A boy is running at a speed of p km/h to cover a distance of 1 km but due to the slippery ground, his speed is reduced by q km/h (where, p>q). If he takes r hours to cover the distance, then

Options

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

Answer

To solve the problem step by step, we will derive the relationship between the variables involved: speed, time, and distance.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The boy runs a distance of 1 km.
- His initial speed is \( p \) km/h.
- Due to slippery ground, his effective speed is reduced by \( q \) km/h.
- Therefore, his effective speed becomes \( (p - q) \) km/h.
- He takes \( r \) hours to cover the 1 km distance.

2. **Using the Formula for Distance**:
- The basic formula relating distance, speed, and time is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
- In this case, the distance is 1 km, the speed is \( (p - q) \) km/h, and the time is \( r \) hours.

3. **Setting Up the Equation**:
- Plugging in the values into the distance formula:
\[ 1 = (p - q) \times r \]

4. **Rearranging the Equation**:
- To isolate \( r \), we can rearrange the equation:
\[ r = \frac{1}{p - q} \]

5. **Conclusion**:
- The relationship we derived shows that the time \( r \) taken to cover the distance of 1 km is inversely proportional to the effective speed \( (p - q) \).
- Thus, we can express \( \frac{1}{r} \) as:
\[ \frac{1}{r} = p - q \]

### Final Result:
The final expression we derived is:
\[ \frac{1}{r} = p - q \]