Answer :
To determine which number, when added to [tex]\(\frac{1}{5}\)[/tex], results in a rational number, we need to recognize the properties of rational and irrational numbers.
1. Rational Numbers are numbers that can be expressed as the quotient of two integers (e.g., [tex]\(\frac{1}{5}\)[/tex] and [tex]\(-\frac{2}{3}\)[/tex]).
2. Irrational Numbers are numbers that cannot be expressed as the quotient of two integers (e.g., [tex]\(\pi\)[/tex], [tex]\(1.41421356 \ldots \text{(which approximates \(\sqrt{2}\)[/tex])}\), and [tex]\(\sqrt{11}\)[/tex]).
Adding two rational numbers always results in a rational number. Adding a rational number and an irrational number results in an irrational number.
Let's analyze each option:
### Option A: [tex]\(-\frac{2}{3}\)[/tex]
[tex]\[ \frac{1}{5} + \left( -\frac{2}{3} \right) = \frac{1}{5} - \frac{2}{3} = \frac{1 \cdot 3 - 2 \cdot 5}{5 \cdot 3} = \frac{3 - 10}{15} = \frac{-7}{15} \][/tex]
[tex]\(\frac{-7}{15}\)[/tex] is a rational number, as it is the quotient of two integers.
### Option B: [tex]\(\pi\)[/tex]
[tex]\[ \frac{1}{5} + \pi \text{ is irrational because }\pi \text{ is an irrational number.} \][/tex]
### Option C: [tex]\(-1.41421356 \ldots\)[/tex] (approximately [tex]\(-\sqrt{2}\)[/tex])
[tex]\[ \frac{1}{5} - 1.41421356 \ldots \text{ is irrational because } -\sqrt{2} \text{ is an irrational number.} \][/tex]
### Option D: [tex]\(\sqrt{11}\)[/tex]
[tex]\[ \frac{1}{5} + \sqrt{11} \text{ is irrational because } \sqrt{11} \text{ is an irrational number.} \][/tex]
Hence, the only option that produces a rational number when added to [tex]\(\frac{1}{5}\)[/tex] is:
[tex]\[ \boxed{-\frac{2}{3}} \][/tex]
1. Rational Numbers are numbers that can be expressed as the quotient of two integers (e.g., [tex]\(\frac{1}{5}\)[/tex] and [tex]\(-\frac{2}{3}\)[/tex]).
2. Irrational Numbers are numbers that cannot be expressed as the quotient of two integers (e.g., [tex]\(\pi\)[/tex], [tex]\(1.41421356 \ldots \text{(which approximates \(\sqrt{2}\)[/tex])}\), and [tex]\(\sqrt{11}\)[/tex]).
Adding two rational numbers always results in a rational number. Adding a rational number and an irrational number results in an irrational number.
Let's analyze each option:
### Option A: [tex]\(-\frac{2}{3}\)[/tex]
[tex]\[ \frac{1}{5} + \left( -\frac{2}{3} \right) = \frac{1}{5} - \frac{2}{3} = \frac{1 \cdot 3 - 2 \cdot 5}{5 \cdot 3} = \frac{3 - 10}{15} = \frac{-7}{15} \][/tex]
[tex]\(\frac{-7}{15}\)[/tex] is a rational number, as it is the quotient of two integers.
### Option B: [tex]\(\pi\)[/tex]
[tex]\[ \frac{1}{5} + \pi \text{ is irrational because }\pi \text{ is an irrational number.} \][/tex]
### Option C: [tex]\(-1.41421356 \ldots\)[/tex] (approximately [tex]\(-\sqrt{2}\)[/tex])
[tex]\[ \frac{1}{5} - 1.41421356 \ldots \text{ is irrational because } -\sqrt{2} \text{ is an irrational number.} \][/tex]
### Option D: [tex]\(\sqrt{11}\)[/tex]
[tex]\[ \frac{1}{5} + \sqrt{11} \text{ is irrational because } \sqrt{11} \text{ is an irrational number.} \][/tex]
Hence, the only option that produces a rational number when added to [tex]\(\frac{1}{5}\)[/tex] is:
[tex]\[ \boxed{-\frac{2}{3}} \][/tex]
