MathRevolution wrote:
[GMAT math practice question]
(number properties) There are \(6\) bags. Each bag contains \(18, 19, 21, 23, 25\), and \(34\) beads, respectively. All the beads in one bag are broken, and no other bags have any broken beads. How many beads are broken?
1) Adam has \(3\) bags, and Betty has \(2\) bags, and no one has the bag of broken beads.
2) Adam has twice as many beads as Betty has.
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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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Since there are \(6\) bags, we have many variables and \(0\) equations, and 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)
Assume \(a\) and \(b\) are the numbers of beads that Adam and Betty have, respectively.
Since Adam chooses \(3\) bags and Betty chooses \(2\) bags, one bag is not chosen by Adam or Betty.
Since we have \(a = 2b, a + b = 2b + b = 3b\), then \(a + b\) is a multiple of \(3.\)
\(18 + 19 + 21 + 23 + 25 + 34 = 140\) has a remainder \(2\) when \(140\) is divided by \(3\). When we subtract the number of beads in \(1\) of the \(6\) bags from \(140\), it must be a multiple of \(3\) and only \(23\) satisfies this condition, since \(140 – 23 = 117\), which is a multiple of \(3.\)
\(23\) is the number of broken beads.
Since both conditions together yield a unique solution, they are sufficient.
Since this question is a statistics 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)
There are \(6\) possibilities for the bag with broken beads.
Since condition 1) does not yield a unique solution, it is not sufficient.
Condition 2)
Since condition 2) doesn’t give us any information about the number of broken beads, it is obviously not sufficient.
Therefore, C is the 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.
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