Sure! Let's break this down step-by-step.
First, we need to convert the mixed fraction [tex]\(5 \frac{1}{6}\)[/tex] into an improper fraction.
To convert a mixed fraction to an improper fraction:
1. Multiply the whole number part by the denominator of the fractional part.
2. Add the numerator of the fractional part.
3. Write the result over the original denominator.
For [tex]\(5 \frac{1}{6}\)[/tex]:
1. [tex]\(5 \times 6 = 30\)[/tex]
2. [tex]\(30 + 1 = 31\)[/tex]
3. So, [tex]\(5 \frac{1}{6} = \frac{31}{6}\)[/tex].
Next, we need to multiply this improper fraction by [tex]\(-\frac{2}{5}\)[/tex].
To multiply two fractions:
1. Multiply the numerators together.
2. Multiply the denominators together.
3. Simplify if necessary.
So,
1. Multiply the numerators: [tex]\(31 \times -2 = -62\)[/tex]
2. Multiply the denominators: [tex]\(6 \times 5 = 30\)[/tex]
Thus, we get [tex]\(\frac{-62}{30}\)[/tex].
Finally, we simplify the fraction if needed. The fraction [tex]\(\frac{-62}{30}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
[tex]\[
\frac{-62}{30} = \frac{-62 \div 2}{30 \div 2} = \frac{-31}{15}
\][/tex]
Therefore, the answer is
[tex]\[
\boxed{-31/15}
\][/tex]
