Question

If x is a positive number, then which of the following fractions has the greatest value?

• A. \(\frac{x}{x}\)
• B. \(\frac{x+1}{x}\)
• C. \(\frac{x}{(x+1)}\)
• D. \(\frac{(x+2)}{(x+3)}\)

Answer 1

The Correct Option is B

The correct option is (B): \(\frac{x+1}{x}\)
Explanation: The given options are:
- \( \frac{x}{x} = 1 \)

- \( \frac{x+1}{x} = 1 + \frac{1}{x} \)

- \( \frac{x}{x+1} \)

- \( \frac{x+2}{x+3} \)

Since \( x \) is a positive number, let's compare the values of these fractions:
- \( \frac{x}{x} = 1 \)

- \( \frac{x+1}{x} = 1 + \frac{1}{x} \), which is always greater than 1.

- \( \frac{x}{x+1} \), which is always less than 1 for any positive \( x \).

- \( \frac{x+2}{x+3} \), which is also less than 1 for any positive \( x \).

Clearly, \( \frac{x+1}{x} \) has the greatest value because it is always greater than 1, while the other fractions are less than or equal to 1.

Thus, the correct answer is B: \( \frac{x+1}{x} \).