If the average of the numbers 18, 15, x, 22, 28 and 31 is 31, what is

Question

If the average of the numbers 18, 15, x, 22, 28 and 31 is 21, what is the value of x?

(A) 12
(B) 14 
(C) 17
(D) 16
(E) 18

AnujSheth · Answer

I dont understand, for the avg to be 31, x has to be greater than 31, and there is not option for that. Am I missing something?

GMATinsight · Answer

Average of {18, 15, x, 22, 28 and 31} = 21

CONCEPT : If the Average is added/subtracted in average of numbers then the average doesn't change
e.g. If you add/subtract 10 in  average of several numbers with average 10  then the new average will also be 10

i.e. AVerage of {18, 15, x, 22, 28, 31} = 21

i.e.  \frac{18+15+x+22+28+31}{6} = 21

i.e.  114+x = 21*6=126

i.e.  x = 12

Answer: Option A

Bunuel · Answer

_________________________
Fixed the issue. Thank you!

sujoykrdatta · Answer

Method 1:  
Average = Total of the terms ÷ number of terms

=> (18+15+x+22+28+31)/6 = 21
=> 114+x = 126
=> x = 12

Method 2:

If we compute the deviations of each of the terms about the average, the SUM of those deviations MUST be ZERO
(This is a very useful concept)

Let us use that:
Deviation of 18 about 21 = 18-21 = -3
Deviation of 15 about 21 = 15-21 = -6
Deviation of 22 about 21 = 22-21 = +1
Deviation of 28 about 21 = 28-21 = +7 
Deviation of 31 about 21 = 31-21 = +10

Sum of the above deviations = 9
Hence, deviation of the last number, x, must be -9
Thus, the number x = 21 + (-9) = 12

Answer A

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Archit3110 · Answer

sum of numbers = total numbers * avg
in this case we have
x= (21*6)-(18+15+22+28+31)
x=126-114
x=12

CrackverbalGMAT · Answer

Solution:

Difference of the numbers from the mean are (18-21) , (15-21), (22-21),(28-21),(31-21)

= -3 , -6 , +1 , +7 , +10

Sum of the differences = -9 +18 = +9

The sum of differences of the elements from the mean is always 0.

{For ex- If we have 1,2,3,4,5 and we replace every element with the mean of these 5 numbers that is 3 then the differences are -2,-1,0,1,2 and the sum of the differences is 0}

=> Thus x must be creating a difference of -9 from 21 in order to make the sum 0

=> x = 21-9

=12  (option a)

Devmitra Sen
GMAT SME

If the average of the numbers 18, 15, x, 22, 28 and 31 is 31, what is

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