A person goes to his office at one third of the speed at which he returns from his office. If the average speed for the entire to-and-fro trip is 12 km/h, what is his speed while going to the office (in km/h)?

Introduction / Context:
This question deals with average speed over a round trip where the speeds in the two directions are different. It tests your understanding of how to compute average speed when the distance is the same in each direction but the speeds are unequal. The relationship between the speeds and the given average speed must be used to determine the speed while going to the office.

Given Data / Assumptions:
- Let the speed while going to the office be v km/h.
- The speed while returning is three times this, that is 3v km/h, because going speed is one third of returning speed.
- The distance from home to office is the same in both directions and is positive, say d km one way.
- The average speed for the whole trip (going and returning) is 12 km/h.
- Motion is at constant speeds with no intermediate stops considered for calculation.

Concept / Approach:
Average speed for a round trip with unequal speeds is not the simple arithmetic mean of the speeds. Instead, average speed is defined as total distance divided by total time. For a round trip over the same distance d each way, total distance is 2d, and total time is d/v + d/(3v). Using the definition of average speed, we set up an equation with the given average speed (12 km/h) and then solve for v.

Step-by-Step Solution:
Step 1: Let one-way distance be d km.Speed while going = v km/h, speed while returning = 3v km/h.Step 2: Total distance for the round trip = 2d km.Step 3: Time taken to go = d / v hours. Time taken to return = d / (3v) hours.Total time for the round trip = d / v + d / (3v) = d * (1 / v + 1 / (3v)).Step 4: Compute the expression inside the brackets.1 / v + 1 / (3v) = (3 + 1) / (3v) = 4 / (3v).So total time = d * 4 / (3v) = 4d / (3v).Step 5: Average speed is total distance divided by total time.Average speed = 2d / (4d / (3v)) = 2d * (3v / 4d) = (6v / 4) = 3v / 2.Step 6: We are given that average speed is 12 km/h, so 3v / 2 = 12.Solve for v: 3v = 24 ⇒ v = 8 km/h.

Verification / Alternative check:
If v = 8 km/h, then the return speed is 3 * 8 = 24 km/h. Let d = 24 km as an example. Time to go = 24 / 8 = 3 hours. Time to return = 24 / 24 = 1 hour. Total distance = 48 km, total time = 4 hours, so average speed = 48 / 4 = 12 km/h, which matches the given data. Thus, v = 8 km/h is correct.

Why Other Options Are Wrong:
- 10 km/h and 6 km/h do not satisfy the equation 3v / 2 = 12 when substituted as v, and the resulting average speeds do not match 12 km/h.
- “Cannot be determined” is incorrect because the given information is sufficient to form a unique equation and determine v exactly.

Common Pitfalls:
A frequent error is to take the simple average of the two speeds v and 3v, which would give (v + 3v) / 2 = 2v. Using the condition that 2v = 12 would wrongly lead to v = 6. This is incorrect because average speed over equal distances depends on the harmonic mean, not the arithmetic mean. Always use the definition of average speed as total distance divided by total time when dealing with unequal speeds in each direction.

Final Answer:
The person's speed while going to the office is 8 km/h.