12 Days of Christmas GMAT Competition - Day 5: If p, q are each greate

Answer is option A. Only I is true.

If p, q are each greater than 0 but less than 1, which of the following may be true?

I. p/q > 1
II. pq > 1
III. q – p > 1

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

I) take p=1/2, q=1/3. p/q=1.5>1 (True)
II) pq >1 ? product of two proper fraction is always less than 1. (False)
III) This will never happen because max val of q is <1, and min val of p is >0. (False)

Hence A is the answer

Range of p,q is between 0 and 1.

Q. asks "may be true"

1. If p=0.9 and =0.1 then yes so b out
2. pq > 1 the product of 2 no.s between this range will always be <1 . Try the same no's as 1 C,E out
3. q – p > 1 If all the no's are <1 then how can the difference be >1? You can try some no and check D out

Ans: A

0<p<1
0<q<1
product of any two numbers less than 1 can never be greater than 1. Therefore, statement II is definitely false.
0.99999........9 will be the greatest number which is less than 1 and when you subtract anything (greater than 0) from it. You can never get a number greater than or equal to 1. Therefore, statement III is definitely false.\
Statement I can be or cannot be true depending on the value of p & q
p=1/2 and q=1/4;p/q=2
p=1/4 and q=1/2;p/q=1/2
Therefore, option A

Asked: If p, q are each greater than 0 but less than 1, which of the following may be true?

I. p/q > 1 : If p=1/2 ; q = 1/3; p/q = 3/2 >1: MAY BE TRUE
II. pq > 1 : 0<p<1 ; 0<q<1; 0<pq<1: NEVER TRUE
III. q – p > 1 : 0<q<1: -1<-p<0; -1<q-p<1: NEVER TRUE

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

IMO A
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Only possible is I.

II.. multiplication of below 1 will always result in less than 1.

III... Subtraction of less than 1 amounts/ numbers will answer in less than .

Answer is I as only Denominator lower than numerator will give results more than 1 if both numerator and denominator are between 0 and 1 . It satisfies the given .

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I. p/q > 1 Possible for value of p=0.6 and q=0.2

II. pq > 1 Not possible p can have value as 0.9 and q as 0.8 product is 0.72

III. q – p > 1 Difference of number p,q in the range 0<p,q<1 will always be less than 1.

IMO A.

from given info we have following possibilities

0<p < q <1
or 0<q<p<1
1/2 , 1/3 can be used to test values
which of following may be true

I. p/q > 1
1/3 / 1/2 ; NO
1/2 / 1/3 ; yes


II. pq > 1
1/3 * 1/2 ; 1/6 <1
not possible

III. q – p > 1

1/3-1/2 ; (-1/6)
1/2-1/3 ; ( 1/6)
not possible
I only
OPTION A

I. p/q > 1 Yes if p=1/2 and q=1/4
II. pq > 1 No since fractions are proper fractions
III. q – p > 1 No as the distance can never be > 1 as they are both b/w 0 and 1

Ans A

We know both p and q is a fraction whose value is between 0 and 1. We need to identify at least for some cases, if option I, II, and III are true.

I. p/q > 1
Let p = 1/2 and q = 1/4.
Then p/q = 2.
For the above example p/q > 1.
So keep options that include I. So, eliminate B.

II. pq > 1

Multiplying 2 fractions both 0<p,q < 1, can only diminish the product result.

Say both p=q= 9/10 = 0.9

Product = pxq = (9/10) *(9/10) = 81/100 = 0.81.

0.81< 0.9, proves that the product will diminishes the value further as both are less than 1 and greater than zero.

Eliminate choices that have option II. So, B, C, E is gone, and only A and D remains.


III. q – p > 1

Even if we assume q is larger than p, substraction will result in diminishing the number.

Example, let's assume q = 9/10 and p = 1/10.

q-p = 8/10.
So, never the result will be greater than 1. Eliminate options that include III. D is gone.

So, the best answer choice is A.

Q-p<1 since qmax=1 and pmin=0
Let p=.75 and q=.25. p/q=3
Pq<1
Hence answer is A

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Let's suppose
p=0.8 q=0.5
I. 0.8/0.5>1
II. 0.8*0.5<1
III. 0.5-0.8<1

So A must be true.

If p, q are each greater than 0 but less than 1, which of the following may be true?

I. p/q > 1
Possible. p = 0.3, q = 0.2
p/q = 3/2 = 1.5 > 1

II. pq > 1
Never. 0 < p < 1 and 0 < q < 1. eg: p = 0.99999999, q = 0.99999999 would still not make pq > 1

III. q – p > 1
Not possible. Take close to largest value for q = 0.9999999999 and close to smallest for p = 0.0000000000001
If you subtract, the difference is still not greater than 1 because q itself is smaller than 1 and p is greater than 0.

Answer: A. Only I

This task is rather straightforward - we need to deal with the options one by one.

I. Whenever \(p>q\), we will always get \(\frac{p}{q}>1\)
For instance, \(\frac{0.8}{0.5} = 1.6\)
Therefore, option I may be true.

II. We have two decimals below 1. And when we multiply any number by a decimal less than 1,
we always decrease this number, no exceptions: for instance, \(10*0.9=9\), where (obviously) \(9<10\).
Therefore, by multiplying one number below 1 by another similar number will only further decrease the first multiple.
That is to say, it is impossible to go beyond 1 by multiplying two number less than 1.
Thus, option II may not be true.

III. To see the solution to this option, let's compare the new data to the original task:
We know that \(q<1\). Now we are told that \(q-p>1\), or \(q>1+p\).
Can this inequality be true? \(1+p < q < 1\)
Of course not, unless P is negative (which is not the case).
Therefore, option III may not be true.

So, the answer is A. I only.

I. p/q > 1
when p=1/2 and q=1/4 p/q=2 so its true

II. pq > 1 not true
p=0.99 & q=0.99 pq<1
III. q – p > 1
since both q&P less than 1 & higher than 0 this case is also not true

so And A. I only

If p, q are each greater than 0 but less than 1, which of the following may be true?

I. p/q > 1
II. pq > 1
III. q – p > 1

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

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IMO A

p/q >1 , eg - p=3/5, q=2/7, p/q = 21/10

pq cannot be greater than 1 as the numerator in both the cases are less than the denominator. so their product will also be less than 1

q - p cannot be greater than 1 as both the numbers are less than 1 and their difference can either be another fraction or a negative number.

I- Let, p=0.5 , q=0.25
so, p/q=2>1.
Hence, I may be true.

II- When two numbers between 0 and 1 are multiplied the result will always be a number between 0 and 1.

III- Difference between two numbers between 0 and 1 will always be a number between 0 and 1.

Hence, IMO A.

Given that 0<p<1 and 0<q<1

Option 1
p/q >1

If p = 0.5 and q = 0.1
p/q = 5, >1
This may be possible

Option 2
pq > 1
Let us consider the largest value possible for both p and q to get the maximum value for the product
p = q = 0.999 (cannot be negative)
pq = 0.998.. <1
Will never be>1

Option 3
p - q >1
Even this is impossible
Consider the maximum value for P and the minimum value for q (q cannot be negative)
p = 0.999 and q = 0.001
even the p-q not > 1

I only

If p, q are each greater than 0 but less than 1, which of the following may be true?

I. p/q > 1

valid when p=.2 and q=.1
II. pq > 1

cant possible since multiplication will bring a lower value
III. q – p > 1

this is also not possible

oa:A