List $T$ consist of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is $S.$ The

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 amFollowed by: 3 members

by VJesus12 » Fri Sep 04, 2020 5:46 am

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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

Answer: B

Source: Official Guide

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Source: Beat The GMAT — Problem Solving | Fri Sep 04, 2020 5:46 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

Re: List $T$ consist of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is $S.$ Th

by Brent@GMATPrepNow » Tue Jul 06, 2021 6:09 am

VJesus12 wrote: ↑
Fri Sep 04, 2020 5:46 am
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

Answer: B

Source: Official Guide

Given: 10 of the values must have an EVEN tenths digit, and the remaining 20 values must have an ODD tenths digit.

Let's try to determine the MAXIMUM value of E - S and the MINIMUM value of E - S
Once we do that, we'll know the range of possible values of E - S

IMPORTANT: To make things easier, let's consider decimals in the form 0.something

MAXIMUM value of E - S
In order to MAXIMIZE the value of E - S, we must MINIMIZE the value of S
We can do this by making the decimals with an ODD tenths digit 0.1, and by making the decimals with an EVEN tenths digit 0.01
So List T consists of twenty 0.1's and ten 0.01's
This means S = 20(0.1) + 10(0.01) = 2.1

When we complete our ROUNDING, we get: twenty 0's and ten 1's
So, E = 20(0) + 10(1) = 10

So, the MAXIMUM value of E - S = 10 - 2.1 = 7.9

MINIMUM value of E - S
In order to MINIMIZE the value of E - S, we must MAXIMIZE the value of S
We can do this by making the decimals with an ODD tenths digit 0.99999..., and by making the decimals with an EVEN tenths digit 0.89999...
So List T consists of twenty 0.9999....'s and ten 0.89999....'s
This means S ≈ 20(1) + 10(0.9) ≈ 29
ASIDE: We need not get super crazy about how many 9's we add to our decimals. Let's just look for an APPROXIMATE value.

When we complete our ROUNDING, we get: twenty 0's and ten 1's
So, E = 20(0) + 10(1) = 10

So, the MINIMUM value of E - S = 10 - 29 = -19

So, the value of E - S can range from (approximately) -19 to 7.9

Since -16 and 6 fall within this range, the correct answer is B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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swerve: Legendary Member; Posts: 2568; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

Re: List $T$ consist of 30 positive decimals, none of which is an integer, and the sum of the 30 decimals is $S.$ Th

by swerve » Wed Jul 07, 2021 6:24 am

VJesus12 wrote: ↑
Fri Sep 04, 2020 5:46 am
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

Answer: B

Source: Official Guide

We have whole numbers in the answer choices, so we don't need to go through all the $3,0000001$ cases. We limit here to the tenth!

Round Up -Even
Max: $3,2 \Rightarrow 4$ so we get $0,8\cdot 10=8$
Min: $3,8 \Rightarrow 4$ so we get $0,2\cdot 10=2$

Round down - Odd
Max: $3,9 \Rightarrow 3$ so we get $-0,9\cdot 20=-18$
Min: $3,1 \Rightarrow 3$ so we get $-0,1\cdot 20=-2$

Now we can manipulate those numbers:
I. $-16 \Rightarrow -18 + 2 = -16$ OK $\color{green}\checkmark$
II. $6 \Rightarrow -2 + 8 = 6$ OK $\color{green}\checkmark$
III. $10 X$ You can not get $10$ because the max positive Value is $8.$

Therefore, the correct answer is B

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