Why does price elasticity of demand formula work? Hi there,

I understand how to use the price elasticity of demand formula (mid-point method), shown in the picture above. 

However, I'm not too sure why exactly it works.

Why are dividing the change in quantity and price by the averages of the quantity and price change? Is that a way of calculating the percentage change?

And why is elasticity a fraction and not a ratio? Would it be better to write: as price changes by 5 percent, quantity decreases by 10 percent, and write that as 5:10?
Why are dividing the change in quantity and price by the averages of the quantity and price change? Is that a way of calculating the percentage change?


Yes, elasticity is a measure of proportional response.  The math you've shown is the "midpoint" method.  It's basically a linear approximation of elasticity, and you'll abandon it once calculus starts showing up in your learning.  I would suggest not spending a huge amount of time trying to understand it, but only for this reason.  As long as you understand that elasticity of demand says something about how consumers respond to price changes at a certain price level, that's the key message.


  And why is elasticity a fraction and not a ratio? 


It's really neither - it's a quotient of quotients, which yields a pure (that is, unitless) number. You can check this for yourself if you explicitly write out the units for the things you're looking at.  This is called "dimensional analysis" and can be extremely useful for interpreting complicated expressions.  The way you do it is to replace each quantity with the thing it "counts" in [square brackets]:

$\frac{2(Q_2 - Q_1)}{Q_1 + Q_2} \longrightarrow \frac{[purenumber]([apples]-[apples])}{[apples]+[apples]} \longrightarrow \frac{[purenumber][apples]}{[apples]} \longrightarrow [purenumber]$


  Would it be better to write: as price changes by 5 percent, quantity decreases by 10 percent, and write that as 5:10?


In general I would caution against using ratio notation except when playing with statistics (and even then, I personally find them tricky to work with). Here, you may risk confusing a ratio of absolute changes (not what you want to express) for a ratio of percentage changes (this is what elasticity actually is).