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# If negative integer a is multiplied by b and the result is greater tha

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Math Expert
Joined: 02 Sep 2009
Posts: 49320
If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 02:12
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35% (medium)

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70% (01:21) correct 30% (01:29) wrong based on 182 sessions

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If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

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Joined: 02 Aug 2009
Posts: 6802
If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 03:14
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Bunuel wrote:
If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

If a*b>0, where a<0, so b is also <0

If ab<|a|, square both sides
$$a^2b^2<a^2.......a^2b^2-a^2<0....a^2(b^2-1)<0$$
Since a^2>0, so $$b^2-1<0........b^2<1$$
Range of b is -1<b<1....
But b is negative, so -1<b<0

Other way
Substitute a as -2
So -2b<|-2|.............-2b<2..........divide both sides by -2...
$$\frac{-2b}{-2}>\frac{2}{-2}...............b>-1$$
Since b<0....-1<b<0

C
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 08:05
1
Bunuel wrote:
If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

I think conceptual understanding is a must to work out this question. Few points can be made based on the information given:

1. a is negative . b must also be negative.
2. Absolute value of a is greater than the product of a and b.
3. we know that negative times negative yields positive result.

Let assume the value of a and b.

a = -4 and b = -2

so, ab = 8 but it is stated in the question that |a| > ab . So, b can't be a integer rather it's a fraction.

a = -4
b = -$$\frac{1}{2}$$

a*b = (-4)*(-1/2)
= 2
and |a| > 2.

Thus , b is a negative fraction.

Option C is the best answer.
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Joined: 08 Oct 2017
Posts: 24
Re: If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 09:05
Keep in mind - a -ve and |a|>ab>0
implies -- b has to be -ve
For a*b to be less than |a| - b has to be -ve fraction < 1
Senior SC Moderator
Joined: 22 May 2016
Posts: 1979
If negative integer a is multiplied by b and the result is greater tha  [#permalink]

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29 Jun 2018, 10:04
1
Bunuel wrote:
If negative integer a is multiplied by b and the result is greater than 0 but less than |a|, then which of the following must be true of b?

A. $$b > 1$$

B. $$0 < b < 1$$

C. $$-1 < b < 0$$

D. $$b < a$$

E. $$|b| < a$$

Use a negative integer for $$a$$
in order to simplify the arithmetic

Let $$a=-3$$
$$|-3|= 3$$

(1) the result of $$a*b$$ is greater than 0
$$a*b>0$$
$$(-3*b)>0$$
= $$0<(-3*b)$$

(2) the result of $$a*b$$ is less than $$|a|$$
$$a*b<|a|$$
$$(-3*b)<3$$

(3) Combine:
$$0<(-3*b)<3$$
Divide all terms by -3, flip the signs
$$0>b>-1$$
= $$-1<b<0$$

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If negative integer a is multiplied by b and the result is greater tha &nbs [#permalink] 29 Jun 2018, 10:04
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