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# List T consist of 30 positive decimals, none of which is an integer

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Veritas Prep GMAT Instructor
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Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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15 Dec 2017, 00:35
shamanth25 wrote:
List T consist of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is S. The estimated sum of the 30 decimals, E, is defined as follows. Each decimal in T whose tenths digit is even is rounded up to the nearest integer, and each decimal in T whose tenths digits is odd is rounded down to the nearest integer. If 1/3 of the decimals in T have a tenths digit that is even, which of the following is a possible value of E - S ?

I. -16
II. 6
III. 10

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

Check out our video solution to this very tricky problem:
https://www.veritasprep.com/gmat-soluti ... olving_224
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Karishma
Veritas Prep | GMAT Instructor
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Director Status: Professional GMAT Tutor Affiliations: AB, cum laude, Harvard University (Class of '02) Joined: 10 Jul 2015 Posts: 646 Location: United States (CA) Age: 38 GMAT 1: 770 Q47 V48 GMAT 2: 730 Q44 V47 GMAT 3: 750 Q50 V42 GRE 1: 337 Q168 V169 WE: Education (Education) Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 18 Dec 2017, 20:49 Top Contributor Attached is a visual that should help. To keep it simple, I kept all my positive decimals between 0 and 1; since the question refers to the relative distance between the actual (S) and estimated (E) sets of sums, making them larger doesn't have any effect on the range of E - S. _ Attachments Screen Shot 2017-12-18 at 7.47.28 PM.png [ 203.95 KiB | Viewed 490 times ] _________________ Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching, both in-person (San Diego, CA, USA) and online worldwide, since 2002. One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong). You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y8zh6qby Date of Birth: 09 December 1979. GMAT Action Plan and Free E-Book - McElroy Tutoring PS Forum Moderator Joined: 25 Feb 2013 Posts: 1142 Location: India GPA: 3.82 Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 23 Dec 2017, 22:49 shamanth25 wrote: List T consist of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is S. The estimated sum of the 30 decimals, E, is defined as follows. Each decimal in T whose tenths digit is even is rounded up to the nearest integer, and each decimal in T whose tenths digits is odd is rounded down to the nearest integer. If 1/3 of the decimals in T have a tenths digit that is even, which of the following is a possible value of E - S ? I. -16 II. 6 III. 10 A. I only B. I and II only C. I and III only D. II and III only E. I, II, and III it is clear that in T 10 decimals have even tenths digit and 20 decimals have odd tenths digit. S is the sum of numbers in T Now let's try to find E-S(max) and E-S(min) to get the ranges. Here S is constant only E will change Now E-S(max) will occur when there will be a maximum increment in E. This will happen when Even decimals are rounded up the max and odd decimals when rounded down has the least impact So lowest tenth digit Even decimals could be 0 for eg. 2.01 when rounded up becomes 3, an increment of 0.99 or 1 to be approx Hence increase from even decimals = 1*10=10 points Odd decimals has to be 0.1, when rounded down becomes 0, a decrease of -0.1 Hence decrease from Odd decimals = -0.1*20=-2 Hence Net change i.e $$E-S(max)=10-2=8$$ Thus 6 is a possibility (if you take even 0.2 and odd as 0.1, you will get exact 6) and as 10>8 so 10 is not possible II holds true Now E-S(min) will be when even decimals increment is as low as possible and odd decimals are deceased the most So Even decimals has to be 0.899999 when rounded up becomes 1, an increase of 0.100001 or apprx 0.1 Hence increase from even decimals = 0.1*10=1 points Odd decimals has to be 0.99999, when rounded down becomes 0, a decrease of -0.999999 or approx -1 Hence decrease from Odd decimals = -1*20=-20 Hence Net change i.e $$E-S(min)=1-20=-19$$ so -16 is possible (if you take even to be 0.8 & odd to be 0.9, you will get exact -16) (I holds true) Hence Answer is B Director Status: Professional GMAT Tutor Affiliations: AB, cum laude, Harvard University (Class of '02) Joined: 10 Jul 2015 Posts: 646 Location: United States (CA) Age: 38 GMAT 1: 770 Q47 V48 GMAT 2: 730 Q44 V47 GMAT 3: 750 Q50 V42 GRE 1: 337 Q168 V169 WE: Education (Education) Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 23 Dec 2017, 23:03 Top Contributor niks18 wrote: So Even decimals has to be 0.2 when rounded up becomes 1, an increment of 0.8 Hence increase from even decimals = 0.8*10=8 points The even decimals rounded up can actually equal (nearly) 10, not just 8, because 0.01 also has an even tenths digit (don't forget that zero, not 2, is the smallest even digit). _________________ Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching, both in-person (San Diego, CA, USA) and online worldwide, since 2002. One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong). You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y8zh6qby Date of Birth: 09 December 1979. GMAT Action Plan and Free E-Book - McElroy Tutoring PS Forum Moderator Joined: 25 Feb 2013 Posts: 1142 Location: India GPA: 3.82 Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 23 Dec 2017, 23:45 mcelroytutoring wrote: niks18 wrote: So Even decimals has to be 0.2 when rounded up becomes 1, an increment of 0.8 Hence increase from even decimals = 0.8*10=8 points The even decimals rounded up can actually equal (nearly) 10, not just 8, because 0.01 also has an even tenths digit (don't forget that zero, not 2, is the smallest even digit). Hi mcelroytutoring Yes 0.001 will have even tenths digit. actually the intention of my solution was to arrive at exact 6 & -16 to prove the option B is correct as this is a could be true question. I guess I should re-word the solution to avoid ambiguity. Thanks for highlighting. Director Status: Professional GMAT Tutor Affiliations: AB, cum laude, Harvard University (Class