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# In which one of the following choices must p be greater than

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In which one of the following choices must p be greater than [#permalink]

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15 Feb 2012, 11:19
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In which one of the following choices must p be greater than q?

(A) 0.9^p = 0.9^q
(B) 0.9^p = 0.9^2q
(C) 0.9^p > 0.9^q
(D) 9^p < 9^q
(E) 9^p > 9^q

I need an explanation on this tricky problem.

A) p=q so is not the answer

B) p=2q is not sufficient to answer to our question, because p and q can be negatice or positive; hence q > p or p > q.

C) p>q but in this case we can have for instance 2 > 3 that is false. But if we have for example -1/2 > -1/3 or -2 > -3 are true. So is unclear if p > q.

For D ) and E ) I have applied the similar reasoning.

Thanks GMAT club.
[Reveal] Spoiler: OA

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Re: In this exponent problem P > Q ???? [#permalink]

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15 Feb 2012, 11:32
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In which one of the following choices must p be greater than q?

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

A. 0.9^p = 0.9^q --> $$p=q$$. Discard;
B. 0.9^p = 0.9^2q --> $$p=2q$$ --> if $$p=-2$$ and $$q=-1$$ then $$p<q$$. Discard;
C. 0.9^p > 0.9^q --> if $$p=1$$ and $$q=2$$ then then $$p<q$$. Discard;
D. 9^p < 9^q --> $$9^{p-q}<1$$ --> $$p-q<0$$ --> $$p<q$$. Discard;
E. 9^p > 9^q --> $$9^{p-q}>1$$ --> $$p-q>0$$ --> $$p>q.$$ Correct.

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Re: In this exponent problem P > Q ???? [#permalink]

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15 Feb 2012, 12:04
Bunuel wrote:
In which one of the following choices must p be greater than q?

Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

A. 0.9^p = 0.9^q --> $$p=q$$. Discard;
B. 0.9^p = 0.9^2q --> $$p=2q$$ --> if $$p=-2$$ and $$q=-1$$ then $$p<q$$. Discard;
C. 0.9^p > 0.9^q --> if $$p=1$$ and $$q=2$$ then then $$p<q$$. Discard;
D. 9^p < 9^q --> $$9^{p-q}<1$$ --> $$p-q<0$$ --> $$p<q$$. Discard;
E. 9^p > 9^q --> $$9^{p-q}>1$$ --> $$p-q>0$$ --> $$p>q.$$ Correct.

Bunuel thanks for the explanation very clear. This was a tricky problem and sometimes I have the propensity to over calculate: I'm wrong ??' do you have a suggestion ?? of course I guess to have idea in which direction I have to go but is not enough to beat problems at high level.....
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Re: In which one of the following choices must p be greater than [#permalink]

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16 Feb 2012, 04:07
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carcass wrote:
In which one of the following choices must p be greater than q?

(A) 0.9^p = 0.9^q
(B) 0.9^p = 0.9^2q
(C) 0.9^p > 0.9^q
(D) 9^p < 9^q
(E) 9^p > 9^q

I need an explanation on this tricky problem.

A) p=q so is not the answer

B) p=2q is not sufficient to answer to our question, because p and q can be negatice or positive; hence q > p or p > q.

C) p>q but in this case we can have for instance 2 > 3 that is false. But if we have for example -1/2 > -1/3 or -2 > -3 are true. So is unclear if p > q.

For D ) and E ) I have applied the similar reasoning.

Thanks GMAT club.

This question is tricky. You might end up losing lots of valuable time trying to plug in numbers in each option and getting lost mid-way. The trick is to use exponents theory. I like to use number plugging only where it is apparent that it is useful (e.g. in options A and B above to rule them out in 10 sec) or where I can't think of anything else. The specific theory which helps in dealing with this and similar questions:

If the base is positive, no matter what power you give to it, it will not become negative.
e.g. 5^n must be positive no matter what the value of n is.
The reason is that 5^n is just 5 multiplied by itself n number of times, whatever n is. So this number can never be negative.

