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If a and b are real numbers, is a < b?

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Joined: 20 Nov 2017
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If a and b are real numbers, is a < b?  [#permalink]

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17 Jul 2018, 19:35
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29% (02:27) correct 71% (01:48) wrong based on 41 sessions

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If a and b are real numbers, is a < b?

(1) $$a^b < b^a$$

(2) $$\frac{a}{b} > 1$$
Math Expert
Joined: 02 Aug 2009
Posts: 6554
If a and b are real numbers, is a < b?  [#permalink]

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17 Jul 2018, 21:08
1
If a and b are real numbers, is a < b?

(1) $$a^b < b^a$$..
a) let a be 4 and b be 3......a>b
$$a^b<b^a........4^3<3^4........64<81$$
b) let a be -2 and b be -1......a<b
$$(-2)^{-1}<(-1)^{-2}...........-1/2<1$$
So insufficient

(2) $$\frac{a}{b} > 1$$
This means |a|>|b|...
a=4, and b=3..........4/3>1 and a>b
a=-4, and b=-1..........(-4)/(-1)=4>1 and a<b
So insufficient

Combined
Both cases remain..
a=4, and b=3......a>b
a=-2, and b =-1.....a<b
Insufficient

E
_________________

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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Joined: 11 Mar 2018
Posts: 65
Re: If a and b are real numbers, is a < b?  [#permalink]

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17 Jul 2018, 21:45
Karthik200 wrote:
If a and b are real numbers, is a < b?

(1) $$a^b < b^a$$

(2) $$\frac{a}{b} > 1$$

(1) $$a^b < b^a$$
$$a=2$$, $$b=3$$ $$a<b$$
$$8<9$$ True

$$a=-3$$, $$b=-2$$ $$a<b$$
$$9<\frac{-1}{8}$$ False

Insufficient.

(2) $$\frac{a}{b} > 1$$

$$\frac{a}{b} > 1$$
$$a>b$$
However
$$\frac{-a}{-b} >1$$
$$-a<-b$$
Eg: $$a=+-5$$, $$b=+-4$$

Insufficient.

(1) and (2) Insufficient.

Hence, E
_________________

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Intern
Joined: 20 Nov 2017
Posts: 26
Location: India
GRE 1: Q158 V150
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WE: Consulting (Consumer Electronics)
If a and b are real numbers, is a < b?  [#permalink]

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18 Jul 2018, 02:17
chetan2u wrote:
If a and b are real numbers, is a < b?

(1) $$a^b < b^a$$..
a) let a be 4 and b be 3......a>b
$$a^b<b^a........4^3<3^4........64<81$$
b) let a be -2 and b be -1......a<b
$$(-2)^{-1}<(-1)^{-2}...........-1/2<1$$
So insufficient

(2) $$\frac{a}{b} > 1$$
This means |a|>|b|...
a=4, and b=3..........4/3>1 and a>b
a=-4, and b=-1..........(-4)/(-1)=4>1 and a<b
So insufficient

Combined
Both cases remain..
a=4, and b=3......a>b
a=-2, and b =-1.....a<b
Insufficient

E

Hi chetan2u,

In these kinds of problems where plugging in of numbers required, I am finding it difficult to try out all the possibilities to disprove the given statements under two minutes. Is there any strategy to plug in the smart numbers that work best, especially for number property DS questions? I hope it will benefit many others as well.

Thanks & Regards,
Karthik
Math Expert
Joined: 02 Aug 2009
Posts: 6554
Re: If a and b are real numbers, is a < b?  [#permalink]

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18 Jul 2018, 02:26
Karthik200 wrote:
chetan2u wrote:
If a and b are real numbers, is a < b?

(1) $$a^b < b^a$$..
a) let a be 4 and b be 3......a>b
$$a^b<b^a........4^3<3^4........64<81$$
b) let a be -2 and b be -1......a<b
$$(-2)^{-1}<(-1)^{-2}...........-1/2<1$$
So insufficient

(2) $$\frac{a}{b} > 1$$
This means |a|>|b|...
a=4, and b=3..........4/3>1 and a>b
a=-4, and b=-1..........(-4)/(-1)=4>1 and a<b
So insufficient

Combined
Both cases remain..
a=4, and b=3......a>b
a=-2, and b =-1.....a<b
Insufficient

E

Hi chetan2u,

In these kinds of problems where plugging in of numbers required, I am finding it difficult to try out all the possibilities to disprove the given statements under two minutes. Is there any strategy to plug in the smart numbers that work best, especially for number property DS questions? I hope it will benefit many others as well.

Thanks & Regards,
Karthik

If I were to do this problem, I'll pick up the statement II as it is more friendly...
a/b>1 means a and b are of same sign and numeric value of a is more than that of b.. hence |a|>|b|
And |a|>|b| means a>b if both positive and a<b if both negative

So check for the same cases in statement I too..
Both negative and a<b.....-4<-3....(-4)^{-3}<(-3)^{-4}.....-1/64<1/81
Both positive and a>b......4>3.....4^3<3^4.........64<81
So insufficient
_________________

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

GMAT online Tutor

Intern
Joined: 20 Nov 2017
Posts: 26
Location: India
GRE 1: Q158 V150
GPA: 3.9
WE: Consulting (Consumer Electronics)
Re: If a and b are real numbers, is a < b?  [#permalink]

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18 Jul 2018, 02:42
chetan2u wrote:
Karthik200 wrote:
chetan2u wrote:
If a and b are real numbers, is a < b?

(1) $$a^b < b^a$$..
a) let a be 4 and b be 3......a>b
$$a^b<b^a........4^3<3^4........64<81$$
b) let a be -2 and b be -1......a<b
$$(-2)^{-1}<(-1)^{-2}...........-1/2<1$$
So insufficient

(2) $$\frac{a}{b} > 1$$
This means |a|>|b|...
a=4, and b=3..........4/3>1 and a>b
a=-4, and b=-1..........(-4)/(-1)=4>1 and a<b
So insufficient

Combined
Both cases remain..
a=4, and b=3......a>b
a=-2, and b =-1.....a<b
Insufficient

E

Hi chetan2u,

In these kinds of problems where plugging in of numbers required, I am finding it difficult to try out all the possibilities to disprove the given statements under two minutes. Is there any strategy to plug in the smart numbers that work best, especially for number property DS questions? I hope it will benefit many others as well.

Thanks & Regards,
Karthik

If I were to do this problem, I'll pick up the statement II as it is more friendly...
a/b>1 means a and b are of same sign and numeric value of a is more than that of b.. hence |a|>|b|
And |a|>|b| means a>b if both positive and a<b if both negative

So check for the same cases in statement I too..
Both negative and a<b.....-4<-3....(-4)^{-3}<(-3)^{-4}.....-1/64<1/81
Both positive and a>b......4>3.....4^3<3^4.........64<81
So insufficient

Thanks chetan2u. I will try to apply this from my next DS problem onwards.
Re: If a and b are real numbers, is a < b? &nbs [#permalink] 18 Jul 2018, 02:42
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