Sheila is making a regular-hexagon-shaped sign with side length $ 1$. Let $ABCDEF$ be the regular hexagon, and let $R, S,T$ and U be the midpoints of $FA$, $BC$, $CD$ and $EF$, respectively. Sheila splits the hexagon into four regions of equal width: trapezoids $ABSR$, $RSCF$ , $FCTU$, and $UTDE$. She then paints the middle two regions gold. The fraction of the total hexagon that is gold can be written in the form $m/n$ , where m and n are relatively prime positive integers. Compute $m + n$.
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1. **Calculate the area of the entire hexagon:**
   The area \( A \) of a regular hexagon with side length \( s \) is given by the formula:
   \[
   A = \frac{3s^2 \sqrt{3}}{2}
   \]
   Given \( s = 1 \), the area of the hexagon is:
   \[
   A = \frac{3(1^2) \sqrt{3}}{2} = \frac{3\sqrt{3}}{2}
   \]

2. **Determine the area of one of the shaded trapezoids:**
   The hexagon is divided into four regions of equal width. Each region is a trapezoid. The midpoints \( R, S, T, \) and \( U \) divide the hexagon into these trapezoids. The height of each trapezoid is \( \frac{\sqrt{3}}{2} \) because the height of the entire hexagon is \( \sqrt{3} \) and it is divided into two equal parts.

   The bases of the trapezoid are the sides of the hexagon and the segments connecting the midpoints. The length of the longer base is \( 1 \) (side of the hexagon), and the length of the shorter base is \( \frac{1}{2} \) (half the side of the hexagon).

   The area \( A_{\text{trapezoid}} \) of a trapezoid is given by:
   \[
   A_{\text{trapezoid}} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}
   \]
   Substituting the values:
   \[
   A_{\text{trapezoid}} = \frac{1}{2} \times \left(1 + \frac{1}{2}\right) \times \frac{\sqrt{3}}{2} = \frac{1}{2} \times \frac{3}{2} \times \frac{\sqrt{3}}{2} = \frac{3\sqrt{3}}{8}
   \]

3. **Calculate the total area of the two shaded trapezoids:**
   Since there are two shaded trapezoids, the total shaded area is:
   \[
   A_{\text{shaded}} = 2 \times \frac{3\sqrt{3}}{8} = \frac{6\sqrt{3}}{8} = \frac{3\sqrt{3}}{4}
   \]

4. **Determine the fraction of the hexagon that is shaded:**
   The fraction of the hexagon that is shaded is the ratio of the shaded area to the total area of the hexagon:
   \[
   \text{Fraction shaded} = \frac{A_{\text{shaded}}}{A_{\text{hexagon}}} = \frac{\frac{3\sqrt{3}}{4}}{\frac{3\sqrt{3}}{2}} = \frac{3\sqrt{3}}{4} \times \frac{2}{3\sqrt{3}} = \frac{1}{2}
   \]

5. **Express the fraction in the form \( \frac{m}{n} \) and compute \( m + n \):**
   The fraction \( \frac{1}{2} \) is already in its simplest form, where \( m = 1 \) and \( n = 2 \). Therefore:
   \[
   m + n = 1 + 2 = 3
   \]

The final answer is \( \boxed{3} \)