**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

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

## 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:

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.

In the practice question #2 above, under the second condition which says j+k is greater than 0, the points j,k may also lie in quadrant 1 right? For example, if j is 2 and k is 3 and since positive + positive is positive, j+k is still greater than 0 right?

Hi Sukanya 🙂

Yes, you’re correct that, given Statement 2, the point (j,k) could be in quadrant 1, since the sum j+k would be positive in that case 🙂 The video explanation simply shows that since, the point (j,k) could be in at least two different quadrant and still satisfy the condition given in Statement 2, the statement is not sufficient to answer the question in the prompt.

Hope this helps 🙂

one of the most beautiful post.

thanks Mikey

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

Rahul,

Yes! Thank you for pointing out this typo — I just corrected it. Thank you very much for your kind words.

Mike 🙂

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

Thank you for your kind words. Best of luck to you.

Mike 🙂

Thank you so much I was so confused about the xy >0

Quadrant IV is shown as III ?

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

Best,

Margarette

Hi Margaretta,

Quardrant IV is still shown as III.

Thanks,

Suresh

Suresh,

We fixed that. Thanks for pointing it out.

Mike 🙂