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k, m and n are positive integers. Is their average equal to

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by Max@Math Revolution » Tue Jul 02, 2019 11:42 pm

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[GMAT math practice question]

k, m and n are positive integers. Is their average equal to their median?

1) The median of k, m and n is 11.
2) The range of k, m and n is 13
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Source: — Data Sufficiency |
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by Max@Math Revolution » Thu Jul 04, 2019 11:37 pm
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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. We should simplify the conditions, if necessary.

Without loss of generality, we may assume k ≤ m ≤ n.
If their average and their median are equal, then ( k + m + n ) / 3 = m or k + m + n = 3m, so n + k = 2m and n-k=2m-2k. So, the range of the numbers must be even.
Thus, condition 2) yields the unique answer 'no', and is sufficient by CMT 1).

Condition 1)
If k = 10, m = 11 and n = 12, then their average and median are equal, and the answer is 'yes'.
If k = 10, m = 11 and n = 13, then their average and their median are not equal, and the answer is 'no'.
Condition 1) is not sufficient since it doesn't yield a unique solution.

Therefore, B is the answer.
Answer: B
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