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If a > b and if c > d, then which of the following must be true?

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If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post Updated on: 20 Jan 2019, 21:24
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If a > b and if c > d, then which of the following must be true?

A. a-b > c+d
B. a-c > b-d
C. c+d < a-b
D. b+d < a+c
E. a-c < b+d

here the answer is E but i wonder if is incorrect because we add unequal quantities but should not be a + c > b+d ???

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Originally posted by carcass on 19 Mar 2012, 05:39.
Last edited by Bunuel on 20 Jan 2019, 21:24, edited 2 times in total.
Edited the question and the OA
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 19 Mar 2012, 05:49
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4
carcass wrote:
If a > b and if c > d , then wich of the following must be true ??

1) a-b > c+d
2) a-c > b-d
3) c+d < a-b
4) b+d < a+c
5) a-c < b+d

here the answer is E but i wonder if is incorrect because we add unequal quantities but should not be a + c > b+d ???


You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

BACK TO THE ORIGINAL QUESTION:
If a > b and if c > d , then which of the following must be true?
A. a-b > c+d
B. a-c > b-d
C. c+d < a-b
D. b+d < a+c
E. a-c < b+d

Since the signs of the inequalities are in the same direction we can add them: \(a+c>b+d\), which is the same as the option D: \(b+d < a+c\). So, the correct answer is D, not E.

Answer: D.

Hope its' clear.
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 19 Mar 2012, 05:58
I think answer E is wrong here.

Is a > b and if c > d , then wich of the following must be true ??

1) a-b > c+d
2) a-c > b-d
3) c+d < a-b
4) b+d < a+c
5) a-c < b+d

so i picked 100 > 1 and 3 > 2 to test E

100-3<1+2 (FALSE)

I think D is answer
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 19 Mar 2012, 07:02
Bunuel wrote:
carcass wrote:
If a > b and if c > d , then wich of the following must be true ??

1) a-b > c+d
2) a-c > b-d
3) c+d < a-b
4) b+d < a+c
5) a-c < b+d

here the answer is E but i wonder if is incorrect because we add unequal quantities but should not be a + c > b+d ???


You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

BACK TO THE ORIGINAL QUESTION:
If a > b and if c > d , then which of the following must be true?
A. a-b > c+d
B. a-c > b-d
C. c+d < a-b
D. b+d < a+c
E. a-c < b+d

Since the signs of the inequalities are in the same direction we can add them: \(a+c>b+d\), which is the same as the option D: \(b+d < a+c\). So, the correct answer is D, not E.

Answer: D.

Hope its' clear.



Infact Bunuel as you can see from my question above I pick the right answer D but something make me suspicious and I thought to ask here to you.


This shows how is important to rely solely and exclusively on good gmat material.

Thanks Mod.
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 31 Oct 2012, 11:47
1
Algebraic Approach
a>b ==> a-b>0 (1)
c>d ==> c-d>0 (2)
(1) + (2) ==> a-b + c-d > 0 (a+c) -(b+d)>0 ==> (b+d) < (a+c)
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 31 Jul 2015, 10:54
If a>b and c<d, which of the following MUST be true?
a)a−d>c−b
b)a+d>b
c)b+c>a−d
d)b−d<a−c
e)a2+d2>b2+c2
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 31 Jul 2015, 16:01
Hi All,

To address the original prompt....In questions that include inequalities in this fashion, it's important to note that we know NOTHING about how A and B relate to C and D. For example, it could be that A is the biggest number, but it might be second biggest (behind C) or third biggest (behind C and D). As such, what MUST be true will likely involve combining the bigger variables from each pair and the smaller variables in each pair.

Since A > B and C > D, adding A and C will lead to a sum that MUST be greater than the sum of B and D.

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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 06 Jan 2019, 23:49
Bunuel wrote:
carcass wrote:
If a > b and if c > d , then wich of the following must be true ??

1) a-b > c+d
2) a-c > b-d
3) c+d < a-b
4) b+d < a+c
5) a-c < b+d

here the answer is E but i wonder if is incorrect because we add unequal quantities but should not be a + c > b+d ???


You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

BACK TO THE ORIGINAL QUESTION:
If a > b and if c > d , then which of the following must be true?
A. a-b > c+d
B. a-c > b-d
C. c+d < a-b
D. b+d < a+c
E. a-c < b+d

Since the signs of the inequalities are in the same direction we can add them: \(a+c>b+d\), which is the same as the option D: \(b+d < a+c\). So, the correct answer is D, not E.

Answer: D.

Hope its' clear.


I was solving this question by taking some values: a=2; b=1; c=1; d=100
Then b+d=101 & a+c=3 ; Clearly b+d < a+c doesn't hold in this case. Please share your thoughts. :)
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 07 Jan 2019, 02:10
AVNISH2123 wrote:
Bunuel wrote:
carcass wrote:
If a > b and if c > d , then wich of the following must be true ??

1) a-b > c+d
2) a-c > b-d
3) c+d < a-b
4) b+d < a+c
5) a-c < b+d

here the answer is E but i wonder if is incorrect because we add unequal quantities but should not be a + c > b+d ???


You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

BACK TO THE ORIGINAL QUESTION:
If a > b and if c > d , then which of the following must be true?
A. a-b > c+d
B. a-c > b-d
C. c+d < a-b
D. b+d < a+c
E. a-c < b+d

Since the signs of the inequalities are in the same direction we can add them: \(a+c>b+d\), which is the same as the option D: \(b+d < a+c\). So, the correct answer is D, not E.

Answer: D.

Hope its' clear.


I was solving this question by taking some values: a=2; b=1; c=1; d=100
Then b+d=101 & a+c=3 ; Clearly b+d < a+c doesn't hold in this case. Please share your thoughts. :)


The question says that: c > d.
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Re: If a > b and if c > d, then which of the following must be true?  [#permalink]

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New post 07 Jan 2019, 02:24
I was solving this question by taking some values: a=2; b=1; c=1; d=100
Then b+d=101 & a+c=3 ; Clearly b+d < a+c doesn't hold in this case. Please share your thoughts. :)[/quote]

The question says that: c > d.[/quote]
Apologies!
While assigning the values, I was looking at the below question (from Veritas prep) which is very similar to the above stated one:

If a>b and c<d, which of the following MUST be true?

(A) a-d>c-b
(B) a+d>b
(C) b+c>a-d
(D) b-d<a-c
(E) a^2 + d^2 > b^2 + c^2

Veritas Prep Ans- D

And the assigned values holds true for (D).

Thanks Bunuel for your response! :)
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Re: If a > b and if c > d, then which of the following must be true?   [#permalink] 07 Jan 2019, 02:24
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