The Lowly Mathematician

On reddit today, there was a comment about a radio personality who argued with a Mathematician about how percentages work, specifically about how percentages are combined. I replied with a short explaination of how 35% + 15% = 55.25% and not latex 50%. Shortly thereafter, there were half-a-dozen comments, ranging from “That explaination was longwinded/overcomplicated”, to “Thanks, that really made the idea clear.” I realized that, while I don’t hold a Teaching License, or Advanced degree in math, I am pretty good at it, and maybe teaching a bit of basic, useful math or programming that anyone can understand and use might be a pretty popular topic for blog posts, enter the new Sporadic Series “Basic Math, how to use number any day.” (To bastardize the tagline from Numb3rs, which is an excellent show, by the way).

The posts will be aimed at laypersons with no mathematical experience beyond 10th grade or so, we’ll talk about geometry and algebra, basic statistics and numerical stuff, maybe even some more abstract stuff, but always presented from a practical point of view.

For the inagural post, Let’s talk about percentages, we’ll aim to look at percentages from a qualitative, practical viewpoint.

First, what is a percentage? Basically, a percentage is just a fraction, it represents a portion of a whole. The basic meaning of a percentage is simply that, for some numbers x and y, finding x-percent of y is the same as saying “calculate latex y/100x.” For instance, heres a simple word problem.

You’ve got a “pay two-thirds-price” coupon on your favorite coffee, Moroccan French Roast, it usually costs 7.99$ for a 1-lb bag of these delicious beans, but you only have a five in your pocket, can you afford the coffee?

So first, how do we say “pay two-thirds-price” in the language of Percentages? Remember that every percentage is really a fraction, and two-thirds is just the fraction 2/3. So we can easily convert this fraction using the definition I gave above, namely that:

x% of y = y/100x.

So, we can represent 2/3 = 0.6666… in decimal. So if we apply the definition, the percentage equivalent to two-thirds is 66.7% percent, or so. All that is useful, but when push comes to shove, we really can’t use this notation for percentages, it’s a nice way to look at them, but as far actual usefulness goes, we know alot more about the fractional representation, so we like to use that.

Back to our problem, we are simply asking how much two-thirds of 7.99 is, and whether that is more or less than 5 dollars. we can evaluate this quickly by using the following trick. 7.99 is the same as (8.00 - 0.01), now when we try to take 2/3s of 7.99, we can instead take 2/3s of 8 and 2/3s of 0.01, which is 5.33 and 0.006, we then take the latter value from the former, and we find that we have 5.27, or so. So no coffee for us.

Thats the fundamental problem in every percentage question. How do we convert to a fraction, and then how do we multiply quickly? The interesting stuff comes when we start to combine percentages, like stacking coupons, or finding out how well you did on a test, and similar.

Now lets start taking an algebraic slant. Think about the following problem.

You’re back at the store, and this time, you’re prepared, you have four 10% off coupons for you coffee, and that same fiver. The 1-lb bag still costs 7.99$, can you afford the coffee now?

So, the first question, what does “10% off” mean? Intuitively, it means that the store will subtract 10% of the price of the item if you have the coupon. So for any x%-off coupon, what we really have is that you’re going to pay (100-x)% of the price. So 10% off really means 90% of the price. Now heres the interesting part, with some number n 10% off coupons, you might suspect that you’ll get 10n% off, but this isn’t always the case, most places that will give you 10%-off coupons chain the coupons together, meaning they apply the first coupon to the original price, the second to the now discounted price, and so on. So instead of getting 60% off the price of our coffee, we’d be getting 10% off of 10% off of 10% and so on off of the original price.

What does this mean for our total discount? Well, converting the chain above, we need to find 90% of 90% of 90% of 90% of 90% of 90% of 7.99. Note that 100% of the price is 7.99, so we could write this as a product of fractions: 9/10 * 9/10 * … * 9/10. That’s basic algebra though, we just compute 9^4 / 10^4, using a calculator, we find that that is equal to 6561/10000, which is equal to the percentage 65.61% of the original price, which is equivalent to (100 - 65.61)% = 34.39% off. Again, using a calculator, we use the fractional representation of 65.61% of 7.99$, and find that it’s equal to: 5.24$. Damn, no coffee.

So, depending on how your local store combines coupons, you may be getting a bigger or smaller discount.

I’ll leave you folks with an exercise, Imagine that you’re taking the SATs. As many of you may know, the SAT’s recently redesigned their system to include three tests, with a possible 800 points each. You want to compare your scores with that of your older brother, who scored 1460 on the old “double-800’s” version[1]. You want to at least tie your older brothers score, how well do you have to do on the SAT’s to match your brothers score?

Remember that the SAT’s score in 10s, that is, you can’t score a 1423.52, you can only score in a multiple of ten.

The answer will appear in the next post in this series. Stay tuned!

[1] Yes, that is my real score on the SATs, I took them twice, that was the higher of the two (the first time I took them, I got a 1320, but I don’t like to talk about that.