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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]  17 Dec 2012, 05:16
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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) 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]  17 Dec 2012, 05:21
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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]  17 Dec 2012, 05:33
Expert's post
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]  21 Jan 2013, 10:46
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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]  22 May 2014, 21:29
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]  23 May 2014, 06:25
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]  17 Jun 2014, 21:03
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]  18 Jun 2014, 00:27
Expert's post
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]  29 Aug 2015, 01:05
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Re: If p/q < 1, and p and q are positive integers, which of the   [#permalink] 29 Aug 2015, 01:05
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