The given expression is:

$-\left[ \frac{1 \frac{4}{5}}{0.3 \times 1.2} \right] - \frac{5}{6}$

Let's break this down step by step.

1. Convert mixed number: $1 \frac{4}{5} = \frac{9}{5}$

2. Simplify the multiplication inside the brackets: $0.3 \times 1.2 = 0.36$

3. Division: $\frac{9}{5} \div 0.36 = \frac{9}{5} \times \frac{1}{0.36}$ Converting $0.36$ into a fraction: $0.36 = \frac{9}{25}$ So: $\frac{9}{5} \times \frac{25}{9} = \frac{9 \times 25}{5 \times 9} = \frac{25}{5} = 5$

4. Apply the negative: $-[5] = -5$

5. Subtract the second term: $-5 - \frac{5}{6}$ To subtract these, convert $-5$ into a fraction with a denominator of 6: $-5 = \frac{-30}{6}$ So: $\frac{-30}{6} - \frac{5}{6} = \frac{-35}{6}$

6. Convert to a mixed number: $\frac{-35}{6} = -5 \frac{5}{6}$

Final Answer:

The value of the expression is: $-5 \frac{5}{6}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you convert a mixed number to an improper fraction?
2. What is the process for dividing fractions by decimals?
3. How do you subtract fractions with different denominators?
4. Why is it important to convert results to mixed numbers?
5. Can you explain the rules for applying negative signs in expressions?

Tip: Always simplify fractions at each step to avoid dealing with large numbers later.