First we balance the metals P:phosphorus
[tex]\text{ HClO}_4\text{ + P}_2\text{O}_5\text{ }\rightarrow\text{ Cl}_2\text{O}_7\text{ + 2HPO}_3[/tex]then we balance the non-metals Cl:chlorine
[tex]\text{ 2HClO}_4\text{ + P}_2\text{O}_5\text{ }\rightarrow\text{ Cl}_2\text{O}_7\text{ + 2HPO}_3[/tex]Then we balance the hydrogens and oxygens
If we look at the hydrogens and oxygens they are already balanced because there are 2 hydrogens in the reactants and 2 hydrogens in the products. Similarly, There are 13 oxygens in the reactants and 13 oxygens in the products.
The answer is:
[tex]2\text{ HClO}_4\text{ + P}_2\text{O}_5\text{ }\rightarrow\text{ Cl}_2\text{O}_7\text{ + 2HPO}_3[/tex]