Which of the following is always true?

(I) The average (arithmetic mean) of five negative consecutive integers is equal to the average of the largest and smallest of them.

(II) The average (arithmetic mean) of four positive consecutive integers is equal to the average of the largest and smallest of them.

(III) The average (arithmetic mean) of five consecutive integers is equal to the average of the largest and smallest of them.

A. I and III
B. I, II and III
C. II
D. I and II
E. None

Hi lheiannie07,
Let's take a look at your question.

We will take some random values to check each statement.

Let's start from statement (I).
Let 5 negative consecutive integers be -1, -2, -3, -4, -5.
Average of five numbers = (-1-2-3-4-5)/5 = -15/5 = -3
Average of largest and smallest number = (-1-5) / 2 = -6/2 =-3
It means Statement (I) is correct.

We will now check statement (II).
Let 4 positive consecutive integers be 1, 2, 3, 4.
Average of four numbers = (1+2+3+4)/4 = 10/4 = 2.5
Average of largest and smallest number = (1+4) / 2 = 5/2 =2.5
Hence, statement (II) is also true.

Now check statement (III).
Let 5 consecutive integers be -1, 0, 1, 2, 3.
Average of five numbers = (-1+0+1+2+3)/5 = 5/5 = 1
Average of largest and smallest number = (-1+3) / 2 = 2/2 =1
Hence, statement (III) is also true.

Therefore, option B is correct, because statement I, II and III are true.

Hope it helps.
I am available if you'd like any follow up.