Question

given:
q = a positive terminating decimal
m = a square root of a non - perfect square whole number
choose whether each statement is \always,\ \sometimes\, or
ever\ true.

statement	true?
$\frac{1}{\sqrt{5}} \times m$ is rational	always	sometimes	never

Explanation:

Step1: Analyze \( 3 \times q \)

A positive terminating decimal \( q \) is a rational number (since terminating decimals can be expressed as fractions, e.g., \( 0.25=\frac{1}{4} \)). The product of two rational numbers (3 is rational, \( q \) is rational) is rational. So \( 3 \times q \) is always rational.

Step2: Analyze \( \frac{1}{\sqrt{5}} \times m \)

\( m \) is the square root of a non - perfect square whole number, so \( m \) is irrational (e.g., \( \sqrt{2},\sqrt{3} \)). \( \frac{1}{\sqrt{5}} \) is also irrational. The product of two irrational numbers: sometimes it can be rational (e.g., if \( m = \sqrt{5} \), then \( \frac{1}{\sqrt{5}}\times\sqrt{5}=1 \), which is rational) and sometimes it can be irrational (e.g., if \( m=\sqrt{2} \), then \( \frac{1}{\sqrt{5}}\times\sqrt{2}=\frac{\sqrt{10}}{5} \), which is irrational). So \( \frac{1}{\sqrt{5}}\times m \) is sometimes rational.

Answer:

For the statement " \( 3 \times q \) is rational", the answer is Always.
For the statement " \( \frac{1}{\sqrt{5}} \times m \) is rational", the answer is Sometimes.