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:

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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

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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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