Hydrogen peroxide is H2O2, while water is H2O and oxygen (a diatomic gas) is O2. The (unbalanced) reaction is:
H2O2 --> H2O + O2
Notice that the H2O2 has 2 H atoms, and so does H2. This means that both must have the same coefficients, and we can adjust the coefficient of O2. Since H2O2 has 2 O atoms, and H2O has 1, we multiply O2 by 1/2:
H2O2 --> H2O + (1/2)O2
This has an equivalent number of H and O atoms on either side, but we want the coefficients to be whole numbers, so we multiply everything by 2:
2H2O2 --> 2H2O + O2
