If 0 < x < 1, What is the Median of the Values x, x^-1, x^2 GMAT Problem Solving

Question: If 0 < x < 1, what is the median of the values x,\(x{^-{^1}},x^2,\sqrt{x}, and x^3\)?

1. \(x\)
2. \(x{^-{^1}}\)
3. \(x^2\)
4. \(\sqrt{x}\)
5. \(x^3\)

Solution and Explanation:
Approach Solution (1):
x is a positive fraction. Pick any number, say\(\frac{1}{4}\)

Then
x =\(\frac{1}{4}\)

\(x{^-{^1}}=\frac{1}{x}=4\)

\(x^2=\frac{1}{16}\)

\(\sqrt{x}=x^{1/2}=\frac{1}{16}\)

\(x^3=\frac{1}{64}\)

Rearranging in ascending order:

\(x^3,x^2,x,x^{\frac{1}{2}},x{^-{^1}}\)
Median = x

Correct option: A

Approach Solution (2):

To find the median, we can let x be any value (fraction or decimal) between 0 and 1.
So let’s let x = \(\frac{1}{4}\) therefore:

\(x{^-{^1}}=\frac{1}{x}=4\)

\(x^2=\frac{1}{16}\)

\(\sqrt{x}=x^{1/2}=\frac{1}{16}\)

\(x^3=\frac{1}{64}\)

From the smallest to the largest, the order is:

\(x^3=\frac{1}{64}\)

\(x^2=\frac{1}{16}\)

\(x=\frac{1}{4}\)

\(\sqrt{x}=x^{1/2}=\frac{1}{16}\)

\(x{^-{^1}}=\frac{1}{x}=4\)

Therefore, median = x
Note: We used ¼ as the value of x since it’s easy to take the square root of ¼, but any value between 0 and 1 will behave similarly, and x will always be the median.

Correct option: A

Approach Solution (3):
To find the median one has to arrange the numbers in ascending or descending order and choose the middle one
\(x^3\)will be the least, followed by
\(x^2\), then by
\(x\)
\(\sqrt{x}\), and
\(x{^-{^1}}\)
So, x is the median

Correct option: A

“If 0 < x < 1, what is the median of the values x,\(x{^-{^1}},x^2,\sqrt{x}, and x^3\)?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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