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# Operations on Rational Numbers

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Intern
Joined: 27 Jul 2016
Posts: 1

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09 Apr 2017, 04:40
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This a GMAT OG question which i didn't understand.
List T Consists of 30 positive decimals, none of which is an integer, and the sum of 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 upto the nearest integer, and each decimal in T whose tenths digit is odd is rounded down to the nearest integer;E is the sum of the resulting integers. if 1/3 of the decimals in T have 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,III
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4888
GPA: 3.82

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09 Apr 2017, 19:24
pramodmallya1 wrote:
This a GMAT OG question which i didn't understand.
List T Consists of 30 positive decimals, none of which is an integer, and the sum of 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 upto the nearest integer, and each decimal in T whose tenths digit is odd is rounded down to the nearest integer;E is the sum of the resulting integers. if 1/3 of the decimals in T have 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,III

Let $$T = \{ a_1, a_2, ... , a_{30} \}$$ and $$a_1 = n_1 + h_1, ... , a_{30} = n_{30} + h_{30}$$ where $$n_i$$'s are integers and $$0 < h_i < 1$$ for $$i \in \{1,...,30\}$$.
Assume that $$a_1, ... , a_{10}$$ have even tenths digits and $$a_{11}, ... , a_{30}$$ have odd tenths digits.

Then we have followings.
$$S = a_1 + ... + a_{30} = n_1 + ... + n_{30} + h_1 + ... + h_{30}$$.
$$E = ( n_1 - 1 ) + ... + ( n_{10} - 1 ) + ( n_{11} + 1 ) + ... + ( n_{30} - 1 ) = n_1 + ... + n_{30} - 10$$.
$$E - S = n_1 + ... + n_{30} - 10 - ( n_1 + ... + n_30 ) - ( h_1 + ... + h_{30} ) = -10 - ( h_1 + ... + h_{30} )$$

Since $$0 < h_i < 1$$ for all $$i \in \{ 1, ... , 30 \}$$, we have $$0 < h_1 + ... + h_{30} < 30$$.
Thus $$-40 < -10 - (h_1 + ... + h_{30}) < -10$$.

There among the statements, only I is an answer.

Therefore, the correct choice is A.
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Joined: 02 Sep 2009
Posts: 43828

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12 Apr 2017, 06:49
pramodmallya1 wrote:
This a GMAT OG question which i didn't understand.
List T Consists of 30 positive decimals, none of which is an integer, and the sum of 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 upto the nearest integer, and each decimal in T whose tenths digit is odd is rounded down to the nearest integer;E is the sum of the resulting integers. if 1/3 of the decimals in T have 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,III

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 43828
Re: Operations on Rational Numbers [#permalink]

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12 Apr 2017, 06:50
pramodmallya1 wrote:
This a GMAT OG question which i didn't understand.
List T Consists of 30 positive decimals, none of which is an integer, and the sum of 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 upto the nearest integer, and each decimal in T whose tenths digit is odd is rounded down to the nearest integer;E is the sum of the resulting integers. if 1/3 of the decimals in T have 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,III

_________________
Re: Operations on Rational Numbers   [#permalink] 12 Apr 2017, 06:50
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