A man travels $800\text{ km}$ partly by rail and partly by steamer. He spends $7\text{ hours}$ more time on the steamer. If the velocity of the steamer is $20\text{ km/hr}$ and the velocity of the rail is $55\text{ km/hr}$, how much distance does he cover by steamer?

Let the time spent traveling by rail be $t$ hours.Since he spends $7$ hours more on the steamer, the time spent on the steamer is $(t+7)$ hours.We are given:Speed of rail = $55\text{ km/hr}$ Speed of steamer = $20\text{ km/hr}$ Total distance = $800\text{ km}$ Using the formula $\text{Distance}=\text{Speed}\times \text{Time}$:
$$\text{Distance by rail}+\text{Distance by steamer}=\text{Total Distance}$$
$$55t+20(t+7)=800$$
$$55t+20t+140=800$$
$$75t=800-140$$
$$75t=660$$
$$t=\frac{660}{75}=8.8\text{ hours}$$
Now, find the distance traveled by the steamer:
$$\text{Distance by steamer}=20\times (t+7)$$
$$\text{Distance by steamer}=20\times (8.8+7)=20\times 15.8=316\text{ km}$$