Fraction Between Two Fractions

 In this applet, we model fractions as slopes of line segments. 
  1. Form two fractions by dragging the blue points. One fraction is the ratio of the lengths of the red segments. The ratio of the lengths of the green segments is the second fraction.
  2. Drag the slider to construct segments with slopes equal to the given fractions.
  3. Try the construction with other fractions by dragging the blue points.
What is the relationship between the original fractions and the fraction formed by adding their numerators and the denominators? In other words, how the slope of the blue segment relates to the slopes of the red and the green one?