The length of train X is equal to the combined length of trains Y and Z. The length of train Z is 20\% more than the length of train Y. Train Y runs at a speed of $50 ext{ m/s}$ and crosses a pole in $6$ seconds. Train X runs at a speed of $40 ext{ m/s}$ and crosses train Z in $17$ seconds while running in opposite directions. What is the speed of train Z (in meters/second)?

Question

The length of train X is equal to the combined length of trains Y and Z. The length of train Z is 20\% more than the length of train Y. Train Y runs at a speed of $50 	ext{ m/s}$ and crosses a pole in $6$ seconds. Train X runs at a speed of $40 	ext{ m/s}$ and crosses train Z in $17$ seconds while running in opposite directions. What is the speed of train Z (in meters/second)?

Accepted Answer

**Solution:**

1. Calculate the length of train Y:
   - Speed of train Y = $50 	ext{ m/s}$
   - Time to cross a pole = $6$ seconds
   - Length of train Y = Speed $	imes$ Time = $50 	imes 6 = 300 	ext{ meters}$

2. Find the length of train Z:
   - Length of Z is 20\% more than Y
   - Length of Z = $300 + 0.20 	imes 300 = 300 + 60 = 360 	ext{ meters}$

3. Length of train X:
   - Given that length of X = length of Y + length of Z
   - Length of X = $300 + 360 = 660 	ext{ meters}$

4. Let speed of train Z be $v 	ext{ m/s}$.

5. Since trains X and Z are moving in opposite directions, their relative speed = $(40 + v) 	ext{ m/s}$.

6. Time taken to cross each other = $17$ seconds.

7. Total distance covered when crossing each other = sum of their lengths = $660 + 360 = 1020 	ext{ meters}$.

8. Using formula: Distance = Speed $	imes$ Time
   $$1020 = (40 + v) 	imes 17$$

9. Solve for $v$:
   $$40 + v = \frac{1020}{17} = 60$$
   $$v = 60 - 40 = 20 	ext{ m/s}$$

**Answer:** $20$

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