Is (2a - b)(b - 3c) 0?

Question

Is (2a - b)(b - 3c) > 0?

1) 2ab - 6ac > b^2 - 3bc
2) 2ab + 9c^2 = 3bc + 6ac

Source: GMAT Course

aniteshgmat1101 · Answer

I doubt the solution provided. Problem expects an answer in Yes or No.
Answer should be D.
Statement 1: 2ab - 6ac > b^2 - 3bc.
After taking factor and keeping all the element on L.H.S, we directly (2a - b)(b - 3c) > 0. So, Sufficient.

Statement 2: 2ab + 9c^2 = 3bc + 6ac.
After taking factors, we get (b - 3c) (2a - 3c) = 0.
Now, if b - 3c =0, then given equation (2a - b)(b - 3c) = 0.
Again, if 2a - 3c =0, then 2a = 3c. Putting the value of 2a in given equation we get (-(b - 3c)^2). This value could be -ve or zero (not > 0). So, Sufficient.

stavya · Answer

can anyone post an oe for this.

thefibonacci · Answer

Don't know a proper approach to this one, but this is how i did it..

AD/BCE

For options AD to go away we need to prove that this cannot be solved by option 1.

1) 2ab - 6ac > b^2 - 3bc
put a=b=0 and c=-1 
above inequality holds true and we get NO as an answer to question stem.

put a=2,b=1, and c=-1
above inequality holds true and we get YES as an answer to question stem.

hence, 1 is insufficient.

2) 2ab + 9c^2 = 3bc + 6ac
or 2ab + 9c^2 - 3bc - 6ac = 0 
or (2a-3c)(b-3c) = 0 
for this to hold true 
2a=b or (and) b=3c

if we put just b=3c in question stem, we get answer as NO 
if we put just 2a=3c in question stem, it gives (b-3c)^2 < 0 which cannot be true as square can never be negative (so invalid) 
so we get a definite answer as NO from this statement.

hence, 2 is sufficient. B.

EMPOWERgmatRichC · Answer

Hi thefibonacci,

You've made a slight error in your work. The above example does NOT "fit" the given inequality in Fact 1.

GMAT assassins aren't born, they're made,
Rich

thefibonacci · Answer

thanks for pointing that out...not able to think of any other combination at the moment....can you pls suggest ?

EMPOWERgmatRichC · Answer

Hi thefibonacci,

Here's another factor to consider when dealing with this question...

DS questions are based heavily on patterns, so when a "strange looking" question shows up, it's NOT by accident. This prompt asks if (2A - B)(B - 3C) > 0..... and THAT is a weird question to ask.....

If we FOIL out the question, we get....

Is 2AB - 6AC - B^2 + 3BC > 0?

Now, take good look at Fact 1 again (and do a bit of algebra to "move around" the pieces.....

Fact 1: 2AB - 6AC > B^2 - 3BC

What do you end up with here?.... And how does it relate to the question that is asked?....

GMAT assassins aren't born, they're made,
Rich

stavya · Answer

not getting this at all...
shouldnt the oa be A..

ashakil3 · Answer

I don't under stand how the answer is B.

From simplifying the question, we are essentially being asked, is b>3c?

it is clear the (1) is not sufficient since it doesn't provide any new information.
(2) does not seem clear to me since b=3c and/or 2a=3c. If 2a=3c then it can hold true if b<3c or b>3c, so we still don't have a definitive answer to the question stem.

Therefore, the answer should be E.

Bunuel  - can you help with this please.

ENGRTOMBA2018 · Answer

Ideally, this question should have an OA of D. Statement 1 is sufficient because it is directly telling you that  2b-b^2-6ac+3bc>0  (this is the exact question asked!). So the answer based on statement 1 alone is a definite "yes".

For statement 2, you are given that  2ab-6ac+9c^2-3bc=0 ---> (2a-3c)(b-3c)=0  ---> 2 cases

Case 1: either 2a=3c ---> substituting this in the original expression you get,  2b-b^2-6ac+3bc>0 ---> (2a-b)(b-3c) = (3c-b)(b-3c)=-(b-3c)^2 and this can never be >0 . So you get a "NO" for the question asked.

Case 2: b=3c ---> this makes the original expression  2b-b^2-6ac+3bc>0 ---> (2a-b)(b-3c) = 0  and still you get a "NO" for >0. Thus for both the cases, you are getting the same answer of "no" making this statement sufficient as well.

Thus, D should be the correct answer.

Your reasoning of eliminating statement 1 alone is not correct. If a statement directly provides the 'same' information as the question asked, then it should be sufficient alone.

