Gọi \(n_{CaCO_3}=x\left(mol\right);n_{MgCO_3}=y\left(mol\right)\)
\(m_{Al_2O_3}=\dfrac{100x+84y}{10}\)
Bảo toàn Ca \(\Rightarrow n_{CaO}=n_{CaCO_3}=x\left(mol\right)\)
Bảo toàn Mg \(\Rightarrow n_{MgO}=n_{MgCO_3}=y\left(mol\right)\)
\(\Rightarrow m_Y=m_{CaO}+m_{MgO}+m_{Al_2O_3}\)\(=56x+40y+\dfrac{100x+84y}{10}\)
\(\Rightarrow56x+40y+\dfrac{100x+84y}{10}=56,8\%.m_X=56,8\%.\dfrac{11}{10}.\left(100x+84y\right)\)
\(=\dfrac{781}{1250}.\left(100x+84y\right)\)\(\Leftrightarrow56x+40y=\dfrac{328}{625}\left(100x+84y\right)\)
\(\Leftrightarrow x=\dfrac{29}{25}y\)
\(\%m_{CaCO_3}=\dfrac{100x}{\dfrac{11}{10}.\left(100x+84y\right)}.100\%=\dfrac{100.\dfrac{29}{25}y}{\dfrac{11}{10}.\left(100.\dfrac{29}{25}y+84y\right)}.100\%\approx52,73\left(\%\right)\)
\(\%m_{MgCO_3}=\dfrac{84y}{\dfrac{11}{10}.\left(100x+84y\right)}.100\%=\dfrac{84y}{\dfrac{11}{10}.\left(100.\dfrac{29}{25}y+84y\right)}.100\%\approx38,18\left(\%\right)\)
\(\Rightarrow\%m_{Al_2O_3}\approx9,09\left(\%\right)\)