of '02) Joined: 10 Jul 2015 Posts: 646 Location: United States (CA) Age: 38 GMAT 1: 770 Q47 V48 GMAT 2: 730 Q44 V47 GMAT 3: 750 Q50 V42 GRE 1: 337 Q168 V169 WE: Education (Education) Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 24 Dec 2017, 00:12 Top Contributor Hi niks18, Sure, that makes sense--both strategies work. _________________ Harvard grad and 99% GMAT scorer, offering expert, private GMAT tutoring and coaching, both in-person (San Diego, CA, USA) and online worldwide, since 2002. One of the only known humans to have taken the GMAT 5 times and scored in the 700s every time (700, 710, 730, 750, 770), including verified section scores of Q50 / V47, as well as personal bests of 8/8 IR (2 times), 6/6 AWA (4 times), 50/51Q and 48/51V (1 question wrong). You can download my official test-taker score report (all scores within the last 5 years) directly from the Pearson Vue website: https://tinyurl.com/y8zh6qby Date of Birth: 09 December 1979. GMAT Action Plan and Free E-Book - McElroy Tutoring Intern Joined: 18 Jan 2017 Posts: 49 Location: India Concentration: General Management, Entrepreneurship GPA: 4 Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 07 Jan 2018, 12:43 1 1 Total 30. the value of E: 1/3 of the decimals in T have a tenths digit that is even so 10 numbers have an even tenths digit 20 numbers have an odd tenths digit. now for any decimal with a even tenths digits would be 0.2,.04,0.6, 0.8, 1.2, 1.4, 1.6, 1.8 and so on! for evens we round up! so they'll become 1,1,1,1,2,2,2 and so on! now for any decimal with a odd tenths digits would be 0.1,0.3,0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 1.7 and so on! for odds we round down! so they'll become 0,0,0,0,0,1,1,1,1 and so on! For E : E = (sum of all 30 integer parts) +10(1)-20(1) =(sum of all 30 integer parts)-10 or simply 10! (if you consider all decimals are between 0 and 1 then sum of evens = 10(1) and sum of odds become 20(0) Now for "S": The maximum possible value of S occurs when ten numbers have '8' as tenths digit and remaining 20 numbers have '9' as tenths digit. Smax = (sum of all 30 integer parts) +10(0.8)+20(0.9) = (sum of all 30 integer parts)+26 or simply .8(10)+0.9(20) = 26 The minimum possible value of S occurs when ten numbers have '2' as tenth digit and remaining 20 numbers have '1' as tenth digit. Smin = (sum of all 30 integer parts) +10(0.2)+20(0.1) = (sum of all 30 integer parts)+4 or simple 0.2(10)+0.1(20) = 4 MAX S - E= 26-10 =16 MIN S-E = 4-10= -6 Intern Joined: 18 Sep 2017 Posts: 3 Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 21 Feb 2018, 10:13 I can assume all of the 30 decimals are in the form of 0. something. In this case, E = 10, S could never be 0 as all numbers are positive. And S could be, 10 * 0,2 + 20 * 0.1 as a minimum. So E-S = 6. And S could be, 10 * 0,8 + 20 * 0,9. So E-S = -16 Intern Joined: 14 Dec 2016 Posts: 11 Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 04 Mar 2018, 03:28 VeritasPrepKarishma wrote: shamanth25 wrote: List T consist of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is S. The estimated sum of the 30 decimals, E, is defined as follows. Each decimal in T whose tenths digit is even is rounded up to the nearest integer, and each decimal in T whose tenths digits is odd is rounded down to the nearest integer. If 1/3 of the decimals in T have a tenths digit that is even, which of the following is a possible value of E - S ? I. -16 II. 6 III. 10 A. I only B. I and II only C. I and III only D. II and III only E. I, II, and III what is the best way to solve this question. many thanks S This is how I would solve it: Even tenth digit - Round up - 10 numbers Odd tenth digit - Round down - 20 numbers E - S can take many values so how do we figure which ones it cannot take? We need to find the range of E - S - the minimum value it can take and the maximum value it can take. Minimum value of E - S => E is much less than S. How do we make E much less than S? By doing 2 things: 1. When I round up, the difference between actual and estimate should be little. Say the numbers are something like 3.8999999 (very close to 3.9) and they will be rounded up to 4 i.e. the estimate gains 0.1 per number. Since there are 10 even tenth digit numbers, the estimate will be apprx .1*10 = 1 more than actual 2. When I round down, the difference between actual and estimate should be huge. Say the numbers are something like 3.999999 (very close to 4) and they will be rounded down to 3 i.e. the estimate loses apprx 1 per number. Since there are 20 such numbers, the estimate is 1*20 = 20 less than actual. Overall, the estimate will be apprx 20 - 1 = 19 less than actual E - S = -19 Maximum value of E - S => E is much greater than S. How do we make E much greater than S? By doing 2 things: 1. When we round up, the difference between actual and estimate should be very high. Say the numbers are something like 3.000001 (very close to 3) and they will be rounded up to 4 i.e. the estimate gains 1 per number. Since there are 10 even tenth digit numbers, the estimate will be apprx 1*10 = 10 more than actual 2. When we round down, the difference between actual and estimate should be very little. Say the numbers are 3.1. They will be rounded down to 3 i.e. the estimate loses apprx 0.1 per number. Since there are 20 such numbers, the estimate is 0.1*20 = 2 less than actual. Maximum value of E - S = 10 - 2 = 8 So 10 cannot be the value of E - S. 1. When we round up, the difference between actual and estimate should be very high. Say the numbers are something like 3.000001 (very close to 3) and they will be rounded up to 4 i.e. the estimate gains 1 per number. Since there are 10 even tenth digit numbers, the estimate will be apprx 1*10 = 10 more than actua 0 is not a even number.. should'nt it be min 3.29999999 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8102 Location: Pune, India Re: List T consist of 30 positive decimals, none of which is an integer [#permalink] ### Show Tags 04 Mar 2018, 04:09 qazi11 wrote: 0 is not a even number.. should'nt it be min 3.29999999 0 is an even integer. It is neither negative nor positive but it is even. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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07 Apr 2018, 08:44
Easier than it looks like:

