Answer
$P_{soln} = 50.0\text{ torr}$
Work Step by Step
First, we solve for the number of moles of C$_3$H$_8$O$_3$ using the given mass of the compound,
Mol C$_3$H$_8$O$_3$ $= 164 \ g \times \dfrac{1 \ mol}{92.09 \ g} = $ 1.78 mol C$_3$H$_8$O$_3$
Then we solve for the number of moles of H$_2$O using the volume of the solvent,
Mol H$_2$O $= 338 \ mL \times \dfrac{0.992 \ g }{mL} \times \dfrac{1 \ mol}{18.02 \ g} = $ 18.6 mol H$_2$O
Then we solve for the vapor pressure of the solution by multiplying the pressure of the solvent with its mole fraction,
P$_{soln} = \chi_{H_2O} \ P^\circ_{H_2O} = \dfrac{18.6 \ mol}{(1.78 + 18.6) \ mor} \times 54.74 \ torr = 0.913 \times 54.74 \ torr = 50.0 \ torr$
