.
• 窓を開けっぱなしにした。 = I left the window open carelessly / just left it open.

In your sentence, 開けっぱなし would also work:

• 蛇口を開けっぱなしでは、水道代が高くなってしまいます。

That version sounds a bit more like leaving it running carelessly.

蛇口 refers to the faucet or tap, especially the part you turn to let water flow.

Depending on dialect or region in English:

• American English: faucet
• British English: tap

So both are good translations.

In everyday Japanese:

• 蛇口をひねる = turn the tap
• 蛇口を開ける = open the faucet
• 蛇口を閉める = close the faucet

There is no explicit subject stated, and that is very normal in Japanese.

The sentence means something like:

• If you leave the faucet open, the water bill will go up.
• If the faucet is left open, the water bill will end up being high.

The doer could be:

• you
• someone in the household
• people in general

Japanese often leaves this kind of subject unstated when it is obvious from context or when the point is the general consequence, not who does it.

Using なります would be grammatically correct:

• 蛇口を開けたままでは、水道代が高くなります。

That sounds more neutral and factual:

• the water bill goes up

Using なってしまいます adds a stronger warning tone:

• the water bill will end up going up
• the water bill will unfortunately become high

So the original sentence feels more like advice or caution.

Usually, no. It is more naturally understood as a general result or future consequence:

• If you leave the faucet open, the water bill will become high.

It describes what happens under that condition, not necessarily what has already happened right now.

That said, depending on context, it could also imply:

• That’s why the water bill ends up high
• That causes high water bills

But the most straightforward reading is a warning about consequence.

It can function as both, depending on context.

As a general statement:

• If a faucet is left open, the water bill will go up.

As a warning to someone nearby:

• If you leave the tap running, the water bill will go up, you know.

The grammar itself is neutral enough for either use. The polite ending しまいます makes it sound like a gentle warning or explanatory statement rather than a harsh command.

They are different.

• 蛇口を開けたまま = leave it open / it remains open
• 蛇口を開けておく = open it in advance / keep it open intentionally for later convenience

〜ておく often implies preparation or deliberate arrangement.

Examples:

• 窓を開けておく。 = I’ll leave the window open (on purpose).
• 資料を読んでおく。 = I’ll read the materials beforehand.

So with a faucet, 開けておく might sound like deliberately keeping it open, while 開けたまま simply describes the faucet being left open. In your sentence, 開けたまま is the natural choice because the focus is the state and its consequence.

Yes, in many contexts that is a very natural translation.

Possible translations:

• If you leave the tap open, the water bill will go up.
• With the tap left open, the water bill will end up being high.
• If the faucet is left running, the water bill will become expensive.

A small nuance:

• 開けたまま literally focuses on the faucet being left open
• English often says leave the tap running, which emphasizes the water flow

In real translation, left running is often the most natural English if water is clearly flowing.