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

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BrentGMATPrepNow
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If p and q are positive integers, and p < q, then which of [#permalink] New post   18 May 2017, 06:52
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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
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B
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Re: If p and q are positive integers, and p < q, then which of [#permalink] New post   18 May 2017, 11:27
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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..

Answer is B
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Re: If p and q are positive integers, and p < q, then which of [#permalink] New post   26 May 2017, 14:26
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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

Answer:
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B


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Re: If p and q are positive integers, and p < q, then which of [#permalink] New post   18 May 2017, 08:50
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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
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If p and q are positive integers, and p < q, then which of [#permalink] New post   Updated on: 14 Jul 2017, 00:48
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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
Answer B.

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.
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If p and q are positive integers, and p < q, then which of [#permalink] New post   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.
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Re: If p and q are positive integers, and p < q, then which of [#permalink] New post   11 Jul 2017, 04:25
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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?
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If p and q are positive integers, and p < q, then which of [#permalink] New post   12 Jul 2017, 16:34
Expert Reply
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
Answer B.


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 ...?
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Re: If p and q are positive integers, and p < q, then which of [#permalink] New post   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
Answer B.


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)
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Re: If p and q are positive integers, and p < q, then which of [#permalink] New post   30 Oct 2023, 23:02
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.
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Re: If p and q are positive integers, and p < q, then which of [#permalink]
30 Oct 2023, 23:02
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