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# If a, b and c are integers such that b > a, is b+c > a

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If a, b and c are integers such that b > a, is b+c > a [#permalink]

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26 May 2009, 02:46
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If a, b and c are integers such that b > a, is b+c > a ?

(1) c > a
(2) abc > 0
[Reveal] Spoiler: OA

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Last edited by Bunuel on 10 Aug 2012, 00:22, edited 1 time in total.
Edited the question.
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Re: If a, b and c are integers such that b > a, is b+c > a [#permalink]

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10 Aug 2012, 01:15
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If a, b and c are integers such that b > a, is b+c > a ?

Question: is $$b+c > a$$ ? --> or: is $$b+c-a > 0$$?

(1) c > a. If $$a=1$$, $$b=2$$ and $$c=3$$, then the answer is clearly YES but if $$a=-3$$, $$b=-2$$ and $$c=-1$$, then the answer is NO. Not sufficient.

(2) abc > 0. Either all three unknowns are positive (answer YES) or two unknowns are negative and the third one is positive. Notice that in the second case one of the unknowns that is negative must be $$a$$ (because if $$a$$ is not negative, then $$b$$ is also not negative so we won't have two negative unknowns). To get a NO answer for the second case consider $$a=-3$$, $$b=1$$ and $$c=-4$$ and . Not sufficient.

(1)+(2) We have that $$b > a$$, $$c > a$$ ($$c-a>0$$) and that either all three unknowns are positive or $$a$$ and any from $$b$$ and $$c$$ is negative. Again for the first case the answer is obviously YES. As for the second case: say $$a$$ and $$c$$ are negative and $$b$$ is positive, then $$b+(c-a)=positive+positive>0$$ (you can apply the same reasoning if $$a$$ and $$b$$ are negative and $$c$$ is positive). So, we have that in both cases we have an YES answer. Sufficient.

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27 May 2009, 11:27
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why cant it be C?

1. b>a and c>a and abc>0:

If a<0, b can be less than or greater than 0.

If a<0, b<0 then c>0 (a=-6, b=-4, c=2) => b+c >a
If a<0, b>0, c<0 (a=-3, b=5, c=-1) => b+c >a

If all are positive, then also b+c >a.

Hence, C.

Are there any assumptions which do not satisfy both conditions simultaneously?

What is the OA?
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26 May 2009, 11:34
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sondenso wrote:
If a, b and c are integers such that b > a, is b+c > a ?
(1) c > a
(2) abc > 0

b>a <==> 0>a-b <==> a-b is negative
Question:(b+c>a)?
Question:(c>a-b)?
Question:(c >= 0)?

(1) Insufficient, as a-b<0 could mean that a<0 or a>0, which means c<0 or c>=0.

(2) We have an even # of negatives, ie 0/2. If we have 0 negatives, then sufficient. Again if we have 2 insufficient.

I can see the pattern, the answer is E.

Here are the Yes/No cases:

Yes: 4 5 7
No: -9 3 -10

Final Answer, $$E$$.
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27 May 2009, 14:20
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statement 1 does work for positives, but not for negatives. a= -2 b= -1 c= -1.

statement 2 doesnt work on its own because c could be way less than a, negating b,
a= -3 b=2 c= -5

however, both together mean that there can only be two negatives, and c must be larger than A.
for any value A, negative or not, two values that are greater than it with one positive are going to together be greater than A.
if A is positive = a=2, c=3, b=3, a=1 b=2 c=2
if A is negative = a= -3, c= -2, b=1, a= -2 b=-1 c=1

There is no solution with (1) and (2) true that doesn't end up with B + C > A

gmatprep is right
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26 May 2009, 07:18
sondenso wrote:
If a, b and c are integers such that b > a, is b+c > a ?
(1) c > a
(2) abc > 0

E

a = -4 or 3
b = -3 or 4

The above satifies b>a, using - or + as examples.

(1) c>a
c = -2 or 4
Insuff
(2) abc>0
This means either 2 of the intergers are negative or all of the intergers are positive.
Insuff
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27 May 2009, 09:39
For me its straight A)
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27 May 2009, 14:22
Has anybody considered taking both the conditions 1 and 2 together? What is the OA for this? I still think its C.
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27 May 2009, 20:18
b>a
stmt 1 :

c>a so b+c>2a
we cannot tell whether b+c >a may or may not. So insufficient.

stmt 2 :
abc>0 and b >a
a = +ve b = +ve c =+ve so definetly b+c > a
a = -ve b = -ve c = +ve so definetly b+c > a. ( since c =+ve and b>a)
a = -ve b = +ve c = -ve ( here it is just the opposite u know b =+ve but do not know c>a).so u cant tell whether b+c > a

Combing : yes u know you have got what u wanted so C.

