If p and q are positive integers, and p

Question

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

BrentGMATPrepNow · Accepted Answer

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

Answer:  B

RELATED VIDEOS
 https://www.youtube.com/watch?v=718lE2Hvlnk

https://www.youtube.com/watch?v=hnpJiUoFY_Q

rohit8865 · Answer

(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

sobby · Answer

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..

Answer is B

FB2017 · Answer

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.

BrentGMATPrepNow · Answer

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 ?

shashankism · Answer

Given : p 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 p+1<=q, So, (p–1)/q <= (p–1)/p+1 Hence only I & II are correct Answer B.

generis · Answer

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 ...?

shashankism · Answer

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)

sachin_27 · Answer

We can take p=1 and q=2 ( positive integers where p<q) 
Substitute these values one by one in all 3 options 
Hence 1 and 2 holds the condition.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions