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

 It is currently 18 Aug 2018, 09:17

### 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

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

# If S is the sum of the reciprocals of the consecutive

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

### Hide Tags

Manager
Joined: 10 Dec 2005
Posts: 93
If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

23 Dec 2005, 02:19
8
44
00:00

Difficulty:

55% (hard)

Question Stats:

59% (00:55) correct 41% (00:53) wrong based on 1302 sessions

### HideShow timer Statistics

If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

_________________

JAI HIND!

Math Expert
Joined: 02 Sep 2009
Posts: 47983
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

25 Nov 2013, 08:43
32
39
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

Given that $$S=\frac{1}{91}+\frac{1}{92}+\frac{1}{93}+\frac{1}{94}+\frac{1}{95}+\frac{1}{96}+\frac{1}{97}+\frac{1}{98}+\frac{1}{99}+\frac{1}{100}$$. Notice that 1/91 is the larges term and 1/100 is the smallest term.

If all 10 terms were equal to 1/91, then the sum would be 10/91, but since actual sum is less than that, then we have that S<1/91.

If all 10 terms were equal to 1/100, then the sum would be 10/100=1/10, but since actual sum is more than that, then we have that S>1/10.

Therefore, 1/10 < S < 10/91.

Also, notice that 10/91 < 1/9 < 1/8, thus we have that 1/10 < S < 10/91 < 1/9 < 1/8.

Therefore only 1/10 is less than S.

Similar questions to practice:
m-is-the-sum-of-the-reciprocals-of-the-consecutive-integers-143703.html
if-k-is-the-sum-of-reciprocals-of-the-consecutive-integers-145365.html
_________________
Manager
Joined: 12 Nov 2005
Posts: 78

### Show Tags

23 Dec 2005, 23:44
18
4
This question is a repetition here.
Anyways lemme explain the solution

1/91+1/92+..........1/100 > 1/100+1/100......10 times
or Summation S >10/100
i.e S>1/10

Similarly
1/91+1/92+..........1/100 < 1/91+1/91......10 times
Summation < 10/91 <1/9

Hence summation is only greater than 1/10.........Hence C
##### General Discussion
Director
Joined: 17 Dec 2005
Posts: 536
Location: Germany

### Show Tags

23 Dec 2005, 02:47
13
1
I think it's C, but I'm not quite sure.

Since we summarize the reciprocals from 100 to 91, we can say also that we add ten numbers who are all (with one exception 1/100) greater than 1/100, so that the sum must be greater than 1/10.

On the other side we can say that we add the reciprocals from 91 to 100, so that the sum has to be less than the sum of ten times 1/91.

We can conclude that the sum has to be less than 1/9 but more than 1/10. That leaves us C as the only possible answer.
VP
Joined: 06 Jun 2004
Posts: 1029
Location: CA
Re: PS: Summation  [#permalink]

### Show Tags

23 Dec 2005, 03:14
3
2
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

S = 1/91 + 1/92 + 1/93.....+ 1/100
S = (1/91 + 1/100)(10/2) = 191/1820 = approx 0.1049.....

I. 1/8 = 0.12...
II. 1/9 = 0.11....
III. 1/10 = 0.10
_________________

Don't be afraid to take a flying leap of faith.. If you risk nothing, than you gain nothing...

Senior Manager
Joined: 15 Apr 2005
Posts: 411
Location: India, Chennai
Re: PS: Summation  [#permalink]

### Show Tags

23 Dec 2005, 05:26
10
1
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

From the answer choice we get 1) 1/8 = .125
2) 1/9 = .11 (Approx)
3) 1/10 = .1

1/100 = .01 The other numbers from 1/91 to 1/99 will also be close to .01 but little greater than that . so approx .01 * 10 = .1 So its C.
Intern
Status: I'm trying to GMAT?
Joined: 12 Feb 2013
Posts: 24
Location: United States
Concentration: Finance, General Management
GMAT Date: 06-22-2013
WE: Engineering (Consulting)
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

10 Jan 2014, 11:40
2
1
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

We know that there are 10 numbers in the sum: (100-91)+1=10
Take the mean of the sum and times it by 10 to get our sum: (1/ ((100+91)/2)) x 10 = 10/95.5 = 1/9.55

