If 5 kg of haematite gives 2.78 kg of iron what is percentage purity of haematite?

Mass fraction of Fe in Fe₂O₃ (hematite) = (55.85×2 g/mol) / [(55.85×2 + 16.00×3) g/mol] = 111.7/159.7

Mass of Fe in the sample = 2.78 kg
Mass of Fe₂O₃ in the sample = (2.78 kg) ÷ (111.7/159.7) = (2.78 kg) × (159.7/111.7) = 3.97 kg

Percentage purity of hematite = (3.97/5) × 100(%) = 79.4(%)

You need to know the % iron by mass in pure haematite ... which you could calculate if you knew the chemical formula for it.

Mass of haematite in the sample in kg would then be 2.78 x 100/% Fe