Introduction / Context:
This problem combines speed, distance and proportional reasoning. A car travels faster after being repaired compared to before repairing. We know how far it goes in a certain time after repair, and we must find the time it would need to cover three times that distance at its old, slower speed. Problems like this are common in aptitude tests and help build comfort with manipulating ratios and the basic speed formula.
Given Data / Assumptions:
- Speed after repair = 72 km/h.
- Speed before repair = 54 km/h.
- After repair, the car travels X km in 6 hours.
- We need the time to travel 3X km at 54 km/h.
- Speeds are constant in each phase and the same distance unit (km) is used throughout.
Concept / Approach:
We use the formula:
speed = distance / time
From this we get:
distance = speed * time
and
time = distance / speed
First we determine X using the faster speed and then compute 3X and the time needed at the slower speed.
Step-by-Step Solution:
Step 1: Find X using the speed after repair.
Speed after repair = 72 km/h, time = 6 hours.
X = speed * time = 72 * 6 km.
72 * 6 = 432, so X = 432 km.
Step 2: Compute the new distance 3X.
3X = 3 * 432 = 1296 km.
Step 3: Calculate the time at the old speed (54 km/h).
time = distance / speed = 1296 / 54 hours.
Compute 1296 / 54: 54 * 20 = 1080, 1296 - 1080 = 216, and 54 * 4 = 216.
So 1296 / 54 = 24 hours.
Therefore, at 54 km/h the car takes 24 hours to cover 3X km.
Verification / Alternative check:
We can check using proportional reasoning. At 72 km/h, the car covers 72 km each hour. In 6 hours, it covers 72 * 6 = 432 km, matching X. At 54 km/h, the time required to cover 432 km would be 432 / 54 = 8 hours. For 3X = 3 * 432 = 1296 km, the time is 3 * 8 = 24 hours. This confirms our calculation is consistent.
Why Other Options Are Wrong:
18 hours corresponds to a distance of 54 * 18 = 972 km, which is less than 3X.
20 hours corresponds to 54 * 20 = 1080 km, still less than 1296 km.
21 hours gives 54 * 21 = 1134 km, also too small.
16 hours gives 864 km, which is even smaller.
Only 24 hours gives exactly the required 1296 km.
Common Pitfalls:
A common mistake is to assume that since the speed reduced from 72 to 54, the time will change in the same simple ratio without carefully tracking the change in distance from X to 3X. Another error is miscalculating products such as 72 * 6 or 1296 / 54. Learners may also mix up which speed applies to which distance. Writing each step clearly and checking intermediate computations reduces these errors.
Final Answer:
The car will take
24 hours to cover 3X km at the old speed of 54 km/h.
Discussion & Comments