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.


Chicago – Sunday, 07/26/2009

July 28, 2009

We had a tough time getting up Sunday morning, being exhausted from walking many, many miles on concrete and asphalt the day before. I think by 8:00am we managed to be out of bed. The first order of business was breakfast, but this time I chose the Blueberry Hill Cafe (in historic LaGrange, IL), having been discovered by my keenly observant mother a few weeks earlier. You see, Mom, my sister Sarah, and my nephew Alec were visiting me in early July. Sarah and I wanted breakfast at IHOP. On the way, Mom spotted the Blueberry Hill Cafe with a decidedly subordinate comment like “Oh, that looks like a nice place to eat” as we drove past it. After we failed to locate a local IHOP, Mom asserted the Blueberry Hill Cafe again, knowing from her long experience that it’s better to eat at a local (vs. large chain) restaurant when traveling. I couldn’t agree with her more.

Sharing in Mom’s wisdom, I took Dad to the Blueberry Hill Cafe.  He loved the restaurant on the way in, but got an omelette he didn’t like, spoiling his impression.  We made it to the train station and headed for the Museum of Science & Industry.  On the basis of the transit map alone, we decided the shortest way to the museum was to take the green line towards Cottage Grove all the way to the end.

Exiting the train station and finding ourselves in the projects, I realized there’s a lot more to getting around a city than train stops and street names. You’d think I would have remembered this lesson from when I lived in the D.C. suburbs, but apparently not. We made a hair-raising 2 mile walk through this war zone, replete with gold-toothed thugs wearing wife-beaters and pants sagging down to their knees. We got more than a few hostile stares from street goons, but managed to avoid any tolls or beatings. Both of us were sweating bullets. Dilapidated and crumbling structures with barred doors and broken windows boxed us in at every turn, while chain link fences with barbed wire rounded out the scenery.

Emerging suddenly from the projects was the University of Chicago, which was at once both a visual and psychological safe-haven, but the damage was done to any desires I might have about a postdoc there.  It’s too close to the projects (literally, one street away).  Although the campus was lush and green and seemed safe, one needs only to stray down a wrong street or two to find oneself in trouble. Finding ourselves out of the danger area, our heart and perspiration rates returned to normal. My brain emerged from fight-or-flight mode, and its cognitive abilities were freed up enough for me to generate a good idea: Mark regions of high crime rate on maps, especially tourist maps. Somebody should do this. Maybe they already have.

A quick walk past UChicago and Hyde Park, we arrived at the Museum of Science and Industry, the largest science museum in the western hemisphere. Flashing our CityPass booklets again, we skipped lines and made it into the exhibit halls. Confronting us head-on was a Foucalt pendulum, and I couldn’t help but think of the Coriolis effect and noninertial reference frames.

Focault Pendulum

Focault Pendulum

I surprised Dad by telling him I could calculate the length of the pendulum by just measuring the period of its swing (the period is the time for the pendulum to make one full swing back and forth). Indeed, the period of a simple pendulum is given by T=2\pi\sqrt{\frac{L}{g}} where L is the length of the pendulum and g is the magnitude of the gravitational acceleration at the pendulum’s location. Solving this equation for L we get L=\frac{gT^2}{4\pi^2}. I measured the period T of the pendulum over several swings and averaged my result, getting T=8.9\pm 0.1\textrm{~s}. Taking g=9.81\textrm{~m/s}^2 we get L=19.7\pm 0.2\textrm{~m}. Thus the length of the pendulum shown in the picture is 64 feet 7 inches, with an uncertainty of about 7 inches. Thus just by measuring the period alone, I am able to calculate how long the wire is that connects the pendulum to the ceiling, several stories above. Even though this is freshman physics, I still think it’s pretty cool.

Our first major exhibit was the Henry Crown Space Center, where was saw the original Apollo 8 space capsule and the Apollo 11 training mock-up. Next were the giant LEGO models of Adam Reed Tucker and the Navy, Auto, and Ships exhibits. Passing by a whispering gallery, I mentioned the reflection property of an ellipse. But the real gem of our trip was the U-505 submarine. This is the german U-boat captured by the United States on June 4, 1944. We only toured a few compartments of the U-boat, but it was enough to give you a palpable feel for life in those conditions. The tour was timed and had visual and audio cues, which I thought might be hokey at first but were pretty effective. Das Boot was on my mind during the tour. If you haven’t seen it, you should rent it. English subtitles will get you through this compelling movie.

Exiting the U-boat, I saw some of the original German Enigma machines and told Dad a little about Alan Turing and the contributions made by mathematicians that helped end World War II, since this was largely neglected from the exhibit. I guess math isn’t sexy enough for museums, even a science one. After briefly seeing a few more exhibits, like the Genetics Lab, Chick Hatchery, and Earth Revealed, we focused our attention on the Transportation Gallery, which has a replica of the 1903 Wright Flyer. But the gallery has even more more important originals, such as one of the only two remaining German Stukas in existence, a Spitfire, the Piccard Gondala, and a Boeing 727, whereupon Dad couldn’t help but reciprocate by lecturing me about jet engines and telling me all about the instrument panels in the flight deck.

Transportation Gallery

Transportation Gallery

After finishing our visit to the museum, we opted to take the #10 bus for the trip back to the loop, avoiding our harrowing experience earlier, which no doubt would be worse as dusk was coming on fast. Despite having to stand on the bus on account of crowdedness, it was well worth it in terms of personal safety.

Having had such a good experience the night before at Elephant & Castle, we decided not to risk anything new and repeated ourselves. We weren’t disappointed. Dinner conversation centered on organized religion again, and whether or not faith is a virtue (no need to say which way I argued that point). We left downtown around 6:30pm. Making it back to my car, I drove Dad through Westmont, IL and then on to Argonne National Laboratory, my reason for being in Chicago this summer. At 9:00pm we were back near the hotel, and Dad, with his characteristic generosity, insisted on filling my car up with gas despite having paid for almost everything so far the entire trip. Against my protestations to the contrary, he still filled it up. As the Borg say, resistance is futile.

Finally, I had the impulse to drive down the famous Lake Shore Drive at night, a scenic drive splitting Lake Michigan and the Chicago skyline. So we did.  The city was gorgeous at night time.  The trip up Lake Shore Drive was so nice, I surprised Dad by doing it twice. We made it home by around midnight again, and stayed up an hour or so talking, before fading to sleep.


Props to Skulls in the Stars

April 2, 2009

After commenting on a nice post about infinite series, gg suggested I start my own blog, a prospect I’ve not been too eager to pursue… until now.  Wordpress’ \LaTeX support persuaded me! Don’t expect to be enthralled by my posts. They’ll probably cover standard ground for me, e.g. math, physics, skepticism, and of course the occasional rant.