Thanks gmatprep09 and dk94588.
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27 May 2009, 21:02
I agree C is the answer.
If a,b and c and positive,it is possible.
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27 May 2009, 21:42
tkarthik wrote -
c>a so b+c>2a
we cannot tell whether b+c >a may or may not. So insufficient

If b+c>2a how in the world can it not be > a !!!! I m confused.Can you pls. elaborate or give example numbers.
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27 May 2009, 23:12
Hi mdfrahim

you need to take some negative value.

Say a,b,c are +ve then b+c > a and b+c>2a

say b = -2 c = -3 a =-4 ( c> a and b > a)

b + c = -5 which less than a but will be greater than 2a.

I meant to say this only. Hope i have made it clear.
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28 May 2009, 11:53

If you look at both together,

we have that b>a & c>a.

from (1) we get b+c>2a and we're almost sufficient, but only if 2a>a.
Well if a<0, no, but if a>0, yes.

(2) tells us that 0 or 2 are negative. If all 3 are positive, sufficient.
Now if 2 are negative, then a has to be negative as well. Suppose a were positive-- then b/c would be negative, and b>a & c>a would be false. Hence a has to be positive.

(C)

Very tricky
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Re: Property of Integers, Inequalities [#permalink]

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23 Oct 2009, 00:02
(1) because b and c both can be negative, thus (1) is insufficient

(2) insufficient for example, a = -2, b = 1, c = -4

Both are sufficient because either b or c is always greater than a.
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Re: If a, b and c are integers such that b > a, is b+c > a [#permalink]

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09 Aug 2012, 23:40
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Re: If a, b and c are integers such that b > a, is b+c > a [#permalink]

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05 Jul 2013, 02:44
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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If a, b and c are integers such that b > a, is b+c > a [#permalink]

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21 Oct 2014, 22:12
sondenso wrote:
If a, b and c are integers such that b > a, is b+c > a ?

(1) c > a
(2) abc > 0

The question stem says that b > a. There are following possibilities for this
a. Both Positive Lets say $$b = 5 and a = 2$$

b. Both Negative, lets say $$b = -3 and a = -5$$

c. One positive one negative, lets say $$b = 3 and a = -1$$

d. Either of the two zero

1. c > a

C can be positive or negative and can fit in any of the above mentioned scenarios.

Insufficient

2. abc > 0

This means either all are positive or any two are negative. Since b > a, there are numerous possibilities for c.

So, Insufficient.

c > a and abc > 0

This would mean that either all are positive - b + c > a

or two negatives and one one positive - a has to be negative, either of b and c has to be negative and the other one positive. This would also mean that c + b > a

Ans - C
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Re: If a, b and c are integers such that b > a, is b+c > a [#permalink]

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07 Mar 2016, 14:51
Bunuel wrote:
If a, b and c are integers such that b > a, is b+c > a ?

Question: is $$b+c > a$$ ? --> or: is $$b+c-a > 0$$?

(1) c > a. If $$a=1$$, $$b=2$$ and $$c=3$$, then the answer is clearly YES but if $$a=-3$$, $$b=-2$$ and $$c=-1$$, then the answer is NO. Not sufficient.

(2) abc > 0. Either all three unknowns are positive (answer YES) or two unknowns are negative and the third one is positive. Notice that in the second case one of the unknowns that is negative must be $$a$$ (because if $$a$$ is not negative, then $$b$$ is also not negative so we won't have two negative unknowns). To get a NO answer for the second case consider $$a=-3$$, $$b=1$$ and $$c=-4$$ and . Not sufficient.

(1)+(2) We have that $$b > a$$, $$c > a$$ ($$c-a>0$$) and that either all three unknowns are positive or $$a$$ and any from $$b$$ and $$c$$ is negative. Again for the first case the answer is obviously YES. As for the second case: say $$a$$ and $$c$$ are negative and $$b$$ is positive, then $$b+(c-a)=positive+positive>0$$ (you can apply the same reasoning if $$a$$ and $$b$$ are negative and $$c$$ is positive). So, we have that in both cases we have an YES answer. Sufficient.

Bunuel could you please explain how you can solve the problem with the approach above in 2 min?

It took me almost 3,5...

Thanks!
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If a, b and c are integers such that b > a, is b+c > a [#permalink]

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16 Mar 2016, 22:41
If a, b and c are integers such that b > a, is b+c > a ?

(1) c > a
(2) abc > 0
Either two are negative and one positive or all are positive.

Individually both the statements are not sufficient.
Combining 1 and 2 Sufficient.

Hence C.
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Re: If a, b and c are integers such that b > a, is b+c > a [#permalink]

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18 Mar 2016, 08:08
Bunuel

In this problem can we subtract the following inequalities?

b>a
b+c>a

and arrive at 'is c>0??'
Re: If a, b and c are integers such that b > a, is b+c > a   [#permalink] 18 Mar 2016, 08:08

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