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Hi Zeke,
I took a GMAT Prep Practice test the other day and I struggled with a couple of question. Please help:

Question 1:
If ab != 0 and points (-a,b) and (-b,a) are in the same quadrant of the xy-plane, is point (-x,y) in this same quadrant?
(1) xy > 0
(2) ax > 0

Question 2:
Juan bought some paperback books that cost \$8 each and some hardcover books that cost \$25 each.
If Juan bought more than 10 paper books, how many hardcover books did he buy?

(1) total cost of hardcover books that Juan bought was at least \$150
(2) total cost of the books that Juan bought was less than \$260

Thanks,
xxxxxx

Modupe,

Question 1:
It looks like this question is about signs.
ab!=0 tells us that neither a nor b is zero. On a graph, the points a and b do not lie on either the x axis or y axis. It’s everything else that it can be.

Let’s see what our possibilities are, we’ll run 4 scenarios for a and b:
positive, positive
positive, negative
negative, negative
negative, positive

Let’s take a=5, b=3.
(-a, b) is (-5, 3) – upper left quadrant
(-b, a) is (-3, 5) – upper left quadrant.

However, if we have a=5 b=-3 then
(-5, -3) and (3, -5) are different quadrants. Not possible in our question.

If a=-5 and b=-3
(5, -3) and (3, -5) then both in quadrant 4.

If a=-5 and b=3 then
(5, 3) and (-3, 5) are different quadrants. Not possible in our question.

The question we are asked is whether point (-x, y) is in that same quadrant as where the common quadrant is (which is 2 or 4)

Without any additional information, point (-x, y) can be any quadrant since there is no limitation on x and y yet. So we do not have sufficient information to answer the question just yet. Let’s take the (1) and (2) we are given:

(1) tells us that x and y are the same sign. Either both positive or both negative

(2) tells us that both a and x are the same sign. Either both positive or both negative

From (1) we see that (-x, y) is going to be in quadrants 2 and 4.
If x and y are both positive, the point is in quadrant 2.
If x and y are both negative, the point is in quadrant 4.

We can’t be 100% sure that this information answers our question. Originally point (-x, y) can be any quadrant – since there is no restriction on x or y.

With (1) we know they are both the same sign, and this limits the quadrants to 2 and 4. We still dont’ know which one exactly it’s going to be.

From (2), we see that a and x are the same sign. However, this does not help us limit what quadrants (-x, y) are NOR does it help us limit what quadrants (-a, b) and (-b, a) are.

However, if we combine (1) and (2), we can use transitive property and say that x, y and a all have the same sign.

x = y
y = z
x = y = z

Same concept.

x same sign as y
y same sign as z
x same sign as y same sign as z

Now we know that if x is positive, then a must also be positive and so points (-a, b) and (-b, a) are in quadrant 2. At the same time, point (-x, y) is ALSO quadrant 2. We know definitively that quadrant 2 is the common quadrant here.

If x is negative, then a must also be negative. From above, the only possibility is for b to be negative. And so points (-a, b) and (-b, a) are in quadrant 4.
At the same time, point (-x, y) is ALSO quadrant 4. (both x and y are negative).

Combining (1) and (2) we can get a definitive answer so (C) is the answer.

You might ask, how do I answer this question in real-time during the actual exam.

This question is about signs. Quickly draw that (1) gives us that x and y are the same sign. If they are the same sign, do I have enough info to answer the question? That question being that (-x, y) is in the same quadrant as the original condition proposed in the question. It doesn’t seem like the information helps us narrow down so (1) is insufficient.

Quickly draw from (2) that a and x are the same sign. This also doesn’t narrow down the quadrants for us.

Combine (1) and (2) – notice the overlap is in x. Draw that a, x, and y all are the same sign.
THen run scenarios where a,x, and y are positive =>quardrant 2
THen run scenarios where a,x, and y are negative => quadrant 4.

We got definitive answers so (C) helps us get there.

Question 2:
Juan bought some paperback books that cost \$8 each and some hardcover books that cost \$25 each.
If Juan bought more than 10 paper books, how many hardcover books did he buy?

(1) total cost of hardcover books that Juan bought was at least \$150
(2) total cost of the books that Juan bought was less than \$260

8x + 25y
x >10

(1) Total cost of harder was at least \$150. That means he bought at least 6 hardcover books at \$25 each. We don’t know how many hardcover books he bought exactly, just that it was more than 6.

(2) Well, if he spent at least 10*\$8 = \$80 on paperbacks – the remainder goes to hardcover books. At most, \$260 – \$80 = \$180 was spent on hardcovers. This could have been 1 hardcover – 2, 3, 4, 5, 6, or 7 hardcovers. We don’t have the definitive answer.

What if we combined (1) and (2).

More than 6 hardcovers…the only option is 7 hardcovers for \$175. It’s a definitive answer.
Does this meet requirements for (1) and (2)? It does. Answer (C)

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