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x is a positive number. What is the median of x, √x and x^

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by Max@Math Revolution » Wed Dec 05, 2018 12:56 am

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[Math Revolution GMAT math practice question]

x is a positive number. What is the median of x, √x and x^2?

1) x^2=x
2) x^2+x+1=3x
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by GMATGuruNY » Wed Dec 05, 2018 4:01 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

x is a positive number. What is the median of x, and x^2?

1) x^2=x
2) x^2+x+1=3x
To determine the median of x, √x and x², we need to know the value of x.

Statement 1:
Since x is positive, we can safely divide both sides by x:
x²/x = x/x
x = 1
SUFFICIENT.

Statement 2:
x² + x + 1 = 3x
x² - 2x + 1 = 0
(x-1)² = 0
x = 1
SUFFICIENT.

The correct answer is D.
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x is a positive number. What is the median of x, √x and x^

by fskilnik@GMATH » Wed Dec 05, 2018 7:19 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

x is a positive number. What is the median of x, √x and x^2?

1) x^2=x
2) x^2+x+1=3x
$$x > 0$$
$$? = {\rm{Med}}\left( {x,\sqrt x ,{x^2}} \right)$$

$$\left( 1 \right)\,\,x = {x^2}\,\,\,\,\mathop \Rightarrow \limits^{:\,\,x\,\left( { \ne 0} \right)\,} \,\,1 = x\,\,\,\,\, \Rightarrow \,\,\,? = {\rm{Med}}\left( {1,1,1} \right) = 1\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}{\rm{.}}$$

$$\left( 2 \right)\,\,\,\, \Rightarrow \,\,\,\,0 = {x^2} - 2x + 1 = {\left( {x - 1} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,x = 1\,\,\,\, \Rightarrow \,\,\,\,{\rm{idem}}\,\,\,\left( 1 \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}{\rm{.}}\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Max@Math Revolution » Thu Dec 06, 2018 11:44 pm
=>

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 the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

If x >1, then √x < x < x^2 and x is their median.
If 0 < x <1, then √x > x > x^2 and x is their median.
If x = 1, then √x = x = x^2 and x is their median.
Thus, the question asks for the value of x.

Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
x^2=x
=> x^2-x=0
=> x(x-1)=0
=> x = 0 or x = 1
Since x is positive, x = 1.
Condition 1) is sufficient.

Condition 2)
x^2+x+1=3x
=> x^2-2x+1=0
=> (x-1)^2=0
=> x = 1.
Condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

FYI, Tip 1) of the VA method states that D is most likely to be the answer if conditions 1) and 2) provide the same information.

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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