Question

What is the average of the integers 1, 2, 3, 4, and \(x\)? Statement (I): \(x\) is an even integer. Statement (II): \(x<10\)

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

For Data Sufficiency questions, always ask: “Can I obtain one unique answer?” If more than one value satisfies the statements, the information is insufficient.

Answer 1

The Correct Option is D

Concept: The average of a set of numbers is obtained by dividing the sum of all observations by the number of observations. \[ \text{Average}=\frac{\text{Sum of observations}}{\text{Number of observations}} \] In Data Sufficiency questions, the objective is not necessarily to compute the exact answer immediately, but to determine whether the given statements provide enough information to uniquely determine the answer.

Step 1: Form the required expression for the average. The five integers are: \[ 1,\;2,\;3,\;4,\;x \] Their sum is \[ 1+2+3+4+x=10+x \] Hence the average is \[ \frac{10+x}{5}. \] Therefore, finding the average requires knowing the exact value of \(x\).

Step 2: Analyze Statement (I). Statement (I) says: \[ x \text{ is an even integer}. \] Possible values include \[ 2,\;4,\;6,\;8,\;10,\;12,\ldots \] Different values of \(x\) give different averages. For example, \[ x=2 \Rightarrow \text{Average}=\frac{12}{5} \] and \[ x=8 \Rightarrow \text{Average}=\frac{18}{5}. \] Since multiple averages are possible, Statement (I) alone is not sufficient.

Step 3: Analyze Statement (II). Statement (II) says \[ x<10. \] Possible values are numerous. For example, \[ x=1,\;2,\;3,\ldots,9. \] Again, different values produce different averages. Hence Statement (II) alone is not sufficient.

Step 4: Analyze both statements together. Combining: \[ x \text{ is even and } x<10. \] Possible values are \[ 2,\;4,\;6,\;8. \] Corresponding averages are \[ \frac{12}{5},\; \frac{14}{5},\; \frac{16}{5},\; \frac{18}{5}. \] Since more than one average is possible, even the two statements together do not determine a unique answer. Therefore, the data is insufficient.