`{:("Column A","p and q are two postive integers and p/q = 7.5" ,"Column B"),("q", ,"15"):}`

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 relationship between the positive integers \( p \) and \( q \) given that \( \frac{p}{q} = 7.5 \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation:
\[ \frac{p}{q} = 7.5 \]
This can be rewritten as:
\[ p = 7.5q \]

2. **Expressing \( p \) in terms of \( q \)**:
Since \( p \) and \( q \) are both positive integers, we can express \( p \) as:
\[ p = \frac{75}{10}q = \frac{15}{2}q \]
This means that \( p \) must be a multiple of \( \frac{15}{2} \).

3. **Finding Integer Values**:
For \( p \) to be an integer, \( q \) must be even (since \( \frac{15}{2}q \) needs to yield an integer). Let’s denote \( q = 2k \) where \( k \) is a positive integer.

4. **Substituting \( q \)**:
Substituting \( q \) in terms of \( k \):
\[ p = \frac{15}{2}(2k) = 15k \]
Thus, we have:
\[ p = 15k \quad \text{and} \quad q = 2k \]

5. **Analyzing Values of \( q \)**:
Now, we can express \( q \) as:
\[ q = 2k \]
As \( k \) takes positive integer values (1, 2, 3, ...), \( q \) will take the values 2, 4, 6, 8, 10, 12, 14, 16, 18, ...

6. **Comparing with Column B**:
Column B has the value 15. We need to compare \( q \) with 15:
- If \( k = 1 \), \( q = 2 \) (Column B is greater).
- If \( k = 2 \), \( q = 4 \) (Column B is greater).
- If \( k = 3 \), \( q = 6 \) (Column B is greater).
- If \( k = 4 \), \( q = 8 \) (Column B is greater).
- If \( k = 5 \), \( q = 10 \) (Column B is greater).
- If \( k = 6 \), \( q = 12 \) (Column B is greater).
- If \( k = 7 \), \( q = 14 \) (Column B is greater).
- If \( k = 8 \), \( q = 16 \) (Column A is greater).

7. **Conclusion**:
Since we have found cases where \( q < 15 \) and cases where \( q > 15 \), we conclude that there is not enough information to determine which column is larger.

### Final Answer:
The correct option is **4: There is not enough information to decide**.