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.

 

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  • Mike MᶜGarry

    Mike served as a GMAT Expert at Magoosh, helping create hundreds of lesson videos and practice questions to help guide GMAT students to success. He was also featured as "member of the month" for over two years at GMAT Club. Mike holds an A.B. in Physics (graduating magna cum laude) and an M.T.S. in Religions of the World, both from Harvard. Beyond standardized testing, Mike has over 20 years of both private and public high school teaching experience specializing in math and physics. In his free time, Mike likes smashing foosballs into orbit, and despite having no obvious cranial deficiency, he insists on rooting for the NY Mets. Learn more about the GMAT through Mike's Youtube video explanations and resources like What is a Good GMAT Score? and the GMAT Diagnostic Test.