Introduction / Context:
This problem is a variant of speed and distance questions where a vehicle moves with one speed after repair and a different speed before repair. We know how far it travels after repair and in how much time. From this we must find how long it would take to travel a related distance at the old speed. The question checks your understanding of distance = speed * time and how to relate distances in different scenarios using proportional reasoning.
Given Data / Assumptions:
- Speed after repair = 84 km/h.
- Speed before repair = 56 km/h.
- After repair, the bus covers 2X km in 8 hours.
- We need the time required to cover X km before repair.
- Speeds are constant and distances are measured in kilometres.
Concept / Approach:
We first determine the value of 2X using the known speed and time after repair. From that we find X. Once we know X, we can calculate the time taken at the older, slower speed using:
time = distance / speed
Step-by-Step Solution:
Step 1: Calculate 2X using the after-repair speed.
Speed = 84 km/h, time = 8 hours.
distance covered = speed * time = 84 * 8 km.
84 * 8 = 672 km.
So 2X = 672 km.
Step 2: Find X.
X = 672 / 2 = 336 km.
Step 3: Compute time to cover X km at 56 km/h.
Speed before repair = 56 km/h.
time = distance / speed = 336 / 56 hours.
56 * 6 = 336, so time = 6 hours.
Therefore, it takes 6 hours to cover X km at the old speed.
Verification / Alternative check:
We can check using proportional reasoning. At 84 km/h, the bus covers 84 km every hour. Over 8 hours, that is 672 km, which is 2X. At the slower speed of 56 km/h, the time to cover 336 km is 336 / 56 hours. Dividing gives 6. This double-check confirms our arithmetic is correct.
Why Other Options Are Wrong:
5 hours would cover 56 * 5 = 280 km, which is less than 336 km.
7 hours would cover 56 * 7 = 392 km, which is more than X.
8 hours would cover 56 * 8 = 448 km, again too much distance.
9 hours would cover 56 * 9 = 504 km, even farther from X.
Only 6 hours corresponds exactly to travelling X km at 56 km/h.
Common Pitfalls:
One common mistake is to treat 2X as if it were just X and divide by the speed prematurely. Another error is miscalculating the product 84 * 8 or mis-dividing 336 by 56. Learners may also confuse which speed applies to which distance, especially when both before and after repair conditions are present. Writing each stage clearly, first solving for the unknown distance and then for time, helps avoid these difficulties.
Final Answer:
The bus will take
6 hours to cover X km at its older speed of 56 km/h.
Discussion & Comments