Practice GMAT Data Sufficiency Question

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What is the value of x?

The average (arithmetic mean) of 5, x², 2, 10x, and 3 is -3.
The median of 109, -32, -30, 208, -15, x, 10, -43, 7 is -5.

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: D

Evaluate Statement (1) alone.
1. Based upon the formula for the average, you know that:
  (5 + x² + 2 + 10x + 3)/5 = -3
  x² + 10x + 5 + 2 + 3 = -15
  x² + 10x + 5 + 2 + 3 + 15 = 0
  x² + 10x + 25 = 0
  (x + 5)² = 0
  x = -5
2. Statement (1) alone is SUFFICIENT.
Evaluate Statement (2) alone.
1. Order the numbers in ascending order without x:
  -43, -32, -30, -15, 10, 7, 109, 208
2. Consider the possible placements for x and whether these would make the median equal to -5:
  Case (1): x, -43, -32, -30, -15, 10, 7, 109, 208
  Median: -15
  Not a possible case since the median is not -5.
  
  Case (2): -43, x, -32, -30, -15, 10, 7, 109, 208
  Median: -15
  Not a possible case since the median is not -5.
  
  Case (3): -43, -32, x, -30, -15, 10, 7, 109, 208
  Median: -15
  Not a possible case since the median is not -5.
  
  Case (4): -43, -32, -30, x, -15, 10, 7, 109, 208
  Median: -15
  Not a possible case since the median is not -5.
  
  Case (5): -43, -32, -30, -15, x, 10, 7, 109, 208
  Median: x
  A possible case since the median is x, which can legally be -5.
  In this case, x must be -5 in order for the median of the set to be -5, which must be according to Statement (2).
  
  Case (6): -43, -32, -30, -15, 10, x, 7, 109, 208
  Median: 10
  Not a possible case since the median is not -5.
  
  Case (7): -43, -32, -30, -15, 10, 7, x, 109, 208
  Median: 10
  Not a possible case since the median is not -5.
  
  Case (8): -43, -32, -30, -15, 10, 7, 109, x, 208
  Median: 10
  Not a possible case since the median is not -5.
  
  Case (9): -43, -32, -30, -15, 10, 7, 109, 208, x
  Median: 10
  Not a possible case since the median is not -5.
3. Since Statement (2) tells us that the median must be -5, we know that x must be a value such that the median is -5. This can only happen in Case 5. Specifically, it can only happen when x = -5. Since the median must equal -5 and this can only happen when x = -5, we know that x = -5.
4. Statement (2) alone is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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