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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.
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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 conditions if necessary.

Since the sum of all the differences between the weights and their average is zero, we have 4 + (-7) + 6 + (-5) + x = 0 or x = 2.
Thus, we don’t need condition 2), since the table already tells us that x = 2.

Let’s look at the condition 1).

Assume the new average, including Fred, is m’.
We have m’ = [5m + (m + 12)] / 6, m’ = (6m + 12) / 6, m’ = m + 2. We also m’ = 1.04m, since we are told the average increases by 4%.
Putting the 2 equations together gives us m + 2 = 1.04m.
Then we have 0.04m = 2 or m = 50.
Thus, the lightest weight is Betty’s weight m – 7 = 43.

The answer is unique, and the condition is sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.

Condition 1) ALONE is sufficient.
Therefore, A is the answer.
Answer: A