Also,
If n is positive, 5^n > 1
If n = 0, 5^n = 1
If n is negative, 0 < 5^n < 1

Another thing, the positive base, a, can lie between one of two ranges - 'between 0 and 1' and 'greater than 1'. We saw what happens if the base is greater than 1 above (in 5^n example)

Let's see what happens when the base lies between 0 and 1:
If n is positive, a^n < 1
If n = 0, a^n = 1
If n is negative, a^n > 1

These relations hold the other way around too. If a is between 0 and 1, and a^n > 1, n must be negative etc.
Plug in values to understand them.

Now look at this question:

In these three options, each term is positive so we can take the term on the right to left hand side so that we have to deal with a single base.

(C)$$0.9^p > 0.9^q$$
$$(0.9)^{p-q} > 1$$
Here, p-q must be negative i.e. p-q < 0 or p<q (from red above)

(D) $$9^p < 9^q$$
$$9^{p-q} < 1$$
p-q < 0 (from blue above) or p < q

(E)$$9^p > 9^q$$
$$9^{p-q} > 1$$
So p-q > 0 (from green above) or p > q
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17815 [4], given: 235 Manager Joined: 25 Aug 2011 Posts: 198 Kudos [?]: 384 [0], given: 11 Location: India GMAT 1: 730 Q49 V40 WE: Operations (Insurance) Re: In this exponent problem P > Q ???? [#permalink] ### Show Tags 22 Feb 2012, 23:07 if x^2 >x^3 : we write x^2-x^3 >0 ie we subtract and dont divide in this case we are dividinge is that right? Bunuel wrote: In which one of the following choices must p be greater than q? Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer. A. 0.9^p = 0.9^q --> $$p=q$$. Discard; B. 0.9^p = 0.9^2q --> $$p=2q$$ --> if $$p=-2$$ and $$q=-1$$ then $$p<q$$. Discard; C. 0.9^p > 0.9^q --> if $$p=1$$ and $$q=2$$ then then $$p<q$$. Discard; D. 9^p < 9^q --> $$9^{p-q}<1$$ --> $$p-q<0$$ --> $$p<q$$. Discard; E. 9^p > 9^q --> $$9^{p-q}>1$$ --> $$p-q>0$$ --> $$p>q.$$ Correct. Answer: E. Kudos [?]: 384 [0], given: 11 Math Expert Joined: 02 Sep 2009 Posts: 42264 Kudos [?]: 132781 [0], given: 12372 Re: In this exponent problem P > Q ???? [#permalink] ### Show Tags 22 Feb 2012, 23:30 devinawilliam83 wrote: if x^2 >x^3 : we write x^2-x^3 >0 ie we subtract and dont divide in this case we are dividinge is that right? Bunuel wrote: In which one of the following choices must p be greater than q? Note that we are asked "which of the following MUST be true, not COULD be true. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer. A. 0.9^p = 0.9^q --> $$p=q$$. Discard; B. 0.9^p = 0.9^2q --> $$p=2q$$ --> if $$p=-2$$ and $$q=-1$$ then $$p<q$$. Discard; C. 0.9^p > 0.9^q --> if $$p=1$$ and $$q=2$$ then then $$p<q$$. Discard; D. 9^p < 9^q --> $$9^{p-q}<1$$ --> $$p-q<0$$ --> $$p<q$$. Discard; E. 9^p > 9^q --> $$9^{p-q}>1$$ --> $$p-q>0$$ --> $$p>q.$$ Correct. Answer: E. If its given that x^2<x^3 then since x^2>0 then we can reduce by it and write x>1; But if its given that x>x^3 then since we don't know the sign of x then we can not reduce by it and should rewrite: x(x^2-1)<0 --> (x+1)x(x-1)<0 --> x<-1 or 0<x<1. Check this for more - Solving Inequalities: x2-4x-94661.html#p731476 inequalities-trick-91482.html data-suff-inequalities-109078.html range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535 everything-is-less-than-zero-108884.html?hilit=extreme#p868863 As for $$9^p<9^q$$. Since $$9^q$$ is positive for any value of q we can divide by it and write $$9^{p-q}<1$$ --> $$p-q<0$$. Hope it helps. _________________ Kudos [?]: 132781 [0], given: 12372 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7738 Kudos [?]: 17815 [0], given: 235 Location: Pune, India Re: In this exponent problem P > Q ???? [#permalink] ### Show Tags 23 Feb 2012, 02:48 devinawilliam83 wrote: if x^2 >x^3 : we write x^2-x^3 >0 ie we subtract and dont divide in this case we are dividinge is that right? If the variable is given to be non zero, you can divide in an equation and if you also know the sign of the variable, you can divide in an inequality too. (and adjust the inequality according to the sign of the variable) e.g. $$x^3 = x$$. What values can x take? x can be 0 so we don't divide. $$x^3 - x = 0$$ $$x(x^2 - 1) = 0$$ $$x = 0, 1, -1$$ Here, if you forget about 0 and divide by x, you get x^2 = 1 so x = +1 or -1. You lose out one solution: x = 0 But in case you have $$x^3 = x$$. What values can x take if x is non zero? Now you can divide and you will get x = 1 or -1 Similarly, say you have an inequality: $$x^2 > x^3$$ If you divide by x^2 thinking it's non negative, you get the range x < 1 which is incorrect since you have to specify that 'x cannot be 0'. It holds for every value less than 1 other than 0. $$x^2 > x^3$$ does not hold for x = 0. So instead, you can choose to solve in two ways: 1. Take cases and use division: $$x^2 > x^3$$ If $$x \neq 0$$, we can divide. We get x < 1. If x = 0, the inequality does not hold. Hence, $$x \neq 0$$ Answer: x < 1, $$x \neq 0$$ 2. Subtract: x^3 - x^2 < 0 x^2 (x - 1) < 0 One of the terms must be negative and the other positive. Since x^2 cannot be negative, x - 1 must be negative (so x < 1) and x^2 must be positive so $$x \neq 0$$ Answer: x < 1, $$x \neq 0$$ Either way, you will get the same answer (obviously!) In this question, as Bunuel said, $$9^q$$ must be positive. It cannot take value 0 or negative for any value of q so there are no complications at all. You can easily divide. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: In this exponent problem P > Q ???? [#permalink]

