A student was asked to divide the half of a certain number by 6 and the other half by 4 and then to add the two quantities so obtained. Instead of doing so the student divided the number by 5 and the result fell short by 4. The given number was ?

Let the number be x
Then,
$$\eqalign{ & \Leftrightarrow \left[ {\frac{{\left( {\frac{x}{2}} \right)}}{6} + \frac{{\left( {\frac{x}{2}} \right)}}{4}} \right] - \frac{x}{5} = 4 \cr & \Leftrightarrow \frac{x}{{12}} + \frac{x}{8} - \frac{x}{5} = 4 \cr & \Leftrightarrow \frac{{10x + 15x - 24x}}{{120}} = 4 \cr & \Leftrightarrow x = 4 \times 120 \cr & \Leftrightarrow x = 480 \cr} $$