Data Sufficiency

[GMAT math practice question]

If the median and average (arithmetic mean) of a set of 4 different numbers are both 10, what is the smallest number?

1) The range of the 4 numbers is 10
2) The sum of the smallest and the largest numbers is 20

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

If the median and average (arithmetic mean) of a set of 4 different numbers are both 10, what is the smallest number?

1) The range of the 4 numbers is 10
2) The sum of the smallest and the largest numbers is 20

In ascending order, let the four numbers = a, b, c, d.
Since the average of the 4 numbers = 10, we get:
(a+b+c+d)/4 = 10
a+b+c +d = 40
Since the median of the 4 numbers = 10, we get:
(b+c)/2 = 10
b+c = 20
Subtracting the blue equation from the red equation, we get:
(a+b+c+d) - (b+c) = 40-20
a+d = 20

Statement 1:
d-a = 10.
Since we have two variables (a and d) and two distinct linear equations (a+d=20 and d-a=10), we can solve for the two variables.
Thus, the value of a can be determined.
SUFFICIENT.

Statement 2:
The prompt on its own implies that a+d=20.
Since Statement 2 offers no new information, INSUFFICIENT.

The correct answer is A.

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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 a, b, c and d be the 4 numbers, and suppose a < b < c < d.
Then ( a + b + c + d ) / 4 = 10 and ( b + c ) / 2 = 10.
Since b + c = 20 and a + b + c + d = 40, we must have a + d = 20.

Condition 1)
Since d - a = 10 by condition 1), we can figure out the values of a and d. Thus, condition 1) is sufficient.

Condition 2)
a + d = 20 can be deduced from the original condition as shown above.
So, condition 2) provides no additional information.
If a = 1, b = 9, c = 11 and d = 19, then the smallest number is 1.
If a = 2, b = 9, c =11 and d = 18, then the smallest number is 2.
Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A