Question : 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?
Option 1: 44 sec
Option 2: 45 sec
Option 3: 40 sec
Option 4: 60 sec
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Correct Answer: 44 sec
Solution :
2 cases will be formed in this situation.
Case 1:
Vehicle P gives vehicle Q a start of 200 m, and P wins the race by 8 sec.
⇒ Distance traveled by P = 1500 m
And, Distance travel by Q = 1500 – 200 = 1300 m
⇒ Speed of vehicle P = $\frac{1500}{t}$
And, Speed of vehicle Q = $\frac{1300}{t+8}$
Case 2:
Vehicle P gives vehicle Q a start of 400 m, the race ends in a dead heat.
⇒ Distance travel by P = 1500 m
And, Distance travel by Q = 1500 – 400 = 1100 m
As the race ends at a dead heat, they finish it at an equal time.
⇒ Time taken by vehicle P = Time taken by vehicle Q
⇒ $\frac{1500}{\frac{1500}{t}}=\frac{1100}{\frac{1300}{t+8}}$
⇒ $t=\frac{11(t+8)}{13}$
⇒ $13t = 11t + 88$
⇒ $2t = 88$
$\therefore t = 44$ sec
Hence, the correct answer is 44 sec.
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