I can prove the Rule of 100 algebraically, below. But my school kids are hankering after intuition, and a plainer explanation.
Follow the Rule of 100
Should discounts be percentages or absolutes?
Consider a \$150 blender. Should you offer 20% off? Or an equivalent \$30 off?
Answer:
- Over \$100? Give absolutes (e.g., \$30)
- Under \$100? Give percents (e.g., 20%)
In both cases, you show the higher numeral. For a \$50 blender, 20% off is the same as \$10 off — yet 20% is more persuasive because it’s a higher numeral. For a \$150 blender, the absolute discount (\$30 off) is a higher numeral (González, Esteva, Roggeveen, & Grewal, 2016).
References
González, E. M., Esteva, E., Roggeveen, A. L., & Grewal, D. (2016). Amount off versus percentage off—when does it matter?. Journal of Business Research, 69(3), 1022-1027.
Let d = discount, p = price. Then $d, p > 0$ because there is no free lunch and the quotation is expatiating on discounts. Then
\$ off vs. % off $\iff p - d \quad \text{ vs } \quad p - (d/100)p \iff -d \quad \text{ vs } \quad -dp/100 \iff 1 \quad \text{ vs } \quad p/100 \iff p \quad \text{ vs } \quad 100. $
A simpler way of going about it is to describe it from the opposite perspective, looking at the discounts instead of the prices. If you’re offering $a$ dollars off and the price is $p,$ then the numeral for the flat discount is $a$ and the numeral for the percentage is $\frac{a}{p} \cdot 100,$ which is the same as $a \cdot \frac{100}p.$
So, if $p < 100$ then $\frac{100}p > 1,$ so $a \cdot \frac{100}p > a,$ so the percentage number is bigger, and if $p > 100$ then $\frac{100}p < 1,$ so $a \cdot \frac{100}p < a$ and the flat number is bigger.
For a slightly looser explanation, we can just say that when we multiply by $\frac{100}p,$ when $p > 100$ the denominator is bigger, so the fraction is smaller than $1$ and multiplying by it decreases the value, and when $p < 100$ the numerator is bigger so it’s more than $1$ and multiplying by it increases the value.