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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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17 Dec 2012, 06:16
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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) $$\frac{p}{q^2}$$

(C) $$\frac{p}{2q}$$

(D) $$\frac{q}{p^2}$$

(E) $$\frac{q}{p}$$
[Reveal] Spoiler: OA

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

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17 Dec 2012, 06:21
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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.

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.

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

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17 Dec 2012, 06:33
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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

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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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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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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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, $$\frac{q}{p} > 1$$

Rgds,
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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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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18 Jun 2014, 01:27
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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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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16 Jun 2016, 06:03
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.

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

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01 Aug 2016, 08:01
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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

IMPORTANT: For questions like this, where you need to test each answer choice, the test-makers will often make D or E the correct answer (because they want to eat up your valuable time ). So, in these situations, always begin with E and work your way up.

E. Is q/p > 1?
Well, we're told that p/q < 1.
Since q is a positive integer, we can multiply both sides by q to get: p < q
Since p is a positive integer, we can now divide both sides by p to get: 1 < q/p
So, answer choice E must be true.

[Reveal] Spoiler:
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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25 Aug 2017, 06:30
Hello Moderators,

Do you it would be better to have all the answer choices math friendly. Specially B and D?

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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25 Aug 2017, 08:27
susheelh wrote:
Hello Moderators,

Do you it would be better to have all the answer choices math friendly. Specially B and D?

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

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21 Sep 2017, 07:04
0<p/q<1

For example: 2/3

Numerator<Denominator

When denominator>Numerator, the number would be greater than 1.

q/p

For example: 3/2

(Note: We know from the question stem that p and q are not equal to each other. If that was the case, then p/q= 1 not<1).
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If p/q < 1, and p and q are positive integers, which of the   [#permalink] 21 Sep 2017, 07:04
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