[UG]

I'm not sure what you mean, but I'll have a guess, on the left there are two O₃'s wich make six and on the right there are three O₂'s which also make six so the total amount of atoms do not change.

[Evaldas]

I'm not sure what you mean, but I'll have a guess, on the left there are two O₃'s wich make six and on the right there are three O₂'s which also make six so the total amount of atoms do not change.

Well then why we need a ratio 2:3? Why not 1:1?

[UG]

Er, can you expand on that?  I'm lost 

[Evaldas]

I don't understand why when there's an equation that has different coeficients or different indexes on both sides it must have some different counting... like: 2Au₂O₃  4Au + 3O₂

[Borek]

I don't understand why when there's an equation that has different coeficients or different indexes on both sides it must have some different counting... like: 2Au₂O₃  4Au + 3O₂

No idea what you men by different counting.

2 molecules of Au₂O₃ decompose into 4 atoms of gold and 3 diatomic molecules of oxygen.

You start with 2 molecules of Au₂O₃, each containing 2 atoms of gold, that makes 4 atoms of gold on the left. There are 4 atoms of gold on the right - so it is the same number of atoms of gold in reactants and in products.

You start with 2 molecules of Au₂O₃, each containing 3 atoms of oxygen, that makes 6 atoms of oxygen. There are 3 molecules of diatomic molecules of oxygen on the right, that makes 6 atoms of oxygen - so there are identical numbers of oxygen atoms on both sides.

Thsi equation can be also read in a slightly different way, that is more convenient for stoichiometric calculations. See this page:

http://www.chembuddy.com/?left=balancing-stoichiometry&right=stoichiometric-calculations