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

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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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.

Answer:

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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.

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

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!

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

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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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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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).

Bunuel niks18 chetan2u amanvermagmat

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

I did not understand highlighted part. It could also be possible that p/q > 0.

My approach was same as Marcab and hope it is valid approach.
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Hi,

You can multiply both sides of the inequality by any positive number because by doing so it will not change the inequality sign. hence when stem inequality is multiplied by q/p on both sides you get 1 in LHS (p/q*q/p) and q/p in RHS (1*q/p)

You can also cross multiply here without worrying about the sign because p & q are positive. So your approach is correct

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Hi All,

This question can be solved in a variety of different ways. Every so often, the GMAT offers a question that looks far more complex than it actually is. Each of the other explanations provided in this string is correct. Here's a fairly quick way of looking at it though…

Since P and Q are POSITIVE INTEGERS and P/Q < 1, inverting the fraction will invert the relationship…so Q/P > 1. The question ask which of the 5 options is greater than 1….

That's obviously

GMAT assassins aren't born, they're made,
Rich

Same here, any idea why p=1, q=2 wouldn't work? I got stuck between D and E as well since I chose (1,2).

Hi vishnusheth,

When TESTing VALUES, you will occasionally have to test more than one example to find the definitive correct answer. The example you chose (P=1, Q=2) DOES fit the correct answer, but it also matches an answer that is 'sometimes' correct. Thus, your example is fine, but it's not enough on its own to define which of the five answers is ALWAYS correct. From a mathematical-standpoint, the reason why it's not enough is because when P=1, answer choices D and E will produce the exact same result (re: Q/1 and Q/1).

GMAT assassins aren't born, they're made,
Rich

\(\sqrt{\frac{q}{p}} > 1\) ??

Hi gmatzpractice,

Do you have a specific question regarding Answer Choice A?

Based on the information in the prompt, we know that P/Q will be a POSITIVE FRACTION (0 < P/Q < 1). When taking the square-root of a positive fraction, the result is a BIGGER POSITIVE FRACTION. For example, the square-root of 1/4 is 1/2. Thus, Answer A will NEVER be greater than 1.

GMAT assassins aren't born, they're made,
Rich

Here, plugging in will make the question very lengthy.
If p/q < 1, numerator is less than denominator.
Thus q/p > 1 as numerator is greater than denominator.

Through question stem we know that 0 < p/q <1 and p ,q both are positive integers, means it could be anything between 0 and 1
and this could be only possible when numerator P is less than denominator Q , (p<q).

we dont have to check each option as we already know that bigger the denominator, smaller the value of fraction. we need here an option which is not making denominator bigger but smaller and only E is the option which matches our requirement.

on other hand if we test by taking p=2, q=3
so, it gives, p/q = 2/3 = 0.666
but when we take (option E) = q/p= 3/2 = 1.5 which is greater then 1 and hence E is our answer.
Bunuel, please critique if my approach is wrong, i am in process of improving my score. Thanks