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If p/q < 1, and p and q are positive integers, which of the

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If p/q < 1, and p and q are positive integers, which of the [#permalink]

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If p/q < 1, and p and q are positive integers, which of the following must be greater than 1 ?

(A) \(\sqrt{\frac{p}{q}}\)
(B) p/q^2
(C) p/2q
(D) q/p^2
(E) q/p
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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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If p/q < 1, and p and q are positive integers, which of the following must be greater than 1 ?

(A) \(\sqrt{\frac{p}{q}}\)
(B) p/q^2
(C) p/2q
(D) q/p^2
(E) q/p

Since p and q are positive integers, then q/p>0. Now, multiply p/q < 1 by q/p to get 1 < q/p.

Answer: E.

Or: since this is a must be true question, then even if we find only one example for which an option is not true, it'll mean that this option is not always true, thus not a correct answer.

Say p=2 and q=3. In this case, no option is correct but E.

Answer: E.
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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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New post 17 Dec 2012, 06:33
Expert's post
Walkabout wrote:
If p/q < 1, and p and q are positive integers, which of the following must be greater than 1 ?

(A) \(\sqrt{\frac{p}{q}}\)
(B) p/q^2
(C) p/2q
(D) q/p^2
(E) q/p


Given:
\(p/q <1\). Since p and q are positive integers, we can multiply by q on both sides.
The relation becomes:
\(p<q\).

Now if we divide the entire relation by p, then:
\(p/p < q/p\) or \(1< q/p\)

Hence \(q/p > 1\).
+1 E.
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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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New post 21 Jan 2013, 11:46
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If p/q < 1 then q>p.

A) \sqrt{\frac{p}{q}} will obviously be less than 1.
(B) p/q^2 Will increase the denominator and still makes it less than 1.
(C) p/2qWill increase the denominator and still makes it less than 1.
(D) q/p^2Still less than 1. Suppose p =2 and q =3
(E) q/p Since q > p, q/p >1
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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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New post 22 May 2014, 22:29
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When we know the signs of fraction in an inequality and want to take the reciprocal here is a handy rule: flip the inequality when taking reciprocal unless both sides have different signs.
Thus we can flip the inequality in the given relation p/q<1, to get q/p>1. E it is!
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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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New post 23 May 2014, 07:25
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Not sure if it is solved this way:

\(\frac{p}{q} < 1\),

As p and q are positive, Denominator should be greater than numerator.

Therefore, \(q>p\)

Hence, \(\frac{q}{p} > 1\)

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Rajat
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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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New post 17 Jun 2014, 22:03
Did anyone solve this with smart numbers? Used p = 1 and q = 2 which narrowed it down to D and E and then tested those with p = 5 and q = 9? This gave me E.
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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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New post 18 Jun 2014, 01:27
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carolinanmd wrote:
Did anyone solve this with smart numbers? Used p = 1 and q = 2 which narrowed it down to D and E and then tested those with p = 5 and q = 9? This gave me E.


The second approach in my solution here: if-p-q-1-and-p-and-q-are-positive-integers-which-of-the-144262.html#p1156616 uses p=2 and q=3 to discard all options but E.

Hope it helps.
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Resources:
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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Re: If p/q < 1, and p and q are positive integers, which of the [#permalink]

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New post 16 Jun 2016, 06:03
Walkabout wrote:
If p/q < 1, and p and q are positive integers, which of the following must be greater than 1 ?

(A) \(\sqrt{\frac{p}{q}}\)
(B) p/q^2
(C) p/2q
(D) q/p^2
(E) q/p


We can use some actual numbers to solve this problem. We see that p/q is less than 1. It follows that we have a positive proper fraction, where q is greater than p. Let's let p = 1 and q = 4; thus p/q = 1/4.

We now consider each answer choice:

Choice A: √(p/q) = √ (1/4) = ½, which is less than 1. Choice A is not correct.

Choice B: p/q^2 = 1/4^2 = 1/16, which is less than 1. Choice B is not correct.

Choice C: p/(2q) = 1/(2 x 4) = 1/8, which is less than 1. Choice C is not correct.

Choice D: q/p^2 = 4/1^2 = 4, which is greater than 1. This could be the answer.

Choice E: q/p = 4/1 = 4, which is greater than 1. This could be the answer.

Choices D and E work for the fraction ¼.
Let's now consider additional values for the original fraction p/q, such as p = 3 and q = 4.

Choice D now becomes q/p^2 = 4/3^2 = 4/9, which is less than 1. Thus, we can now eliminate Choice D.

Choice E now becomes q/p = 4/3, which is greater than 1. Thus, Choice E is correct. In fact, we can see that no matter what the values of p and q are, if p/q is less than 1, then its reciprocal, q/p will always be greater than 1.

Answer: E
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Re: If p/q < 1, and p and q are positive integers, which of the   [#permalink] 16 Jun 2016, 06:03
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