A water tap fills a tub in 'p' hours and a sink at the bottom empties it in 'q' hours. If `p lt q` and both tap and sink are open, the tank is filled in 'r' hours, then

Options

A. `(1)/(r )=(1)/(p)+(1)/(q)`
B. `(1)/(r )=(1)/(p) -(1)/(q)`
C. `r=p+q`
D. `r=p-q`

Answer

To solve the problem step by step, we need to analyze the information given and derive the relationship between the variables \( p \), \( q \), and \( r \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A water tap fills a tub in \( p \) hours.
- A sink at the bottom empties the tub in \( q \) hours.
- Given that \( p < q \), which means the tap fills the tub faster than the sink empties it.
- When both the tap and the sink are open, the tub is filled in \( r \) hours.

2. **Finding the Filling Rate of the Tap**:
- The rate at which the tap fills the tub is \( \frac{1}{p} \) of the tub per hour. This means in one hour, the tap fills \( \frac{1}{p} \) of the tub.

3. **Finding the Emptying Rate of the Sink**:
- The rate at which the sink empties the tub is \( \frac{1}{q} \) of the tub per hour. This means in one hour, the sink empties \( \frac{1}{q} \) of the tub.

4. **Combined Rate When Both are Open**:
- When both the tap and the sink are open, the effective rate of filling the tub is the filling rate minus the emptying rate:
\[ \text{Effective rate} = \frac{1}{p} - \frac{1}{q} \]
- This effective rate fills the tub in \( r \) hours, so we can express this as:
\[ \frac{1}{r} = \frac{1}{p} - \frac{1}{q} \]

5. **Rearranging the Equation**:
- To find the relationship, we can rearrange the equation:
\[ \frac{1}{r} = \frac{1}{p} - \frac{1}{q} \]
- This can be rewritten as:
\[ \frac{1}{r} = \frac{q - p}{pq} \]

6. **Conclusion**:
- The relationship between \( p \), \( q \), and \( r \) is given by:
\[ \frac{1}{r} = \frac{1}{p} - \frac{1}{q} \]

### Final Answer:
The correct option is:
\[ \frac{1}{r} = \frac{1}{p} - \frac{1}{q} \]