If p and q are in direct proportion, then `1/p` and `1/q` are _______

Options

A. in direct proportion
B. in inverse proportion
C. neither in direct nor in inverse proportion
D. sometimes in direct and sometimes in inverse proportion.

Answer

To solve the problem, we need to understand the concept of direct proportion and how it relates to the reciprocals of the variables involved.

### Step-by-Step Solution:

1. **Understanding Direct Proportion**:
- When we say that \( p \) and \( q \) are in direct proportion, it means that \( p \) can be expressed as \( k \cdot q \) for some constant \( k \). This implies that if \( p \) increases, \( q \) also increases, and if \( p \) decreases, \( q \) decreases.

2. **Expressing the Relationship**:
- Since \( p \) is directly proportional to \( q \), we can write:
\[ p = k \cdot q \]
- Rearranging this gives us:
\[ q = \frac{p}{k} \]

3. **Taking Reciprocals**:
- Now, we want to find the relationship between \( \frac{1}{p} \) and \( \frac{1}{q} \). From the equation \( p = k \cdot q \), we can express \( \frac{1}{p} \) in terms of \( \frac{1}{q} \):
\[ \frac{1}{p} = \frac{1}{k \cdot q} \]
- This can be rewritten as:
\[ \frac{1}{p} = \frac{1}{k} \cdot \frac{1}{q} \]

4. **Conclusion**:
- The equation \( \frac{1}{p} = \frac{1}{k} \cdot \frac{1}{q} \) shows that \( \frac{1}{p} \) is directly proportional to \( \frac{1}{q} \) because they are related by a constant factor \( \frac{1}{k} \).
- Therefore, we conclude that if \( p \) and \( q \) are in direct proportion, then \( \frac{1}{p} \) and \( \frac{1}{q} \) are also in direct proportion.

### Final Answer:
If \( p \) and \( q \) are in direct proportion, then \( \frac{1}{p} \) and \( \frac{1}{q} \) are **directly proportional**.