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23 Feb 2012, 02:54
VeritasPrepKarishma wrote:
devinawilliam83 wrote:
if x^2 >x^3 : we write x^2-x^3 >0 ie we subtract and dont divide
in this case we are dividinge is that right?

If the variable is given to be non zero, you can divide in an equation and if you also know the sign of the variable, you can divide in an inequality too. (and adjust the inequality according to the sign of the variable)

e.g. $$x^3 = x$$. What values can x take?
x can be 0 so we don't divide.
$$x^3 - x = 0$$
$$x(x^2 - 1) = 0$$
$$x = 0, 1, -1$$

Here, if you forget about 0 and divide by x, you get x^2 = 1 so x = +1 or -1. You lose out one solution: x = 0

But in case you have
$$x^3 = x$$. What values can x take if x is non zero?
Now you can divide and you will get x = 1 or -1

Similarly, say you have an inequality: $$x^2 > x^3$$
If you divide by x^2 thinking it's non negative, you get the range x < 1 which is incorrect since you have to specify that 'x cannot be 0'. It holds for every value less than 1 other than 0.
$$x^2 > x^3$$ does not hold for x = 0.

So instead, you can choose to solve in two ways:

1. Take cases and use division: $$x^2 > x^3$$
If $$x \neq 0$$, we can divide. We get x < 1.
If x = 0, the inequality does not hold. Hence, $$x \neq 0$$
Answer: x < 1, $$x \neq 0$$

2. Subtract: x^3 - x^2 < 0
x^2 (x - 1) < 0
One of the terms must be negative and the other positive. Since x^2 cannot be negative, x - 1 must be negative (so x < 1) and x^2 must be positive so $$x \neq 0$$
Answer: x < 1, $$x \neq 0$$

Either way, you will get the same answer (obviously!)

