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

It is currently 20 Sep 2018, 08:13

LIVE NOW:

Q&A Session on Executive MBA Apps - Learn More about EMBA Programs, Ask your Questions to Expert. Join HERE


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

If (5/4)^(-n) < 16^(-1). What is the least integer value of n?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 49271
If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 01 Aug 2017, 01:06
4
1
21
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

32% (02:36) correct 68% (02:29) wrong based on 225 sessions

HideShow timer Statistics

Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 6795
If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 01 Aug 2017, 06:07
3
4
Bunuel wrote:
If \((\frac{5}{4})^{-n} < 16^{-1}\). What is the least integer value of n?

A. 12
B. 13
C. 14
D. 15
E. 16


if you counter such Q, an approximation way would be..

\((\frac{5}{4})^{-n} < 16^{-1}..........(\frac{4*2}{5*2})^n<\frac{1}{16}......2^{3n}*16<10^n....2^{3n+4}<10^n\)

now\(2^{10}=1024\) and \(10^3 = 1000\) ..nearly equal so lets convert ..

If you could get some even higher close values, the answer will be even closer
\(1024^{\frac{3n+4}{10}}<1000^{\frac{n}{3}}\)...

so \(\frac{3n+4}{10}<\frac{n}{3}........9n+12<10n....n>12\)
next integer value after 12 is 13....
so ans 13
B
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

General Discussion
Director
Director
User avatar
P
Joined: 13 Mar 2017
Posts: 619
Location: India
Concentration: General Management, Entrepreneurship
GPA: 3.8
WE: Engineering (Energy and Utilities)
If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 03 Sep 2017, 11:39
chetan2u wrote:
Bunuel wrote:
If \((\frac{5}{4})^{-n} < 16^{-1}\). What is the least integer value of n?

A. 12
B. 13
C. 14
D. 15
E. 16


if you counter such Q, an approximation way would be..

\((\frac{5}{4})^{-n} < 16^{-1}..........(\frac{4*2}{5*2})^n<\frac{1}{16}......2^{3n}*16<10^n....2^{3n+4}<10^n\)

now\(2^{10}=1024\) and \(10^3 = 1000\) ..nearly equal so lets convert ..

If you could get some even higher close values, the answer will be even closer
\(1024^{\frac{3n+4}{10}}<1000^{\frac{n}{3}}\)...

so \(\frac{3n+4}{10}<\frac{n}{3}........9n+12<10n....n>12\)
next integer value after 12 is 13....
so ans 13
B


The options are so close, How can one say that on taking 1024 ~ 1000 , one will get correct answer..
one might get n>13 or n>11 instead of n>12 in some other question of this type....

We need to find some other approach..
_________________

CAT 99th percentiler : VA 97.27 | DI-LR 96.84 | QA 98.04 | OA 98.95
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".

Intern
Intern
avatar
B
Joined: 16 Apr 2017
Posts: 18
Re: If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 25 Sep 2017, 09:54
I would say taking logs on both sides would be better.

On taking logs, we get

(n+2)*log4 < n*log 5

=> (n+2)/n < log5/log4

Now log5 = 1-log2 = 1-.301 =.699
log4 = 2*log2 =0.602

=> (n+2)/n = 0.699/0.602

Putting n=12, we know that the inequality is not satisfied. Because LHS = 7/6 whereas RHS is less than 7/6 (0.699/0.602 is less than 0.7/0.6). Hence next integral value of n is correct
PS Forum Moderator
avatar
D
Joined: 25 Feb 2013
Posts: 1212
Location: India
GPA: 3.82
GMAT ToolKit User Premium Member Reviews Badge
Re: If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 25 Sep 2017, 11:02
Bunuel wrote:
If \((\frac{5}{4})^{-n} < 16^{-1}\). What is the least integer value of n?

A. 12
B. 13
C. 14
D. 15
E. 16


The question stem can be written as \((\frac{4}{5})^n < \frac{1}{16}\)

Now we know that \(16 = 2^4\) so if we can make \((\frac{4}{5})\) some approximate form of \(2\), then our job might become easier.

\((\frac{4}{5})^3 = \frac{64}{125}\) this is slightly than greater \(\frac{1}{2}\)

Hence \((\frac{4}{5})^3 = \frac{1}{2}\),

and \((\frac{1}{2})^4 = \frac{1}{16}\)

Substitute the value of \(\frac{1}{2}\) to get \((\frac{4}{5})^{12}\) but this will be greater than \(\frac{1}{16}\).

Hence to make it lower we will have to multiply \((\frac{4}{5})^{12}\) by a lower value \(\frac{4}{5}\)

Therefore \((\frac{4}{5})^{13}<\frac{1}{16}\)

So \(n=13\)

Option B

P.S: This is a very tough question and probably beyond the scope of GMAT. Unless you have solved similar questions, it will be very difficult to solve it in real GMAT within 2 minutes.

