calculation of characteristic expansion time of the universe Hubble's constant is estimated at 15km/s at a distance of million light years, without considering effects of mutual gravity, relativity, etc.
I am trying to calculate the characteristic expansion time of the universe.
Its formula is given as reciprocal of the Hubble constant, because time is distance by velocity. But:

$$v=kx$$

This implies that velocity and distance are exponential functions of time and so time cannot simply be calculated by dividing 1 million light years by 15km/s.
So what am I missing? Why is the characteristic expansion time simply the reciprocal of Hubble's constant?
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.