GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 26 Apr 2019, 01:08

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If p and q are positive integers, and p < q, then which of

Author Message
TAGS:

### Hide Tags

CEO
Joined: 12 Sep 2015
Posts: 3595
If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

18 May 2017, 06:52
Top Contributor
5
00:00

Difficulty:

45% (medium)

Question Stats:

68% (02:02) correct 32% (01:59) wrong based on 182 sessions

### HideShow timer Statistics

If p and q are positive integers, and p < q, then which of the following MUST be true?

I) $$\frac{p}{q} < \frac{p+1}{q+1}$$

II) $$\frac{p–1}{q} < \frac{p+1}{q}$$

III) $$\frac{p–1}{q} < \frac{p–1}{p+1}$$

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

*kudos for all correct solutions

_________________
Test confidently with gmatprepnow.com
Director
Joined: 05 Mar 2015
Posts: 999
Re: If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

18 May 2017, 08:50
1
GMATPrepNow wrote:
If p and q are positive integers, and p < q, then which of the following MUST be true?

I) $$\frac{p}{q} < \frac{p+1}{q+1}$$

II) $$\frac{p–1}{q} < \frac{p+1}{q}$$

III) $$\frac{p–1}{q} < \frac{p–1}{p+1}$$

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

*kudos for all correct solutions

(1) p/q - (p+1)/(q+1) <0
-( p+q)/q^2+1 < 0
-ve / +ve <0
always true

(2) (p-1)/q - (p+1)/q <0
p-1-p-1/ q <0
since q is +ve
numerator = p-1-p-1 = -2 <0
always true

(3) 1/q < 1/(p+1) ----cancelling p-1 both sides
(p+1-q) / q(p+1) <0
as denominator is +ve
numerator must be <0
p+1-q <0
p-q < -1-------------(a)
if p= 1 , q=2 then NO
if p=1 , q=3 then YES
Not true

satisfied condition only (1) & (2)

Ans B
Director
Joined: 14 Nov 2014
Posts: 624
Location: India
GMAT 1: 700 Q50 V34
GPA: 3.76
Re: If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

18 May 2017, 11:27
2
GMATPrepNow wrote:
If p and q are positive integers, and p < q, then which of the following MUST be true?

I) $$\frac{p}{q} < \frac{p+1}{q+1}$$

II) $$\frac{p–1}{q} < \frac{p+1}{q}$$

III) $$\frac{p–1}{q} < \frac{p–1}{p+1}$$

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

*kudos for all correct solutions

Equation 1 can be re written as:
pq + p < q + pq
p<q -----------true.(given in question stem)

Equation 2
qp - q < pq + q
-q < q
true as q is positive-------------2

Equation 3
(p-1)(p+1) < q(p-1)
(p-1)(p+1-q) < 0
No ..take p=1 and q=2
it will violate the equation .
so , this is not true..

CEO
Joined: 12 Sep 2015
Posts: 3595
Re: If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

26 May 2017, 14:26
2
Top Contributor
1
GMATPrepNow wrote:
If p and q are positive integers, and p < q, then which of the following MUST be true?

I) $$\frac{p}{q} < \frac{p+1}{q+1}$$

II) $$\frac{p–1}{q} < \frac{p+1}{q}$$

III) $$\frac{p–1}{q} < \frac{p–1}{p+1}$$

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

*kudos for all correct solutions

Statement I: There's a nice rule that says "If we add the same positive value to the numerator and denominator of a positive fraction, the resulting fraction is closer to one than the original fraction was.
For example (23+8)/(50+8) is closer to 1 than is 23/50

Since p < q, we know that p/q is less than 1
By the above rule, we know that (p+1)/(q+1) is closer to 1 than is p/q, which means p/q < (p+1)/(q+1) < 1
Statement I is TRUE

Statement II: The positive denominators are the same, but the numerator p+1 is greater than p-1
So, it must be the case that (p-1)/q < (p+1)/q
Statement II is TRUE

Statement III: This time the numerators are the same, but the denominators are different (q and p+1)
We can right away that if p = 1, then the two sides are EQUAL
In other words, it is NOT the case that (p–1)/q < (p–1)/(p+1)
Statement III need NOT be true

RELATED VIDEOS

_________________
Test confidently with gmatprepnow.com
Manager
Joined: 18 Jun 2017
Posts: 59
If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

11 Jul 2017, 00:53
If we substitute each equation with p=4 and q=8
the outcome is
1. 1/2<5/9=-1<-4
2. 3/8<5/8 = -5<-3
3. 3/8<1/3 = -5<-2
So conclusion should be options 2 & 3.

Please advise why do we get different values on plugging values and directly solving the algebraic expression.

These (-)ve values were assumed as they were considered as integers.
CEO
Joined: 12 Sep 2015
Posts: 3595
Re: If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

11 Jul 2017, 04:25
Top Contributor
FB2017 wrote:
If we substitute each equation with p=4 and q=8
the outcome is
1. 1/2<5/9=-1<-4
2. 3/8<5/8 = -5<-3
3. 3/8<1/3 = -5<-2
So conclusion should be options 2 & 3.

Please advise why do we get different values on plugging values and directly solving the algebraic expression.

