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Which of the following is always true?

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by Vincen » Wed Feb 21, 2018 2:19 am
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

The OA is the option B.

How can I show that all three statements are true? I am interested in knowing how can I solve this PS question. Experts, I wait for your help.
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by Jay@ManhattanReview » Fri Mar 09, 2018 12:29 am
Vincen wrote: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

The OA is the option B.

How can I show that all three statements are true? I am interested in knowing how can I solve this PS question. Experts, I wait for your help.
Let us first take statement (III). If (III) is correct, then (I) and (II) are true as they are the sub-sets of (III).

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

Say the five consecutive integers are -1, 0, 1, 2, & 3. The average (arithmetic mean) of five consecutive integers = 1 and the average of the largest and smallest = (-1 + 3)/2 = 1. (III) is true.

Thus, all the statements are true.

You may take few examples to get this better. For example, {-2, -1, 0, 1, & 2}; All positive: {3, 4, 5, 6, 7, & 8}; All negative: {-10, -9, -8, -7, & -6}.

The correct answer: B

Hope this helps!

-Jay
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by Jay@ManhattanReview » Fri Mar 09, 2018 12:35 am
Vincen wrote: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

The OA is the option B.

How can I show that all three statements are true? I am interested in knowing how can I solve this PS question. Experts, I wait for your help.
Let's take an Algebraic route to understand this.

Say the five consecutive integers are x, (x + 1), (x + 2), (x + 3), & (x + 4), where they can be negative, 0, and positive.

The average (arithmetic mean) of five consecutive integers = [x + (x + 1) + (x + 2) + (x + 3) + (x + 4)]/5 = (5x + 10)/5 = x + 2

The average of the largest and smallest integers = [x + (x + 4)]/2 = (2x + 4)/2 = x + 2

We see that the results are equal.

This is a generic case and is applicable to all the three statements. Thus, all the three statements are true.

The correct answer: B

Hope this helps!

-Jay
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