calculation of characteristic expansion time of the universe
Here is some remarks on the issue, straight from Ryden:


  If galaxies are currently moving away from each other, then it implies
  they were closer together in the past. Consider a pair of galaxies
  currently separated by a distance $r$, with a velocity $v = H_0r$
  relative to each other. If there are no forces acting to accelerate or
  decelerate their relative motion, then their velocity is constant, adn
  the time that has elapsed since they were in contact is
  
  $t_0 = \frac{r}{v} = \frac{r}{H_0r} = H_0^{-1}$, 
  
  independent of their current separation $r$.


Which part of this are you confused about? I can edit my answer once I better understand your question.