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# The average (arithmetic mean) of the four distinct positive

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Joined: 06 Sep 2013
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The average (arithmetic mean) of the four distinct positive  [#permalink]

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01 Feb 2014, 15:19
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Difficulty:

75% (hard)

Question Stats:

55% (02:07) correct 45% (02:00) wrong based on 196 sessions

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The average (arithmetic mean) of the four distinct positive integers p, q, s and t is x. If p and q are odd, and s and t are even, which of the following must be true?

I. x is an integer
II. (x−0.5) is an integer.
III. 2x is an odd integer.

A. I only
B. II only
C. III only
D. II and III only
E. None of the above
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Re: The average (arithmetic mean) of the four distinct positive  [#permalink]

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02 Feb 2014, 17:25

Given :

p + q = EVEN (odd + odd =even)
p + q + s + t = EVEN (even + even = even) nothing more....

(p + q + s + t) / 4 = x

I.

x may or may not be integer, we only know sum of 4 variables is even not necessarily multiple of 4. I is out.

II.

Similarly, x an integer if sum of 4 variables is multiple of 4 but not an integer if it is not multiple of 4, we only know it is even. II is out.

III.

As in II, if X is integer 2x is Even, if X has decimal part (a.5), 2x is odd. III is out.

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Re: The average (arithmetic mean) of the four distinct positive  [#permalink]

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02 Feb 2014, 23:56
2
jlgdr wrote:
The average (arithmetic mean) of the four distinct positive integers p, q, s and t is x. If p and q are odd, and s and t are even, which of the following must be true?

I. x is an integer
II. (x−0.5) is an integer.
III. 2x is an odd integer.

A. I only
B. II only
C. III only
D. II and III only
E. None of the above

Using the information given to us:
p = 2a+1
q = 2b+1
s = 2c
t = 2d
$$x = \frac{p+q+s+t}{4} = \frac{2a + 2b + 2c + 2d + 2}{4} = \frac{2(a+b+c+d) + 2}{4} = \frac{a+b+c+d+1}{2}$$
Now, whether x is even or odd depends on the values of a, b, c and d. We have no restrictions on these as in they can be even or odd. If the sum (a+b+c+d) is odd, the numerator will become even and x will be an integer. If the sum (a+b+c+d) is even, then the numerator will become odd and x will be 0.5 more than an integer.
Hence, neither I nor II is essential.
III is not essential either since 2x will be odd only when the sum (a+b+c+d) is even.
So none of the three MUST be true.

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Re: The average (arithmetic mean) of the four distinct positive  [#permalink]

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25 Sep 2015, 13:09
1
Hi All,

This question can be solved by TESTing VALUES.

We're given a number of Facts to work with:
1) P, Q, S and T are all DISTINCT POSITIVE INTEGERS
2) P and Q are ODD
3) S and T are EVEN
4) The average of the 4 integers is X

We're asked which of the following MUST be TRUE...

Let's TEST....
P = 1
Q = 3
S = 2
T = 6

I. X is an integer

The average of the above 4 integers is 13/4 = 3.25
Roman Numeral 1 is NOT always true

II. (X - 0.5) is an integer

With this X (3.25), 3.25 - .5 = 2.75
Roman Numeral 2 is NOT always true

III. 2X is an ODD integer

With this X (3.25), 2X is 6.5, so it's not even an integer (much less an ODD one).
Roman Numeral 3 is NOT always true

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Re: The average (arithmetic mean) of the four distinct positive  [#permalink]

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04 Jan 2016, 19:06
I actually tried to solve by picking number:

I. x is an integer
p=1, q=3, s=2, q=4 => 1+3+2+4=10. 10/4 not integer.

II. (x−0.5) is an integer.
p=1, q=5, s=2, q=4. 1+5+2+4=12. 12/4=3. 3-0.5 - not integer.

III. 2x is an odd integer.
1+5+2+4=12/4=3. 3*2 = 6. not an odd integer.

E. None of the above
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The average (arithmetic mean) of the four distinct positive  [#permalink]

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05 Jan 2016, 05:53
jlgdr wrote:
The average (arithmetic mean) of the four distinct positive integers p, q, s and t is x. If p and q are odd, and s and t are even, which of the following must be true?

I. x is an integer
II. (x−0.5) is an integer.
III. 2x is an odd integer.

A. I only
B. II only
C. III only
D. II and III only
E. None of the above

E.

plugged in values and tried.

(1+3+2+4)/4 = 2.5 [I becomes false]

(3+5+4+8)/4 = 5 [II becomes false]

between the above 2 instances 2x is odd as well as even [III becomes false]
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Re: The average (arithmetic mean) of the four distinct positive  [#permalink]

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18 Dec 2016, 14:00
Great Question.

Here is what i did in this Question=>

p
q
s
t
are integers that are distinct and positive.
p=odd
q=odd
s=even
t=even

We can use test cases here -->

Statement 1=>
Lets think of cases where x is not an integer =>

3
5
8
10

Hence This statement is not always true.

Statement 2-->
Lets think of cases where x is an integer =>
3
5
8
16

Hence this statement is not always true.

Statement 3-->
For 2x to be odd x must be of the form -> x.5 for integer x

Suing test cases -->
3
5
8
16

2x will be even as mean is 8

Hence this statement is not always true.

Hence E

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Re: The average (arithmetic mean) of the four distinct positive  [#permalink]

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25 Mar 2019, 17:16
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Re: The average (arithmetic mean) of the four distinct positive   [#permalink] 25 Mar 2019, 17:16
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