A bag contains only black and white balls. What is the proba

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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.

The first step of VA(Variable Approach) method is modifying the original condition and the question, and rechecking the number of variables and the number of equations.

We can modify the original condition and question as follows.

Assume b and w are the numbers of black and white balls, respectively.
Then b + w = 12.
The question asks for the value of b / ( b + w ).

Since we have 2 variables and 0 equations, C is most likely to be 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:
b + w = 12
w = 3b

Combining these equations yields b + 3b = 4b = 12 or b = 3.
So, w = 9.
Thus, b / ( b + w ) = 3 / 12 = 1/4
Both conditions together are sufficient.

Since this question is a probability 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) : b + w = 12
Since we can't determine the value of b, we can't determine b / ( b + w ) = b / 12.
Condition 1) is not sufficient.

Condition 2) : w = 3b
b / ( b + w ) = b / ( b + 3w ) = b / 4b = 1/4.
Condition 2) is sufficient.

Therefore, B is the answer.

Answer: B

When a question asks for a ratio, if one condition includes a ratio and the other condition just gives a number, the condition including the ratio is most likely to be sufficient. This tells us that B is most likely to be the answer to this question.

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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