# How can school children intuit why over 100, D is larger? But under 100, D% is larger?

#### 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

percentagesorabsolutes?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. $

## 1 answer

(I’m going to do this post in pounds so that I don’t have to escape dollar signs everywhere. But the currency doesn’t matter.)

“Per cent” is, literally, “for each hundred”. Imagine making a literal pile of money from the cost of the item, and splitting it up into stacks of £100. So if the item costs £500, you have five stacks.

Now, a discount that’s “ten pounds” means just that: take £10 out of the pile (any stack, it doesn’t matter) and put it back in your wallet.

But “ten per cent” means £10 out of *every single one* of those stacks. For five stacks, that’s £50 I get to keep! I’d rather have that discount.

What if the price is only £100, though? In that case, there’s only one stack, and “£10 from one stack” and “£10 from every stack” mean the same thing.

*Below* £100, the intuition is a little bit trickier, because you have to imagine the 10% case as taking “part of” £10 from a stack that’s “part of” £100. But it’s not an insurmountable hurdle to envision that, and grasp that it won’t be as good as taking out a full £10.

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