E = 10 --> 1/3 odd (rounded up) - 1*10 + 2/3 even (rounded down) 0*20

E-S:
I)-16 Possible -> E-S=10 - 10*0.6 - 20*0.5 = -16
II) 6 Possible -> E-S = 10 - 10*(0.2) - 20*0.1 = 6
III) Not possible -> E-S = 10 - a positive number = cannot be 10!!!

Hope it helps
Matt
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Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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07 Apr 2018, 11:27
Schawjibb wrote:
IMO, the best solution is shown by M Dabral of GMAT Quantum in one of its video explanations.

Here is the link: http://www.gmatquantum.com/og13/218-pro ... ition.html

Although NOT most of GMAT Quantum's video explanations/solutions are up to the par, I found that this one along with some others (e.g., PS 178) is the best video explanation floating out there in the internet.

thanks a lot for sharing the video explanation. Video explanations are a great resource for quicker preparation.
I wish all 700+ level gmatclub questions have video explanations to them.
Intern
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Posts: 1
Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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14 Jun 2018, 18:25
a: the sum of 30 integers
b: the sum of 30 decimals
So : E=a+10,0
S= a,b
E-S= 10 + (0-b)

b max: 08*10 + 0.9 *20 = 26
b min: 0.2*10 + 0.1 * 20 = 6

So the range of E-S = [-16,6]
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Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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17 Jun 2018, 14:25
I've read through most of the solutions here, but nobody has really stated the obvious rule that actually makes this problem solvable:

1. The laws of this problem are absolute. If you find any solution for any set of numbers, it applies to every possible combination of numbers. Therefore, if you find that E-S=10 for one set of example numbers, E-S=10 is a solution for the entire problem space. No exceptions.

2. Therefore, since #1 is true, choose the simplest set of example numbers you can.

That's why the solution is easiest by choosing the following numbers:
Round Up: 0.2, 0.4, 0.6, 0.8
Round Down: 0.1, 0.3, 0.5, 0.7, 0.9

Because whatever differences you find using the above numbers, will apply to every possible combination of other numbers! 1.1, 1000.1, 123413546.1 - it doesn't matter!

If you can understand that, now solve the problem.
Re: List T consist of 30 positive decimals, none of which is an integer   [#permalink] 17 Jun 2018, 14:25

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