$Mn{O}_2$  on ignition converts into $M{n}_3{O}_4$ . A sample of pyrolusite having 75% $Mn{O}_2$ ,20% inert impurities and rest water is ignited in air to constant mass . What is the % of Mn in the ignited sample?(atomic mass of Mn=55)

Let the mass of sample is x gms mass of $Mn{O}_2$  is 0.75 x gm. Moles of  $Mn{O}_2=\frac{0.75x}{87}$,mass of 
$Mn=\frac{0.75x}{87}\times 55=0.47x$
$3Mn{O}_2\to M{n}_3{O}_4+O_2$
Mass of $M{n}_3{O}_4$ produced  $=\frac{0.75}{87}\times \frac{1}{3}\times 229=0.658xgm$
Total mass of residue $=0.658x+0.2x$ (mass unpurified)
% Mn in residue  $=\frac{0.474x}{0.858x}\times 100=55.24$