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Sagot :
To determine which number produces a rational number when added to [tex]\(\frac{1}{5}\)[/tex], let's evaluate each option step by step.
### Option A: [tex]\(\sqrt{11}\)[/tex]
[tex]\[ \frac{1}{5} + \sqrt{11} \][/tex]
Since [tex]\(\sqrt{11}\)[/tex] is an irrational number, the sum of a rational number ([tex]\(\frac{1}{5}\)[/tex]) and an irrational number ([tex]\(\sqrt{11}\)[/tex]) will always be irrational. Hence, this option does not produce a rational number.
### Option B: [tex]\(\pi\)[/tex]
[tex]\[ \frac{1}{5} + \pi \][/tex]
[tex]\(\pi\)[/tex] is also an irrational number, so the sum of [tex]\(\frac{1}{5}\)[/tex] and [tex]\(\pi\)[/tex] will be irrational. Therefore, this option does not produce a rational number.
### Option C: [tex]\(-\frac{2}{3}\)[/tex]
[tex]\[ \frac{1}{5} + (-\frac{2}{3}) \][/tex]
When we add these, we are dealing with the sum of two rational numbers. Compute the sum:
[tex]\[ \frac{1}{5} + \left( -\frac{2}{3} \right) = \frac{1}{5} - \frac{2}{3} \][/tex]
We need a common denominator to subtract these fractions. The least common multiple of 5 and 3 is 15. Convert each fraction:
[tex]\[ \frac{1}{5} = \frac{3}{15} \quad \text{and} \quad -\frac{2}{3} = -\frac{10}{15} \][/tex]
Now, subtract:
[tex]\[ \frac{3}{15} - \frac{10}{15} = \frac{3 - 10}{15} = \frac{-7}{15} \][/tex]
Since [tex]\(\frac{-7}{15}\)[/tex] is a rational number, this option works.
### Option D: [tex]\(-1.41421356 \ldots\)[/tex]
[tex]\[ \frac{1}{5} + (-1.41421356 \ldots) \][/tex]
The number [tex]\(-1.41421356 \ldots\)[/tex] (which is approximately [tex]\(-\sqrt{2}\)[/tex]) is an irrational number. The sum of a rational number and an irrational number will be irrational. So, this option does not produce a rational number.
### Conclusion
After evaluating each option, the correct answer is:
C. [tex]\(-\frac{2}{3}\)[/tex].
Adding [tex]\(-\frac{2}{3}\)[/tex] to [tex]\(\frac{1}{5}\)[/tex] produces a rational number, specifically [tex]\(\frac{-7}{15}\)[/tex].
### Option A: [tex]\(\sqrt{11}\)[/tex]
[tex]\[ \frac{1}{5} + \sqrt{11} \][/tex]
Since [tex]\(\sqrt{11}\)[/tex] is an irrational number, the sum of a rational number ([tex]\(\frac{1}{5}\)[/tex]) and an irrational number ([tex]\(\sqrt{11}\)[/tex]) will always be irrational. Hence, this option does not produce a rational number.
### Option B: [tex]\(\pi\)[/tex]
[tex]\[ \frac{1}{5} + \pi \][/tex]
[tex]\(\pi\)[/tex] is also an irrational number, so the sum of [tex]\(\frac{1}{5}\)[/tex] and [tex]\(\pi\)[/tex] will be irrational. Therefore, this option does not produce a rational number.
### Option C: [tex]\(-\frac{2}{3}\)[/tex]
[tex]\[ \frac{1}{5} + (-\frac{2}{3}) \][/tex]
When we add these, we are dealing with the sum of two rational numbers. Compute the sum:
[tex]\[ \frac{1}{5} + \left( -\frac{2}{3} \right) = \frac{1}{5} - \frac{2}{3} \][/tex]
We need a common denominator to subtract these fractions. The least common multiple of 5 and 3 is 15. Convert each fraction:
[tex]\[ \frac{1}{5} = \frac{3}{15} \quad \text{and} \quad -\frac{2}{3} = -\frac{10}{15} \][/tex]
Now, subtract:
[tex]\[ \frac{3}{15} - \frac{10}{15} = \frac{3 - 10}{15} = \frac{-7}{15} \][/tex]
Since [tex]\(\frac{-7}{15}\)[/tex] is a rational number, this option works.
### Option D: [tex]\(-1.41421356 \ldots\)[/tex]
[tex]\[ \frac{1}{5} + (-1.41421356 \ldots) \][/tex]
The number [tex]\(-1.41421356 \ldots\)[/tex] (which is approximately [tex]\(-\sqrt{2}\)[/tex]) is an irrational number. The sum of a rational number and an irrational number will be irrational. So, this option does not produce a rational number.
### Conclusion
After evaluating each option, the correct answer is:
C. [tex]\(-\frac{2}{3}\)[/tex].
Adding [tex]\(-\frac{2}{3}\)[/tex] to [tex]\(\frac{1}{5}\)[/tex] produces a rational number, specifically [tex]\(\frac{-7}{15}\)[/tex].
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