In this question, as Bunuel said, $$9^q$$ must be positive. It cannot take value 0 or negative for any value of q so there are no complications at all. You can easily divide.

One little thing Karishma.

If its given that $$x^2<x^3$$ then it's clear that $$x\neq{0}$$ thus $$x^2>0$$ and you can safely reduce by it and write $$x>1$$.
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Re: In this exponent problem P > Q ???? [#permalink]

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23 Feb 2012, 04:38
Bunuel wrote:
One little thing Karishma.

If its given that $$x^2>x^3$$ then it's clear that $$x\neq{0}$$ thus $$x^2>0$$ and you can safely reduce by it and write $$x>1$$.

Sure Bunuel, I understand that one can see that when $$x^2>x^3$$, x cannot be 0. But since here, you get x < 1, a range that includes x = 0, we need to specify that x can take any value less than 1 except 0. The point I am making here is that if you only reduce it and write 'x<1', it's not correct. I think you got a typo 'x > 1' above.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 17815 [0], given: 235 Math Expert Joined: 02 Sep 2009 Posts: 42264 Kudos [?]: 132781 [0], given: 12372 Re: In this exponent problem P > Q ???? [#permalink] ### Show Tags 23 Feb 2012, 04:49 VeritasPrepKarishma wrote: Bunuel wrote: One little thing Karishma. If its given that $$x^2>x^3$$ then it's clear that $$x\neq{0}$$ thus $$x^2>0$$ and you can safely reduce by it and write $$x>1$$. Sure Bunuel, I understand that one can see that when $$x^2>x^3$$, x cannot be 0. But since here, you get x < 1, a range that includes x = 0, we need to specify that x can take any value less than 1 except 0. The point I am making here is that if you only reduce it and write 'x<1', it's not correct. I think you got a typo 'x > 1' above. Actually I do have a typo but not the one you pointed above: in 4 posts above I considered x^2<x^3 (not x^2>x^3) which when reduced gives x>1. So, it should read: if its given that $$x^2<x^3$$ then it's clear that $$x\neq{0}$$ thus $$x^2>0$$ and you can safely reduce by it and write $$x>1$$. _________________ Kudos [?]: 132781 [0], given: 12372 Intern Joined: 10 Jan 2012 Posts: 42 Kudos [?]: 11 [0], given: 11 Location: United States Concentration: Finance, Entrepreneurship Schools: Jones '15 (M) Re: In which one of the following choices must p be greater than [#permalink] ### Show Tags 24 Feb 2012, 12:20 Awesome post VeritasPrepKarishma and Bunuel! Thanks! Kudos [?]: 11 [0], given: 11 Non-Human User Joined: 09 Sep 2013 Posts: 15637 Kudos [?]: 283 [0], given: 0 Re: In which one of the following choices must p be greater than [#permalink] ### Show Tags 29 Jan 2014, 14:11 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Kudos [?]: 283 [0], given: 0 Non-Human User Joined: 09 Sep 2013 Posts: 15637 Kudos [?]: 283 [0], given: 0 Re: In which one of the following choices must p be greater than [#permalink] ### Show Tags 05 Apr 2015, 21:19 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Kudos [?]: 283 [0], given: 0 Intern Joined: 07 May 2015 Posts: 8 Kudos [?]: [0], given: 2 In which one of the following expressions must M be greater than N? [#permalink] ### Show Tags 09 May 2015, 08:51 In which one of the following expressions must M be greater than N. A. $$0.9^m = 0.9^n$$ B. $$0.9^m = 0.9^2^n$$ C. $$0.9^m > 0.9^n$$ D. $$9^m < 9^n$$ E. $$9^m > 9^n$$ [Reveal] Spoiler: The answer is E but I don't understand why. If M = -4 and N = 2 then $$9^m > 9^n$$ is still correct but M is not bigger than N right? Kudos [?]: [0], given: 2 Math Expert Joined: 02 Sep 2009 Posts: 42264 Kudos [?]: 132781 [0], given: 12372 Re: In which one of the following