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09 Dec 2020, 03:49
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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.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the value of \(x + y.\)
Follow the second and the third step: From the original condition, we have \(2\) variables (\(x\) and \(y\)). To match the number of variables with the number of equations, we need 2 equations. Since conditions (1) and (2) will provide 1 equation each, C would most likely be the answer.
Recall 3 Principles and choose C as the most likely answer. Let’s look at both conditions (1) and (2) together.
Conditions (1) and (2) tell us that \(x\) and \(y\) are positive integers and \(\frac{1}{x}+\frac{1}{y}=\frac{1}{5}\), from which we get \(\frac{1}{x}+\frac{1}{y}=\frac{1}{5}, \frac{y}{xy} + \frac{x}{xy} = \frac{1}{5}, \frac{y + x}{xy}=\frac{1}{5}, 5y + 5x = xy, xy – 5x – 5y = 0\), or \(xy – 5x – 5y + 25 = 25.\) We can factor \(xy – 5x – 5y + 25 = 25\) as follows: \((xy – 5x) + (-5y + 25) = 25, x(y – 5) + (-5)(y – 5) = 25\), which is equal to \((x - 5)(y - 5) = 25.\) We have \(3\) possible cases: \(x – 5 = 25\), and \(y – 5 = 1 / x – 5 = 5\), and \(y – 5 = 5 / x – 5 = 1\), and \(y – 5 = 25,\) since \(x\) and \(y\) are positive integers from condition (1).
Thus, the possible pairs of \(x\) and \(y\) are \(x = 6\), and \(y = \frac{30 }{ x} = 10\), and \(y = 10\) / and \(x = 30\), and \(y = 6.\)
If \(x = 6\) and \(y = 30\), then \(x + y = 36.\)
If \(x = 10\) and \(y = 10\), then \(x + y = 20.\)
The answer is not unique, and both conditions (1) and (2) combined are not sufficient according to Common Mistake Type 2, which states that the number of answers must be only one.
Both conditions (1) and (2) together are not sufficient.
Therefore, E is the correct answer.
Answer: E
Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B, or D is the answer. This occurs in Common Mistake Types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D, or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
12 Dec 2020, 03:52
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MathRevolution
Solution:
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.
Now we will solve this DS question using the Variable Approach.
Let’s apply the 3 steps suggested previously.
Follow the first step of the Variable Approach by modifying and rechecking the original condition and the question.
We have to find the value of f(2) + f(3) + f(5).
Follow the second and the third step: From the original condition, we have many variables to determine a function f(x). To match the number of variables with the number of equations, we need many equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.
Recall 3 Principles and choose E as the most likely answer.
Let’s look at both conditions 1) & 2) together.
Since f(1) = 1, we have f(2) = f(1+1) = f(1) + f(1) + 1·1 = 1 + 1 + 1 = 3 using condition 2).
Then we have f(3) = f(2+1) = f(2) + f(1) + 2·1 = 3 + 1 + 2 = 6.
f(5) = f(3+2) = f(3) + f(2) + 3·2 = 6 + 3 + 6 = 15.
Thus, we have f(2) + f(3) + f(5) = 3 + 6 + 15 = 24.
The answer is unique, so both conditions 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.
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.
14 Dec 2020, 04:43
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.
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.
Condition 1)
Since \(x > y\) and the average of \(x\) and \(y\) is \(9\), we have \(x > 9 > y.\)
Thus, the median of \(x, 9, 9\) and \(y\) is \(9\).
Since condition 1) yields a unique solution, it is sufficient.
Condition 2)
Since \(\frac{( x + y + 18 )}{3} = 12\) or \(x + y + 18 = 36\), the average of \(x\) and \(y\) is \(9.\)
Condition 2) is sufficient by the same reasoning as condition 1).
Therefore, D is the answer.
Answer: D
Note: Tip 1) of the VA method states that D is most likely to be the answer if condition 1) gives the same information as condition 2).
This question is a CMT4(B) question: condition 1) is easy to work with and condition 2) is difficult to work with. For CMT4(B) questions, D is most likely to be the answer.
