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# M14-33

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Math Expert
Joined: 02 Sep 2009
Posts: 53709

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16 Sep 2014, 00:54
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Difficulty:

55% (hard)

Question Stats:

58% (01:53) correct 42% (01:43) wrong based on 52 sessions

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If $$ab \ne 0$$ and $$|a| \lt |b|$$, which of the following must be negative?

A. $$\frac{a}{b} - \frac{b}{a}$$
B. $$\frac{a - b}{a + b}$$
C. $$a^b - b^a$$
D. $$a \frac{b}{a - b}$$
E. $$\frac{b - a}{b}$$

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Math Expert
Joined: 02 Sep 2009
Posts: 53709

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16 Sep 2014, 00:54
2
Official Solution:

If $$ab \ne 0$$ and $$|a| \lt |b|$$, which of the following must be negative?

A. $$\frac{a}{b} - \frac{b}{a}$$
B. $$\frac{a - b}{a + b}$$
C. $$a^b - b^a$$
D. $$a \frac{b}{a - b}$$
E. $$\frac{b - a}{b}$$

$$|a| \lt |b|$$ means that $$a^2 \lt b^2$$, which can be written as $$a^2 - b^2 \lt 0$$ or as $$(a - b)(a + b) \lt 0$$. So, $$a - b$$ and $$a + b$$ have the opposite signs, which means that $$\frac{a - b}{a + b}$$ will always be negative.

To discard other options consider $$a=-1$$ and $$b=2$$.

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Joined: 26 Jun 2012
Posts: 4
Concentration: Strategy, General Management
GMAT 1: 730 Q49 V41
WE: Engineering (Computer Hardware)

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10 Nov 2014, 04:58
1
Bunnel,

option e: (b-a)/b = 1 - (a/b)
Case i: if either a or b is negative , a/b is positive , so the 1 - (a/b) is positive
case ii: if both are of the same sign, since |a|<|b|, 0< a/b < 1 => 1 - (a/b) is always positive
Hence option e.

What am I missing?

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 53709

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10 Nov 2014, 05:42
private wrote:
Bunnel,

option e: (b-a)/b = 1 - (a/b)
Case i: if either a or b is negative , a/b is positive , so the 1 - (a/b) is positive
case ii: if both are of the same sign, since |a|<|b|, 0< a/b < 1 => 1 - (a/b) is always positive
Hence option e.

What am I missing?

Thanks

The question asks which of the following must be negative, not positive.
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Intern
Joined: 26 Jun 2012
Posts: 4
Concentration: Strategy, General Management
GMAT 1: 730 Q49 V41
WE: Engineering (Computer Hardware)

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10 Nov 2014, 05:49
2

Thanks
Manager
Joined: 01 Jun 2013
Posts: 104
GMAT 1: 650 Q50 V27

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10 Nov 2015, 04:15
|a|<|b| is possible in below scenarios; (I don't like using smart numbers, but in that case I could not avoid using them)

a=2 b=3
a=-2 b=-3
a=2 b=-3
a=-2 b=3

If you try each of them individually for each choice, you will get negative result for the equation in choice B.
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Senior Manager
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Location: India
Concentration: Operations, Finance
GMAT 1: 670 Q48 V34
GPA: 3.8
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20 Aug 2016, 04:53
I think this is a high-quality question and I agree with explanation.
Intern
Joined: 22 Nov 2018
Posts: 22

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21 Feb 2019, 08:12
I think this is a high-quality question and I agree with explanation.
Re M14-33   [#permalink] 21 Feb 2019, 08:12
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# M14-33

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