5, 9, 13, x, 19, 21, 1 The median of these seven integers is the sam

Question

5, 9, 13, x, 19, 21, 1

The median of these seven integers is the same as the mean. What is the sum of all the possible values of x?

A. 9

B. 17

C. 18

D. 24

E. 28

rajtare · Accepted Answer

Three scenario exist here:
a) When x is the smaller than median
x,1,5,9,13,19,21
mean=median=9
thus,  \frac{x+68}{7}  = 9
or x+68=63
or x=-5
b) When x is the middle term

1,5,9,x,13,19,21
mean=median=x
 \frac{x+68}{7}  = x
6x=68
x=68/7

x cant be in the middle as x will take only integer values

c) when x is larger than median
in this case median=mean=13
1,5,9,13,19,21,x
 \frac{x+68}{7}  = 13
x+68=91
x=23

Thus sum of possible values of x = 23-5 =18

hence correct answer should be C

gmatophobia · Answer

Visualization is key to solving this question.

Number of integers = 7.

So let's visualize 7 places as shown below

_ _ _    _    _ _ _

The median is the dash in red.

Now, lets see which number / numbers can be the median.

Say if 1 is the median, the data set should contain

three numbers which are less than or equal to 1 : These numbers will lie on the left of 1 (i.e. median) 
 three numbers which are greater than or equal to 1 : These numbers will lie to the right of 1 (i.e. median)

However, the data set already has 5 numbers which are greater than 1, so  1 cannot be the median .

Similarly  5 cannot be the median  by the same reasoning.

Now lets see if 9 can be the median

_ _ _    9    _ _ _

If 9 is the median, 1 and 5 will lie to the left of 9, and 13, 19 and 21 will lie to the right of y. So if, we can place x to the left of y, 9 could possibly be the median.

We also know that 9 is the arithmetic mean of the data set.

So  \frac{9+5+13+x+19+21+1}{7} = 9

\frac{68 + x }{7} = 9

x = -5 ; well this is valid, because -5 lies to the left of 9.

Similarly for 13 : The three places to the left of 13 are taken by 1, 5 and 9, so x should be  \geq{13}

_ _ _    13    _ _ _

\frac{68 + x }{7} = 13

x = 23

This is also valid.

Now  19 and 21 cannot be the median  (same reason for which we eliminated 1 and 5).

So the only possible values of x = {-5 , 23 }

Sum = 23 - 5 = 18

Option C

btsaami · Answer

Avg = (68+x)/7
Simplify it, Avg = 9 + (x+5)/7
Min value of avg will be 9 when x = -5.
Additionally, when x=-5 median =9 which fulfills the condition median=avg

a) Consider x<=9
Next value of x can -5+7 = 2, avg = 9+1=10 but 10 cannot be the median when x=2 hence it fails.
Next Avg = 9+(9+5) = 11; 11 is not a median. Hence rejected.

b) 9<x<=13
No value gives integer avg which will be equal to median.

c)13<x<=21
Integer avg can be found when x=16 but it cannot be the median.

d) x>21
Avg = 9 + 23+5/7 = 9+4 = 13 when arranging the numbers in ascending order we find that median = 13. Hence for value of x=23, median = avg.

From above we have sum of x which satisfies median=avg is 23-5 =18. Hence, C

5, 9, 13, x, 19, 21, 1 The median of these seven integers is the sam

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5, 9, 13, x, 19, 21, 1 The median of these seven integers is the sam

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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Perfect 805 Score Debrief by Julia

My Rewards