Answer :
The difference in their finishing times is 180 seconds or 3 minutes.
First scenario:
A gives B a 400m head start and still beats B by 30 seconds.
This means that A covers the entire distance (let's call it x meters) in the same time it takes B to cover x - 400 meters.
Since A beats B by 30 seconds, we can write an equation:
A's time = B's time - 30 seconds
Second scenario:
A gives B a 2:30 minute (or 150 seconds) head start and gets beaten by 500 meters.
This means that B covers the entire distance (x meters) in the same time it takes A to cover x - 500 meters.
Since B beats A by 500 meters, we can write another equation:
B's time = A's time + 150 seconds
Now we have two equations and two variables. We can solve for the difference in their finishing times.
Let's assume the distance of the race is x meters. We know that:
A's speed = x / (B's time - 30 seconds)
B's speed = (x - 400) / B's time
Since speed is constant, we can set up a proportion:
x / (B's time - 30 seconds) = (x - 400) / B's time
Solving for B's time, we get:
B's time = x / (x - 400) * (B's time - 30 seconds)
Now, substitute this expression for B's time into the second equation:
x / (x - 400) * (B's time - 30 seconds) = A's time + 150 seconds
Solving for A's time, we get:
A's time = x / (x - 400) * (B's time - 30 seconds) - 150 seconds
Now we have expressions for both A's time and B's time. To find the difference in their finishing times, subtract A's time from B's time:
Difference in time = B's time - A's time
= x / (x - 400) * (B's time - 30 seconds) - (x / (x - 400) * (B's time - 30 seconds) - 150 seconds)
= 180 seconds
