Difference between price elasticity of demand and arc price elasticity of demand

Question

I am very confused between the definitions of price elasticity of demand. The actual definition I was given is $$E=\frac{\Delta Q \backslash Q}{\Delta P\backslash P}$$ I was once asked the following : find the price elasticity of demand supposing the price of a product increases from 12\$ to 20\$ ($\frac{200}3$% rise) and the quantity demanded falls from 55 to 45 ($\frac{-200}{11}$% fall). So it should have an elasticity of $E=-0.27$ approximately. Simple application of the formula I was given, nothing crazy. This is exactly the same procedure for what is being done here.

However, the right answer was $-0.4$... After a quick look on internet, this seems to be the arc price elasticity of demand : $$E=\frac{(Q_1-Q_0)(Q_1+Q_2)}{(P_1-P_0)(P_1+P_0)}=\frac{(45-55)(45+55)}{(20-12)(20+12)}=-0.4 $$

So these two concepts are different things right ?

Answer 1

They are both related concepts and they are both price elasticities.

Price elasticity can be derived at a single point, for that we would use the point price elasticity of demand formula:

$$e= \frac{dQ/Q}{dP/P}= dQ/dP \cdot P/Q $$

An alternative to point elasticity is the arc elasticity which tells you what the elasticity is between the two points.

This is usually calculated using the midpoint formula.

$$e = \frac{(Q_2-Q_1)/((Q_2+Q_1)/2)}{(P_2-P_1)/((P_2+P_1)/2)}$$

If someone just states calculate price elasticity of demand it could be either the point or arc method, depending on context.