Troye and Daniella are running at constant speeds in opposite directions around a circular track. Troye completes one lap every 56 seconds and meets Daniella every 24 seconds. How many seconds does it take Daniella to complete one lap?
(A) 32
(B) 36
(C) 40
(D) 48
(E) 42
Let's say that time equals 0 seconds when Troye and Daniella first meet. Then at time 24 seconds, they will meet again.

In 24 seconds, how far does Troye get around the track? Since it takes her 56 seconds to complete one lap, then she has made it $\frac{24}{56}=\frac{3}{7}$ of the way around the track.

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-225.jpg?height=364&width=455&top_left_y=1344&top_left_x=1366)

Since Daniella is running in the opposite direction, then she will go $\frac{4}{7}$ of the way around the track in 24 seconds, and so one complete lap will take her $\frac{7}{4}(24)=42$ seconds.

ANSWER: (E)