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?

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.