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

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BTGmoderatorLU: Moderator; Posts: 2558; Joined: Sun Oct 15, 2017 1:50 pmFollowed by: 6 members

by BTGmoderatorLU » Thu Jun 13, 2019 6:26 pm

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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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Source: Beat The GMAT — Problem Solving | Thu Jun 13, 2019 6:26 pm

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri Jun 14, 2019 4:33 am

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

Brent Hanneson - Creator of GMATPrepNow.com

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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Fri Jun 14, 2019 1:27 pm

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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Scott@TargetTestPrep: GMAT Instructor; Posts: 8155; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Fri Jun 21, 2019 6:50 pm

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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