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

### Concept & Speed-Time Relation The fundamental algebraic relationship between distance, time, and speed states that speed equals distance divided by time. Set up the equation using the effective speed. $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ ### Step-by-Step Solution * **Determine Effective Speed:** The original speed is $p$, but it is reduced by $q$. Therefore, the new effective speed is $(p - q)$ kmph. * **Set Up Equation:** The distance to cover is given as 1 km, and the time taken is $r$ hours. * Substitute these into the formula: $r = \frac{1}{p - q}$. * **Rearrange:** Multiply both sides by $(p - q)$ and divide by $r$ (or take the reciprocal of both sides) to match the given options. $$ \frac{1}{r} = p - q $$ ### Exam Strategy & Shortcut Effective Speed = Distance / Time. The effective speed is directly $(p - q)$ and Distance/Time is $1/r$. Equating them gives $(p - q) = 1/r$ instantly. ### Common Pitfall Selecting option (b) $r = p - q$, carelessly confusing the position of the time variable with the reciprocal of time in the formula. ### Final Answer Therefore, the correct answer is **$\frac{1}{r} = p - q$**.