The molarity of the original calcium hydroxide solution will be 0.176 molar.
First, we will write the chemical equations for both reactions:
 HBr + Ca(OH)₂  →  H₂O + CaBr₂
 HBr + Ba(OH)₂  →  H₂O + BaBr₂
Balance the chemical equations
 2HBr + Ca(OH)₂  →  2H₂O + CaBr₂
 2HBr + Ba(OH)₂  →  2H₂O + BaBr₂
Concentration of hydrobromic acid (HBr) = 59.6 ml x 0.303 M / 1000
Concentration of hydrobromic acid (HBr) = 0.0181 mol
Hence
 0.5 equivalent Ca or Ba hydroxide = 0.0181 mol x 0.5 mol
 0.5 equivalent Ca or Ba hydroxide = 0.00905 Moles
 Concentration of Barium hydroxide = 21.2 ml x 0.186 / 1000
 Concentration of Barium hydroxide = 0.00394 mol
 Concentration of Calcium hydroxide = 0.00905 - 0.00394
 Concentration of Calcium hydroxide = 0.00511 mol
Molar solution of Calcium hydroxide = 0.00511 x 1000 / 29.0 ml
Molar solution of Calcium hydroxide = 0.176 molar calcium hydroxide solution is needed.
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