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# Is |a-c| + |a| = |c|?

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Is |a-c| + |a| = |c|?  [#permalink]

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14 Feb 2019, 03:24
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Question Stats:

43% (02:35) correct 57% (02:13) wrong based on 297 sessions

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Is |a-c| + |a| = |c|?

(1) ab > bc
(2) ab < 0

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Is |a-c| + |a| = |c|?  [#permalink]

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Updated on: 12 Jun 2019, 15:17
8
|a-c| + |a| = |c|
|a-c| = |c| - |a|
to satisfy "|a-c| = |c| - |a|", there are 3 options:
1. a = c
2. a = 0
3. |c| > |a| (c>a in magnitude) but have similar signs.

statement 1: ab > bc
eliminates option 1 and 2, but leaves us with option 3 which can't be known as the sign of b is unknown.

statement 2: ab < 0
gives no information except that a and b $$\neq{0}$$ and that a and b has different signs.

by combining statement 1 and 2: (see the testing table)
0>ab>bc while a and b have different signs, so a and c must have similar signs
so if b>0, then both a and c are negative and c>a in magnitude (|c| > |a|)
so if b<0, then both a and c are positive and c>a in magnitude (|c| > |a|)
these criteria satisfies the tested equation

so C
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Originally posted by MahmoudFawzy on 14 Feb 2019, 05:43.
Last edited by MahmoudFawzy on 12 Jun 2019, 15:17, edited 1 time in total.
##### General Discussion
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Re: Is |a-c| + |a| = |c|?  [#permalink]

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14 Feb 2019, 07:48
Ans (C). Both statements together are sufficient to answer the question.
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Re: Is |a-c| + |a| = |c|?  [#permalink]

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14 Mar 2019, 13:46
3
2
In my opinion,
Such type of question (mentioned in main question) ask u a simple thing I.e. the two variables are in same directions or not.

For example
(I)
{|x|+|y|} is either EQUAL or GREATER than {|x+y|}.
When the two numbers (x & y) are in SAME DIRECTION (both positive or both negatives or both zero), both expressions [(|x|+|y|) & |x+y|] are EQUAL.

When the two numbers (x & y) are in OPPSITE DIRECTION (one positive and another one negative), |x|+|y| is GREATER than |x+y|.

(ii)
{|x|-|y|} is either EQUAL or LESS than {|x-y|}.
When the two numbers (x & y) are in SAME DIRECTION (both positive or both negatives or both zero) & x has greater ABSOLUTE value, both expressions [(|x|-|y|) & |x-y|] are EQUAL.

When the two numbers (x & y) are in OPPSITE DIRECTION (one positive and another one negative), |x|-|y| is LESS than |x-y|.

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Re: Is |a-c| + |a| = |c|?  [#permalink]

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14 Mar 2019, 14:28
2
1
In this question,
|a-c| + |a| = |c|?
or |a-c| = |c| - |a|?
or |c-a| = |c| - |a|?
That is (ii) situation of my previous post.

Thus, the question is asking : IS a & c in same direction & is the absolute value of a is greater than that of C?

statement 1 & 2 don’t guarantee anything about same direction of a&c.
Only combination of of c says the both requirement (same direction & absolute value of c).
So I would go for C.

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Re: Is |a-c| + |a| = |c|?  [#permalink]

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14 Mar 2019, 14:43
3
Number picking (I think better approach).

Statement 1: Insufficient
Case 1: a=1, b=-2 & c=3 (yes to main question)
Case 2: a=-1, b=2, &c=-3 (yes to main question)
Case 3: a=1,b=2 & c=-3 (no to main question)

Statement 2: insufficient. Coz no info about c.

Combined:
Case 3 from statement 1 is out coz statement 2 says ab is negative.
So remaining case 1&2 say YES to main question.

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Re: Is |a-c| + |a| = |c|?  [#permalink]

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14 Mar 2019, 15:02
5
1
Absolute value concept.

|a| or |a-0| is the distance between 0 and a.

|a-c| is same thing as |c-a| which means the distance between a and c.

|c| is the distance between 0 and c.

Thus the question says
Is the sum of (I) DISTANCE between (0&a) & (ii) DISTANCE between (a&c) is EQUAL to DISTANCE between (0 &c)?

This is only possible if
I) a&c are on the same side of zero and
(ii) a is closer to zero and c is distant from zero.
[this is possible in positive and negative sides, as shown in attached photo.]

