A man travels a total distance of 430 kilometres, partly by rail at a speed of 65 km/h and partly by steamer at a speed of 25 km/h. He spends 10 hours more time on the steamer than on the train. How many kilometres does he cover by steamer?

Introduction / Context:
This question involves travelling a fixed total distance using two different modes of transport with different speeds and a known difference in travel times. It combines the basic distance speed time formula with an additional condition comparing time spent on each mode. The objective is to determine how much of the total distance is covered by steamer, which is a standard type of algebraic time and distance problem.

Given Data / Assumptions:

Total distance travelled = 430 kilometres.
Speed by rail = 65 km/h.
Speed by steamer = 25 km/h.
Let distance covered by rail be x kilometres and by steamer be 430 − x kilometres.
Time spent on steamer is 10 hours more than time spent on rail.
Speeds are constant and there are no breaks during each part of the journey.

Concept / Approach:
Time taken on each mode is distance / speed. Time by rail is x / 65 hours and time by steamer is (430 − x) / 25 hours. The given condition is (430 − x) / 25 = x / 65 + 10. This equation can be solved for x, the distance by rail, and then 430 − x gives the distance by steamer. Careful algebraic manipulation and solving of linear equations are required.

Step-by-Step Solution:
Step 1: Let the distance covered by rail be x kilometres. Step 2: Then the distance covered by steamer is 430 − x kilometres. Step 3: Time by rail = x / 65 hours. Time by steamer = (430 − x) / 25 hours. Step 4: Given that time on steamer is 10 hours more than time on rail: (430 − x) / 25 = x / 65 + 10. Step 5: Multiply through by the common denominator 325 to clear fractions: 13 * (430 − x) = 5x + 3250. Step 6: Expand: 13 * 430 = 5590, so 5590 − 13x = 5x + 3250. Step 7: Combine like terms: 5590 − 3250 = 5x + 13x, so 2340 = 18x, hence x = 2340 / 18 = 130 kilometres. Step 8: Distance by steamer = 430 − 130 = 300 kilometres.

Verification / Alternative check:
Check the times with these distances. Time by rail = 130 / 65 = 2 hours. Distance by steamer = 300 km, so time by steamer = 300 / 25 = 12 hours. The difference in times is 12 − 2 = 10 hours, which matches the condition given in the question. Therefore the distance travelled by steamer is correctly found as 300 km.

Why Other Options Are Wrong:
If the man travelled 360 km by steamer, then the rail distance would be 70 km and the times would not differ by 10 hours. Values like 450 km or 540 km exceed the total distance of 430 km and are impossible. A value of 280 km would give a different time difference, not equal to 10 hours. Only 300 km satisfies both the total distance and the time difference constraint.

Common Pitfalls:
A common mistake is to misplace x and 430 − x with rail and steamer, or to forget that the time difference condition is steamer time minus rail time equals 10, not the other way around. Errors in clearing fractions and simplifying the equation also occur. Writing each step clearly and verifying that the final distances add up to 430 km helps to avoid these issues.

Final Answer:
The man covers 300 kilometres by steamer.