Last visit was: 24 Jul 2024, 12:45 It is currently 24 Jul 2024, 12:45
Close
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
Your Progress

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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Date
User avatar
Manager
Manager
Joined: 25 Aug 2011
Posts: 116
Own Kudos [?]: 1526 [247]
Given Kudos: 11
Location: India
GMAT 1: 730 Q49 V40
WE:Operations (Insurance)
Send PM
Most Helpful Reply
Math Expert
Joined: 02 Sep 2009
Posts: 94609
Own Kudos [?]: 643615 [98]
Given Kudos: 86737
Send PM
Tutor
Joined: 16 Oct 2010
Posts: 15148
Own Kudos [?]: 66847 [7]
Given Kudos: 436
Location: Pune, India
Send PM
General Discussion
User avatar
Manager
Manager
Joined: 25 Aug 2011
Posts: 116
Own Kudos [?]: 1526 [0]
Given Kudos: 11
Location: India
GMAT 1: 730 Q49 V40
WE:Operations (Insurance)
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
So the best strategy is to pick numbers in these ques?
Math Expert
Joined: 02 Sep 2009
Posts: 94609
Own Kudos [?]: 643615 [6]
Given Kudos: 86737
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
5
Kudos
1
Bookmarks
Expert Reply
devinawilliam83 wrote:
So the best strategy is to pick numbers in these ques?


Number picking is a good strategy for such kind of questions, though as you can see we proved that I is true with algebra (so to prove that a statement MUST be true you might need an algebraic or logical/conceptual approach).

Generally it really depends on the problem to pick the way of handling it. Check the link in my previous post to see bunch of Must or Could be True Questions solved using different approaches.
User avatar
Manager
Manager
Joined: 07 Sep 2010
Posts: 222
Own Kudos [?]: 5366 [0]
Given Kudos: 136
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
Hello Experts,

Can we evaluate the value of rs by below method

Given:
0<r<1 ---(1)
1<s<2 ---(2)

It can be seen that both r and s are positives, hence multiplying r and s will not have any effect on inequality

Hence,
Multiplying 1 and 2
0<rs<2 ---(3)

Am I right in concluding equation (3). Please clarify.

Thanks
Manager
Manager
Joined: 29 Aug 2013
Posts: 51
Own Kudos [?]: 168 [5]
Given Kudos: 26
Location: United States
Concentration: Finance, International Business
GMAT 1: 590 Q41 V29
GMAT 2: 540 Q44 V20
GPA: 3.5
WE:Programming (Computer Software)
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
5
Kudos
imhimanshu wrote:
Hello Experts,

Can we evaluate the value of rs by below method

Given:
0<r<1 ---(1)
1<s<2 ---(2)

It can be seen that both r and s are positives, hence multiplying r and s will not have any effect on inequality

Hence,
Multiplying 1 and 2
0<rs<2 ---(3)

Am I right in concluding equation (3). Please clarify.

Thanks


Though I am not an expert, the answer to your question is Yes,
And, in fact we can solve all the equations in I, II, III by this method.

If 0<r<1 and 1<s<2
then 0<r/s<1 ----- I) True

Again as you suggested 0<rs<2 --- II) Hence False since it can take values more than 1 as well.

For III) multiply r inequality by -1
we get -1 < -r < 0 and 1<s<2
adding the inequalities

0 < s-r < 2 ------ Again this inequality can take value more than 1 hence False..

Consider Kudos if it helped :)
avatar
Intern
Intern
Joined: 21 Apr 2016
Posts: 22
Own Kudos [?]: 20 [4]
Given Kudos: 11
Location: United States
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
4
Kudos
Curious if this can be solved this way:

Given: 0<r<1<s<2

1. r/s <1
= r<s (yes)

2. rs<1
= r<1/s (no)

3. s-r<1
= s<r (no)

So only 1 must be true?

Bunuel wrote:
If 0 < r < 1 < s < 2, which of the following must be less than 1?
I. r/s
II. rs
III. s - r

A. I only
B. II only
C. III only
D. I and II
E. I and III

Notice that we are asked "which of the following MUST be lees than 1, not COULD be less than 1. For such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

Given: \(0 < r < s\) --> divide by \(s\) (we can safely do this since we know that \(s>0\)) --> \(\frac{r}{s}<1\), so I must be true;

II. rs: if \(r=\frac{9}{10}<1\) and \(1<(s=\frac{10}{9})<2\) then \(rs=1\), so this statement is not alway true;

III. s-r: if \(r=0.5<1\) and 1\(<(s=1.5)<2\) then \(s-r=1\), so this statement is not alway true.

Answer: A (I only).

Check our new Must or Could be True Questions to practice more: search.php?search_id=tag&tag_id=193

Hope it helps.
Manager
Manager
Joined: 24 Feb 2017
Status:wake up with a purpose
Posts: 173
Own Kudos [?]: 397 [2]
Given Kudos: 114
Location: Bangladesh
Concentration: Accounting, Entrepreneurship
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
1
Kudos
1
Bookmarks
Here, you have to think about extreme values like:
r can be 0.001, or 0.999 and
s can be 1.001 or 1.99 etc.
Intern
Intern
Joined: 29 Mar 2017
Posts: 1
Own Kudos [?]: 0 [0]
Given Kudos: 8
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
Bunuel

Since 0<r<1 and 1<s<2 can we multiply the inequalities to prove (II)?

(1)*(0)<R*S<(1)*(2)

0<RS<2

Thanks
GMATWhiz Representative
Joined: 07 May 2019
Posts: 3401
Own Kudos [?]: 1860 [1]
Given Kudos: 68
Location: India
GMAT 1: 740 Q50 V41
GMAT 2: 760 Q51 V40
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
1
Bookmarks
Expert Reply
Since this a MUST be question, we use the constraints to set up cases, disprove the statements and eliminate the answer options.

0 < r < 1 < s < 2.

Let r = ½ and s = \(\frac{3}{2}\).

Clearly, s- r =\( \frac{3}{2}\) – ½ = 1. So, we found a case where s - r is not less than 1. Statement III does not represent an expression that is always less than 1.

Eliminate answer options containing statement III viz., option C and option E.

Let r = \(\frac{3}{5}\) and s = \(\frac{5}{3}\).

Clearly, r * s = \(\frac{3}{5}\) * \(\frac{5}{3}\) = 1. Statement II does not represent an expression that is always less than 1.

Eliminate answer options containing statement II viz., option B and option D.

The answer option left out i.e. option A MUST be true.
The correct answer option is A.
Manager
Manager
Joined: 12 Oct 2023
Posts: 103
Own Kudos [?]: 79 [0]
Given Kudos: 134
Send PM
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
The best strategy here is to use limits or approximation for r and s:

r/s will have highest value when r is highest , s is smallest ( and still r will be less than 1 and s greater than 1 ) hence 0 < max( r/s ) < 1

rs max when r max and s max ( r can be a little less than 1, s a little less than 2 ) hence 2 > max( rs ) > 1

s - r max when s max and r min ( r can be little more than 0 , s a little less than 2 ) hence 0 < max(r - s) < 2

Only r/s will be always less than 1
GMAT Club Bot
Re: If 0 < r < 1 < s < 2. Which of the following must be less than 1. [#permalink]
Moderator:
Math Expert
94609 posts