The pitfalls of rates

To avoid the pitfalls of percentages, it is essential to follow the calculations meticulously. If we add 10% to €100, we add €10 and obtain €110. That is what matters, not the rate itself. The rule for percentages is multiplicative, not additive. Adding 10% to an amount means multiplying it by 1.1. If we add another 10%, we multiply the new amount by 1.1. The original amount has therefore been multiplied by 1.1². But 1.1² is not 1.2; it is 1.21. Thus, adding 10% twice does not add 20%, but 21%. Similarly, reducing a price by 20% amounts to multiplying it by 0.8. A further reduction of the same 20% applies the same rate once again. Ultimately, the original price is multiplied by 0.8², or 0.64, corresponding to a reduction of 36%, not 40%. Rates are not added; they are multiplied! In mathematics, everyday language can be a source of confusion: although it is perfectly legitimate to speak of adding one rate to another, this does not mean that the corresponding mathematical operation is addition. Sometimes adding can mean multiplying!

The same phenomenon occurs when we want to recover the original price after a reduction. Consider a situation that often arises at private art shows. Since there is no intermediary to pay, the artist will often agree to reduce gallery prices by one third. What does this tell us about gallery commissions? Take a painting sold for €1,200 in a gallery. The painter sells it directly for one third less (€400 less), or €800. To go from €800 to €1,200, the multiplier is 1,200 divided by 800, or 1.5. The gallery's commission is therefore 50% (!).

The explanation is much the same as before. Applying a one-third reduction means multiplying the original price by 2 / 3. To recover that price, we must multiply the final price by the reciprocal of 2 / 3, namely 3 / 2, which gives our 50% increase.

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