In a 1500 m race, if vehicle P gives vehicle Q a start of 200 m, then vehicle P wins the race by 8 sec. Alternatively, if vehicle P gives vehicle Q a start of 400 m, the race ends in a dead heat. How long does vehicle P take to run 1500 m?

Answer: Option A

Solution

Answer: Option A

Solution:

$$\begin{array}{rl}& \frac{1300}{\text{Q}}-\frac{1500}{\text{P}}=8\text{. }\text{. }\text{. }\text{. }\left(\text{i}\right)\\ & \text{and, }\frac{1100}{\text{Q}}=\frac{1500}{\text{P}}\text{. }\text{. }\text{. }\text{. }\left(\text{ii}\right)\\ & \frac{\text{P}}{\text{Q}}=\frac{15}{11}=\frac{15x}{11x}\\ & \text{From equation }\left(\text{i}\right)\\ & \frac{1300}{11x}-\frac{1500}{15x}=8\\ & 19500-16500=8\times 11x\times 15x\\ & 3000=8\times 11\times 15x\\ & x=\frac{25}{11}\\ & \text{Time of P to travel 1500m}=\frac{1500}{15x}\\ & =\frac{1500\times 11}{15\times 25}\\ & =44\sec \\ & \\ & \mathbf{Alternate}\mathbf{solution}:\end{array}$$

$$\begin{array}{rl}& 11\mu \to 25\\ & 1\mu \to \frac{25}{11}\\ & 15\mu \to \frac{15\times 25}{11}\\ & \text{Speed of Q}=\frac{200}{8}=25\\ & \text{Time of P}=\frac{1500\times 11}{15\times 25}=44\sec \end{array}$$