Let's analyze each of the options to determine which number produces an irrational number when multiplied by \(\frac{1}{4}\).
### Option A: \(\sqrt{12}\)
1. The square root of 12 is \(\sqrt{12}\).
2. When we multiply \(\sqrt{12}\) by \(\frac{1}{4}\), we get:
[tex]\[
\frac{\sqrt{12}}{4}
\][/tex]
3. Since \(\sqrt{12}\) is an irrational number (it cannot be expressed as a simple fraction), \(\frac{\sqrt{12}}{4}\) is also an irrational number, because multiplying an irrational number by a rational one (non-zero) remains irrational.
### Option B: \(-\sqrt{36}\)
1. The square root of 36 is \(\sqrt{36} = 6\), so \(-\sqrt{36} = -6\).
2. When we multiply \(-6\) by \(\frac{1}{4}\), we get:
[tex]\[
-6 \times \frac{1}{4} = -\frac{6}{4} = -\frac{3}{2}
\][/tex]
3. \(-\frac{3}{2}\) is a rational number (it can be expressed as a simple fraction).
### Option C: \(0.444444 \ldots\)
1. Express \(0.444444 \ldots\) as a fraction, which is \(\frac{4}{9}\) (since it is a repeating decimal).
2. When we multiply \(\frac{4}{9}\) by \(\frac{1}{4}\), we get:
[tex]\[
\frac{4}{9} \times \frac{1}{4} = \frac{4 \times 1}{9 \times 4} = \frac{4}{36} = \frac{1}{9}
\][/tex]
3. \(\frac{1}{9}\) is a rational number (it can be expressed as a simple fraction).
### Option D: \(\frac{4}{3}\)
1. \(\frac{4}{3}\) is already in fractional form.
2. When we multiply \(\frac{4}{3}\) by \(\frac{1}{4}\), we get:
[tex]\[
\frac{4}{3} \times \frac{1}{4} = \frac{4 \times 1}{3 \times 4} = \frac{4}{12} = \frac{1}{3}
\][/tex]
3. \(\frac{1}{3}\) is a rational number (it can be expressed as a simple fraction).
### Conclusion
Among the given options, the number that produces an irrational number when multiplied by \(\frac{1}{4}\) is \(\sqrt{12}\).
Thus, the correct answer is:
[tex]\[
\boxed{1 \text{ (A)}}
\][/tex]
