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

I. p/q > 1
II. pq > 1
III. q – p > 1

given that 0<p,q<1
suppose p = 0.5, q=0.3
hence, p/q = 5/3 which is >1 hence I can be true

II) suppose p=0.1, q = 0.9 pq=0.09 <1 therefore II is not true
III) suppose q= 0.9, p = 0.01 q-p = 0.89 which is not greater than 1 therefore III is not true

Hence, 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

Normally I interpret the question, but not much to interpret here, just read carefully. Moving on to consider each option.

I. p/q IS > 1 for positive numbers (including fractions less than zero) whenever p is greater than q. In this, fractions less than 1 do not behave differently than any other positive numbers. (This falls apart if negatives are included.)
This MAY BE TRUE.

II. pq >1 is NEVER TRUE for (positive) fractions less than 1. A fraction (less than one) times a fraction is the same as saying you will end up with only a fraction OF what you started with. Thus, a fraction of a fraction is never greater than one.
This is NOT true.

III. q - p > 1 is NEVER TRUE for (positive) fractions less than one. If you start out with a number less than one and subtract (a positive value) you can never arrive at a number higher than you started with.
This is NOT true!

A is the answer

If 0<p<1 and 0<q<1, which of the following may be true?

I. p/q > 1 May be true when numerator is greater than denominator: \(\frac{1}{2}:\frac{1}{10}=5\)
II. pq > 1 Can not be true, since product of any two numbers that are between 0 and 1 is always between 0 and 1
III. q – p > 1 Can not be true as the difference of any positive numbers that are less than 1 can not be equal to 1 or exceed it

Answer A

Lets take p =9/10, q=1/10
In this case I p/q > 1 is true.

pq cannot be >1 as both p & q are <1. So II cannot be true.

q-p can't be >1 as the max difference b/w them is <1.


So only I is possible.

IMO A

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

I. p/q > 1 eg. 0.5/0.2 =2.5 hence satisfies
II. pq > 1 eg. 0.5*0.2 = 0.1

III. q – p > 1 eg. 0.2-0.5=-0.3

Therefore A is the answer

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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

0 < p , q < 1

pq < 1

p/q > 1 may be true p/q = 0.4/0.2 = 2 >1

q-p <1

Only I may be true

A

p and q are positive proper fractions less than one

I. p/q > 1 : May be true
If \(p = \frac{1}{2}\) and \(q = \frac{1}{4} \)then \(\frac{p}{q } = 2 \)

II. pq > 1 : Not Possible
If \(0<p<1\) and \(0<q<1\) then \(0<pq<p\) and \(0<pq<q\)
therefore \( 0<pq<1\)

III. q – p > 1 : Not Possible
Since both p and q both are within a range of 0 and 1 subtracting q - p will not result in a value greater than one.

Ans. A

p and q both have the range of 0<p,q>1 let's come to statements
1. Assume p=0.8 and q=0.2 so p/q= 4, this may be true
2. Both are less then 1 so product should always less the 1 so condition unsatisfied
3. Difference of both never goes more then 1 because here are no value more then 1

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0 < p, q < 1
We need to find statement which MAY BE TRUE i.e., at least one combination of p and q must make the statement true.

I. p/q > 1
=> If p > q then statement 1 is true. e.g., if p = 1/2 and q = 1/3 then p/q > 1. Statement 1 may be true.

II. pq > 1
=> Since both p and q are positive and less than 1 then p*q can never be > 1. Statement 2 can never be true.

III. q – p > 1
=> Since both p and q are positive and less than 1 then q-p can never be > 1. Statement 3 can never be true.

Answer: A

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

I. p/q > 1

Let p = 0.4
Let q = 0.2
Therefore, p/q = 2

Option I is Possible

II. pq > 1

The maximum value of p = 0.9999…
The maximum value of q = 0.9999…

Therefore, pq will ALWAYS be less than 1 and NEVER greater than 1

Option II is NOT Possible

III. q – p > 1

The maximum value of q = 0.9999…
The minimum value of p = 0.0000…..0001

Therefore, q-p will ALWAYS be less than 1 and NEVER greater than 1

Option III is NOT Possible

Answer: CHOICE A

I. P= 0.9, q=0.1
p/q = 0.9/0.1 = 9
p/q>1

II. when two number less than 1, but greater than zero are multiplied their product is always less than 1

III. Difference of two numbers less than 1, but greater than zero is always less than 1

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

p, q are each greater than 0 but less than 1
I. p/q > 1
if p=0.9 and q=0.1
then p/q=9>1
May be true

II. pq > 1
p and q both lie between 0 and 1
Product will always be less than 1
Does not hold, Eliminate

III. q – p > 1
the difference between p and q can not be more than 1 because both p, q are each greater than 0 but less than 1
Does not hold, Eliminate

Ans : A

Solution:

Given, 0<p<1
& 0<q<1
Question asks us which may be true.

Statement I: p/q > 1
For p=1/2, q=1/3,
p/q= 3/2 > 1.
Hence, POSSIBLE.


Statement II: pq > 1
Even after taking max. possible values i.e. 0.99 for each p and q.
pq = 0.99*0.99 < 1.
Hence, NOT POSSIBLE.

Statement III: q – p > 1
Even after taking boundary values, i.e. maximizing q and minimizing p by taking q=0.99 & p=0.00001
q-p=0.99-0.00001<1.
Hence, NOT POSSIBLE.

ANSWER A.

If 0<p,q<1,

I. p/q > 1 - may be true as p=1/2 & Q=1/3 then P/Q=3/2 >1

II. pq > 1 - Not True as if maximum value of PQ is when P=.9999 & Q=.9999 then PQ<1

III. q – p > 1 - Not True as if maximum value of q-p when P=.00001 & Q=.9999 then q-p <1

Only I is correct hence Option A

OA) A

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

based on the question 0<p<1 & 0<q<1 given

option I) :
if 0<q<p<1 , so p > q then, p/q>1 so option I will be true in this case

option II):
0<p<1 & 0<q<1 then definitely 0<pq<1
so option II cannot be true

option III):
q-p > 1 the q > 1+p & p>0 so q >1 which cannot be true based on the given conditions
so options III also cannot be true

p, q are fractions between 0 and 1

I. p/q > 1
p = 1/2 q=1/5
p/q = 5/2>1
Possible

II. pq > 1
This cannot be possible since both values are less than 1
0.9 * 0.7<1
0.2*0.1<1

III. q – p > 1
This is not possible as well.
If both are between 0 and 1 then the highest possible difference will be 0.9-0.1 =0.8<1

Ans: A

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

I. p/q > 1
= p>q (It is true when p=0.2 and q=0.3)

II. pq > 1 (It can't be true, multiplying decimal never yield greater than 1)

III. q – p > 1
q>1+p (It is false, q can't be greater than 1+p)


Answer: A

I. p/q > 1
If p = 1/4 and q = 1/8
=> p/q = 1/4 divided by 1/8 = 2
Possible

II. pq > 1
Multiplying two fractions between 0 and would never result in a number greater than 1
Not Possible

III. q – p > 1
Since the range of p and q is less than 1, it is not possible for p - q > 1
Not Possible

IMHO Option A

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

If p and q lie between 0 and 1; then

(I) p/q > 1 for p=0.99 and q = 0.01..Possible
(II) pq is always less than 1 ..NOT possible
(III) q-p is always less than 1..NOT possible

IMO, (A) will be the answer