The unbalanced redox reaction is,
\(\mathrm{MnO}_{4}^{-}+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}+\mathrm{H}^{+} \rightarrow \mathrm{Mn}^{2+}+\mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}\)
Balance C atoms,
\(\mathrm{MnO}_{4}^{-}+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}+\mathrm{H}^{+} \rightarrow \mathrm{Mn}^{2+}+2 \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}\)
The oxidation number of \(\mathrm{Mn}\) decreases from \(+7\) to \(+2\).
The decrease in the oxidation number is \(7-2=5\)
The oxidation number of \(C\) increases from \(+3\) to \(+4\).
The increase in the oxidation number for \(1 \mathrm{C}\) atom is \(4-3=1\)
The increase in the oxidation number for \(2 C\) atom is \(2 \times 1=2\)
To balance the increase in the oxidation number with decrease in the oxidation number, multiply Mn containing species with 2 and C containing species with 5.
\(2 \mathrm{MnO}_{4}^{-}+5 \mathrm{C}_{2} \mathrm{O}_{4}^{2-} \rightarrow 2 \mathrm{Mn}^{2+}+10 \mathrm{CO}_{2}\)
Balance \(\mathrm{H}\) atoms by adding \(16 \mathrm{H}^{+}\)ions to the reactants side.
\(2 \mathrm{MnO}_{4}^{-}+5 \mathrm{C}_{2} \mathrm{O}_{4}^{2-}+16 \mathrm{H}^{+} \rightarrow 2 \mathrm{Mn}^{2+}+10 \mathrm{CO}_{2}+8 \mathrm{H}_{2} \mathrm{O}\)
This is the balanced equation.