5/39.(39/5.5/3+25/3.39/5)+(-30/7.7/11+7/11.16/3).5-17/3-(268/15-24/7)-(42/15-4)-(145/69.36/29):5/23+25.0,375.(-10).1

Let's break down the given expression step by step to solve it accurately.

The expression is:

\[
\frac{5}{39} \cdot \left(\frac{39}{5.5 \cdot 3} + \frac{25}{3} \cdot \frac{39}{5}\right) + \left(-\frac{30}{7} \cdot \frac{7}{11} + \frac{7}{11} \cdot \frac{16}{3}\right) \cdot 5 - \frac{17}{3} - \left(\frac{268}{15} - \frac{24}{7}\right) - \left(\frac{42}{15} - 4\right) - \frac{145}{\frac{69 \cdot 36}{29}} \cdot \frac{5}{23} + 25 \cdot 0.375 \cdot (-10) \cdot 1
\]

### Step 1: Simplify each part of the expression

1. \(\frac{39}{5.5 \cdot 3}\) and \(\frac{25}{3} \cdot \frac{39}{5}\)
2. \(-\frac{30}{7} \cdot \frac{7}{11} + \frac{7}{11} \cdot \frac{16}{3}\)
3. \(\frac{268}{15} - \frac{24}{7}\)
4. \(\frac{42}{15} - 4\)
5. \(\frac{145}{\frac{69 \cdot 36}{29}}\)
6. \(25 \cdot 0.375 \cdot (-10) \cdot 1\)

### Step 2: Perform the calculations

Let's compute the values.

#### Part 1: \(\frac{39}{5.5 \cdot 3} + \frac{25}{3} \cdot \frac{39}{5}\)

\[
\frac{39}{5.5 \cdot 3} = \frac{39}{16.5} = 2.3636
\]

\[
\frac{25}{3} \cdot \frac{39}{5} = 8.3333 \cdot 7.8 = 65.0007
\]

So, \(\frac{39}{5.5 \cdot 3} + \frac{25}{3} \cdot \frac{39}{5} = 2.3636 + 65.0007 = 67.3643\).

#### Part 2: \(-\frac{30}{7} \cdot \frac{7}{11} + \frac{7}{11} \cdot \frac{16}{3}\)

\[
-\frac{30}{7} \cdot \frac{7}{11} = -\frac{30}{11} = -2.7273
\]

\[
\frac{7}{11} \cdot \frac{16}{3} = \frac{112}{33} = 3.3939
\]

So, \(-2.7273 + 3.3939 = 0.6666\).

#### Part 3: \(\frac{268}{15} - \frac{24}{7}\)

\[
\frac{268}{15} = 17.8667, \quad \frac{24}{7} = 3.4286
\]

\[
17.8667 - 3.4286 = 14.4381
\]

#### Part 4: \(\frac{42}{15} - 4\)

\[
\frac{42}{15} = 2.8, \quad 2.8 - 4 = -1.2
\]

#### Part 5: \(\frac{145}{\frac{69 \cdot 36}{29}}\)

\[
69 \cdot 36 = 2484, \quad \frac{2484}{29} = 85.6552
\]

\[
\frac{145}{85.6552} = 1.6933
\]

#### Part 6: \(25 \cdot 0.375 \cdot (-10) \cdot 1\)

\[
25 \cdot 0.375 = 9.375, \quad 9.375 \cdot (-10) = -93.75
\]

### Step 3: Plug these back into the main expression

\[
\frac{5}{39} \cdot 67.3643 + 0.6666 \cdot 5 - \frac{17}{3} - 14.4381 - (-1.2) - 1.6933 \cdot \frac{5}{23} + (-93.75)
\]

### Step 4: Final Calculation

Compute the above expression to get the final answer:

1. \(\frac{5}{39} \cdot 67.3643 = 8.640\)
2. \(0.6666 \cdot 5 = 3.333\)
3. \(\frac{17}{3} = 5.667\)
4. \(\frac{5}{23} \cdot 1.6933 = 0.368\)

Final Calculation:

\[
8.640 + 3.333 - 5.667 - 14.438 + 1.2 - 0.368 - 93.75 = -101.050
\]

### Final Answer:
\(\boxed{-101.05}\)

Would you like more details, or do you have any other questions?

---

1. What are the basic rules for simplifying fractions?
2. How do you handle calculations involving mixed operations (addition, subtraction, multiplication, division)?
3. What is the significance of order of operations in complex expressions?
4. How do you manage fractional coefficients in equations?
5. How do you approach simplifying complex algebraic expressions?
6. What methods can be used to check the accuracy of a solution in algebra?
7. How do you simplify expressions involving both positive and negative numbers?
8. What are common mistakes to avoid when simplifying mathematical expressions?

**Tip:** Always ensure to perform calculations step by step, following the correct order of operations to avoid mistakes.

Learn how to solve a complex fractional expression step by step with detailed calculations and explanations. This problem involves fractions, mixed operations, negative numbers, and follows the order of operations. Suitable for high school students in Grades 9-12.

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