Igniting \( \mathrm{MnO}_{2} \) converts it quantitatively to \( \mathrm{Mn}_{3} \mathrm{O}_{4} ...

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Igniting \( \mathrm{MnO}_{2} \) converts it quantitatively to
\( \mathrm{Mn}_{3} \mathrm{O}_{4} \). A sample of pyrolusite is of the following composition :
\[
\mathrm{MnO}_{2}=80 \%, \mathrm{SiO}_{2} \text { and other inert contents }=15 \% \text {, }
\]
rest being water.
The sample is ignited in air to constant weight.
What is the percentage of \( \mathrm{Mn} \) in the ignited sample?
(At. wt. of \( \mathrm{Mn}=55 \) ).
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