1. Given data

• Vapor pressure of pure benzene at $27^\circ \mathrm{C}$: $P^\circ_{\text{benzene}} = 80\ \text{Torr}$
• Vapor pressure of pure methyl benzene at $27^\circ \mathrm{C}$: $P^\circ_{\text{methyl benzene}} = 24\ \text{Torr}$
• Liquid mixture is equimolar, so in liquid phase: $x_{\text{benzene}} = x_{\text{methyl benzene}} = 0.5$

Since the solution is ideal, we apply Raoult's law.

2. Partial pressures in vapor phase

For benzene: $p_{\text{benzene}} = x_{\text{benzene}} P^\circ_{\text{benzene}} = 0.5 \times 80 = 40\ \text{Torr}$

For methyl benzene: $p_{\text{methyl benzene}} = x_{\text{methyl benzene}} P^\circ_{\text{methyl benzene}} = 0.5 \times 24 = 12\ \text{Torr}$

3. Total vapor pressure

$P_{\text{total}} = 40 + 12 = 52\ \text{Torr}$

4. Mole fraction of methyl benzene in vapor phase

Using Dalton’s law, $y_{\text{methyl benzene}} = \frac{p_{\text{methyl benzene}}}{P_{\text{total}}} = \frac{12}{52} = \frac{3}{13}$

$\frac{3}{13} \approx 0.2308$

So, $y_{\text{methyl benzene}} \approx 23.08 \times 10^{-2}$

Nearest integer: $\boxed{23}$

5. Comparison with stored answer

Stored correct answer = $23$

Our derived answer = $23$

So the answer matches the stored correct answer.