Question : Having started from the same point simultaneously, two runners P and Q are running around a circular track of length 500 m in opposite directions with speeds of 6 m/s and 10 m/s, respectively. If they exchange their speeds after meeting for the first time, who will reach the starting point first?
Option 1: Q
Option 2: P
Option 3: Both P and Q will reach at the same time
Option 4: No one of the P and Q
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Correct Answer: Both P and Q will reach at the same time
Solution : Two runners P and Q run in a circular track of length with speeds 6 m/s and 10 m/s. The time they meet for the first time $=\frac{\text{Length of Circular Track}}{\text{Relative Speed}}$ $= \frac{500}{10 + 6} = \frac{500}{16} = \frac{125}{4}$ sec Now, Distance covered by P $= 6\times \frac{125}{4}=\frac{375}{2}$ m And distance covered by Q $=10\times \frac{125}{4}=\frac{625}{2}$ m After interchanging their speeds, Speed of P = 10 m/s Speed of Q = 6 m/s The remaining distance for P and Q will be equal to (500 – Distance covered respectively) Time taken by P to reach the starting point $= (500-\frac{375}{2})\times\frac{1}{10}$ $= \frac{625}{20}=\frac{125}{4}$ sec Time taken by Q to reach the starting point $=(500-\frac{625}{2})\times\frac{1}{6}$ $= \frac{375}{12} = \frac{125}{4}$ sec $\therefore$ They will reach the starting point at the same time. Hence, the correct answer is both P and Q will reach at the same time.
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