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The given grid seems to follow a pattern involving fractions. Let's analyze and find the missing values step by step.

First, let's observe the given values and identify any patterns, especially considering how they relate to each other both horizontally and vertically.

### First Row Analysis:
- \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{5}\), \(\frac{19}{20}\)

### Second Row Analysis:
- ?, \(\frac{2}{7}\), \(\frac{2}{7}\), \(\frac{19}{21}\)

### Third Row Analysis:
- \(\frac{2}{4}\), ?, \(\frac{3}{5}\), ?

### Fourth Row Analysis:
- \(\frac{4}{3}\), \(\frac{101}{84}\), ?, \(\frac{507}{140}\)

Let's start by solving one step at a time.

#### Third Row:
- We know the first value: \(\frac{2}{4} = \frac{1}{2}\)
- To identify a pattern, we can see if there is a connection horizontally.

#### Vertical Relationships:
- Consider the first column: \(\frac{1}{2}\), ?, \(\frac{2}{4}\), \(\frac{4}{3}\)
  - \(\frac{1}{2}\), \(\frac{2}{4}\), and \(\frac{4}{3}\) suggest a possible relationship in doubling the numerator.

#### Let's complete the second column vertically:
- We have \(\frac{1}{4}\), \(\frac{2}{7}\), ?, \(\frac{101}{84}\)
  - We see a fractional increment in the numerator and denominator.

Let's start solving each missing cell:

### Second Row, First Column:
Given: \(\frac{1}{2}\), ?, \(\frac{2}{4}\), \(\frac{4}{3}\)

The fractions increase in both numerator and denominator; a pattern emerges:
- Multiply numerator and denominator by 2 to move from \(\frac{1}{2}\) to \(\frac{2}{4}\)
- Thus the second row should be \(\frac{1 \cdot 2}{2 \cdot 2} = \frac{2}{4} = \frac{1}{2}\)

However, this seems too straightforward and suggests further relationships might be more complex.

### Second Column:
- \(\frac{1}{4}\), \(\frac{2}{7}\), ?, \(\frac{101}{84}\)
- The increase seems irregular, indicating a potentially different logic or relationship.

### Row-wise Pattern:
- Third row: Simplified, each fraction may have a horizontal pattern or relationship with each other.

#### Revising the relationships based on horizontal and vertical sequences:
- \(\frac{4}{3}\), \(\frac{101}{84}\), (101 + 4), 84 + 3 \(\frac{105}{87}\), simplification might be needed.

### Let's Try a Comprehensive Approach:
- Each element follows an arithmetic pattern or algebraic rule that needs recognition.

#### Solving Using Ratios:
- Utilize known fractions to find common ratios to deduce unknowns.

Let's start with each point systematically, considering patterns more comprehensively:

**Details or questions regarding these fractions and solving process needed, please provide:**
1. Could you provide more about the step-by-step approach needed to clarify?
2. Do you have specific arithmetic patterns in mind?

Here are five example math questions for additional practice:
1. How do we find the least common multiple (LCM) of two numbers?
2. What is the formula for solving quadratic equations?
3. How can we calculate the area of a triangle?
4. What is the principle of mathematical induction?
5. How do we perform polynomial long division?

**Tip:** Always cross-check your results by simplifying fractions and ensuring consistent patterns throughout rows and columns.

Explore the solution to a mathematical problem involving fraction patterns and arithmetic sequences. This problem covers concepts such as fractions, arithmetic sequences, and common mathematical formulas, including the LCM, formula for solving quadratic equations, area of a triangle, principle of mathematical induction, and polynomial long division. Suitable for students in Grades 10-12.

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