From here we know that the only number which will be smaller than our sum must be divisible by >9.55

Hence only III satisfies. Answer: C
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12189
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

21 Feb 2015, 23:50
2
Hi All,

The other explanations in this thread have properly explained the "math" behind this prompt - it's essentially about figuring out the "minimum" and "maximum" value that the sum COULD be, then realizing the sum is between those two values. Any time you find yourself reading a Quant question and you think "the math will take forever", then you're probably right AND there should be another way to get to the correct answer. The Quant section of the GMAT is NOT a "math test" (at least not in the way that you might be used to thinking about it). Yes, you will do plenty of small calculations and use formulas, but the Quant section is there to test you on LOTS of other non-math related skills: organization, accuracy, attention to detail, ability to prove that you're correct, pattern-matching, pacing, etc. To maximize your performance on Test Day, you have to be a stronger 'strategist' and 'pattern-matcher' than 'mathematician.'

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ ***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*********************** Manager Joined: 22 Jan 2014 Posts: 172 WE: Project Management (Computer Hardware) Re: If S is the sum of the reciprocals of the consecutive [#permalink] ### Show Tags 22 Mar 2015, 12:10 JAI HIND wrote: If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S? I. 1/8 II. 1/9 III. 1/10 A. None B. I only C. III only D. II and III only E. I, II, and III S = 1/91 + 1/92 + ... + 1/100 or S > 1/100 + 1/100 +... + 1/100 S > 1/10 or S < 10/91 S < 0.109 => 0.1 < S < 0.109 hence C. _________________ Illegitimi non carborundum. Intern Joined: 31 Oct 2015 Posts: 33 Re: If S is the sum of the reciprocals of the consecutive [#permalink] ### Show Tags 20 Jan 2016, 18:30 1 See attachment for response. Attachments SmartSelectImage_2016-01-20-20-29-20.png.png [ 56.1 KiB | Viewed 24125 times ] Retired Moderator Status: I Declare War!!! Joined: 02 Apr 2014 Posts: 242 Location: United States Concentration: Finance, Economics GMAT Date: 03-18-2015 WE: Asset Management (Investment Banking) Re: If S is the sum of the reciprocals of the consecutive [#permalink] ### Show Tags 17 Jul 2016, 04:27 hey! callig all experts... bunnuel / chetan4u / egmat etc to help in solving this with some less math richi do help plz EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12189 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: If S is the sum of the reciprocals of the consecutive [#permalink] ### Show Tags 18 Jul 2016, 12:10 1 Hi Celestial09, Based on the wording of the prompt, you might think that you should add up the fractions 1/91 + 1/92 + .... 1/100, but the GMAT would NEVER require that you do that math. Instead, lets do some real basic estimation of what that sum would be LESS than and GREATER than.... There are 10 total fractions and 9 of them are GREATER than 1/100. So, at the 'lower end', let's just say that all 10 fractions are equal to 1/100.... (10)(1/100) = 10/100 = 1/10 Thus, we know that the sum of those 10 fractions will be GREATER than 1/10. Similarly, we know that all 10 of those fractions are LESS than 1/90. So, at the 'higher end', let's just say that all 10 fractions are equal to 1/90... (10)(1/90) = 10/90 = 1/9 Thus, we know that the sum of those 10 fractions will be LESS than 1/9. With those two deductions, there's only one answer that 'fits'... Final Answer: GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

VP
Joined: 07 Dec 2014
Posts: 1069
If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

18 Jul 2016, 14:55
1
1/91+1/100=.021
.021/2=.011
10*.011=.11
only 1/10<.11
C. III only
Manager
Joined: 26 Mar 2016
Posts: 77
Location: Greece
GMAT 1: 710 Q51 V34
GPA: 2.9
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

18 Jul 2016, 16:41
1
1/91 + 1/92 +...+ 1/100..
I took 1/95 as the median of the sequence (approximately, the real median is 1/95.5).
So 10*1/95= approximately 1/9.5.
So the only answer possible is C.
It took me like 15 seconds for this question approaching the problem this way.
_________________

+1 Kudos if you like the post

Senior Manager
Joined: 15 Sep 2011
Posts: 342
Location: United States
WE: Corporate Finance (Manufacturing)
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

18 Jul 2016, 18:10
Top Contributor
C. The sum of the reciprocals is 10/955, which is derived from the sum of consecutive integers formula n(n+1)/2, and that number is less than 1/8 and 1/9, both. Tested via cross multiplication.

