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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

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

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18 Dec 2017, 19:49

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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. _

Attachments

Screen Shot 2017-12-18 at 7.47.28 PM.png [ 203.95 KiB | Viewed 144 times ]

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

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23 Dec 2017, 21:49

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)

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

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23 Dec 2017, 22:03

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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 [#permalink]

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23 Dec 2017, 22:45

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).

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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GPA: 4

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

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07 Jan 2018, 11:43

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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