Hope this helps.

ccooley · Answer

1. Simplify the question: The GMAT will rarely give you a question in the most useful 'format', especially when you're doing a hard problem. Since the statements are both in 'multiplied out' form, simplify the question by getting it into the same form.

Remember to retain the question mark.

Is (2a - b)(b - 3c) > 0?
Is 2ab - 6ac - b^2 + 3bc > 0?

2. Pick a statement to start with. They both look about the same, so I'll start with statement 1. Does statement 1 give me enough information to answer the question? I'm not sure, so I'll simplify it to see what it tells me:

2ab - 6ac > b^2 - 3bc
2ab - 6ac - b^2 + 3bc > 0

The statement tells me the above, so I can answer the question with a "yes". Yes, if we have the info from statement 1, then the quantity in the question is positive. This is a trick you'll see occasionally with tough rephrasing problems: both the question and the statement simplify to essentially the same thing. Since the statement  tells  you information, while the question  asks  you about the exact same information, the statement is sufficient.

3. Move to the next statement. On a first glance, this one is giving me an equation, rather than an inequality. It's interesting, because it contains  some  of the same pieces as the question, but not all of them. That makes me think that I should simplify it by putting everything that looks similar to the question on one side, and everything that doesn't on the other.

2ab + 9c^2 = 3bc + 6ac
2ab - 6ac + 9c^2 = 3bc
2ab - 6ac = 3bc - 9c^2

Now, I have something that appears in the question on the left side of the equation. Since the two sides are equal, I can replace one with the other, and still be asking essentially the same question.

Question: Is (2ab - 6ac) - b^2 + 3bc > 0?
Is (3bc - 9c^2) - b^2 + 3bc > 0? 
Is 6bc - 9c^2 - b^2 > 0?

This kind of looks like a special quadratic. With a little manipulation, I managed to factor it.

Is (3c - b)(b - 3c) > 0?

At this point, I noticed that one of the factors was the 'opposite' of the other. To make everything look even simpler, I replaced "3c-b" with "-(b-3c)", which is mathematically the same. 
 
Is -(b - 3c)(b - 3c) > 0?
Is -(b - 3c)^2 > 0?

Using the rule for negatives and inequalities, I flip the sign and remove the negative:

Is (b - 3c)^2 < 0?

This is still a rephrase of the original question - don't forget. If it has only one answer, then 2 is sufficient. If it has more than one answer, then 2 is insufficient. In this case, it has only one possible answer. A perfect square is never negative. The answer is always "no". 2 is sufficient, and the answer is D.

That makes this a bad/invalid GMAT question, since on the test, the two statements will never contradict each other. That is, if one gives you a definite answer of "yes", then the other will never give you a definite answer of "no". It'll either give you a definite answer of "yes", or it won't give you enough information to come up with a specific answer. That isn't a very useful thing on the test in most situations, although I can think of a few OG problems where you can use that fact to save a couple of seconds.

DigitsnLetters · Answer

The OA is showing B as the correct answer. However, Statement-1 is also sufficient. Statement-1 can be factorized to directly obtain the question stem and hence is sufficient. The discussion and comments above also point out the same. Since, both statements are sufficient to answer the question, the OA should be D and not B.

Moderators, please edit the OA.

Bunuel · Answer

This is a poor quality question (https://gmatclub.com/forum/is-2a-b-b-3c ... l#p1648415). Moving to the archive.

hellosanthosh2k2 · Answer

Hi Bunuel,

Why this is a bad question.?

This is how approached it.

(2a - b) (b - 3c) > 0? or rephrasing a > (b/2) && b > 3c?

Statement 1: re-arranging this, yields to question asked. So suff

Statement 2: 2ab + 9c^2 = 3bc + 6ac

case 1: suppose if b = 3c, substituting in statement 2,
2a(3c) + 9c^2 = 3(3c)c + 6ac => 6ac + 9c ^ 2 = 9c ^ 2 + 6ac => satisfies, 
 in which case given Q: (2a - b) (b - 3c) == 0

case 2: suppose if b = 2c, substituting in statement 2,
2a(2c) + 9c^2 = 3(2c)c + 6ac => 4ac + 9c^2 = 6c^2 + 6ac => 3c^2 = 2ac => c == 0 or a = (3/2c)

if a = (3/2c) and b = 2c, given Q: (2a - b) (b - 3c) => (3c - 2c)(2c - 3c) is 8ac + 9c^2 = 12c^2 + 6ac => 3c^2 = 2ac => c == 0 or a = (3/2c)
same as case 3

So combining all 3 cases, statement2 is suff, as (2a - b) (b - 3c) <= 0 , answering the question as NO

So answer should not be D?. Please correct me if wrong

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Is (2a - b)(b - 3c) > 0?

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