MathRevolution wrote:
[GMAT math practice question]
(Number Properties) Adam is a teenage boy and his age is \(x\) and his father’s age is \(y\). We have a four-digit number \(S\) by putting \(x\) after \(y\). What is the value of \(x\)?
1) \(y - x = 27\)
2) \(S( - (y - x)) = 4289\)
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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 we put \(x\) after \(y\), we have \(S = 100y + x.\)
And we have \(13 ≤ x ≤ 9\) and \(13 ≤ x ≤ y ≤ 99\) since \(x\) and \(y\) are two-digit numbers.
Since we have \(3\) variables (\(x, y\), and \(S\)) and \(1\) equation, C 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)
We can rearrange the equation \(S - (y - x) = 4289\) from condition 1) to get \(S = 4289 + (y - x).\) Substituting \(y - x = 27\) from condition 2) we get \(S = 4289 + 27 = 4316.\) Then \(y = 43\) and \(x = 16\).
Since both conditions together yield a unique solution, they are sufficient.
Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.
Condition 1)
We have many possible pairs of \(x\) and \(y\) from \(y – x = 27.\)
Since condition 1) does not yield a unique solution, it is not sufficient
Condition 2)
Substituting \(S = 100y + x\), into \(S – (y-x)\) we get \(100y + x – (y - x) = 99y + 2x = 4289.\)
If we make \(y = 10a + b\) and \(x = 10 + c\) where \(1 ≤ a ≤ 9, 0 ≤ b ≤ 9\) and \(3 ≤ c ≤ 9.\)
Then we have \(S = 99y + 2x = 990a + 99b + 20 + 2c = 4289\) or \(990a + 99b + 2c = 4269.\)
When we substitute integers between \(1\) and \(9\) for the variable a one by one, we notice \(a = 4\) is the unique value in order to have units digits \(b\) and \(c\).
Then we have \(99b + 2c = 4269 – 3960 = 309.\)
When we substitute integers between \(1\) and \(9\) for the variable \(b\) one by one, we notice \(b = 3\) is the unique value in order to have a units digit \(c\).
Then we have \(c = 6.\)
Thus the boy’s age is \(16\) and his father’s age is \(43.\)
Since condition 2) yields a unique solution, it is sufficient.
Therefore, B is the answer.
Answer: B
This question is a CMT 4 (A) question: When we easily get C as an answer, consider A and B as an answer.
If the question has both C and B as its answer, then B is an answer rather than C by the definition of DS questions. Also this question is a 50/51 level question and can be solved by using the relationship between the Variable Approach and Common Mistake Type 3 and 4 (A or B).
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.
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