Data Sufficiency

[Math Revolution GMAT math practice question]

What is the median of 10 numbers?

1) 6 of the ten numbers are less than or equal to 10.
2) 6 of the ten numbers are greater than or equal to 10.

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

What is the median of 10 numbers?

1) 6 of the ten numbers are less than or equal to 10.
2) 6 of the ten numbers are greater than or equal to 10.

Statement 1:
Case 1: X, X, X, X, 10, 10, X, X, X, X
Here, the 6 blue values are all less than or equal to 10.
In this case, the median = (10+10)/2 = 10
Case 2: X, X, X, X, 8, 10, X, X, X, X
Here, the 6 red values are all less than or equal to 10.
In this case, the median = (8+10)/2 = 9.
Since the median can be different values, INSUFFICIENT.

Statement 2:
Case 1 also satisfies Statement 2.
Case 1: X, X, X, X, 10, 10, X, X, X, X
Here, the 6 blue values are all greater than or equal to 10.
In this case, the median = 10.
Case 3: X, X, X, X, 10, 12, X, X, X, X
Here, the 6 red values are all greater than or equal to 10.
In this case, the median = (10+12)/2 = 11.
Since the median can be different values, INSUFFICIENT.

Statements 1 and 2:
Only Case 1 satisfies both statements.
In Case 1, the median = 10.
SUFFICIENT.

The correct answer is C.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Let x1, x2, ... , x10 be the 10 numbers, written in increasing order: x1 ≤ x2 ≤ ... ≤ x10.
The median of 10 numbers is the average of the 5th number, x5 and the 6th number, x6.

By condition 1), x5 ≤ x6 ≤ 10, and by condition 2), 10 ≤ x5 ≤ x6.
Thus, x5 = x6 = 10.
The median is ( x5 + x6 ) / 2 = 10.
Both conditions together are sufficient.

Therefore, the answer is C.

Answer: C