problem: In a $100$ meter track event, Alice runs at a constant speed and crosses the finish line $5$ seconds before Beatrice does. If it took Beatrice $25$ seconds to complete the race, how many seconds would it take Alice to run $50$ meters?
solution: Alice took $25 - 5 = 20$ seconds to complete the race. If she can run $100$ meters in $20$ seconds, by direct proportionality, she must be able to run $\frac{100}{2} = 50$ meters in $\frac{20}{2} = \boxed{10}$ seconds.