## Innumeracy

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.