# Thread: Race and time problem.

1. ## Race and time problem.

In a km race, if A gives B a 40 m start, A wins by 19 seconds, but if A gives B 30 seconds start, B wins by 40 m. Find the time that each takes to run a km?

2. ## Re: Race and time problem.

This is just algebra, isn't it? Let $t_A$ be A's time for a kilometer and $t_B$ be B's time for a kilometer. Then the first condition is:

$t_A = 0.96t_B - 19$

The second condition is:

$0.96t_A = t_B-30$

So the solution is $t_B=150$, $t_A=125$, i.e. A takes 125 seconds to run a kilometer and B takes 150 seconds to run a kilometer.

- Hollywood

3. ## Re: Race and time problem.

Sorry for the late response. can you explain in details how do you get to those equations.

4. ## Re: Race and time problem.

It's pretty much just translating the words to equations. It's actually not easy - you have to think about what's happening in these races:

"if A gives B a 40 m start, A wins by 19 seconds"
so A runs 1km in 19 seconds less time than B runs 0.96km
$t_A = 0.96t_B - 19$

"if A gives B 30 seconds start, B wins by 40 m"
so A runs 0.96km in 30 seconds less time than B runs 1km
$0.96t_A = t_B-30$

- Hollywood