Discussion
P and Q run a Kilometers and P win by 1 minutes. P and R run a kilometre and P wins by 375 m. Q and R run a kilometre and Q wins By 30 Seconds. Find the time taken by each runner to run a kilometre.
Answer: A
Let the three times over a kilometre be \( p \), \( q \) and \( r \).
P beats Q by 1 minute, and Q beats R by 30 seconds:
\[ q = p + 60, \qquad r = q + 30 = p + 90 \]
P beats R by 375 m, so R covers only 625 m in P's time:
\[ \frac{625}{1000} = \frac{p}{r} \quad\Longrightarrow\quad r = 1.6p \]
\[ 1.6p = p + 90 \quad\Longrightarrow\quad 0.6p = 90 \quad\Longrightarrow\quad p = 150 \]
\[ q = 210 \text{ s}, \qquad r = 240 \text{ s} \]
Checking: R covers \( 1000 \times \tfrac{150}{240} = 625 \) m in P's 150 seconds, exactly 375 m short.
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