To determine which number produces a rational number when added to [tex]\(\frac{1}{5}\)[/tex], consider the following step-by-step solution:
1. Identify the given number:
The given number is [tex]\(\frac{1}{5}\)[/tex], which is a rational number.
2. Evaluate the options provided:
A. [tex]\(-1.41421356 \ldots\)[/tex] is an approximation of [tex]\(-\sqrt{2}\)[/tex], which is an irrational number. Adding an irrational number to [tex]\(\frac{1}{5}\)[/tex] will yield an irrational number.
B. [tex]\(-\frac{2}{3}\)[/tex] is a rational number. Adding two rational numbers will always result in a rational number. Let's perform the addition:
[tex]\[
\frac{1}{5} + \left(-\frac{2}{3}\right) = \frac{1}{5} - \frac{2}{3}
\][/tex]
To add these fractions, find a common denominator, which is 15:
[tex]\[
\frac{1}{5} = \frac{3}{15}, \quad \frac{2}{3} = \frac{10}{15}
\][/tex]
Thus,
[tex]\[
\frac{3}{15} - \frac{10}{15} = \frac{3 - 10}{15} = \frac{-7}{15}
\][/tex]
The result, [tex]\(\frac{-7}{15}\)[/tex], is a rational number.
C. [tex]\(\pi\)[/tex] is an irrational number. Adding an irrational number to [tex]\(\frac{1}{5}\)[/tex] will yield an irrational number.
D. [tex]\(\sqrt{11}\)[/tex] is an irrational number. Adding an irrational number to [tex]\(\frac{1}{5}\)[/tex] will yield an irrational number.
3. Conclusion:
The only option that produces a rational number when added to [tex]\(\frac{1}{5}\)[/tex] is:
[tex]\[
\boxed{B. -\frac{2}{3}}
\][/tex]
