A train completely overtakes two persons, walking in the same direction with speeds 3 km/hr and 4 km/hr in 9 seconds and \frac{75}{8} seconds respectively. What is the length of the train?

Explanation

Convert speeds to m/s: u_1 = 3 \times \frac{5}{18} = \frac{5}{6} m/s and u_2 = 4 \times \frac{5}{18} = \frac{10}{9} m/s. Let train speed be v and length L. From the overtakes: \frac{L}{v-5/6} = 9 \implies L = 9v - 7.5. And \frac{L}{v-10/9} = \frac{75}{8} \implies L = \frac{75}{8}v - \frac{125}{12}. Equating L: 9v - 7.5 = \frac{75}{8}v - \frac{125}{12} \implies v(\frac{75}{8} - 9) = \frac{125}{12} - \frac{15}{2} \implies v(\frac{3}{8}) = \frac{35}{12} \implies v = \frac{70}{9} m/s. Length L = 9(\frac{70}{9}) - 7.5 = 70 - 7.5 = 62.5 m.

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