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Formulas And The GRE

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Occasionally, we’ll get an algebra problem in which a pre-defined formula is given for some phenomenon, and then we’re told to manipulate that formula in some way. People often find these quite tough because we typically didn’t see questions like this in school.

Let’s try this problem first (© Manhattan Prep) from our GRE Algebra Strategy Guide.

 Life expectancy is defined by the formula 2SB/G, where S = shoe size, B = average monthly electric bill in dollars, and G = GRE score. If Melvin’s GRE score is twice his monthly electric bill, and his life expectancy is 50, what is his shoe size?

(There are no multiple choice answers for this one. Also, yes, we’re being a little silly with this problem.

GRE formulas But don’t the big story problems feel like this sometimes? Just having a little fun while we learn : ) )

Many students will tell me, It doesn’t seem like we can solve this one at all. There are four variables and they only give us the value for one of them. How can we possibly figure out his shoe size?

Appearances can be deceiving. Start writing. Let’s call life expectancy L:

L = 2SB/G

The problem also told us that Melvin’s GRE score is twice his monthly electric bill. Translate that into an equation:

G = 2B

Now, what can we do with that? First, we can substitute:

L = 2SB/2B

And they gave us a value for L:

50 = 2SB/2B

And here’s where we can begin to get excited. That B on the top of the fraction is the exact same B on the bottom of the fraction “ in other words, we can cancel out those two variables. Bingo! We’re down to one variable.

50 = 2S/2

The 2’s on the top and bottom of the fraction also cancel, and we’re left with S = 50. Melvin has some seriously large feet.

Okay, now try this one that’s more like a real GRE problem; again, there are no multiple choice answers for this one. (© Manhattan Prep from our GRE Algebra Strategy Guide)

If the radius of a circle is tripled, what is the ratio of the area of half the original circle to the area of the whole new circle?

Yeah, that’s more like it. That feels like a proper GRE question, doesn’t it? Also, yuck.

First, they’re talking about the area of a circle, and we know a formula for that: A = Πr2. Let’s write that down on our scrap paper.

Notice that the problem never says anything about how big the original radius is, or the new one, or really anything solid about either circle. This means we get to pick our own numbers.

Let’s say that the original circle has a radius of 1, which gives us a nice and easy A = Πr2 = Π(1)2 = Π.

Now, the problem says if the radius is tripled, so let’s see what happens when we do that: A = Πr2 = Π(3)2 = 9Π. That’s the new area. So the ratio is Π:9Π, or 1:9, right?

Not so fast! Read the question again. We were asked for the ratio of the area of half of the original circle to the area of the new circle. Argh. The original circle’s area was Π, so half the area is Π/2. The area of the new circle is 9Π, so the ratio is Π/2:9Π. We’re not supposed to have fractional values in ratios, though, so we need to multiply both sides by 2: Π:18Π = 1:18. (You can also write this as a fraction.)

The ratio of half the area of the old circle to the entire area of the new circle is 1:18, or 1/18.

Takeaways for formulas:

(1) Sometimes they’ll give us weird formulas, as we saw on our first problem, and sometimes they’ll ask us to do weird manipulations to real formulas that we already know.

(2) As with other problems, if they give us real numbers, we need to use those. If they don’t, then we can test the problem using our own numbers, as we did in our second example.

(3) These are just another form of algebra problem, even though we didn’t often see questions like this in school. Familiarize yourself with them.