To solve the expression [tex]\(\frac{2}{3y} + \frac{1}{6y}\)[/tex], we need to find a common denominator for the two fractions.
### Step-by-Step Solution:
1. Identify the denominators:
The denominators are [tex]\(3y\)[/tex] and [tex]\(6y\)[/tex].
2. Find the least common denominator (LCD):
To add fractions, the denominators need to be the same. The least common multiple of [tex]\(3y\)[/tex] and [tex]\(6y\)[/tex] is [tex]\(6y\)[/tex].
3. Rewrite each fraction with the common denominator:
- For the fraction [tex]\(\frac{2}{3y}\)[/tex], we need to adjust the denominator to [tex]\(6y\)[/tex]. We can do this by multiplying the numerator and denominator by 2:
[tex]\[
\frac{2}{3y} = \frac{2 \cdot 2}{3y \cdot 2} = \frac{4}{6y}
\][/tex]
- The fraction [tex]\(\frac{1}{6y}\)[/tex] already has the denominator of [tex]\(6y\)[/tex], so it remains the same:
[tex]\[
\frac{1}{6y} = \frac{1}{6y}
\][/tex]
4. Add the fractions:
Now that both fractions have a common denominator, we can add them by adding their numerators:
[tex]\[
\frac{4}{6y} + \frac{1}{6y} = \frac{4 + 1}{6y} = \frac{5}{6y}
\][/tex]
5. Conclusion:
The result of the addition [tex]\(\frac{2}{3y} + \frac{1}{6y}\)[/tex] is [tex]\(\frac{5}{6y}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{5}{6y}}
\][/tex] which corresponds to option D.
