To solve the given problem, [tex]\( 5 \frac{1}{6} \cdot \left(-\frac{2}{5}\right) \)[/tex], let's follow these steps:
1. Convert the mixed number to an improper fraction:
The mixed number is [tex]\( 5 \frac{1}{6} \)[/tex].
- Multiply the whole number part (5) by the denominator of the fraction part (6): [tex]\( 5 \times 6 = 30 \)[/tex].
- Add the result to the numerator of the fraction part: [tex]\( 30 + 1 = 31 \)[/tex].
- Put this sum over the original fraction’s denominator: [tex]\( \frac{31}{6} \)[/tex].
So, [tex]\( 5 \frac{1}{6} = \frac{31}{6} \)[/tex].
2. Identify the fraction to be multiplied:
The other fraction is [tex]\( -\frac{2}{5} \)[/tex].
3. Multiply the fractions:
- Multiply the numerators: [tex]\( 31 \times -2 = -62 \)[/tex].
- Multiply the denominators: [tex]\( 6 \times 5 = 30 \)[/tex].
So, [tex]\(\frac{31}{6} \cdot \left(-\frac{2}{5}\right) = \frac{-62}{30} \)[/tex].
4. Simplify the result:
Ideally, we should simplify the fraction to its simplest form.
- Find the greatest common divisor (GCD) of 62 and 30, which is 2.
- Divide both the numerator and the denominator by their GCD: [tex]\( \frac{-62 \div 2}{30 \div 2} = \frac{-31}{15} \)[/tex].
Therefore, the final answer is:
[tex]\[ \boxed{-31/15} \][/tex]
