Solution: To save time and improve accuracy on DS questions in GMAT, learn, and apply Variable Approach.
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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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 ‘x – where ‘x’ and ‘y’ are positive integers.
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 combined together.
Condition (1) tells us that \(3^x\ +\ 5^y\) = 134
Condition (2) tells us that y = 3
=> \(3^x\ +\ 5^3\) = 134
=> \(3^x\) +125 = 134
=> \(3^x\) = 134 - 125 = 9
=> \(3^x\) = \(3^2\)
Therefore, x = 2
The answer is a unique value; both conditions combined together are sufficient according to Common Mistake Type 2 which states that the answer should be a unique value. So, C seems to be the answer.
However, since this question is an integer question, which is also one of the key questions, we should apply CMT 4(A), which means that if an answer C is found too easily, either A or B should be considered as the answer. Let’s look at each condition separately.
Condition (1) tells us that \(3^x\ +\ 5^y\) = 134
Since the exponents of 5 would reach 134 faster than the exponents of 3, we need to try with the exponents of 5 so that we can get the answer(s) in the least possible trials.
If y = 1, then
=> \(3^x\ +\ 5^1\) = 134
=> \(3^x\) + 5 = 134
=> \(3^x\) = 134 - 5 = 129 [129 cannot be expressed as an exponent of 3, so y ≠ 1]
If y = 2, then
=> \(3^x\ +\ 5^2\) = 134
=> \(3^x\) + 25 = 134
=> \(3^x\) = 134 - 25 = 109 [109 cannot be expressed as an exponent of 3, so y ≠ 2]
But if y = 3, then
=> \(3^x\ +\ 5^3\) = 134
=> \(3^x\) + 125 = 134
=> \(3^x\) = 134 - 125 = 9 [9 can be expressed as an exponent of 3, so y = 2]
=> \(3^x\) = \(3^2\)
Therefore, x = 2
The answer is a unique value; condition (1) alone is sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.
Condition (2) tells us that y = 3
=> But ‘x’ is still unknown
The answer is not a unique value; condition (2) alone is not sufficient according to Common Mistake Type 2 which states that the answer should be a unique value.
If the question has both C and A as its answer, then A is the answer rather than C according to the definition of DS questions.
Condition (1) alone is not sufficient.
Therefore, A is the correct answer.
Answer: A
