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Last time, we talked about how to make story problems real; if you haven’t read that article yet, go take a look before you continue with this one.

I’ve got another one for you that’s in that same vein: the math topic is different, but the “story” idea still hold in general. This one has something extra though: you need to know how a certain math topic (standard deviation) works in general. Otherwise, you won’t be able to think your way through the problem.

Try this GMATPrep® problem:

* ” During an experiment, some water was removed from each of 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?

“(1) For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.

“(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.”

Standard deviation! Ugh. : )

Okay, it’s no accident that they’re using a DS-format problem for this one. It’s not possible to calculate a standard deviation in 2 minutes without a calculator (unless, perhaps, that standard deviation is zero!). They never expect us to calculate standard deviation on this test, but they do want to know whether we understand the concept in general.

So what is standard deviation? Try to answer—aloud—in your own words before you continue reading.

(Why did I say “aloud”? Often, we tell ourselves that we can explain something, but not until we actually try do we realize that we need a refresher on the concept. Giving an explanation aloud forces you to prove that you really do know how to explain the concept. If you don’t, you’ll hear your uncertainty in your own explanation.)

Standard deviation is the measure of how spread apart a set of data points is. For example, let’s say you have the following 5 numbers in a set: {3, 3, 3, 3, 3}. The standard deviation is zero because the numbers are all exactly the same—there is no “spread” at all in the set.

Which of the following two sets has a larger standard deviation?

{1, 2, 3, 4, 5}

{1, 10, 20, 80, 2,000}

The second one! The numbers are much more spread apart than in the first set.

Right now, some of you are wondering: okay, but what’s the actual standard deviation of those two sets?

I don’t know. I could calculate it—I’m sure there are many online “standard deviation” calculators I could use. But I don’t care. The real test is never going to make me calculate this! (And that’s why I haven’t gotten into the actual calculation method here… nor will I.)

There are a few concepts that we should know, though, in terms of how changes to sets can affect the standard deviation. Read more

I’ve been on a story problem kick lately. People have a love / hate relationship with these. On the one hand, it’s a story! It should be easier than “pure” math! We should be able to figure it out!

On the other hand, we have to figure out what they’re talking about, and then we have to translate the words into math, and then we have to come up with an approach. That’s where story problems start to go off the rails.

You know what I mean, right? Those ones where you think it’ll be fine, and then you’re about 2 minutes in and you realize that everything you’ve written down so far doesn’t make sense, but you’re sure that you can set it up, so you try again, and you get an answer but it’s not in the answer choices, and now you’re at 3.5 minutes or so… argh!

So let’s talk about how to make story problems REAL. They’re no longer going to be abstract math problems. You’re riding Train X as it approaches Train Y. You’re the store manager figuring out how many hours to give Sue so that she’ll still make the same amount of money now that her hourly wage has gone up.

Try this GMATPrep® problem:

* ” Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?

“(A) 2

“(B) 3

“(C) 4

“(D) 6

“(E) 8”

Yuck. A work problem.

Except… here’s the cool thing. The vast majority of rate and work problems have awesome shortcuts. This is so true that, nowadays, if I look at a rate or work problem and the only solution idea I have is that old, annoying RTD (or RTW) chart… I’m probably going to skip the problem entirely. It’s not worth my time or mental energy.

This problem is no exception—in fact, this one is an amazing example of a complicated problem with a 20-second solution. Seriously—20 seconds!

You own a factory now (lucky you!). Your factory has 6 machines in it. At the beginning of the first day, you turn on all 6 machines and they start pumping out their widgets. After 12 continuous days of this, the machines have produced all of the widgets you need, so you turn them off again.

Let’s say that, on day 1, you turned them all on, but then you turned them off at the end of that day. What proportion of the job did your machines finish that day? They did 1/12 of the job.

Now, here’s a key turning point. Most people will then try to figure out how much work one machine does on one day. (Many people will even make the mistake of thinking that one machine does 1/12 of the job in one day.) But don’t go in that direction in the first place! If you were really the factory owner, you wouldn’t start writing equations at this point. You’d figure out what you need by testing some scenarios. Read more