Bunuel Do you have repository of similar kind of questions?
Manager
Manager
avatar
B
Joined: 24 Jun 2017
Posts: 122
Re: If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 25 Sep 2017, 11:11
One method (gmat style :))
(5/4)^(−n)<16^(−1)
(4/5)^n<1/2^4
so left side should be less than 1/2

we raise to cube
(4/5)^3=64/125 and multiply right side by 64 just to compare, makes 64/128, so the inequality is still not satisfied but just a bit, it's clear that if multiply the left side by 4/5 one more time the inequality will be respected
But as we have 4 power on the right 1/2^4
then for the left
(4/5)^12 *4/5 which is 13
Manager
Manager
avatar
B
Joined: 24 Jun 2017
Posts: 122
Re: If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 25 Sep 2017, 14:05
1
And another approach for math geeks, solving through logs, I always recommend to use ln (natural log) just memorize 6 numbers by heart (ln for 2 3 5 6 7 and 10) you will cove most of that kind of questions https://gmatclub.com/forum/which-of-the-following-is-the-largest-163644.html#p1930561
(5/4)^(−n)<16^(−1)
ln (4/5)^(n)< ln 2^(−4)
as per log property
n *ln (4/5)< -4 * ln 2
another property
n *(ln 4 - ln5) < -4 * ln 2
n* (1.4 - 1.6) < - 4 * 0.7
n * (-0.2) < - 2.8
then
n < - 2.8/ -0.2 =>> n < 14 so 13
Manager
Manager
avatar
S
Joined: 04 Apr 2015
Posts: 185
Reviews Badge
Re: If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 03 Oct 2017, 21:07
mikemcgarry can you please explain this Magoosh question?
Magoosh GMAT Instructor
User avatar
G
Joined: 28 Dec 2011
Posts: 4667
Re: If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 05 Oct 2017, 09:20
StrugglingGmat2910 wrote:
mikemcgarry can you please explain this Magoosh question?

Dear StrugglingGmat2910,

I'm happy to respond. :-)

My friend, this was posted as a Magoosh question, but I don't believe it's one that we still have in our Product. This is a very hard question, one that is probably best solved with math that is completely beyond the GMAT. I am going to say that you don't need to worry about this question at all for the GMAT.

Does this make sense?
Mike :-)
_________________

Mike McGarry
Magoosh Test Prep

Image

Image

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Manager
Manager
avatar
S
Joined: 04 Apr 2015
Posts: 185
Reviews Badge
Re: If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 05 Oct 2017, 09:27
mikemcgarry wrote:
StrugglingGmat2910 wrote:
mikemcgarry can you please explain this Magoosh question?

Dear StrugglingGmat2910,

I'm happy to respond. :-)

My friend, this was posted as a Magoosh question, but I don't believe it's one that we still have in our Product. This is a very hard question, one that is probably best solved with math that is completely beyond the GMAT. I am going to say that you don't need to worry about this question at all for the GMAT.

Does this make sense?
Mike :-)

thank you sir for your reply
Senior Manager
Senior Manager
avatar
G
Joined: 02 Apr 2014
Posts: 477
GMAT 1: 700 Q50 V34
If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 07 Jan 2018, 19:20
\((4/5)^n < 1/16\)

as both 4/5 and 1/16 are positive

we take 4th root on both sides

\((4/5)^{n/4} < 1/2\)

let \(k = n/4\)

when k = 3, \((4/5)^3\) = \(64/125\) is slightly greater than 1/2,

so least value of n for which \((4/5)^n < 1/16\) is n = 3(4) + 1 = 13
Intern
Intern
User avatar
B
Joined: 07 Jun 2018
Posts: 20
Location: India
Concentration: Marketing, International Business
GPA: 3.9
WE: Marketing (Insurance)
GMAT ToolKit User
If (5/4)^(-n) < 16^(-1). What is the least integer value of n?  [#permalink]

Show Tags

New post 14 Sep 2018, 09:04
Bunuel wrote:
If \((\frac{5}{4})^{-n} < 16^{-1}\). What is the least integer value of n?

A. 12
B. 13
C. 14
D. 15
E. 16


\((\frac{5}{4})^{-n} < 16^{-1}\)
=> A = \((\frac{4}{5})^n*2^4 < 1\)

For n = 0, A =16; n = 1, A = 16*0.8.
So, as the value of n increases, A decreases and as n is increased by 1, A decreases by 80 %. We need to look for the value of n for which A >1 and A*80% less than 1.

As the power of 2 is 4, I was looking to test multiple of 4 for n while solving. So, I decided to first test with n = 12; from the list of option. As it will bit simplify A.
For n = 12, A = \((\frac{4}{5})^{12}*2^4\) = \((\frac{4^3}{5^3})^4*2^4\) = \((\frac{64}{125})^4*2^4\)
= \((\frac{128}{125})^4\)
A will be slightly greater than 1. But, definitely 80% of A will be just greater than 0.8. So, if n=13, value of A will be around 0.8 and it is the least integer value for which the inequality satisfies.
Hence, B. I might be a bit lucky. :cool: as the question was definitely not an easy one to crack.
_________________

Give a Kudo, If I deserve it.



Target 750+
  • 17th June 2018 - Diagnostic Test - Kaplan Premier 2015 - 570
  • 10th Aug 2018 - Veritas 1 - 650 (Q51,V29)
  • 17th Aug 2018 - Veritas 2 - 670 (Q49,V33)
  • 24th Aug 2018 - Veritas 3 - 650 (Q49,V30)
  • 1st Sep 2018 - Veritas 4 - 680 (Q51,V33)
  • 12th Aug 2018 - Veritas 5 - 690 (Q51,V34)

What do you think? Will I achieve my target score?

If (5/4)^(-n) < 16^(-1). What is the least integer value of n? &nbs [#permalink] 14 Sep 2018, 09:04
Display posts from previous: Sort by

If (5/4)^(-n) < 16^(-1). What is the least integer value of n?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  

Events & Promotions

PREV
NEXT


GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.