If p, q, r, and s are consecutive integers, with p

Question

Tough and Tricky questions: Inequalities.

If  p ,  q ,  r , and  s  are consecutive integers, with  p \lt q \lt r \lt s , is  pr \lt qs ?

(1)  pq \lt rs

(2)  ps \lt qr

Kudos for a correct  solution .

Bunuel · Accepted Answer

Official Solution:

If  p ,  q ,  r , and  s  are consecutive integers, with  p \lt q \lt r \lt s , is  pr \lt qs ? 
 
We can solve this problem either by algebra or by number-plugging. Let's use algebra. All four variables can be expressed in terms of just one variable, since they are consecutive integers and we know their order. If we keep  p  as the basic variable, then  q = p + 1 ,  r = p + 2 , and  s = p + 3 .
 
Now we can rephrase the question:
 
Is  pr \lt qs ? 
 
Is  p(p + 2) \lt (p + 1)(p + 3) ?
 
Is  p^2 + 2p \lt p^2 + 4p + 3 ? 
 
Is  2p \lt 4p + 3 ?
 
Is  0 \lt 2p + 3 ?
 
Is  -3 \lt 2p ?
 
Is  -\frac{3}{2} \lt p ?
 
Since  p  is an integer, the question is answered "yes" if  p = -1  or greater, and "no" if  p = -2  or less. 
 
Statement (1): SUFFICIENT.
 
We rephrase the statement similarly.
  pq \lt rs 
  p(p + 1) \lt (p + 2)(p + 3) 
  p^2 + p \lt p^2 + 5p + 6 
  0 \lt 4p + 6 
  0 \lt 2p + 3 
  -\frac{3}{2} \lt p 
 
This is precisely the same condition as asked in the question. Thus, we can answer the question definitively.
 
Statement (2): INSUFFICIENT.
 
Again, we rephrase the statement similarly.
  ps \lt qr 
  p(p + 3) \lt (p + 1)(p + 2) 
  p^2 + 3p \lt p^2 + 3p + 2 
  0 \lt 2 
 
Since 0 is always less than 2, no matter the value of  p , the statement is always true. Thus, we do not gain any information that would help us answer the question.
 
Answer: A.

manpreetsingh86 · Answer

let p=k-1, q=k, r=k+1, s=k+2 now question states is pr-1/2 or the question becomes is k>-1/2 st.1 pq-2 or k>-1/2 hence statement 1 alone is sufficient st.2 ps

kinghyts · Answer

For the set of consecutive integers, p,q,r,s, we have find out whether pr < qs is true
Statement 1: pq < rs
This statement is true if 
 a) All the integers are positive. In that case, pr < qs is true
 b) When the numbers are p= -1, q= 0, r= 1, s= 2. In this case also, pr < qs is true
 
There are not other sets of consecutive integers that fulfills the condition of pq<rs. Since, from this condition we always get pr < qs as true, this statement is sufficient.

Statement 2 : ps < qr
Here are the cases:
 1. p= -1, q= 0, r= 1, s= 2 ( pr < qs is true ) 
 2. p= -2, q= -1, r= 0, s= 1 ( pr < qs is false) 
 3. p= -3, q= -2, r= -1, s= 0 ( pr < qs is false) 
 .......
Since, we get both true and false from this condition. This condition is insufficient.

Hence A) is the answer

amitshri15 · Answer

lets have consecutive numbers be: p, p+1,p+2,p+3

now pr < qs => p(p+2) < (p+1) (p+3) => p > -3/2

Statement1: p(p+1) < (p+2) (p+3) => p> -3/2 Sufficient

Statement2: p(p+3) < (p+1) (P+2) => which eliminates all P, so we can not conclude anything.

Hence answer is A.

akumar5 · Answer

Given that, p , q , r & s are consecutive integers & is pr<qs ?

SO solving pr<qs
 p(q+s)/2<s(p+r)/c ; since they are consecutive hence can be simplified to their mean.
 pq+ps<ps+sr
 hence, pq<rs.............A

1) pq<rs,
 This statement proves equation A. SUFFICIENT.

2) ps<qr
 There is no fruitful link with equation A.INSUFFICIENT.

Answer is A

mvictor · Answer

picked A...didn't think that much..
if the consecutive numbers start with negative values and reach positive values, then we can't know for sure anything...
1. clearly states that all the numbers are positive - sufficient.
2. not much offered.

A

HappyQuakka · Answer

I have a question about how to approach it at an actual exam.

The above explanations are very mathematical and accurate, but I don't think I will have time to really think it through like that.

My method of getting through this problem was
1. noticing that they could be negative or positive (all or in part)
2. plug in two sets of numbers to explore sufficiency of each option.

Am I putting myself at a risk of missing something crucial by plugging it in?

chetan2u · Answer

Hi..

for saving time, one should exactly do the way you have mentioned...

the MAIN point of these Q where consecutive integers are involved is  they could be positive or negative and plugging 2-3 sets of number should tell you..  
the center TWo would always be greater than product of !st and 4th, so statement II would not tell us whether the numbers are positive or negative..

a set of number in each positive integers and negative integers would tell you that statement I will give you a clearcut answer.

Having said that, it is not always that such method will work. If these were not given as consecutive integers, there would have been many options possible. there may have been some other info then in statements to get to your answer

hk111 · Answer

if I assume p=1, q=2, r=3, s=4 is it wrong?

bumpbot · Answer

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If p, q, r, and s are consecutive integers, with p < q < r < s, is pr

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