We are currently putting together some sales reports for our company, and noticed some oddities when we have multiple tax rates, and it's bugging me as to why this is happening.
| Tax Rate | Retail Price (R) | Tax Amount (retail) | Cost Price (C) | Tax Amount (cost) |
|---|---|---|---|---|
| 0.2 | 1492.925 | 298.63 | 1036.68 | 207.31 |
| 0.05 | 1632.27619 | 81.63 | 1159.66 | 57.98 |
When we calculate the percentages and figures in isolation, the values are all correct.
However, when we add them all together, the percentage of tax between retail and cost is different:
Retail:
(298.63 + 81.63) / (1492.93 + 1632.28) * 100 == 12.17
Cost:
(207.31 + 57.98) / (1036.68 + 1159.66) * 100 == 12.08
I would have expected the two blended percentages to have the same value. Shouldn't both calculations give the same result?
Since $\dfrac {C_2}{C_1}\ne\dfrac {R_2}{R_1}$ (that is, since items/rows 1 and 2 have different profit margins), there is no basis for assuming that quantities (a) and (b) should have the same value.
Quantity (a) is the average tax rate of the two items, weighted by their retail prices.
Quantity (b) is the average tax rate of the two items, weighted by their cost prices.
Quantities (a) and (b) are not the average tax rate of the two items, which is just $12.5\%$ (the average of $20\%$ and $5\%$). You'd get this value with these alternative computations analogous to (a) and (b): $$\frac12\left(\frac{298.63}{1492.93} + \frac{81.63}{1632.28}\right)=12.5\%$$ and $$\frac12\left(\frac{207.31}{1036.68} + \frac{57.98}{1159.66}\right)=12.5\%.$$
Fuller explanation at Average of ratios versus Ratio of averages.
Ambiguous terms like "blended percentage" and "retail percentage of tax" are not precise enough to distinguish among the various related quantities, and give rise to misunderstandings like this.