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

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

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New post 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



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

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New post 18 Dec 2017, 20:49
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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.
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Screen Shot 2017-12-18 at 7.47.28 PM.png
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Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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

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New post 23 Dec 2017, 23:03
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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).
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Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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

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New post 24 Dec 2017, 00:12
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Hi niks18,

Sure, that makes sense--both strategies work.
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Re: List T consist of 30 positive decimals, none of which is an integer [#permalink]

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

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

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

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

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

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New post 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.
Re: List T consist of 30 positive decimals, none of which is an integer   [#permalink] 07 Apr 2018, 11:27

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