A train moving at two thirds of its normal speed reaches its destination 20 minutes late. What is the normal time, in hours, that the train usually takes to cover this journey?

Introduction / Context:
This question examines the relationship between speed and time for a fixed distance, a common theme in quantitative reasoning. If speed is reduced, time increases proportionally. We are told that the train runs at two thirds of its normal speed and therefore arrives 20 minutes late. Using proportionality, we can compute the normal travel time for the journey.

Given Data / Assumptions:

• Normal speed of the train = v (some value in km/h).
• Normal time for the journey = T hours (unknown).
• Reduced speed = (2/3) * v.
• At the reduced speed, the train arrives 20 minutes late.
• Distance of the journey is fixed.

Concept / Approach:
For a fixed distance, speed and time are inversely proportional: speed * time = constant. If speed is multiplied by some factor, time is divided by the same factor. Here, speed becomes two thirds of normal, so time becomes three halves of normal. The difference between the new time and the original time is given as 20 minutes. Setting up an equation in T using this relationship allows us to solve for the normal time.

Step-by-Step Solution:
Step 1: Let normal time be T hours. Step 2: Normal speed is v, so distance = v * T. Step 3: Reduced speed = (2/3) * v. Step 4: New time with reduced speed = distance / reduced speed = (v * T) / [(2/3) * v] = T * (3/2) hours. Step 5: Increase in time = new time - normal time = (3/2) * T - T = (1/2) * T. Step 6: This increase is given as 20 minutes = 20/60 = 1/3 hour. Step 7: So (1/2) * T = 1/3. Step 8: Multiply both sides by 2: T = 2/3 hours.

Verification / Alternative check:
If normal time is 2/3 hours (which is 40 minutes), then new time at reduced speed is (3/2) * 2/3 = 1 hour = 60 minutes. The delay is 60 - 40 = 20 minutes, which matches the given condition. This confirms that the computed normal time is consistent with the situation described.

Why Other Options Are Wrong:
Values such as 4/3, 3/2, or 1/4 hours would produce different time differences when multiplied by 3/2 and then subtracted from the original. None of them lead to an exact 20 minute difference except 2/3 hours. The distractor 1 hour clearly gives no delay when scaled, contradicting the problem statement.

Common Pitfalls:
A typical mistake is to misinterpret the factor relating speed and time, for example assuming time changes by two thirds instead of three halves. Another error is failing to convert 20 minutes into hours correctly, which leads to incorrect equations. Carefully using the idea that speed * time is constant for a fixed distance avoids these errors.

Final Answer:
The normal time for the journey is 2/3 hrs (that is, 40 minutes).