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The problem in the first image shows a proportion that needs to be solved for the missing term, represented by dots. Here's the given proportion:

\[
\frac{\frac{4 \frac{1}{3}}{\frac{1}{9}}} = \frac{\dots}{\frac{4 \frac{2}{3}}{5}}
\]

### Step-by-Step Solution:

#### 1. Convert mixed numbers to improper fractions:
- \( 4 \frac{1}{3} = \frac{13}{3} \)
- \( 4 \frac{2}{3} = \frac{14}{3} \)

So, the proportion becomes:

\[
\frac{\frac{\frac{13}{3}}{\frac{1}{9}}} = \frac{\dots}{\frac{\frac{14}{3}}{5}}
\]

#### 2. Simplify the complex fractions:
- \( \frac{\frac{13}{3}}{\frac{1}{9}} = \frac{13}{3} \times \frac{9}{1} = \frac{13 \times 9}{3 \times 1} = \frac{117}{3} = 39 \)
- \( \frac{\frac{14}{3}}{5} = \frac{14}{3} \times \frac{1}{5} = \frac{14 \times 1}{3 \times 5} = \frac{14}{15} \)

Now the equation becomes:

\[
39 = \frac{\dots}{\frac{14}{15}}
\]

#### 3. Solve for the missing term (\(\dots\)):
- Multiply both sides by \( \frac{14}{15} \) to isolate the missing term:

\[
\dots = 39 \times \frac{14}{15}
\]

#### 4. Calculate the missing term:
- \( 39 \times \frac{14}{15} = \frac{39 \times 14}{15} = \frac{546}{15} = 36.4 \)

### Final Answer:
The missing term is \( 36.4 \).

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions to explore this concept further:
1. How do you simplify complex fractions?
2. What is the process of converting mixed numbers to improper fractions?
3. How can proportions be applied in real-world problems?
4. What is the significance of cross-multiplication in solving proportions?
5. What are some alternative methods for solving proportions?

**Tip:** When solving proportions involving mixed numbers, always convert them to improper fractions first to make calculations easier.

Solve the proportion for the missing value represented by a letter (represented as dots in the image). The proportion involves mixed fractions and is displayed as: (4 1/3)/(1/9) = (5/...)/(4 2/3).

Learn how to solve a proportion involving mixed numbers and complex fractions. The problem includes converting mixed numbers to improper fractions, simplifying complex fractions, and solving for the unknown term using cross-multiplication. Suitable for students in Grades 8-10.

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