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Let's analyze the new table with missing fractions to determine the correct values. Here is the table:

\[
\begin{array}{|c|c|c|c|}
\hline
\frac{1}{3} & \frac{1}{2} & \frac{1}{3} & \frac{7}{6} \\
\hline
\text{?} & \frac{2}{5} & \frac{2}{5} & \frac{29}{30} \\
\hline
\frac{2}{3} & \text{?} & \frac{3}{5} & \text{?} \\
\hline
\frac{7}{6} & \frac{47}{30} & \text{?} & \frac{61}{15} \\
\hline
\end{array}
\]

We will analyze the pattern in each column and row to determine the missing fractions.

### First Column
\[
\frac{1}{3}, \text{?}, \frac{2}{3}, \frac{7}{6}
\]
- Looking at these fractions, it seems they might follow a proportional or sequential pattern.

### Second Column
\[
\frac{1}{2}, \frac{2}{5}, \text{?}, \frac{47}{30}
\]
- Here, it is necessary to look at the relationship between the numbers to deduce the missing fraction.

### Third Column
\[
\frac{1}{3}, \frac{2}{5}, \frac{3}{5}, \text{?}
\]

### Fourth Column
\[
\frac{7}{6}, \frac{29}{30}, \text{?}, \frac{61}{15}
\]

#### Solving for the missing fractions:

1. **First Column**:
   - Let's analyze the possible relationship between the fractions.
   - If we look at the fractions:
     - \(\frac{1}{3}, \frac{2}{3}, \frac{7}{6}\) seems to indicate an increasing pattern.

   Let's try a logical approach:
   - \(\frac{1}{3}\)
   - Intermediate might be double:
   - \(\frac{1 \cdot 2}{3} = \frac{2}{3}\)

2. **Second Column**:
   - Given \(\frac{1}{2}, \frac{2}{5}, \frac{47}{30}\):
     - Analyzing pattern.
     - Simplifying into pattern: \(\frac{1}{2}, \frac{2}{5}\)

3. **Third Column**:
   - Seems logical progression of numerator increasing \(\frac{1}{3}, \frac{2}{5}, \frac{3}{5}\)

4. **Fourth Column**:
   - Fractions indicate \(\frac{7}{6}, \frac{29}{30}, \frac{61}{15}\)
   - Analyzing similar logical:

Cross-verifying intermediate values:

Thus, the filled table would be:

\[
\begin{array}{|c|c|c|c|}
\hline
\frac{1}{3} & \frac{1}{2} & \frac{1}{3} & \frac{7}{6} \\
\hline
\frac{2}{3} & \frac{2}{5} & \frac{2}{5} & \frac{29}{30} \\
\hline
\frac{2}{3} & \frac{3}{5} & \frac{3}{5} & \frac{47}{30} \\
\hline
\frac{7}{6} & \frac{47}{30} & \frac{61}{15} & \frac{61}{15} \\
\hline
\end{array}
\]

Would you like further detailed explanation or additional calculations?

### Questions you can ask:
1. How to determine the sequence of fraction values?
2. Can you explain how to deduce the pattern in fractions?
3. How to simplify complex fractions effectively?
4. What are techniques to find missing values in fraction tables?
5. How to verify fraction sequences accurately?

### Tip:
When dealing with sequences of fractions, try to identify whether the progression is arithmetic, geometric, or involves some common ratio or difference. This helps in predicting and verifying missing values.

Learn how to analyze patterns in fractions to solve for missing values in a table. This problem involves identifying proportional or sequential patterns in fractions and applying logical reasoning to deduce missing values. Suitable for students in Grades 7-9.

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