Calcium is in the second group of the periodic table, so its oxidation number will always be . This is true for all alkaline earth metals.

As for chromium, the dichromate anion () has an overall charge of . Each oxygen atom will always have an oxidation number of  as well, so the oxygens would have a total negative charge of . The sum of the oxidation numbers have to equal the overall charge, so chromium must be used to balance the negative charges from the oxygen. To bring the overall charge to , the two chromiums have to have total charge of , giving each an oxidation number of .

Manganese is a transition metal, so its oxidation number is more variable than the other elements of the compound. Oxygen is almost always . Potassium is almost always . The sum of the oxygens and the potassium is .

The compound is neutral, so the manganese is  to balance the net molecular charge.

To solve this question we need to calculate the oxidation number of oxygen in both molecules. The formula for water is . The oxidation number of hydrogen is +1. Since there are two of them, the hydrogen atoms contribute to a charge of +2. The water molecule is neutral; therefore, the oxygen must have an oxidation number of  to balance the charge. The formula for hydrogen peroxide is . Using the same logic as water, we can determine that hydrogen contributes +2. We have two oxygen atoms in this case; therefore, each oxygen atom will have an oxidation number of  to give a charge of . This will balance the charges and provide a neutral hydrogen peroxide molecule.

Recall that have a negative charge suggests that an atom has extra valence electrons. A charge of  suggests one extra valence electron and a charge of  suggests two extra valence electrons. An oxygen typically has six valence electrons. The oxygen in water has  oxidation number; therefore, it will have two extra valence electrons (eight total). On the other hand, oxygen in hydrogen peroxide will have one extra valence electron (seven total); therefore, oxygen in hydrogen peroxide has one less valence electron than oxygen in water.

Potassium bromide has a formula of . This molecule is made up of an alkali metal (potassium) and a halogen (bromine). Alkali metals have one valence electron that they readily lose to obtain octet whereas halogens have seven valence electrons and they readily gain an electron to obtain octet. Recall that losing an electron will give you a  oxidation number whereas gaining an electron will give you a  oxidation number. This means that alkali metals always have an oxidation number of  whereas halogens always have an oxidation number of ; therefore, potassium has an oxidation number of .

Manganese(III) oxide has a formula of . The oxidation number of oxygen is . Since there are three of them, the oxygen atoms contribute to a charge of . To maintain the neutrality of the molecule, manganese must balance the  charge (meaning contribute  charge). There are two manganese atoms; therefore, each manganese atom will have an oxidation number of  and contribute  charge.

The reaction stated in this question is as follows.

This is a double replacement reaction. Both products contain a metal (an alkali metal) and a nonmetal (a halogen). The oxidation number of all alkali metals (first column of periodic table) is  and of all halogens (seventh column of periodic table) is . The results stated in the question seem invalid because a halogen can never have an oxidation number of .

Generally, oxygen has an oxidation number of , and hydrogen has an oxidation number of . If we multiply  by the number of oxygen atoms in the compound (), we get an overall charge of  on the oxygen. If we multiply  by the number of hydrogen atoms in the compound (), we get an overall charge of  on the hydrogen. The sum of  and  is . We can see that  is a neutral molecule, so the charge on  would have to balance out the  charge on the rest of the molecule. To obtain a charge of , the oxidation number on  must then be .