A powder contains mainly calcium oxide but also contains calcium nitrate and calcium sulfate. $15\mathrm{.}0g$ of this powder is used for a thermolysis experiment. The gases formed are passed through water to produce a $250-mL$ solution $($solution A$).$ The following two experiments are performed independently with this solution:

$($a$)50ml$ of solution $A$ is titrated with $0\mathrm{.}095M$ barium hydroxide. $26\mathrm{.}3ml$ of barium hydroxide is required to react all the acid present.

$($b$)$ To $25ml$ of solution A$,100ml$ of $2\mathrm{.}0M$ barium chloride is added. The precipitate formed is $0\mathrm{.}233g.$

Calculate the percentage composition of the original powder. Also determine the mole fraction of each component in this powder.