Answer :
To determine which number produces a rational number when multiplied by \(\frac{1}{3}\), we need to examine each option individually. Let's analyze each option step-by-step.
### Option A: \(\sqrt{12}\)
The number \(\sqrt{12}\) represents the square root of 12.
- The square root of any number that is not a perfect square is an irrational number.
- \( \sqrt{12} \) is not a perfect square since 12 can only be factored into \(\sqrt{4 \cdot 3} = 2\sqrt{3}\), and \(\sqrt{3}\) is irrational.
Hence, multiplying an irrational number by \(\frac{1}{3}\) will not yield a rational number.
### Option B: \(2.236067978 \ldots\)
The number \(2.236067978 \ldots\) is a decimal approximation.
- Specifically, this approximation is very close to \(\sqrt{5}\), which is an irrational number.
- Because \(\sqrt{5}\) is irrational, the exact value cannot be expressed as a simple fraction.
Multiplying an irrational number by \(\frac{1}{3}\) results in another irrational number.
### Option C: \(\frac{3}{7}\)
The number \(\frac{3}{7}\) is already presented as a fraction, which is a rational number.
- Rational numbers are numbers that can be expressed as the quotient of two integers.
- Multiplying two rational numbers results in another rational number.
In this case, multiplying \(\frac{3}{7}\) by \(\frac{1}{3}\):
[tex]\[ \frac{1}{3} \times \frac{3}{7} = \frac{1 \times 3}{3 \times 7} = \frac{3}{21} = \frac{1}{7} \][/tex]
Since \(\frac{1}{7}\) is a rational number, Option C produces a rational number when multiplied by \(\frac{1}{3}\).
### Option D: \(\pi\)
The number \(\pi\) (pi) is a well-known irrational number.
- \(\pi\) cannot be expressed as a precise fraction of two integers.
Multiplying an irrational number by \(\frac{1}{3}\) results in another irrational number.
### Conclusion
Among the given options, the only number that remains rational when multiplied by \(\frac{1}{3}\) is Option C, \(\frac{3}{7}\).
Therefore, the answer is C. [tex]\(\frac{3}{7}\)[/tex].
### Option A: \(\sqrt{12}\)
The number \(\sqrt{12}\) represents the square root of 12.
- The square root of any number that is not a perfect square is an irrational number.
- \( \sqrt{12} \) is not a perfect square since 12 can only be factored into \(\sqrt{4 \cdot 3} = 2\sqrt{3}\), and \(\sqrt{3}\) is irrational.
Hence, multiplying an irrational number by \(\frac{1}{3}\) will not yield a rational number.
### Option B: \(2.236067978 \ldots\)
The number \(2.236067978 \ldots\) is a decimal approximation.
- Specifically, this approximation is very close to \(\sqrt{5}\), which is an irrational number.
- Because \(\sqrt{5}\) is irrational, the exact value cannot be expressed as a simple fraction.
Multiplying an irrational number by \(\frac{1}{3}\) results in another irrational number.
### Option C: \(\frac{3}{7}\)
The number \(\frac{3}{7}\) is already presented as a fraction, which is a rational number.
- Rational numbers are numbers that can be expressed as the quotient of two integers.
- Multiplying two rational numbers results in another rational number.
In this case, multiplying \(\frac{3}{7}\) by \(\frac{1}{3}\):
[tex]\[ \frac{1}{3} \times \frac{3}{7} = \frac{1 \times 3}{3 \times 7} = \frac{3}{21} = \frac{1}{7} \][/tex]
Since \(\frac{1}{7}\) is a rational number, Option C produces a rational number when multiplied by \(\frac{1}{3}\).
### Option D: \(\pi\)
The number \(\pi\) (pi) is a well-known irrational number.
- \(\pi\) cannot be expressed as a precise fraction of two integers.
Multiplying an irrational number by \(\frac{1}{3}\) results in another irrational number.
### Conclusion
Among the given options, the only number that remains rational when multiplied by \(\frac{1}{3}\) is Option C, \(\frac{3}{7}\).
Therefore, the answer is C. [tex]\(\frac{3}{7}\)[/tex].
