A car travels a certain distance at 32 km/h and returns over the same distance at 68 km/h. What is the average speed of the car for the entire journey?

Introduction / Context:
Average speed for a round trip with different speeds in each direction is a very common type of aptitude question. Many candidates mistakenly average the two speeds arithmetically, which is incorrect. Instead, we must use the harmonic mean, derived from the formula for total distance divided by total time. Here, a car travels one way at 32 km/h and returns at 68 km/h. We must find the average speed for the whole journey.

Given Data / Assumptions:

• One-way speed going from starting point to destination = 32 km/h.
• Return speed on the same route = 68 km/h.
• Distances in both directions are equal.
• We must find the average speed over the entire journey.

Concept / Approach:
Let the one-way distance be D km. Then total distance for the round trip is 2D. Corresponding times are:
time_1 = D / 32 hours time_2 = D / 68 hours Total time = time_1 + time_2. Average speed is:
average speed = total distance / total time = 2D / (D / 32 + D / 68) The D cancels out, leaving a pure expression in terms of the two speeds.

Step-by-Step Solution:
Step 1: Express the total distance. Total distance = D (going) + D (return) = 2D. Step 2: Express the total time. time_1 = D / 32, time_2 = D / 68. Total time = D / 32 + D / 68. Factor out D: total time = D * (1 / 32 + 1 / 68). Step 3: Compute the average speed. average speed = 2D / [D * (1 / 32 + 1 / 68)]. Cancel D: average speed = 2 / (1 / 32 + 1 / 68). Step 4: Simplify the denominator. 1 / 32 + 1 / 68 = (68 + 32) / (32 * 68) = 100 / (2176). So average speed = 2 / (100 / 2176) = 2 * (2176 / 100). 2 * 2176 = 4352, so average speed = 4352 / 100 = 43.52 km/h. Therefore, the average speed for the entire journey is 43.52 km/h.
Verification / Alternative check:
As an alternative, assume a convenient one-way distance, such as D = 2176 km. Then:
time_1 = 2176 / 32 = 68 hours. time_2 = 2176 / 68 = 32 hours. Total distance = 4352 km, total time = 100 hours. Average speed = 4352 / 100 = 43.52 km/h. This matches the previous computation, confirming the result.

Why Other Options Are Wrong:
50 km/h is the simple arithmetic mean of 32 and 68, which is not correct for average speed over equal distances with different speeds.
37.04 km/h, 40 km/h and 56.48 km/h result from incorrect manipulations of the formula or from mixing up distances and times.
Only 43.52 km/h comes from the correct calculation using total distance divided by total time.

Common Pitfalls:
The biggest pitfall is averaging the speeds directly using (32 + 68) / 2. Average speed problems with equal distances require the harmonic mean: 2ab / (a + b), not the simple average. Another common error is to assume different distances for each leg of the trip, which is not given in the problem. Also, computational mistakes with fractions can lead to wrong decimals. Carefully applying the formula and checking with a concrete distance helps avoid these errors.

Final Answer:
The average speed of the car for the entire journey is 43.52 km/h.