To find the least common denominator (LCD) of the fractions [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{4}{6}\)[/tex], and [tex]\(\frac{1}{2}\)[/tex], we need to determine the smallest common multiple of their denominators.
Step-by-Step Solution:
1. Identify the denominators:
The denominators of the fractions are 3, 6, and 2.
2. List the multiples of each denominator:
- Multiples of 3: [tex]\( 3, 6, 9, 12, 15, \ldots \)[/tex]
- Multiples of 6: [tex]\( 6, 12, 18, \ldots \)[/tex]
- Multiples of 2: [tex]\( 2, 4, 6, 8, 10, 12, \ldots \)[/tex]
3. Find the common multiples:
- Common multiples of 3, 6, and 2 are the numbers that appear in all lists. The first common multiple they share is 6.
4. Determine the smallest common multiple:
The smallest common multiple of 3, 6, and 2 is 6.
Thus, the least common denominator for the fractions [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{4}{6}\)[/tex], and [tex]\(\frac{1}{2}\)[/tex] is [tex]\(6\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{6}
\][/tex]
