To find the product of [tex]\(5 \frac{1}{6} \cdot \left(-\frac{2}{5}\right)\)[/tex], follow these steps:
1. Convert the mixed number [tex]\(5 \frac{1}{6}\)[/tex] to an improper fraction.
[tex]\[5 \frac{1}{6} = 5 + \frac{1}{6}\][/tex]
To express [tex]\(5\)[/tex] as a fraction with a denominator of 6:
[tex]\[5 = \frac{30}{6}\][/tex]
Therefore,
[tex]\[5 \frac{1}{6} = \frac{30}{6} + \frac{1}{6} = \frac{31}{6}\][/tex]
2. Multiply the improper fraction [tex]\(\frac{31}{6}\)[/tex] by the negative fraction [tex]\(-\frac{2}{5}\)[/tex]:
[tex]\[\frac{31}{6} \cdot \left(-\frac{2}{5}\right) = \frac{31 \cdot (-2)}{6 \cdot 5} = \frac{-62}{30}\][/tex]
3. Simplify the resulting fraction. The greatest common divisor of 62 and 30 is 2, so divide both the numerator and the denominator by 2:
[tex]\[\frac{-62}{30} = \frac{-62 \div 2}{30 \div 2} = \frac{-31}{15}\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{\frac{-31}{15}}
\][/tex]
