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GRE Algebra

There are common algebraic patterns that the new GRE wants you to recognize. Before we discuss these, I need to mention the FOIL method.

The FOIL method is one that almost everybody remembers learning at some point circa middle school. Though you may have forgotten the details, with a little practice (and you definitely want to become adept at the foil method), you should be able to use it effectively.

First off , FOIL stands for First, Outer, Inner, and Last, and refers to the position of numbers and/or variables within parenthesis. Let’s have a look:

(x - y)*(x + y) = ?

Remember, parenthesis stand for multiplication. The tricky part is how to multiply together a bunch of x’s and y’s. The answer: the FOIL method.

  • F (First): The first term in each parentheses is x, so we multiply the x’s together to get: x^2
  • O (Outer): The term on the outside of the left parenthesis is ‘x’ and on the outside of the right parenthesis is y. We multiply the two together to get: xy.
  • I (Inner): Now we multiply the inner terms in each parenthesis: ({-y})(x) = -xy
  • L (Last): Finally, we multiply the terms that are the rightmost to get ({-y})(y) = -y^2

Now we add together our results F + O + I + L: x^2 - xy + xy - y^2 = x^2 - y^2


So (x - y) (x + y) = x^2 - y^2.

Memorize this. Do not spend time on the test actually completing the steps above.


Other important algebraic expressions to memorize are:

(x-y)(x -y) = x^2 - 2xy + y^2

(x + y)(x + y) = x^2 + 2xy + y^2.


Here are some examples in which we apply the above.

  1. (x - 5) (x + 5) = x^2 - 25
  2. (a + 3)(a + 3) = a^2 + 6a + 9
  3. (b - 2c)(b - 2c) = b^2 - 4bc + 4c^2


Reversing FOIL and Solving for Roots of Equation

In other instances, you will take an equation in which you have to turn into parentheses.


Other applications of FOIL

(98)(102) = ?

(79)(81) = ?


These questions appear as though they would not relate to the FOIL method. But upon closer inspection, we can see that these numbers aren’t random.

If we add them, instead of multiplying, we get 200, for the first question, and 160, for the second. Or 100 + 100 and 80 + 80.

Let’s focus on the pair of hundreds: 100 * 100 vs. 98 * 102.

Notice that (98)(102) can be written as (100 - 2)(100 + 2). Now you should see the (x - y)(x + y) form, which expressed correctly is 100^2 - 2^2 = 9,996.

Solving in this way is much more effective, because 100^2 = 10,000, a number you should know of the top of your head.


Other Tips

Compare the following:

(x - 2)^2 = 4 vs. 2 = sqrt{x - 2}

In the first case, there are two solutions: 0 and 4. Remember when you square a negative number, as in (-2)^2 = 4, you get a positive number.

With the equation on the right side, the one in which x is under the square root sign, if you get a negative number, you cannot take the square root of it (at least in GRE world, where imaginary numbers do not come into play).

In the case of 2 = sqrt{x-2}, we square both sides to get 4 = x - 2, so x = 6.


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9 Responses to GRE Algebra

  1. Basia August 11, 2012 at 11:37 am #

    Can this be made clearer with an explanation of what the numbers refer to with regard to the question? Is there another way to approach the problem?

    • Chris Lele
      Chris August 13, 2012 at 1:41 pm #

      Hi Basia,

      I am not sure as to what you are asking? Which question were you referring to?

      • Basia August 13, 2012 at 2:11 pm #

        This solution by Krupa is what I would like further explicated:

        Krupa July 20, 2012 at 10:55 am #

        In response to your question,
        You can solve this by unitary method:
        3% (18%-15%) corresponds to the amount of alcohol being removed i.e. 18% of 8litres,
        Then 100% corresponds to how much?
        i.e. 100/3*0.18*8=48.

        The answer is C.

        Math is not my forte. Sorry if it seems obvious to most.

        • Chris Lele
          Chris August 13, 2012 at 3:59 pm #

          Upon doing this problem, I am getting a wildly different answer. I agree that the explanation is vague. After all if 3% corresponds to 8liters, wouldn’t 100% corresponds to something much closer to 250 liters.

          I think the problem with this question is it should never have been asked in the first place on this post, because the post is about FOIL (sadly, nobody seems to have commented on FOIL in the thread).

          So Basia, don’t worry about this question :). In terms of mixture it is more difficult than any mixture problem you are likely to see test day (I had to use a system of equations to solve it).

  2. Sneha July 20, 2012 at 3:37 pm #

    Thanks a lot Krupa . Got the answer.

  3. Krupa July 20, 2012 at 10:55 am #

    In response to your question,
    You can solve this by unitary method:
    3% (18%-15%) corresponds to the amount of alcohol being removed i.e. 18% of 8litres,
    Then 100% corresponds to how much?
    i.e. 100/3*0.18*8=48.

    The answer is C.

    • Chris Lele
      Chris July 23, 2012 at 10:29 am #

      Actually, I used a system of equations and got a very different answer. Seems like a very difficult question, even for the GRE. Wouldn’t worry about this level of question :).

  4. Sneha July 13, 2012 at 3:37 pm #

    Hi Chris,

    Can u solve this problem for me ??
    A certain amount of solution contains 18% alcohol. 8 litres of the solution is taken out and replaced with water. The resultant solution contains 15% alcohol. Find the volume(in litres) of the solution

    A. 40

    B. 50

    C. 48

    D. 44

    E. None of these

    • Chris Lele
      Chris July 23, 2012 at 10:29 am #

      Hi Sneha,

      I’m actually getting a very different answer then any provided. Also, I had to use a system of equations to do so, something you won’t see in GRE word problems (it’s more likely to show up on the GMAT). Anyhow, I wouldn’t worry about this level of question for the GRE.

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