In a km race if A gives B a start of 100 m, A wins by 20 sec and if A gives B a start of 50 sec, B wins by 30 m....

Question

A and B run a km race. If A gives B a start of 100 m, A wins by 20 sec and if A gives B a start of 50 sec, B wins by 30 m. Then what is the time taken by A to run a km?

Answer 1

It takes A 196.9 seconds to run 1000 m.
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We have two races.
Race 1 is 1000 meter for A and 900 meter for B. A wins by 20 seconds.
Race 2 is 970 meter for A and 1000 meter for B.
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t = time (sec), v = speed (m/s), a and b are for A and B. And time = distance / speed
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This gives us the following equations:
For race 1:
t1 = time for B to finish the race
t1 - 20 = 1000 / va
t1 = 900 / vb
Thus: 900 / vb = 1000 / va + 20
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For race 2:
t2 = time for B to finish the race (excluding the head start)
t2 = 970 / va
t2 + 50 = 1000 / vb
Thus: 970 / va = 1000 / vb - 50
This can be rewritten to: 900 / vb = 873 / va + 45
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Now we have 900 / vb = 1000 / va + 20 = 873 / va + 45
Rewrite to: 127 / va = 25 => va = 127 / 25 = 5.08 meter/second.
vb then is 4.15 m/s
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Let's check:
Race 1: A runs the 1000 m in 197 seconds. B runs the 900 m in 217 seconds => 20 second difference.
Race 2: B runs the 1000 m in 241 seconds. Subtracting the head starts means A runs for 191 seconds. In that time A runs 970 meter and loses by 30 meter.