Gautam goes from his home to his office at 12 km/h and returns from office to home at 10 km/h by the same route. What is his average speed for the entire journey in km/h?

Introduction / Context:

This problem is about average speed for a round trip when the speeds in each direction are different but the distance is the same. Many candidates mistakenly take the simple arithmetic mean of the two speeds, but the correct average speed in such cases is the harmonic mean because the time spent at each speed is different. This question helps test and reinforce that important concept.

Given Data / Assumptions:

• Speed from home to office = 12 km/h.
• Speed from office to home = 10 km/h.
• The route and distance each way are the same.
• Let the one way distance be D kilometres.
• Motion is uniform at each stated speed.

Concept / Approach:

Average speed is defined as total distance divided by total time. For a journey where the distance each way is the same but speeds differ, a convenient formula is harmonic mean: average speed = 2 * v1 * v2 / (v1 + v2). This comes directly from computing total distance 2D and total time D / v1 + D / v2. After simplifying, D cancels out and we obtain the harmonic mean expression. We apply this formula with v1 = 12 km/h and v2 = 10 km/h.

Step-by-Step Solution:

Verification / Alternative check:

We can also compute using an assumed distance. Suppose D = 60 km. Time going = 60 / 12 = 5 hours. Time returning = 60 / 10 = 6 hours. Total distance = 120 km and total time = 11 hours, so average speed = 120 / 11 ≈ 10.9 km/h. The value matches the harmonic mean formula and confirms the result. Importantly, the answer is closer to the lower speed because more time is spent at the slower speed.

Why Other Options Are Wrong:

Option 11 km/h is close but slightly higher than the correct harmonic mean. Option 22 km/h is obviously too high, roughly double the individual speeds. Option 12.5 km/h is the simple average of 12 and 10, but that formula does not apply when distances are equal and speeds differ because the times are not equal. Therefore only 10.9 km/h is consistent with distance and time calculations.

Common Pitfalls:

The most common error is to compute (12 + 10)/2 and select 11 km/h. Another pitfall is forgetting that average speed depends on total distance and total time, not just the numerical values of speeds. Some learners also make algebraic mistakes while simplifying D / 12 + D / 10. Always factor D and carefully combine fractions before simplifying.

Final Answer:

Gautam’s average speed for the whole trip is approximately 10.9 km/h.