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What is the value of the integer a?

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What is the value of the integer a?

by Max@Math Revolution » Wed Aug 21, 2019 11:41 pm

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[GMAT math practice question]

What is the value of the integer a?

1) x - (2/3)(x-4a) = 7 has a positive integer solution
2) a is positive
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Source: — Data Sufficiency |
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by deloitte247 » Sun Aug 25, 2019 5:14 am

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Here, we are to find the value of a?
$$Statement\ 1=>x-\left(\frac{2}{3}\right)\left(x-4a\right)=7$$
$$x-\frac{2}{3}x+\frac{8a}{3}=7$$
$$\frac{8a}{3}=7-x+\frac{2x}{3}$$
Multiply through by 3, we have;
$$8a=21-3x+2x$$
$$8a=21-x$$
The value of 'x' is unknown, so, we cannot arrive at the definite value forinteger 'a'. Hence, statement 1 is NOT SUFFICIENT.

Statement 2=> a is positive
This means that 0 < a < infinity. i.e, integer 'a' can be any number between 0 and infinity. So, the information provided is not enough to arrive at a definite answer for statement 2. Hence, statement 2 is NOT SUFFICIENT too.

Combining both statements together
Statement 1: 8a = 21 - x
Statement 2: a is positive.
The information available does not provide the value of 'x' to evaluate 'a' neither does it gives a definite value for an integer 'a'.
Hence, both statements together are NOT SUFFICIENT. So, option E is the correct answer.

Hope this helps? Thanks
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by Max@Math Revolution » Sun Aug 25, 2019 5:33 pm

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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.

Since we have 2 variables (x and a) 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)
x - (2/3)(x - 4a) = 7 is equivalent to 3x - 2(x-4a) = 21 or x = 21 - 8a.
The possible pairs (x,a) are (13,1) and (5,2).
Since both conditions together don't yield a unique solution, they are not sufficient.

Therefore, E is the 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.
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