To simplify the given expression [tex]\(\frac{2}{5} + \frac{1}{6} + \frac{3}{5}\)[/tex] using the commutative property, we can reorder the terms in any way that makes the calculation easier.
We will combine the fractions with the same denominators first:
1. Observe that [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] have the same denominator. So, let's add these two first:
[tex]\[
\frac{2}{5} + \frac{3}{5} = \frac{2 + 3}{5} = \frac{5}{5} = 1
\][/tex]
2. Now, add [tex]\(\frac{1}{6}\)[/tex] to the result:
[tex]\[
1 + \frac{1}{6} = 1 \frac{1}{6}
\][/tex]
Thus, the simplified expression is [tex]\(1 \frac{1}{6}\)[/tex].
Let's verify the options:
A. [tex]\(\frac{2}{5}+\left(\frac{1}{6}+\frac{3}{5}\right)=\frac{2}{5}+\frac{23}{30}=\frac{35}{30}=1 \frac{1}{6}\)[/tex]
B. [tex]\(\frac{2}{5}+\frac{3}{5}+\frac{1}{6}=1+\frac{1}{6}=1 \frac{1}{6}\)[/tex]
C. [tex]\(\left(\frac{2}{5}+\frac{1}{6}\right)+\frac{3}{5}=\frac{17}{30}+\frac{18}{30}=\frac{35}{30}=1 \frac{1}{6}\)[/tex]
D. [tex]\(\frac{1}{30}(12+5+18)=\frac{1}{30}(35)=\frac{35}{30}=1 \frac{1}{6}\)[/tex]
All options correctly simplify to [tex]\(1 \frac{1}{6}\)[/tex]. Using the commutative property to combine the fractions effectively leads us to the answer [tex]\(1 \frac{1}{6}\)[/tex].
