Let's solve the expression step by step. The expression we need to evaluate is:
[tex]\[
\left(\frac{6}{5} - \frac{1}{3}\right) \cdot \frac{2}{3}
\][/tex]
### Step 1: Evaluate the subtraction within the parentheses
First, we need to find a common denominator to subtract [tex]\(\frac{1}{3}\)[/tex] from [tex]\(\frac{6}{5}\)[/tex]. The least common denominator (LCD) of 5 and 3 is 15. So, we rewrite the fractions with a denominator of 15:
[tex]\[
\frac{6}{5} = \frac{6 \times 3}{5 \times 3} = \frac{18}{15}
\][/tex]
[tex]\[
\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}
\][/tex]
Now we perform the subtraction:
[tex]\[
\frac{18}{15} - \frac{5}{15} = \frac{18 - 5}{15} = \frac{13}{15}
\][/tex]
The result of the subtraction is [tex]\(\frac{13}{15}\)[/tex].
### Step 2: Multiply the result by [tex]\(\frac{2}{3}\)[/tex]
Next, we need to multiply this result by [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\left(\frac{13}{15}\right) \cdot \frac{2}{3} = \frac{13 \cdot 2}{15 \cdot 3} = \frac{26}{45}
\][/tex]
So, the result of the expression is [tex]\(\frac{26}{45}\)[/tex].
### Summary of Numerical Values
Let's summarize the numerical values:
1. [tex]\(\frac{6}{5} = 1.2\)[/tex]
2. [tex]\(\frac{1}{3} \approx 0.3333333333333333\)[/tex]
3. [tex]\(\frac{2}{3} \approx 0.6666666666666666\)[/tex]
4. [tex]\(\frac{13}{15} \approx 0.8666666666666667\)[/tex]
5. [tex]\(\frac{26}{45} \approx 0.5777777777777777\)[/tex]
Thus, the final evaluated result of the expression is:
[tex]\[
\left(\frac{6}{5} - \frac{1}{3}\right) \cdot \frac{2}{3} \approx 0.5777777777777777
\][/tex]
Therefore, the detailed numerical result of the operation is approximately [tex]\(0.5777777777777777\)[/tex].
