Solución 2012, 5.1

Sea $x$ el número buscado, entonces:
$$\begin{eqnarray} \frac{x}{5} + \frac{3x}{8} & = & 49 + 2 \left(\frac{x}{6} - \frac{x}{12}\right)\\ \frac{8x+15x}{40} & = & 49 + \frac{x}{3} - \frac{x}{6}\\ \frac{23x}{40} & = & 49 + \frac{2x-x}{6}\\ \frac{23x}{40} - \frac{x}{6} & = & 49\\ \frac{69x-20x}{120} & = & 49\\ \frac{49x}{120} & = & 49\\ x & = & 120 \end{eqnarray}$$