To determine which number produces an irrational number when multiplied by 0.5, let's evaluate each option step-by-step.
Option A: [tex]\(\sqrt{16}\)[/tex]
1. Compute [tex]\(\sqrt{16}\)[/tex]:
[tex]\[
\sqrt{16} = 4
\][/tex]
2. Multiply by 0.5:
[tex]\[
0.5 \times 4 = 2
\][/tex]
The result is 2, which is a rational number.
Option B: [tex]\(0.555 \ldots\)[/tex]
1. Given [tex]\(0.555 \ldots\)[/tex], which is actually 0.555 repeated (or simply, a rational number).
2. Multiply by 0.5:
[tex]\[
0.5 \times 0.555 \ldots = 0.2775
\][/tex]
The result is 0.2775, which is a rational number.
Option C: [tex]\(\frac{1}{3}\)[/tex]
1. Compute [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{1}{3} \approx 0.333 \ldots
\][/tex]
2. Multiply by 0.5:
[tex]\[
0.5 \times \frac{1}{3} = \frac{1}{6} \approx 0.1667
\][/tex]
The result is [tex]\(\frac{1}{6}\)[/tex], which is also a rational number.
Option D: [tex]\(\sqrt{3}\)[/tex]
1. Compute [tex]\(\sqrt{3}\)[/tex]:
[tex]\[
\sqrt{3} \approx 1.732
\][/tex]
2. Multiply by 0.5:
[tex]\[
0.5 \times \sqrt{3} \approx 0.5 \times 1.732 \approx 0.866
\][/tex]
The result is [tex]\(0.5 \times \sqrt{3}\)[/tex], which is still an irrational number because the product of a rational number (0.5) and an irrational number ([tex]\(\sqrt{3}\)[/tex]) is irrational.
Thus, the option that when multiplied by 0.5 produces an irrational number is:
[tex]\[
\boxed{4}
\][/tex]
