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?

I. At least one student had more than 60 correct answers.

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

III. At least one student had less than 40 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!

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