Introduction / Context:
This is a classic combined journey problem. A traveller covers a fixed total distance partly by one mode of transport (rail) and partly by another (steamer). Each mode has a different constant speed. Additionally, we know the difference in time spent on each mode. From this information, we must determine the part of the journey completed by steamer. Such questions test the ability to translate word statements into algebraic equations involving speed, distance and time.
Given Data / Assumptions:
- Total distance travelled = 400 km.
- Speed by rail = 70 km/h.
- Speed by steamer = 30 km/h.
- Time on steamer is 9 hours more than time on rail.
- We must find the distance travelled by steamer.
Concept / Approach:
Let the distance travelled by steamer be S km. Then the distance by rail is (400 - S) km. Time is distance divided by speed, so:
Time on steamer = S / 30 hours.
Time on rail = (400 - S) / 70 hours.
We are told that time on steamer is 9 hours more than time on rail:
S / 30 - (400 - S) / 70 = 9
We solve this equation for S.
Step-by-Step Solution:
Step 1: Write the time difference equation.
S / 30 - (400 - S) / 70 = 9.
Step 2: Use a common denominator for 30 and 70.
The least common multiple of 30 and 70 is 210.
S / 30 = (7S) / 210.
(400 - S) / 70 = (3 * (400 - S)) / 210 = (1200 - 3S) / 210.
Step 3: Substitute back into the equation.
(7S / 210) - (1200 - 3S) / 210 = 9.
Combine numerators: (7S - 1200 + 3S) / 210 = 9.
(10S - 1200) / 210 = 9.
Step 4: Clear the denominator.
10S - 1200 = 9 * 210.
9 * 210 = 1890.
So 10S - 1200 = 1890.
10S = 1890 + 1200 = 3090.
S = 3090 / 10 = 309 km.
Therefore, the man travels 309 km by steamer.
Verification / Alternative check:
Time on steamer = 309 / 30 = 10.3 hours.
Time on rail = (400 - 309) / 70 = 91 / 70 ≈ 1.3 hours.
Difference in times = 10.3 - 1.3 = 9 hours, which matches the given condition. Also, 309 km + 91 km = 400 km, so the distances add up correctly. This confirms that the calculation is consistent.
Why Other Options Are Wrong:
371 km: Then rail distance is only 29 km, leading to a very small time on rail and a time difference different from 9 hours.
280 km: Rail distance becomes 120 km; the resulting time difference is not 9 hours.
464 km or 556 km exceed or distort the total journey of 400 km when combined with the implied rail distances.
Only 309 km satisfies both the total distance of 400 km and the 9-hour time difference condition.
Common Pitfalls:
A typical mistake is to mix up which time is greater and write the equation in reverse, leading to a negative or incorrect solution. Another issue is mishandling the algebraic manipulation when dealing with fractions, especially forgetting to use a common denominator or making arithmetic errors in multiplication. Some learners may also assume equal distances instead of using the given time difference. Carefully setting up the equation and solving step by step is crucial.
Final Answer:
The man covers
309 km of the journey by steamer.
Discussion & Comments