A train 'A' of length $350$ meters passes a pole in $17.5$ seconds and passes another train 'B' running in the opposite direction in $15.4$ seconds. If the length of train 'B' is $420$ meters, find the time taken by train 'A' to pass train 'B' when both trains run in the same direction.

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**Solution:**

1. Calculate the speed of train A:
   - Length of train A = $350$ meters
   - Time taken to pass a pole = $17.5$ seconds
   - Speed of train A, $v_A = \frac{350}{17.5} = 20 	ext{ m/s}$

2. Let speed of train B be $v_B$ m/s.

3. When trains A and B move in opposite directions, they pass each other in $15.4$ seconds.
   - Combined length = $350 + 420 = 770$ meters
   - Relative speed when opposite = $v_A + v_B$
   - Using formula: $	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}}$, we get
     $$15.4 = \frac{770}{v_A + v_B} \implies v_A + v_B = \frac{770}{15.4} = 50 	ext{ m/s}$$

4. Since $v_A = 20$ m/s, then
   $$v_B = 50 - 20 = 30 	ext{ m/s}$$

5. When trains run in the same direction, relative speed = $|v_B - v_A| = 30 - 20 = 10$ m/s

6. Time taken to pass each other when running in the same direction:
   $$	ext{Time} = \frac{	ext{Sum of lengths}}{	ext{Relative speed}} = \frac{770}{10} = 77 	ext{ seconds}$$

**Answer:** $77 	ext{ seconds}$

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