A certain distance is covered by a cyclist at a certain speed. If a jogger covers half the distance in double the time, the ratio of the speed of the jogger to that of the cyclist is :

### Concept & Formula Speed is directly proportional to distance and inversely proportional to time. To find a ratio of speeds, substitute the relative distances and times into the speed formula. $$Speed = \frac{Distance}{Time}$$ ### Step-by-Step Solution 1. **Define Cyclist Variables:** * Let the distance covered by the cyclist be $D$. * Let the time taken by the cyclist be $T$. * Speed of cyclist, $S_c = \frac{D}{T}$. 2. **Define Jogger Variables:** * Distance covered by jogger = $\frac{D}{2}$ (half the distance). * Time taken by jogger = $2T$ (double the time). * Speed of jogger, $S_j = \frac{\frac{D}{2}}{2T} = \frac{D}{4T}$. 3. **Calculate the Ratio:** * Ratio = $S_j : S_c$ * Ratio = $\frac{D}{4T} : \frac{D}{T}$ * Cancel out the common $\frac{D}{T}$ term. * Ratio = $\frac{1}{4} : 1$ * Multiply by $4$ to clear the fraction: $1 : 4$. ### Exam Strategy & Shortcut Use fractional multipliers directly on the speed formula $S = \frac{D}{T}$. For the jogger, Distance is $\times \frac{1}{2}$ and Time is $\times 2$. So, new Speed multiplier = $\frac{\frac{1}{2}}{2} = \frac{1}{4}$. The jogger is $\frac{1}{4}$ the speed of the cyclist, meaning the ratio is $1:4$. ### Common Pitfall Mixing up the order of the ratio. The question asks for "jogger to cyclist", which is $1:4$. Reversing it to $4:1$ is a very common error. ### Final Answer Therefore, the correct answer is **1 : 4**.