Question

5/6th of the way across a narrow railroad bridge, Anoop hears a train coming towards him. He knows that the train's speed is 35 miles per hour and that he has just enough time to reach safety at either end of the bridge. How fast can Anoop run (in miles per hour)?

Answer 1

See diagram.

let the bridge be L miles long.  Let Anoop run v miles per hour.

Hence by the time Anoop runs to EITHER END of the railroad bridge, the train just comes to the SAME end of the bridge.
 
1)  Anoop runs L - 5L/6 = L/6 distance,  to the nearer end of the bridge.
     The train has to run through the distance x.

$\frac{L/6}{v} = \frac{x}{35}\ \ \ \ => \ \ v = \frac{35 L}{6 x}, \ \ \ - equation 1$

2)  Anoop runs 5L/6 to the farther end of the bridge.

$\frac{L+x}{35} = \frac{5L/6}{v},\ \ \ => \ v = \frac{35*5 L}{6 (L+x)}, \ - equation\ 2$

   divide equation 2  by equation 1  =>
 $1 = \frac{5 x}{L+x},\ \ => \ L + x = 5 x,\ \ \ L=4\ x,\ \ \ - equation3$ 

   
     substitute in equation 1,
 $v = \frac{35 * 4 x}{6 * x} = \frac{140}{6} = 23.66 mph$

Anoop can run at 23.66 mph