We have to look at this in terms of moles. If 1.0 g of Mn2P2O7 is obtained, this equates to 1/283.9 = 0.00352 moles of Mn2P2O7. Since the Mn2P2O7 originates from MnO2, then we divide by two to get the moles of MnO2 that this originated from. We know this...

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We have to look at this in terms of moles. If 1.0 g of Mn2P2O7 is obtained, this equates to 1/283.9 = 0.00352 moles of Mn2P2O7. Since the Mn2P2O7 originates from MnO2, then we divide by two to get the moles of MnO2 that this originated from. We know this from the subscripts by the Mn element in each formula. So that's 0.00176 moles of MnO2. Multiply by the molecular weight of MnO2 to get the grams of MnO2 from the sample: 0.00176 * 86.9 = 0.115 g MnO2. Divide by the total mass of the sample of the ore to get the percent composition: 0.153 g / 0.75 g = 20.4%.

If you want the actual percent of Mn (not MnO2), then multiply the moles by the molecular weight of Mn: 0.00176 * 54.9 = 0.097 g Mn. Divide by 0.75 g to get the percent Mn: 0.097 g / 0.75 g = 12.9%.