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Can you solve this equation and show the complete detailed solution 

The image contains a mathematical expression that needs to be simplified. Let's analyze and solve it.

The expression appears to be:

\[
\frac{\frac{1}{7}}{\frac{1}{4}} + \frac{\frac{2}{5}}{\frac{1}{2}} - \frac{\frac{3}{4}}{\frac{1}{6}} + \frac{\frac{1}{2}}{\frac{3}{2}}
\]

### Step 1: Simplify each fraction division
To simplify each term, recall that dividing by a fraction is equivalent to multiplying by its reciprocal.

1. \(\frac{\frac{1}{7}}{\frac{1}{4}} = \frac{1}{7} \times \frac{4}{1} = \frac{4}{7}\)
2. \(\frac{\frac{2}{5}}{\frac{1}{2}} = \frac{2}{5} \times \frac{2}{1} = \frac{4}{5}\)
3. \(\frac{\frac{3}{4}}{\frac{1}{6}} = \frac{3}{4} \times \frac{6}{1} = \frac{18}{4} = \frac{9}{2}\)
4. \(\frac{\frac{1}{2}}{\frac{3}{2}} = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3}\)

### Step 2: Combine the simplified fractions
Now, sum and subtract the simplified fractions:

\[
\frac{4}{7} + \frac{4}{5} - \frac{9}{2} + \frac{1}{3}
\]

### Step 3: Find the common denominator
The denominators are 7, 5, 2, and 3. The least common denominator (LCD) for these numbers is 210.

\[
\text{Convert each fraction to have the denominator 210}:
\]

1. \(\frac{4}{7} = \frac{4 \times 30}{7 \times 30} = \frac{120}{210}\)
2. \(\frac{4}{5} = \frac{4 \times 42}{5 \times 42} = \frac{168}{210}\)
3. \(\frac{9}{2} = \frac{9 \times 105}{2 \times 105} = \frac{945}{210}\)
4. \(\frac{1}{3} = \frac{1 \times 70}{3 \times 70} = \frac{70}{210}\)

### Step 4: Combine the fractions
Now, we can combine all the fractions:

\[
\frac{120}{210} + \frac{168}{210} - \frac{945}{210} + \frac{70}{210} = \frac{(120 + 168 - 945 + 70)}{210}
\]

Simplify the numerator:

\[
120 + 168 + 70 = 358
\]

\[
358 - 945 = -587
\]

\[
\frac{-587}{210}
\]

### Step 5: Simplify the result
The final simplified fraction is \(\frac{-587}{210}\), which is already in its simplest form.

### Final Answer
The simplified expression is:

\[
\boxed{\frac{-587}{210}}
\]

Would you like to see details about any particular step or have any other questions?

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Here are five related questions:

1. How can we simplify complex fractions without finding the least common denominator?
2. What is the process of finding the least common denominator for fractions with large numbers?
3. How can we simplify fractions when dealing with negative numerators or denominators?
4. What strategies can help in identifying the best order of operations when simplifying expressions?
5. How can we apply this method to solving algebraic fractions?

**Tip:** Always look for the least common denominator when adding or subtracting fractions to simplify the process.

Learn how to solve a complex fraction expression step-by-step with detailed explanations and examples. This problem covers fraction simplification and arithmetic operations. Suitable for students in Grades 6-8.

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