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Quadrants on the GMAT: The Cartesian Plane

Learn to master an area the GMAT loves to test. 

The Cartesian Plane (a.k.a. the x-y plane) is a favorite GMAT topic, especially on GMAT Data Sufficiency.  It’s a simple topic, but with just enough subtlety that the testmakers can spin endless questions from it.

Quadrants

quadrants

The quadrants begin with I, where both x and y are positive, and rotate counterclockwise from there.  Notice

quadrant sign list

 

Mixing in arithmetic operations

OK, still relatively simple, but now what happens when you multiply or divide the x- and y-coordinates?  Both (positive)*(positive) and (negative)*(negative) are positive, but if one factor is positive and one is negative, the product is negative.  Similarly, (positive)/(positive) and (negative)/(negative) are positive, but if dividend is positive & divisor negative, or vice versa, the quotient is negative.  Thus:

quadrant products & quotients

 

Already, that’s rich fodder for GMAT Data Sufficiency.  Now, what happens when we add the x- and y-coordinates?  (positive) + (positive) = (positive), and (negative) + (negative) = (negative), but (positive) + (negative) — hmmm — the sign of the sum depends on which has a larger absolute value.  Thus:

In QI, x + y > 0

In QII, the sign of x + y is unclear (i.e. it depends on the values of x & y)

In QIII, x + y < 0

In QIV, the sign of x + y is unclear (i.e. it depends on the values of x & y)

Even worse: subtraction!  We know a (positive) – (negative) must be positive, and a (negative) – (positive) must be negative, but if we have either a (positive) – (positive) or (negative) – (negative), then the sign of the difference depends on which has a larger absolute value.  Thus:

In QI, the signs of (y – x) and (x – y) are unclear (i.e. it depends on the values of x & y)

In QII, y – x > 0 and x – y < 0

In QIII, the signs of (y – x) and (x – y) are unclear (i.e. it depends on the values of x & y)

In QIV, y – x < 0 and x – y > 0

Now, you probably have some appreciation of how many potential Data Sufficiency Questions the GMAT could concoct simply on the quadrants and the signs of x & y coordinates.  A simple topic, but one worth thinking through thoroughly so you are ready on test day.

Practice questions:

1)  Is the point (x, y) in the fourth quadrant?

(1) xy > 0

(2) y > 0

 

2) http://gmat.magoosh.com/questions/1031

The question at that link will be followed by a video explaining the solution.

 

Practice question explanations

1) This DS question is a yes/no question.  We don’t actually need to determine the quadrant of (x, y), only whether it’s in QIV.

Statement #1: xy > 0

This inequality means that the (x,y) is either in QI or QIII, as we saw above.  We don’t know which, but in either case, we can definitively say: no, it’s not in QIV.  Because the statement allows us to arrive at a definitive answer — and it doesn’t matter one peep whether it’s a yes or no answer, as long as it’s definitive — the statement is sufficient.  We got a definitive no answer to the question, so Statement #1 is sufficient.

Statement #2: y > 0

This inequality means that the point (x,y) is north of the x-axis, in either QI or QII.  We don’t know which, but in either case, we can definitively say: no, it’s not in QIV.  Again, definitive answer to the question, so Statement #2 is sufficient.

Both statements sufficient: answer = D.

 

The solution & explanation to question #2 are available at the link above.

 

About the Author

Mike McGarry is a Content Developer for Magoosh with over 20 years of teaching experience and a BS in Physics and an MA in Religion, both from Harvard. He enjoys hitting foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Follow him on Google+!

8 Responses to Quadrants on the GMAT: The Cartesian Plane

  1. Rahul Sehgal August 26, 2013 at 10:10 pm #

    Mike – As I was going through this great article, I noticed one thing – under ‘even worse – subtraction’ section, at one point, it says (negative) – (positive) will always be positive. But, it should always be negative.

    Please make the update.

    As always, I simply like reading your articles.

    Rahul

    • Mike
      Mike August 27, 2013 at 10:03 am #

      Rahul,
      Yes! Thank you for pointing out this typo — I just corrected it. Thank you very much for your kind words.
      Mike :-)

  2. news feed February 10, 2013 at 11:30 pm #

    I really appreciate this post. I’ve been looking all over for this! Thank goodness I found it on Bing. You have made my day! Thank you again

    • Mike
      Mike February 11, 2013 at 9:27 am #

      Thank you for your kind words. Best of luck to you.
      Mike :-)

  3. Suresh October 28, 2012 at 6:49 am #

    Quadrant IV is shown as III ?

    • Margarette
      Margarette October 30, 2012 at 10:10 am #

      Hi, Suresh! Sorry about that– it should be fixed now :)

      Best,
      Margarette

      • Suresh August 27, 2013 at 11:29 am #

        Hi Margaretta,

        Quardrant IV is still shown as III.

        Thanks,
        Suresh

        • Mike
          Mike August 27, 2013 at 2:38 pm #

          Suresh,
          We fixed that. Thanks for pointing it out.
          Mike :-)


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