Identifying Irrational Numbers: MCQ Breakdown

The question asks to identify which of the given options is an irrational number. An irrational number cannot be expressed as a simple fraction of two integers ($p/q$, where $q \neq 0$), and its decimal representation neither terminates nor repeats.

Analysis of Options

• Option A: $\frac{6}{15}$
  This is a ratio of two integers (6 and 15). Therefore, it is a rational number. It simplifies to $\frac{2}{5}$.
• Option B: $\frac{\pi}{\sqrt{3}}$
  Pi ($\pi$) is a well-known irrational number. The square root of 3 ($\sqrt{3}$) is also an irrational number. The quotient of $\pi$ and $\sqrt{3}$ results in an irrational number.
• Option C: -1
  This is an integer. Integers are rational numbers because they can be written as a fraction with a denominator of 1 (e.g., $\frac{-1}{1}$).
• Option D: $\frac{4}{15}$
  This is a ratio of two integers (4 and 15). Therefore, it is a rational number.
• Option E: (Empty)
  This option is empty and does not represent a number.

Conclusion on Irrationality

Based on the analysis:

• Options A, C, and D are confirmed as rational numbers.
• Option B, $\frac{\pi}{\sqrt{3}}$, contains the irrational number $\pi$ and is divided by another irrational number, $\sqrt{3}$. The result is an irrational number.

Thus, $\frac{\pi}{\sqrt{3}}$ is the irrational number among the choices.