The goal is to determine which number, when added to [tex]\(\frac{1}{5}\)[/tex], results in a rational number.
A rational number is a number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex].
Let's evaluate each option:
A. [tex]\(\sqrt{11}\)[/tex]
The square root of a non-perfect square (like 11) is an irrational number. Adding an irrational number to [tex]\(\frac{1}{5}\)[/tex], a rational number, will result in an irrational number.
B. [tex]\(\pi\)[/tex]
[tex]\(\pi\)[/tex] is a well-known irrational number. Adding [tex]\(\pi\)[/tex] to [tex]\(\frac{1}{5}\)[/tex] will result in an irrational number.
C. [tex]\(-\frac{2}{3}\)[/tex]
Let's add this to [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[
\frac{1}{5} + \left(-\frac{2}{3}\right) = \frac{1}{5} - \frac{2}{3}
\][/tex]
To perform this subtraction, we need a common denominator. The least common multiple of 5 and 3 is 15.
[tex]\[
\frac{1}{5} = \frac{3}{15}
\][/tex]
[tex]\[
-\frac{2}{3} = -\frac{10}{15}
\][/tex]
Now we subtract:
[tex]\[
\frac{3}{15} - \frac{10}{15} = \frac{3 - 10}{15} = \frac{-7}{15}
\][/tex]
Since [tex]\(\frac{-7}{15}\)[/tex] is a quotient of two integers, it is a rational number.
D. [tex]\(-1.41421356 \ldots\)[/tex]
This number approximates the negative square root of 2, which is irrational. Adding this to [tex]\(\frac{1}{5}\)[/tex] will result in an irrational number.
Thus, the only number from the options given that produces a rational number when added to [tex]\(\frac{1}{5}\)[/tex] is:
C. [tex]\(-\frac{2}{3}\)[/tex]
