MathRevolution wrote:
[GMAT math practice question]
(Number Property) \(x\) and \(y\) are positive integers. What is the difference between \(x\) and \(y\)?
1) \((x - 8)^2 = -|y - 36|\)
2) \((x + y)^2 + 3x + y = 1996\)
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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.
Thus, look at condition (1). It tells us that \(x = 8\) and \(y = 36\) for the following reason.
\((x - 8)^2 = -|y - 36|\)
⇔ \((x - 8)^2 + |y - 36| = 0\)
⇔ \(x = 8\) and \(y = 36\) since \((x - 8)^2 ≥ 0, |y - 36| ≥ 0\)
Then we have the difference \(y – x = 36 – 8 = 28.\)
It is exactly what we are looking for. 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 2) tells us that \(x = 8\) and \(y = 36\) for the following reason.
\((x + y)^2 < (x + y)^2 + 3x + y = 1996 < 45^2\), since \(x\) and \(y\) are positive
⇔ \(x + y ≤ 44\)
Case 1: \(x + y = 44\)
⇔ \((x + y)^2 + 3(x + y) – 2y = 1996\)
⇔ \((44)^2 +3(44) – 2y = 1996 \)
⇔ \(1936 + 132 – 2y = 1996\)
⇔ \(2068 – 2y = 1996\)
⇔ \(-2y = -72\)
⇔ \(y = 36\)
Substituting \(y = 36\) into \(x + y = 44\) gives us \(x + 36 = 44\) and \(x = 8. \)
Thus, we have \(y = 36\) and \(x = 8.\)
Case 2: \(x + y ≤ 43\)
\(2y = (x + y)^2 + 3(x + y) – 1996 ≤ 43^2 + 129 – 1996 = -18.\)
We don’t have a solution in this case, since \(y\) is a positive integer.
Thus, we have a unique solution for \(x\) and \(y\), which is \(x = 8\) and \(y = 36.\)
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.
Each condition ALONE is sufficient
Also, according to Tip 1, it is about 95% likely that D is the answer when condition (1) = condition (2).
This question is a CMT 4(B) question: condition 1) is easy to work with, and condition 2) is hard to work with. For CMT 4(B) questions, D is most likely the answer.
Therefore, D is the correct answer.
Answer: D
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