Discussion
In a 500-m race, A gives B a headstart of 30 m and wins by 20 sec. If A gives B a headstart of 40 m, the race ends in a dead heat. What will be the time taken by A to complete the race?
Answer: D
A takes 920 seconds.
Let A take t seconds. When A gives B a 40 m start the race is a dead heat, so B runs 460 m in t seconds. With a 30 m start B runs 470 m, which takes 20 seconds more than t. At B's speed 460/t, the time for 470 m is 470t/460.
\[ \frac{470t}{460}=t+20 \]
\[ \frac{10t}{460}=20 \Rightarrow t=920 \]
325, 417 and 266 seconds do not satisfy these equations.
The key idea is that the 10 m extra headstart changes the result from B finishing 20 s behind to a dead heat, so B covers 10 m in 20 s at the speed 460 / t. This gives B's speed of 0.5 m/s, and the 460 m B covers takes 920 seconds, which is A's time over 500 m.
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