Inequalities

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by Brent@GMATPrepNow » Fri Sep 20, 2013 2:58 pm
jose.mario.amaya wrote:Is ad > bc ?

(1) a/b > c/d
(2) b/d >0

Target question: Is ad > bc ?

Statement 1: a/b > c/d
IMPORTANT: We must avoid the temptation of multiplying or dividing both sides by one of the variables. Since we don't know whether each variable is positive or negative, we wouldn't know whether to reverse the direction of the inequality.

Okay, There are several values of a, b, c and d that satisfy this condition. Here are two:
Case a: a = 2, b = 1, c = 1 and d = 1, in which case ad > bc
Case b: a = 2, b = 1, c = 1 and d = -1, in which case ad < bc
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b/d > 0
No information about a or d, so there's no way to determine whether ad > bc
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: If b/d > 0 (statement 2), then EITHER b and d are both positive, OR b and d are both negative.
Let's examine each case and how it affects statement 1.

case a: b and d are both positive
Statement 1: a/b > c/d
Multiply both sides by b to get: a > cb/d
Multiply both sides by d to get: ad > cb

case b: b and d are both negative
Statement 1: a/b > c/d
Multiply both sides by b to get: a < cb/d [reversed inequality, since b is negative]
Multiply both sides by d to get: ad > cb [reversed inequality, since d is negative]

In both possible cases, the conclusion is that ad > cb.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent
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by theCodeToGMAT » Fri Sep 20, 2013 8:15 pm
Jose, i will solve this question by the method with "Mitch" once shared

for ad > bc

(1) a/b > c/d

IF b & d both have same signs, simply cross multiply : ad > cb
BUT, we are not sure of the sign of "b" and "d".
INSUFFICIENT

Statement 2: b/d > 0
this implies that "b" and "d" have same signs...
But, this statement doesn't provide any good info other than this.
So, INSUFFICIENT

Combining...
b & d have same signs, simply cross multiply : ad > cb
SUFFICIENT

ANSWER [spoiler][C][/spoiler]
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by mevicks » Fri Sep 20, 2013 8:21 pm
Brent@GMATPrepNow wrote: Statements 1 and 2 combined
IMPORTANT: If b/d > 0 (statement 2), then EITHER b and d are both positive, OR b and d are both negative.

Cheers,
Brent
Hi Brent,

I stopped after reaching the conclusion that both b,d can be negative or positive. And in order to save time quickly chose [spoiler]E (incorrect answer here)[/spoiler] "hoping" that they have no role in the combined statements since a and c are still unknown.

However after reviewing my error I did understand that my mistake was not going that extra step to check and verify the solutions together. I'm prone to similar problems in complicated DS questions where we have to take that extra leap, for eg:
Stem : What is x ?
St.1: |x+3| = 4x - 3

Ans: St. 1 is sufficient since x = 2 or x = 0, but x = 0 does not satisfy the equation so x = 2 is the only answer, thus sufficient.

Same is the situation with deadly DS+Quadratic equations.

My question is do we make it a rule of thumb to recheck and reverify ALL DS answers by plugging in the final values if possible?

Thanks & Regards,
Vivek

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by Brent@GMATPrepNow » Sat Sep 21, 2013 5:36 am
mevicks wrote:
Brent@GMATPrepNow wrote: Statements 1 and 2 combined
IMPORTANT: If b/d > 0 (statement 2), then EITHER b and d are both positive, OR b and d are both negative.

Cheers,
Brent
Hi Brent,

I stopped after reaching the conclusion that both b,d can be negative or positive. And in order to save time quickly chose [spoiler]E (incorrect answer here)[/spoiler] "hoping" that they have no role in the combined statements since a and c are still unknown.

However after reviewing my error I did understand that my mistake was not going that extra step to check and verify the solutions together. I'm prone to similar problems in complicated DS questions where we have to take that extra leap, for eg:
Stem : What is x ?
St.1: |x+3| = 4x - 3

Ans: St. 1 is sufficient since x = 2 or x = 0, but x = 0 does not satisfy the equation so x = 2 is the only answer, thus sufficient.

Same is the situation with deadly DS+Quadratic equations.

My question is do we make it a rule of thumb to recheck and reverify ALL DS answers by plugging in the final values if possible?

Thanks & Regards,
Vivek
I'd say don't confuse intermediate goals with the actual target question.

When we encounter a DS question, we often begin imagining what kind of information would be sufficient to answer the target question, and we can get into trouble by confusing this intermediate goal with the actual target question.

For example, if a target question asks us to find the value of x², we might think, "To answer this question, it would be sufficient if I knew the value of x" (So, the intermediate goal is to find the value of x, even though this is actually more information than is really needed). From here, you might check one of the statements, and it leads you to the conclusion that x equals EITHER 3 or -3. At this point, you might think, "That's not enough information, because I want to find the value of x." This is what I mean by confusing an intermediate goal with the actual target question. If the goal were to find the value of x, then the statement would, indeed, be insufficient. However, since the goal is to find the value of x², the statement is sufficient (since 3²=9 and (-3)²=9).

Likewise, for the original question, we have two possible scenarios for the variables b and d, but this does not necessarily mean we have two possible (and different) answers to the target question.

I hope that helps.

Cheers,
Brent
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by mevicks » Sat Sep 21, 2013 6:41 am
Thanks a ton Brent for pointing out the exact problem in my DS related approach, indeed I was losing sight of the Target Question. I'd surely try to incorporate your pointers when solving DS questions in future, hopefully avoiding some more silly mistakes :oops:

Thanks & Regards,
Vivek