Is there a discrepancy between the ratio and total expenditure methods of measuring elasticity? According to the Expenditure method of measuring price elasticity of demand, if the two expenditures in comparison are constant (in spite of changes in price), the demand is considered to be unitary elastic. 
However, there are several examples where this does not seem to hold true. 
For instance, 

Price: 10 ; Demand: 12 ; Expenditure: 120

Price: 08 ; Demand: 15 ; Expenditure: 120

Here, though the expenditure is equal, if we calculate the percentage changes in price and demand, they come to be 20% and 25% respectively. So as per the ratio (or percentage method), the elasticity would be equal to 25% / 20%, which is not equal to 1. Then, how can we conclude that the expenditure method gives the right results?
If something rises by $25\%$ then you then need a $20\%$ reduction in the new larger number to get back where you started: $1.25 \times 0.8=1$. This asymmetry means comparing large percentage changes can produced oddities

Another approach, closer to the idea of approaching the derivative as a limit, is to use logarithms and calculate the elasticity as a ratio of logarithms, something like $$\frac{\log\left(\frac{Q_2}{Q_1}\right)}{\log\left(\frac{P_2}{P_1}\right)}= \frac{\log(Q_2)-\log(Q_1)}{\log(P_2)-\log(P_1)}$$

and with your example this would give $\frac{\log(15)-\log(12)}{\log(8)-\log(10)} = \frac{\log(1.25)}{\log(0.8)} =-1$ as expected