## Saturday, April 18, 2009

### 30-20-10 Pricing

Department store pricing has always bothered me.

The other day, I faced a dilemma when assessing whether it was worth buying a green shirt. The shirt was \$50 dollars, but 30 percent off for that day's sale, an additional 20 percent off for using a particular credit card, and 10 percent off because the item is green.

I had no trouble figuring out the final price -- I'm pretty good at math. In fact, I often enjoy the challenge of figuring out department store prices. It just bothers me that retail stores try to hide the price from customers. Ultimately, people will find out what the shirt really costs (the money leaves their possession at some point), so why put them through three flaming hoops of grade school math?

Honestly, I don't know of any good reason to torture the mathematically-challenged and that's what bothers me. But, in this post, I want to show why 30 percent off + 20 percent off + 10 percent off does not equal 60 percent off.

Let's take my shirt example. Most grade schools teach that 30 percent off of \$50 is like multiplying by 0.7. Similarly, taking an additional 20 percent off is like multiplying the result by 0.8. Take another 10 percent off by multiplying again by 0.9. Most people do this in three steps:

1. 30% off of \$50 = (1-0.3)*50= 0.7*50 (\$35)
2. 20% more off = 0.8*35 (\$28)
3. 10% more off = 0.9*28 (\$25.20)
Now, that's a lot of mental exercise if you don't know how to use your cell phone calculator. But, notice one sleezy thing about this pricing. The final price (\$25.20) is only 49.6 percent off of the original price (\$50). It is not 60 percent off as the store would want you to believe.

Why is this? As we saw above, the right calculation is at each step to multiply by the percentage paid, not add the percentage off. Therefore, no matter what the original price is, the percentage paid equals (0.7*0.8*0.9)*100=50.4 percent. That's 49.6 percent off!

This is not addition like the word "additional" suggests, so there's no reason to believe that adding percentages off will get the right answer. That's how a store can take 75 percent off and then an additional 40 percent off AND still take some of your money.

Surprisingly, the percentage off does not depend on the order in which the discounts are taken. To see this, rewrite the final price of my green shirt as 0.9*[0.8*(0.7*50)]. This is mathematically equivalent to taking 30 percent, then 20 percent, then 10 percent off of that. But, we could have just as well taken 20, then 30, then 10 and we would have gotten the same result because 0.9*0.8*0.7=0.8*0.9*0.7 (multiplication commutes).

So, the next time you are in a department store, remember your grade school math, and you'll easily know what you pay at the register.