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8, 5, x, 6 The median of the list of positive integers

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8, 5, x, 6 The median of the list of positive integers

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Source: Manhattan Prep

\(8, 5, x, 6\)

The median of the list of positive integers above is 5.5. Which of the following could be the average (arithmetic mean) of the list?

A. 3
B. 5.5
C. 6.25
D. 7
E. 7.5

The OA is B

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BTGmoderatorLU wrote:
Source: Manhattan Prep

\(8, 5, x, 6\)

The median of the list of positive integers above is 5.5. Which of the following could be the average (arithmetic mean) of the list?

A. 3
B. 5.5
C. 6.25
D. 7
E. 7.5

The OA is B
Since there is an even number of values in the list, the median will be the AVERAGE of the two middlemost values (when all values are listed in ASCENDING order)

Let's the answer choices...

A) If x = 3, then the values are {3, 5, 6, 8}, which means the median = (5 + 6)/2 = 5.5
NO GOOD. We want the median to be greater than 5.5
ELIMINATE A

B) If x = 5.5, then the values are {5, 5.5, 6, 8}, which means the median = (5.5 + 6)/2 = 5.75
Perfect! The median IS greater than 5.5

Answer: B

Cheers,
Brent

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Median = 5.5, find the mean
$$Median\ =\ \frac{\left(5+x\right)}{2}$$
Median = 5,5 ; $$x\ must\ be\ \le5$$
$$list\ =\ x,\ 5,\ 6,\ 8$$
$$median\ =\ \frac{\left(5+6\right)}{2}=5.5$$
$$x=\left\{5,\ 4,\ 3,\ 2,\ or\ 1\right\}$$
If x = 5
$$mean\ =\ \frac{\left(5+5+6+8\right)}{4}=\frac{24}{4}=6\ \left\{wrong\right\}$$
if x = 4
$$mean\ =\frac{\left(4+5+6+8\right)}{4}=\frac{23}{4}=5.75\ \left\{wrong\right\}$$
if x = 3
$$mean\ =\frac{\left(3+5+6+8\right)}{4}=\frac{22}{4}=5.5$$

$$5.5\ is\ in\ Option\ B$$

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BTGmoderatorLU wrote:
Source: Manhattan Prep

\(8, 5, x, 6\)

The median of the list of positive integers above is 5.5. Which of the following could be the average (arithmetic mean) of the list?

A. 3
B. 5.5
C. 6.25
D. 7
E. 7.5

The OA is B
Let’s analyze the given choices.

A) 3

If the average is 3, then we have:

(8 + 5 + x + 6)/4 = 3

19 + x = 12

x = -7

We see that the list {-7, 5, 6, 8} does have a median of 5.5, however, the problem says all the integers in the list are positive. So x can’t be -7, which means the average can’t be 3 either.

B) 5.5

If the average is 5.5, then we have:

(8 + 5 + x + 6)/4 = 5.5

19 + x = 22

x = 3

We see that the list {3, 5, 6, 8} does have a median of 5.5 and all the integers in the list are positive. So x can be 3, which means the average can be 5.5 also.

Answer: B

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Scott Woodbury-Stewart
Founder and CEO
scott@targettestprep.com



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