For real numbers, A, B, C, and D on the number line

Question

https://gmatclub.com/forum/download/file.php?id=21935 For real numbers, A, B, C, and D on the number line above, is |A| + |B| < |C| + |D| ? (1) |B| < |C| (2) The product of any two of the numbers A, B, C and D is between 0 and 1. download.jpg

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=21935

For real numbers, A, B, C, and D on the number line above, is |A| + |B| < |C| + |D| ?

Notice that from the diagram we have that A<B<C<D.

(1) |B| < |C|. This one is clearly insufficient. Consider: A=1, B=2, C=3, and D=4 for an YES answer and A=-10, B=2, C=3, and D=4 for a NO answer. Not sufficient.

(2) The product of any two of the numbers A, B, C and D is between 0 and 1. So, we have that the product of ANY two of the numbers is positive. This implies either that all the numbers are negative or that all the numbers are positive. In the first case the answer is NO and in the second case the answer is YES. Not sufficient.

(1)+(2) If all the numbers were negative, then because we also have that B < C, then we would have that |B| > |C|, which contradicts the first statement, thus we have the second case: all the numbers are positive. Therefore since A<B<C<D, then |A| + |B| < |C| + |D|. Sufficient.

Answer: C.

Hope it's clear.

pahujanavdeep · Answer

Hi Bunuel,

I have been reading your posts quite regularly, and they are very helpful. I am really grateful to you for providing assistance to people like us.

As far as this question is concerned i have a doubt on s(2). As you have mentioned above "that all the numbers are negative or that all the numbers are positive." 
I have been working on one or two examples and came across this one
Suppose B and C are -1/2 and -1/3. Their product is 1/6, which is between 0 and 1. Now the possible values can be 
A<B<C<D :- -1 < -1/2 < -1/3 <1 . In this case |A| + |B| IS NOT LESS THAN |C| + |D| insufficient

Suppose A and B are -1/2 and -1/3. Again their product is 1/6, which is between 0 and 1. Now the possible values can be 
A<B<C<D :- -1/2 < -1/3 < 1 < 2. In this case |A| + |B| < |C| + |D| sufficient.

So, how can we say "that all the numbers are negative or that all the numbers are positive." ? and it doesn't say that The product of   every   two of the numbers A, B, C and D is between 0 and 1.

Thanks

EMPOWERgmatRichC · Answer

Hi pahujanavdeep,

Unfortunately, your example does NOT fit the "restriction" that is described in Fact 2:

Fact 2: The product of any two of the numbers A, B, C and D is between 0 and 1.

This tells us that if we take ANY two of the 4 values and multiply them....then the product will be a POSITIVE fraction.

Your example:
A = -1
B = -1/2
C = -1/3
D = 1

If one of the two numbers chosen is D, then the product will be a NEGATIVE fraction. This is NOT a viable option given what Fact 2 tells us. The ONLY ways for the product of ANY two of those values to be a POSITIVE fraction is if ALL 4 values are negative OR ALL 4 are positive.

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

pahujanavdeep · Answer

Thank you Rich

I got confused with the wording. But it make sense now. Appreciate it :-D

Regards Navdeep

nycgirl212 · Answer

where does the diagram tell us that?

EMPOWERgmatRichC · Answer

Hi nycgirl212,

The prompt specifically refers to the number line that's presented. On a number line, numbers increase from left to right. Thus, based on the given number line, A < B < C < D.

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

nycgirl212 · Answer

so even though it doesn't explicitly state that the numbers increase from left to right, we are automatically supposed to assume that they do when we see a number line?

Bunuel · Answer

Please check the highlighted part below.

OFFICIAL GUIDE :

Problem Solving 
 Figures:  All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight.  The positions of points, angles, regions, etc., exist in the order shown , and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency: 
 Figures: 
• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.
• All figures lie in a plane unless otherwise indicated.

Foreheadson · Answer

Hi Bunuel,

I am thinking of thr following numbers. lets go straight to test choice C.

so let the numbers be: A=-100000, B=0.00000000002, C=0.000000000003, D=0.000000000004. 
statement A and B are satisfied here. B<C and product of any two numbers is betweem 0 and one. but A+B is greater than B+C bcs of module symbol.

another case:
A=0.0000000001, B=0.00000000002, C=0.0000000003, D=10000000000

In this case all conditions are also met, but A+B is less than C+D.

Am I missing anything here?

Posted from my mobile device

Bunuel · Answer

I think your reading of the second statement is not correct.

" The product of any two of the numbers A, B, C and D is between 0 and 1 " means that the product of ANY two of the four numbers is between 0 and 1. So,

0 < AB < 1
0 < AC < 1
0 < AD < 1
0 < BC < 1 
0 < BD < 1
0 < CD < 1

Now, if A = -100000, B=0.00000000002, C=0.000000000003, D=0.000000000004, then AB, AC and AD are negative not between 0 and 1.

Foreheadson · Answer

Oh, I see the mistake now. I was confused with |x| symbols and assumed the same here for no reason.

Thanks a lot for a quick and comprehensive reply. I really appreciate it.

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

For real numbers, A, B, C, and D on the number line

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

For real numbers, A, B, C, and D on the number line

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards