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To solve the given proportion, we need to break down and simplify both sides of the equation. Let's rewrite the equation:

\[
\frac{x}{10.5 \cdot 0.24 - 2.02} = \frac{9 \cdot \left(\frac{11}{20} - 0.945 : 0.9\right)}{\frac{3}{40} - \frac{3}{8} \cdot 7}
\]

### Step 1: Simplify the denominator on the left-hand side

First, calculate \( 10.5 \cdot 0.24 \):

\[
10.5 \cdot 0.24 = 2.52
\]

Then subtract 2.02 from 2.52:

\[
2.52 - 2.02 = 0.5
\]

So, the left-hand side of the equation becomes:

\[
\frac{x}{0.5}
\]

### Step 2: Simplify the numerator on the right-hand side

First, let's solve inside the parentheses \( \left(\frac{11}{20} - 0.945 : 0.9\right) \).

Calculate \( 0.945 : 0.9 \) (which is division):

\[
0.945 \div 0.9 = 1.05
\]

So, the expression inside the parentheses becomes:

\[
\frac{11}{20} - 1.05
\]

Convert \(\frac{11}{20}\) to a decimal:

\[
\frac{11}{20} = 0.55
\]

Now subtract:

\[
0.55 - 1.05 = -0.5
\]

Thus, the numerator becomes:

\[
9 \cdot (-0.5) = -4.5
\]

### Step 3: Simplify the denominator on the right-hand side

Calculate \( \frac{3}{8} \cdot 7 \):

\[
\frac{3}{8} \cdot 7 = \frac{21}{8} = 2.625
\]

Now subtract \( 2.625 \) from \( \frac{3}{40} \):

Convert \( \frac{3}{40} \) to a decimal:

\[
\frac{3}{40} = 0.075
\]

Now subtract:

\[
0.075 - 2.625 = -2.55
\]

So, the right-hand side of the equation becomes:

\[
\frac{-4.5}{-2.55}
\]

### Step 4: Simplify the right-hand side

Finally, simplify the expression:

\[
\frac{-4.5}{-2.55} \approx 1.7647
\]

### Step 5: Set up the proportion

Now we have:

\[
\frac{x}{0.5} = 1.7647
\]

### Step 6: Solve for \(x\)

Multiply both sides by 0.5 to solve for \(x\):

\[
x = 1.7647 \cdot 0.5
\]

\[
x \approx 0.88235
\]

Thus, the solution is:

\[
x \approx 0.882
\]

Would you like more details on any part of the solution or have any questions?

### Related Questions:
1. How do you convert fractions to decimals?
2. What are the steps to simplify complex expressions?
3. How can you solve a proportion involving multiple operations?
4. What methods can be used to check the solution of an equation?
5. How do rounding and approximation affect the final answer?

### Tip:
Always carefully check each arithmetic step when simplifying complex expressions to avoid calculation errors.

Learn how to solve a complex proportion problem involving decimal and fraction arithmetic. Follow a detailed step-by-step solution to understand the calculation process. Suitable for high school students.

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