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# On Monday, 9 students each took a test with 100 questions. The average

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Intern
Joined: 14 Sep 2017
Posts: 34
Location: Italy
On Monday, 9 students each took a test with 100 questions. The average [#permalink]

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01 Mar 2018, 10:18
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On Monday, 9 students each took a test with 100 questions. The average (arithmetic mean) number of correct answers was 50, and the median number of correct answers was 40. With of the following statements must be true?

II. At least one student had more than 40 and less than 50 correct answers.

A) I only
B) II only
C) III only
D) I and III
E) II and III
[Reveal] Spoiler: OA
BSchool Forum Moderator
Joined: 26 Feb 2016
Posts: 2256
Location: India
GPA: 3.12
Re: On Monday, 9 students each took a test with 100 questions. The average [#permalink]

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01 Mar 2018, 12:29
Fedemaravilla wrote:
On Monday, 9 students each took a test with 100 questions. The average (arithmetic mean) number of correct answers was 50, and the median number of correct answers was 40. With of the following statements must be true?

II. At least one student had more than 40 and less than 50 correct answers.

A) I only
B) II only
C) III only
D) I and III
E) II and III

Imagine the score that the 9 students took in a test to be values of a set

Here, 40 is the median of the set and the mean(average) is 50.
The 5th element(middle element) is 40 - the median.
Therefore, the other 4 elements before the median take a minimum value of 40.

To offset this difference of 10(from the 5 elements of the set), at least one of the elements
in the set will have to greater than 60, in order to attain the average of 50!

Illustrating with the help of an example

40 40 40 40 40 60 60 60 x is one possibility for the set.

$$\frac{40*5 + 60*3 + x}{9} = 50$$ -> $$200 + 180 + x = 450$$ -> $$x = 450 - 380 = 70$$

Here, the minimum value that x will take is 70, making Option A(I only) correct.
This example also proves that case II and III are wrong!
_________________

Stay hungry, Stay foolish

Intern
Joined: 18 Jun 2017
Posts: 13
On Monday, 9 students each took a test with 100 questions. The average [#permalink]

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02 Mar 2018, 23:10
let us take one option at a time. Let none of the scores are more than 60. Then the maximum score that can be obtained will be 40+40+40+40+40+59+59+59+59 = 436. This is clearly not possible since the average is 50 (the sum must be 450). So, the option I is correct. This automatically eliminates option II but not option III. How about having 39+39+39+39+40+62+62+62+68? Option III does not cap the number of correct answers on the upper side. Am I missing something here?

pushpitkc wrote:
Fedemaravilla wrote:
On Monday, 9 students each took a test with 100 questions. The average (arithmetic mean) number of correct answers was 50, and the median number of correct answers was 40. With of the following statements must be true?

II. At least one student had more than 40 and less than 50 correct answers.

A) I only
B) II only
C) III only
D) I and III
E) II and III

Imagine the score that the 9 students took in a test to be values of a set

Here, 40 is the median of the set and the mean(average) is 50.
The 5th element(middle element) is 40 - the median.
Therefore, the other 4 elements before the median take a minimum value of 40.

To offset this difference of 10(from the 5 elements of the set), at least one of the elements
in the set will have to greater than 60, in order to attain the average of 50!

Illustrating with the help of an example

40 40 40 40 40 60 60 60 x is one possibility for the set.

$$\frac{40*5 + 60*3 + x}{9} = 50$$ -> $$200 + 180 + x = 450$$ -> $$x = 450 - 380 = 70$$

Here, the minimum value that x will take is 70, making Option A(I only) correct.
This example also proves that case II and III are wrong!
BSchool Forum Moderator
Joined: 26 Feb 2016
Posts: 2256
Location: India
GPA: 3.12
Re: On Monday, 9 students each took a test with 100 questions. The average [#permalink]

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03 Mar 2018, 00:29
sandeepkonda wrote:
let us take one option at a time. Let none of the scores are more than 60. Then the maximum score that can be obtained will be 40+40+40+40+40+59+59+59+59 = 436. This is clearly not possible since the average is 50 (the sum must be 450). So, the option I is correct. This automatically eliminates option II but not option III. How about having 39+39+39+39+40+62+62+62+68? Option III does not cap the number of correct answers on the upper side. Am I missing something here?

pushpitkc wrote:
Fedemaravilla wrote:
On Monday, 9 students each took a test with 100 questions. The average (arithmetic mean) number of correct answers was 50, and the median number of correct answers was 40. With of the following statements must be true?

II. At least one student had more than 40 and less than 50 correct answers.

A) I only
B) II only
C) III only
D) I and III
E) II and III

Imagine the score that the 9 students took in a test to be values of a set

Here, 40 is the median of the set and the mean(average) is 50.
The 5th element(middle element) is 40 - the median.
Therefore, the other 4 elements before the median take a minimum value of 40.

To offset this difference of 10(from the 5 elements of the set), at least one of the elements
in the set will have to greater than 60, in order to attain the average of 50!

Illustrating with the help of an example

40 40 40 40 40 60 60 60 x is one possibility for the set.

$$\frac{40*5 + 60*3 + x}{9} = 50$$ -> $$200 + 180 + x = 450$$ -> $$x = 450 - 380 = 70$$

Here, the minimum value that x will take is 70, making Option A(I only) correct.
This example also proves that case II and III are wrong!

Hey sandeepkonda

If you read the question which reads "With of the following statements must be true"

As already written in the previously mentioned case, it is not always necessary that the statement III is always right
If we have 40 40 40 40 40 60 60 60 60 70, the average is 50 and the median is 40, but none of the students

Hope this helps you!
_________________

Stay hungry, Stay foolish

Intern
Joined: 18 Jun 2017
Posts: 13
Re: On Monday, 9 students each took a test with 100 questions. The average [#permalink]

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03 Mar 2018, 00:41
Thanks!!! I missed the "must be true" part of the question.

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Re: On Monday, 9 students each took a test with 100 questions. The average   [#permalink] 03 Mar 2018, 00:41
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