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To solve the given expression for \( Q \):

\[
Q = -0.25 - \frac{1}{3} \left[ -\frac{1}{3} + 5 \cdot \frac{1}{3} - 8 \right] + (3.5) \div \frac{8}{9}
\]

we'll break it down step-by-step:

1. **Evaluate inside the brackets \([- \frac{1}{3} + 5 \cdot \frac{1}{3} - 8]\):**

   \[
   -\frac{1}{3} + 5 \cdot \frac{1}{3} = -\frac{1}{3} + \frac{5}{3} = \frac{4}{3}
   \]

   Now, subtract 8:

   \[
   \frac{4}{3} - 8 = \frac{4}{3} - \frac{24}{3} = \frac{4 - 24}{3} = -\frac{20}{3}
   \]

2. **Multiply \(-\frac{1}{3}\) with the result:**

   \[
   -\frac{1}{3} \cdot \left(-\frac{20}{3}\right) = \frac{20}{9}
   \]

3. **Calculate \( (3.5) \div \frac{8}{9} \):**

   Dividing by a fraction is equivalent to multiplying by its reciprocal:

   \[
   3.5 \div \frac{8}{9} = 3.5 \times \frac{9}{8} = \frac{7}{2} \times \frac{9}{8} = \frac{63}{16}
   \]

4. **Now substitute back into the original equation:**

   \[
   Q = -0.25 + \frac{20}{9} + \frac{63}{16}
   \]

   Convert \(-0.25\) to a fraction:

   \[
   -0.25 = -\frac{1}{4}
   \]

5. **Find a common denominator for \(-\frac{1}{4}\), \(\frac{20}{9}\), and \(\frac{63}{16}\):**

   The common denominator is 144.

   \[
   -\frac{1}{4} = -\frac{36}{144}, \quad \frac{20}{9} = \frac{320}{144}, \quad \frac{63}{16} = \frac{567}{144}
   \]

6. **Add them together:**

   \[
   Q = -\frac{36}{144} + \frac{320}{144} + \frac{567}{144}
   \]

   Combine the fractions:

   \[
   Q = \frac{-36 + 320 + 567}{144} = \frac{851}{144}
   \]

   Convert it to a decimal:

   \[
   Q \approx 5.91
   \]

Therefore, the value of \( Q \approx 5.91 \).

Would you like more details or have any questions?

### Further Questions:
1. How can we simplify complex fractions?
2. What are the rules for order of operations when dealing with brackets?
3. How to convert between fractions and decimals?
4. What are some common methods to find the least common denominator?
5. How can we check our work in calculations involving multiple operations?

### Tip:
Always remember to handle operations inside brackets first and work your way outward, following the order of operations.

Learn how to solve a complex arithmetic expression step-by-step with detailed explanations and calculations. This problem covers arithmetic operations, fraction operations, and the order of operations rule. Suitable for students in Grades 7-9.

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