Train P of length $540$ meters crosses a bridge in $22$ seconds and crosses a tree in $18$ seconds. The ratio of the speeds of train P to train Q is $5:2$, and train Q crosses a pole in $20$ seconds. What is the length of the bridge as a percentage of the length of train Q?

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**Solution:**

1. Calculate the speed of train P:
   - Length of train P = $540$ meters
   - Time to cross a tree (which is equivalent to length of train) = $18$ seconds
   - Speed of train P = $\frac{540}{18} = 30$ m/s

2. Calculate the length of the bridge:
   - Time to cross the bridge = $22$ seconds
   - Distance covered while crossing the bridge = length of train + length of bridge
   - Speed of train P = $30$ m/s
   - Total distance = speed $	imes$ time = $30 	imes 22 = 660$ meters
   - Length of bridge = total distance - length of train = $660 - 540 = 120$ meters

3. Calculate the speed of train Q:
   - Ratio of speeds of P to Q = $5:2$
   - Speed of train P = $30$ m/s
   - So, speed of train Q = $\frac{2}{5} 	imes 30 = 12$ m/s

4. Calculate the length of train Q:
   - Train Q crosses a pole in $20$ seconds
   - Length of train Q = speed $	imes$ time = $12 	imes 20 = 240$ meters

5. Find the length of the bridge as a percentage of the length of train Q:
   $$\frac{	ext{Length of bridge}}{	ext{Length of train Q}} 	imes 100 = \frac{120}{240} 	imes 100 = 50\%$$

**Answer:** $50\%$

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