Intuition: Difference in price elasticities of demand due to different bases Source: pp 91-92, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw


  If you try calculating the price elasticity of demand between two points on a
  demand
  curve, you will quickly notice an annoying problem: The elasticity from
  point A to point B seems different from the elasticity from point B to point A. For
  example, consider these numbers:
  Point A: Price = \$4, Quantity = 120
  Point B: Price = \$6, Quantity = 80
  
  Going from point A to point B, the price rises by 50 percent and the quantity falls by
  33 percent, indicating that the price elasticity of demand is 33/50, or 0.66.
  By contrast,
  going from point B to point A, the price falls by 33 percent and the quantity
  rises by 50 percent, indicating that the price elasticity of demand is 50/33, or 1.5.     $\color{darkred } { \text { This
difference}}$ arises because the $\color{darkgreen } { \text {percentage changes are calculated from a different base.}   }$


Though I calculated these 2 different price elasticities of demand and understood the last sentence, the big pictures or the deeper intuitions elude me? How can I naturalise this such difference? 

What's the intuition behind  $\color{darkred } { \text { This
difference}}$, caused by $\color{darkgreen } { \text {percentage changes [that] are calculated from a different base}   }$ ?
The solution to this issue is called "Arc Elasticity", used for "large" absolute changes. On the surface, the formula for Arc Elasticity is the familiar one, for example for price elasticity of demand

$$\eta_{arc}= \dfrac{ \text{
Percentage change in quantity demanded }} {  \text{ Percentage change in price } }$$

But the thing is how these percentage changes are calculated: If we went from quantity $q_a$ to quantity $q_b$, then while we "usually" calculate the percentage change as $(q_b-q_a)/q_a$ for the Arc Elasticity we divide by the mid-point of the interval -not by the point we started with. Same for price. So

$$\eta_{arc}= \dfrac{ \text{
Percentage change in quantity demanded }} {  \text{ Percentage change in price } } = \frac {\frac {q_b-q_a}{(q_b+q_a)/2}}{\frac {p_b-p_a}{(p_b+p_a)/2}}$$

$$\implies \eta_{arc} = \frac {q_b-q_a}{p_b-p_a}\cdot \frac{p_b+p_a}{q_b+q_a}$$

Re-arrange the terms at will to suit your own mnemonic triggers.

The Arc Elasticity has the convenient properties

(1) it is symmetric with respect to the two prices and quantities,
(2) it is independent of the units of measurement,
(3) it yields a value of unity if the total revenues (price times quantity) at the two points are equal. 

See Allen (1933).

P.S. to OP: Use this for your other question.