Answered

Which number produces an irrational number when multiplied by [tex]\frac{1}{4}[/tex]?

A. [tex]\sqrt{12}[/tex]
B. [tex]-\sqrt{36}[/tex]
C. [tex]0.444444 \ldots[/tex]
D. [tex]\frac{4}{3}[/tex]



Answer :

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]