To determine which number produces an irrational number when multiplied by 0.4, let's evaluate each of the provided options:
1. Option A: [tex]\( 0.444 \ldots \)[/tex]
- [tex]\( 0.444 \ldots \)[/tex] (a repeating decimal) is a rational number since it can be expressed as a fraction.
- Multiplying a rational number by another rational number (0.4) results in a rational number.
- Therefore, [tex]\( 0.4 \times 0.444 \ldots \)[/tex] is rational.
2. Option B: [tex]\( 3 \pi \)[/tex]
- [tex]\( \pi \)[/tex] is an irrational number.
- Multiplying an irrational number by a rational number, such as [tex]\( 0.4 \)[/tex], results in an irrational number.
- Therefore, [tex]\( 0.4 \times 3 \pi \)[/tex] is irrational.
3. Option C: [tex]\( \frac{2}{7} \)[/tex]
- [tex]\( \frac{2}{7} \)[/tex] is a rational number because it is expressed as a fraction.
- Multiplying two rational numbers results in a rational number.
- Therefore, [tex]\( 0.4 \times \frac{2}{7} \)[/tex] is rational.
4. Option D: [tex]\( \sqrt{9} \)[/tex]
- [tex]\( \sqrt{9} \)[/tex] equals 3, which is a rational number.
- Multiplying two rational numbers results in a rational number.
- Therefore, [tex]\( 0.4 \times 3 \)[/tex] is rational.
After evaluating all the options, we find that only Option B produces an irrational number when multiplied by 0.4. Hence, the correct answer is:
B. [tex]\( 3 \pi \)[/tex]
