`{:("Column A","a and b are positive".(a +6) : (b +6) = 5 : 6 ,"Column B"),((a + 10)/(b + 10),,1):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the problem, we need to analyze the given ratios and derive the relationship between the two columns.

### Step-by-Step Solution:

1. **Understanding the Given Ratio**:
We have the ratio \((a + 6) : (b + 6) = 5 : 6\). This means that:
\[ \frac{a + 6}{b + 6} = \frac{5}{6} \]

2. **Cross-Multiplying**:
We can cross-multiply to eliminate the fraction:
\[ 6(a + 6) = 5(b + 6) \]
Expanding both sides gives:
\[ 6a + 36 = 5b + 30 \]

3. **Rearranging the Equation**:
Rearranging the equation to isolate \(a\) and \(b\):
\[ 6a - 5b = 30 - 36 \]
Simplifying this results in:
\[ 6a - 5b = -6 \]

4. **Adding 24 to Both Sides**:
To facilitate the next steps, we add 24 to both sides:
\[ 6a - 5b + 24 = 18 \]
This can be rearranged as:
\[ 6(a + 4) - 5b = 18 \]

5. **Dividing by \(b + 10\)**:
We want to express \(\frac{a + 10}{b + 10}\). First, we can express \(a + 10\) in terms of \(b + 10\):
\[ 6(a + 10) = 6a + 60 \]
From our previous equation, we can substitute \(6a\):
\[ 6a = 5b + 30 \]
Thus:
\[ 6(a + 10) = 5b + 30 + 60 = 5b + 90 \]

6. **Finding the Expression**:
Now, we can express \(\frac{a + 10}{b + 10}\):
\[ \frac{a + 10}{b + 10} = \frac{5b + 90}{b + 10} \]

7. **Simplifying the Expression**:
We can simplify this expression:
\[ \frac{5b + 90}{b + 10} = 5 + \frac{40}{b + 10} \]

8. **Analyzing the Result**:
Since \(b\) is positive, \(b + 10\) is also positive. Therefore, \(\frac{40}{b + 10}\) will always be positive. This means:
\[ \frac{a + 10}{b + 10} = 5 + \frac{40}{b + 10} > 5 \]

9. **Comparing with Column B**:
Column B is simply 1. Since \(5 + \frac{40}{b + 10} > 5\), it is clear that:
\[ \frac{a + 10}{b + 10} > 1 \]

### Conclusion:
Thus, we conclude that Column A is greater than Column B.