the mass of a certain mixture of BaO and CaO is found to be exactly one half that of the mixed sulfates formed from it by the action of H2SO4. calculate the percentage of BaO and CaO in the mixture.

Answer 1

An interesting problem.
Three equation and three unknowns but we can work it around with just two since these are related.
Let X = mass BaO
and Y = mass CaO
and mm stand for molar mass.
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eqn 1 is X + Y = mass whatever that may be.

Convert to sulfates and you have eqn 2.
(mm BaSO4/mm BaO)*X + (mm CaSO4/mm CaO)*Y = 2*mass whatever that is.

Make up a convenient number for mass X + Y and it really makes no difference what number you choose. I chose 1 to make it easier.
Solve for X and Y; then
%BaO = (X/1)*100 = ? (I obtained about 47%)
%CaO = (Y/1)*100 = ?(I obtained about 53%) but these were just quick calculations. You need to go thorought it more carefully. By the way, most students "shudder" at the thought of making up a number to use for mass BUT try it. Use it with 1 for mass, then change it to 2 or 3 or 4 or whatever you choose and work the problem. X and Y will always come out different, of course, BUT the problem doesn't ask for BaO and CaO. It asks for PERCENT and the percent will always come out the same.