choices must p be greater than [#permalink] ### Show Tags 09 May 2015, 08:55 MarcvR wrote: In which one of the following expressions must M be greater than N. A. $$0.9^m = 0.9^n$$ B. $$0.9^m = 0.9^2^n$$ C. $$0.9^m > 0.9^n$$ D. $$9^m < 9^n$$ E. $$9^m > 9^n$$ [Reveal] Spoiler: The answer is E but I don't understand why. If M = -4 and N = 2 then $$9^m > 9^n$$ is still correct but M is not bigger than N right? Merging topics. Please refer to the discussion above. _________________ Kudos [?]: 132781 [0], given: 12372 Intern Joined: 07 May 2015 Posts: 8 Kudos [?]: [0], given: 2 Re: In which one of the following choices must p be greater than [#permalink] ### Show Tags 09 May 2015, 09:06 Thanks for merging the topics Bunuel. I am new to this forum and still need to find my way around here! About the question: even when I read the whole explanation it still does not makes sense to me.. I am having a real hard time mastering the maths section for the G-mat. My background in maths is far from average sadly. But from what I have learned so far I would say that when M = -4 and N = 2 the expression is still right but M is not bigger than N. Please help me on this one! Kudos [?]: [0], given: 2 GMAT Tutor Joined: 24 Jun 2008 Posts: 1340 Kudos [?]: 2004 [0], given: 6 Re: In which one of the following choices must p be greater than [#permalink] ### Show Tags 09 May 2015, 09:14 MarcvR wrote: The answer is E but I don't understand why. If M = -4 and N = 2 then $$9^m > 9^n$$ is still correct but M is not bigger than N right? No, that is not correct. If M = -4, then 9^M = 9^(-4) = 1/9^4, which is a positive number extremely close to zero. It is certainly less than 9^2 = 81. In general, the bigger you make x, the larger the value of 9^x will be. This is true whenever your base is greater than 1. So if 9^y > 9^z, then y > z, and vice versa. _________________ GMAT Tutor in Toronto If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com Kudos [?]: 2004 [0], given: 6 Intern Joined: 07 May 2015 Posts: 8 Kudos [?]: [0], given: 2 In which one of the following choices must p be greater than [#permalink] ### Show Tags 09 May 2015, 09:17 Wait I am making a huge mistake here.. $$-2^3$$ doesnt equal $$2^-3$$ of course. So when an exponent is negative the result will go towards zero and can be written as a fraction right? Edit: You are right Ianstewart! Thanks for the quick answer. I am still having a real hard time mastering the math section! But I will get there! Kudos [?]: [0], given: 2 SVP Joined: 06 Nov 2014 Posts: 1905 Kudos [?]: 541 [0], given: 23 Re: In which one of the following choices must p be greater than [#permalink] ### Show Tags 14 May 2015, 12:24 In which one of the following choices must p be greater than q? (A) 0.9^p = 0.9^q (B) 0.9^p = 0.9^2q (C) 0.9^p > 0.9^q (D) 9^p < 9^q (E) 9^p > 9^q Since the bases are equal, p and q must be the same for A and B. Those choices can be eliminated. At this point I would substitute values for p and q in choices C and D, keeping mind to also try positive and negative values for p and q. You will find that p has to be less than q for the inequalities in C and D to hold true. _________________ # Janielle Williams Customer Support Special Offer:$80-100/hr. Online Private Tutoring
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Re: In which one of the following choices must p be greater than [#permalink]

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21 May 2016, 13:06
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Re: In which one of the following choices must p be greater than   [#permalink] 21 May 2016, 13:06

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