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The ratio table given has two missing entries in the first column, corresponding to the values \( x \) and \( y \) for the rows where the second column contains 7 and 14, respectively.

To find the missing values, observe that the entries in the table follow a constant ratio. Let's focus on the relationship between the numbers in the second column. The ratio between the entries 7 and 14 is consistent with the other entries. 

1. The ratio between corresponding entries in both columns should remain the same. That is, if we take one entry from the first column and divide it by the corresponding entry in the second column, the result should always be the same constant.

Using the known entries:
\[
\frac{54}{63} = \frac{6}{7}
\]

Thus, the constant ratio between the two columns is \( \frac{6}{7} \).

### Step 1: Finding the first missing value (corresponding to 7)
\[
\frac{x}{7} = \frac{6}{7}
\]
Solving for \( x \), we multiply both sides by 7:
\[
x = 6
\]

### Step 2: Finding the second missing value (corresponding to 14)
\[
\frac{y}{14} = \frac{6}{7}
\]
Solving for \( y \), we multiply both sides by 14:
\[
y = 12
\]

### Completed Table:
|   |   7   |  14   |  63  |  84  |  91  |
|---|-------|-------|------|------|------|
|   |   6   |  12   |  54  |  72  |  78  |

The missing values are \( 6 \) and \( 12 \).

Would you like more details or explanations on how this ratio process works?

Here are 5 related questions to deepen your understanding:
1. How would the ratio change if the values in the second column were doubled?
2. Can you find the missing values if the ratio is \( \frac{5}{9} \) instead of \( \frac{6}{7} \)?
3. How can we check if a given table follows a constant ratio?
4. What if the ratio was non-constant? How would that affect the solution?
5. How can we use this method to solve ratio problems in real-life scenarios?

**Tip:** Always ensure that you maintain the ratio consistency when dealing with tables or problems involving proportional relationships!

Complete the ratio table and find the missing values.

Learn how to complete a ratio table with missing values using proportional relationships. This example helps reinforce understanding of ratios and algebraic methods. Suitable for middle school students, Grades 6-8.

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