For a certain set of 3 numbers, is the average(arithmetic mean) equal

For a certain set of 3 numbers, is the average (arithmetic mean) equal to the median?

Let the 3 numbers in ascending order be: a, b, and c.

The range of a set is the difference between the largest and the smallest numbers of a set, so Range=Largest-Smallest=c-a;
The median of a set with odd # of numbers is the middle number (when arranged in ascending or descending order):, so median=b.

Question: is (a+b+c)/3=b --> is a+c=2b?

(1) The range of the set is double the difference between the smallest number in the set and the middle number --> c-a=2(b-a) --> c+a=2b. Sufficient.

(2) The set consist of three numbers 10, 16 and 22 --> directly gives the numbers of the set so we can get the average as well as the median. Sufficient.

Answer: D.

Hope it's clear.

let three numbers be x,y,z. Also we know that for an evenly spaced numbers mean= median.

suppose x,y,z are evenly spaced i.e. differene between first and second, and second and third is same. thus we have
y-x=z-y
or 2y=x+z---------------------1)

let's consider st.1:
z-x=2(y-x)
z-x=2y-2x
2y=z+x
as can be seen from 1, thus these numbers are evenly spaced. hence mean=median. therefore statement 1 alone is sufficient.

st.2 again clearly sufficient. as value of x,y,z are given to us.

hence answer should be D

Hi All,

For anyone interested - a slightly tougher variation of this question appears in the new OG2016 (DS #135) - it's one of the *new* questions and it showcases the same logic.

GMAT assassins aren't born, they're made,
Rich

Say three numbers are X, Y , z in increasing order. Median = Y, hence range =Z-X ------equation(1)

take statement1: range =2(Y-X)----------equation(2)
From equation(1) and equation(2)

2(Y-X)= Z-X .... 2Y-2X=Z-X this gives Z=2Y-X ----------equation(3)
average = \(\frac{(X+Y+Z)}{3}\) .... put the value from equation 3
average = Y = Median

Hence sufficient
take statement2: median =16
\(average = \frac{(10+16+22)}{3}\) = \(\frac{48}{3}\) =16
Hence sufficient

Ans: D