## Innumeracy

August 24, 2010

So I just saw an article about Justin Bieber’s newly released song “U Smile 800% Slower.” This song was purportedly obtained by slowing down Bieber’s original “U Smile” song by 800%. Read that last sentence again and see if you can make sense of it. That’s what I thought. Slowing something down by 800% is gibberish, unless you allow for going in reverse.

See the original article for yourself by clicking here.

To be fair, what they mean is that the new version has been “stretched out” to be 8 times longer than the old version. Equivalently, its new speed is one eighth its old speed. But this is not the same as being slowed down by 800%. In fact, if your new speed is one eighth your old speed, then you’ve slowed down by 87.5%, not 800%, a fact which I’ll prove below.

To understand what’s going on, first consider increasing a quantity by a certain percentage. What does it mean to increase something up by 50%? Well, it’s new value should be 50% more than its old value. For example, if you increase 100 by 50%, what’s the new value? Answer: 150. What did you do? You added 50% of 100 to itself. More generally, if $x$ is the quantity we’re increasing by $p$ percent, its new value is $x+(p/100)x=(1+p/100)x$. Thus, to increase 100 by 75% is to multiply it by $1+75/100=1.75$, so its new value is 175, right? Right.

Now consider reducing something by a certain percentage. Say you’re driving down the highway at 100 mph. If you slow down by 50%, how fast are you going? Answer: 50 mph. If you slow down instead by 75% how fast are you going? Answer: 25 mph. What did you do? Answer: you subtracted the given percentage of the original quantity from itself. Generally, if $x$ is the quantity to be reduced by $p$ percent, its new value is $x-(p/100)x=(1-p/100)x$. In other words, to find the new (reduced) value, you multiply the original value by $1-p/100$. In the 75% case for example, the new value of $x$ is $(1-75/100)x=0.25x$. Similarly, if you slow down by 100% how fast are you going? Answer: 0 mph, since $(1-100/100)x=0x=0$. You’ve stopped.

Now, if you slow down by 800% how fast are you going? Answer: You’re going in reverse at 700 mph, since $x(1-800/100)=-7x$.

In any case, the new song is 1/8th as fast as the original song. If $x$ is the speed of the original song, its new speed is $(1/8)x$. By what percentage was the song slowed down? According to the formula developed above, we need to find a $p$ such that $(1/8)x=(1-p/100)x$. Solving for $p$ gives $p=87.5$. That is, the song was slowed down by 87.5%, not 800%.

Got it? Good.

## Truth in advertising

August 19, 2009

My campus bookstore made a dramatic claim about reduced textbook pricing. Karen Natale, the “Bookstore & Licensing Program Manager” has this to say about the new bookstore and reduced prices:

In case you don’t have time to watch the video, here’s the relevant quote:

As you know, publishers set the prices, the bookstore doesn’t, on [new] textbooks. But bookstores add a margin, and the margin covers overhead, salaries, shipping and handling, all the things we need to do to stay in business. In the past, our margin was 23%, which is what the national average is. Many of the UNC campuses are at 25%. We are now, thanks to our new contract [with Barnes & Noble], at 18%. So that’s a significant savings that students are going to see staying in their pockets.

Wow! A margin of only 18%, when the national average is 23% and some UNC campuses are charging as much as 25%, according to her. Sounds great, right?

Suppose the publisher sells a book to the bookstore for the wholesale price of $105. With an advertised 18% margin, what do you think the shelf price should be? Pause for a moment and calculate this yourself. You probably came up with$123.90. You took $105 and multiplied it by 1.18, right? Unfortunately, that’s not how it’s done. What you’ve done (and what advertisers are counting on) is to confuse margin with markup. There’s quite a difference. To be brief, if you start with the sale price, the margin is percentage you *discount* to get the cost, whereas if you start with the cost, the markup is the percentage you *add* to determine the sale price. Thus markup and margin are “dual” to each other in some sense. To be definite, suppose $C$ is the wholesale cost of the item to the reseller. Let $M$ represent the percent margin or markup desired, with $0\leq M<1$. If $M$ represents markup, then the price $P$ of the item is $P=C(1+M)$. If $M$ represents margin, the formula is $P=C\frac{1}{1-M}$ instead. To summarize, $P_\text{markup}=C(1+M)$ $P_\text{margin}=C\frac{\displaystyle 1}{\displaystyle 1-M}$. So, back to our textbook example. Its wholesale cost to the bookstore was$105.00. Since they have an 18% margin, the price listed on the shelf will be $105\frac{1}{1-.18}=105\frac{1}{.82}=105(1.22)=128.05$. Did you notice? An 18% margin became a 22% markup!

In general, if the margin is $M$ then the markup is $\frac{M}{1-M}$. Here’s a short table showing the relationship in increments of 10%.

0% margin = 0% markup
10% margin = 11.1% markup
20% margin = 25% markup
30% margin = 42.9% markup
40% margin = 66.7% markup
50% margin = 100% markup
60% margin = 150% markup
70% margin = 233% markup
80% margin = 400% markup
90% margin = 900% markup

Assuming $0\leq M<1$, we may Taylor expand the margin factor as follows: $\frac{1}{1-M}=1+M+M^2+M^3+\cdots$. Thus we see that

$P_\text{markup}=C(1+M)$

$P_\text{margin}=C(1+M+M^2+M^3+\cdots)$

So the markup formula is really just a first order Taylor approximation to the margin formula. In using the margin formula, they’re benefitting from keeping all those higher order terms, and they can really add up over many, many sales.

What’s worse, the difference between the markup formula and the margin formula becomes more dramatic as the margin goes up. The formulas are identical for $M=0$, but the margin formula blows up to infinity as $M$ approaches 1 (i.e. 100%). Meanwhile, the honest markup formula remains “honest” no matter how high $M$ goes.

By the way, I don’t place any blame on Karen Natale. She’s been excellent in working with professors and publishers to get low prices and correct errors.

I’m guessing resellers advertise margin instead of markup since, for a fixed profit, the advertised margin will be smaller than the markup, making it sound like you’re getting a better deal than you really are. I suppose this tactic is par for the advertising world. Oh well. Now you know.