the voltmeter shows 1V I need to find the voltage of the battery, I can not understand how do the fact the voltmeter shows 1V can help.

I know that the 20 and 40 capacitors have the same charge and so does the 10 and 30

 

Work the problem backwards. If you knew what the voltage of the battery was, could you find the voltage on the meter?

 

So Just to use KVL and charge conservation?

 

Pretty much.

A fine point is that we are assuming the meter has infinite resistance.

A related problem could be the following:

The above circuit uses a meter that has finite resistance and is left as shown for a long time. The meter is then removed. Finally the battery is disconnected. If the meter is once again connected, what is it's initial reading?

But don't try this now -- wait until after you get this problem solved.

 

@WBahn
Well, i solved the problem given by the TS; but i don't get the problem you gave (and how it is related):

Just tell me if i am wrong:
If the meter has a finite resistance, there will be a current flowing through it, if let like this for a long time (steady state), ZERO current will flow through it, which means there will be no potential difference between the two points of the meter (finite resistance), which means it will indicate ZERO ? if this is true, then removing the meter won't change anything to the circuit, and disconnecting the battery also won't change anything (because no current is flowing anywhere); if we put back the meter, it will still indicate ZERO ?

Are there any flaws in my thinking ?

 

No, I left a step out. After removing the battery you short the terminals (or turn the "battery" down to 0V).

 

@WBahn
If we short the battery and place again the meter, its initial reading will be the opposite of the first reading, is it right ?
If yes, can we guess this result without making calculations ?