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Re: Is |a-c| + |a| = |c|?  [#permalink]

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20 Mar 2019, 08:16
is this problem from gmatprep? show me the screen, pls
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Re: Is |a-c| + |a| = |c|?  [#permalink]

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22 Mar 2019, 20:51
1
to have /a-c/+/a/=/c/
we have /a-c/=/c/-/a/
c and a must be the same sign

from both condition
b(a-c)>0
ab<0

case 1
b<0 and a<c
b<0 then a>0, this mean c also >0
this is good
case 2
b>0 and a>c
b>0 the a<0 , the mean c also < 0
this is good

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Re: Is |a-c| + |a| = |c|?  [#permalink]

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12 Jun 2019, 10:55
1
Mahmoudfawzy83 wrote:
|a-c| + |a| = |c|
|a-c| = |c| - |a|
to satisfy "|a-c| = |c| - |a|", there are 3 options:
1. a = c
2. a = 0
3. |c| > |a| (c>a in magnitude) but have similar signs.

statement 1: ab > bc
eliminates option 1 and 2, but leaves us with option 3 which can't be known as the sign of b is unknown.

statement 2: ab < 0
gives no information except that a and b $$\neq{0}$$ and that a and b has different signs.

by combining statement 1 and 2: (see the testing table)
0>ab>bc while a and b have different signs, so a and c must have similar signs
so if b>0, then both a and b are negative and c>a in magnitude (|c| > |a|)
so if b<0, then both a and b are positive and c>a in magnitude (|c| > |a|)
these criteria satisfies the tested equation

so C

Small correction here.
so if b>0, then both a and c are negative and c>a in magnitude (|c| > |a|)
so if b<0, then both a and c are positive and c>a in magnitude (|c| > |a|)
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Re: Is |a-c| + |a| = |c|?  [#permalink]

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15 Jun 2019, 09:27
i do not understand this question at all.. can we get some expert reply on this please?
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Re: Is |a-c| + |a| = |c|?  [#permalink]

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26 Jun 2019, 17:02
Bunuel

what is the correct answer to this question?

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Re: Is |a-c| + |a| = |c|?  [#permalink]

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26 Jun 2019, 22:23
Giro2345 wrote:
Bunuel

what is the correct answer to this question?

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Re: Is |a-c| + |a| = |c|?  [#permalink]

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02 Jul 2019, 01:25
This question follows the rules of the general form:
|x-y|=|x| - |y|
if:
a. y = 0; or
b. x and y have the same sign, and x>y

So reorganising the stem.
is |a-c| + |a| =|c|

is |a-c| =|c| - |a|
reshuffle by multiply by -1

is -|a-c| = |a| - |c|

For this to be true c must equal 0 or a and c must have the same sign and a must be greater than c
(1) ab > bc
++ >+(-) a and c could have different signs; or--> No
++>++ a and c could have the same signs; or -->Yes
++>+(0) c could be equal to zero -->Yes

Insufficient

(2) ab<0
could be +- or -+
either way we knowing about c
Insufficient

Combined (1+2)
ab<0
therefore
ab > bc
(-) > bc

Possible cases:
ab > bc
(-)+> +(-) a and c are both negative, but a must be greater than c to satisfy the constraint
+(-)>(-)+ a and c are both positive, but a must be greater than c to satisfy the constraint

In either case, a must be greater than c and must have the same sign as c, therefore combined statements are sufficient.

We can test numbers to validate the theory.
-|a-c| = |a| - |c|

condition 1: a and c are both negative and a > c
-|-2-(-3)| = |-2|-|-3|
-|1| = 2-3
-1 = -1

condition 2: a and c are both positive and a > c
-|3-2|=|3|-|2|
-|1| = |3|-|2|
-1 = 1 is not possible, therefore a and c must both be negative

Bunuel, sorry to pester but is my working out correct?
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Is |a-c| + |a| = |c|?  [#permalink]

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24 Sep 2019, 09:18
RashedVai wrote:
Is |a-c| + |a| = |c|?

(1) ab > bc
(2) ab < 0

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Is |a-c| + |a| = |c|?
Q. |c-a| = |c|-|a|

(1) ab > bc
ab - bc > 0
b (a-c) > 0
If b>0; a-c>0; a>c
If b<0; a-c<0; a<c
NOT SUFFICIENT

(2) ab < 0
If b>0; a<0
If b<0; a>0
No information provided for c
NOT SUFFICIENT

(1) + (2)
(1) ab > bc
ab - bc > 0
b (a-c) > 0
If b>0; a-c>0; a>c
If b<0; a-c<0; a<c
(2) ab < 0
If b>0; a<0
If b<0; a>0
If b>0; a-c>0; a>c; c<a<0; |c-a| = |c| - |a|
c------a-----------0
If b<0; a-c<0; a<c; c>a>0; |c-a| = |c| - |a|
0-------a------------c
SUFFICIENT

IMO C
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Is |a-c| + |a| = |c|?   [#permalink] 24 Sep 2019, 09:18
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