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?
This is just algebra, isn't it? Let $\displaystyle t_A$ be A's time for a kilometer and $\displaystyle t_B$ be B's time for a kilometer. Then the first condition is:
$\displaystyle t_A = 0.96t_B - 19$
The second condition is:
$\displaystyle 0.96t_A = t_B-30$
So the solution is $\displaystyle t_B=150$, $\displaystyle t_A=125$, i.e. A takes 125 seconds to run a kilometer and B takes 150 seconds to run a kilometer.
- Hollywood
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
$\displaystyle t_A = 0.96t_B - 19$
"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
$\displaystyle 0.96t_A = t_B-30$
- Hollywood