Sent from my iPhone using Tapatalk
Intern
Joined: 23 Jul 2016
Posts: 7
Location: India
GMAT 1: 620 Q49 V26
WE: Medicine and Health (Health Care)
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

01 Apr 2017, 01:02
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

we can solve this problem using the Harmonic progression rule

(highest value)(# of terms) > (sum of all terms in the HP) > (lowest value)(# of terms)
=> (1/91)(10) > sum of terms > (1/100)(10)
therefore, (1/8) > (1/9) > (1/9.1) > sum of the HP > (1/10)

Manager
Joined: 23 Dec 2013
Posts: 203
Location: United States (CA)
GMAT 1: 710 Q45 V41
GMAT 2: 760 Q49 V44
GPA: 3.76
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

22 Jul 2017, 18:17
1
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

This problem is a max/min one in disguise. We know that 1/91 > 1/100. So if all of the numbers were 1/91, then the upper bound for addition would be 10/91

10/91 < 1/9 = 10/90 because the denominator is larger. So 10/91 < 1/9. So we know that S, the sum, is less than 1/9. 1/9 is smaller than 1/8, so I. and II. are out.

Lastly, we know that the smallest possible sum is 10*1/100 = 10/100 = 1/10 = III. We know that the actual sum is larger than 1/10, so 1/10<S<10/91<1/9<1/8. C is the right answer.
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 3176
Location: United States (CA)
Re: If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

11 Dec 2017, 07:29
1
JAI HIND wrote:
If S is the sum of the reciprocals of the consecutive integers from 91 to 100, inclusive, which of the following is less than S?

I. 1/8
II. 1/9
III. 1/10

A. None
B. I only
C. III only
D. II and III only
E. I, II, and III

Let's first analyze the question. We are trying to find a potential range for S, and S is equal to the sum of the reciprocals from 91 to 100, inclusive. Thus, S is:

1/91 + 1/92 + 1/93 + …+ 1/100

The easiest way to determine the RANGE of S is to use easy numbers that can be quickly manipulated.

Note that 1/90 is greater than each of the addends and that 1/100 is less than or equal to each of the addends. Therefore, instead of trying to add together 1/91 + 1/92 + 1/93 + …+ 1/100, we are instead going to first add 1/90 ten times and then add 1/100 ten times. These two sums will give us a high estimate of S and a low estimate of S, respectively. Again, we are adding 1/90 and then 1/100, ten times, because there are 10 numbers from 1/91 to 1/100, inclusive.

Instead of actually adding each of these values ten times, we will simply multiply each value by 10:

1/100 x 10 = 1/10

1/90 x 10 = 1/9

We see that S is between 1/10 and 1/9, i.e., 1/10 < S < 1/9. Of the three numbers given in the Roman numerals, only 1/10 is less than S.

_________________

Scott Woodbury-Stewart
Founder and CEO

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Manager
Joined: 01 Feb 2017
Posts: 155
If S is the sum of the reciprocals of the consecutive  [#permalink]

### Show Tags

19 Dec 2017, 06:57
We need to apply Minimax concept in this question.

Sum of this Series= 1/91 + 1/92....1/100

To Maximize the sum of this series, we must consider the lowest possible denominator as common to all terms= 91
Hence, Sum Maximized= 1/91 + 1/91.....1/91= 10/91= 1/9.1

To Minimize the sum of this series, we must consider the highest possible denominator as common to all terms= 100
Hence, Sum Minimized= 1/100 + 1/100.....1/100= 10/100= 1/10

Hence, 1/9.1 >Sum of series >1/10

Therefore, Only option III matches with above range.

So, correct Ans choice: C
If S is the sum of the reciprocals of the consecutive &nbs [#permalink] 19 Dec 2017, 06:57
Display posts from previous: Sort by

# If S is the sum of the reciprocals of the consecutive

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

# Events & Promotions

 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®.