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May 0608:30 AM PDT
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In Episode 3 of our GMAT Ninja Critical Reasoning series, we tackle Discrepancy, Paradox, and Explain an Oddity questions. You know the feeling: the passage gives you two facts that seem completely contradictory....- May 08
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Join the special YouTube live-stream for selecting the winners of GMAT Club MBA Scholarships sponsored by Juno live. Watch who gets these coveted MBA scholarships offered by GMAT Club and Juno.
24 Mar 2021, 04:02
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MathRevolution
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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.
Visit https://www.mathrevolution.com/gmat/lesson for details.
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.
Condition 2)
Condition 2) tells us that \(abc = 6\). For three numbers to multiply to \(6\), the numbers must be \(1, 2,\) and \(3\) and the order does not matter. We can substitute these numbers into the given equation to get:
\(2(a^4 + b^4 + c^4) = 2(1^4 + 2^4 + 3^4) = 2(1 + 16 + 81) = 2*98 = 196 = 14^2,\) which is a perfect square.
Thus condition 2) is sufficient.
Condition 1)
\((a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)\)
Then, we have \(a^2 + b^2 + c^2 = -2(ab + bc + ca)\) since \(a + b + c = 0.\)
We have \((a^2 + b^2 + c^2)^2 = (-2(ab + bc + ca))^2\) when we square both sides.
Its left-hand side is
\((a^2 + b^2 + c^2)^2 \)
\(= (a^2 + b^2 + c^2) (a^2 + b^2 + c^2)\)
\(= a^4 + a^2b^2 +a^2c^2 + a^2b^2 + b^4 + b^2c^2 + a^2c^2 + b^2c^2 + c^4\)
\(= a^4 + b^4 + c^4 + 2a^2b^2 + 2b^2c^2 + 2c^2a^2.\)
Its right-hand side is
\((-2(ab + bc + -ca))^2 \)
\(= 4(ab + bc + ca)(ab + bc + ca)\)
\(= 4(a^2b^2+ ab^2c + a^2bc + ab^2c + b^2c^2 + abc^2 + a^2bc + abc^2 + c^2a^2 \)
\(= 4(a^2b^2 + b^2c^2 + c^2a^2 + 2a^2bc + 2ab^2c + 2abc^2)\)
\(= 4(a^2b^2 + b^2c^2 + c^2a^2) + 2abc(a + b + c))\)
\(= 4(a^2b^2 + b^2c^2 + c^2a^2),\) since \(a + b + c = 0\)
We have \(a^4 + b^4 + c^4 = 2(a^2b^2 + b^2c^2 + c^2a^2).\)
Then \(2(a^4 + b^4 + c^4) = 4(a^2b^2 + b^2c^2 + c^2a^2) = (-2(ab + bc + ca))^2 \)
Thus \(2(a^4 + b^4 + c^4)\) is a square of \(-2(ab + bc + ca).\)
We realize we have only used condition 1), but haven’t used condition 2).
It means condition 1) alone is sufficient.
Thus, condition 1) is sufficient.
Therefore, D is the answer.
Answer: D
This question is a CMT 4(B) question: condition 2) is easy to work with, and condition 1) is difficult to work with. For CMT 4(B) questions, D is most likely the answer.
In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
27 Mar 2021, 01:24
Kudos
Bookmarks
MathRevolution
=>
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.
Visit https://www.mathrevolution.com/gmat/lesson for details.
Since we have \(2\) variables (\(a\) and \(b\)) and \(1\) equation, D is most likely the answer. So, we should consider each condition on its own first.
Let’s look at condition 1). It tells us that it is not sufficient since we don’t have any information regarding a.
The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Let’s look at condition 2). It tells us that it is not sufficient since we don’t have any specific numbers for a or b.
The answer is not unique, and the condition is not sufficient, according to Common Mistake Type 2, which states that the number of answers must be only one.
Conditions 1) & 2) together give us that they are not sufficient.
If \(a = 30\) and \(b = 2\), we have \(a + b = 32.\)
If \(a = 15\) and \(b = 4\), we have \(a + b = 19.\)
The answer is not unique, and both conditions 1) and 2) together are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions 1) & 2) together are not sufficient.
Therefore, E is the correct answer.
Answer: E
If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations,” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A, B, C, or E.
30 Mar 2021, 02:58
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MathRevolution
=>
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 we have \(4\) variables (\(A, B, C,\) and \(X\)) and \(1\) equation, E is most likely the answer. So, we should consider conditions 1) & 2) together first After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Conditions 1) & 2) together give us \(x = \frac{-5}{2}.\)
Assume \(x = [X].\)
We have \([A○B] = \frac{[(-5) + 7] }{ 2} = \frac{2}{2} = 1\), since we have \([A] = -5\) and \([B] = 7.\)
We have \([C○X] = \frac{[(9}{2) + x] / 2} = \frac{9}{4} + \frac{x}{2}.\)
Thus, we have \(\frac{9}{4} + \frac{x}{2} = 1\) or \(\frac{x}{2} = 1 – (\frac{9}{4}) = \frac{-5}{4}.\)
Then we have \(x = \frac{-5}{2.}\)
The answer is unique, and conditions 1) and 2) together are sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions 1) & 2) together are sufficient.
Therefore, C is the correct answer.
Answer: C
In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B, or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
