Math Series #3 (Cutting up fractions)

If a number $x$ has three factors $a$ , $b$ , and $c$ , $\frac {1}{x} = \frac {a}{x \times (a + b + c)} + \frac {b}{x \times (a + b + c)} + \frac {c}{x \times (a + b + c)}$ . Since $x = 24$ and it can be expressed as $2 \times 3 \times 4$ , $\frac {1}{24} = \frac {2}{24 \times (2 + 3 + 4)} + \frac {3}{24 \times (2 + 3 + 4)} + \frac {4}{24 \times (2 + 3 + 4)} = \boxed{\frac {1}{108} + \frac {1}{72} + \frac {1}{54}}$ .