Ram walks from his house to school. At a speed of 3 km/h he reaches 6 minutes late, but at a speed of 4 km/h he reaches 4 minutes early. What is the distance in kilometres between his house and the school?

Introduction / Context:

This is a classic time and distance problem where early and late arrival times are given for two different speeds. Such questions test your ability to set up equations based on the relationship between time, speed and distance. Instead of giving the distance directly, the problem describes how the arrival time changes when Ram changes his walking speed. From this information, we must deduce the distance between his home and school.

Given Data / Assumptions:

• Speed of Ram in the first case = 3 km/h and he is 6 minutes late.
• Speed of Ram in the second case = 4 km/h and he is 4 minutes early.
• Let the scheduled or correct time to reach school be T hours.
• Let the distance between home and school be D kilometres.
• The walking route is the same in both cases and speed remains constant during each walk.

Concept / Approach:

Time, speed and distance are connected by distance = speed * time, or time = distance / speed. In this problem there are two forms of the time taken: one when Ram walks slower and arrives late, and another when he walks faster and arrives early. The difference between these times equals the total difference between late and early arrival. We form two equations for time using D and the two speeds, relate them through the scheduled time T, and then solve for D.

Step-by-Step Solution:

Verification / Alternative check:

Now we can calculate the actual times and see if the early and late conditions are satisfied. At 3 km/h, time = 2 / 3 hour which is 40 minutes. At 4 km/h, time = 2 / 4 hour which is 30 minutes. The difference between these times is 10 minutes. Since Ram is 6 minutes late in the first case and 4 minutes early in the second case, the total difference is 6 + 4 = 10 minutes. This matches our calculation, confirming that D = 2 kilometres is correct.

Why Other Options Are Wrong:

If the distance were 1.8 km, the times would not differ by exactly 10 minutes. A distance of 3.5 km or 4 km would produce even larger time differences and would not satisfy both the late and early arrival conditions together. Only 2 km leads to 40 and 30 minutes, which perfectly match the description in the problem.

Common Pitfalls:

Learners often forget to convert minutes into hours before forming equations. Another common mistake is to think that 6 minutes late minus 4 minutes early equals only 2 minutes, but in reality the total difference between the two arrival times is 10 minutes. Some students also attempt to guess distances without forming equations, which can easily go wrong. Always use the relationship time = distance / speed carefully.

Final Answer:

The distance between Ram’s house and his school is 2 kilometres.