The speeds of a car, a train and a bus are in the ratio 5 : 9 : 4. The average of their speeds taken together is 72 km/h. What is the average speed of the car and the train taken together?

Introduction / Context:
This question combines averages with ratios of speeds. You are given the ratio of the speeds of three vehicles and the average of all three speeds. From this information, you must first determine each individual speed in actual km/h and then find the average speed of just two of the vehicles, the car and the train.

Given Data / Assumptions:

Speeds of car, train and bus are in the ratio 5 : 9 : 4.
The average of the three speeds is 72 km/h.
We need the average speed of the car and the train together.
Speeds are assumed constant and measured in km/h.

Concept / Approach:
Let the speeds of the car, train and bus be 5k, 9k and 4k respectively. The average of the three speeds is the sum of the speeds divided by 3, and this is given as 72 km/h. Using this we can solve for k. Once we know k, we find the actual speed of the car and the train and then compute their average in the usual way by adding the two speeds and dividing by 2.

Step-by-Step Solution:
Step 1: Represent the speeds using a common multiplier. Let speed of car = 5k, train = 9k, bus = 4k. Step 2: Use the given average speed of all three vehicles. Average speed of three vehicles = (5k + 9k + 4k) / 3 = 72. So, (18k) / 3 = 72. This simplifies to 6k = 72. Step 3: Solve for k. k = 72 / 6 = 12. Step 4: Compute individual speeds. Car speed = 5k = 5 * 12 = 60 km/h. Train speed = 9k = 9 * 12 = 108 km/h. Bus speed = 4k = 4 * 12 = 48 km/h, which is not directly needed but confirms the ratio. Step 5: Find the average speed of car and train. Average speed of car and train = (60 + 108) / 2 = 168 / 2 = 84 km/h.

Verification / Alternative Check:
To verify, we can check that the average of all three speeds is still 72 km/h. Sum of speeds = 60 + 108 + 48 = 216 km/h. Average = 216 / 3 = 72 km/h, which matches the given condition. This ensures that the value of k and the derived speeds are correct, so the computed average of 84 km/h for the car and train is reliable.

Why Other Options Are Wrong:
Option 96 km/h would mean the sum of car and train speeds is 192 km/h, inconsistent with the correct individual speeds derived from the ratio and the overall average.
Option 72 km/h is simply the average of all three vehicles, not of only the car and train.
Option 60 km/h would correspond to the speed of the car alone, not the joint average of car and train.

Common Pitfalls:
Students may forget that the overall average is based on three speeds and incorrectly set 5k + 9k + 4k = 72 instead of dividing by 3. Another common error is to mis-handle the ratio or miscalculate k, which affects all subsequent speeds. Some may also attempt to directly average the ratio numbers 5 and 9 without scaling them properly, which is incorrect.

Final Answer:
The average speed of the car and the train together is 84 km/h.