Certainly! Let's evaluate the given expression step-by-step. The given mathematical expression is:
[tex]\[
\frac{-1}{6} \times \frac{4}{7} + \frac{1}{2} \times \frac{-3}{7} \times \frac{1}{6}
\][/tex]
First, let's handle each fraction multiplication separately.
### First Term:
The first term is:
[tex]\[
\frac{-1}{6} \times \frac{4}{7}
\][/tex]
Multiplying the numerators:
[tex]\[
-1 \times 4 = -4
\][/tex]
Multiplying the denominators:
[tex]\[
6 \times 7 = 42
\][/tex]
So, the first term becomes:
[tex]\[
\frac{-4}{42} = \frac{-2}{21} \approx -0.09523809523809523
\][/tex]
### Second Term:
The second term involves calculating the product of three fractions. Let's first find:
[tex]\[
\frac{-3}{7} \times \frac{1}{6}
\][/tex]
Multiplying the numerators:
[tex]\[
-3 \times 1 = -3
\][/tex]
Multiplying the denominators:
[tex]\[
7 \times 6 = 42
\][/tex]
So, this part becomes:
[tex]\[
\frac{-3}{42} = \frac{-1}{14} \approx -0.07142857142857142
\][/tex]
Now we need to multiply this result by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
\frac{1}{2} \times \frac{-1}{14}
\][/tex]
Multiplying the numerators:
[tex]\[
1 \times -1 = -1
\][/tex]
Multiplying the denominators:
[tex]\[
2 \times 14 = 28
\][/tex]
So, this entire term becomes:
[tex]\[
\frac{-1}{28} \approx -0.03571428571428571
\][/tex]
### Adding the Two Terms:
Now we need to add the results of the two terms:
[tex]\[
\frac{-2}{21} + \frac{-1}{28}
\][/tex]
First, let's convert them to a common denominator. The least common multiple of 21 and 28 is 84.
Converting [tex]\(\frac{-2}{21}\)[/tex]:
[tex]\[
\frac{-2}{21} = \frac{-2 \times 4}{21 \times 4} = \frac{-8}{84}
\][/tex]
Converting [tex]\(\frac{-1}{28}\)[/tex]:
[tex]\[
\frac{-1}{28} = \frac{-1 \times 3}{28 \times 3} = \frac{-3}{84}
\][/tex]
Now, adding these fractions:
[tex]\[
\frac{-8}{84} + \frac{-3}{84} = \frac{-8 - 3}{84} = \frac{-11}{84} \approx -0.13095238095238093
\][/tex]
Thus, the final result of the given expression is:
[tex]\[
\boxed{-0.13095238095238093}
\][/tex]
This result matches our initial calculations.
