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The average (arithmetic mean) of numbers is given by the formula

We are dealing with four numbers: a, b, c and d. This tells us the number of values is 4. Therefore, if we have the sum of a, b, c, and d, we can compute the average as

What is a + b + c + d equal to?

Statement 1:

The average of a, b and c is 6

There is no information about d, so there is no way to determine the sum a + b + c + d. 

We can still write this information as an equation if we like.

6 = 

 a + b + c = 18 ...... (I)

This alone is not sufficient to answer the question “what is the average (arithmetic mean) of a, b, c and d?”.

Statement 2:

The average of b, c and d is 6

There is no information about a, so there is no way to determine the sum a + b + c + d.

We can still write this information as an equation if we like.

6 = 

 b + c + d = 18 ...... (II)

This alone is not sufficient to answer the question “what is the average (arithmetic mean) of a, b, c and d?”.

Statements 1 & 2:

We can consider the information from both statements in equation form:

a + b + c = 18……(I)

b + c + d = 18……(II)

Subtracting equation (II) from equation (I) gives

(a + b + c) − (b + c + d) = 18 − 18

Or, a + b + c − b − c − d = 0

Or, a − d = 0

Or, a = d

But that is not enough to allow us to calculate a single correct sum a + b + c + d (or by extension, a single correct average).

We can prove this through counterexamples as well. We can pick sets of numbers for a, b, c, and d that satisfy both statements but provide different average values. For example,

Statement 1 and 2 TOGETHER are NOT sufficient to answer the question.

Answer: (E)