These (-)ve values were assumed as they were considered as integers.

I'm confused with the steps you took to make the conclusions in blue above

For example, how does 1/2<5/9 turn into -1<-4?
_________________
Test confidently with gmatprepnow.com
Director
Joined: 13 Mar 2017
Posts: 724
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

Updated on: 14 Jul 2017, 00:48
1
GMATPrepNow wrote:
If p and q are positive integers, and p < q, then which of the following MUST be true?

I) $$\frac{p}{q} < \frac{p+1}{q+1}$$

II) $$\frac{p–1}{q} < \frac{p+1}{q}$$

III) $$\frac{p–1}{q} < \frac{p–1}{p+1}$$

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

*kudos for all correct solutions

Given : p<q
-> p/q<1

I. p/q . When a constant is added to nr and dr, the fraction tends to move towards 1.
Since p/q<1 so (p+1)/(q+1) will move towards 1.
Hence p/q < (p+1)/(q+1)

II. (p-1)/q < (p+1)/q.. Since dr. is same and Nr. on right side is greater than left side.

III. (p–1)/q < (p–1)/p+1 .. NOT TRUE . Since p<q -> p+1<=q,
So, (p–1)/q <= (p–1)/p+1

Hence only I & II are correct
_________________
CAT 2017 (98.95) & 2018 (98.91) : 99th percentiler
UPSC Aspirants : Get my app UPSC Important News Reader from Play store.

MBA Social Network : WebMaggu

Appreciate by Clicking +1 Kudos ( Lets be more generous friends.)

What I believe is : "Nothing is Impossible, Even Impossible says I'm Possible" : "Stay Hungry, Stay Foolish".

Originally posted by shashankism on 11 Jul 2017, 05:35.
Last edited by shashankism on 14 Jul 2017, 00:48, edited 1 time in total.
Senior SC Moderator
Joined: 22 May 2016
Posts: 2654
If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

12 Jul 2017, 16:34
shashankism wrote:
GMATPrepNow wrote:
If p and q are positive integers, and p < q, then which of the following MUST be true?

I) $$\frac{p}{q} < \frac{p+1}{q+1}$$

II) $$\frac{p–1}{q} < \frac{p+1}{q}$$

III) $$\frac{p–1}{q} < \frac{p–1}{p+1}$$

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

*kudos for all correct solutions

Given : p<q
-> p/q<1

I. p/q . When a constant is added to nr and dr, the fraction tends to move towards 1.
Since p/q<1 so (p+1)/(q+1) will move towards 1.
Hence p/q < (p+1)/(q+1)

II. (p-1)/q < (p+1)/q.. Since dr. is same and Nr. on right side is greater than left side.

III. (p–1)/q < (p–1)/p+1 .. NOT TRUE . Since p>q -> p+1>q,
So, p–1)/q > (p–1)/p+1

Hence only I & II are correct

shashankism , I can't follow the bolded part. p isn't greater than q, and in analysis of I you note that, so I think something went wrong here ...?
_________________
Listen, are you breathing just a little, and calling it a life?
-- Mary Oliver

For practice SC questions with official explanations that were posted and moderated by the SC Team,
go to SC Butler here: https://gmatclub.com/forum/project-sc-butler-get-2-sc-questions-everyday-281043.html
Director
Joined: 13 Mar 2017
Posts: 724
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
Re: If p and q are positive integers, and p < q, then which of  [#permalink]

### Show Tags

14 Jul 2017, 00:50
genxer123 wrote:
shashankism wrote:
GMATPrepNow wrote:
If p and q are positive integers, and p < q, then which of the following MUST be true?

I) $$\frac{p}{q} < \frac{p+1}{q+1}$$

II) $$\frac{p–1}{q} < \frac{p+1}{q}$$

III) $$\frac{p–1}{q} < \frac{p–1}{p+1}$$

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

*kudos for all correct solutions

Given : p<q
-> p/q<1

I. p/q . When a constant is added to nr and dr, the fraction tends to move towards 1.
Since p/q<1 so (p+1)/(q+1) will move towards 1.
Hence p/q < (p+1)/(q+1)

II. (p-1)/q < (p+1)/q.. Since dr. is same and Nr. on right side is greater than left side.

III. (p–1)/q < (p–1)/p+1 .. NOT TRUE . Since p>q -> p+1>q,
So, (p–1)/q > (p–1)/p+1

Hence only I & II are correct

shashankism , I can't follow the bolded part. p isn't greater than q, and in analysis of I you note that, so I think something went wrong here ...?

Yes i didi a mistake over there. I have corrected my solution
Basically p < q
so p+1<=q
and hence
(p-1)/q <= (p-1)/(p+1)
_________________
CAT 2017 (98.95) & 2018 (98.91) : 99th percentiler
UPSC Aspirants : Get my app UPSC Important News Reader from Play store.

MBA Social Network : WebMaggu

Appreciate by Clicking +1 Kudos ( Lets be more generous friends.)

What I believe is : "Nothing is Impossible, Even Impossible says I'm Possible" : "Stay Hungry, Stay Foolish".
Re: If p and q are positive integers, and p < q, then which of   [#permalink] 14 Jul 2017, 00:50
Display posts from previous: Sort by