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< x,y > denotes x + y/2. What is the value of x?

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< x,y > denotes x + y/2. What is the value of x?

by Max@Math Revolution » Mon Sep 16, 2019 10:50 pm

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[GMAT math practice question]

< x,y > denotes x + y/2. What is the value of x?

1) < x,y > = y + x/2
2) < 2x,2y >+1=< y,x > - 2
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Source: — Data Sufficiency |
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by Max@Math Revolution » Wed Sep 18, 2019 11:54 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.
Visit https://www.mathrevolution.com/gmat/lesson for details.

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.

When we simply condition 2), we have
< 2x, 2y >+1=< y, x >-2
=> 2x + 2y/2 + 1 = y + x/2 - 2
=> 2x + y + 1 = y + x/2 - 2
=> (3/2)x = -3
=> x = -2.
Condition 2) is sufficient.

Condition 1)
Since we have < x,y > = y + x/2 = x + y/2, we have y/2 = x/2 or x = y.
Condition 1) is not sufficient, since it does not yield a unique solution.

Therefore, B is the answer.
Answer: B
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by deloitte247 » Sat Sep 21, 2019 9:43 am

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Statement 1: x = 3/2
$$\left[\left(x+y\right)\left(2x-y\right)\right]\left[\left(x-y\right)\left(2x+y\right)\right]$$
Where x = 3/2
$$\left[\left(\frac{3}{2}+y\right)\left(\left(2\cdot\frac{3}{2}\right)-y\right)\right]\left[\left(\frac{3}{2}-y\right)\left(2\left(\frac{3}{2}\right)+y\right)\right]$$
Since the value of y is unknown, then statement 1 is NOT SUFFICIENT.

Statement 2: xy=2
x = 2/y
y = 2/x
We are still left with unknown variables, so, statement 2 is NOT SUFFICIENT.

Combining both statements together
x=3/2
xy=2
3/2y = 2
$$y=\frac{2}{\left(\frac{3}{2}\right)}=\frac{4}{3}$$
Since the value of x and y are now known, it will be easy to evaluate the expression in the question.
Hence, both statements combined ARE SUFFICIENT.

Answer = option C
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