Establish the ratio of angles

Given $3\angle P = 4\angle Q = 6\angle R = k$. Expressing each angle in terms of k: $\angle P = k/3$, $\angle Q = k/4$, $\angle R = k/6$. The ratio $\angle P: \angle Q: \angle R$ is $\frac{1}{3}: \frac{1}{4}: \frac{1}{6}$, which simplifies to $4: 3: 2$.

$\angle P: \angle Q: \angle R = 4: 3: 2$

Determine individual angle values

Let the angles be $4x$, $3x$, and $2x$. The sum of angles in a triangle is $180°$, so $4x + 3x + 2x = 180°$, which means $9x = 180°$ or $x = 20°$. Thus, $\angle P = 80°$, $\angle Q = 60°$, and $\angle R = 40°$.

$x = 20°$

Calculate the final expression

Substitute the angles into the required expression: $\frac{2\angle P - \angle Q + 3\angle R}{5} = \frac{2(80°) - 60°+ 3(40°)}{5}$. This equals \frac{160°- 60°+ 120°{5} = \frac{220°{5} = 44°.

$\frac{2(80) - 60 + 3(40)}{5} = 44°$