https://r2.erweima.ai/mathbot/image/1717951438860u89lfmvn.png

The extracted text seems a bit unclear due to OCR inaccuracies. However, I can analyze the visible fractions from the image and fill in the missing values.

Here is the extracted table with identified fractions and spaces for missing values:

\[
\begin{array}{|c|c|c|c|c|}
\hline
\frac{1}{4} & \frac{1}{2} & \frac{1}{2} & \frac{5}{4} \\
\hline
\frac{1}{2} & \frac{2}{7} & \frac{1}{4} & \frac{103}{140} \\
\hline
\frac{2}{5} & \frac{3}{4} & & \frac{29}{28} \\
\hline
\frac{17}{20} & \frac{9}{7} & \frac{7}{4} & \frac{509}{140} \\
\hline
\end{array}
\]

We can infer the pattern and fill in the missing value. The missing value seems to be in the third row, third column.

To determine the pattern, we analyze each column from top to bottom. 

1. **First Column**: \( \frac{1}{4}, \frac{1}{2}, \frac{2}{5}, \frac{17}{20} \)
   - Each number increases slightly in the numerator and denominator.
   - It might be multiplying by a factor or adding specific fractions.

2. **Second Column**: \( \frac{1}{2}, \frac{2}{7}, \frac{3}{4}, \frac{9}{7} \)
   - Here, it looks like a sequence but less obvious.

3. **Third Column**: \( \frac{1}{2}, \frac{1}{4}, \text{Missing Value}, \frac{7}{4} \)
   - Given the missing value is part of this pattern.

4. **Fourth Column**: \( \frac{5}{4}, \frac{103}{140}, \frac{29}{28}, \frac{509}{140} \)
   - This column also needs a closer look.

Analyzing each column:

- In the **third column**, the pattern isn't immediately clear. The fractions are changing in a non-linear manner.

Let's check the ratio of the given fractions:

For the first row and third column:
- \(\frac{1}{2} = 0.5\)

For the second row and third column:
- \(\frac{1}{4} = 0.25\)

For the last row:
- \(\frac{7}{4} = 1.75\)

The third row should fit a logical sequence or geometric progression.

If we consider the second column progression:
- \(\frac{1}{2}, \frac{2}{7}, \frac{3}{4}, \frac{9}{7}\)
- The second fraction \(\frac{2}{7}\) and the fourth fraction \(\frac{9}{7}\) is a hint that they might be cumulative.

To guess the third row, let's check intermediate arithmetic or geometric sequence logic:

Given the pattern of fractions isn't immediately obvious, we can assume a proportional pattern:

Let's hypothesize:
- Since \(\frac{7}{4} = 1.75\), and next value logically continuing from \(\frac{1}{4}\).

We'll solve intermediate missing value with arithmetic progression might be:
- \(\frac{1}{4}, \text{Missing Value}, \frac{7}{4}\).

Let's calculate:
\[
\frac{7}{4} - \frac{1}{4} = \frac{6}{4} = 1.5
\]
Divided intermediate:
\[
\frac{1}{4} + \frac{1.5}{2} = \frac{1}{4} + 0.75 = 1
\]

Hence, the missing value is logically:

**The missing value seems to be \( \frac{1}{1} = 1 \)**

Thus, the table filled in looks like:
\[
\begin{array}{|c|c|c|c|c|}
\hline
\frac{1}{4} & \frac{1}{2} & \frac{1}{2} & \frac{5}{4} \\
\hline
\frac{1}{2} & \frac{2}{7} & \frac{1}{4} & \frac{103}{140} \\
\hline
\frac{2}{5} & \frac{3}{4} & \frac{1}{1} & \frac{29}{28} \\
\hline
\frac{17}{20} & \frac{9}{7} & \frac{7}{4} & \frac{509}{140} \\
\hline
\end{array}
\]

Would you like further analysis on how the values were derived or any additional questions? 

### Questions you can ask:
1. How to verify a sequence of fractions?
2. What is the simplest way to find missing fractions in a series?
3. Can you explain how to convert a fraction to a decimal and back?
4. How to identify patterns in a sequence of fractions?
5. How do I solve similar problems with fractions?

### Tip:
When filling in missing fractions, always look for arithmetic or geometric sequences as a starting point for identifying patterns.

Learn how to fill in missing fractions in a series using arithmetic or geometric progression logic. This problem covers key mathematical concepts such as fractions, arithmetic progression, and geometric progression. Suitable for students in Grades 9-12.

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